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    <fileDesc><titleStmt><title>A Mathematical and Philosophical Dictionary</title><author>Charles Hutton</author></titleStmt>
<publicationStmt>
<p>Berlin: Max-Planck-Institut f&#xFC;r Wissenschaftsgeschichte, 2000
<!-- archimedes_locator:078.xml: -->
      </p></publicationStmt><sourceDesc default="false"> <p>London, 1796.
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    <front>
      <pb/>
      <titlePage>
        <docTitle>
          <titlePart rend="center" type="main">A
MATHEMATICAL <hi rend="smallcaps">AND</hi> PHILOSOPHICAL
DICTIONARY:</titlePart>
          <titlePart rend="center" type="main">CONTAINING
<hi rend="italics">AN EXPLANATION OF THE TERMS, AND AN ACCOUNT OF THE SEVERAL SUBJECTS,</hi></titlePart>
          <titlePart rend="center" type="main">COMPRIZED UNDER THE HEADS
MATHEMATICS, ASTRONOMY, <hi rend="smallcaps">AND</hi> PHILOSOPHY
BOTH NATURAL AND EXPERIMENTAL:
WITH AN
HISTORICAL ACCOUNT OF THE RISE, PROGRESS, AND PRESENT STATE OF THESE SCIENCES:
ALSO
MEMOIRS OF THE LIVES AND WRITINGS OF THE MOST EMINENT AUTHORS,
BOTH ANCIENT AND MODERN,
<hi rend="italics">WHO BY THEIR DISCOVERIES OR IMPROVEMENTS HAVE CONTRIBUTED TO THE ADVANCEMENT OF THEM.</hi>
IN TWO VOLUMES.
WITH MANY CUTS AND COPPER-PLATES.</titlePart>
        </docTitle>
        <byline rend="center"><hi rend="smallcaps">By</hi> <docAuthor>CHARLES HUTTON</docAuthor>, LL.D.
F. R. SS. OF LONDON AND EDINBURGH, AND OF THE PHILOSOPHICAL SOCIETIES OF HAARLEM AND AMERICA;
AND PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY, WOOLWICK.</byline>
        <docTitle>
          <titlePart rend="center" type="main">VOL. I.</titlePart>
        </docTitle>
        <docImprint rend="center"><pubPlace rend="italics">LONDON:</pubPlace>
PRINTED BY J. DAVIS,
FOR J. JOHNSON, IN ST. PAUL'S CHURCH-YARD; AND G. G. AND J. ROBINSON,
IN PATERNOSTER-ROW.
<docDate>M.DCC.XCVI.</docDate></docImprint>
      </titlePage>
      <div1 part="N" org="uniform" sample="complete" type="preface"><pb/><pb/><head>PREFACE.</head><p>AMONG the Dictionaries of Arts and Sciences which have been published, of late
years, in various parts of Europe, it is matter of surprise that Philosophy and Mathematics
should have been so far overlooked as not to be thought worthy of a separate
Treatise, in this form. These Sciences constitute a large portion of the present stock
of human knowledge, and have been usually considered as possessing a degree of importance
to which few others are entitled; and yet we have hitherto had no distinct
Lexicon, in which their constituent parts and technical terms have been explained, with
that amplitude and precision, which the great improvements of the Moderns, as well as
the rising dignity of the Subject, seem to demand.</p><p><hi rend="smallcaps">The</hi> only works of this kind in the English language, deserving of notice, are Harris's
Lexicon Technicum, and Stone's Mathematical Dictionary; the former of which,
though a valuable performance at the time it was written, is now become too dry and
obsolete to be referred to with pleasure or satisfaction: and the latter, consisting only
of one volume in 8vo, must be regarded merely as an unfinished sketch, or brief
compendium, extremely limited in its plan, and necessarily desieient in useful in
formation.</p><p><hi rend="smallcaps">It</hi> became, therefore, the only resource of the Reader, in many cases where explanation
was wanted, to have recourse to Chambers's Dictionary, in four large Volumes folio,
or to the Encyclop&#xE6;dia Britannica, now in eighteen large volumes 4to, or the still more,
stupendous performance of the French Encyclopedists; and even here his expectations
might be frequently disappointed. These great and useful works, aiming at a
general comprehension of the whole circle of the Sciences, are sometimes very delicient
in their descriptions of particular branches; it being almost impossible, in
such
<pb n="vi"/>
extensive undertakings, to appreciate, with exactness, the due value of every article:
They are, besides, so voluminous and heterogeneous in their nature, as to render
a frequent reference to them  extremely inconvenient; and even if this were not the
case, their high price puts them out of the reach of the generality of readers.</p>
<p><hi rend="smallcaps">WITH</hi> a view to obviate these defects, the Public are here presented with a Dic-
tionary of a moderate size and price, which is devoted solely to Philosophical and Ma-
thematical subjects. It is a work for which materials have been collecting through a
course of many years ; and is the result of great labour and reading. Not only most of
the Encyclopedias already extant, and the various publications of the Learned Societies
throughout Europe, have been carefully consulted, but also all the original works, of
any reputation, which have hitherto appeared upon these subjects, from  the earliest
writers down to the present times.</p>
<p><hi rend="smallcaps">FROM</hi> the latter of there sources, in particular, a considerable portion of information
has been obtained, which the curious reader will find, in many cases, to be highly in-
teresting and important. The History of Algebra, for instance, which is detailed at
considerable length in the First Volume, under the head of that Article, will afford
sufficient evidence to shew in what a superficial and partial way the inquiry has been
hitherto investigated, even by professed writers on the subject; the principal of whom
are M. Montucla, our countryman the celebrated Dr. Wallis, and the Abb&#xE9; De Gua,
a late French author, who has pretended to correct the Doctor's errors and misrepresen-
tations.</p>
<p><hi rend="smallcaps">REGULAR</hi> historical details are in like manner given of the origin and progress of
each of these Sciences, as well as of the inventions and improvements by which they
have been gradually brought from their first rude beginnings to their present ad-
vanced state.</p>
<p><hi rend="smallcaps">IT</hi> is also to be observed, that besides the articles common to the generality of
Dictionaries of this kind, an interesting Biographical Account is here introduced
of the most celebrated Philosophers and Mathematicians, both ancient and modern;
among which will be found the Lives of many eminent characters, who have hi-
therto been either wholly overlooked, or very imperfectly recorded. Complete lists of
their works are also subjoined to each Article, where they could be procured; which
cannot but prove highly acceptable to that class of readers, who are desirous of ob-
taining the most satisfactory information upon the subjects of their particular enqui-
ries and pursuits. On the head of Biography however the Author has still to
<pb n="vii"/>
to his list of authors, not having been able to procure any circumstantial accounts of
their lives. He could have wished to have comprised in his list, the lives of all
such public literary characters as the University Professors of Astronomy, Philosophy and
Mathematics, as well as those of the other more respectable class of Authors on those
Sciences. He will therefore thankfully receive the communication of any such memoirs
from the hands of gentlemen possessed of them; as well as hints and information on
such useful improvements in the sciences as may have been overlooked in this Dictionary,
or any articles that may here have been imperfectly or incorrectly treated; that
he may at some future time, by adding them to this work, render it still more complete
and deserving the public notice.</p><p><hi rend="smallcaps">As</hi> this work is an attempt to separate the words in the sciences of Astronomy, Mathematics,
and Philosophy, from those of other arts or sciences, in several of which there
are already separate Dictionaries; as in Chemistry, Geography, Music, Marine and
Naval affairs, &amp;c; words sometimes occurred which it was rather doubtful whether
they could be considered as properly belonging to the present work or not;
in which case many of such words have been here inserted. But such as appeared
clearly and peculiarly to belong to any of those other subjects, have been either wholly
omitted, or else have had a very short account only given of them. The readers of this
work therefore, recollecting that it is not a General Dictionary of all the Arts and
Sciences, will not expect to find all sorts of words and subjects here treated of; but
such only as peculiarly appertain to the proper matter of the work. And therefore,
although some few words may inadvertently have been omitted; yet when the Reader
does not immediately find every word which he wishes to consult, he will not always
consider them as omissions of the Author, but for the most part as relating to some other
science foreign to this Dictionary.</p><p><hi rend="smallcaps">In</hi> all cases where it could be conveniently done, the necessary figures and diagrams
are inserted in the same page with the subjects which they are designed to
elucidate; a method which will be found much more commodious than that of
putting them in separate plates at the end of each volume, but, which has added
very considerably to the expence of the undertaking: where the subjects are of
such a nature that they could not be otherwise well represented, they are engraved on
Copperplates.</p><p><hi rend="smallcaps">As</hi> the whole of this work was written before it was put to the press, the Reader
will find it of an equal and uniform nature and construction throughout; in
which respect many publications of this kind are very defective, from the subjects
being diffusely treated under the first letters of the alphabet, while articles
<pb n="viii"/>
of equal importance in the latter part are so much abridged as to be rendered al-
most useless, in order that the whole might be comprised in a limited number of
sheets, according to proposals made before the works were composed. The pre-
sent Dictionary having been completed without any of there unfavourable circum-
stances, will be found in most cases equally instructive and useful, and may be
consulted with no less advantage by the Man of Science than the Student.

</p></div1>
    </front>
    <body><pb/>

<!-- <C>A
PHILOSOPHICAL and MATHEMATICAL
DICTIONARY.</C> -->
<div0 part="N" n="A" org="uniform" sample="complete" type="alphabetic letter"><head>A</head><cb/><div1 part="N" n="ABACIST" org="uniform" sample="complete" type="entry"><head>ABACIST</head><p>, an Arithmetician. In this sense we
find the word used by William of Malmesbury,
in his History <hi rend="italics">de Gestis Anglorum,</hi> written about the
year 1150; where he shews that one Gerbert, a learned
monk of France, who was afterwards made pope of
Rome in the year 998 or 999, by the name of Silvester
the 2d, was the first who got from the Saracens the
abacus, and that he taught such rules concerning it, as
the Abacists themselves could hardly understand.</p></div1><div1 part="N" n="ABACUS" org="uniform" sample="complete" type="entry"><head>ABACUS</head><p>, <hi rend="italics">in Arithmetic,</hi> an ancient instrument used
by most nations for casting up accounts, or performing
arithmetical calculations: it is by some derived from the
Greek <foreign xml:lang="greek">a&lt;*&gt;ac</foreign>, which signifies a cupboard or beaufet, perhaps
from the similarity of the form of this instrument;
and by others it is derived from the Ph&#x153;nician <hi rend="italics">abak,</hi>
which signifies dust or powder, because it was said that
this instrument was sometimes made of a square board
or tablet, which was powdered over with fine sand or dust,
in which were traced the figures or characters used in
making calculations, which could thence be easily defaced,
and the abacus refitted for use. But Lucas Paciolus,
in the sirst part of his second distinction, thinks it
is a corruption of Arabicus, by which he meant their
Algorism, or the method of numeral computation received
from them.</p><p>We find this instrument for computation in use, under
some variations, with most nations, as the Greeks, Romans,
Germans, French, Chinese, &amp;c.</p><p>The Grecian abacus was an oblong frame, over
which were stretched several brass wires, strung with
little ivory balls, like the beads of a necklace; by the
various arrangements of which all kinds of computa-
<cb/>
tions were easily made. Mahudel, in Hist. Acad. R.
Inscr. t. 3. p. 390.</p><p>The Roman Abacus was a little varied from the Grecian,
having pins sliding in grooves, instead of strings or
wires and beads. Philos. Trans. No. 180.</p><p>The Chinese Abacus, or Shwan-pan, like the Grecian,
consists of several series of beads strung on brass
wires, stretched from the top to the bottom of the instrument,
and divided in the middle by a cross piece
from side to side. In the upper space every string has
two beads, which are each counted for 5; and in the
lower space every string has five beads, of different
values, the first being counted as 1, the second as 10,
the third as 100, and so on, as with us. See S<hi rend="smallcaps">HWANPAN.</hi></p><p>The Abacus chiefly used in European countries, is
nearly upon the same principles, though the use of it is
here more limited, because of the arbitrary and unequal
divisions of money, weights, and measures, which, in
China, are all divided in a tenfold proportion, like our
scale of common numbers. This is made by drawing
any number of parallel lines, like paper ruled for music,
at such a distance as may be at least equal to twice the
diameter of a calculus, or counter. Then the value of
these lines, and of the spaces between them, increases,
from the lowest to the highest, in a tenfold proportion.
Thus, counters placed upon the first line, signify so many
units or ones; on the second line 10's, on the third
line 100's, on the fourth line 1000's, and so on: in
like manner a counter placed in the first space, between
the first and second line, denotes 5, in the second space
50, in the third space 500, in the fourth space 5000,
<pb n="2"/><cb/>
and so on. So that there are never more than four
counters placed on any line, nor more than one placed
in any space, this being of the same value as five counters
on the next line below. So the counters on the
Abacus, in the figure here below, express the number
or sum 47382.
<figure/></p><p>Besides the above instruments of computation, there
have been several others invented by different persons;
as <hi rend="italics">Napier's rods</hi> or <hi rend="italics">bones,</hi> deseribed in his Rabdologia,
which see under the word <hi rend="smallcaps">Napier;</hi> also the <hi rend="italics">Abacus
Rhabdologicus,</hi> a variation of Napier's, which is described
in the first vol. of <hi rend="italics">Machines et Inventions approuv&#xE9;es
par l'Academie Royale des Sciences.</hi> An ingenious
and general one was also invented by Mr. Gamaliel
Smethurst, and is described in the Philosophical
Transactions, vol. 46; where the inventor remarks that
computations by it are much quicker and easier than by
the pen, are less burthensome to the memory, and can
be performed by blind persons, or in the dark as well as
in the light. A very comprehensive instrument of this
kind was also contrived by the late learned Dr. Nicholas
Saunderson, by which he performed very intricate
calculations: an account of it is prefixed to the
first volume of his Algebra, and it is there by the editor
called <hi rend="italics">Palpable Arithmetic:</hi> which see.</p><div2 part="N" n="Abacus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Abacus</hi></head><p>, <hi rend="italics">Pythagorean,</hi> so denominated from its inventor,
Pythagoras; a table of numbers, contrived for
readily learning the principles of arithmetic; and was
probably what we now call the multiplication-table.</p></div2><div2 part="N" n="Abacus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Abacus</hi></head><p>, or <hi rend="smallcaps">Abaciscus</hi>, in <hi rend="italics">Architecture,</hi> the upper
part or member of the capital of a column; serving as
a crowning both to the capital and to the whole column.
Vitruvius informs us that the <hi rend="italics">Abacus</hi> was originally
intended to represent a square flat tile laid over
an urn, or a basket; and the invention is ascribed to
Calimachus, an ingenious statuary of Athens, who, it is
said, adopted it on observing a small basket, covered
with a tile, over the root of an Acanthus plant, which
grew on the grave of a young lady; the plant shooting
up, encompassed the basket all around, till meeting
with the tile, it curled back in the form of scrolls:
Calimachus passing by, took the hint, and immediately
executed a capital on this plan; representing the tile by
the Abacus, the leaves of the acanthus by the volutes or
scrolls, and the basket by the vase or body of the capital.
See <hi rend="smallcaps">Acanthus.</hi></p><p><hi rend="italics">Abacus</hi> is also used by Scamozzi for a concave moulding
in the capital of the Tuscan pedestal. And the
word is used by Palladio for other members which he
describes. Also, in the ancient architecture, the same
term is used to denote certain compartments in the incrustation
or lining of the walls of state-rooms, mosaicpavements,
and the like. There were <hi rend="italics">Abaci</hi> of marble,
<cb/>
porphyry, jasper, alabaster, and even glass; variously
shaped, as square, triangular, and such-like.</p><p><hi rend="smallcaps">Abacus</hi> <hi rend="italics">Logisticus</hi> is a right angled triangle, whose
sides, about the right angle, contain all the numbers
from 1 to 60; and its area the products of each two of
the opposite numbers. This is also called a <hi rend="italics">canon of
sexagesimals,</hi> and is no other than a multiplication-table
carried to 60 both ways.</p><p><hi rend="smallcaps">Abacus</hi> <hi rend="italics">&amp; Palmul&#xE6;,</hi> in the Ancient Music, denote
the machinery by which the strings of the polyplectra,
or instruments of many strings, were struck, with a
plectrum made of quills.</p><p><hi rend="smallcaps">Abacus</hi> <hi rend="italics">Harmonicus</hi> is used by Kircher for the structure
and disposition of the keys of a musical instrument,
either to be touched with the hands or feet.</p></div2><div2 part="N" n="Abacus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Abacus</hi>, in <hi rend="italics">Geometry</hi></head><p>, a table or slate upon which
schemes or diagrams are drawn.</p></div2></div1><div1 part="N" n="ABAS" org="uniform" sample="complete" type="entry"><head>ABAS</head><p>, a weight used in Persia for weighing pearls;
and is an eighth part lighter than the European carat.</p></div1><div1 part="N" n="ABASSI" org="uniform" sample="complete" type="entry"><head>ABASSI</head><p>, a silver coin current in Perlia, deriving
its name from Schaw Abbas II. King of Persia, and is
worth near eighteen pence English money.</p></div1><div1 part="N" n="ABATIS" org="uniform" sample="complete" type="entry"><head>ABATIS</head><p>, or <hi rend="smallcaps">Abattis</hi>, from the French <hi rend="italics">abattre,</hi> to
throw down, or beat down, in the Military Art, denotes
a kind of retrenchment made by a quantity of whole
trees cut down, and laid lengthways beside each other,
the closer the better, having all their branches pointed
towards the enemy, which prevents his approach, at the
same time that the trunks serve as a breast-work before
the men. The Abattis is a very useful work on most
occasions, especially on sudden emergencies, when trees
are near at hand; and has always been practised with
considerable success, by the ablest commanders in all
ages and nations.</p><p>ABBREVIATE; to abbreviate fractions in arithmetic
and algebra, is to lessen proportionally their terms,
or the numerator and denominator; which is performed
by dividing those terms by any number or quantity,
which will divide them without leaving a remainder.
And when the terms cannot be any farther so divided,
the fraction is said to be in its least terms.</p><p>So ,
by dividing the terms continually by 2.</p><p>And ,
by dividing by 2, 3, and 7.</p><p>Also ,
by dividing by 3 and by 2.</p><p>And , by dividing by 4 <hi rend="sup"><hi rend="italics">ax.</hi></hi></p><p>And , by dividing by <hi rend="italics">a</hi>+<hi rend="italics">x.</hi></p></div1><div1 part="N" n="ABBREVIATION" org="uniform" sample="complete" type="entry"><head>ABBREVIATION</head><p>, of fractions, in Arithmetic
and Algebra, the reducing them to lower terms.</p></div1><div1 part="N" n="ABERRATION" org="uniform" sample="complete" type="entry"><head>ABERRATION</head><p>, in <hi rend="italics">Astronomy,</hi> an apparent motion
of the celestial bodies, occasioned by the progressive
motion of light, and the earth's annual motion
in her orbit.</p><p>This effect may be explained and familiarized by the
<pb n="3"/><cb/>
motion of a line parallel to itself, much after the manner
that the composition and resolution of forces are explained.
If light have a progressive motion, let the
<figure/>
proportion of its velocity to that
of the earth in her orbit, be as
the line BC to the line AC;
then, by the composition of these
two motions, the particle of
light will seem to describe the
line BA or DC, instead of its
real course BC; and will appear
in the direction AB or CD, instead
of its true direction CB.
So that if AB represent a tube,
carried with a parallel motion by
an observer along the line AC,
in the time that a particle of light
would move over the space BC,
the different places of the tube being AB, <hi rend="italics">ab, cd,</hi> CD;
and when the eye, or end of the tube, is at A, let a
particle of light enter the other end at B; then when
the tube is at <hi rend="italics">ab,</hi> the particle of light will be at <hi rend="italics">e,</hi> exactly
in the axis of the tube; and when the tube is at <hi rend="italics">cd,</hi> the
particle of light will arrive at <hi rend="italics">f,</hi> still in the axis of the
tube; and lastly, when the tube arrives at CD, the particle
of light will arrive at the eye or point C, and consequently
will appear to come in the direction DC of
the tube, instead of the true direction BC. And so
on, one particle succeeding another, and forming a continued
stream or ray of light in the apparent direction
DC. So that the apparent angle made by the ray of
light with the line AE, is the angle DCE, instead of
the true angle BCE; and the difference, BCD or
ABC, is the quantity of the aberration.</p><p>M. de Maupertuis, in his Elements of Geography,
gives also a familiar and ingenious idea of the aberration,
in this manner: &#x201C;It is thus,&#x201D; says he, &#x201C;concerning
the direction in which a gun must be pointed
to strike a bird in its flight; instead of pointing it
straight to the bird, the fowler will point a little before
it, in the path of its flight, and that so much the more
as the flight of the bird is more rapid, with respect to
the flight of the shot.&#x201D; In this way of considering the
matter, the flight of the bird represents the motion of
the earth, or the line AC, in our scheme above, and
the flight of the shot represents the motion of the ray
of light, or the line BC.</p><p>Mr. Clairaut too, in the Memoires of the Academy of
Sciences for the year 1746, illustrates this effect in a
familiar way, by supposing drops of rain to fall rapidly
and quickly after each other from a cloud, under which
a person moves with a very narrow tube; in which case
it is evident that the tube must have a certain inclination,
in order that a drop which enters at the top, may
fall freely through the axis of the tube, without touching
the sides of it; which inclination must be more or less
according to the velocity of the drops in respect to that
of the tube: then the angle made by the direction of
the tube and of the falling drops, is the aberration arising
from the combination of those two motions.</p><p>This discovery, which is one of the brightest that have
been made in the present age, we owe to the accuracy
and ingenuity of the late Dr. Bradley, Astronomer
Royal; to which he was occasionally led by the result
<cb/>
of some accurate observations which he had made with
another view, namely, to determine the annual parallax
of the fixed stars, or that which arises from the motion
of the earth in its annual orbit about the sun.</p><p>The annual motion of the earth about the sun had
been much doubted, and warmly contested. The defenders
of that motion, among other proofs of the reality
of it, conceived the idea of adducing an incontestable
one from the annual parallax of the fixed stars,
if the stars should be within such a distance, or if instruments
and observations could be made with such accuracy,
as to render that parallax sensible. And with
this view various attempts have been made. Before the
observations of M. Picard, made in 1672, it was the
general opinion, that the stars did not change their position
during the course of a year. Tycho Brahe and
Ricciolus fancied that they had assured themselves of it
from their observations; and from thence they concluded
that the earth did not move round the sun, and that
there was no annual parallax in the fixed stars. M. Picard,
in the account of his <hi rend="italics">Voyage d'Uranibourg,</hi> made
in 1672, says that the pole star, at different times of
the year, has certain variations which he had observed
for about 10 years, and which amounted to about 40&#x2033;
a year: from whence some who favoured the annual
motion of the earth were led to conclude that these variations
were the effect of the parallax of the earth's
orbit. But it was impossible to explain it by that parallax;
because this motion was in a manner contrary to
what ought to follow only from the motion of the earth
in her orbit.</p><p>In 1674 Dr. Hook published an account of observations
which he said he had made in 1669, and by
which he had found that the star <foreign xml:lang="greek">g</foreign> Draconis was 23
more northerly in July than in October: observations
which, for the present, seemed to favour the opinion of
the earth's motion, although it be now known that
there could not be any truth or accuracy in them.</p><p>Flamsteed having observed the pole star with his
mural quadrant, in 1689 and the following years, found
that its declination was 40&#x2033; less in July than in December;
which observations, although very just, were
yet however improper for proving the annual parallax:
and he recommended the making of an instrument of
15 or 20 feet radius, to be firmly fixed on a strong
foundation, for deciding a doubt which was otherwise
not soon likely to be brought to a conclusion.</p><p>In this state of uncertainty and doubt, then, Dr.
Bradley, in conjunction with Mr. Samuel Molineux,
in the year 1725, formed the project of verifying, by
a series of new observations, those which Dr. Hook
had communicated to the public almost 50 years
before. And as it was his attempt that chiefly
gave rise to this, so it was his method in making the
observations, in some measure, that they followed; for
they made choice of the same star, and their instrument
was constructed upon nearly the same principles: but
had it not greatly exceeded the former in exactness,
they might still have continued in great uncertainty as
to the parallax of the fixed stars. And this was chiefly
owing to the accuracy of the ingenious Mr. George
Graham, to whom the lovers of astronomy are also indebted
for several other exact and convenient instruments.
<pb n="4"/><cb/></p><p>The success then of the intended experiment, evidently
depending very much on the accuracy of the instrument,
that leading object was first to be well secured.
Mr. Molineux's apparatus then having been
completed, and fitted for observing, about the end of
November 1725, on the third day of December following,
the bright star in the head of Draco, marked <foreign xml:lang="greek">g</foreign> by
Bayer, was for the first time observed, as it passed near
the zenith, and its situation carefully taken with the instrument.
The like observations were made on the
fifth, eleventh, and twelfth days of the same month;
and there appearing no material difference in the place
of the star, a farther repetition of them, at that season,
seemed needless, it being a time of the year in which
no sensible alteration of parallax, in this star, could soon
be expected. It was therefore curiosity that chiefly
urged Dr. Bradley, being then at Kew, where the in
strument was fixed, to prepare for observing the star
again on the 17th of the same month; when, having
adjusted the instrument as usual, he perceived that it
passed a little more southerly this day than it had done
before. Not suspecting any other cause of this appearance,
they first concluded that it was owing to the uncertainty
of the observations, and that either this, or
the foregoing, was not so exact as they had before supposed.
For which reason they proposed to repeat the
observation again, to determine from what cause this difference
might proceed: and upon doing it, on the 20th of
December, the doctor found that the star passed still more
southerly than at the preceding observation. This sensible
alteration surprised them the more, as it was the
contrary way from what it would have been, had it
proceeded from an annual parallax of the star. But
being now pretty well satisfied, that it could not be entirely
owing to the want of exactness in the observations,
and having no notion of any thing else that could
cause such an apparent motion as this in the star; they
began to suspect that some change in the materials, or
fabric of the instrument itself, might have occasioned it.
Under these uncertainties they remained for some time;
but being at length fully convinced, by several trials, of
the great exactness of the instrument; and finding, by
the gradual increase of the star's distance from the pole,
that there must be some regular cause that produced it;
they took care to examine very micely, at the time of
each observation, how much the variation was; till
about the beginning of March 1726, the star was found
to be 20&#x2033; more southerly than at the time of the first
observation: it now indeed seemed to have arrived at its
utmost limit southward, as in several trials, made about
this time, no sensible difference was observed in its situation.
By the middle of April it appeared to be returning
back again towards the north; and about the
beginning of June, it passed at the same distance from
the zenith, as it had done in December, when it was
first observed.</p><p>From the quick alteration in the declination of the
star about this time, increasing about one second in
three days, it was conjectured that it would now proceed
northward, as it had before gone southward, of its
present situation; and it happened accordingly; for
the star continued to move northward till September
following, when it became stationary again; being then
near 20&#x2033; more northerly than in June, and upwards of
<cb/>
39&#x2033; more northerly than it had been in March. From
September the star again returned towards the south,
till, in December, it arrived at the same situation in
which it had been observed twelve months before, allowing
for the difference of declination on account of the
precession of the equinox.</p><p>This was a sufficient proof that the instrument had
not been the cause of this apparent motion of the star;
and yet it seemed difficult to devise one that should be
adequate to such an unusual effect. A nutation of the
earth's axis was one of the first things that offered itself
on this occasion; but it was soon found to be insufficient;
for though it might have accounted for the
change of declination in <foreign xml:lang="greek">g</foreign> Draconis, yet it would not
at the same time accord with the phenomena observed
in the other stars, particularly in a small one almost opposite
in right ascension to <foreign xml:lang="greek">g</foreign> Draconis; and at about
the same distance from the north pole of the equator:
for though this star seemed to move the same way, as a
nutation of the earth's axis would have made it; yet
changing its declination but about half as much as <foreign xml:lang="greek">g</foreign>
Draconis in the same time, as a peared on comparing
the observations of both made on the same days, at different
seasons of the year, this plainly proved that the
apparent motion of the star was not occasioned by a
real nutation; since, had that been the case, the alteration
in both stars would have been nearly equal.</p><p>The great regularity of the observations left no room
to doubt, but that there was some uniform cause by
which this unexpected motion was produced, and which
did not depend on the uncertainty or variety of the
seasons of the year. Upon comparing the observations
with each other, it was discovered that, in both the
stars above mentioned, the apparent difference of declination
from the <hi rend="italics">maxima,</hi> was always nearly proportional
to the versed sine of the sun's distance from the equinoctial
points. This was an inducement to think that
the cause, whatever it was, had some relation to the
sun's situation with respect to those points. But not
being able to frame any hypothesis, sufficient to account
for all the phenomena, and being very desivous to search
a little farther into this matter, Dr. Bradley began to
think of erecting an instrument for himself at Wanstead;
that, having it always at hand, he might with the more
ease and certainty enquire into the laws of this new
motion. The consideration likewise of being able, by
another instrument, to confirm the truth of the observations
hitherto made with that of Mr. Molineux,
was no small inducement to the undertaking; but the
chief of all was, the opportunity he should thereby have
of trying in what manner other stars should be affected
by the same cause, whatever it might be. For Mr. Molineux's
instrument being originally designed for observing
<foreign xml:lang="greek">g</foreign> Draconis, to try whether it had any sensible parallax, it
was so contrived, as to be capable of but little alteration
in its direction; not above seven or eight minutes of a
degree: and there being but few stars, within half that
distance from the zenith of Kew, bright enough to be
well observed, he could not, with his instrument, thoroughly
examine how this cause affected stars that were
differently situated, with respect to the equinoctial and
solsticial points of the ecliptic.</p><p>These considerations determined him; and by the
contrivance and direction of the same ingenious person,
<pb n="5"/><cb/>
Mr. Graham, his instrument was fixed up the 19th of
August 1727. As he had no convenient place where
he could make use of so long a telescope as Mr. Molineux's,
he contented himself with one of but little
more than half the length, namely of 12 feet and a half,
the other being 24 feet and a half long, judging from
the experience he had already had, that this radius would
be long enough to adjust the instrument to a sufficient
degree of exactness: and he had no reason afterwards
to change his opinion; for by all his trials he was very
well satisfied, that when it was caresully rectisied, its
situation might be securely depended on to half a
second. As the place where his instrument was hung,
in some measure determined its radius; so did it also
the length of the arc or limb, on which the divisions
were made, to adjust it: for the arc could not conveniently
be extended farther, than to reach to about 6 1/4
degrees on each side of his zenith. This however was
sufficient, as it gave him an opportunity of making
choice of several stars, very different both in magnitude
and situation; there being more than two hundred, inserted
in the British Catalogue, that might be observed
with it. He needed not indeed to have extended the
limb so far, but that he was willing to take in <hi rend="italics">Capella,</hi>
the only star of the first magnitude that came so near his
zenith.</p><p>His instrument being fixed, he immediately began to
observe such stars as he judged most proper to give him
any light into the cause of the motion already mentioned.
There was a sufficient variety of small ones,
and not less than twelve that he could observe through
all seasons of the year, as they were bright enough to
be seen in the day-time, when nearest the sun. He
had not been long observing, before he perceived that
the notion they had before entertained, that the stars
were farthest north and south when the sun was near
the equinoxes, was only true of those stars which are
near the solsticial colure. And after continuing his
observations a few months, he discovered what he then
apprehended to be a general law observed by all the
stars, namely, that each of them became stationary, or
was farthest north or south, when it passed over his
zenith at six of the clock, either in the evening or
morning. He perceived also that whatever situation
the stars were in, with respect to the cardinal points of
the ecliptic, the apparent motion of every one of them
tended the same way, when they passed his instrument
about the same hour of the day or night; for they all
moved southward when they passed in the day, and
northward when in the night; so that each of them
was farthest north, when it came in the evening about
six of the clock, and farthest south when it came about
six in the morning.</p><p>Though he afterwards discovered that the maxima,
in most of these stars, do not happen exactly when they
pass at those hours; yet, not being able at that time to
prove the contrary, and supposing that they did, he endeavoured
to find out what proportion the greatest
alterations of declination, in different stars, bore to each
other; it being very evident that they did not all change
their declination equally. It has been before noticed,
that it appeared from Mr. Molineux's observations, that
<foreign xml:lang="greek">g</foreign> <hi rend="italics">Draconis</hi> changed its declination above twice as much
as the before-mentioned small star that was nearly op-
<cb/>
posite to it; but examining the matter more nicely, he
found that the greatest change in the declination of
these stars, was as the sine of the latitude of each star
respectively. This led him to suspect that there might
be the like proportion between the <hi rend="italics">maxima</hi> of other
stars; but finding that the observations of some of them
would not perfectly correspond with such an hypothesis,
and not knowing whether the small difference he met
with might not be owing to the uncertainty and error
of the observations, he deferred the farther examination
into the truth of this hypothesis, till he should be sarther
furnished with a series of observations made in all parts
of the year; which would enable him not only to determine
what errors the observations might be liable to,
or how far they might safely be depended on; but also
to judge, whether there had been any sensible change in
the parts of the instrument itself.</p><p>When the year was completed, he began to examine
and compare his observations; and having pretty well
satisfied himself as to the general laws of the phenomena,
he then endeavoured to sind out the cause of
them. He was already convinced that the apparent
motion of the stars was not owing to a nutation of the
earth's axis. The next that occurred to him, was an
alteration in the direction of the plumb-line, by which
the instrument was constantly adjusted; but this, upon
trial, proved insufficient. Then he considered what refraction
might do; but here also he met with no satisfaction.
At last, through an amazing sagacity, he
conjectured that all the phenomena hitherto mentioned,
proceeded from the progressive motion of light, and the
earth's annual motion in her orbit: for he perceived,
that if light were propagated in time, the apparent
place of a fixed object would not be the same when the
eye is at rest, as when it is moving in any other direction
but that of the line passing through the object and
the eye; and that when the eye is moving in different
directions, the apparent place of the object would be
different.</p><p>He considered this matter in the following manner.
He imagined CA to be a ray of
<figure/>
light, falling perpendicularly upon the
line BD: then, if the eye be at rest at
A, the object must appear in the direction
AC, whether light be propagated
in time, or in an instant. But
if the eye be moving from B towards
A, and light be propagated in time,
with a velocity that is to the velocity
of the eye, as AC to AB; then,
light moving from C to A, whilst the
eye moves from B to A, that particle
of it by which the object will be discerned,
when the eye in its motion
comes to A, is at C when the eye is
at B. Joining the points B, C, he
supposed the line BC to be a tube,
inclined to the line BD in the angle DBC, and of
such a diameter as to admit of but one particle of light:
then it was easy to conceive, that the particle of light
at C, by which the object must be seen when the eye
arrives at A, would pass through the tube BC, so inclined
to the line BD, and accompanying the eye in
its motion from B to A; and that it would not come
<pb n="6"/><cb/>
to the eye, placed behind such a tube, if it had any
other inclination to the line BD. If, instead of supposing
BC so small a tube, we conceive it to be the
axis of a larger; then, for the same reason, the particle
of light at C cannot pass through that axis, unless it be
inclined to BD in the same angle DBC.</p><p>In the like manner, if the eye move the contrary
way, from D towards A, with the same velocity; then
the tube must be inclined in the angle BDC. Although
therefore the true or real place of an object, be perpendicular
to the line in which the eye is moving, yet the
visible place will not be so; since that must doubtless
be in the direction of the tube. But the difference between
the true and apparent place, will be, <hi rend="italics">c&#xE6;teris paribus,</hi>
greater or less, according to the different proportions
between the velocity of light and that of the
eye: so that if we could suppose light to be propagated
in an instant, then there would be no difference between
the real and visible place of an object, although the eye
were in motion; for in that case, AC being infinite
with respect to AB, the angle ACB, which is the
difference between the true and visible place, vanishes.
But if light be propagated in time, which was then
allowed by most philosophers, then it is evident from
the foregoing considerations, that there will always be
a difference between the true and visible place of an object,
except when the eye is moving either directly towards,
or from the object. And in all cases, the sine
of the difference between the true and visible place of
the object, will be to the sine of the visible inclination
of the object to the line in which the eye is moving, as
the velocity of the eye, is to the velocity of light.</p><p>If light moved only 1000 times faster than the eye,
and an object, supposed to be at an infinite distance,
were really placed perpendicularly over the plane in
which the eye is moving; it follows, from what has
been sa&#xED;d, that the apparent place of such object will
always be inclined to that plane, in an angle of 89&#xB0;
56&#x2032; 1/2; so that it will constantly appear 3&#x2032; 1/2 from its true
place, and will seem so much less inclined to the plane,
that way towards which the eye tends. That is, if
AC be to AB or AD, as 1000 to 1, the angle
ABC will be 89&#xB0; 56&#x2032; 1/2, and the angle ACB 3&#x2032; 1/2, and
BCD or 2ACB will be 7&#x2032;, if the direction of the motion
of the eye be contrary at one time to what it is at
another.</p><p>If the earth revolve about the sun annually, and the
velocity of light were to the velocity of the earth's motion
in its orbit, as 1000 is to 1; then it is easy to
conceive, that a star really placed in the pole of the
ecliptic, would to an eye carried along with the earth,
seem to change its place continually; and, neglecting
the small difference on account of the earth's diurnal
revolution on its axis, it would seem to describe a circle
about that pole, every where distant from it by 3&#x2032; 1/2.
So that its longitude would be varied through all the
points of the ecliptic every year, but its latitude would
always remain the same. Its right ascension would also
change, and its declination, according to the different
situation of the sun in respect of the equinoctial points;
and its apparent distance from the north pole of the
equator, would be 7&#x2032; less at the autumnal, than at the
vernal equinox.
<cb/></p><p>The greatest alteration of the place of a star, in the
pole of the ecliptic, or, which in effect amounts to the
same, the proportion between the velocity of light and
the earth's motion in its orbit, being known, it will
not be difficult to find what would be the difference, on
this account, between the true and apparent place of
any other star at any time; and, on the contrary, the
difference between the true and apparent place being
given, the proportion between the velocity of light, and
the earth's motion in her orbit, may be found.</p><p>After the history of this curious discovery, related
by the author nearly in the terms above, he gives the
results of a multitude of accurate observations, made on
a great number of stars, at all seasons of the year.
From all which observations, and the theory as related
above, he found that every star, in consequence of the
earth's motion in her orbit and the progressive motion
of light, appears to describe a small ellipse in the
heavens, the transverse axis of which is equal to the
same quantity for every star, namely 40&#x2033;, nearly; and
that the conjugate axis of the ellipse, for different stars,
varies in this proportion, namely, as the right sine of
the star's latitude; that is, radius is to the sine of the
star's latitude, as the transverse axis to the conjugate
axis: and consequently a star in the pole of the ecliptic,
its latitude being there 90&#xB0;, whose sine is equal to the
radius, will appear to describe a small circle about that
pole as a centre, whose radius is equal to 20&#x2033;. He
also gives the following law of the variation of the
star's declination: if A denote the angle of position,
or the angle at the star made by two great circles drawn
from it through the poles of the ecliptic and equator,
and B another angle, whose tangent is to the tangent
of A, as radius is to the sine of the star's latitude;
then B will be equal to the difference of longitude between
the sun and the star, when the true and apparent
declination of the star are the same. And if
the sun's longitude in the ecliptic be reckoned from that
point in which it is when this happens; then the difference
between the true and apparent declination of
the star, will be always as the sine of the sun's longitude
from that point. It will also be found that the greatest
difference of declination that can be between the true
and apparent place of the star, will be to 20&#x2033;, the
semitransverse axis of the ellipse, as the sine of A to the
sine of B.</p><p>The author then shews, by the comparison of a number
of observations made on different stars, that they exactly
agree with the theory deduced from the progressive
motion of light, and that consequently it is highly probable
that such motion is the cause of those variations
in the situation of the stars. From which he infers,
that the parallax of the fixed stars is much smaller, than
hath been hitherto supposed by those, who have pretended
to deduce it from their observations. He
thinks he may venture to say, that in the stars he had
observed, the parallax does not amount to 2&#x2033;; nay,
that if it had amounted to 1&#x2033;, he should certainly have
perceived it, in the great number of observations that
he made, especially of <foreign xml:lang="greek">g</foreign> Draconis; which agreeing
with the hypothesis, without allowing any thing for
parallax, nearly as well when the sun was in conjunction
with, as in opposition to, this star, it seems very pro-
<pb n="7"/><cb/>
bable that the parallax of it is not so much as one
single second; and consequently that it is above 400000
times farther from us than the sun.</p><p>From the greatest variation in the place of the stars,
namely 40&#x2033;, Dr. Bradley deduces the ratio of the
velocity of light in comparison with that of the earth
in her orbit. In the preceding figure, AC is to AB,
as the velocity of light to that of the earth in her or
bit, the angle ACB being equal to 20&#x2033;; so that the
ratio of those velocities is that of radius to the tangent
of 20&#x2033;, or of radius to 20&#x2033;, since the tangent has no
sensible difference from so small an are: but the radius
of a circle is equal to the arc of 57&#xB0; 3/10 nearly, or equal
to 206260&#x2033;; therefore the velocity of light is to the
velocity of the earth, as 206260 to 20, or as 10313
to 1.</p><p>And hence also the time in which light passes over
the space from the sun to the earth, is easily deduced;
for this time will be to one year, as AB or 20&#x2033; to 360&#xB0;
or the whole circle; that is, 360&#xB0;: 20&#x2033; :: 365 1/4 days:
8<hi rend="sup">m</hi> 7<hi rend="sup">s</hi>, namely, light will pass from the sun to the earth
in the time of 8 minutes, 7 seconds; and this will be
the same, whatever the distance of the sun is.</p><p>Dr. Bradley having annexed to his theory the rules
or formul&#xE6; for computing the aberration of the fixed
stars in declination and right ascension; these rules have
been variously demonstrated, and reduced to other practical
forms, by Mr. Clairaut in the Memoirs of the
Academy of Sciences for 1737; by Mr. Simpson in his
Essays in 1740; by M. Fontaine des Crutes in 1744;
and several other persons. The results of these rules
are as follow: Every star appears to describe in the
course of a year, by means of the aberration, a small
ellipse, whose greater axis is 40&#x2033;, and the less axis, perpendicular
to the ecliptic, is equal to 40&#x2033; multiplied by
the sine of the star's latitude, the radius being 1.
The eastern extremity of the longer axis, marks the
apparent place of the star, the day of the opposition;
and the extremity of the less axe, which is farthest
from the ecliptic, marks its situation three months
after.</p><p>The greatest aberration in longitude, is equal to 20&#x2033;
divided by the cosine of its latitude. And the aberration
for any time, is equal to 20&#x2033; multiplied by the
cosine of the elongation of the star found for the same
time, and divided by the cosine of its latitude. This
aberration is subtractive in the first and last quadrant of
the argument, or of the difference between the longitudes
of the sun and star; and additive in the second
and third quadrants. The greatest aberration in latitude,
is equal to 20&#x2033; multiplied by the sine of the star's
latitude. And the aberration in latitude for any time,
is equal to 20&#x2033; multiplied by the sine of the star's latitude,
and multiplied also by the sine of the elongation.
The aberration is subtractive before the opposition, and
additive after it.</p><p>The greatest aberration in declination, is equal to 20&#x2033;
multiplied by the sine of the angle of position A, and
divided by the sine of B the difference of longitude
between the sun and star when the aberration in declination
is nothing. And the aberration in declination
at any other time, will be equal to the greatest aberration
multiplied by the sine of the difference between the
sun's place at the given time and his place when the
<cb/>
aberration is nothing. Also the sine of the latitude of
the star is to radius, as the tangent of A the angle of
position at the star, is to the tangent of B, the difference
of longitude between the sun and star when the
aberration in declination is nothing. The greatest aberration
in right-ascension, is equal to 20&#x2033; multiplied by
the cosine of A the angle of position, and divided by
the sine of C the difference in longitude between the
sun and star when the aberration in right ascension is
nothing. And the aberration in right-ascension at any
other time, is equal to the greatest aberration multiplied
by the sine of the difference between the sun's place
at the given time, and his place when the aberration is
nothing. Also the sine of the latitude of the star is
to radius, as the cotangent of A the angle of position,
to the tangent of C.</p><p><hi rend="smallcaps">Aberration</hi> <hi rend="italics">of the Planets,</hi> is equal to the geocentric
motion of the planet, the space it appears to move
as seen from the earth, during the time that light employs
in passing from the planet to the earth. Thus,
in the sun, the aberration in longitude is constantly 20&#x2033;,
that being the space moved by the sun, or, which is the
same thing, by the earth, in the time of 8<hi rend="sup">m</hi> 7<hi rend="sup">s</hi>, which is
the time in which light passes from the sun to the earth,
as we have seen in the foregoing article. In like manner,
knowing the distance of any planet from the earth,
by proportion it will be, as the distance of the sun is to
the distance of the planet, so is 8<hi rend="sup">m</hi> 7<hi rend="sup">s</hi> to the time of
light passing from the planet to the earth: then computing
the planet's geocentric motion in this time,
that will be the aberration of the planet, whether it be
in longitude, latitude, right-ascension, or declination.</p><p>It is evident that the aberration will be greatest in
the longitude, and very small in latitude, because the
planets deviate very little from the plane of the ecliptic,
or path of the earth; so that the aberration in the latitudes
of the planets, is commonly neglected, as insensible;
the greatest in Mercury being only 4&#x2033; 1/3, and much
less in the other planets. As to the aberrations in declination
and right-ascension, they must depend on the
situation of the planet in the zodiac. The aberration
in longitude, being equal to the geocentric motion, will
be more or less according as that motion is; it will
therefore be least, or nothing at all, when the planet is
stationary; and greatest in the superior planets Mars,
Jupiter, Saturn, &amp;c, when they are in opposition to the
sun; but in the inferior planets Venus and Mercury, the
aberration is greatest at the time of their superior conjunction.
These maxima of aberration for the several
planets, when their distance from the sun is least, are as
below: viz, for
<table><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" role="data">27&#x2033;</cell><cell cols="1" rows="1" role="data">.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">.2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Moon</cell><cell cols="1" rows="1" role="data">&#xA0;2/3</cell><cell cols="1" rows="1" role="data"/></row></table>
And between these numbers and nothing the aberrations
of the planets, in longitude, vary according to their
situations. But that of the sun varies not, being constantly
20&#x2033;, as has been before observed. And this
may alter his declination by a quantity, which varies
from 0 to near 8&#x2033;; being greatest or 8&#x2033; about the
equinoxes, and vanishing in the solstices.
<pb n="8"/><cb/></p><p>The methods of computing these, and the formulas
for all cases, are given by M. Clairaut in the Memoirs
of the Academy of Sciences for the year 1746, and
by M. Euler in the Berlin Memoirs, vol. 2, for 1746.</p><p><hi rend="italics">Optic</hi> <hi rend="smallcaps">Aberration</hi>, the deviation or dispersion of
the rays of light, when reflected by a speculum, or refracted
by a lens, by which they are prevented from
meeting or uniting in the same point, called the geometrical
focus, but are spread over a small space, and
produce a confusion of images. Aberration is either
lateral or longitudinal: the lateral aberration is measured
by a perpendicular to the axis of the speculum
or lens, drawn from the focus to meet the refracted or
reflected ray: the longitudinal aberration is the distance,
on the axis, between the focus and the point
where the ray meets the axis. The aberrations are
very amply treated in Smith's Complete System of
Opties, in 2 volumes 4to.</p><p>There are two species of aberration, distinguished
according to their different causes: the one arises from
the figure of the speculum or lens, producing a geometrical
dispersion of the rays, when these are perfectly
equal in all respects; the other arises from the unequal
refrangibility of the rays of light themselves; a discovery
that was made by Sir Isaac Newton, and for this
reason it is often called the Newtonian aberration. As
to the former species of aberration, or that arising
from the sigure, it is well known that if rays issue from
a point at a given distance; then they will be reflected
into the other focus of an ellipse having the given luminous
point for one focus, or directly from the other
focus of an hyperbola; and will be variously dispersed
by all other figures. But if the luminous point be
infinitely distant, or, which is the same, the incident
rays be parallel, then they will be reflected by a parabola
into its focus, and variously dispersed by all other
figures. But those figures are very difficult to make,
and therefore curved specula are commonly made spherical,
the figure of which is generated by the revolution
of a circular arc, which produces an aberration
of all rays, whether they are parallel or not, and therefore
it has no accurate geometrical focus which is common
to all the rays. Let BVF represent a concave spherical
speculum, whose centre is C; and let AB, EF
be incident rays parallel to the axis CV. Because the
angle of incidence is equal to the angle of reflection in
<figure/>
all cases, therefore if the radii CB, CF be drawn to the
points of incidence, and thence BD making the angle
CBD equal to the angle CBA, and FG making the
angle CFG equal to the angle CFE; then BD, FG
will be the reflected rays, and D, G, the points where
<cb/>
they meet the axis. Hence it appears that the point
of coincidence with the axis is equally distant from the
point of incidence and the centre: for because the angle
CBD is equal to the angle CBA, which is equal to
the alternate angle BCD, therefore their opposite
sides CD, DB are equal: and in like manner, in any
other, GF is equal to GC. And hence it is evident
that when B is indefinitely near the vertex V, then D
is in the middle of the radius CV; and the nearer the
incident ray is to the axis CV, the nearer will the reflected
ray come to the middle point D; and the contrary.
So that the aberration DG of any ray EFG,
is always more and more, as the incident ray is farther
from the axis, or the incident point F from the vertex
V; till when the distance VI is 60 degrees, then the
reflected ray falls in the vertex V, making the aberration
equal to the whole length DV. And this shews
the reason why specula are made of a very small segment
of a sphere, namely, that all their reflected rays may
arrive very near the middle point or focus D, to produce
an image the most distinct, by the least aberration of
the rays. And in like manner for rays refracted through
lenses.</p><p>In spherical lenses, Mr. Huygens has demonstrated
that the aberration from the figure, in different lenses,
is as follows.</p><p>1. In all plano-convex lenses, having their plane surface
exposed to parallel rays, the longitudinal aberration
of the extreme ray, or that remotest from the axis, is
equal to 9/2 of the thickness of the lens.</p><p>2. In all plano-convex lenses, having their convex
surface exposed to parallel rays, the longitudinal aberration
of the extreme ray, is equal to 7/6 of the thickness
of the lens. So that in this position of the same planoconvex
lens, the aberration is but about one-fourth
of that in the former; being to it only as 7 to 27.</p><p>3. In all double convex lenses of equal spheres, the
aberration of the extreme ray, is equal to 5/3 of the
thickness of the lens.</p><p>4. In a double convex lens, the radii of whose spheres
are as 1 to 6, if the more convex surface be exposed to
parallel rays, the aberration from the figure is less than
in any other spherical lens; being no more than 15/14 of
its thickness.</p><p>But the foregoing species of aberration, arising from
the figure, is very small, and easily remedied, in comparison
with the other, arising from the unequal refrangibility
of the rays of light; insomuch that Sir
Isaac Newton shews in his Optics, pa. 84 of the 8vo.
edition, that if the object-glass of a telescope be planoconvex,
the plane side being turned towards the object,
and the diameter of the sphere, to which the convex
side is ground, be 100 seet, the diameter of the aperture
being 4 inches, and the ratio of the sine of incidence
out of glass into air, be to that of refraction, as
20 to 31; then the diameter of the circle of aberrations
will in this case be only 961/72000000 parts of an inch:
while the diameter of the little circle, through which
the same rays are scattered by unequal refrangibility,
will be about the 55th part of the aperture of the object-glass,
which here is 4 inches. And therefore the
error arising from the spherical figure of the glass, is to
the error arising from the different refrangibility of the
rays, as 961/72000000 to 4/55, that is as 1 to 5449.
<pb n="9"/><cb/></p><p>So that it may seem strange that objects appear
through telescopes so distinct as they do, considering
that the error arising from the different refrangibility, is
almost incomparably larger than that of the figure.
Newton however solves the difficulty by observing that
the rays, under their various aberrations, are not scattered
uniformly over all the circular space, but collected
insinitely more dense in the centre than in any other
part of the circle; and that, in the way from the centre
to the circumference, they grow more and more
rare, so as at the circumference to become infinitely
rare; and, by reason of their rarity, they are not
strong enough to be visible, unless in the centre, and
very near it.</p><p>In consequence of the discovery of the unequal refrangibility
of light, and the apprehension that equal
refractions must produce equal divergencies in every
sort of medium, it was supposed that all spherical objectglasses
of telescopes would be equally affected by the
different refrangibility of light, in proportion to their
aperture, of whatever materials they might be constructed:
and therefore that the only improvement that
could be made in refracting telescopes, was that of increasing
their length. So that Sir Isaac Newton, and
other persons after him, despairing of success in the use
and fabric of lenses, directed their chief attention to
the construction of reflecting telescopes.</p><p>However, about the year 1747, M. Euler applied
himself to the subject of refraction; and pursued a
hint suggested by Newton, for the design of making
object-glasses with two lenses of glass inclosing water
between them; hoping that, by constructing them of
different materials, the refractions would balance one
another, and so the usual aberration be prevented. Mr.
John Dollond, an ingenious optician in London, minutely
examined this scheme, and found that Mr.
Euler's principles were not satisfactory. M. Clairaut
likewise, whose attention had been excited to the same
subject, concurred in opinion that Euler's speculations
were more ingenious than useful. This controversy,
which seemed to be of great importance in the science of
optics, engaged also the attention of M. Klingenstierna
of Sweden, who was led to make a careful examination
of the 8th experiment in the second part of Newton's
Optics, with the conclusions there drawn from it. The
consequence was, that he found that the rays of light,
in the circumstances there mentioned, did not lose their
colour, as Sir Isaac had imagined. This hint of the
Swedish philosopher led Mr. Dollond to re-examine the
same experiment: and after several trials it appeared,
that different substances caused the light to diverge very
differently, in proportion to their general refractive
powers. In the year 1757 therefore he procured
wedges of different kinds of glass, and applied them
together so that the refractions might be made in contrary
directions, that he might discover whether the
refraction and divergency of colour would vanish together.
The result of his sirst trials encouraged him to
persevere; for he discovered a difference far beyond his
hopes in the qualities of different kinds of glass, with
respect to their divergency of colours. The Venice
glass and English crown glass were found to be nearly
allied in this respect: the common English plate glass
made the rays diverge more; and the English flint
<cb/>
glass most of all. But without enquiring into the
cause of this difference, he proceeded to adapt wedges
of crown glass, and of white flint glass, ground to different
angles, to each other, so as to refract in different
directions; till the refracted light was entirely free from
colours. Having measured the refractions of each
wedge, he found that the refraction of the white glass
was to that of the crown glass, nearly as 2 to 3: and
he hence concluded in general, that any two wedges
made in this proportion, and applied together so as to
refract in contrary directions, would refract the light
without any aberration of the rays.</p><p>Mr. Dollond's next object was to make similar trials
with spherical glasses of different materials, with the
view of applying his discovery to the improvement of
telescopes: and here he perceived that, to obtain a refraction
of light in contrary directions, the one glass
must be concave, and the other convex; and the
latter, which was to refract the most, that the rays
might converge to a real focus, he made of crown
glass, the other of white flint glass. And as the
refractions of spherical glasses are inversely as their
focal distances, it was necessary that the focal distances
of the two glasses should be inversely as the ratios of
the refractions of the wedges; because that, being thus
proportioned, every ray of light that passes through
this compound glass, at any distance from its axis,
will constantly be refracted, by the difference between
two contrary refractions, in the proportion required;
and therefore the different refrangibility of the light
will be entirely removed.</p><p>But in the applications of this ingenious discovery
to practice, Mr. Dollond met with many and great
difficulties. At length, however, after many repeated
trials, by a resolute perseverance, he succeeded so far as
to construct refracting telescopes much superior to any
that had hitherto been made; representing objects with
great distinctness, and in their true colours.</p><p>Mr. Clairaut, who had interested himself from the
beginning in this discovery, now endeavoured to ascertain
the principles of Mr. Dollond's theory, and to lay
down rules to facilitate the construction of these new
telescopes. With this view he made several experiments,
to determine the resractive power of different kinds
of glass, and the proportions in which they separated
the rays of light: and from these experiments he deduced
several theorems of general use. M. D'Alembert
made likewise a great variety of calculations to
the same purpose; and he shewed how to correct the
errors to which these telescopes are subject, sometimes
by placing the object-glasses at a small distance from
each other, and sometimes by using eye-glasses of different
refractive powers. But though foreigners were
hereby supplied with the most accurate calculations,
they were very defective in practice. And the English
telescopes, made, as they imagined, without any precise
rule, were greatly superior to the best of their construction.</p><p>M. Euler, whose speculations had sirst given occasion
to this important and useful enquiry, was very reluctant
in admitting Mr. Dollond's improvements, because
they militated against a pre-conceived theory of his
own. At last however, after several altercations, being
convinced of their reality and importance by M. Clair-
<pb n="10"/><cb/>
aut, he assented; and he soon after received farther satisfaction
from the experiments of M. Zeiher, of Petersburgh.</p><p>M. Zeiher shewed by experiments that it is the lead,
in the composition of glass, which gives it this remarkable
property, namely, that while the refraction of the
mean rays is nearly the same, that of the extreme rays
considerably differs. And, by increasing the lead, he
produced a kind of glass, which occasioned a much
greater separation of the extreme rays than that of the
flint glass used by Mr. Dollond, and at the same time
considerably increased the mean refraction. M. Zeiher,
in the course of his experiments, made glass of minium
and lead, with a mixture also of alkaline salts; and he
found that this mixture greatly diminished the mean
refraction, and yet made hardly any change in the dispersion:
and he at length obtained a kind of glass
greatly superior to the flint glass of Mr. Dollond for
the construction of telescopes; as it occasioned three
times as great a dispersion of the rays as the common
glass, whilst the mean refraction was only as 1.61 to 1.</p><p>Other improvements were also made on the new or
achromatic telescopes by the inventor Mr. John Dollond,
and by his son Peter Dollond; which may be
seen under the proper words. For various dissertations
on the subject of the aberration of light, colours, and
the figure of the glass, see Philos. Trans. vols. 35, 48,
50, 51, 52, 55, 60; Memoirs of the Academy of
Sciences of Paris, for the years 1737, 1746, 1752,
1755, 1756, 1757, 1762, 1764, 1765, 1767, 1770;
the Berlin Ac. 1746, 1762, 1766; Swed. Mem. vol.
16; Com. Nov. Petripol. 1762; M. Euler's Dioptrics;
M. d'Alembert's Opuscules Math.; M. de Rochon
Opuscules; &amp;c, &amp;c.</p></div1><div1 part="N" n="ABRIDGING" org="uniform" sample="complete" type="entry"><head>ABRIDGING</head><p>, <hi rend="italics">in Algebra,</hi> is the reducing a compound
equation, or quantity, to a more simple form of
expression. This is done either to save room, or the
trouble of writing a number of symbols; or to simplisy
the expression, either to ease the memory, or to render
the formula more easy and general.</p><p>So the equation , by putting
<hi rend="italics">p</hi> = <hi rend="italics">a, q</hi> = <hi rend="italics">ab,</hi> and <hi rend="italics">r</hi> = <hi rend="italics">abc,</hi> becomes </p><p>And the equation , by putting
, and , becomes .</p></div1><div1 part="N" n="ABSCISS" org="uniform" sample="complete" type="entry"><head>ABSCISS</head><p>, <hi rend="smallcaps">Abscisse</hi>, or <hi rend="smallcaps">Abscissa</hi>, is a part or
segment cut off a line, terminated at some certain point,
by an ordinate to a curve; as AP or BP.
<figure/></p><p>The absciss may either commence at the vertex of
the curve, or at any other fixed point. And it may be
taken either upon the axis or diameter of the curve,
or upon any other line drawn in a given position.
<cb/></p><p>Hence there are an infinite number of variable abscisses,
terminated at the same fixed point at one end, the
other end of them being at any point of the given line
or diameter.</p><p>In the common parabola, each ordinate PQ has but
<figure/>
one absciss AP; in the ellipse or circle, the ordinate
has two abscisses AP, BP lying on the opposite sides
of it; and in the hyperbola the ordinate PQ has also
two abscisses, but they lie both on the same side of it.
That is, in general, a line of the second kind, or a
curve of the first kind, may have two abscisses to each
ordinate. But a line of the third order may have three
abscisses to each ordinate; a line of the fourth order
may have four; and so on.</p><p>The use of the abscisses is, in conjunction with the
ordinates, to express the nature of the curves, either
by some proportion or equation including the abfcifs
and its ordinate, with some other fixed invariable line
or lines. Every different curve has its own peculiar
equation or property by which it is expressed, and different
from all others: and that equation or expression
is the same for every ordinate and its abscisses, whatever
point of the curve be taken. So, in the circle, the
square of any ordinate is equal to the rectangle of its
two abscisses, or AP.PB = PQ<hi rend="sup">2</hi>; in the parabola,
the square of the ordinate is equal to the rectangle of
the absciss and a certain given line called the parameter;
in the ellipse and hyperbola, the square of the ordinate
is always in a certain constant proportion to the
rectangle of the two abscisses, namely, as the square of
the conjugate to the square of the transverse, or as the
parameter is to the transverse axis; and so other properties
in other curves.</p><p>When the natures or properties of curves are expressed
by algebraic equations, any general absciss, as
AP, is commonly denoted by the letter <hi rend="italics">x,</hi> and the ordinate
PQ by the letter <hi rend="italics">y;</hi> the other or constant lines
being represented by other letters. Then the equations
expressing the nature of these curves are as follow;
namely, for the
circle , where <hi rend="italics">d</hi> is the diameter AB;
parabola - <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi> , where <hi rend="italics">p</hi> is the parameter;
<hi rend="brace"><note anchored="true" place="unspecified">ellipse - <hi rend="italics">t</hi><hi rend="sup">2</hi> : <hi rend="italics">c</hi><hi rend="sup">2</hi> :: <hi rend="italics">tx</hi> - <hi rend="italics">x</hi><hi rend="sup">2</hi> : <hi rend="italics">y</hi><hi rend="sup">2</hi>,
hyperbola <hi rend="italics">t</hi><hi rend="sup">2</hi> : <hi rend="italics">c</hi><hi rend="sup">2</hi> :: <hi rend="italics">tx</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi> : <hi rend="italics">y</hi><hi rend="sup">2</hi>,</note>
where <hi rend="italics">t</hi> is the transverse,
&amp; <hi rend="italics">c</hi> the conjugate axis.</hi></p></div1><div1 part="N" n="ABSIS" org="uniform" sample="complete" type="entry"><head>ABSIS</head><p>, ABSIDES. See <hi rend="smallcaps">Apsis, Apsides.</hi>
<pb n="11"/><cb/></p><p>ABSOLUTE <hi rend="smallcaps">Equation</hi>, <hi rend="italics">in Aftronomy,</hi> is the
sum of the optic and excentric equations. The apparent
inequality of a planet's motion, arising from its
not being equally dislant from the earth at all times,
is called its optic equation; and this would subsist even
if the planet's real motion were uniform. The excentric
inequality is caused by the planet's motion being
not uniform. To illustrate this, conceive the sun to
move, or to appear to move, in the circumference of a
circle, in whose centre the earth is placed. It is manifest,
that if the sun move uniformly in this circle,
then he must appear to move uniformly to a spectator
at the earth; and in this case there will be no optic nor
excentric equation. But suppose the earth to be placed
out of the centre of the circle; and then, though the
sun's motion should be really uniform, it would not appear
to be so, being seen from the earth; and in this
case there would be an optic equation, without an excentric
one. Imagine farther, the sun's orbit to be,
not circular, but elliptical, and the earth in its focus:
it will be full as evident that the sun cannot appear to
have an uniform motion in such ellipse; so that his
motion will then be subject to two equations; that is,
the optic equation, and the excentric equation. <hi rend="italics">See</hi>
<hi rend="smallcaps">Equation</hi>, and <hi rend="smallcaps">Optic Inequality.</hi></p><p><hi rend="smallcaps">Absolute</hi> <hi rend="italics">Number,</hi> in Algebra, is that term or
member of an equation that is completely known, and
which is equal to all the other, or unknown terms,
taken together; and is the same as what Vieta calls
the <hi rend="italics">homogeneum comparationis.</hi> So, of the equation
, or , the absolute
number, or known term, is 36.</p><p><hi rend="smallcaps">Absolute</hi> <hi rend="italics">Gravity, Motion, Space, Time, &amp;c.</hi> See
the respective substantives.</p><p>ABSTRACT <hi rend="smallcaps">Mathematics</hi>, otherwise called
pure mathematics, is that which treats of the properties
of magnitude, figure, or quantity, absolutely and
generally confidered, without restriction to any species
in particular: such as Arithmetic and Geometry. In
this sense, abstract or pure mathematics, is opposed
to mixed mathematics, in which simple and abstract
properties, and the relations of quantities, primitively
considered in pure mathematics, are applied to sensible
objects; as in astronomy, hydrostatics, optics, &amp;c.</p><p><hi rend="smallcaps">Abstract</hi> <hi rend="italics">Number,</hi> is a number, or collection of
units, considered in itself, without being applied to
denote a collection of any particular and determinate
things. So, for example, 3 is an abstract number, so
far as it is not applied to something: but when we say
3 feet, or 3 persons, the 3 is no longer an abstract, but
a concrete number.</p></div1><div1 part="N" n="ABSURD" org="uniform" sample="complete" type="entry"><head>ABSURD</head><p>, or <hi rend="smallcaps">Absurdum</hi>, a term commonly used
in demonstrating converse propositions; a mode of demonstration,
in which the proposition intended is not
proved in a direct manner, by principles before laid
down; but it proves that the contrary is absurd or impossible;
and so indirectly as it were proves the proposition
itself. The 4th proposition in the first book of
Euclid, is the first in which he makes use of this mode
of proof; where he shews that if the extremities of
two lines coincide, those lines will coincide in all their
parts, otherwise they would inclose a space, which is
absurd or contrary to the 10th axiom. Most converse
<cb/>
propositions are proved in this way, which mode of
proof is called <hi rend="italics">reductio ad absurdum.</hi></p><p>ABUNDANT <hi rend="smallcaps">Number</hi>, in <hi rend="italics">Arithmetic,</hi> is a number
whose aliquot parts, added all together, make a
sum which is greater than the number itself. Thus 12
is an abundant number, because its aliquot parts,
namely 1, 2, 3, 4, 6, when added together, make 16,
which is greater than the number 12 itself.</p><p>An abundant number is opposed to a deficient one,
which is less than the sum of its aliquot parts taken together,
as the number 14, whose aliquot parts 1, 2, 7,
make no more than 10; and to a perfect number, which
is exactly equal to the sum of all its aliquot parts, as
the number 6, which is equal to the sum of 1, 2, 3,
which are its aliquot parts.</p></div1><div1 part="N" n="ACADEMICIAN" org="uniform" sample="complete" type="entry"><head>ACADEMICIAN</head><p>, a member of a society called
an academy, instituted for the promotion of arts, sciences,
or natural knowledge in general.</p></div1><div1 part="N" n="ACADEMICS" org="uniform" sample="complete" type="entry"><head>ACADEMICS</head><p>, an ancient sect of philosophers,
who followed the doctrine of Socrates and Plato, as to
the uncertainty of knowledge, and the incomprehensibility
of truth.</p><p><hi rend="italics">Academic,</hi> in this sense, amounts to much the same
with Platonist; the difference between them being only
in point of time. Those who embraced the system of
Plato, among the ancients, were called <hi rend="italics">academici,</hi> academician
or academic; whereas those who did the same
since the restoration of learning, have assumed the denomination
of Platonists.</p><p>We usually reckon three sects of academics; though
some make five. The ancient academy was that of
which Plato was the chief.</p><p>Arcessilas, one of Plato's successors, introducing
some alterations into the philosophy of this sect, founded
what they call the second academy.</p><p>The establishment of the third, called also the new
academy, is attributed to Lacydes, or rather to Carneades.</p><p>Some authors add a fourth, founded by Philo; and
a fifth, by Antiochus, called the Antiochan, which
tempered the ancient academy with Stoicism.</p><p>The ancient academy doubted of every thing; and
carried this principle so far as to make it a doubt, whether
or no they ought to doubt. It was a kind of a
principle with them, never to be certain or satisfied of
any thing; never to affirm or to deny any thing, either
for true or false.</p><p>The new academy was somewhat more reasonable;
they acknowledged several things for truths, but without
attaching themselves to any with entire assurance.
These philosophers had found that the ordinary commerce
of life and society was inconsistent with the absolute
and universal doubtfulness of the ancient academy:
and yet it is evident that they looked upon things
rather as probable, than as true and certain: by this
amendment thinking to secure themselves from those
absurdities into which the ancient academy had fallen.</p></div1><div1 part="N" n="ACADEMIST" org="uniform" sample="complete" type="entry"><head>ACADEMIST</head><p>, the same as Academician.</p></div1><div1 part="N" n="ACADEMY" org="uniform" sample="complete" type="entry"><head>ACADEMY</head><p>, <hi rend="smallcaps">Academia</hi>, in <hi rend="italics">Antiquity,</hi> a fine villa
or pleasure house, in one of the submbs of Athens,
about a mile from the city; where Plato, and the wise
men who followed him, held assemblies for disputes
<pb n="12"/><cb/>
and philosophical conference; which gave the name to
the sect of Academics.</p><p>The house took its name, <hi rend="italics">Academy,</hi> from one Academus,
or Ecademus, a citizen of Athens, to whom it
originally belonged: he lived in the time of Theseus;
and here he used to have gymnastic sports or exercises.</p><p>The academy was farther improved by Cimon, and
adorned with fountains, trees, shady walks, &amp;c, for the
convenience of the philosophers and men of learning,
who here met to confer and dispute for their mutual
improvement. It was surrounded with a wall by Hipparchus,
the son of Pisistratus; and it was also used as
the burying-place for illustrious persons, who had deserved
well of the republic.</p><p>It was here that Plato taught his philosophy; and
hence it was that all public places, destined for the assemblies
of the learned and ingenious, have been since
called <hi rend="italics">Academies.</hi></p><p>Sylla facrificed the delicious walks and groves of the
academy, which had been planted by Cimon, to the
ravages of war; and employed those very trees in constructing
machines to batter the walls of the city which
they had adorned.</p><p>Cicero too had a villa, or country retirement, near
Puzzuoli, which he called by the same name, <hi rend="italics">Academia.</hi>
Here he used to entertain his philosophical
friends; and here it was that he composed his Academical
Questions, and his books <hi rend="italics">De Natur&#xE2; Deorum.</hi></p><div2 part="N" n="Academy" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Academy</hi></head><p>, among the moderns, denotes a regular
society or company of learned persons, instituted under
the protection of some prince, or other public authority,
for the cultivation and improvement of arts or
sciences.</p><p>Some authors confound Academy with University;
but though much the same in Latin, they are very different
things in English. An university is properly a
body composed of graduates in the feveral faculties;
of professors, who teach in the public schools; of regents
or tutors, and students who learn under them,
and aspire likewise to degrees. Whereas an academy
is not intended to teach, or profess any art or science,
but to improve it: it is not for novices to be instructed
in, but for those that are more knowing; for persons
of learning to confer in, and communicate their lights
and discoveries to each other, for their mutual benefit
and improvement.</p><p>The first modern academy we read of, was established
by Charlemagne, by the advice of Alcuin, an English
monk: it was composed of the chief geniuses of the
court, the emperor himself being a member. In their
academical conferences, every person was to give some
account of the ancient authors he had read; and each
one assumed the name of some ancient author, that
pleased him most, or some celebrated person of antiquity.
Alcuin, from whose letters we learn these particulars,
took that of Flaccus, the surname of Horace;
a young lord, named Augilbert, took that of Homer;
Adelard, bishop of Corbie, was called Augustin; Recluse,
bishop of Mentz, was Dametas; and the king
himself, David.</p><p>Since the revival of learning in Europe, academies
have multiplied greatly, most nations being furnished
with several, and from their communications the chief
<cb/>
improvements have been made in the arts and sciences,
and in cultivating natural knowledge. There are now
academies for almost every art, or species of knowledge;
but I shall give a short account only of those
institutions of this kind, which regard the cultivation of
subjects mathematical or philosophical, which are the
proper and peculiar objects of our undertaking.</p><p>Italy abounds more in academies than all the world
besides; there being enumerated by Jarckius not less
than sive hundred and fifty in all; and even to the
amount of twenty-five in Milan itself. These are however
mostly of a private and inferior nature; the consequence
of their too great number.</p><p>The first academy of a philosophical kind was established
at Naples, in the house of Baptista Porta, about
the year 1560, under the name of <hi rend="italics">Academy Secretorum
Natur&#xE6;;</hi> being formed for the improvement of natural
and mathematical knowledge. This was succeeded by
the</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">of Lyncei,</hi> founded at Rome by prince
Frederick Cesi, towards the end of the same century.
It was rendered famous by the notable discoveries made
by several of its members; among whom was the celebrated
Galileo Galilei.</p><p>Several other academies contributed also to the advancement
of the sciences; but it was by speculations
rather than by repeated experiments on the phenomena
of nature: such were the academy of Bessarian at Rome,
and that of Laurence de Medicis at Florence, in the
15th century; and in the 16th were that of Infiammati
at Padua, of Vegna Juoli at Rome, of Ortolani at
Placentia, and of Umidi at Florence. The first of these
studied fire and pyrotechnia, the second wine and vineyards,
the third pot-herbs and gardens, the fourth
water and hydraulics. To these may be added that of
Venice, called La Veneta, and sounded by Frederick
Badoara, a noble Venetian; another in the same city,
of which Campegio, bishop of Feltro, appears to have
been the chief; also that of Cosenza, or La Consentina,
of which Bernadin Telesio, Sertorio Quatromanni,
Paulus Aquinas, Julio Cavalcanti, and Fabio Cicali,
celebrated philosophers, were the chief members. The
compositions of all these academies, of the 16th century,
were good in their kind; but none of them comparable
to those of the Lyncei.</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">del Cimento,</hi> that is, of Experiments, arose
at Florence, some years after the death of Torricelli,
namely in the year 1657, under the protection of prince
Leopold of Tuscany, afterwards cardinal de Medicis,
and brother to the Grand Duke Ferdinand the Second.
Galileo, Toricelli, Aggiunti, and Viviani had prepared
the way sor it: and some of its chief members were
Paul del Buono, who in 1657 invented the instrument
for trying the incompressibility of water, namely a thick
globular shell of gold, having its cavity filled with
water; then the globe being compressed by a strong
screw, the water came through the pores of the gold
rather than yield to the compression: also, Alphonsus
Borelli, well known for his ingenious treatise <hi rend="italics">De Motu
Animalium,</hi> and other works; Candide del Buono,
brother of Paul; Alexander Marsili, Vincent Viviani,
Francis Rhedi, and the Count Laurence
Magalotti, secretary of this academy, who pub-
<pb n="13"/><cb/>
lished a volume of their curious experiments in 1667,
under the title of <hi rend="italics">Saggi di Naturali Esperienze;</hi> a
copy of which being presented to the Royal Society,
it was translated into English by Mr. Waller, and published
at London, in 4to, 1684: A curious collection
of tracts, containing ingenious experiments on the
pressure of the air, on the compressing of water, on
cold, heat, ice, magnets, electricity, odours, the motion
of sound, projectiles, light, &amp;c, &amp;c. But we
have heard little or nothing more of the academy since
that time. It may not be improper to observe here,
that the Grand Duke Ferdinand, above mentioned,
was no mean philosopher and chemist, and that he
invented thermometers, of which the construction and
use may be seen in the collection of the academy del
Cimento.</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">degl' Inquieti</hi> at Bologna, incorporated
afterwards into that <hi rend="italics">della traccia</hi> in the same city, followed
the example of that <hi rend="italics">del Cimento.</hi> The members
met at the house of the abbot Antonio Sampieri;
and here Geminiano Montanari, one of the chief members,
made excellent discourses on mathematical and
philosophical subjects, some parts of which were published
in 1667, under the title of Pensieri Fisico-Mathematici.
This academy afterwards met in an apartment
of Eustachio Manfredi; and then in that of Jacob
Sandri; but it arrived at its chief lustre while its assemblies
were held in the palace Marsilli.</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">of Rossano,</hi> in the kingdom of Naples,
called <hi rend="italics">La Societa Scientifica Rossanese degl' Incuriosi,</hi>
was founded about the year 1540, under the name of
<hi rend="italics">Naviganti;</hi> and was renewed under that of <hi rend="italics">Spensierati</hi>
by Camillo Tuscano, about the year 1600. It was
then an academy of belles-lettres, but was afterwards
transformed into an academy of sciences, on the solicitation
of the learned abbot Don Giacinto Gimma;
who, being made president under the title of promotergeneral,
in 1695, gave it a new set of regulations. He
divided the academists into several classes, namely, grammarians,
rhetoricians, poets, historians, philosophers,
physicians, mathematicians, lawyers, and divines; with
a separate class for cardinals and persons of quality. To
be admitted a member, it was necessary that the candidate
have degrees in some faculty. Members, in the
beginning of their books, are not allowed to take the
title of <hi rend="italics">academist</hi> without a written permission from the
president, which is not granted till the work has been
examined by the censors of the academy. This permission
is the highest honour the academy can confer;
since they hereby, as it were, adopt the work, and engage
to answer for it against any criticisms that may
be made upon it. The president himself is not exempt
from this law: and it is not permitted that any academist
publish any thing against the writings of another,
without leave obtained from the society.</p><p>There have been several other academies of sciences
in Italy, but which have not subsisted long, for want
of being supported by the princes. Such were at
Naples that of the <hi rend="italics">Investiganti,</hi> founded about the year
1679, by the marquis d'Arena, Don Andrea Concubletto;
and that which, about the year 1698, met in
the palace of Don Lewis della Cerda, the duke de
Medina, and viceroy of Naples: at Rome, that of
<hi rend="italics">Fisico-Matematici,</hi> which in 1686 met in the house of
<cb/>
Signior Ciampini: at Verona, that of <hi rend="italics">Aletosili,</hi> founded
the same year by Signior Joseph Gazola, and which
met in the house of the count Serenghi della Cucca:
at Brescia, that of <hi rend="italics">Filesotici,</hi> founded the same year
for the cultivation of philosophy and mathematics, and
terminated the year following: that of F. Francisco
Lana, a jesuit of great skill in these sciences: and lastly
that of Fisico-Critici at Sienna, founded in 1691, by
Signior Peter Maria Gabrielli.</p><p>Some other academies, still subsisting in Italy, repair
with advantage the loss of the former. One of the
principal is the academy of Filarmonici at Verona, supported
by the marquis Scipio Maffei, one of the most
learned men in Italy; the members of which academy,
though they cultivate the belles lettres, do not
neglect the sciences. The academy of <hi rend="italics">Ricovrati</hi> at
Padua still subsists with reputation; in which; from
time to time, learned discourses are held on philosophical
subjects. The like may be said of the academy of the
<hi rend="italics">Muti di Reggio,</hi> at Modena. At Bologna is an academy
of sciences, in a flourishing condition, known by the
name of <hi rend="italics">The Institute of Bologna;</hi> which was founded
in 1712 by count Marsigli, for cultivating physics,
mathematics, medicine, chemistry, and natural history.
The history of it is written by M. de Limiers, from
memoirs furnished by the founder himself. Among the
new academies, the first place, after the Institute of
Bologna, is given to that of the Countess Donna
Clelio Grillo Boromeo, one of the most learned ladies
of the age, to whom Signior Gimma dedicates his
literary history of Italy. She had lately established an
academy of experimental philosophy in her palace at
Milan; of which Signior Vallisnieri was nominated
president, and had already drawn up the regulations for
it, though we do not find it has yet taken place. In the
number of these academies may also be ranked the assembly
of the learned, who of late years met at Venice
in the house of Signior Cristino Martinelli, a noble
Venetian, and a great patron of learning.</p><p><hi rend="smallcaps">Academia</hi> <hi rend="italics">Cosmografica,</hi> or that of the Argonauts,
was instituted at Venice, at the instance of F. Coronelli,
for the improvement of geography; the design
being to procure exact maps, geographical, topographical,
hydrographical, and ichnographical, of the
celestial as well as terrestrial globe, and their several
regions or parts, together with geographical, historical,
and astronomical descriptions accommodated to them:
to promote which purposes, the several members oblige
themselves, by their subscription, to take one copy or
more of each piece published under the direction of
the academy; and to advance the money, or part of
it, to defray the charge of publication. To this end
there are three societies settled, namely at Venice,
Paris, and Rome; the first under F. Moro, provincial
of the Minorites of Hungary; the second under the
abbot Laurence au Rue Payenne au Marais; the third
under F. Ant. Baldigiani, jesuit, professor of mathematics
in the Roman college; to whom those address
themselves who are willing to engage in this design. The
Argonauts number near 200 members in the different
countries of Europe; and their device is the terraqueous
globe, with the motto <hi rend="italics">Plus ultra.</hi> All the globes, maps,
and geographical writings of F. Coronelli have been
published at the expence of this academy.
<pb n="14"/><cb/></p><p><hi rend="smallcaps">The Academy</hi> <hi rend="italics">of Apatists,</hi> or Impartial Academy,
deserves to be mentioned on account of the extent of its
plan, including universally all arts and sciences. It
holds from time to time public meetings at Florence,
where any person, whether academist or not, may read
his works, in whatever form, language, or subject; the
academy receiving all with the greatest impartiality.</p><p>In France there are many academies for the improvement
of arts and sciences. F. Mersenne, it is
said, gave the first idea of a philosophical academy in
France, about the beginning of the seventeenth century,
by the conferences of mathematicians and naturalists,
held occasionally at his lodgings; at which Des Cartes,
Gassendus, Hobbes, Roberval, Pascal, Blondel, and
others, assisted. F. Mersenne proposed to each of them
certain problems to examine, or certain experiments to
be made. These private assemblies were succeeded by
more public ones, formed by M. Monmort, and M. Thevenot,
the celebrated traveller. The French example
animated several Englishmen of rank and learning to
erect a kind of philosophical academy at Oxford, towards
the close of Cromwell's administration; which
after the restoration was erected, by public authority,
into a Royal Society: an account of which see under
the word. The English example, in its turn,
animated the French. In 1666 Louis XIV, assisted by
the counsels of M. Colbert, founded an academy of
fciences at Paris, called the</p><p><hi rend="smallcaps">Academie</hi> <hi rend="italics">Royale des Sciences, or Royal Academy of
Sciences,</hi> for the improvement of philosophy, mathematics,
chemistry, medicine, belles-lettres, &amp;c. Among
the principal members, at the commencement in 1666,
were the respectable names of Carcavi, Huygens, Roberval,
Frenicle, Auzout, Picard, Buot, Du Hamel
the Secretary, and Mariotte. There was a perfect
equality among all the members, and many of them received
salaries from the king, as at present. By the
rules of the academy, every class was to meet twice a
week; the philosophers and geometricians were to
meet, separately, every Wednesday, and then both together
on the Saturday, in a room of the king's library,
where the philosophical and mathematical books were
kept: the history class was to meet on the Monday
and Thursday in the room of the historical books; and
the class of belles-lettres on the Tuesday and Friday:
and on the first Thursday of every month all the classes
met together, and by their secretaries made a mutual
report of what had been transacted by each class during
the preceding month.</p><p>In 1699, on the application of the president, the
abb&#xE9; Bignon, the academy received, under royal authority
and protection, a new form and constitution; by
the articles of which, the academy was to consist of four
sorts of members, namely honorary, pensionary, associates,
and eleves. The honorary class to consist of ten
persons, and the other three classes of twenty persons
each. The president to be chosen annually out of the
honorary class, and the secretary and treasurer to be
perpetual, and of the pensionary class. The meetings
to be twice a week, on the Wednesday and Saturday;
besides two public meetings in the year.</p><p>Of the pensionaries, or those who receive salaries,
three to be geometricians, three astronomers, three mechanists,
three anatomists, three botanists, and three
<cb/>
chemists, the other two being the secretary and treasurer.
Of the twenty associates, of which twelve to be
French, and eight might be foreigners, two were to
cultivate geometry, two astronomy, two mechanics,
two anatomy, two botany, and two chemistry. Of the
twenty eleves, one to be attached to each pensionary,
and to cultivate his peculiar branch of science. The
pensionaries and their eleves to reside at Paris. No regulars
nor religious to be admitted, except into the
honorary class: nor any person to be admitted a pensioner
who was not known by some considerable work,
or some remarkable discovery.</p><p>In 1716 the Duke of Orleans, then regent of France,
by the king's authority made some alteration in their
constitution. The class of eleves was suppressed; and
instead of them were instituted twelve adjuncts, two to
each of the six classes of pensioners. The honorary
members were increased to twelve: and a class of fix
free associates was made, who were not under the obligation
of cultivating any particular branch of science,
and in this class only could the regulars or religious be
admitted. A president and vice-president to be appointed
annually from the honorary class, and a director
and sub-director annually from that of the pensioners.
And no person to be allowed to make use of
his quality of academician, in the title of any of his
books that he published, unless such book were first approved
by the academy.</p><p>The academy has for a device or motto, <hi rend="italics">Invenit &amp;
perficit.</hi> And the meetings, which were formerly held
in the king's library, have since the year 1699 been
held in a fine hall of the old Louvre.</p><p>Finally, in the year 1785 the king confirmed, by
letters patent, dated April 23, the establishment of the
academy of sciences, making the sollowing alterations,
and adding classes of agriculture, natural history, mineralogy,
and physics; incorporating the associates and
adjuncts, and limiting to six the members of each class,
namely three pensioners and three associates; by which
the former receive an increase of salary, and the latter
approach nearer to becoming pensioners.</p><p>By the articles of this instrument it is ordained, that
the academy shall consist of eight classes, namely, that
of geometry, 2d astronomy, 3d mechanics, 4th general
physics, 5th anatomy, 6th chemistry and metallurgy,
7th botany and agriculture, and 8th natural history
and mineralogy. That each class shall remain
irrevocably sixed at six members; namely, three pensioners
and three associates, independent however of &lt;*&gt;
perpetual secretary and treasurer, of twelve free-associates
and of eight associate strangers or foreigners, the
same as before, except that the adjunct-geographer for
the future be called the associate-geographer.</p><p>The classes at first to be filled by the following
persons, namely, that of geometry by Messieurs de
Borda, Jeaurat, Vandermonde, as pensioners; and
Messieurs Cousin, Meusnier, and Charles, as associates:
that of astronomy by Messieurs le Monnier,
de la Lande, and le Gentil, as pensioners; and Messieurs
Messier, de Cassini, and Dagelat, as associates:
that of mechanics by Messieurs l'abbe Bossut, Pabbe
Rochon, and de la Place, as pensioners; and Messieurs
Coulomb, le Gendre, and Perrier, as associates: that
of general physics by Messieurs Leroy, Brisson, and
<pb n="15"/><cb/>
Bailly, as pensioners; and Messieurs Monge, Mechain,
and Quatremere, as associates: that of anatomy by
Messieurs Daubluton, Tenon, and Portal, as pensioners;
and Messieurs Sabatier, Vicq-d'azir, and
Broussonet, as associates: that of chemistry and metallurgy
by Messieurs Cadet, Lavoisier, and Beaume, as
pensioners; and Messieurs Cornette, Bertholet, aud
Fourcroy, as associates: that of botany and agriculture
by Messieurs Guettard, Fougeroux, and Adanson,
as pensioners; and Messieurs de Jussieu, de la
Marck, and Desfontaines, as associates: and that of
natural history and mineralogy by Messieurs Desmaretz,
Saye, and l'abbe de Gua, as pensioners; and Messieurs
Darcet, l'abbe Haui, and l'abbe Tessier, as associates.
All names respectable in the common-wealth of letters;
and from whom the world might expect still farther
improvements in the several branches of science.</p><p>The late M. Rouille de Meslay, counsellor of the
parliament of Paris, founded two prizes, the one of
2500 livres, the other of 2000 livres, which the academy
distributed alternately every year: the subjects of
the former prize respecting physical astronomy, and of
the latter, navigation and commerce.</p><p>The world is highly indebted to this academy for the
many valuable works they have executed, or published,
both individually and as a body collectively, especially
by their memoirs, making upwards of a hundred
volumes in 4to, with the machines, indexes, &amp;c. in
which may be found most excellent compositions in
every branch of science. They publish a volume of
these memoirs every year, with the history of the academy,
and eloges of remarkable men lately deceased:
also a general index to the volumes every ten years.
An alteration was introduced into the volume for 1783,
which it seems is to be continued in future, by omitting,
in the history, the minutes or extracts from the
registers, containing some preliminary account of the
subjects of the memoires; but still however retaining
the eloges of distinguished men, lately deceased.</p><p>M. l'abbe Rozier also has published in four 4to
volumes, an excellent index of the contents of all the
volumes, and the writings of all the members, from the
beginning of their publications to the year 1770; with
convenient blank spaces for continuing the articles in
writing.</p><p>Their history also, to the year 1697, was written by
M. Du Hamel; and after that time continued from
year to year by M. Fontenelle, under the following titles,
Du Hamel Histori&#xE6; Regi&#xE6; Academi&#xE6; Scientiarum,
Paris, 4to. Histoire de l'Academie Royale des Sciences,
avec les Memoires de Mathematique &amp; de Physique,
tirez des Registres de l'Academie, Paris, 4to.
Histoire de l'Academie Royale des Sciences depuis son
etablissement en 1666, jusqu'en 1699, en 13 tomes, 4to.
A new history, from the institution of the academy, to
the period from whence M. de Fontenelle commences,
has been formed; with a series of the works published
under the name of this academy, during the first
interval.</p><p>Since the foregoing account was written, it is said the
Academy has been suppressed and abolished, by the present
convention of France.</p><p>Besides the academies in the capital, there are a
great many in other parts of France. The
<hi rend="smallcaps">Academie</hi> <hi rend="italics">Royale,</hi> at Caen, was established by letters
<cb/>
patent in the year 1705; but it had its rise fifty years
earlier in private conferences, held first in the house of
M. de Brieux. M. de Segrais retiring to this city, to
spend the rest of his days, restored and gave new lustre
to their meetings. In 1707 M. Foucault, intendant of
the generality of Caen, procured the king's letters
patent for erecting them into a perpetual academy, of
which M. Foucault was to be protector for the time,
and the choice afterwards left to the members, the
number of whom was fixed to thirty, chosen for this
time by M. Foucault. But besides the thirty original
members, leave&lt;*&gt;was given to add six supernumerary members,
from the ecclesiastical communities in that city.</p><p>At Toulouse is the <hi rend="italics">Academie des jeux floraux,</hi> composed
of forty persons, the oldest of the kingdom: besides
an academy of sciences and belles-lettres, founded
in 1750.</p><p>At Montpelier is the royal society of sciences, which
since 1708 makes but one body with the royal academy
of sciences at Paris.</p><p>There are also other academies at Bourdeaux, founded
in 1703, at Soissons in 1674, at Marseilles in 1726,
at Lyons in 1700, at Pau in Bearn in 1721, at Montauban
in 1744, at Angers in 1685, at Amiens in 1750,
at Villefranche in 1679, at Dijon in 1740, at Nimes in
1682, at Besan&#xE7;on in 1752, at Chalons in 1775, at
Rochelle in 1734, at Beziers in 1723, at Rouen in
1744, at Metz in 1760, at Arras in 1773, &amp;c.
The number of these academies is continually augmenting;
and even in such towns as have no academies,
the literati form themselves into literary societies, having
nearly the same objects and pursuits.</p><p>In Germany and other parts of Europe, there are
various academies of sciences, &amp;c. The</p><p><hi rend="smallcaps">Academie</hi> <hi rend="italics">Royale des Sciences &amp; des Belles Lettres</hi> of
Prussia, was founded at Berlin, in the year 1700, by
Frederic I. king of Prussia, of which the famous
M. Leibnitz was the first president, and its great promoter.
The academy received a new form, and a new
set of statutes in 1710; by which it was ordained, that
the president shall be one of the counsellors of state; and
that the members be divided into four classes; the first
to cultivate physics, medicine, and chemistry; the
second, mathematics, astronomy, and mechanics; the
third, the German language, and the history of the
country; and the fourth, oriental learning, particularly
as it may concern the propagation of the gospel among
infidels. That each class elect a director for themselves,
who shall hold his post for life. That they meet
in the castle called the New Marshal, the classes to meet
in their turns, one each week. And that the members
of any of the classes have free access into the assemblies
of the rest. Several volumes of their transactions have
been published in Latin, from time to time, under the
title of Miscellanea Berolinensia.</p><p>In 1743 the late famous Frederic II. king of Prussia,
made great alterations and improvements in the academy.
Instead of a great lord or minister of state,
who had usually presided over the academy, he wisely
judged that office would be better filled by a man of
letters; and he honoured the French academy of
sciences by fixing upon one of its members for a president,
namely M. Maupertuis, a distinguished character
in the literary world, and whose conduct in improving
the academy was a proof of the sound judgment of the
<pb n="16"/><cb/>
king, who at the same time made new regulations for
the academy, and took the title of its Protector.
From that time the transactions of the academy have
been published, under the title of Histoire de l'Academie
Royale des Sciences et Belles Lettres &#xE0; Berlin, much
in the manner of those of the French academy of sciences,
and in the French language; and the volumes
are now commonly published annually. Besides the ordinary
private meetings of the academy, it has two
public ones in the year, in January and May, at the
latter of which is given a prize gold medal, of the value
of 50 ducats, or about 20 guineas. The subject of the
prize is successively physics, mathematics, metaphysics,
and general literature. For the academy has this peculiar
circumstance, that it embraces also metaphysics,
logic, and morality; having one class particularly appropriated
to these objects, called the class of Speculative
Philosophy.</p><p><hi rend="italics">Imperial</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Petersburgh.</hi> This academy
was projected by the Czar Peter I, commonly called
Peter the Great, who in so many other instances also
was instrumental in raising Russia from the state of barbarity
in which it had been immerged for so many ages.
Having visited France in 1717, and among other things
informed himself of the advantages of an academy of
arts and sciences, he resolved to establish one in his new
capital, whither he had drawn by noble encouragements
several learned strangers, and made other preparations,
when his death prevented him from fully accomplishing
that object, in the beginning of the year 1725. Those
preparations and intentions however were carried into
execution the same year, by the establishment of the
academy, by his consort the czarina Catherine, who
succeeded him. And soon after the academy composed
the first volume of their works, published in 1728,
under the title of Commentarii Academi&#xE6; Scientiarum
Imperialis Petropolitan&#xE6;; which they continued almost
annually till 1746, the whole amounting to 14 volumes,
which were published in Latin, and the subjects divided
and classed under the following heads, namely mathematics,
physics, history, and astronomy. Their device
a tree bearing fruit not ripe, with the modest motto
<hi rend="italics">paullatim.</hi></p><p>Most part of the strangers who composed this academy
being dead, or having retired, it was rather in a
languishing state at the beginning of the reign of the
empress Elizabeth, when the count Rasomowski was
happily appointed president, who was instrumental in
recovering its vigour and labours. This empress renewed
and altered its constitution, by letters patent
dated July 24, 1747, giving it a new form and regulations.
It consists of two chief parts, an academy, and
a university, having regular professors in the several saculties,
who read lectures as in our colleges. The ordinary
assemblies are held twice a week, and public or
solemn ones thrice in the year; in which an account is
given of what has been done in the private ones. The
academy has a noble building for their meetings, &amp;c,
with a good apparatus of instruments, a sine library,
observatory, &amp;c. Their first volume, after this renovation,
was published for the years 1747 and 1748, and
they have been fince continued from year to year, now
to the amount of near thirty volumes, under the title of
Novi Commentarii Academi&#xE6; Scientiarum Imperialis
<cb/>
Petropolitan&#xE6;. They are printed in the Latin language,
and contain many excellent compositions in all
the sciences, especially the mathematical papers of the
late excellent M. L. Euler, which always made a considerable
portion of every volume. The subjects are
classed under heads in the following order, mathematics,
physico-mathematics, physics, which include botany,
anatomy, &amp;c, and astronomy; the whole prefaced by
historical extracts, or minutes, relating to each paper
or memoir, after the manner of the volumes of the
French academy; but wanting however the eloges of
deceased eminent men. Their device is a heap of ripe
fruits piled on a table, with the motto <hi rend="italics">En addit fructus
&#xE6;tate recentes.</hi></p><p><hi rend="italics">Imperial and Royal</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Sciences and Belles
Lettres, at Brussels.</hi> This academy was founded in the
year 1773; and several volumes of their memoirs have
been published.</p><p><hi rend="italics">Royal</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Sciences,</hi> at Stockholm, was instituted
in 1739, and since that time it has published
about sixty volumes of transactions, quarterly, in 8vo,
in the Swedish language.</p><p>For an account of the Royal Society of London,
and several other similar institutions, see the words
Journal, Society, &amp;c.</p><p><hi rend="italics">American</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Arts and Sciences,</hi> was established
in 1780 by the council and house of representatives
in the province of Massachuset's Bay, for
promoting the knowledge of the antiquities of America,
and of the natural history of the country; for determining
the uses to which its various natural productions
may be applied; for encouraging medicinal
discoveries, mathematical disquisitions, philosophical
enquiries and experiments, astronomical, meteorological,
and geographical observations, and improvements
in agriculture, manufactures, and commerce;
and, in short, for cultivating every art and science,
which may tend to advance the interest, honour, dignity,
and happiness, of a free, independent, and virtuous
people. The members of this academy are never
to be less than forty, nor more than two hundred.</p><p><hi rend="smallcaps">Academy</hi> is also used among us for a kind of collegiate
school, or seminary; where youth are instructed
in the liberal arts and sciences in a private way: now
indeed it is used for all kinds of schools.</p><p>Frederic 1, king of Prussia, established an academy
at Berlin in 1703, for educating the young nobility of
the court, suitable to their extraction. The expence
of the students was very moderate, the king having
undertaken to pay the extraordinaries. This illustrious
school, which was then called the academy of
princes, has now lost much of its first splendour.</p><p>The Romans had a kind of military academies established
in all the cities of Italy, under the name of
<hi rend="italics">Campi Martis.</hi> Here the youth were admitted to be
trained sor war at the public expence. And the
Greeks, besides academies of this kind, had military
professors, called <hi rend="italics">Tactici,</hi> who taught all the higher
offices of war, &amp;c.</p><p>We have two royal academies of this kind in England,
the expences of which are defrayed by the government;
the one at Woolwich, for the artillery and
military engineers; and the other at Portsmouth, for
the navy. The former was established by his late
<pb n="17"/><cb/>
majesty king George II, by warrants dated April the 30th
and November the 18th, 1741, for instructing persons
belonging to the military part of the ordnance, in the
several branches of mathematics, fortification, &amp;c,
proper to qualify them for the service of artillery
and the office of engineers. This institution is under
the direction of the master-general and board of
ordnance for the time being: and at first the lectures
of the masters in the academy were attended by the
practitioner-engineers, with the officers, serjeants, corporals,
and private men of the artillery, besides the
eadets. At present however none are educated there
but the gentlemen cadets, to the number of 90 or 100,
where they receive an education perhaps not to be obtained
or purchased for money in any part of the
world. The master-general of the ordnance is always
captain of the cadets' company, and governor of the
academy; under him are a lieutenant-governor, and an
inspector of studies. The masters have been gradually
increased, from two or three at first, now to the number
of twelve, namely, a professor of mathematics, and two
other mathematical masters, a professor of fortification,
and an assistant, two drawing masters, two French
masters, with masters for fencing, dancing, and chemistry.
This institution is of the greatest consequence
to the state, and it is hardly credible that so important
an object should be accomplished at so trifling an expence.
It is to be lamented however that the academy
is fixed in so unhealthy a situation; that the lecture
rooms and cadets' barracks are so small as to be insufsicient
for the purposes of the institution; and that the
salaries of the professors and masters should be so inadequate
to their labours, and the benefit of their
services.</p><p>The Royal Naval Academy at Portsmouth was
founded by George I, in 1722, for instructing young
gentlemen in the sciences useful for navigation, to
breed officers for the royal navy. The establishment
is under the direction of the board of admiralty, who
give salaries to two masters, by one of whom the students
are boarded and lodged, the expence of which
is defrayed by their own friends, nothing being supplied
by the government but their education.</p></div2></div1><div1 part="N" n="ACANTHUS" org="uniform" sample="complete" type="entry"><head>ACANTHUS</head><p>, <hi rend="italics">in Architecture,</hi> the leaves of a plant
which forms the ornament of the capital of the Corinthian
order. Vitruvius ascribes the use of it to the
following accident. A young girl dying, her nurse
was desirous of consecrating to her manes certain toys
which she was fond of in her life-time; which the
good woman carried in a little basket, covered with a
square tile, and placed it among some green plants
which grew on her grave. One of these, which happened
to be the acanthus, as it grew up, invironed and
in a manner embraced the basket; which Callimachus,
a noted Greek sculptor, casting his eyes upon, from
thence took the hint of this elegant ornament. See
<hi rend="smallcaps">Abacus.</hi></p><p>ACCELERATED <hi rend="italics">Motion,</hi> is that which receives
fresh accessions of velocity; and the acceleration may be
either equably or unequably: if the accessions of velocity
be always equal in equal times, the motion is
said to be equably or uniformly accelerated; but if the
<cb/>
accessions, in equal times, either increase or decrease,
then the motion is unequably or variably accelerated.</p><p>Acceleration is directly opposite to retardation,
which denotes a diminution of velocity.</p><p><hi rend="smallcaps">Acceleration</hi> comes chiefly under consideration
in physics, in the descent of heavy bodies, tending or
falling towards the centre of the earth, by the force of
gravity.</p><p>That bodies are accelerated in their natural descent,
is evident both to the sight, and from observing that
the greater height they fall from, the greater force they
strike with, and the deeper impressions they make in
soft substances.</p><p>The acceleration of falling bodies has been ascribed
to various causes, by different philosophers. Some
have attributed it to the pressure of the air downwards:
the more a body descends, the longer and
heavier, say they, must be the column of atmosphere
incumbent upon it; to which they add, that the whole
mass of fluid pressing by an infinity of right-lines all
ultimately meeting in the earth's centre, such central
point must support, as it were, the pressure of the
whole mass; and that consequently the nearer a body
approaches to it, the more must it receive of the
pressure of a multitude of lines tending to unite in
the central point.</p><p>Mr. Hobbes endeavours to account for this acceleration
from a new impression of the cause which makes
bodies fall; in which he is so far right. But then he
as far mistakes, as to the cause of the fall, which he
thinks is the air: at the same time, says he, that one
particle of air ascends, another descends; for in consequence
of the earth's motion being two-fold, that is
circular and progressive, the air must at once both
ascend and circulate; whence it follows, that a body
falling in this medium, and receiving a new pressure
every instant, must have its motion accelerated.</p><p>But to both these systems it may be answered, that
the air is quite out of the question; for it is very evident
that bodies fall, and in falling have their motion
accelerated, in vacuo, as in open air, and even more
than in the air, in as much as this opposes and somewhat
retards their fall.</p><p>The Gassendists assign another reason for the acceleration:
they pretend that there are continually issuing
out of the earth certain attractive corpuscles, directed
in an infinite number of rays; those, say they,
afcend and then descend, in such sort that the nearer a
body approaches to the earth's centre, the more of
these attractive rays press upon it, in consequence of
which its motion becomes more accelerated.</p><p>The peripatetics endeavour to explain the matter
thus: the motion of heavy bodies downward, arises,
say they, out of an intrinsic principle that causes a
tendency in them to the centre, as the place appropriated
to their element; where, when they can once
arrive, they will be at perfect rest; and therefore, continue
they, the nearer bodies approach to it, the more
the velocity of their motion is increased: a notion too
idle to merit confutation.</p><p>The Cartesians account for acceleration, by reiterated
impulses of their <hi rend="italics">materia subtilis,</hi> acting continually
<pb n="18"/><cb/>
on falling bodies, and propelling them downwards: a
conceit equally unintelligible and absurd with the
former.</p><p>But, leaving all such visionary causes of acceleration,
and only admitting the existence of such a force as gravity,
so evidently inherent in all bodies, without regard
to what may be the cause of it, the whole mystery of
acceleration will be cleared up. Consider gravity then,
with Galileo, only as a cause or force which acts continually
on heavy bodies; and it will be easy to conceive
that the principle of gravitation, which determines
bodies to descend, must by a necessary consequence
accelerate them in falling.</p><p>A body then having once begun to descend, through
the impulse of gravity; the state of descending is now,
by Newton's first law of nature, become as it were natural
to it; insomuch that, were it left to itself, it
would for ever continue to descend, even though the
first cause of its descent should cease. But besides this
determination to descend, impressed upon it by the first
cause of motion, which would be sufficient to continue
to infinity the degree of motion already begun, new
impulses are continually superadded by the same cause;
which continues to act upon the body already in motion,
in the same manner as if it had remained at rest.
There being then two causes of motion, acting both in
the same direction; it necessarily follows, that the motion
which they unitedly produce, must be more considerable
than what either could produce separately.
And as long as the velocity is thus continued, the same
cause still subsisting to increase it more, the descent
must of necessity be continually accelerated.</p><p>Supposing then that gravity, from whatever principle
it arises, acts uniformly upon all bodies at the
same distance from the centre of the earth: dividing
the time which the heavy body takes up in falling to
the earth, into indefinitely small equal parts; gravity
will impel the body toward the centre of the earth, in
the first indefinitely short instant of the descent. If
after this we suppose the action of gravity to cease, the
body will continue perpetually to advance uniformly
toward the earth's centre, with an indefinitely small
velocity, equal to that which resulted from the first
impulse.</p><p>But then if we suppose that the action of gravity
still continues the same after the first impulse; in the
second instant, the body will receive a new impulse toward
the earth, equal to that which it received in the
first instant; and consequently its velocity will be
doubled; in the third instant, it will be tripled; in the
fourth, quadrupled; in the fifth, quintupled; and so
on continually: for the impulse made in any preceding
instant, is no ways altered by that which is made in the
following one; but they are, on the contrary, always
accumulated on each other.</p><p>So that the instants of time being supposed indefinitely
small, and all equal, the velocity acquired by
the falling body, will be, in every instant, proportional
to the times from the beginning of the descent; and
consequently the velocity will be proportional to the
time in which it is produced. So that if a body, by
this constant force, acquire a velocity of 16 1/12 feet suppose
in one second of time; it will acquire a velocity
<cb/>
of 32 1/6 feet in two seconds, 48 1/4 feet in 3 seconds,
64 1/3 in 4 seconds, and so on. Nor ought it to seem
strange that all bodies, small or large, acquire, by the
force of gravity, the same velocity in the same
time. For every equal particle of matter being endued
with an equal impelling force, namely its gravity or
weight, the sum of all the forces, in any compound
mass of matter, will be proportional to the sum of all
the weights, or quantities of matter to be moved; consequently,
the forces and masses moved, being thus constantly
increased in the same proportion, the velocities
generated will be the same in all bodies, great or small.
That is, a double force moves a double mass of matter,
with the same velocity that the single force moves the
single mass; and so on. Or otherwise, the whole compound
mass falls all together with the same velocity,
and in the same manner, as if its particles were not
united, but as if each fell by itself, separated all from one
another. And thus all being let go at once, they would
fall together, just as if they were united into one mass.</p><p>The foregoing law of the descent of falling bodies,
namely that the velocities are always proportional to
the times of descent, as well as the following laws concerning
the spaces passed over, &amp;c, were first discovered
and taught by the great Galileo, and that nearly in the
following manner.</p><p>Because the constant velocity with which any body
moves, or the space it passes over in a given time, as
suppose one second, being multiplied by the time, or
number of seconds it is in motion, expresses the space
passed over in that time; and the area or space of a
rectangular figure being denoted by the length multiplied
by the breadth; therefore the space so run over,
may be considered as a rectangle compounded of the
time and velocity, that is a rectangle of which the time
denotes the length, and the velocity the breadth. Suppose
then A to be the heavy body which descends, and
AB to denote the whole time of any descent; which
<figure/>
let be divided into a certain number of equal parts,
denoting intervals or portions of the given time, as
AC, CD, DE, &amp;c. Imagine the body to descend,
during the time expressed by the first of the divisions
AC, with a certain uniform velocity arising from the
force of gravity acting on it, which let be denoted by
AF, the breadth of the rectangle CF; then the space
run through during the time denoted by AC, with the
velocity denoted by AF, will be expressed by the rectangular
space CF.</p><p>Now the action of gravity having produced, in the
first moment, the velocity AF, in the body, before at
rest; in the first two moments it will produce the velocity
CG, the double of the former; in the third moment,
to the velocity CG will be added one degree
<pb n="19"/><cb/>
more, by which means will be produced the velocity
DH, triple of the first; and so of the rest; so that
during the whole time AB, the body will have acquired
the velocity BK. Hence, taking the divisions
of the line AB at pleasure; for example, the divisions
AC, CD, &amp;c, for the times; the spaces run through
during those times, will be as the areas or rectangles
CF, DG, &amp;c; and so the space described by the
moving body during the whole time AB, will be
equal to all the rectangles, that is, equal to the whole
indented space ABKIHGF. And thus it will
happen if the increments of velocity be produced, as
we may say, all at once, at the end of certain portions
of finite time; for instance at C, at D, &amp;c; so that the
degree of motion remains the same to the instant that a
new acceleration takes place.</p><p>By conceiving the divisions of time to be shorter,
for example but half as long as the former, the indentures
of the figure will be proportionably more
contracted, and it will approach nearer to a triangle;
and so much the nearer as the divisions of time are
shorter: and if these be supposed infinitely small; that
is, if increments of the velocity be supposed to be acquired
continually, and at each indivisible particle of
time, which is really the case, the rectangles so successively
produced, will form a true triangle, as ABC;
the whole time AB consisting of minute portions A 1,
<figure/>
12, 23, &amp;c; and the area of the triangle ABC, of
all the minute surfaces, or minute trapeziums, which
answer to the divisions of the times; the area of the
whole triangle ABC, denoting the space run through
during the whole time AB; and the area of any
smaller triangle A 7 <hi rend="italics">g,</hi> denoting the space run through
during the corresponding time A 7. Bnt the triangles
A 1 <hi rend="italics">a,</hi> A 7 <hi rend="italics">g,</hi> &amp;c, being similar, have their areas to
each other as the squares of their like sides A 1, A 7,
&amp;c; and consequently the spaces gone through, in
any times counted from the beginning, are to each
other as the squares of the times.</p><p>Hence, in any right-angled triangle, as ABC, the
one side AB represents the time, the other side BC
the velocity acquired in that time, and the area of the
triangle the space described by the falling body.</p><p>From the preceding demonstration is also drawn
this other general theorem in motions that are uniformly
accelerated; namely, that a body descending
with a uniformly accelerated motion, describes in the
whole time of its descent, a space, which is exactly the
half of that which it would describe uniformly in the
same time, with the velocity it has acquired at the end
of its accelerated fall. For it has been shewn that the
whole space which the falling body has run through in
the time AB, is represented by the triangle ABC,
the last velocity being BC; and the space which the
<cb/>
body would run through uniformly in the same time
AB, constantly with the said greatest velocity BC, is
represented by the rectangle ABCD: but it is well
known that the rectangle ABCD is double the
triangle ABC; and therefore the latter space run
through, is double the former; that is, the space run
through by the accelerated motion, is just half of that
which the body would describe in the same time, moving
uniformly with the velocity acquired at the end of
its accelerated fall.</p><p>Hence then, from the foregoing considerations are
deduced the following general laws of uniformly accelerated
motions, namely,</p><p>1st. That the velocities acquired, are constantly proportional
to the times; in a double time a double velocity,
&amp;c.</p><p>2d. That the spaces described in the whole times,
each counted from the commencement of the motion,
are proportional to the squares of the times, or to the
squares of the velocities; that is, in twice the time,
the body will describe 4 times the space; in thrice the
time, it will describe 9 times the space; in quadruple
the time, 16 times the space; and so on. In short, if
the times are proportional
to the numbers 1, 2, 3, 4, 5, &amp;c,
the spaces will be as 1, 4, 9, 16, 25, &amp;c,
which are the squares of the former. So that if a
body, by the natural force of gravity, fall through the
space of 16 1/12 feet in the first second of time; then in
the first two seconds of time it will fall through four
times as much, or 64 1/3 feet; in the first three seconds
it will fall nine times as much, or 144 3/4 feet; and so
on. And as the spaces fallen through are as the
squares of the times, or of the velocities; therefore the
times, or the velocities, are proportional to the square
roots of the spaces.</p><p>3d. The spaces described by falling bodies, in a
series of equal instants or intervals of time, will be as
the odd numbers 1, 3, 5, 7, 9, &amp;c,
<hi rend="brace"><note anchored="true" place="unspecified">1, 4, 9, 16, 25, &amp;c,</note>
which are the differences of
the squares or whole spaces</hi>
that is, the body which has run through 16 1/12 feet in the
firft second, will in the next second run through 48 1/3
feet, in the third second 80 3/12, and so on.</p><p>4th. If the body fall through any space in any time,
it acquires a velocity equal to double that space; that
is, in an equal time, with the last velocity acquired, if
uniformly continued, it would pass over just double the
space. So if a body fall through 16 1/12 feet in the first
second of time, then it has acquired a velocity of 32 1/6
feet in a second; that is, if the body move uniformly
for one second, with the velocity acquired, it will pass
over 32 1/6 feet in this one second: and if in any time
the body fall through 100 feet; then in another equal
time, if it move uniformly with the velocity last acquired,
it will pass over 200 feet. And so on.</p><p>But, as the method of demonstration used by Galileo,
by means of infinitely small parts forming a regular
triangle, is not approved of by many persons, the same
laws may be otherwise demonstrated thus: let the
whole time of a body's free descent be divided into
any number of parts, calling each of these parts 1; and
let <hi rend="italics">a</hi> denote the velocity acquired at the end of the first
<pb n="20"/><cb/>
part of time; then will 2<hi rend="italics">a,</hi> 3<hi rend="italics">a,</hi> 4<hi rend="italics">a,</hi> &amp;c, represent
the velocities at the end of the 2d, 3d, 4th, &amp;c, part of
time, because the velocities are as the times; and for
the same reason 1/2<hi rend="italics">a,</hi> 3/2<hi rend="italics">a,</hi> 5/2<hi rend="italics">a,</hi> &amp;c, will be the
 velocities at
the middle point of the first, second, third, &amp;c, part of
time. But now as the velocities increase uniformly,
the space described in any one of these parts of time,
may be considered as uniformly deseribed with its
middle velocity, or the velocity in the middle of that
part of time; and therefore multiplying those mean
velocities each by their common time 1, we have the
same fractions 1/2<hi rend="italics">a,</hi> 3/2<hi rend="italics">a,</hi> 5/2<hi rend="italics">a,</hi> &amp;c, for the spaces
 passed
over in the successive parts of the time; that is, the
space 1/2<hi rend="italics">a</hi> in the first time, 3/2<hi rend="italics">a</hi> in the second, 5/2<hi rend="italics">a</hi>
 in the
third, and so on: then add these spaces successively to
one another, and we obtain 1/2<hi rend="italics">a,</hi> 4/2<hi rend="italics">a,</hi> 9/2<hi rend="italics">a,</hi>
 16/2<hi rend="italics">a,</hi> &amp;c, for
the whole spaces described from the beginning of the
motion to the end of the first, second, third, &amp;c, portion
of time; namely 1/2<hi rend="italics">a</hi> space in one time, 4/2<hi rend="italics">a</hi> in 2
times, 9/2<hi rend="italics">a</hi> in 3 times, and so on: and it is evident that
these spaces are as the numbers 1, 4, 9, 16, &amp;c, which
are as the squares of the times.</p><p>And from this mode of demonstration, all the properties
above mentioned evidently flow: such as that
the whole spaces 1/2<hi rend="italics">a,</hi> 4/2<hi rend="italics">a,</hi> 9/2<hi rend="italics">a,</hi> &amp;c,
are as the squares of the times 1, 2, 3, &amp;c,
that the separate spaces 1/2<hi rend="italics">a,</hi> 3/2<hi rend="italics">a,</hi> 5/2<hi rend="italics">a,</hi> &amp;c,
<hi rend="brace"><note anchored="true" place="unspecified">1, 3, 5, &amp;c,</note>
described in the successive times,
are as the odd numbers</hi>
and that the velocity <hi rend="italics">a</hi> acquired in any time 1, is
double the space 1/2<hi rend="italics">a</hi> described in the same time.</p><p>As the laws of acceleration are very important, I
shall here insert the two following propositions, sent
me by my learned friend Mr. Abram Robertson, of
Christ Church College Oxford, in which those laws are
demonstrated in a manner somewhat different.
<hi rend="center">&#x201C;<hi rend="smallcaps">Ppoposition</hi> 1.</hi></p><p>If from the point P in the straight line AB, the
points M, N begin to move at the same time, namely,
M towards A with a motion, uniformly retarded, and
N from rest towards B with a motion uniformly accelerated;
and if the velocity of M decreases as much as
the velocity of N increases in the same time; then the
space MN is generated by an uniform motion, equal
to the velocity with which M begins to move.
<figure/></p><p>For, by hypothesis, whatever is lost in the velocity
of M by retardation, is added to the velocity of N by
acceleration: the joint velocities, therefore, of M and N
must always be equal. But it is by the joint velocities
of M and N that the space MN is generated. Consequently
MN is generated by an uniform motion,
which is evidently equal to the velocity with which
M begins to move.
<hi rend="center">&#x201C;<hi rend="smallcaps">Proposition</hi> II.</hi></p><p>If a point begins to move in the direction of a
straight line, and continues to move in the same di-
<cb/>
rection with a velocity uniformly aocelerated; the
space passed over in any given time, will be equal to
half the space passed over in the same time with the
velocity with which the acceleration ends.</p><p>Let the point D begin to move from A towards B,
along the straight line AB, with a motion unisormly
accelerated; the space AD passed over, is equal to
half the space which the point would pass over, in the
same time with the acquired velocity at D.
<figure/></p><p>Let the points M, N begin to move in the straight
line GH, at the same time, with equal velocities uniformly
accelerated; M beginning to move from G,
and N from P; and at the same time that M comes to
the point P, let N come to H. Then as M and N
<figure/>
move with equal velocities, uniformly accelerated, it is
evident that the spaces, which they pass over in the
same time, are equal to one another; consequently the
space GP is equal to the space PH. Now as M begins
to move from G with a velocity uniformly accelerated,
it will arrive at P with an acquired velocity.
Hence it is evident, if it be supposed to begin to move
from P with this acquired velocity, and proceed toward
G with a velocity uniformly retarded in the same degree
that it was accelerated when it began to move
from G, that it will pass over the same space GP in
the same time. Wherefore, supposing the two points
M, N to begin to move from P at the same time,
namely the point M beginning to move with the acquired
velocity mentioned above, and proceeding towards
G with the velocity uniformly retarded, described
above; and the point N as before with the
velocity uniformly accelerated: then as the acceleration
and retardation are uniform, they will be equal in equal
spaces of time. Again, as M is retarded in the same
degree that it was accelerated when it began to move
from G, that is, in the same degree that N is accelerated,
by the former prop. MN is generated
by an uniform velocity. But when the point M
arrives at G, its velocity becomes equal to o or
nothing; and at the time that M arrives at G, N
arrives at H with the acquired velocity. Wherefore,
as the velocities of M and N taken jointly are equal,
and consequently uniform, the space GH is passed
over with the velocity of N at H, in the same time
that PH is passed over by N beginning to move from
P with a velocity uniformly accelerated to H. But
PH is half of GH. &#x201C;Hence the prop. is manifest.&#x201D;</p><p>And hence the other laws of the spaces, before&lt;*&gt;
mentioned, easily follow.</p><p>Since the spaces descended are as the squares of the
times, and the abscisses of a parabola are as the squares
of the ordinates, therefore the relation of the times and
spaces descended may be very well represented by the
ordinates and abscisses of that figure. Thus if AB be
the axis of the parabola A<hi rend="italics">bdfh,</hi> and AC a tangent
<pb n="21"/><cb/>
<figure/>
at the vertex perpendicular to
the axis, divided into any number
of equal parts A<hi rend="italics">a, ac, ce,</hi>
&amp;c, for the times; and if there
be drawn <hi rend="italics">ab, cd, ef,</hi> &amp;c, parallel
to the axis: hence if <hi rend="italics">ab</hi>
be the space descended in the
time A<hi rend="italics">a,</hi> then <hi rend="italics">cd</hi> will be the
space descended in the time
A<hi rend="italics">c,</hi> and <hi rend="italics">ef</hi> the space defcended
in the time A<hi rend="italics">e,</hi> and so on continually.</p><p>From the properties above-demonstrated, are derived
the following practical formulas or theorems for use.
Namely, if <hi rend="italics">g</hi> denote the space passed over in the first
second of time, by a body urged by any constant force,
denoted by 1, and <hi rend="italics">t</hi> denote the time or number of seconds
in which the body passes over any other space <hi rend="italics">s,</hi>
and <hi rend="italics">v</hi> the velocity acquired at the end of that time;
then from the foregoing laws we have <hi rend="italics">v</hi> = 2<hi rend="italics">gt,</hi> and
<hi rend="italics">s</hi> = <hi rend="italics">gt</hi><hi rend="sup">2</hi>; and from these two equations result the
following general formulas:</p><p>And here, when the constant force 1, is the natural
force of gravity, then the distance <hi rend="italics">g</hi> descended in the
first second, in the latitude of London, is 16 1/12 feet:
but if it be any other constant force, the value of <hi rend="italics">g</hi>
will be different, in proportion as the force is more or
less.</p><p>The motion of an ascending body, or of one that
is impelled upwards, is diminished or retarded by the
same principle of gravity, acting in a contrary direction,
after the same manner that a falling body is accelerated.</p><p>A body projected upwards, ascends until it has lost
all its motion; which it does in the same space of time,
that the body would have taken up in acquiring, by
falling, a velocity equal to that with which the falling
body began to be projected upwards. And consequently
the heights to which bodies ascend, when projected
upwards with different velocities, are to each
other as the squares of those velocities.</p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Bodies on Inclined Planes.</hi>
The same general laws obtain here, as in bodies falling
freely, or perpendicularly; namely, that the velocities
are as the times, and the spaces descended down the
planes as the squares of the times, or of the velocities.
But those velocities are less, according to the sine of
the plane's inclination; and the spaces less, according
to the square of the sine. See <hi rend="smallcaps">Inclined</hi> <hi rend="italics">Plane.</hi></p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Pendulums.</hi> See P<hi rend="smallcaps">ENDULUM.</hi></p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Projectiles.</hi> See P<hi rend="smallcaps">ROJECTILE.</hi></p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Compressed Bodies,</hi> in ex-
<cb/>
panding or restoring themselves. See <hi rend="smallcaps">Dilatation,
Compression</hi>, and <hi rend="smallcaps">Elasticity.</hi></p><p><hi rend="smallcaps">Accelerating Force</hi>, in Physics, is the force
that accelerates the motion or velocity of bodies;
and it is equal to, or expressed by, the quotientarising
from the motive or absolute force, divided by the
mass or weight of the body that is moved. In
treating of physical considerations respecting forces,
velocities, times, and spaces gone over, the first inquiry
is the accelerating or accelerative force. This force is
greater or less in proportion to the velocity it generates
in the same time, and by this velocity it is measured. All
accelerating forces are equal, and generate equal velocities,
that have the motive forces directly proportional
to the quantities of matter: so a double motive force
will move a double quantity of matter with the same
velocity, as also a triple motive force a triple quantity,
a quadruple force a quadruple quantity, &amp;c, all with the
same velocity. And this is the reason why all bodies
fall equally swift by the force of gravity; for the motive
force is exactly proportional to their weight or
mass. In general, the accelerating force is in the direct
ratio of the motive force, and inverse ratio of the
quantity of matter. When a body is let fall freely, to
descend by the force of its natural gravity, it has been
found by experiment that it falls through 16 1/12 feet in
one second of time, and requires a velocity of 32 1/6 feet in
that time: but if the quantity of matter be doubled, and
the motive force remain the same as before, by connecting
the falling body to another of equal weight by
means of a thread, this other body being laid on a horizontal
plane, and the falling body hanging down off
the plane, and drawing the other equal body along the
plane after it; then the accelerating force will be only
half of what it was before, and the space fallen in one
second will be only 8 1/24 feet, and the velocity acquired
16 1/12: and if the quantity of matter be tripled, or
the body drawn along the plane doubled; then the accelerating
force will be only one-third of what it was
at first, and the space descended in one second, and velocity
acquired, each one-third of the sirst: and so on.</p><p>But accelerating forces are sometimes variable, as
well as sometimes constant; and the variation may be
either increasing or decreasing.</p><p>The nature of constant and variable accelerating
forces, may be illuftrated in the following manner.
Let two weights W, <hi rend="italics">w,</hi> be connected by a thread
<figure/>
passing over a pully at A, B, or C; and let the weight
W descend perpendicularly down, while it draws the
smaller weight <hi rend="italics">w</hi> up the line AD, or BE, or CF, the
<pb n="22"/><cb/>
first being a straight inclined plane, and the other two
curves, the one convex and the other concave to the
perpendicular. Then the small weight <hi rend="italics">w</hi> will always make
some certain resistance to the free descent of the large
weight W, and that resistance will be constantly the
same in every part of the plane AD, the difficulty to
draw it up being the same in every point of it, because
every part of it has the same inclination to the horizon,
or to the perpendicular; and consequently the accessions
to the velocity of the descending weight W, will
be always equal in equal times; that is, in this case W
descends by a uniformly accelerating force. But in
the two curves BE, CF, the resistance or opposition
of the small weight <hi rend="italics">w</hi> will be constantly altering as it
is drawn up the curves, because every part of them has
a different inclination to the horizon, or to the perpendicular:
in the former curve, the direction becomes
more and more upright, or nearer perpendicular, as the
small weight <hi rend="italics">w</hi> ascends, and the opposition it makes to
the descent of W, becomes more and more; and consequently
the accessions to the velocity of W will be
always less and less in equal times; that is, W descends
by a decreasing accelerating force: but in the latter
curve CF, as <hi rend="italics">w</hi> ascends, the direction of the curve becomes
less and less upright, and the opposition it makes
to the descent of W, becomes always less and less; and
consequently the accessions to the velocity of W will
be always more and more in equal times; that is, W
descends by an increasing accelerating force. So that
although the velocity continually increases in all these
cases, yet whilst it increases in a constant ratio to
the times of motion, in the plane AD; the velocity
increases in a less ratio than the time it ascended up
BE, and in a greater ratio than the time increases in
the other curve CF.</p><p>Now the relations between the times and velocities
in all these cases, may be very well represented by the
relations between the abscisses and ordinates of certain
lines. Thus let AB and AC be two straight lines,
<figure/>
making any angle BAC; and AD, AE two curves,
the former concave, and the latter convex towards AB:
divide AB into any parts A<hi rend="italics">a,</hi> A<hi rend="italics">b,</hi> &amp;c, representing
the times of motion; and draw the perpendiculars
<hi rend="italics">acde, bfgh,</hi> &amp;c, representing the velocities. Then
in the right line AC, the ordinates <hi rend="italics">ad, bg,</hi> being as
the abscisses A<hi rend="italics">a,</hi> A<hi rend="italics">b,</hi> this represents the case of uniformly
accelerated motion, in which the velocities are
always as the times: but in the curve AD, the ordinates
<hi rend="italics">ac, bf</hi> increase in a less ratio than the abscisses
A<hi rend="italics">a,</hi> A<hi rend="italics">b;</hi> and therefore this represents the case of
decreasing acceleration, in which the velocities increase
<cb/>
in a less ratio than the times: and in the other curve
AE, the ordinates <hi rend="italics">ae, bh</hi> increase in a greater ratio
than the abscisses; and therefore this represents the
case of increasing acceleration, in which the velocities
increase in a greater ratio than the times.</p><p>The several algebraic formulas or theorems, respecting
the time, velocity, space, for constant accelerating
forces, are delivered above, at the article <hi rend="italics">Accelerated Motion,</hi>
where the value of each circumstance is expressed in
finite determinate quantities. But in the cases of variably
accelerated motions, the formulas will require the
help of the method of fluxions to express, not those
general relations themselves, but the fluxions of them;
and consequently, taking the fluents of those expressions,
in particular cases, the relations of time,
space, velocity, &amp;c, are obtained.</p><p>Now if <hi rend="italics">t</hi> denote the time in motion,
<hi rend="italics">v</hi> the velocity generated by any force,
<hi rend="italics">s</hi> the space passed over,
and 2<hi rend="italics">g</hi> the variable force at any part of the motion,
or the velocity the force would generate in one second
of time, if it should continue invariable, like the force
of gravity, during that one second; and therefore the
value of this velocity 2<hi rend="italics">g,</hi> will be in proportion to 32 1/6
feet, as that variable force, is to 1 the force of gravity.
Then because the force may be supposed constant during
the indefinitely small time <hi rend="italics">t,</hi> and that in uniform
motions the spaces and velocities are proportional to the
times, we from thence obtain these two general fundamental
porportions,</p><p>From which are derived the four formulas below, in
which the value of each quantity is expressed in terms
of the rest.</p><p>And these theorems equally hold good for the destruction
of motion and velocity, by means of retarding
forces, as for the generation of the same by means
of accelerating forces.</p><div2 part="N" n="Acceleration" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Acceleration</hi>, in <hi rend="italics">Mechanics</hi></head><p>, the increase of velocity
in a moving body.</p><p><hi rend="smallcaps">Acceleration.</hi> <hi rend="italics">Astron.</hi> The Diurnal Acceleration
of the fixed stars, is the time which the stars, in one diurnal
revolution, anticipate the mean diurnal revolution
of the sun; which is 3<hi rend="sup">m</hi> 55<hi rend="sup">s</hi> 9/10 of mean time, or nearly
3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi>: that is, a star rises, or sets, or passes the meridian,
about 3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi> sooner each day. This acceleration
of the stars, which is only apparent in them, arises
from the real retardation of the sun, owing to his appa-
<pb n="23"/><cb/>
rent motion in his orbit towards the east, which is about
59&#x2032; 8&#x2033; 2/10 of a degree every day. So that the star which
passed the meridian yesterday at the same moment with
the sun, is to-day about 59&#x2032; 8&#x2033; past the meridian to the
west, when the sun arrives at it; which will take him
up about 3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi> of time to pass over; and therefore
the star passes by 3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi> sooner than the sun each day,
or anticipates his motion at that rate. The true quantity
of this anticipation, or acceleration, is found by
this proportion, 360&#xB0; 59&#x2032; 8&#x2033; 1/5 :: 24 hours:
3<hi rend="sup">m</hi> 55<hi rend="sup">s</hi> 9/10, the fourth term of which is the acceleration.</p><p>The diurnal acceleration serves to regulate the lengths
or vibration of pendulums. If I observe a fixed star
set or pass behind a hill, steeple, or such like, when the
pendulum marks for instance 8<hi rend="sup">h</hi> 10<hi rend="sup">m</hi>; and the next day,
the eye being in the same place as before, the passage
be at 8<hi rend="sup">h</hi> 6<hi rend="sup">m</hi> 4<hi rend="sup">s</hi>; I thence conclude that the pendulum
is well regulated, or truly measures mean time.</p><p><hi rend="smallcaps">Acceleration</hi> <hi rend="italics">of a Planet.</hi> A planet is said to be
accelerated in its motion, when its real diurnal motion
exceeds its mean diurnal motion. And, on the other
hand, the planet is said to be retarded in its motion,
when the mean exceeds the real diurnal motion. This
inequality arises from the change in the distance of the
planet from the sun, which is continually varying; the
planet moving always quicker in its orbit when nearer
the sun, and slower when farther off.</p><p><hi rend="smallcaps">Acceleration</hi> <hi rend="italics">of the Moon,</hi> is a term used to express
the increase of the moon's mean motion from the
sun, compared with the diurnal motion of the earth;
by which it appears that, from some uncertain cause, it
is now a little quicker than it was formerly. Dr. Halley
was led to the discovery, or suspicion, of this acceleration,
by comparing the ancient eclipses observed at
Babylon, &amp;c, and those observed by Albategnius in
the ninth century, with some of his own time; as may
be seen in N 218 of the Philosophical Transactions.
He could not however ascertain the quantity of the
acceleration, because the longitudes of Bagdat, Alexandria,
and Aleppo, where the observations were made,
had not been accurately determined. But since his
time the longitude of Alexandria has been ascertained
by Chazelles; and Babylon, according to Ptolemy's
account, lies 50&#x2032; east of Alexandria. From these data,
Mr. Dunthorne, vol. 46 Philos. Transactions, compared
the recorded times of several ancient and modern eclipses,
with the calculations of them by his own tables, and
thereby verified the suspicion that had been started by
Dr. Halley; for he found that the same tables gave the
moon's place more backward than her true place in
ancient eclipses, and more forward than her true place
in later eclipses; and thence he justly inferred that her
motion in ancient times was slower, and in later times
quicker, than the tables give it.</p><p>Not content however with barely ascertaining the
fact, he proceeded to determine, as well as the observations
would allow, the quantity of the acceleration;
and by means of the most authentic eclipse, of which
any good account remains, observed at Babylon in the
year 721 before Christ, he found that the observed beginning
of this eclipse was about an hour and three
quarters sooner than the beginning by the tables; and
that therefore the moon's true place preceded her place
by computation by about 50&#x2032; of a degree at that time.
<cb/>
Then admitting the acceleration to be uniform, and
the aggregate of it as the square of the time, it will
be at the rate of about 10&#x2033; in 100 years.</p><p>Dr. Long, vol. ii. p. 436 of his Astronomy, enumerates
the following causes from some one or more
of which the acceleration may arise. Either 1st, the
annual and diurnal motion of the earth continuing the
same, the moon is really carried about the earth with
a greater velocity than formerly: or, 2dly, the diurnal
motion of the earth, and the periodical revolution of
the moon, continuing the same, the annual motion of
the earth about the sun is retarded; which makes the
sun's apparent motion in the ecliptic a little slower than
formerly; and consequently the moon, in passing from
any conjunction with the sun, takes up a less time before
she again overtakes the sun, and forms a subsequent
conjunction: in both these cases, the motion of the
moon from the sun is really accelerated, and the synodical
month actually shortened: or, 3dly, the annual motion
of the earth, and the periodical revolution of the
moon, continuing the same, the rotation of the earth
upon its axis is a little retarded; in this case, days,
hours, minutes, &amp;c, by which all periods of time must
be measured, appear of a longer duration; and consequently
the synodical month will appear to be shortened,
though it really contain the same quantity of absolute
time as it always did. If the quantity of matter in the
body of the sun be lessened, by the particles of light
continually streaming from it, the motion of the earth
about the sun may become slower: if the earth increases
in bulk, the motion of the moon about the
earth may thereby be quickened.</p><p>ACCELERATIVE <hi rend="smallcaps">Force</hi>, <hi rend="italics">&amp;c,</hi> the same as A<hi rend="smallcaps">CCELERATING.</hi></p></div2></div1><div1 part="N" n="ACCESSIBLE" org="uniform" sample="complete" type="entry"><head>ACCESSIBLE</head><p>, something that may be approached,
or to which we can come. In Surveying, it is such
a place as will admit of having a distance or length of
ground measured from it; or such a height or depth as
can be measured by actually applying a proper instrument
to it. For the means of doing which, see A<hi rend="smallcaps">LTIMETRY,
Longimetry</hi>, or <hi rend="smallcaps">Heights-and-Distances.</hi></p></div1><div1 part="N" n="ACCIDENS" org="uniform" sample="complete" type="entry"><head>ACCIDENS</head><p>, <hi rend="smallcaps">Accident</hi>, <hi rend="italics">Philos.</hi></p><p><hi rend="italics">Per</hi> <hi rend="smallcaps">Accidens</hi> is a term often used among philosophers,
to denote what does not follow from the nature
of a thing, but from some accidental quality of it: in
this sense it stands opposed to <hi rend="italics">per se,</hi> which denotes
the nature and essence of a thing. Thus, fire is said to
burn <hi rend="italics">per se,</hi> or considered as sire, and not <hi rend="italics">per accidens;</hi>
but a piece of iron, though red-hot, only burns <hi rend="italics">per
accidens,</hi> by a quality accidental to it, and not considered
as iron.</p><div2 part="N" n="Accidents" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Accidents</hi>, in <hi rend="italics">Astrology</hi></head><p>, denote the most extraordinary
occurrences in the course of a person's life, either
good or bad: such as a remarkable instance of good
fortune, a signal deliverance, a great sickness, &amp;c.</p></div2></div1><div1 part="N" n="ACCIDENTAL" org="uniform" sample="complete" type="entry"><head>ACCIDENTAL</head><p>, something that partakes of the
nature of an accident; or that is indifferent, or not essential
to its subject.&#x2014;Thus whiteness is accidental to
marble, and sensible heat to iron.</p><p><hi rend="smallcaps">Accidental</hi> <hi rend="italics">Colours,</hi> so called by M. Buffon, are
those which depend on the affections of the eye, in
contradistinction to such as belong to light itself.</p><p>The impressions made upon the eye, by looking stedfastly
on objects of a particular colour, are various
<pb n="24"/><cb/>
according to the single colour, or assemblage of
colours, in the object; and they continue for some
time after the eye is withdrawn, and give a false colouring
to other objects that are viewed during their continuance.
M. Buffon has endeavoured to trace the connection
between these accidental colours, and those that
are natural, in a variety of instances. M. d' Arcy contrived
a machine for measuring the duration of those
impressions on the eye; and from the result of several
trials he inserred, that the effect of the action of light
on the eye continued about eight thirds of a minute.</p><p>The subject has also been considered by M. de la
Hire, and M. Aepinus, &amp;c. See Mem. Acad. Paris
1743, and 1765; Nov. Com. Petrop. vol. 10; also Dr.
Priestley's Hist. of Discoveries relating to Vision,
pa. 631.</p><p><hi rend="smallcaps">Accidental</hi> <hi rend="italics">Point,</hi> in Perspective, is the point in
which a right line drawn from the eye, parallel to another
right line, cuts the picture or perspective plane.
<figure/></p><p>Let AB be the line given to be put into perspective,
CFD the picture or perspective plane, and E the eye:
draw EF parallel to AB; so shall F be the accidental
point of the line AB, and indeed of all lines parallel
to it, since only one parallel to them, namely EF, can
be drawn from the same point E: and in the accidental
point concur or meet the representations of all the parallels
to AB, when produced.</p><p>It is called the accidental point, to distinguish it
from the principal point, or point of view, where a line
drawn from the eye perpendicular to the perspective
plane, meets this plane, and which is the accidental
point to all lines that are perpendicular to the same plane.</p><p><hi rend="smallcaps">Accidental</hi> <hi rend="italics">Dignities,</hi> and <hi rend="italics">Debilities,</hi> in <hi rend="italics">Astrology,</hi>
are certain casual dispositions, and affections, of the
planets, by which they are supposed to be either
strengthened, or weakened, by being in such a house
of the figure.</p></div1><div1 part="N" n="ACCLIVITY" org="uniform" sample="complete" type="entry"><head>ACCLIVITY</head><p>, the slope or steepness of a line or
plane inclined to the horizon, taken upwards; in contradistinction
to <hi rend="italics">declivity,</hi> which is taken downwards.
So the ascent of a hill, is an <hi rend="italics">acclivity:</hi> the descent of the
same, a <hi rend="italics">declivity.</hi></p><p>Some writers on fortification use acclivity for <hi rend="italics">talus:</hi>
though more commonly the word talus is used to denote
the slope, whether in ascending or descending.</p></div1><div1 part="N" n="ACCOMPANYMENT" org="uniform" sample="complete" type="entry"><head>ACCOMPANYMENT</head><p>, in <hi rend="italics">Music,</hi> denotes either
the different parts of a piece of music for the different
instruments, or the instruments themselves which accompany
a voice, to sustain it, as well as to make the
music more full.</p><p>The Accompanyment is used in recitative, as well as
in song; on the stage, as well as in the choir, &amp;c.
<cb/></p><p>The ancients had likewise their accompanyments
on the theatre; and they had even different kinds of
instruments to accompany the chorus, from those which
accompanied the actors in the recitation.</p><p>The accompanyment among the moderns, is often a
different part, or melody, from the song it accompanies.
But it is disputed whether it was so among the ancients.</p><p>Organists sometimes apply the word to several pipes
which they occasionally touch to accompany the treble;
as the drone, the flute, &amp;c.</p><p>ACCOMPT. See <hi rend="smallcaps">Account.</hi></p></div1><div1 part="N" n="ACCORD" org="uniform" sample="complete" type="entry"><head>ACCORD</head><p>, according to the modern French music,
is the union of two or more sounds heard at the same
time, and forming together a regular harmony.</p><p>They divide Accords into <hi rend="italics">persect</hi> and <hi rend="italics">imperfect;</hi> and
again into <hi rend="italics">consonances</hi> and <hi rend="italics">dissonances.</hi></p><p>Accord is more commonly called <hi rend="smallcaps">Concord</hi>, which
see.</p><p><hi rend="smallcaps">Accord</hi> is also spoken of the state of an instrument,
when its fixed sounds have among themselves all the
justness that they ought to have.</p></div1><div1 part="N" n="ACCOUNT" org="uniform" sample="complete" type="entry"><head>ACCOUNT</head><p>, or <hi rend="smallcaps">Accompt</hi>, in <hi rend="italics">Arithmetic,</hi> &amp;c, a
calculation or computation of the number or order of
certain things; as the computation of time, &amp;c.</p><p>There are various ways of accounting; as, by enumeration,
or telling one by one; or by the rules of
arithmetic, addition, subtraction, &amp;c.</p><div2 part="N" n="Account" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Account</hi>, in <hi rend="italics">Chronology</hi></head><p>, is nearly synonymous with
style. Thus, we say the English, the foreign, the Julian,
the Gregorian, the Old, or the New account, or
style.</p><p>We account time by years, months, &amp;c; the
Greeks accounted it by olympiads; the Romans, by
indictions, lustres, &amp;c.</p></div2><div2 part="N" n="Acherner" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Acherner</hi></head><p>, or <hi rend="smallcaps">Acharner</hi>, in <hi rend="italics">Astronomy,</hi> a star of
the first magnitude in the southern extremity of the
constellation Eridanus, marked <foreign xml:lang="greek">a</foreign> by Bayer. Its longitude
for 1761, <figure/> 11&#xB0; 55&#x2032; 1&#x2033;; and latitude south
59&#xB0; 22&#x2032; 4&#x2033;.</p></div2></div1><div1 part="N" n="ACHILLES" org="uniform" sample="complete" type="entry"><head>ACHILLES</head><p>, a name given by the schools to the
principal argument alleged by each sect of philosophers
in behalf of their system. In this sense we say this is
his Achilles; that is, his master-proof: alluding to
the strength and importance of the hero Achilles among
the Greeks.</p><p>Zeno's argument against motion is peculiarly termed
Achilles. That philosopher made a comparison between
the swiftness of Achilles, and the slowness of a tortoise,
pretending that a very swift animal could never overtake
a slow one that was before it, and that therefore
there is no such thing as motion: for, said he, if the
tortoise were one mile before Achilles, and the motion
of Achilles 100 times swifter than that of the tortoise,
yet he would never overtake it; and for this reason,
namely, that while Achilles runs over the mile, the
tortoise will creep over one hundredth part of a mile,
and will be so much the foremost; again while Achilles
runs over this 1/100th part, the tortoise will creep over
the 100th part of that 1/100th part, and will still be this
last part the foremost; and so on continually, according
to an infinite series of 100th parts: from which he
concluded that the swifter could never overtake the
slower in any finite time, but that they must go on ap-
<pb n="25"/><cb/>
proaching to infinity. But this sophism lay in their
considering as an infinite time, the sum of the infinite
series of small times in which Achilles could run over
the infinite series of spaces, 1 + 1/100 + 1/10000 +
1/1000000 &amp;c, not knowing that the sum of this infinite
series is equal to the quantity 1 1/99 of a mile, and that
therefore Achilles will overtake the tortoise when the
latter has crawled over 1/99th of a mile.</p></div1><div1 part="N" n="ACHROMATIC" org="uniform" sample="complete" type="entry"><head>ACHROMATIC</head><p>, in <hi rend="italics">Optics,</hi> without colour; a
term which, it seems, was first used by M. de la Lande,
in his astronomy, to denote telescopes of a new invention,
contrived to remedy aberrations and colours. See
<hi rend="italics">Aberration</hi> and <hi rend="italics">Telescope.</hi></p><p><hi rend="smallcaps">Achromatic Telescope</hi>, a singular species of refracting
telescope, said to be invented by the late Mr.
John Dollond, optician to the king, and since improved
by his son Mr. Peter Dollond, and others.</p><p>Every ray of light passing obliquely from a rarer
into a denser medium, changes its direction towards the
perpendicular; and every ray passing obliquely from a
denser into a rarer medium, changes its direction from the
perpendieular. This bending of the ray, caused by the
change of its direction, is called its refraction; and
the quality of light which subjects it to this refraction,
is called its refrangibility. Every ray of light, before
it is refracted, is white, though it consists of a number
of component rays, each of which is of a different colour.
As soon as it is refracted, it is separated into its
component rays, which, from that time, proceed diverging
from each other, like rays from a centre: and
this divergency is caused by the different refrangibility
of the component rays, in such sort, that the more the
original or component ray is refracted, the more will
the compound rays diverge when the light is refracted
by one given medium only.</p><p>From hence it has been concluded, that any two different
mediums that can be made to produce equal refractions,
will necessarily produce equal divergencies:
whence it should also follow, that equal and contrary
refractions should not only destroy each other, but that
the divergency of the colours caused by one refraction,
should be corrected by the other; and that to produce
refraction that would not be affected by the different
refrangibility of light, is impossible.</p><p>But Mr. Dollond has proved, by many experiments,
that these conclusions are not well founded; from which
experiments it appeared, that a ray of light, after equal
and contrary refractions, was still spread into component
rays differently coloured: in other words, that
two different mediums may cause equal refraction, but
different divergency; and equal divergency, with different
refraction. It follows therefore that refraction
may be produced, which is not affected by the different
refrangibility of light. In other words, that, if the
mediums be different, different refractions may be produced,
though at the same time the divergency caused
by one refraction shall be exactly counteracted by the
divergency caused by the other; and so an object may
be seen through mediums which, together, cause the
rays to converge, without appearing of different colours.</p><p>This is the foundation of Mr. Dollond's improvement
of refracting telescopes. By subsequent experiments he
found, that different sorts of glass differed greatly in their
refractive qualities, with respect to the divergency of colours.
He found that crown glass causes the least diver-
<cb/>
gency, and white flint the most, when they are wrought
into forms that produce equal refractions. He ground a
piece of white flint glass into a wedge, whose angle was
about 25 degrees; and a piece of crown glass to another,
whose angle was about 29 degrees; and these he found
refracted nearly alike, but that their divergency of
colours was very different.</p><p>He then ground several other pieces of crown glass
to wedges of different angles, till he got one that was
equal, in the divergency it produced, to that of a wedge
of flint glass of 25 degrees; so that when they were
put together, in such a manner as to refract in contrary
directions, the refracted light was perfectly free from
colour. Then measuring the fractions of each wedge,
he found that that of the white flint glass, was to that
of the crown glass, nearly as two to three. And hence
any two wedges, made of these two substances, and in
this proportion, would, when applied together so as to
refract in contrary directions, refract the light without
any effect ariling from the different refrangibility of the
component rays.</p><p>Therefore, to make two spherical glasses that refract
the light in contrary directions, one must be concave,
and the other convex; and as the rays, after passing
through both, must meet in a focus, the excess of the
refraction must be in the convex one: and as the convex
is to refract most, it appears from the experiment
that it must be made of crown glass; and as the concave
is to refract least, it must be made of white flint.</p><p>And farther, as the refractions of spherical glasses
are in an inverse ratio of their focal distances, it follows
that the focal distances of the two glasses should be in
the ratio of the refractions of the wedges; for, being
thus proportioned, every ray of light that passes through
this combined glass, at whatever distance from its axis,
will constantly be refracted by the difference between
two contrary refractions, in the proportion required;
and therefore the effect of the different refrangibility
of light will be prevented.</p><p>The removal of this impediment, however, produced
another: for the two glasses, which were thus combined,
being segments of very deep spheres, the aberrations
from the spherical surfaces became so considerable,
as greatly to disturb the distinctness of the image.
Yet considering that the surfaces of spherical glasses
admit of great variations, though the focal distance be
limited, and that by these variations their aberration
might be made more or less at pleasure; Mr. Dollond
plainly saw that it was possible to make the aberrating
of any two glasses equal; and that, as in this case the
refractions of the two glasses were contrary to each
other, and their aberrations being equal, these would
destroy each other.</p><p>Thus he obtained a persect theory of making object
glasses, to the apertures of which he could hardly perceive
any limits: for if the practice could come up to
the theory, they must admit of apertures of great extent,
and consequently bear great magnifying powers.</p><p>The difficulties of the practice are, however, still
very considerable. For first, the focal distances, as
well as the particular surfaces, must be proportioned
with the utmost accuracy to the densities and refracting
powers of the glasses, which vary even in the same sort
of glass, when made at different times. Secondly,
there are four surfaces to be wrought persectly spherical.
<pb n="26"/><cb/>
However, Mr. Dollond could construct refracting telescopes
upon these principles, with fuch apertures and
magnifying powers, under limited lengths, as greatly
exceed any that were produced before, in forming the
images of objects bright, distinct, and uninfected with
colours about the edges, through the whole extent of a
very large field or compass of view; of which he has
given abundant and undeniable testimony. See T<hi rend="smallcaps">ELESCOPE.</hi></p><p>There has lately appeared in the Gentleman's Magazine
(1790, pa. 890) a paper on the refracting telescope,
by an author who signs <hi rend="italics">Veritus,</hi> in which the
invention is ascribed to another person, not heretofore
mentioned; in these words: &#x201C;As the invention has
been claimed by M. Euler, M. Klingenstierna, and
some other foreigners, we ought, for the honour of
England, to assert our right, and give the merit of the
discovery to whom it is due; and therefore, without
farther preface, I shall observe, that the inventor was
Chester More Hall, Esq. of More-hall, in Essex, who,
about 1729, as appears by his papers, considering the
different humours of the eye, imagined they were
placed so as to correct the different refrangibility of
light. He then conceived, that if he could find substances
having such properties as he supposed these humours
might possess, he should be enabled to construct
an object glass that would shew objects colourless.
After many experiments he had the good fortune to
find those properties in two different sorts of glass, and
making them disperse the rays of light in different directions,
he succeeded. About 1733 he completed several
achromatic object glasses (though he did not give
them this name), that bore an aperture of more than
2 1/2 inches, though the focal length did not exceed 20
inches; one of which is now in the possession of the
Rev. Mr. Smith, of Charlotte Street, Rathbone Place.</p><p>This glass has been examined by several gentlemen
of eminence and scientific abilities, and found to possess
the properties of the present achromatic glasses.</p><p>Mr. Hall used to employ the working opticians to
grind his lenses; at the same time he finished them
with the radii of the surfaces, not only to correct the
different refrangibility of rays, but also the aberration
arising from the spherical figure of the lenses. Old Mr.
Bass, who at that time lived in Bridewell precinct, was
one of these working opticians, from whom Mr. Hall's
invention seems to have been obtained.</p><p>In the trial at Westminster hall about the patent for
making achromatio telescopes, Mr. Hall was allowed
to be the inventor; but Lord Mansfield observed, that
&#x201C;It was not the person that locked up his invention in
his scrutoire that ought to profit by a patent for such
an invention, but he who brought it forth for the benefit
of the public.&#x201D; This, perhaps, might be said with
some degree of justice, as Mr. Hall was a gentleman
of property, and did not look to any pecuniary advantage
from his discovery; and, consequently, it is very
probable that he might not have an intention to make
it generally known at that time.</p><p>That Mr. Ayscough, optician on Ludgate Hill,
was in possession of one of Mr. Hall's achromatic telescopes
in 1754, is a fact which at this time will not be
disputed.&#x201D;</p></div1><div1 part="N" n="ACHRONICAL" org="uniform" sample="complete" type="entry"><head>ACHRONICAL</head><p>, or <hi rend="italics">Achronycal.</hi> See <hi rend="smallcaps">Acronychal.</hi>
<cb/></p><p>ACOUSTICS. This term, in physico-mathematical
meaning, signifies the doctrine of hearing, and the art
of assisting that sense by means of speaking trumpets,
hearing trumpets, whispering galleries, and such like.
See <hi rend="smallcaps">Stentrophonic Tube.</hi></p><p>Sturmius, in his Elements of Universal Mechanics,
treating of Acoustics, after examining into the nature
of sounds, describes the several parts of the external
and internal ear, and their several uses and connexions
with each other; and from thence deduces the mechanism
of hearing: and lastly, he treats of the means of
adding an intensity of force to the voice and other
sounds; and explains the nature of echoes, otacoustic
tubes, and speaking trumpets. See <hi rend="smallcaps">Sound, Ear, Voice</hi>,
and <hi rend="smallcaps">Echo.</hi></p><p>Dr. Hook, in the preface to his Micrography, asserts
that the lowest whisper, by certain means, may be
heard at the distance of a furlong; and that he knew a
way by which it is easy to hear any one speak through
a wall of three feet thick; also that by means of an extended
wire, sound may be conveyed to a very great
distance, almost in an instant.</p></div1><div1 part="N" n="ACRE" org="uniform" sample="complete" type="entry"><head>ACRE</head><p>, from the Saxon <hi rend="italics">&#xE6;cre,</hi> or German <hi rend="italics">acker,</hi> a <hi rend="italics">field,</hi>
of the Latin <hi rend="italics">ager.</hi> It is a measure of land, containing,
by the ordinance for measuring land, made in the 33d
and 34th of Edward I, 160 perches or square poles of
land; that is, 16 in length and 10 in breadth, or in that
proportion: and as the statute length of a pole is 5 1/2
yards, or 16 1/2 feet, therefore the acre will contain 4840
square yards, or 43560 square feet. The chain with
which land is commonly measured, and which was invented
by Gunter, is 4 poles or 22 yards in length;
and therefore the acre is just 10 square chains; that is,
10 chains in length and one in breadth, or in that proportion.
Farther, as a mile contains 1760 yards, or
80 chains in length, therefore the square mile contains
640 acres.</p><p>The acre, in surveying, is divided into 4 roods, and
the rood is 40 perches.</p><p>The French acre, <hi rend="italics">arpent,</hi> is equal to 1 1/4 English acre;</p><p>The Strasburg contains about 1/2 an English acre;</p><p>The Welch acre contains about 2 English acres;</p><p>The Irish acre contains 1 ac. 2 r. 19 27/121 p. English.</p><p>Sir William Petty, in his Political Arithmetic,
reckons that England contains 39 million acres:
but Dr. Greve shews, in the Philos. Trans. N&#xB0; 330,
that England contains not less than 46 million acres.
Whence he infers that England is above 46 times as
large as the province of Holland, which it is said contains
but about one million of acres.</p><p>By a statute of the 31st of Elizabeth, it is ordained,
that if any man erect a cottage, he shall annex four
acres of land to it.</p></div1><div1 part="N" n="ACRONYCHAL" org="uniform" sample="complete" type="entry"><head>ACRONYCHAL</head><p>, or <hi rend="smallcaps">Acronycal</hi>, in <hi rend="italics">Astronomy,</hi>
is said of a star or planet, when it is opposite to the
sun. It is from the Greek <foreign xml:lang="greek">axronuxos</foreign>, the point or extremity
of night, because the star rose at sun-set, or the
beginning of night, and set at sun-rise, or the end of
night; and so it shone all the night.</p><p>The acronychal is one of the three Greek poetic
risings and settings of the stars; and stands distinguished
from Cosmical and Heliacal. And by means of
which, for want of accurate instruments, and other observations,
they might regulate the length of their year.
<pb n="27"/><cb/></p></div1><div1 part="N" n="ACROTERIA" org="uniform" sample="complete" type="entry"><head>ACROTERIA</head><p>, or <hi rend="smallcaps">Acroters</hi>, in <hi rend="italics">Architecture,</hi> small
pedestals, usually without bases, placed on pediments,
and serving to support statues.</p><p>Those at the extremities ought to be half the height
of the tympanum; and that in the middle, according
to Vitruvius, one eighth part more.</p><p><hi rend="smallcaps">Acroteria</hi> also are sometimes used to signify
figures, whether of stone or metal, placed as ornaments
or crownings, on the tops of temples, or other
buildings.</p><p>It is also sometimes used to denote those sharp pinacles
or spiry battlements, that stand in ranges about
flat buildings, with rails and balustres.</p></div1><div1 part="N" n="ACTION" org="uniform" sample="complete" type="entry"><head>ACTION</head><p>, in <hi rend="italics">Mechanics</hi> or <hi rend="italics">Physics,</hi> a term used to
denote, sometimes the effort which some body or power
exerts against another body or power, and sometimes
it denotes the effects resulting from such esfort.</p><p>The Cartesians resolve all physical action into metaphysical.
Bodies, according to them, do not act on one
another; the action comes all immediately from the
Deity; the motions of bodies, which seem to be the
cause, being only the occasions of it.</p><p>It is one of the laws of nature, that action and reaction
are always equal, and contrary to each other
in their directions.</p><p>Action is either instantaneous or continued; that is,
either by collition or perc&lt;*&gt;ssion, or by pressure. These
two sorts of action are heterogeneous quantities, and
are not comparable, the smallest action by percussion
exceeding the greatest action of pressure, as the smallest
surface exceeds the longest line, or as the smallest solid
exceeds the largest surface: thus, a man by a small
blow with a hammer, will drive a wedge below the
greatest ship on the stocks, or under any other weight;
that is, the smallest percussion overcomes the pressure
of the greatest weight. These actions then cannot be
measured the one by the other, but each must have a
measure of its own kind, like as solids must be measured
by solids, and surfaces by surfaces: time being
concerned in the one, but not in the other.</p><p>If a body be urged at the same time by equal and
contrary actions, it will remain at rest. But if one of
these actions be greater than its opposite, motion will
ensue towards the part least urged.</p><p>The actions of bodies upon each other, in a space
that is carried uniformly forward, are the same as if
the space were at rest; and any powers or forces that
act upon all bodies, so as to produce equal velocities in
them in the same, or in parallel right lines, have no
effect on their mutual actions, or relative motions.
Thus the motion of bodies on board of a ship that is
carried uniformly forward, are performed in the same
manner as if the ship was at rest. And the motion of
the earth about its axis has no effect on the actions of
bodies and agents at its surface, except in so far as it
is not uniform and rectilineal. In general, the actions
of bodies upon each other, depend not on their absolute,
but relative motion.</p><p><hi rend="italics">Quantity of</hi> <hi rend="smallcaps">Action</hi>, in Mechanics, a name given
by M. de Maupertuis, in the Memoirs of the Academy
of Sciences of Paris for 1744, and in those of Berlin
for 1746, to the continual product of the mass of a
body, by the space which it runs through, and by its
celerity. He lays it down as a general law, that in the
<cb/>
changes made in the state of a body, the quantity of
action necessary to produce such change is the least possible.
This principle he applies to the investigation of
the laws of refraction, and even the laws of rest, as he
calls them; that is, of the equilibrium or equipollency
of pressures; and even to the modes of acting of the
Supreme Being. In this way Maupertuis attempts
to connect the metaphysics of sinal causes with the
fundamental truths of mechanics; to shew the dependence
of the collision of both elastic and hard
bodies, upon one and the same law, which before had
always been referred to separate laws; and to reduce
the laws of motion, and those of equilibrium, to one
and the same principle.</p><p>But this quantity of motion, of Maupertuis, which
is defined to be the product of the mass, the space
passed over, and the celerity, comes to the same thing
as the mass multiplied by the square of the velocity,
when the space passed over is equal to that by which
the velocity is measured; and so the quantity of force
will be proportional to the mass multiplied by the
square of the velocity; since the space is measured by
the velocity continued for a certain time.</p><p>In the same year that Maupertuis communicated the
idea of his principle, professor Euler, in the supplement
to his book, intitled <hi rend="italics">Methodus inveniendi lineas curvas
maximi vel minimi proprietate gaudentes,</hi> demonstrates,
that in the trajectories which bodies describe by central
forces, the velocity multiplied by what the foreign mathematicians
call the element of the curve, always
makes a minimum; which Maupertuis considered as
an application of his principle to the motion of the
planets.</p><p>It appears from Maupertuis's Memoir of 1744, that
it was his reflections on the laws of refractions, that led
him to the theorem above mentioned. The principle
which Fermat, and after him Leibnitz, made use of,
in accounting for the laws of refraction, is sufficiently
known. Those mathematicians pretended, that a particle
of light, in its passage from one point to another,
through two mediums, in each of which it moves with
a different velocity, must do it in the shortest time possible:
and from this principle they have demonstrated
geometrically, that the particle cannot go from the one
point to the other in a right line; but being arrived at
the surface that separates the two mediums, it must
alter its direction in such a manner, that the sine of
its incidence shall be to the sine of its refraction, as
its velocity in the first medium is to its velocity in
the second: whence they deduced the well known law
of the constant ratio of those sines.</p><p>This explanation, though very ingenious, is liable
to this pressing difficulty, namely, that the particle
must approach towards the perpendicular, in that medium
where its velocity is the least, and which consequently
resists it the most: which seems contrary to
all the mechanical explanations of the refraction of
bodies, that have hitherto been advanced, and of the
refraction of light in particular.</p><p>Sir Isaac Newton's way of accounting for it, is the
most satisfactory of any that has hitherto been offered,
and gives a clear reason for the constant ratio of the
sines, by ascribing the refraction to the attractive force
of the mediums; from which it follows, that the densest
<pb n="28"/><cb/>
mediums, whose attraction is the strongest, should
cause the ray to approach the perpendicular; a fact
confirmed by experiment. But the attraction of the
medium could not ca&#xFA;se the ray to approach towards
the perpendicular, without increasing its velocity; as
may easily be demonstrated. Thus then, according to
Newton, the refraction must be towards the perpendicular,
when the velocity is increased: contrary to
the law of Fermat and Leibnitz.</p><p>Maupertuis has attempted to reconcile Newton's
explanation with metaphysical principles. Instead of
supposing, as the aforesaid gentlemen do, that a particle
of light proceeds from one point to another in
the shortest time possible; he contends that a particle
of light passes from one point to another in such a
manner, that the quantity of action shall be the least
possible. This quantity of action, says he, is a real
expence, in which nature is always frugal. In virtue
of this philosophical principle he discovers, that not
only the sines are in a constant ratio, but also that they
are in the inverse ratio of the velocities, according to
Newton's explanation, and not in the direct ratio, as
had been pretended by Fermat and Leibnitz.</p><p>It is remarkable that, of the many philosophers who
have written on refraction, none should have fallen
upon so simple a manner of reconciling metaphysics
with mechanics; since no more is necessary to that,
than making a small alteration in the ealculus founded
upon Fermat's principle. Now according to that
principle, the time, that is, the space divided by the
velocity, should be a minimum; so that calling the
space run through in the first medium S, with the
velocity V, and the space run through in the second
medium <hi rend="italics">s,</hi> with the velocity <hi rend="italics">v,</hi> we shall have  minimum; that is to say, . Now it is easy to perceive, that the sines of incidence
and refraction are to each other, as S<hi rend="sup">.</hi> to-<hi rend="italics">s<hi rend="sup">.</hi></hi>;
whence it follows, that those sines are in the direct
ratio of the velocities V, <hi rend="italics">v</hi>; which is exactly what
Fermat makes it to be. But in order to have those
sines to be in the inverse ratio of the velocities, it is
only supposing ; which gives  a minimum: which is Maupertuis's
principle.</p><p>In the Memoirs of the Academy of Berlin, above
cited, may be seen all the other applications which
Maupertuis has made of this principle. And whatever
may be determined as to his metaphysical basis of it, as
also to the idea he has annexed to the quantity of action,
it will still hold good, that the product of the
space by the velocity is a minimum in some of the
most general laws of nature.</p></div1><div1 part="N" n="ACTIVE" org="uniform" sample="complete" type="entry"><head>ACTIVE</head><p>, the quality of an agent, or of communicating
motion or action to some body. In this sense
the word stands opposed to passive: thus we say an
active cause, active principle, &amp;c.</p><p>Sir Isaac Newton shews that the quantity of motion
in the world must be always deereasing, in consequence
of the vis inerti&#xE6;, &amp;c. So that there is a necessity for
<cb/>
certain active principles to recruit it: such he takes the
cause of gravity to be, and the cause of fermentation;
adding, that we see but little motion in the universe,
except what is owing to these active principles.</p></div1><div1 part="N" n="ACTIVITY" org="uniform" sample="complete" type="entry"><head>ACTIVITY</head><p>, the virtue or faculty of acting. As
the activity of an acid, a poison, &amp;c: the activity of
fire exceeds all imagination.</p><p>According to Sir Isaac Newton, bodies derive their
activity from the principle of attraction.</p><p><hi rend="italics">Sphere of</hi> <hi rend="smallcaps">Activity</hi>, is the space which surrounds
a body, as far as its efficacy or virtue extends to produce
any sensible effect. Thus we say, the sphere of
activity of a loadstone, of an electric body, &amp;c.</p></div1><div1 part="N" n="ACUBENE" org="uniform" sample="complete" type="entry"><head>ACUBENE</head><p>, in <hi rend="italics">Astronomy,</hi> the Arabic name of a
star of the fourth magnitude, in the southern claw of
Cancer, marked <foreign xml:lang="greek">a</foreign> by Bayer. Its longitude for 1761,
<figure/> 10&#xB0; 18&#x2032; 9&#x2033;, south latitude 5&#xB0; 5&#x2032; 56&#x2033;.</p></div1><div1 part="N" n="ACUTE" org="uniform" sample="complete" type="entry"><head>ACUTE</head><p>, or sharp; a term opposed to obtuse.
Thus, <hi rend="smallcaps">Acute</hi> <hi rend="italics">Angle,</hi> in <hi rend="italics">Geometry,</hi> is that which is less
than a right angle; and is measured by less than 90&#xB0;,
or by less than a quadrant of a circle. As the angle
ABC.
<figure/></p><p><hi rend="smallcaps">Acute</hi> <hi rend="italics">angled Triangle,</hi> is that whose three angles
are all acute; and is otherwise called an oxygenous
triangle. As the triangle DEF.</p><p><hi rend="smallcaps">Acute</hi>-<hi rend="italics">angled Cone,</hi> is that whose opposite sides
make an acute angle at the vertex, or whose axis, in a
right cone, makes less than half a right angle with the
side As the cone GHI.</p><p>Pappus, in his Mathematical Collections, says, this
name was given to such a cone by Euclid and the ancients,
before the time of Apollonius. And they
called an</p><p><hi rend="smallcaps">Acute</hi>-<hi rend="italics">angled Section of a Cone,</hi> an Ellipsis, which
was made by a plane cutting both sides of an acuteangled
cone: not knowing that such a section could
be generated from any cone whatever, till it was shewn
by Apollonius.</p><div2 part="N" n="Acute" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Acute</hi>, in <hi rend="italics">Music</hi></head><p>, is understood of a tone, or sound,
which is high, sharp, or shrill, in respect of some other:
in which sense the word stands opposed to grave. And
both these sounds are independent of loudness or force:
so that the tone may be acute or high, without being
loud; and loud without being high or acute. For
both the affections of acute and grave, depend intirely
on the quickness or slowness of the vibrations by which
they are produced.</p><p>Sounds considered as grave and acute, that is, in the
relation of gravity and acuteness, constitute what is
called tune, the soundation of all harmony.</p></div2></div1><div1 part="N" n="ADAGIO" org="uniform" sample="complete" type="entry"><head>ADAGIO</head><p>, in <hi rend="italics">Music,</hi> one of the terms used by the
Italians to express a degree or distinction of time.</p><p>Adagio denotes the slowest time except grave.</p><p>Sometimes the word is repeated, as <hi rend="italics">adagio, adagio,</hi>
to denote a still slower time than the former.
<pb n="29"/><cb/></p><p>Adagio also signifies a slow movement, when used
substantively.</p></div1><div1 part="N" n="ADAMAS" org="uniform" sample="complete" type="entry"><head>ADAMAS</head><p>, in <hi rend="italics">Astrology,</hi> a name given to the moon.</p></div1><div1 part="N" n="ADAR" org="uniform" sample="complete" type="entry"><head>ADAR</head><p>, in the Hebrew <hi rend="italics">Chronology,</hi> is the 6th month
of their civil year, but the 12th of their ecclesiastical
year. It contains only 29 days; and it answers to our
February; but sometimes entering into the month of
March, according to the course of the moon.</p></div1><div1 part="N" n="ADDITION" org="uniform" sample="complete" type="entry"><head>ADDITION</head><p>, the uniting or joining of two or more
things together; or the finding of one quantity equal
to two or more others taken together.</p><div2 part="N" n="Addition" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Addition</hi>, in <hi rend="italics">Arithmetic</hi></head><p>, is the first of the four fundamental
rules or operations of that science; and it
consists in finding a number equal to several others
taken together, or in finding the most simple expression
of a number according to the established notation. The
quantity so found equal to several others taken together,
is named their sum.</p><p>The sign or character of addition is +, and is called
<hi rend="italics">plus.</hi> This character is set between the quantities to
be added, to denote their sum: thus, , that
is, 3 plus 6 are equal to 9; and ,
that is, 2 plus 4 plus 6 are equal to 12.</p><p>Simple numbers are either added as above;
or else by placing them under one another, as
in the margin, and adding them together, one
after another, beginning at the bottom: thus 2
and 4 make 6, and 6 make 12.
<table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row></table></p><p>Compound numbers, or numbers consisting of more
figures than one, are added, by first ranging the numbers
in columns under each other, placing always the
numbers of the same denomination under each other,
that is, units under units, tens under tens, and so on;
and then adding up each column separately, beginning
at the right hand, setting down the sum of each column
below it, unless it amount to ten or some number of
tens, and in that case setting down only the overplus,
and carrying one for each ten to the next column.
Thus, to add 451 and 326,
<table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">451</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">that is</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">400 + 50 + 1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">326</cell><cell cols="1" rows="1" rend="align=right" role="data">300 + 20 + 6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Sum 777</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">=</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">700 + 70 + 7</cell></row></table></p><p>Also to add the numbers ; set them down as
in the margin, and beginning at the lowest
number on the right hand, say 8 and 7
make 15, and 2 make 17, and 9 make
26; set down 6, and carry 2 to the next
column, saying 2 and 4 make 6, and 4
make 10, and 6 make 16, and 2 make
18; set down 8, and carry 1, saying 1 and
3 make 4, and 5 make 9, and 3 make 12;
set down 2, and carry 1, saying 1 and 2
make 3, and 1 make 4, which set down;
then 1 and 2 make 3; and 7 is 7 to set
down: so the sum of all together is
734286. Or it is the same as the sums
of the columns set under one another, as
in the margin, and then these added up in the same
manner.
<cb/></p><p>When a great number of separate sums or numbers
are to be added, as in long accounts, it is easier to
break or separate them into two or more parcels, which
are added up severally, and then their sums added together
for the total sum. And thus also the truth of
the addition may be proved, by dividing the numbers
into parcels different ways, as the totals must be the
same in both cases when the operation is right.</p><p>Another method of proving addition was given by
Dr. Wallis, in his Arithmetic, published 1657, by casting
out the nines, which method of proof extends also
to the other rules of arithmetic. The method is this:
add the figures of each line of numbers together severally,
casting out always 9 from the sums as they
arise in so adding, adding the overplus to the next
figure, and setting at the end of each line what is over
the nine or nines; then do the same by the sum-total,
as also by the former excesses of 9, so shall the last excesses
be equal when the work is right. So the former
example will be proved as below:
<table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">329</cell><cell cols="1" rows="1" rend="rowspan=6" role="data">Excess of 9's</cell><cell cols="1" rows="1" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1562</cell><cell cols="1" rows="1" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20347</cell><cell cols="1" rows="1" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">712048</cell><cell cols="1" rows="1" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">734286</cell><cell cols="1" rows="1" role="data">3</cell></row></table></p><p>When the numbers are of different denominations;
as pounds, shillings, and pence; or yards, seet, and
inches; place the numbers of the same kind under one
another, as pence under pence, shillings under shillings,
&amp;c; then add each column separately, and carry the
overplus as before, from one column to another. As
in the following examples:
<table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">l.</cell><cell cols="1" rows="1" rend="align=center" role="data">s.</cell><cell cols="1" rows="1" rend="align=center" role="data">d.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Yards.</cell><cell cols="1" rows="1" rend="align=center" role="data">Feet.</cell><cell cols="1" rows="1" rend="align=center" role="data">Inches.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">271</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">271</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">94</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">408</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">sums</cell><cell cols="1" rows="1" rend="align=right" role="data">323</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row></table></p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Decimals,</hi> is performed in the same
manner as that of whole numbers, placing the numbers
of the same denomination under each other, in which
case the decimal separating points will range straight
in one column; as in this example, to add together
these numbers .
<table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">371.0496</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25.213&#xA0;</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1.704&#xA0;</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">924.61&#xA0;&#xA0;</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">.0962</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">The sum</cell><cell cols="1" rows="1" rend="align=right" role="data">1322.6728</cell></row></table></p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Vulgar Fractions,</hi> is performed by
bringing all the proposed fractions to a common denominator,
if they have different ones, which is an indispensable
preparation; then adding all the numerators
<pb n="30"/><cb/>
together, and placing their sum over the common denominator
for the sum total required.</p><p>So .</p><p>ADDITION <hi rend="italics">in Algebra,</hi> or the addition of indeterminate
quantities, denoted by letters of the alphabet,
is performed by connecting the quantities together by
their proper signs, and uniting or reducing such as are
susceptible of it, namely similar quantities, by adding
their co-efficients together if the signs are the same, but
subtracting them when different. Thus the quantity
<hi rend="italics">a</hi> added to the quantity <hi rend="italics">b,</hi> makes <hi rend="italics">a</hi> + <hi rend="italics">b</hi>; and <hi rend="italics">a</hi> joined
with-<hi rend="italics">b,</hi> makes <hi rend="italics">a</hi>-<hi rend="italics">b</hi>; also-<hi rend="italics">a</hi> and-<hi rend="italics">b</hi> make-<hi rend="italics">a</hi>-<hi rend="italics">b</hi>;
and 3<hi rend="italics">a</hi> and 5<hi rend="italics">a</hi> make 3<hi rend="italics">a</hi> + 5<hi rend="italics">a</hi> or 8<hi rend="italics">a,</hi> by uniting
the similar numbers 3 and 5 to make 8.
Thus also .</p><p>In the addition of surd or irrational quantities, they
must be reduced to the same denomination, or to the
same radical, if that can be done; then add or unite
the rational parts, and subjoin the common surd.
Otherwise connect them with their own signs.</p><p>So ;
but of &#x221A;5 and &#x221A;6 the sum is set down &#x221A;5 + &#x221A;6,
because the terms are incommensurable, and not reducible
to a common surd.</p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Logarithms.</hi> See Logarithms.
<cb/></p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Ratios,</hi> the same as composition of
ratios; which see.</p></div2></div1><div1 part="N" n="ADDITIVE" org="uniform" sample="complete" type="entry"><head>ADDITIVE</head><p>, denotes something to be added to
another, in contradistinction to something to be taken
away or subtracted. So astronomers speak of additive
equations, and geometricians of additive rations.</p></div1><div1 part="N" n="ADELARD" org="uniform" sample="complete" type="entry"><head>ADELARD</head><p>, or <hi rend="smallcaps">Athelard</hi>, was a learned monk
of Bath, in England, who flourished about the year
1130, as appears by some manuscripts of his in Corpus
Christi, and Trinity Colleges, Oxford. Vossius says he
was universally learned in all the sciences of his time;
and that, to acquire all sorts of knowledge, he travelled
into France, Germany, Italy, Spain, Egypt, and Arabia.
He wrote many books himself, and translated
others from different languages: thus, he translated,
from Arabic into Latin, Euclid's Elements, at a time
before any Greek copies had been discovered; also
Erichiafarim, upon the seven planets. He wrote a
book on the seven liberal arts, another on the astrolabe,
another on the causes of natural compositions, besides
several on physics and on medicine.</p><p>Although Vossius refers to Oxford for some of these
manuscripts, it would yet seem they were not to be
found there in Wallis's time; for the Doctor, speaking
of this author, and other English authors and travellers
about the fame age, says, &#x201C;A particular account of
these travels of Sholley and Morley was a while since
to be seen in two prefaces to two manuscript books
of theirs in the library of Corpus-Christi College in
Oxford, but hath lately (by some unknown hand) been
cut out, and carried away; which prefaces (one or both
of them) did also make mention of the travels of Athelardus
Bathoniensis, and are, to that purpose, cited by
Vossius out of that manuscript copy. Whoever hath
them, would do a kindness (by some way or other) to
restore them, or at leaft a copy of them.&#x201D; Wallis's
Algebra, pa. 6.</p></div1><div1 part="N" n="ADELM" org="uniform" sample="complete" type="entry"><head>ADELM</head><p>, <hi rend="smallcaps">Aldhelmus</hi>, or <hi rend="smallcaps">Althelmus</hi>, a learned
Englishman, who flourished about the year 680. He
was sirst abbot of Malmsbury, and afterward bishop of
Shirburn. He died in the year 709, in the monastery
of Malmsbury.</p><p>Adelm was the son of Kenred or Kenten, who was
the brother of Ina, king of the West Saxons in England.
Beside certain books in theology, he composed
several on the mathematical sciences &amp;c; as Arithmetic,
and Astrology, and librum de philosophorum disciplinis.
See Bede's History, lib. 5. cap. 19. He is also mentioned
by Bale and William of Malmsbury.</p></div1><div1 part="N" n="ADERAIMIN" org="uniform" sample="complete" type="entry"><head>ADERAIMIN</head><p>, or <hi rend="smallcaps">Alderaimin</hi>, the Arabic name
of a star of the third magnitude, in the left shoulder of
Cepheus, marked <foreign xml:lang="greek">a</foreign> by Bayer. Its longitude for 1761,
<foreign xml:lang="greek">g</foreign> 9&#xB0; 30&#x2032; 8&#x2033; north latitude 68&#xB0; 56&#x2032; 20&#x2033;.</p></div1><div1 part="N" n="ADFECTED" org="uniform" sample="complete" type="entry"><head>ADFECTED</head><p>, see <hi rend="smallcaps">Affected.</hi></p></div1><div1 part="N" n="ADHESION" org="uniform" sample="complete" type="entry"><head>ADHESION</head><p>, <hi rend="smallcaps">Adherence</hi>, in <hi rend="italics">Physics,</hi> is the state
of two bodies, joined or fastened together, whether by
mutual attraction, the interposition of their own parts,
or the impulse or pressure of external bodies. See
<hi rend="smallcaps">Cohesion.</hi></p><p>Thus two hollow hemispheres, exhausted of air, are
made to adhere firmly together by the pressure of the
atmosphere on their convex or external surfaces; for
if they are introduced into an exhausted receiver, they
presently fall asunder. Also two very well polished
<pb n="31"/><cb/>
planes adhere firmly together, partly by the external
pressure of the atmosphere, and partly by the attraction
of their parts.</p><p>In No. 389 of the Philos. Trans. Dr. Desaguliers
has given experiments of the adhesion of leaden bullets
to each other: the cause of which he resolves into the
principle of attraction.</p><p>M. Musschenbroeck, in his Essai de Physique, has
given a great many remarks on the adhesion of bodies,
and relates various experiments which he had made
upon this matter, but chiefly relative to the resistance
made by bodies to fracture, in virtue of the adhesion of
their parts; which adhesion he ascribes principally to
their mutual attraction. Common experiments prove
the mutual adhesion of the parts of water to each other,
as well as to the bodies they touch. The same may be
said of the particles of air, on which M. Petit has a
memoir among those of the Paris Academy of Sciences
for the year 1731.</p><p>Some authors however are not willing to admit that
the adhesion of the parts of water, or indeed of bodies
in general, is to be attributed to the attraction of their
parts, and they reason thus: suppose, say they, that
attraction acts at any small distance, as for example to
the distance of one-tenth of an inch, from a particle of
water: and about this particle describe a circle whose
radius is one-tenth of an inch: then the particle of
water will be attracted only by the particles included
within the circle; but as these particles act in contrary
directions, their mutual effects must destroy one another,
and there can be no attraction of the particle, since it
will have no more tendency one way than another.</p></div1><div1 part="N" n="ADHIL" org="uniform" sample="complete" type="entry"><head>ADHIL</head><p>, in <hi rend="italics">Astronomy,</hi> a star, of the sixth magnitude,
upon the garment of Andromeda, under the
last star in her foot.</p></div1><div1 part="N" n="ADJACENT" org="uniform" sample="complete" type="entry"><head>ADJACENT</head><p>, whatever lies immediately by the side
of another.</p><p><hi rend="smallcaps">Adjacent</hi> <hi rend="italics">Angle,</hi> in <hi rend="italics">Geometry,</hi> is said of an angle
when it is immediately contiguous to another, so that
they have both one common side. And the term is
more particularly used when the two angles have not
only one common side, but also when the other two
sides form one continued right line.</p><p><hi rend="smallcaps">Adjacent</hi> <hi rend="italics">bodies, in Physics,</hi> are understood of those
that are near, or next to, some other body.</p></div1><div1 part="N" n="ADJUTAGE" org="uniform" sample="complete" type="entry"><head>ADJUTAGE</head><p>, or rather AJUTAGE; which see.</p></div1><div1 part="N" n="ADSCRIPTS" org="uniform" sample="complete" type="entry"><head>ADSCRIPTS</head><p>, in <hi rend="italics">Trigonometry,</hi> is used by some
mathematicians, for the tangents of arcs. Vieta calls
them also prosines.</p><p>ADVANCE-<hi rend="smallcaps">Fosse</hi>, in Fortification, a ditch thrown
round the esplanade or glacis of a place, to prevent its
being surprised by the besiegers.</p><p>The ditch sometimes made in that part of the lines
or retrenchments nearest the enemy, to prevent him
from attacking them, is also called the advance-fosse.</p><p>The advance-fosse should always be full of water,
otherwise it will serve to cover the enemy from the fire
of the place, if he should become master of the fosse.
Beyond the advance-fosse it is usual to construct lunettes,
redouts, &amp;c.</p></div1><div1 part="N" n="ADVENT" org="uniform" sample="complete" type="entry"><head>ADVENT</head><p>, <hi rend="italics">Adventus,</hi> in the Calendar, the time immediately
preceding Christmas; and was anciently employed
in pious preparation for the <hi rend="italics">adventus,</hi> or coming
on, of the feast of the Nativity.
<cb/></p><p>Advent includes four Sundays, or weeks; commencing
either with the Sunday which falls on
St. Andrew's day, namely the 30th day of November,
or the nearest Sunday to that day, either before or
after.</p><p>&#xC6;OLIPILE, <hi rend="italics">&#xC6;olipile,</hi> in Hydraulics, a hollow ball
of metal, with a very small hole or opening; chiefly
used to shew the convertibility of water into elastic
steam. The best way of fitting up this instrument, is
with a very slender neck or pipe, to screw on and off,
for the convenience of introducing the water into the
inside; for by unscrewing the pipe, and immerging
the ball in water, it readily fills, the hole being pretty
large; and then the pipe is screwed on. But if the
pipe do not screw off, its orifice is too small to force its
way in against the included air; and therefore to expel
most of the air, the ball is heated red hot, and suddenly
plunged with its orifice into water, which will then
rush in till the ball is about two-thirds filled with the
water. The water having been introduced, the ball is
set upon the fire, which gradually heats the contained
water, and converts it into elastic steam, which rushes
out by the pipe with great violence and noise; and thus
continues till all the water is so discharged; though
not with a constant and uniform blast, but by sits:
and the stronger the fire is, the more elastic will the
steam be, and the force of the blast. Care should be
taken that the ball be not set upon a violent fire with
very little water in it, and that the small pipe be not
stopped with any thing; for in such case, the included
elastic steam will suddenly burst the ball with a very
dangerous explosion.</p><p>This instrument was known to the ancients, being
mentioned by Vitruvius, lib. 1. cap. 6. It is also
treated of, or mentioned, by several modern authors,
as Descartes, in his Meteor. cap. 4; and Father Mersennus,
in prop. 29 Ph&#xE6;dom. Pneumat. uses it to weigh
the air, by first weighing the instrument when red hot,
and having no water in it; and afterwards weighing
the same when it becomes cold. But the conclusion
gained by this means, cannot be quite accurate, as
there is supposed to be no air in the ball when it is
red hot; whereas it is shewn by Varenius, in his
Geography, cap. 19, sect. 6, prop. 10, that the air is
raresied but about 70 times; and consequently the
weight obtained by the above process, will be about
one-70th too small, or more or less according to the
intensity of the heat.</p><p>In Italy it is said that the &#xC6;olipile is often used to
cure smoaky chimneys: for being hung over the fire,
the blast arising from it carries up the loitering smoke
along with it.</p><p>And some have imagined that the &#xE6;olipile might
be employed as bellows to blow up a fire, having the
blast from the pipe directed into the fire: but experience
would soon convince them of their mistake; for
it would rather blow the sire <hi rend="italics">out</hi> than <hi rend="italics">up,</hi> as it is not
air, but rarefied water, that is thus violently blown
through the pipe.</p><p>&#xC6;OLUS, in Mechanics, a small portable machine,
not long since invented by Mr. Tidd, for refreshing
and changing the air in rooms which are made too
close.</p><p>The machine is adapted to supply the place of a
<pb n="32"/><cb/>
square of glass in a sash-window, where it works with
little or no noise, on the principle of the sails of a mill,
or a smoke-jack; and thus admitting an agreeable
quantity of air, at a convenient part of the room.</p><p>&#xC6;<hi rend="smallcaps">OLUS</hi>'<hi rend="italics">s Harp,</hi> or <hi rend="italics">&#xC6;olian Harp,</hi> an instrument so
named, from its producing an agreeable melody, merely
by the action of the wind.</p><p>Neither the age nor inventor of this instrument are
very well known. It is not mentioned by Mersennus
in his Harmonics, where he describes most sorts of
musical instruments: and yet the description and use of
it was given soon after, by Kircher, in his book, Magia
Phenotactica &amp; Phonurgia.</p><p>The construction of this instrument is thus; let a
box be made of as thin deal as possible, its length answering
exactly to the width of the window in which it
is to be placed; five or six inches deep, and seven or
eight inches wide. Across the top, and near each
end, glue on a bit of wainscot, about half an inch
high, and a quarter of an inch thick, to serve as two
bridges for the strings to be stretched over, by means
of pins inserted into holes a little behind the bridges,
nearer the ends, half the number being at one end, and
half at the other end: these pins are like those of a
harpsichord; and for their better support in the thin
deal, a piece of beech of about an inch square, and
length equal to the breadth of the box, is glewed on
the inside of the lid, immediately under the place of
the pins, the holes for receiving them being bored
through this piece. It is strung with small catgut,
or blue first fiddle strings, more or less at pleasure, on
the outside and lengthways of the lid, fixing one end
to one of the small pins, and twisting the other end
about the opposite or stretching pin. A couple of
sound-holes are cut in the lid; and the thinner this is,
the better will be the performance.</p><p>When the strings are tuned unison, and the instrument
placed, with the top or stringed side outwards, in
the window to which it is fitted, the air blowing upon
that window, the instrument will give a sound like a
distant choir, increasing or decreasing according to the
strength of the wind.</p><p>&#xC6;RA, in Chronology, is the same as epoch, or
epocha, and means a sixed point of time, from which
to begin a computation of the years ensuing.</p><p>The word is sometimes also written <hi rend="italics">cra</hi> in ancient
authors. Its origin is contested, though it is generally
supposed that it had its rise in Spain. Some imagine
that it is formed from <hi rend="italics">a. er. a.</hi> the abbreviations of the
words, <hi rend="italics">annus erat Augusti,</hi> or from <hi rend="italics">a. e. r. a.</hi> the initials
of the words <hi rend="italics">annus erat regni Augusti,</hi> because the
Spaniards began their computation from the time that
their country came under the dominion of Augustus.
Others derive it from <hi rend="italics">&#xE6;s, brass,</hi> the tribute money with
which Augustus taxed the world. It is also said that
<hi rend="italics">&#xE6;ra</hi> originally signified a number stamped on money to
determine its current value. And that the ancients
used <hi rend="italics">&#xE6;s</hi> or <hi rend="italics">&#xE6;ra</hi> as an article, as we do the word <hi rend="italics">item,</hi> to
each particular of an account; and hence it came to
stand for a sum or number itself.</p><p>&#xC6;<hi rend="smallcaps">RA</hi> also means the way or mode of accounting
time. Thus we say such a year of the Christian
&#xE6;ra, &amp;c.</p><p><hi rend="italics">Spanish</hi> &#xC6;<hi rend="smallcaps">RA</hi>, otherwise called the year of C&#xE6;sar,
<cb/>
was introduced after the second division of the Roman
provinces, between Augustus, Anthony, and Lepidus,
in the 714th year of Rome, the 4676th year of the
Julian period, and the 38th year before Christ. In the
447th year of this &#xE6;ra, the Alani, the Vandals, Suevi,
&amp;c, entered Spain. It is frequently mentioned in the
Spanish affairs; their councils, and other public acts,
being all dated according to it. Some say it was abolished
under Peter IV, king of Arragon, in the year of
Christ 1358, and the Christian &#xE6;ra introduced instead
of it. But Mariana observes that it ceased in the
year of Christ 1383, under John I, king of Castile.
The like was afterwards done in Portugal.</p><p><hi rend="italics">Christian</hi> &#xC6;<hi rend="smallcaps">RA.</hi> It is generally allowed by Chronologers,
that the computation of time from the birth
of Christ, was only introduced in the sixth century
in the reign of Justinian; and it is commonly ascribed
to Dionysius Exiguus. This &#xE6;ra came then
into use in deeds, and such like; before which time
either the olympiads, the year of Rome, or that of the
reign of the emperors, was used for such purposes.</p><p>See an account of the other principal &#xE6;ras under the
word Epoch.</p><p>AERIAL <hi rend="italics">Perspective,</hi> is that which represents bodies
diminished and weakened, in proportion to their
distance from the eye.</p><p>A&#xEB;rial Perspective chiefly respects the colours of
objects, whose force and lustre it diminishes more or
less, to make them appear as if more or less remote.</p><p>It is founded upon this, that the longer the column
of air an object is seen through, the more feebly do the
visual rays emitted from it affect the eye.</p></div1><div1 part="N" n="AEROGRAPHY" org="uniform" sample="complete" type="entry"><head>AEROGRAPHY</head><p>, a description of the air, or atmosphere,
its limits, dimensions, properties, &amp;c.</p></div1><div1 part="N" n="AEROLOGY" org="uniform" sample="complete" type="entry"><head>AEROLOGY</head><p>, the doctrine or science of the air,
and its ph&#xE6;nomena, its properties, good and bad qualities,
&amp;c. It is much the same with the soregoing
word, Aerography.</p></div1><div1 part="N" n="AEROMETRY" org="uniform" sample="complete" type="entry"><head>AEROMETRY</head><p>, <hi rend="italics">Aerometria,</hi> the science of measuring
the air, its powers and properties; comprehending
not only the quantity of the air itself, as a fluid
body, but also its pressure or weight, its elasticity, rarefaction,
condensation, &amp;c.</p><p>The term is not much used at present; this branch
of natural philosophy being usually called pneumatics,
which see. Wolfius, late professor of mathematics at
Hall, having reduced several properties of the air to
geometrical demonstrations, sirst published at Leipsic his
Elements of Aerometry, in the German language,
and afterwards more enlarged in Latin, which have
since been inserted in his <hi rend="italics">Cursus Mathematicus,</hi> in five
volumes in 4to.</p></div1><div1 part="N" n="AERONAUTICA" org="uniform" sample="complete" type="entry"><head>AERONAUTICA</head><p>, the pretended art of sailing
through the air, or atmosphere, in a vessel, sustained as
a ship in the sea.</p></div1><div1 part="N" n="AEROSTATICA" org="uniform" sample="complete" type="entry"><head>AEROSTATICA</head><p>, is properly the doctrine of the
weight, pressure, and balance of the air and atmosphere.</p></div1><div1 part="N" n="AEROSTATION" org="uniform" sample="complete" type="entry"><head>AEROSTATION</head><p>, in its proper and primary sense,
denotes the science of weights suspended in the air;
but in the modern application of the term, it siguifies
the art of navigating or floating in the air, both as to
the practice and principles of it. Hence also the machines
which are employed for this purpose, are called
<pb/><pb/><pb n="33"/><cb/>
<hi rend="italics">aerostats,</hi> or <hi rend="italics">aerostatic</hi> machines; and which, on account
of their round and bell-like shape, are otherwise called
<hi rend="italics">air ballcons.</hi> Also <hi rend="italics">aeronaut</hi> is the name given to the
person who navigates or sloats in the air by means of
such machines.</p><p><hi rend="italics">Principles of</hi> <hi rend="smallcaps">Aerostation.</hi> The fundamental principles
of this art bave been long and generally known,
as well as speculations on the theory of it; but the
successful application of them to practice seems to be
altogether a modern discovery. These principles chiefly
respect the weight or pressure, and elasticity of the air,
with its specific gravity, and that of the other bodies to
be raised or floated in it: the particular detail of which
principles may be seen under the respective words in
this dictionary. Suffice it therefore in this place to
observe, that any body which is specifically, or bulk for
bulk, lighter than the atmosphere, or air encompassing
the earth, will be buoyed up by it, and ascend, like as
wood, or a cork, or a blown bladder, ascends in water.
And thus the body would continue to ascend to the
top of the atmosphere, if the air were every where of
the same density as at the surface of the earth. But as
the air is compressible and elastic, its density decreases
continually in ascending, on account of the diminished
pressure of the superincumbent air, at the higher elevations
above the earth; and therefore the body will
ascend only to such height where the air is of the same
specific gravity with itself; where the body will float,
and move along with the wind or current of air, which
it may meet with at that height. This body then is
an aerostatic machine, of whatever form or nature it
may be. And an air-balloon is a body of this kind,
the whole mass of which, including its covering and
contents, and the weights annexed to it, is of less
weight than the same bulk of air in which it rises.</p><p>We know of no solid bodies however that are light
enough thus to ascend and float in the atmosphere; and
therefore recourse must be had to some fluid or aeriform
substance.</p><p>Among these, that which is called inslammable air
is the most proper of any that have hitherto been discovered.
It is very elastic, and from six to ten or
eleven times lighter than common atmospheric air at
the surface of the earth, according to the different
methods of preparing it. If therefore a sufficient quantity
of this kind of air be inclosed in any thin bag or
covering, the weight of the two together will be less
than the weight of the same bulk of common air;
and, consequently this compound mass will rise in the
atmosphere, and continue to ascend till it attain a height
at which the atmosphere is of the same specific gravity
as itself; where it will remain or float with the current
of air, as long as the inflammable air does not escape
through the pores of its covering. And this is an inflammable
air-balloon.</p><p>Another way is to make use of common air, rendered
lighter by warming it, instead of the inflammable
air. Heat, it is well known, rarefies and expands
common air, and consequently lessens its specific gravity;
and the diminution of its weight is proportional
to the heat applied. If therefore the air, inclosed in
any kind of a bag or covering, be heated, and consequently
dilated, to such a degree, that the excess of
the weight of an equal bulk of common air, above the
<cb/>
weight of the heated air, be greater than the weight
of the covering and its appendages, the whole compound
mass will ascend in the atmosphere, till, by the
diminished density of the surrounding air, the whole
become of the same specific gravity with the air in
which it sloats; where it will remain, till, by the cooling
and condensation of the included air, it shall gradually
contract and descend again, unless the heat is
renewed or kept up. And such is a heated air-balloon,
otherwise called a Montgolfier, from its inventor.</p><p>Now it has been discovered, by various experiments,
that one degree of heat, according to the seale of
Fahrenheit's thermometer, expands the air about one
five-hundredth part; and therefore that it will require
about 500 degrees, or nearer 484 degrees of hent, to expand
the air to just double its bulk. Which is a degree
of heat far above what it is practicable to give it
on such occasions. And therefore, in this respect, common
air heated, is much inferior to inflammable air,
in point of levity and usefulness for aerostatic machines.</p><p>Upon such principles then depends the construction
of the two sorts of air-balloons. But before treating
of this branch more particularly, it will be proper to
give a short historical account of this late-discovered
art.</p><p><hi rend="italics">History of</hi> <hi rend="smallcaps">Aerostation.</hi> Various schemes for
rising in the air, and passing through it, have been devised
and attempted, both by the ancients and moderns,
and that upon different principles, and with various
success. Of these, some attempts have been upon
mechanical principles, or by virtue of the powers of
mechanism: and such are conceived to be the instances
related of the flying pigeon made by Archytas, the
flying eagle and fly by Regiomontanus, and various
others. Again, other projects have been formed for
attaching wings to some part of the body, which were
to be moved either by the hands or feet, by the help of
mechanical powers; so that striking the air with them,
aster the manner of the wings of a bird, the person
might raise himself in the air, and transport himself
through it, in imitation of that animal. But of these
and various other devices of the like nature, a particular
account will be given under the article <hi rend="italics">artisicial flying,</hi>
as belonging rather to that species or principle of motion,
than to our present subject of aerostation, which
is properly the sailing or floating in the air by means
of a machine rendered specifically lighter than that
element, in imitation of aqueous navigation, or the
sailing upon the water in a ship, or vessel, which is
specisically lighter than the water.</p><p>The first rational account that we have upon record,
for this sort of sailing, is perhaps that of our countryman
Roger Bacon, who died in the year 1292. He
not only affirms that the art is feasible, but assures us
that he himself knew how to make an engine, in which
a man sitting might be able to carry himself through the
air like a bird; and he farther affirms that there was
another person who had tried it with success. And
the secret it seems consisted in a couple of large thin
shells, or hollow globes, of copper, exhausted of air;
so that the whole being thus rendered lighter than air,
they would support a chair, on which a person might
sit.</p><p>Bishop Wilkins too, who died in 1672, in several
<pb n="34"/><cb/>
of his works, makes mention of similar ideas being
entertained by divers persons. &#x201C;It is a pretty notion
to this purpose, says he (in his Discovery of a New
World, prop. 14), mentioned by Albertus de Saxonia,
and out of him by Francis Mendoza, that the air is
in some part of it navigable. And that upon this
statick principle, any brass or iron vessel (suppose a
kettle), whose substance is much heavier than that of
the water; yet being filled with the lighter air, it will
swim upon it, and not sink.&#x201D; And again, in his Dedalus,
chap. 6, &#x201C;Scaliger conceives the framing of
such volant automata to be very easy. <hi rend="italics">Volantis columb&#xE6;
machinulam, cujus autorem Archytam tradunt,</hi> vel
facillime <hi rend="italics">profiteri audeo.</hi> Those ancient motions were
thought to be contrived by the force of some included
air: So <hi rend="italics">Gellius, Ita erat scilicet libramentis suspensum, &amp;
aura spiritus inclusa, atque occulta consitum, &amp;c.</hi> As if
there had been some lamp, or other fire within it,
which might produce such a forcible rarefaction, as
should give a motion to the whole frame.&#x201D; From which
it would seem that Bishop Wilkins had some confused
notion of such a thing as a heated air-balloon.</p><p>Again F. Francisco Lana, in his Prodroma, printed
in 1670, proposes the same method with that of Roger
Bacon, as his own thought.</p><p>He considered that a hollow vessel, exhausted of
air, would weigh less than when filled with that fluid;
he also reasoned that, as the capacity of spherical vessels
increases much faster than their surface, if there were
two spherical vessels, of which the diameter of one is
double the diameter of the other; then the capacity
of the former will be equal to 8 times the capacity of
the latter, but the surface of that only equal to 4 times
the surface of this: and the one sphere have its diameter
equal to triple the diameter of the other; then the
capacity of the greater will be equal to 27 times the
capacity of the less, while its surface is only 9 times
greater: and so on, the capacities increasing as the
cubes of the diameters, while the surfaces increase only
as the squares of the same diameters. And from this
mathematical principle, father Lana deduces, that it is
possible to make a spherical vessel of any given matter,
and thickness, and of such a size as, when emptied of
air, it will be lighter than an equal bulk of that air,
and consequently that it will ascend in that element,
together with some additional weight attached to it.
After stating these principles, father Lana computes
that a round vessel of plate brass, 14 feet in diameter,
weighing 3 ounces the square foot, will only weight 1848
ounces; whereas a quantity of air of the same bulk will
weigh 2155 2/3 ounces, allowing only one ounce to the
cubic foot; so that the globe will not only be sustained
in the air, but will also carry up a weight of 307 2/3
ounces: and by increasing the bulk of the globe,
without increasing the thickness of the metal, he adds,
a vessel might be made to carry a much greater weight.</p><p>Such then were the ingenious speculations of learned
men, and the gradual approaches towards this art. But
one thing more was yet wanting: although acquainted
in some degree with the weight of any quantity of
air, considered as a detached substance, it seems they
were not aware of its great elasticity, and the universal
pressure of the atmosphere; by which pressure, a globe
of the dimensions above-described, and exhausted of its
<cb/>
air, would immediately be crushed inwards, for want
of the equivalent internal counter pressure, to be sought
for in some element, much lighter than common air,
and yet nearly of equal pressure or elasticity with it; a
property or circumstance attending common air when
considerably heated. It is evident then that the
schemes of ingenious men hitherto must have terminated
in mere speculation; otherwise they could never
have recorded schemes, which, on the first attempt to
put in practice, must have manifested their own insufficiency,
by an immediate failure of success: For instead
of exhausting the vessel of air, it must either be filled
with common air heated, or with some other equally
elastic and lighter air. So that upon the whole it appears,
that the art of traversing the air, is an invention
of our own time; and the whole history of it is comprehended
within a very short period.</p><p>The rarefaction and expansion of air by heat, is a
property of it that has long been known, not only to
philosophers, but even to the vulgar: by this means it
is, that the smoke is continually carried up our chimneys;
and the effect of heat upon air, is made very
sensible by bringing a bladder, only partly full of air,
near a fire; when the air presently expands with the
heat, and distends the bladder so as almost to burst it:
and so well are the common people acquainted with
this effect, that it is the common practice of those who
kick blown bladders about for foot-balls, to bring them
from time to time to the fire, to restore the spring of
the air, and distension of the ball, lost by the continual
waste of that fluid through the sides of it.</p><p>But the great levity of inflammable air, is a very
modern discovery. As to the inflammable property of
this air itself, it had been long known to miners, and
especially in coal mines, by the dreadful effects it sometimes
produces by its explosions. Among them it is
sometimes vulgarly called sulphur, but more properly
the fire damp, or inflammable damp, to distinguish it
from the choak damp, and other damps, a species of
air sometimes found in deep wells and mines, and
which does not explode nor take fire, but presently
extinguishes candles, and suffocates the persons who
may happen to go into it. But it seems that it was Mr.
Cavendish who first discovered with exactness the specific
gravity of inflammable air; and his experiments
and observations upon it, are published in the 56th
volume of the Philosophical Transactions for the year
1766. Soon after this discovery of Mr. Cavendish, it
occurred to the ingenious Dr. Black of Edinburgh,
that if a bladder, or other vessel, sufficiently light and
thin, were filled with this air, it would form altogether
a mass lighter than the same bulk of atmospheric air,
and consequently that it would ascend in it.</p><p>This idea he mentioned in his chemical lectures in
the year 1767 or 1768; and he farther proposed to
exhibit the experiment, by filling the allantois of a calf
with such air. The allantois however was not prepared
just at the time when he was at that part of his lectures,
and other avocations afterwards prevented his design:
so that, considering it only as an amusing experiment,
and being fully satisfied of the truth of so evident an
effect, he contented himself with barely mentioning
the experiment from time to time in his lectures. About
the year 1777 or 1778 too it occurred to Mr. Cavallo,
<pb n="35"/><cb/>
that it might be possible to construct a vessel, which,
when filled with inflammable air, would ascend in the
atmosphere: and there is no doubt but that similar ideas
would occur to many other persons, of so evident a
consequence of Mr. Cavendish's discovery.</p><p>But it seems to have been Mr. Cavallo who first
actually attempted the experiment, in which however
he succeeded no farther than in being able to raise
soap bubbles of two or three inches diameter: a thing
which had been done by children for their amusement
time immemorial. These experiments Mr. Cavallo made
in the beginning of the year 1782, and an account of
them was read at a public meeting of the Royal Society
on the 20th day of June of that year. From which it
appears that he tried bladders and paper of various sorts.
But the bladders, however thin they were made by
scraping, &amp;c, were still found too heavy to ascend in
the atmosphere, when fully inflated with the inflammable
air: and in using China paper, he found that this
air passed through its pores, like water through a sieve.
And having failed of success by blowing the same air
into a thick solution of gum, thick varnishes, and oil
paint, he was obliged to rest satisfied with soap-balls or
bubbles, which, being filled with inflammable air, by
dipping the end of a small glass tube, connected with a
bladder containing the air, into a thick solution of
soap, and gently compressing the bladder, ascended rapidly
in the atmosphere, and broke against the ceiling
of the room.</p><p>Here however it seems the matter might have rested,
had it not been for experiments made in France soon
after, by the two brothers Stephen and Joseph Montgolfier,
upon principles suggested, not by the levity of
inflammable air, which probably they had never heard
of, but by that of smoke and clouds ascending in the
atmosphere. These two brothers it seems were natives
of Annonay, a town in the Vivarais, about 36 miles
distant from Lyons; and that in their youth, Stephen,
the elder, had assiduously studied the mathematics, but
the other had applied himself more particularly to natural
philosophy and chemistry. They were not intended
for any particular way of business, but the death
of a brother obliged them to put themselves at the head
of a considerable paper manufactory at Annonay. In
the intervals of time allowed by their business they
amused themselves in several philosophical pursuits, and
particularly with the experiments in aerostation, of
which we are now to give some account. It would be
perhaps impossible to know all the particular steps and
ideas which finally produced this discovery: but it
has been said that the real principle, upon which the
effect of the aerostatic machine depends, was unknown
even for a considerable time after its discovery: that
M. Montgolfier attributed the effect of the machine,
not to the rarefaction of the air, which is the true
cause, but to a certain gas, specifically lighter than
common air, which was supposed to be developed srom
burning substances, and which was commonly called
Montgolfier's gas. Be this however as it may, it is well
known that the two brothers began to think of the
experiment of the aerostatic machine about the middle
or the larter part of the year 1782. The natural
ascension of the smoke and the clouds in the atmosphere,
suggested the first idea; and to imitate those bodies,
<cb/>
or to inclose a cloud in a bag, and let the latter be
lifted up by the buoyancy of the former, was the first
project of those celebrated gentlemen.</p><p>Accordingly the first experiment was made at Avignon
by Stephen, the elder brother, about the middle of
November 1782. Having prepared a bag of sine silk,
in the shape of a parallelopipedon, and of about 40
cubic feet in capacity, he applied burning paper to an
aperture in the bottom, which rarefied the air, and
thus formed a kind of cloud in the bag; and when it
became sufficiently expanded, it ascended rapidly to the
ceiling.</p><p>Soon afterwards the experiment was repeated with
the same machine at Annonay, by the two brothers, in
the open air; when the bag ascended to the height of
about 70 feet. Encouraged by this success, they constructed
another machine, of about 650 cubic feet capacity;
which, when inflated as before, broke the cords
which confined it, and after ascending rapidly to the
height of about 600 feet, descended and fell on the adjoining
ground. With another larger machine, of 37
feet diameter, they repeated the experiment on the
25th day of April, which answered exceedingly well: the
machine had such force of ascension, that, breaking
abruptly from its confinement of ropes, it rose to
the height of more than 1000 feet, and then, being
carried by the wind, descended and fell at a place about
three quarters of a mile from the place of its ascension.
The capacity of this machine was equal to above 23
thousand cubic feet, and, being nearly globular, when inflated,
it measured 117 English feet in circumference.
The covering was formed of linen, lined with paper;
and its aperture at the bottom was fixed to a wooden
frame, of about 4 feet square, or 16 feet in surface.
When filled with vapour, which it was conjectured
might be about half as heavy as common air, it was
capable of lifting up about 490 pounds, besides its own
weight, which, together with that of the wooden frame,
was equal to about 500 pounds. With this same machine
the next experiment was publicly performed at
Annonay, on the 5th of June 1783, before a great
multitude of spectators. The flaccid bag was suspended
on a pole 35 feet high; straw and chopped wool
were burned under the opening at the bottom; the vapour,
or rather smoke, soon inflated the bag, so as to
distend it in all its parts; and this enormous mas&lt;*&gt;
ascended in the air with such velocity, that in less than
ten minutes it reached the height of above 6 thousand
feet; when a breeze carried it in an horizontal direction
to the distance of 7668 feet, or near a mile and a
half, where it descended gently to the ground.</p><p>As soon as the news of this experiment reached
Paris, the philosophers of that city, conceiving that a
new species of gas, of about half the weight of common
air, had been discovered by Messrs. Montgolfier;
and knowing that the weight of inflammable air was
but about the eighth or tenth part of the weight of
common air, they justly concluded that inflammable
air would answer the purpose of this experiment better
than the gas of Montgolfier, and accordingly they
resolved to make trial of it.</p><p>A subscription was opened by M. Faujas de St. Fond,
towards defraying the expence of the experiment. A
sufficient sum of money having soon been raised,
<pb n="36"/><cb/>
Messrs. Roberts were appointed to construct the machine,
and M. Charles, professor of experimental philosophy,
to superintend the work. After a considerable
time spent, and surmounting many difficulties in obtaining
a sufficient quantity of inflammable air, and
searching out a substance light enough for the covering,
they at length constructed a globe of the silk called
lutestring, which was rendered impervious to the inclosed
air by a varnish of elastic gum or caeutchouc,
dissolved in some kind of spirit or essential oil. The diameter
of this globe was about 13 feet; and it had only
one aperture, like a bladder, to which a stop-cock was
adapted: and the weight of this covering, when empty,
together with that of the stop-cock, was 25 pounds.</p><p>On the 23d of August 1783, they began to fill the
globe with inflammable air; but this, being their first
attempt, was attended with many obstructions and disappointments,
which took up two or three days to overcome.</p><p>At length however it was prepared for exhibition,
and on the 27th it was carried from the Place des Victoires,
where it had been prepared, to the Champs de
Mars, a spacious open ground in the front of the Military
School, where, after introducing some more inflammable
air, and disengaging it from the cords by
which it was held down, it rose, in less than two minutes,
to the height of 3123 feet: the specific gravity
of the balloon, when it went up, being 35 pounds less
than that of common air. At that height the balloon
entered a cloud, but soon appeared again; and at last
it was lost among other clouds. After floating about
in the air for about three quarters of an hour, it fell in
a field about 15 miles from the place of its ascent;
where, as we may easily imagine, it occasioned great
amazement to the peasants who found it. Its fall was
owing to a rent in the covering, probably occasioned
by the superior elasticity of the inflammable air, over
that of the rare part of the atmosphere to which it
had ascended.</p><p>In consequence of this brilliant experiment, numberless
small balloons were made, mostly of goldbeater's
skin, from 6 and 9 to 18 or 20 inches diameter;
their cheapness putting it in the power of
almost every family to satisfy its curiosity relative to
the new experiment; and in a few days time balloons
were seen flying all about Paris, from whence they
were soon after sent abroad.</p><p>Mr. Joseph Montgolfier repeted an experiment with
a machine of his construction before the commissaries
of the Academy of Sciences, on the 11th and 12th
of September. The machine was about 74 feet high,
and 43 feet in diameter; it was made of canvass,
covered with paper both within and without, and
weighed 1000 pounds. It was filled with rarefied air
in 9 minutes, and in one trial the weight of eight
men was not sufficient to keep it down. It was not
suffered to go up, as it had been intended for exhibition
before the Royal Family, a few days after.
By the violence of the rain, however, which fell about
this time, it was so much spoiled, that he thought
proper to construct another for that purpose, in which
he used so great dispatch, that it was completed in
the short space of four days time. This machine
was constructed of cloth made of linnen and cotton
<cb/>
thread, and painted with water colours both within
and without. Its height was 60 feet, and diameter
43 feet. Having made the necessary preparation for
inflating it, the operation was b&#xE8;gun about one o'elock
on the 19th of the same month; before the king
and queen, the court, and the inhabitants of the place,
as well as all the Parisians who could procure a conveyance
to Versailles. The balloon was soon filled,
and in eleven minutes after the commencement of the
operation, the ropes being cut, it ascended, bearing
up with it a wicker cage, containing a cock, a duck, and
a sheep, the first animals that ever ascended into the
atmosphere with an aerostatic machine. Its power of
ascension, or the weight by which it was lighter than
an equal bulk of common air, allowing for the animals
and their cage, was 696 pounds. The balloon rose
to the height of 1440 feet; and being driven by the
wind for the space of eight minutes, it gradually
descended in consequence of two large rents made in
the covering by the wind, and fell in a wood at the
distance of 10,200 feet, or about two miles from
Versailles. The animals landed again as safe as when
they went up, and the sheep was found feeding.</p><p>The success of this experiment induced M. Pilatre
de Rozier, with a philosophical intrepidity which will
be recorded with applause in the history of aerostation,
to offer himself as the first adventurer in this aerial navigation.
For this purpose M. Montgolfier constructed
a new machine, of an oval shape, in a garden of the
fauxbourg St. Antoine; its diameter being about 48
feet, and height 74 feet. To the aperture in the
lower part was annexed a wicker gallery about three
feet broad, with a ballustrade of three feet high.
From the middle of the aperture an iron grate, or
brazier, was suspended by chains, descending from
the sides of the machine, in which a fire was lighted
for inflating the machine; and towards the aperture
port-holes were opened in the gallery, through which
any person, who might venture to ascend, might feed
the fire on the grate with fuel, and regulate at pleasure
the dilatation of the air inclosed in the machine: the
weight of the whole being upwards of 1600 pounds.
On the 15th of October 1783. the fire being lighted,
and the balloon inflated, M. P. de Rozier placed himself
in the gallery, and, to the astonishment of a multitude
of spectators, ascended as high as the length of
the restraining cords would permit, which was about
84 feet from the ground, and there kept the machine
afloat about four minutes and a half, by repeatedly
throwing straw and wool upon the fire: the machine
then descended gradually and gently, through a medium
of increasing density, to the ground; and the
intrepid adventurer assured the admiring spectators that
he had not experienced the least inconvensence in
this aerial excursion. This experiment was repeted
on the 17th with nearly the same success; and again
several times on the 19th, when M. P. de Rozier, by
a partial ascent and descent, several times repeted,
evinced to the multitude of observers that the machine
may be made to ascend and descend at the pleasure
of the aeronaut, by merely increasing or diminishing
the fire in the grate. The balloon having been
hauled down, by the ropes which always confined it,
M. Gironde de Villette placed himself in the gallery
<pb n="37"/><cb/>
opposite to M. de Rozier, and the machine being
suffered to ascend, it hovered for about 9 minutes
over Paris, in the sight of all its inhabitants, at the
height of 330 feet. And on their descending, the
marquis of Arlandes ascended with M. de Rozier
much in the same manner.</p><p>In consequence of the report of these experiments,
signed by the commissaries of the Academy of Sciences,
it was ordered that the annual prize of 600
livres should be given to Messrs. Montgolfier for the
year 1783.</p><p>In the experiments above-recited, the machine was
always secured by long ropes, to prevent its entire
escape: but they were soon succeeded by unconfined
aerial navigation. For this purpose the same balloon of
74 feet in height was conveyed to La Muette, a royal
palace in the Bois de Boulogne: and all things being
got ready, on the 21st of November 1783, M. P. de
Rozier and the marquis d'Arlandes took their post in
opposite sides of the gallery, and at 54 minutes after
one the machine was absolutely abandoned to the element,
and it ascended calmly and majestically in the
atmosphere. On reaching the height of about 280
feet the intrepid aeronauts waved their hats to the
astonished multitude: but they soon after rose too
high to be distinguished, and it is supposed they rose
to more than 3000 feet in height. At first they were
driven, by a north-west wind, horizontally over the
river Seine and part of Paris, taking care to clear the
steeples and high buildings by increasing the fire;
and in rising they met with a current of air which
carried them southward. Having thus passed the
Boulevard, and finally desisting from supplying the
fire with fuel, they descended very gently in a field
beyond the new Boulevard, about 9000 yards, or a
little more than 5 miles distant from the palace de La
Muette, having been between 20 and 25 minutes in
the air. The weight of the whole apparatus including
that of the two travellers, was between 1600 and 1700
pounds.</p><p>Notwithstanding the rapid progress of aerostation
in France, it is remarkable that we have no authentic
account of any experiments of this kind being attempted
in other countries. Even in our own island,
where all arts and sciences sind an indulgent nursery,
and many their birth, no aerostatic machine was seen
before the month of November 1783. Various speculations
have been made on the reasons of this strange
neglect of so novel and brilliant an experiment. But
none seemed to carry any shew of probability except
that it was said to be discouraged by the leader of a
philosophical society, expressly instituted for the improvement
of natural knowledge, for the reason, as it
was said, that it was the discovery of a neighbouring
nation. Be this however as it may, it is a fact that
the first aerostatic experiment was exhibited in England
by a foreigner unconnected and unsupported. This
was a count Zambeccari, an ingenious Italian, who
happened to be in London about that time. He made
a balloon of oiled-silk, 10 feet in diameter, weighing
only 11 pounds: it was gilt, both for ornament, and
to render it more impermeable to the inflammable air
with which it was to be filled. The balloon, after
being publicly shewn for several days in London, was
<cb/>
carried to the Artillery Ground, and there being filled
about three-quarters with inflammable air, and having
a direction inclosed in a tin box for any person by whom
it should afterwards be found, it was launched about
one o'clock on the 25th of November 1783. At
half past three it was taken up near Petworth in Sussex,
48 miles distant from London; so that it travelled at
the rate of near 20 miles an hour. Its descent was
occasioned by a rent in the silk, which must have been
the effect of the rarefaction of the inflammable air when
the balloon ascended to a rarer part of the atmosphere.</p><p>The French philosophers having executed the first
aerial voyage with a balloon inflated by heated air, resolved
to attempt a similar voyage with a balloon filled
with inflammable air, which seemed to be preserable
to dilated air in every respect, the expence of preparing
it only excepted. A subscription was opened however
to defray that expence, which was estimated at
about ten thousand livres; and the balloon was constructed
by Messrs. Roberts, of gores of silk, varnished
with a solution of elastic gum. Its form was spherical,
and it measured 27 1/2 feet in diameter. The upper
hemisphere was covered by a net, which was fastened
to a hoop encircling its middle, and called its equator.
To this equator was suspended by ropes a car or boat,
covered with painted linen, and beautifully ornamented,
which swung a few feet below the balloon. To prevent
the bursting of the machine by the expansion of
the inflammable air in a rarer medium, or to cause the
balloon to descend, it was furnished with a valve, which
might be opened by means of a string descending from
it, for discharging a part of the internal air, without
admitting the external to enter: And the car was
ballasted with bags of sand, for the purpose of lightening
it occasionally, and causing it to ascend: so
that by letting some of the air escape through the valve,
they might descend; and by discharging some of their
sand ballast, ascend. To this balloon was likewise
annexed a long pipe by which it was filled. The apparatus
for filling it consisted of several casks placed
round a large tub of water, each having a long tin tube,
that terminated under a vessel or funnel which was inverted
into the water of the tub, and communicated
with the long pipe annexed to the lower part of the
balloon. Iron filings and diluted vitriolic acid being
put into the casks, the inflammable air which was produced
from these materials, passed through the tin
tubes, thence through the water of the tubs to the inverted
funnel, and so through the pipe into the balloon.
When inflated, the weight of the common air which
was equal in bulk to the balloon, was 771 1/2 pounds;
also the power of ascension, or weight just necessary
to keep it from ascending, was 20 pounds, and the
weight of the balloon, with its car, passengers, and all
its appendages, was 604 1/2 pounds, which two together
make 624 1/2 pounds: and this taken from 771 1/2 pounds,
the weight of common air displaced, leaves 147 pounds
for the weight of the inflammable air contained in
the balloon, and which is to 771 1/2 pounds, the weight
of the same bulk of common air, nearly as 1 to 5 1/4;
that is, the inflammable air used in this experiment
was 5 1/4 times lighter than common air.</p><p>The first of December was fixed on for the display
of this grand experiment; and every preparation was
<pb n="38"/><cb/>
made for conducting it with advantage. The garden
of the Thuilleries at Paris was the scene of operation;
which was soon crowded and encompassed with a prodigious
multitude of observers. Signals were given,
from time to time, by the siring of cannon, waving
of flags, &amp;c: and a small montgolsier was launched,
for shewing the direction of the wind, and for the
amusement of the people previous to the general display.
At three quarters after one o'clock, M. Charles
and one of the Roberts, having seated themselves in the
boat attached to the balloon, and being furnished with
proper instruments, cloathing, and provisions, left the
ground, and ascended with a moderately accelerated
velocity to the height of about 600 yards; the surrounding
multitude standing silent with fear and
amazement; while the aerial navigators at this height
made signals of their safety. When they left the
ground, the thermometer, according to Fahrenheit's
scale, stood at 59 degrees; and the barometer, at
30.18 inches: and at the utmost height to which
they ascended, the barometer fell to 27 inches; from
which they deduced their height as above to be 600
yards, or one third part of a mile. During the rest
of the vovage the quicksilver in the barometer was
generally between 27 and 27.65 inches, rising and
falling, as part of the ballast was thrown out, or some
of the inflammable air escaped from the balloon. The
thermometer generally stood between 53 and 57 degrees.
Soon after their ascent, they remained stationary
for some time: they then moved horizontally
in the direction north-north-west: and having crossed
the Seine, and passed over several towns and villages,
to the great amazement of the inhabitants, they descended
in a field, about 27 miles distant from Paris,
at three-quarters past 3 o'clock; so that they had
travelled at the rate of near 15 miles an hour, without
feeling the least inconvenience.</p><p>The balloon still containing a considerable quantity
of inflammable air, M. Charles re-ascended alone, and
it was computed he went to the height of 3100
yards, or almost 2 miles, the barometer being then
at 20 English inches: having amused himself in the
air about 33 minutes, he pulled the string of the
valve, and descended at 3 miles distance from the
place of his ascent. All the inconvenience he experienced
in his great elevation, was a dry sharp cold,
with a pain in one of his ears and a part of his
face, which he ascribed to the dilatation of the internal
air: a circumstance that usually happens to persons who
suddenly change the density of their atmosphere, either
by ascending into a rarer, or descending into a denser
one. The small balloon, launched at the beginning by
M. Montgolfier, was found to have moved in a direction
opposite to that of the aeronauts; from which it is inferred
that there were two currents of air at different
heights above the earth.</p><p>In the month of December this year, several experiments
were made at Philadelphia in America with
air balloons, by Messrs. Rittenhouse and Hopkins.
They constructed and filled a great many small balloons,
and connected them together; in which a man went
up several times, and was drawn down again; and
finally, the ropes being cut, he ascended to the height
of 100 feet, and floated to a considerable distance; but,
<cb/>
being afraid, he cut open the balloons with a knife,
and so descended.</p><p>About the close of this year small balloons were sent
up in many places, and were become very common in
some parts of France and England. And in the beginning
of the year following, their number and magnitude
increased considerably; and some of the more
remarkable ones were as follow:&#x2014;On the 19th of
January M. Joseph Montgolfier, accompanied by six
other persons, ascended from Lyons with a rarefied air
balloon, to the height of 1000 yards. This was the
largest machine that had been hitherto made, being 131
feet high, and 104 feet in diameter: it was formed of
a double covering of linen, with three layers of paper
between them; and it weighed, when it went up, 1600
pounds, including the gallery, passengers, &amp;c. It was
at first intended for six passengers; but before it went
up, it was not judged safe to freight it with more than
three: however no authority nor solicitations could
prevail upon any of the six to quit their place, nor even
to cast lots which three should resign their pretensions:
so that the spectators saw them all ascend with terror
and anxiety; and to add to their distress, when the
ropes were cut, and the machine had ascended a foot or
two from the ground, a seventh person suddenly leaped
into the gallery, and the fire being increased, the whole
ascended together. To add to the terror of the scene,
after being in the air about 15 minutes, a large rent
of about 50 feet in length was made by the balloon
taking fire, in consequence of which it descended very
rapidly to the ground, though fortunately without injury
to any of the aeronauts.</p><p>On the 22d of February an inflammable air balloon
was launched from Sandwich in Kent. It was but
a small one, being only 5 feet in diameter; but it was
rendered remarkable by being the first machine that
crossed the sea from England to France. It was found
in a field at Warneton, about 9 miles from Lisle in
French Flanders, two hours and a half after it left
Sandwich, the distance being about 74 miles; so that
it floated at the rate of about 30 miles an hour.</p><p>The chevalier Paul Andreani, of Milan, was the
first aerial traveller in Italy. The chevalier was at the
sole expence of this machine, but was assisted in the
construction by two brothers of the name of Gerli.
They all three ascended together near Milan on the
25th of February, and remained in the atmosphere
about 20 minutes, when they descended, all their fuel
being exhausted. This machine was a montgolfier, of
a spherical shape, and about 68 feet in diameter. From
calculations made on the power of this, and other
machines of the same sort, it appears that the included
air is raresied commonly but about one-third, or that
the included warm air weighs about two-thirds of the
same bulk of the external or common air.</p><p>The next aerial voyage was performed on the 2d of
March 1784, by M. Jean Pierre Blanchard, a man
who has since that time made more voyages than any
other person, and who has rendered himself famous
by being the first who has floated in the air over
the channel from England to France. M. Blanchard
it scems had sor many years been in pursuit of
mechanical means for flying through the air; but on
hearing of the late invented air balloons, he dropped
<pb n="39"/><cb/>
his former pursuits, and turned his attention to them.
He accordingly constructed one of 27 feet diameter,
to which a boat was suspended with two wings, and a
rudder to steer it by, as also a large parachute spread
horizontally between the boat and the balloon, designed
to check the fall in case the balloon should burst. The
machine being filled with inflammable air, he ascended,
from the Champs de Mars at Paris, to the height of
near ten thousand feet, or almost 2 miles; and after
floating in the air for an hour and a quarter, he descended
at Billancourt near Seve, having experienced
by turns heat, cold, hunger, and an excessive drowsiness.
It appears from his own account, and as might have
been expected, that the wings and rudder of his boat
had little or no power in turning the balloon from the
direction of the wind.</p><p>In the course of this year, 1784, aerostatic experiments
and aerial voyages became so frequent, that the
limits of this article will not allow of any thing farther
than mentioning those which were attended with any
remarkable circumstances. On the 25th of April,
Messrs. de Morveau and Bertrand ascended from Dijon,
with an inflammable-air balloon, to the height of thirteen
thousand feet, or near 2 miles and a half, where
the thermometer marked 25 degrees. They were in
the air 1 hour and 25 minutes, in which time they
floated 18 miles.</p><p>On the 20th of May, four ladies and a gentleman
ascended from Paris, in a large montgolfier, above the
highest buildings, and remained suspended there a considerable
time, the balloon being confined by ropes
from flying away.</p><p>On the 23d of May, M. Blanchard, with the same balloon
as before, ascended from Rouen, to such height that
the mercury in the barometer stood at 20.57 inches,
which on the earth had been at 30.16. It was observed
that in this voyage M. Blanchard's wings or oars
could not turn him aside from the direction of the wind.</p><p>On the 4th of June M. Fleurant and Madame
Thible, the first lady who made an aerial voyage, ascended
at Lyons in a machine of 70 feet diameter.
They went to the height of 8500 feet, and floated
about 2 miles in 45 minutes.</p><p>On the 14th of June, M. Coustard de Massi and
M. Mouchet ascended at Nantes to a great height,
with a balloon of 32 1/2 feet diameter, filled with inflammable
air extracted from zink; and they floated
to the distance of 27 miles in 58 minutes.</p><p>On the 23d of June, the first aerial traveller M.
Pilatre de Rozier, accompanied with M. Prouts, ascended
at Versailles, in the presence of the royal family
and the king of Sweden, with a large montgolfier,
whose diameter was 79 feet, and its height 91 seet
and a half. They floated to the distance of 36 miles
in three-quarters of an hour, when they descended,
which is at the rate of 48 miles an hour. In consequence
of this experiment the king granted to M. de
Rozier a pension of 2000 livres.</p><p>On the 15th of July the duke of Chartres, the two
brothers Roberts, and a fourth person ascended from
the park of St. Cloud, with an inflammable-air machine,
of an oblong form, its diameter being 34 feet, and its
length, which went in a direction parallel to the horizon,
was 55 1/2 feet; and they remained in the atmosphere
<cb/>
for 45 minutes in the greatest fear and danger. The
machine contained an interior small balloon, filled with
common air, by means of which it was proposed to
cause the machine to ascend or descend without the loss
of any inflammable air or ballast: and the boat was
furnished with a helm and oars, which were intended
to guide it. Three minutes after ascending, the machine
was lost in the clouds, and involved in a dense
vapour. A violent agitation of the air, resembling a
whirlwind, greatly alarmed the aeronauts, turned the
machine three times round in a moment, and gave it
such shocks as prevented them from using any of their
instruments for managing the machine. After many
struggles, with great difficulty they tore about 7 or 8
feet of the lower part of the covering, by which the inflamable
air escaped, and they descended to the ground
with great rapidity, though without any hurt. At
the place of departure the barometer slood at 30.12
inches, and at their greatest elevation it stood at 24.36
inches; so that their ascent was about 5100 feet, or
near one mile.</p><p>On the 18th of July, M. Blanchard, with a Mr.
Boley, made his third voyage with the same balloon as
he had before, and rose so high as to sink the barometer
from 30.1 to 25.34 inches, answering to a height of
about 4600 feet. In 2 hours and a quarter they floated
45 miles, which is at the rate of 20 miles an hour. In
this voyage M. Blanchard pretended that he was able
to turn the machine with his wings, and to make it ascend
and descend at pleasure. After descending, it
is said the balloon remained all the night at anchor full of
air; and that the next day several ladies amused themselves
by ascending successively to the height of 80
feet, the length of the ropes by which it was anchored.</p><p>In the course of this summer two persons had nearly
lost their lives by ascending with machines of warmed
air. The one in Spain, on the 5th of June, by the
machine taking fire, was much burnt, and so hurt by
the fall that his life was long despaired of. The latter
having ascended a few feet, the machinery entangled
under the eves of a house, which broke the ropes, and
the man fell about twenty feet: the machine presently
took fire, and was consumed. Other montgolfiers were
also burned about London, by taking fire, through the
defects of their construction.</p><p>The first aerial voyage performed in England was
by one Vincent Lunardi, a native of Italy, who ascended
from the Artillery Ground, London, with an
inflammable-air balloon on the 15th of September.
His machine was made of oiled silk, painted in alternate
stripes of blue and red; and its diameter was 33
feet. From a net, which covered about two-thirds of
the balloon, 45 cords descended to a hoop hanging
below the balloon, to which the gallery was attached.
The machine had no valve; and its neck, which terminated
in the form of a pear, was the aperture
through which the inflammable air was introduced, and
through which it might be let out. The balloon was
filled with air produced from zink by means of diluted
vitriolic acid. And when the aeronaut departed, at
2 o'clock, he took up with him a dog, a cat, and a
pigeon. After throwing out some sand to clear the
houses, he ascended to a considerable height; and the
direction of his motion at first was north-west by west;
<pb n="40"/><cb/>
but as the balloon rose higher, it came into another
current of air, which carried it nearly north. In the
course of his voyage the thermometer was as low as
29 degrees, and the drops of water which had collected
round the balloon were frozen. About half after
three he descended very near the ground, and landed
the cat, which was almost dead with cold: then rising,
he prosecuted his voyage, till at 10 minutes past
4 o'clock he landed near Ware in Hertfordshire. He
pretends that he descended by means of his oars or
wings; but other circumstances related by him, strongly
contradict the fact.</p><p>The longest and most interesting voyage performed
about this time, was that of Messrs. Roberts and M.
Colin Hullin, who ascended at Paris, at noon on the 19th
of September, with an aerostat, filled with inflammable
air, which was 27 &lt;*&gt; feet in diameter, and 46 3/4 feet long,
the machine being made to float with its longest part
parallel to the horizon, and having a boat of near 17
feet long attached to it. The boat was fitted up with
several wings or oars, shaped like an umbrella, and they
ascended at 12 o'clock with 450 pounds of sand ballast,
and after various man&#x153;uvres finally descended, at 40
minutes past 6 o'clock, near Arras in Artois, 150 miles
from Paris, having still 200 pounds of ballast remaining
in the boat. In one part they found the current of
air uniform from 600 to 4200 feet high, which it seems
was their greatest height, and the sall of the barometer
had been near 5.6 degrees. They found that by
means of their oars they could accelerate their course a
little in the direction of the wind, when it moved
slowly, which may be true; but there is great reason
to doubt of the accuracy of their experiments by which
they pretended to cause their path to deviate about
22 degrees from the wind, going with a considerable
velocity.</p><p>The second aerial voyage in England, was performed
by Mr. Blanchard, and Mr. Sheldon professor of anatomy
to the Royal Academy, being the first Englishman
who ascended with an aerostatic machine. They
ascended at Chelsea the 16th of October, at 9 minutes
past 12 o'clock. Mr. Blanchard having landed Mr.
Sheldon at about 14 miles from Chelsea, re-ascended
alone, and finally landed near Rumsey in Hampshire,
about 75 miles distant from London, having gone
nearly at the rate of 20 miles an hour. The wings
used on this occasion it seems produced no deviation
from the direction of the wind. Mr. Blanchard said
that he ascended so high as to seel a great difficulty of
breathing: and that a pigeon, which flew away from
the boat, labou&lt;*&gt;ed for some time to sustain itself with
its wings in the rarefied air, but after wandering a good
while, returned, and rested on the side of the boat.</p><p>On the 4th of October, Mr. Sadler, an ingenious
tradesman at Oxsord, ascended at that place with an
inflammable-air balloon of his own construction and
filling. And again on the 12th of the same month he
ascended at Oxford, and floated to the distance of 14
miles in 17 minutes, which is at the rate of near 50
miles an hour.</p><p>The 30th of November this year Mr. Blanchard's
fifth aerial voyage, still with his old machine, was performed
in company with Dr. J. Jeffries, a native of
America. Their voyage was about 21 miles; and it
<cb/>
does not appear that the greatest action of their oars
produced any effect in directing the course of the
balloon.</p><p>On the 4th of January, 1785, a Mr. Harper ascended
at Bitmingham with an inflammable-air balloon, and
went to the distance of 50 miles in an hour and a quarter,
and sussered no other inconvenience than a temporary
deafness, and what might be expected from the
changes of wet and cold. The thermometer descended
from 40 to 28 degrees.</p><p>On the 7th of January, Mr. Blanchard, accompanied
with Dr. Jeffries, performed his sixth aerial voyage, by
actually crossing the British channel from Dover to
Calais, with the same balloon which had sive times before
carried him successfully through the air. They
ascended with only 30 pounds of sand ballast, besides
their provisions, some books, instruments, and other
necessaries. The machine parted with the gas very
rapidly, and their ballast was soon all exhausted; after
which, from time to time they threw out every thing
else in the boat, to prevent themselves from dropping
into the sea. In this way they disposed of all their
provisions, their books and instruments, and finally the
most part of their very clothes themselves. This however
bringing them near the French coast, they gradually
ascended, cleared the cliffs and houses, and landed
in the forest of Guiennes. It is remarkable that a
bottle, being thrown out when they were in danger of
falling into the sea, struck the water with sach force,
that they heard and felt the shock very sensibly on the
car and balloon. In consequence of this voyage the
king of France presented M. Blanchard with a gift of
12000 livres, and granted him a pension of 1200 lives
a year.</p><p>On the 19th of January, Mr. Crosbie ascended at
Dublin in Ireland, with an inflammable-air balloon to
a great height. He rose so rapidly that he was out of
sight in 3 minutes and a half. By suddenly opening
the valve he descended just at the edge of the sea, as
he was driving towards the channel, being unprovided
for properly passing over to England.</p><p>On the 23d of March, Count Zambeccari and Admiral
Sir Edward Vernon ascended at London, and
sailed to Horsham in Sussex, at the distance of 35
miles in less than an hour. The voyage proved very
dangerous, owing to some of the machinery about the
valve being damaged, which obliged them to cut open
some part of the balloon when they were about two
miles perpendicular height above the earth, the barometer
having fallen from 30.4 to 20.8 inches. In
descending they passed through a dense cloud, which
felt very cold, and covered them with snow. The observatione
they made were, that the balloon kept perpetually
turning round its vertical axis, sometimes so
rapidly as to make each revolution in 4 or 5 seconds;
that a peculiar noise, like rustling, was heard among
the clouds, and that the balloon was greatly agitated
in the descent.</p><p>On May the 5th, Mr. Sadler, and William Windham,
esq. member of parliament for Norwich, ascended
at Moulsey-hurst. The machine took a south-east
course, and the current of air was so strong that they
were in great danger of being driven to sea. They
had the good fortune however to descend near the con-
<pb n="41"/><cb/>
flux of the Thames and Medway, not a mile from the
water's edge. By an accident they lost their balloon:
for while the aeronauts were busied in securing their
instruments, the country people, whom they had employed
in holding down the machine, suddenly let go
the cords, when the balloon instantly ascended, and
was driven many miles out to sea, where it fell, and
was taken up by a trading vessel. It was afterwards
restored again, and another voyage made with it from
Manchester to Pontefract, in which Mr. Sadler was
still more unfortunate; for no person being near when
it descended, and not being able to confine it by his
own strength, he was dragged by it over trees and
hedges; and at last was forced to quit it at the utmost
peril of his life; after which it rose, and was out
of sight in a few minutes. It was afterwards found
near Gainsborough.</p><p>On the 12th of May Mr. Crosbie ascended, at Dublin,
as high as the tops of the houses; but soon descended
again with a velocity that alarmed all the
spectators for his safety. On his stepping out of the
car, in an instant Mr. M'Guire, a college youth, sprung
into it, and the balloon ascended with him to the astonishment
of the beholders, and presently he was carried
with great velocity towards the channel in the direction
of Holyhead. This being observed, a crowd of horsemen
pursued full speed the course he seemed to take,
and could plainly perceive the balloon descending into
the sea. Lord H. Fitzgerald, who was among the
foremost, instantly dispatched a swift-sailing vessel
mounted with oars, and all the boats that could be
got, to the relief of the gallant youth; whom they
found almost spent with swimming, just time enough
to save his life.</p><p>The fate of M. Pilatre de Rozier, the first aerial
traveller, and his companion M. Romain, has been
much lamented. They ascended at Boulogne the 15th
of June, with intent to cross the channel to England:
for the first 20 minutes they seemed to take the proper
direction; when presently the whole apparatus was
seen in flames, and the unfortunate adventurers fell to
the ground from the height of more than a thousand
yards, and were killed on the spot, their bones being
broken, and their bodies crushed in a shocking manner.
The machine in which they ascended, consisted of a
spherical balloon, 37 feet in diameter, filled with inflammable
air; and under this balloon was suspended
a small montgolfier, or fire balloon, of 10 feet diameter;
the gallery which suspended the aeronauts, was attached
to the net of the upper balloon by cords, which were
fastened to a hoop rather larger than the montgolsier,
and descended perpendicularly to the gallery. The
montgolfier was intended to promote and prolong the
ascension, by rarefying the atmospheric air, and by
that means gaining levity. It is not certainly known
whether the balloon was actually set on fire by the
montgolfier, or, being over-rarefied by the heat beneath,
burst, and by that means the inflammable air was set in
a blaze.</p><p>On the 19th of July, at 20 minutes past 2 o'clock,
Mr. Crosbie ascended at Dublin, with intent to cross
the channel to Holyhead in England. The usual
form of the boat had been changed, for a capacious
wicker basket, of a circular form, round the upper
<cb/>
edges of which were fastened a great many bladders,
which were intended to render his gallery buoyant, in
case of a disaster at sea. About 300 pounds of ballast
were put into the basket, but the aeronaut discharged
half a hundred on his first rise. At first the current of
air carried him due west; but it soon changed his
course to nearly north-east, pointing nearly towards
Whitehaven. At upwards of 40 miles from the Irish
shore, he found himself within clear sight of both lands,
and he said it was impossible to give any adequate idea
of the unspeakable beauties which the scenery of the
sea, bounded by both lands, presented. He rose
at one time so high, that by the intense cold his ink
was frozen, and the mercury sunk quite into the ball
of the thermometer. He was sick, and felt a strong
prepulsion on the tympanum of the ears. At his utmost
height he thought himself stationary; but on
liberating some gas, he descended to a current of air
blowing north, and extremely rough. He now entered
a thick cloud, and encountered strong blasts of wind,
with thunder and lightning, which brought him rapidly
towards the surface of the water. Here the
balloon made a circuit, but falling lower, the water
entered his car, and he lost his notes of observation.
All his endeavours to throw out ballast were of no avail;
the force of the wind plunged him into the ocean; and
with much difficulty he put on his cork jacket. The
propriety of his idea was now very manifest in the construction
of his boat; as by the admission of the water
into the lower part of it, and the suspension of his
bladders, which were arranged at the top, the water,
added to his own weight, became proper ballast; and
the balloon maintaining its poise, it became a powerful
sail, by means of which, and a snatch block to his car,
he went before the wind as regularly as a sailing
vessel. In this situation he found himself inclined to
eat, and he took a little fowl. At the distance of a
league he discovered some vessels crowding after him;
but as his progress outstripped all their endeavours,
he lengthened the space of the balloon from the car,
which gave a check to the rapidity of his sailing, and
he was at length overtaken and saved by the Dunleary
barge, which took him on board, and steered to Dunleary,
towing the balloon after them.</p><p>A similar accident happened to Major Money, who
ascended at Norwich, on the 22d of July, at 20 minutes
past 4 in the afternoon; when meeting with an improper
current, and not being able to let himself down, on
account of the smallness of the valve, he was driven out
to sea, where, after blowing about for near two hours,
he dropped into the water. Here the struggles were
astonishing which he made to keep the balloon up,
which was torn, and hung only like an umbrella over
his head. A ship was once within a mile, but, he adds,
whether from want of humanity, or by mistaking the
balloon for a sea monster, they sheered off, and left
him to his fate: but a boat chased him for two hours,
till just dark, and then bore away. He now gave up
all hopes, and began to wish that providence had given
him the fate of Pilatre de Rozier, rather than such
a lingering death. Exerting himself however to preserve
life as long as possible, by keeping the balloon
floating over his head, to keep himself out of the
water, into which nevertheless he sunk gradually inch
<pb n="42"/><cb/>
by inch, as it lost its power, till he was at length breast
deep in water, when he was providentially taken up by
a revenue cutter, at half past eleven at night, but so
weak that he was obliged to be lifted out of the car
into the ship.</p><p>About the latter end of August, the longest aerial
voyage hitherto made, was performed by Mr. Blanchard,
who ascended at Lisle, accompanied by the Chevalier
de L'Epinard, and travelled 300 miles in their balloon
before it descended. On this occasion, as on some
former ones, Mr. Blanchard made trial of a parachute,
like a large umbrella, invented to break the fall in case
of an accident happening the balloon: with this machine
he dropped a dog from the car soon after his ascension,
which descended gently and unhurt.</p><p>On September the 8th, Thomas Baldwin, Esq.
ascended from the city of Chester, at 40 minutes past
one o'clock, and descended at Rixton-Moss, at 25
miles distance, after a voyage of 2 hours and a quarter.
The greatest perpendicular altitude ascended was about
a mile and a half, and the aeronaut computed that in
some parts of the voyage he moved at the rate of 30
miles an hour. Mr. Baldwin published a very circumstantial
account of his voyage, with many ingenious
philosophical remarks relating to aerostation, of which
subject his book may be considered as one of the best
treatises yet given to the public.</p><p>October the 5th, Mr. Lunardi made the first aerial
voyage in Scotland. He ascended at Edinburgh, and
after various turnings, landed near Cupar in Fife, having
described a track of 40 miles over the sea, and 10 over
the land, in an hour and a half. He said the mercury
in the barometer sunk as low as 18.3 inches at his
greatest elevation.</p><p>November the 19th the celebrated Blanchard ascended
at Ghent to a great height, and after many
dangers descended at Delft without his car, which he
cut away to lighten the machine when he was descending
too rapidly, and slung himself by the cords
to the balloon, which served him then in the nature
of a parachute. On his first ascent, when he was almost
out of sight, he let down a dog, by means of a
parachute, which came easily to the ground.</p><p>November the 25th Mr. Lunardi ascended at Glasgow,
and in two hours it is said he described a track
of 125 miles. It is further remarkable that, being
overcome with drowsiness, he says he slept for about
20 minutes in the bottom of the car, during this voyage.</p><p>Many other voyages were made in different countries,
and with various success. But since the year 1785,
the rage for balloons has considerably abated, and we
have gradually had less and less of these aerial excuisions,
so that it is now become rather an uncommon
thing to hear of one of them performed in any country
whatever: which speedy decline in this new art is
perhaps to be ascribed chiefly to the following causes;
namely, a less degree of eagerness in people to pursue
such experiments, from their curiosity having been
satisfied; secondly, the trouble, danger, and great expence,
attending them; and lastly, the want of the
means of conducting them, and the small degree of
utility to which they have hitherto been applied. The
failure in the many attempts that have been made to
direct balloons at pleasure through the air, cannot but
<cb/>
be felt as a very discouraging circumstance: and it ito
be feared that it will ever be felt as such, notwithstanding
the pretensions of some persons on this head;
for they never have caused, nor is it to be expected they
ever can cause the machine to deviate sensibly from the
course of the wind, except only in the case when
this moves with a very small celerity. For when the
current blows only at the rate of 10 miles an hour,
which is but a very gentle wind, it may be shewn
that a balloon of 50 feet in diameter will require a
force equal to the pressure of 72 pounds weight, to
cause it to deviate 30 degrees from the course of the
wind; and a force equal to double or triple that
weight, when the wind blows with a double or
triple velocity, that is, at the rate of 20 or 30 miles
an hour; and so in proportion. To obviate the danger
of a fall, arising from any accident happening to
the balloon, some experiments have been made with a
parachute, chiefly by Mr. Blanchard, whose endeavours
and perseverance it seems have continued longer than
in any other person: we still hear of his excursions in
different parts of Europe, and improvements of the
parachute, wings, &amp;c; and have just read accounts of
two voyages lately performed by him; with which,
being very curious, we shall conclude our narration of
these aerial excursions. They will be best related in
Mr. Blanchard's own words, taken from his letter,
dated Leipsick, October the 9th, 1787, to the editors
of the Paris Journal. &#x201C;I did not mention,&#x201D; says he,
&#x201C;in your interesting paper, my ascension at Strasburg
on the 26th of last August: the weather was so horrible
that I mounted only for the sake of contenting
the astonishing crowd of strangers assembled there from
all parts of the country. Every body seemed satisfied
at the attempt, but I assure you, gentlemen, that I
was far from being pleased with so common an experiment.
The only remarkable thing that occurred at
that time, was the following circumstance: At the
height of about 2000 yards, or a mile and half a
quarter, I let down a dog tied to the parachute, who,
instead of descending gently, was forcibly carried, by a
whirlwind, above the clouds. I met him soon after,
bending his course directly downwards, and, as on recollecting
his master, he began to bark a little, I was
going to take hold of the parachute, when another
whirlwind lifted him again to a great height. I lost
him for the space of six minutes, and perceived him
afterwards, with my telescope, as if sleeping in the
cradle or basket belonging to the machine. Continually
agitated, and impetuously tossed through every point of
the compass, by the violence of the different currents of
air, I determined to end my voyage on the other side
of the Rhine, after having passed vertically over Zell.
I descended at a small village, with an intention to be
assisted a little, and about thirty men soon came within
reach of the balloon very a-propos, and fixed me to the
ground. The wind was so violent that anchors or
ropes would have been of no service. I had however
added to the large aerostatic globe a smaller one, of
60 pounds ascensional force, which would have contributed
to fix me, when once I let it loose; but notwithstanding
this precaution, the men's assistance was very
necessary to me. The parachute was still wavering in
the air, and did not come down till 12 minutes after.&#x201D;
<pb n="43"/><cb/></p><p>&#x201C;I performed my 27th ascension at Leipsick the
29th of September, in the midst of an incredible number
of spectators, forming one of the most brilliant assemblies
I ever beheld. The sky was as clear and serene
as possible, and the air so calm that many of
my friends, and multitudes of others, could follow me
on horseback, and even on foot. I was sometimes so
near them that they thought they could reach me, but I
could soon find the means of rising; and once, when
they had actually taken hold of the cords, to see me
float with the strings in their hands, I suddenly cut
them, and mounted again in the air. All these amusing
evolutions were in sight of the town and its environs.
At length I yielded to the earnest solicirations
of the company, and entered the town triumphantly in
my car, followed by a concourse of people transported
with joy, and amidst the acclamations of thousands.
The next day I emptied the inflammable air into another
globe, with which I intended to try some experiments;
and I let it off with a cradle, in which a dog was
fixed. The balloon, having reached a considerable
height, made an explosion in its under part, as I had
imagined it would, having previously disposed it in a
proper manner for that purpose; by which means the
little animal fell gently to the ground.&#x201D;</p><p>&#x201C;The day before yesterday having repeated this
experiment, at the town's request, I prepared the globe
in such a manner as to cause an explosion in its upper
part, and added a parachute with two small dogs fixed
to it. They went so high that, notwithstanding the
serenity of the sky, the balloon was lost in its immense
expanse. Telescopes of the best sort became useless,
and I began to be apprehensive for the death of the
little animals, on account of the severity of the cold.
They descended however about two hours after, quite
safe and well, in the town of Delitzsch, three miles
from Leipsick. I went yesterday to claim them, and
found them again over the town in the air with the
parachute. Such experiments had been already tried
many times in the course of the day, and some officers
had thrown them from the top of a steeple, in the sight
of all the inhabitants of Delitzsch, from whence they
descended safe.&#x201D;</p><p>We have lately heard of Mr. Blanchard's 32d ascension
at Brunswick in the month of August 1788,
in which he much assisted his ascent by means of his
wings.
<hi rend="center">For several figures of balloons, see plate 1.</hi></p><p><hi rend="italics">Practice of</hi> <hi rend="smallcaps">Aerostation.</hi> The first consideration
in the practice of aerostation, is the form and the size of
the machine. Various shapes have been tried and proposed,
but the globular, or the egg-like figure, is the
most proper and convenient, for all purposes; and this
form also will require less cloth or silk than any other
shape of the same capacity; so that it will both come
cheaper, and have a greater power of ascension. The
bag or cover of an inflammable-air balloon, is best made
of the silk stuff called lustring, varnished over. But
for a montgolfier, or heated-air balloon, on account
of its great size, linen cloth has been used, lined within
or without with paper, and varnished. Small balloons
are made either of varnished paper, or simply of paper
unvarnished, or of gold-beater's skin, or such-like
light substances.
<cb/></p><p>With respect to the form of a balloon, it will be necessary
that the operator remember the common proportions
between the diameters, circumferences, surfaces,
and solidities of spheres; for instance, that of
different spheres, the circumferences are as the diameters;
that the surfaces are as the squares of the diameters;
and the solidities as the cubes of the same
diameters: that any diameter is to its circumference as
7 to 22, or as 1 to 3 1/7; and therefore 3 times and 1/7
of any diameter will be its circumserence; so that if
the diameter of a balloon be 35 feet, its circumference
will be 110 feet. And if the diameter be multiplied
by the circumference, the product will be the surface
of the sphere; thus 35 multiplied by 110 gives 3850,
which is the sursace of the same sphere in square feet.
and if this surface be divided by the breadth of the
stuff, in feet, which the balloon is to be made of, the
quotient will be the number of feet in length necessary
to construct the balloon; so if the stuff be 3 feet wide,
then 3850 divided by 3, gives 1283 1/3 feet, or 428
yards nearly, the requisite quantity of stuff of 3 feet
or one yard wide, to form the balloon of 35 feet diameter.
Hence also, by knowing the weight of a given
piece of the stuff, as of a square foot, or square yard,
it is easy to find the weight of the whole bag, namely
by multiplying the surface, in square feet or yards, by
the weight of a square foot or yard: so if each square
yard weigh 16 ounces or 1 pound, then the whole bag
will weigh 428 pounds. Again, the capacity, or solid
contents, of the sphere, will be found by multiplying 1/6
of the surface by the diameter, or by taking 11/22 of
the cube of the diameter; which gives 22458 cubie
feet for the capacity of the said balloon, that is, it
will contain, or displace, 22458 cubic feet of air. From
the content and surface of the balloon, so found, is to
be derived its power or levity, thus: on an average,
a cubic foot of common air weighs 1 1/5 ounce, and therefore
to the number 22458, which is the content of our
balloon, adding its (1/5)th part, we have 26950 ounces,
or 1684 pounds, for the weight of the common air displaced
or occupied by the balloon. From this weight
must be deducted the weight of the bag, namely 428
pounds, and then there remains 1256 pounds levity of
the balloon, without however considering the contained
air, whether it be heated air, or of the inflammable
kind. If inflammable air be used, as it is of different
weights, from 1/4 to 1/10 or 1/12 the weight of common
air, according to the modes of preparing it, let us
suppose for instance that it is 1/6 of the weight of common
air; then 1/6 of 1684 is 261 pounds, which is the
weight of the bag full of that air; which being taken
from 1256, leaves 995 pounds for the levity of the
balloon when so filled with that inflammable air, or the
weight which it will carry up, consisting of the car,
the ropes, the passengers, the necessaries, and ballast.
But if heated air be used; then as it is known from
experiment that, by heating, the contained air is diminished
in density about one-third only, therefore
from 1684, take 1/3 of itself, and there remains 1123
for the weight of the contained warm air; and this
being subtracted from 1256, leaves only 133 pounds
for the levity of the balloon in this case; which being
too small to carry up the car, passengers, &amp;c, it shews
that for those purposes a larger balloon is necessary, on
<pb n="44"/><cb/>
Montgolfier's principles. But if now, from the preceding
computation, it be required to find how much
the size of the balloon must be increased, that its levity,
or power of ascension, may be equal to any given weight,
as suppose 1000 pounds; then because the levities are
nearly as the cubes of the diameters, therefore the diameters
will be nearly as the cube roots of the levities;
but the levities 133 and 1000 are nearly as 1 to 8, the
cube roots of which are as 1 to 2, and consequently
1 : 2 :: 35 : 70 feet, the diameter of a montgolfier,
made of the same thickness of stuff as the former,
capable of lifting 1000 pounds.</p><p>On the same principles we can easily find the size of
a balloon that shall just float in air when made of stuff
of a given thickness or weight, and filled with air of a
given density; the rule for which is this: from the
weight of a cubic foot of common air, subtract that
of a cubic foot of the lighter or contained air; then
divide 6 times the weight of a square foot of the stuff,
by the remainder; and the quotient will be the diameter,
in feet, of the balloon that will just float at the surface
of the earth. Suppose, for instance, that the materials
are as before, namely, the stuff 1 pound to the square
yard, or 16/9 ounces to the square foot, which taken
6 times is 32/3; then the cubic foot of common air
weighing 1 1/5 ounce, and of heated air 2/3 of the same,
whose difference is 2/5; therefore 32/3 divided by 2/5, gives
26 2/3 feet, which is the diameter of a montgolfier that
will just float: but if inflammable air be used of 1/6 the
weight of common air, the difference between 1 1/5 and 1/6
of it, is 1; by which dividing 32/3 or 10 2/3, the quotient
is the same 10 2/3 feet, which therefore is the diameter of
an inflammable-air balloon that will just float. And
if the diameter be more than these dimensions, the
balloons will rise up into the atmosphere.</p><p>The height nearly to which a given balloon will rise
in the atmosphere, may be thus found, having given
only the diameter of the balloon, and the weight which
just balances it, or that is just necessary to keep it from
rising: compute the capacity or content of the globe
in cubic feet, and divide its restraining weight in
ounces by that content, and the
<table rend="border"><row role="data"><cell cols="1" rows="1" role="data">Height
in miles.</cell><cell cols="1" rows="1" role="data">Density.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" role="data">1.200</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1/4</cell><cell cols="1" rows="1" role="data">1.141</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1/2</cell><cell cols="1" rows="1" role="data">1.085</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3/4</cell><cell cols="1" rows="1" role="data">1.031</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">0.980</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1 1/4</cell><cell cols="1" rows="1" role="data">0.932</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1 1/2</cell><cell cols="1" rows="1" role="data">0.886</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1 3/4</cell><cell cols="1" rows="1" role="data">0.842</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">0.800</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2 1/4</cell><cell cols="1" rows="1" role="data">0.761</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2 1/2</cell><cell cols="1" rows="1" role="data">0.723</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2 3/4</cell><cell cols="1" rows="1" role="data">0.687</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">0.653</cell></row></table>
quotient will be the difference between
the density or specific gravity
of the atmosphere at the earth's surface
and that at the height to which
the balloon will rise; therefore subtract
that difference or quotient from
1 1/5 or 1.2, the density at the earth,
and the remainder will be the density
at that height: then the height answering
to that density will be found
sufficiently near in the annexed table.
Thus, in the foregoing examples, in
which the diameter of the balloon is
35 feet, its capacity 22458, and the
levity of the first one 995 pounds, or
15920 ounces, the quotient of the
latter number divided by the former, is .709, which is
the density at the utmost height, and to which in the
tableanswers a little more than 2 1/2 miles, or 2 5/8 miles nearly,
which therefore is the height to which the balloon will ascend.
And when the same balloon was filled with heated
air, its levity was found equal to only 133 pounds, or 2128
<cb/>
ounces, then dividing this by 22458 the capacity, the
quotient .095 taken from 1.200, leaves 1.105 for the
density; to which in the table corresponds almost half
a mile, or nearer 3/8 of a mile. And so high nearly
would these balloons ascend, if they keep the same
figure, and lose none of the contained air: or rather,
those are the heights they would settle at; for their
acquired velocity would first carry them above that
height, so far as till all their motion should be destroyed;
then they would descend and pass below that height,
but not so much as they had gone above; after which
they would re-ascend, and pass that height again, but
not so far as they had gone below it; and so on for
many times, vibrating alternately above and below
that point, but always less and less every time. The
foregoing rule, for finding the height to which the
balloon will ascend, is independent of the different
states of the thermometer at that highest point, and at
the surface of the earth; but for greater accuracy, including
the allowances depending on the different states
of the thermometer, see under the word <hi rend="smallcaps">Atmosphere</hi>,
where the more accurate rules are given at large.</p><p>The best way to make up the whole coating of the
balloon, is by different pieces or slips joined lengthways
from end to end, like the pieces composing the surface
of a geographical globe, and contained between one
meridian and another, or like the slices into which a
melon is usually cut, and supposed to be spread flat
out. Now the edges of such pieces cannot be exactly
described by a pair of compasses, not being circular,
but flatter or less round than circular arches; but if
the slips are sufficiently narrow, or numerous, they will
differ the less from circles, and may be described as
such. But more accurately, the breadths of the slip, at
the several distances from the
<figure/>
point to the middle, where it is
broadest, are directly as the sines
of those distances, radius being
the sine of the half length of the
slip, or of the distance of either
point from the middle of the
slip: that is, If ACBD be one
of the slips, AB being half the
circumference, or AE a quadrant
conceived to be equal to
AC or AD; then will CD
be to <hi rend="italics">a b,</hi> as radius or the sine
of AC, to the sine of A<hi rend="italics">a.</hi> So
that if the quadrant AE or AC
be divided into any number of
equal parts, as here suppose 9,
then divide the quadrant or 90
degrees by the number of parts
9, and the quotient 10 is the
number of degrees in each part;
and hence the arcs AC, A<hi rend="italics">a,</hi> A<hi rend="italics">c,</hi> &amp;c, will be respectively
90&#xB0;, 80&#xB0;, 70&#xB0;, &amp;c; and CD being radius,
the several breadths <hi rend="italics">ab, cd, ef,</hi> &amp;c, will be respectively
the sines of 80&#xB0;, 70&#xB0;, 60&#xB0;, &amp;c, which are here
placed opposite them, the radius being 1. Therefore
when it is proposed to cut out slips for a globe of
a given diameter; compute the circumference, and
make AE or AC a quarter of that circumference, and
CD of any breadth, as 3 feet, or 2 feet, or any other
<pb n="45"/><cb/>
quantity; then multiply each of the decimal numbers,
set opposite the figure, by that quantity, or breadth of
CD, so shall the products be the several breadths <hi rend="italics">ab,
cd, ef,</hi> &amp;c.</p><p>Various schemes have been devised for conducting balloons
in any direction, whether vertical or sideways. As
to the vertical directions, namely upwards or downwards,
the means are obvious, viz. in order to ascend, the
aeronaut throws out some ballast; and that he may
descend, he opens a valve in the top of his machine
by means of a string, to let some of the gas escape;
or if it be a montgolfier, he increases or diminishes the
fire, as he would ascend or descend. But to direct the
machine in a side or horizontal course, is a very difficult
operation, and what has hitherto not been accomplished,
except in a small degree, and when the current
of air is very gentle indeed. The dissiculty of moving
the balloon sideways, arises from the want of wind
blowing upon it; for as it floats along with the current
of air, it is relatively in a calm, and the aeronaut feels
no more wind than if the machine were at rest in a
perfect calm. For this reason, any thing in the nature
of sails can be of no use; and all that can be
hoped for, is to be attempted by means of oars; and
how small the effect of these must be, may easily
be conceived from the rarity of the medium against
which they must act, and the great magnitude of the
machine to be forced through it. We can easily
assign what force is necessary to move a given machine
in the air with any proposed velocity. From
very accurate experiments I have determined, that a
globe of 6 3/8 inches in diameter, and moving with a
velocity of 20 feet per second of time, suffers a resistance
from the air which is just equal to the weight
or pressure of one ounce Averdupois; and farther
that with different surfaces, and the same velocity,
the resistances are directly proportional to the surface
nearly, a double surface having a double resistance,
a triple surface a triple resistance, and so on; and also
that with different velocities, the resistances are proportional
to the squares of the velocities nearly, so that
a double velocity produced a quadruple resistance, and
three times the velocity nine times the resistance, and
so on. And hence we can assign the resistance to move
a given balloon, with any velocity. Thus, take the
balloon as before of 35 feet diameter; then by comparifon
as above it is found that this globe, if moved with
the velocity of 20 feet per second, or almost 14 miles per
hour, will suffer a resistance equal to 271 pounds; to
move it at the rate of 7 miles an hour, the resistance
will be 68 pounds; and to move it 3 1/2 miles an hour,
the resistance will be 17 pounds; and so on: and with
such force must the aeronauts act on the air in a contrary
direction, to communicate such a motion to the
machine. And if the balloon move through a rarer
part of the atmosphere, than that at the surface of the
earth, as 1/3, or 1/4, &amp;c, rarer, and consequently the resistance
be less in the same proportion; yet the force of
the oars will be diminished as much; and therefore the
same difficulty still remains. In general, the aeronaut
must strike the air, by means of his oars, with a force
just equal to the resistance of the air on the balloon,
and therefore he must strike that air with a velocity
which must be greater as the surface of the oar is less
<cb/>
than the resisted surface of the globe, but not in the
same proportion, because the force is as the square of the
velocity.</p><p>Now suppose the aeronaut act with an oar equal to
100 square feet of surface, to move the balloon above
mentioned at the rate of 20 feet per second, or 14 miles
an hour; then must he move this oar with the great
velocity of 62 feet per second, or near 43 miles per
hour: and so in proportion for other velocities of the
balloon. From whence it is highly probable, that it
will never be in the power of man to guide such machine
with any tolerable degree of success, especially
when any considerable wind blows, which is almost always
the case.</p><p>As some aeronauts have thought of using parachutes,
made something like umbrellas, to break their fall, in
case of any accident happening to the balloon, we shall
here consider the principles and power of such a machine.
Let us suppose a person wants to know what
the size of a parachute must be, that he may descend
with it at the uniform rate of 10 feet in a second, which
is nearly equal to the velocity he acquires by falling or
leaping from the height only of 17 inches, and which
it is presumed he may do with safety. Now in order
to descend with any uniform velocity, the resistance of
the air must be equal to the whole weight that descends:
then suppose the weight of the aeronaut to be 150
pounds, and that the parachute is flat, and circular,
and made of such materials as that every square foot of
its surface weighs 2 ounces, and farther that the weight
increases in the same proportion as the surface; then
the diameter of the parachute necessary to descend with
the moderate velocity of 10 feet per second, must be
upwards of 78 feet in diameter: but if the parachute
be not a flat surface, but concave on the lower side, its
power will be rather the greater, and the diameter may
be somewhat less. If it be required to know the power
of a flat circular parachute, or what resistance it meets
with from air of a mean density, when descending with
a given velocity; say as the number 800 is to the square
of the velocity in feet, so is the square of the diameter
in feet, to a fourth number, which will be the resistance
in pounds. And hence, if it be required to know
with what velocity a parachute will descend with a
given weight; say as the given diameter is to the square
root of the weight, so is the number 28 1/3 to a fourth
term, which will be the velocity when the descent is in
air of a mean density. So if the diameter of a balloon
be 50, and its weight together with that of a man be
530 pounds, the square root of which is 23 very nearly;
then as 50 : 23 :: 28 1/3 : 13, so that the man and parachute
will descend with the velocity of 13 feet per second;
which it is presumed he may safely do, as he would
meet with a shock only equal to that of leaping freely
from a height only of 2 feet 2 inches.</p><p>The methods of extracting inflammable air from
various substances, for filling balloons, and for other
purposes, may be seen under the words Air and Gas.
And as to the methods of filling and constructing
balloons, being matters merely mechanical, they are
omitted in this place.</p><p>Ample information however on these, and many
other particulars, may be met with in several books
expressly written on the subject; as in Cavallo's History
<pb n="46"/><cb/>
and Practice of Aerostation, 8vo, 1785; in Baldwin's
Airopaidia, 8vo, 1786; &amp;c.</p><p>It has often been discussed, says the former of these
gentlemen, whether the preference should be given to
machines raised by inflammable air, or to those raised
by heated air. Each of them has its peculiar advantages
and disadvantages; a just consideration of which
seems to decide in favour of those made with inflammable
air. The principal comparative advantages of
the other sort are, that they do not require to be made
of so expensive materials; that they are filled with
little or no expence; and that the combustibles necessary
to fill them are found almost every where; so that
when the stock of fuel is exhausted, the aeronaut may
descend and recruit it again, in order to proceed on his
voyage. But then this sort of machines must be made
larger than the other, to take up the same weight;
and the presence of a fire is a continual trouble, and a
continual danger: in fact, among the many aerial
voyages that have been made and attempted with such
machines, very few have succeeded without an inconvenience,
or an accident; and some indeed have been
attended with dangerous and even fatal consequences;
from which the other sort is in a great measure exempt.
But, on the other hand, the inflammable air balloon
must be made of a substance impermeable to the subtle
gas: the gas itself cannot be produced without a considerable
expence; and it is not easy to find the materials
and apparatus necessary for the production of it
in every place. However, it has been found that an
inflammable-air balloon, of 30 feet in diameter, may be
made so close as to sustain two persons, and a considerable
quantity of ballast, in the air for more than 24
hours, when properly managed; and possibly one man
might be supported by the same machine for three days:
and it is probable that the stuff for these balloons may
be so far improved, as to be quite impermeable to the
gas, or very nearly so; in which case, the machine,
once silled, would continue to float for a long space of
time. At Paris they have already attained to a great
degree of perfection in this point; and small balloons
have been kept floating in a room for many weeks,
without losing any considerable quantity of their levity:
but the difficulty lies in the large machines: for in
these, the weight of the stuff itself, with the weight
and stress of the ropes and boat, and the folding them
up, may easily crack and rub off the varnish, and make
them leaky.</p><p>In regard to philosophical observations, derived from
the new subject of aerostation, there have been very
few made; the novelty of the discovery, and of the
prospect enjoyed from the car of an aerostatic machine,
have commonly distracted the attention of the aeronauts;
not to remark that many of the adventurers were inadequate
to the purpose of making improvements in
philosophy, being mostly influenced either by pecuniary
motives, or the vanity of adding their names to the list
of aerial travellers.&#x2014;The agreeable stillness and tranquillity
experienced aloft in the atmosphere, have been
matter of general observation. Some machines have
ascended to a great height, as far, it has been said, as
two miles; and they have commonly passed through
fogs and clouds, above which they have enjoyed the
clear light and heat of the sun, whilst the earth beneath
<cb/>
was actually covered by dense clouds, which poured
down abundance of rain. In ascending very high, the
aeronauts have often experienced a pain in their ears,
arising, it is supposed, from the internal air being not
of the same density as the air without; but the pain
usually went off in a short time: and it seems that this
effect is similar to what is experienced by persons who
descend by a diving-bell to considerable depths in the
sea: I remember often to have heard the late unfortunate
Mr. Spalding, the celebrated diver, speak of this
effect, with a marked and philosophical accuracy: after
descending two or three fathoms below the surface, he
began to feel a pain in his ears, which gradually increased
to a very great degree if the descent was too
quick; his method was therefore to descend slowly, and
to make a stop for some minutes at the depth of 5 fathom,
which is equal nearly to the pressure of the atmosphere,
and where consequently the air in his bell
was of double the density of common air at the surface;
after resting here awhile, his ears, as he expressed it,
gave a crack, and he was suddenly relieved of the pain.
He then descended 5 fathoms more, with the same
symptoms, and the same effect: and so on continually,
from one five fathoms to another, descending leisurely,
and stopping a little at each stage, to give time for his
constitution to adapt itself to the degree of condensation
of the air; after which he felt no more inconvenience,
till he came to ascend again, which was performed with
the same caution and circumstances. One experiment
is recorded, in which the air of a high region, being
brought down, and examined by means of nitrous air,
was found to be purer than the air below. The temperature
of the upper regions too, it has been found, is
much colder than that of the air near the earth; the
thermometer, in some aerostatic machines, having descended
many degrees below the freezing point of water,
while it was considerably higher than that degree at
the earth's surface.</p><p>&#xC6;STIVAL, see <hi rend="smallcaps">Estival.</hi></p><p>&#xC6;STUARY, or <hi rend="smallcaps">Estuary</hi>, in <hi rend="italics">Geography,</hi> an arm
of the sea, running up a good way into the land. Such
as Bristol channel, many of the friths in Sootland, and
such like.</p><p>&#xC6;THER, see <hi rend="smallcaps">Ether.</hi></p></div1><div1 part="N" n="AFFECTED" org="uniform" sample="complete" type="entry"><head>AFFECTED</head><p>, or <hi rend="smallcaps">Adfected, Equation</hi>, in
<hi rend="italics">Algebra,</hi> is an equation in which the unknown quantity
rises to two or more several powers or degrees. Such,
for example, is the equation , in
which there are three different powers of <hi rend="italics">x,</hi> namely,
<hi rend="italics">x</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">2</hi>, and <hi rend="italics">x.</hi></p><p>The term, affected, is also used sometimes in algebra,
when speaking of quantities that have co-efficients.</p><p>Thus in the quantity 2<hi rend="italics">a, a</hi> is said to be affected with
the co-efficient 2.</p><p>It is also said, that an algebraic quantity is affected
with the sign + or -, or with a radical sign; meaning
no more than that it has the sign + or -, or that it
includes a radical sign.</p><p>The term adfected, or affected, I think, was introduced
by Vieta.</p></div1><div1 part="N" n="AFFECTION" org="uniform" sample="complete" type="entry"><head>AFFECTION</head><p>, in <hi rend="italics">Geometry,</hi> a term used by some
ancient writers, signifying the same as property.</p><p><hi rend="smallcaps">Affection.</hi> <hi rend="italics">Phys.</hi> The affections of a body
are certain modifications occasioned or induced by
<pb n="47"/><cb/>
motion; in virtue of which the body is disposed after
such, or such a manner.</p><p>The affections of bodies, are sometimes divided into
<hi rend="italics">primary</hi> and <hi rend="italics">secondary.</hi></p><p><hi rend="italics">Primary Affections,</hi> are those which arise either out
of the idea of matter, as magnitude, quantity, and
figure; or out of the idea of form, as quality and
power; or out of both, as motion, place, and time.</p><p><hi rend="italics">Secondary,</hi> or <hi rend="italics">derivative Affections,</hi> are such as arise
out of primary ones, as divisibility, continuity, contiguity,
&amp;c, which arise out of quantity; regularity, irregularity,
&amp;c, which arise out of figure, &amp;c.</p><p>AFFIRMATIVE <hi rend="smallcaps">Quantity</hi>, or <hi rend="smallcaps">Positive
Quantity</hi>, one which is to be added, or taken
effectively; in contradistinction to one that is to be subtracted,
or taken defectively.&#x2014;The term affirmative
was introduced by Vieta.</p><p><hi rend="smallcaps">Affirmative Sign</hi>, or <hi rend="smallcaps">Positive Sign</hi>, in <hi rend="italics">Algebra,</hi>
the sign of addition, thus marked +, and is called
<hi rend="italics">plus,</hi> or <hi rend="italics">more,</hi> or <hi rend="italics">added to.</hi> When set before any single
quantity, it serves to denote that it is an affirmative or
a positive quantity; when set between two or more
quantities, it denotes their sum, shewing that the latter
are to be added to the former. So + 6, and + <hi rend="italics">a,</hi> and
+ AB, are affirmative quantities; also + 6 + 8 +
10 denote the sum of 6, 8, and 10, which is 24, and
are read thus, 6 plus 8 plus 10. Also <hi rend="italics">a</hi> + <hi rend="italics">b</hi> + <hi rend="italics">c</hi> denote
the sum of the quantities represented by <hi rend="italics">a, b</hi> and <hi rend="italics">c,</hi>
when added together. It seems now not easy to ascertain
with certainty, when, or by whom, this sign was
first introduced; but it was probably by the Germans,
as I find it first used by Stifelius in his Arithmetic, printed
in 1544.</p><p>The early writers on Algebra used the word <hi rend="italics">plus</hi> in
Latin, or <hi rend="italics">piu</hi> in Italian, for addition, and afterwards
the initial <hi rend="italics">p</hi> only, as a contraction; like as they used
<hi rend="italics">minus,</hi> or <hi rend="italics">meno,</hi> or the initial <hi rend="italics">m</hi> only, for subtraction:
and thus these operations were denoted in Italy by
Lucas de Burgo, Tartalea, and Cardan, while the signs
+ and - were employed much about the same time
in Germany by Stifelius, Scheubelius, and others, for
the same operations.</p></div1><div1 part="N" n="AGE" org="uniform" sample="complete" type="entry"><head>AGE</head><p>, in <hi rend="italics">Chronology,</hi> is used for a century, being
a system or a period of a hundred years.</p><p>Chronologists also divide the time since the creation
of the world into three ages: The first, from Adam
till Moses, which they call the age of nature; the second
from Moses to Jesus Christ, called the age of the
law; and the third, or age of grace, from Jesus Christ
till the end of the world.</p><p><hi rend="smallcaps">Age</hi> <hi rend="italics">of the Moon,</hi> in Astronomy, is the number of
days elapsed since the last new moon. To find the
moon's age, for any time nearly, for ordinary uses;
add together the epact, the day of the current month,
and the number of months from March to the present
month inclusive; the sum is the moon's age: but if
the sum exceed 29, deduct 29 from it in months that
have 30 days, or 30 in those that have 31; and the remainder
will be the age.&#x2014;At the end of 19 years the
moon's age returns upon the same day of the month,
but falls a little short of the same hour of the day.</p></div1><div1 part="N" n="AGENT" org="uniform" sample="complete" type="entry"><head>AGENT</head><p>, <hi rend="italics">Agens,</hi> in <hi rend="italics">Physics,</hi> that by which a thing
is done or effected; or any thing having a power by
<cb/>
which it acts on another, called the patient, or by its
action induces some change in it.</p></div1><div1 part="N" n="AGGREGATE" org="uniform" sample="complete" type="entry"><head>AGGREGATE</head><p>, the sum or result of several things
added together. See <hi rend="italics">Sum.</hi></p></div1><div1 part="N" n="AGITATION" org="uniform" sample="complete" type="entry"><head>AGITATION</head><p>, in <hi rend="italics">Physics,</hi> a brisk intestine motion,
excited among the particles of a body. Thus fire agitates
the subtlest particles of bodies. Fermentations,
and effervescences, are produced by a brisk agitation of
the particles of the fermenting body.</p></div1><div1 part="N" n="AGUILON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">AGUILON</surname> (<foreName full="yes"><hi rend="smallcaps">Francis</hi></foreName>)</persName></head><p>, or <hi rend="smallcaps">Aquilon</hi>, was a jesuit
of Brussels, and professor of philosophy at Doway, and
of theology at Antwerp. He was one of the first
that introduced mathematical studies into Flanders. He
wrote a large work on <hi rend="italics">Optics,</hi> in 6 books, which was
published in folio, at Antwerp, in 1613; and a treatise
of <hi rend="italics">Projections of the Sphere.</hi> He promised also to treat
upon <hi rend="italics">Catoptrics</hi> and <hi rend="italics">Dioptrics,</hi> but this was prevented
by his death, which happened at Seville, in the year
1617.</p></div1><div1 part="N" n="AIR" org="uniform" sample="complete" type="entry"><head>AIR</head><p>, in <hi rend="italics">Physics,</hi> a thin, fluid, transparent, elastic,
compressible, and dilatable body, which surrounds this
terraqueous globe, and covers it to a considerable
height.</p><p>Some of the ancients considered air as an element,
namely, one of the four elements, air, earth, water,
and fire, of which they conceived all bodies to be composed;
and though it be certain that air, taken in the
common acceptation, be far from the simplicity of an
elementary substance, yet some of its parts may properly
be so called. So that air may be distinguished
into proper or elementary, and vulgar or heterogeneous.</p><p><hi rend="italics">Elementary</hi> <hi rend="smallcaps">Air</hi>, or <hi rend="italics">Air</hi> properly so called, is a subtle,
homogeneous, elastic fluid; being the basis, or
fundamental ingredient, of the whole air of the atmosphere,
from which it takes its name. And in this
sense Dr. Hales, and other modern philosophers, consider
it as entering into the composition of most, or
perhaps all bodies; existing in them in a solid state,
devoid of its elasticity, and most of its distinguishing
properties, and serving as their cement; but, by certain
processes, capable of being disengaged from them, recovering
its elasticity, and resembling the air of our atmosphere.</p><p>The particular nature of this aerial matter we know
but little about: what authors have said concerning it
being chiefly conjectural. There is no way of examining
air pure and defecated from the several matters
with which it is mixed; and consequently we cannot
pronounce what are its peculiar properties, abstractedly
from other bodies.</p><p>Dr. Hook, and some others, maintain that it is the
same with the <hi rend="italics">ether,</hi> or that imaginary fine, fluid,
active matter, conceived to be diffused through the
whole expanse of the celestial regions: which comes to
much the same thing as Newton's subtle medium, or
spirit. In this sense it is supposed to be a body <hi rend="italics">sui
generis,</hi> incorruptible, immutable, incapable of being
generated, but present in all places, and in all bodies.</p><p>Other philosophers place its essence in elasticity,
making that its distinctive character. These suppose
that it may be generated, and that it is nothing else
but the matter of other bodies, rendered by the
<pb n="48"/><cb/>
changes it has undergone, susceptible of a permanent
elasticity. Mr. Boyle produces a number of experiments,
which he made on the production of air, that
is, according to him, the extraction of a sensible quantity
of air from a body in which there appeared to be
little or none at all, by whatever means this may be
effected. He observes that among the different methods
for this purpose, the chief are fermentation, corrosion,
dissolution, decomposition, ebullition of water and
other fluids, the reciprocal action of bodies, especially
saline ones, upon one another; he adds, that different
solid and mineral bodies, in the parts of which no
elasticity could be conceived to exist, being plunged
into corrosive mediums, which also are quite unelastic,
will, by the attenuation of their parts from their mutual
collision, produce a considerable quantity of elastic
air.</p><p>Sir Isaac Newton is of the same opinion, according
to whom the particles of a dense compact fixed substance,
adhering to each other by a powerful attractive
force, cannot be separated but by a violent heat,
and perhaps never without fermentation; and these
bodies, raresied by heat and fermentation, are finally
transformed into a truly permanent elastic air. On
these principles, he adds, gunpowder produces air on
explosion. Optics, Qu. 31, &amp;c.</p><p><hi rend="italics">Common,</hi> or <hi rend="italics">heterogeneous</hi> <hi rend="smallcaps">Air</hi>, is an assemblage of
corpuscles of various kinds, which together constitute
one fluid mass, in which we live and move, and which
we constantly breathe; which compound mass altogether,
is called the atmosphere.</p><p>In this popular and extensive meaning of the term,
Mr. Boyle acknowledges that air is the most heterogeneous
body in the universe; and Boerhaave proves that
it is an universal chaos, a mere jumble of all species of
created things. Besides the matter of light or fire,
which continually flows into it from the celestial bodies,
and perhaps the magnetic effluvia of the earth, whatever
fire can volatilize must be found in the air.</p><p>Hence, for instance, 1. All sorts of vegetable matter
must be contained in the air; being either exhaled from
plants growing all over the face of the earth, or rendered
volatile by putrefaction, not excepting even the more
solid and vascular parts of them.</p><p>2. It is no less certain that the air must contain particles
of every substance belonging to the animal kingdom.
For the copious emanations which are perpetually
issuing from the bodies of animals, in the perspiration
constantly kept up by the vital heat, are absorbed
by the air; and in such quantities too, during
the course of an animal life, that, could they be recollected,
they would be sufficient to compose a good
round number of the like animals. And besides, when
a dead animal continues exposed to the air, all its particles
evaporate, and are quickly dissipated; so that the
substance which composed the animal, is almost wholly
incorporated with the air.</p><p>3. The whole fossil kingdom must necessarily be
found in the atmosphere; for all of that kind, as salts,
sulphurs, stones, metals, &amp;c, are convertible into fume,
and must consequently take place among aerial substances.
Gold itself, the most fixed of all natural bodies;
is found among ores, closely adhering to sulphurs
in mines, and so is raised along with the mineral.
<cb/></p><p>Of all the emanations which float in the vast ocean
of the atmosphere, perhaps the principal are such as
consist of saline particles. Many writers suppose that
they are of a nitrous kind; but it is probable that they
are of all sorts, as vitriol, alum, mariue salt, and many
others. And Mr. Boyle thinks that there may be
great quantities of compound salts, not to be met with
on or in the bowels of the earth, formed by the fortuitous
concourse and mixture of different saline spirits.</p><p>We often find the window-glass of old buildings
corroded, as if eaten by worms; though we know of
no particular salt that is capable of producing such an
effect.</p><p>Sulphurs too must make a considerable portion of this
compound mass, on account of the many volcanos, grotts,
caverns, and mines, dispersed over the face of the globe.</p><p>Finally, the various attritions, separations, dissolutions,
and other mutual operations of matter of different
sorts upon one another, may be regarded as the
sources of many other neutral, or anony mous bodies,
unknown to us, which rise and float in the air.</p><p>Air, taken in this extensive sense, is one of the most
general and considerable agents in nature; being concerned
in the preservation of animal and vegetable
life, and in the production of most of the phenomena
that take place in the material world.</p><p>Its properties and effects, having been the principal
objects of the researches and discoveries of modern
philosophers, have been reduced to precise laws and
demonstrations, forming no inconsiderable branch of
mixed mathematics, under the titles of Pneumatics,
Aerometry, &amp;c.</p><p><hi rend="italics">Mechanical Properties and Effects of</hi> <hi rend="smallcaps">Air.</hi> Of these
the most considerable are its fluidity, its weight, and
its elasticity.</p><p>1. <hi rend="italics">Its Fluidity.</hi>&#x2014;The great fluidity of the air is
manifest from the great facility with which bodies traverse
it; as in the propagation of, and easy conveyance
it affords to, sounds, odours and other effluvia and emanations
that escape from bodies: for these effects prove
that it is a body whose parts give way to any force, and
in yielding are easily moved amongst themselves; which
is the definition of a fluid. That the air is a fluid is also
proved from this circumstance, that it is found to exert
an equal pressure in all directions; an effect which
could not take place otherwise than from its extreme
fluidity. Neither has it been found that the air can be
deprived of this property, whether it be kept for many
years together consined in glass vessels, or be exposed
to the greatest natural or artificial cold, or condensed
by the most powerful pressure; for in none of these
circumstances has it ever been reduced to a solid state.
It is true indeed that real permanent air may be extracted
from solid bodies, and may also be absorbed by
them; and we also know that in this case it must be
exceedingly condensed, and reduced to a bulk many
hundred times less than in its natural state: but in what
form it exists in those bodies, or how their particles are
combined together, is a mystery which remains hitherto
inexplicable.</p><p>Those philosophers who, with the Cartesians, make
fluidity to consist in a perpetual intestine motion of the
parts, think they can prove that this character belongs
to air: thus, in a darkened room, where the represen-
<pb n="49"/><cb/>
tations of external objects are introduced by a single
ray, the corpuscles with which the air is replete, are
seen to be in a continual fluctuation. Some moderns
attribute the fluidity of the air, to the fire which is
intermixed with it; without which, say they, the whole
atmosphere would harden into a solid impenetrable
mass: and indeed it must be allowed that the more fire
it contains, the greater will its fluidity, mobility, and
permeability be; and according as the different positions
of the sun augment or diminish the degree of
fire, the air always receives a proportional temperature,
and is kept in a continual reciprocation.</p><p>2. <hi rend="italics">Its Weight or Gravity.</hi>&#x2014;The weight or gravity
of the air, is a property belonging to it as a body;
for gravity is a property essential to matter, or at least
a property found in all bodies. But independent of
this, we have many direct proofs of its gravity from
sense and experiment: thus, the hand laid close upon
the end of a vessel, out of which the air is drawn at
the other end, soon feels the load of the incumbent atmosphere:
thus also, thin glass vessels, exhausted of
their air, are easily crushed to pieces by the weight of
the external air: and so two hollow segments of a
sphere, 4 inches in diameter, exactly fitting each other,
being emptied of air, are, by the weight of the ambient
air, pressed together with a force which requires the
weight of 188 pounds to separate them; and that they
are thus forcibly held together by the pressure of the
air, is made evident by suspending them in an exhausted
receiver, for then they quickly separate of themselves,
and fall asunder. Again, if a tube, close at one end,
be filled with quicksilver, and the open end be immerged
in a bason of the same fluid, and so held upright, the
quicksilver in the tube will be kept raised up in it
to the height of about 30 inches above the surface of
that in the bason, being supported and balanced by the
pressure of the external air upon that surface: and that
this is the cause of the suspension of the quicksilver in
the tube, is made evident by placing the whole apparatus
under the receiver of an air-pump; for then the
fluid will descend in the tube in proportion as the receiver
is exhausted of its air; and then on gradually
letting in the air again, the quicksilver reascends to its
former beight in the tube: and this is what is called,
from its inventor, the Terricellian experiment. Nay
farther, air can actually be weighed like any other
body: for a rigid vessel, full even of common air, by a
nice balance is found to weigh more than when the air
is exhausted from it; and the essect is proportionally
more sensible, if the vessel be weighed full of condensed
air, and more still if it be weighed in a receiver void
of air.</p><p>But although we have innumerable proofs of the
gravitating property of the air, yet the full discovery
of the laws and circumstances of it are certainly due
to the moderns. It cannot indeed be denied, that
several of the ancients had some confused notions about
this property: thus Aristotle says that all the elements
have gravity, and even air itself; and as a proof of it,
says that a bladder inflated with air, weighs more than
the same when empty; and Plutarch and Stob&#xE6;us quote
him as teaching that the air in its weight is between
that of fire and of earth; and farther, he himself,
treating of respiration, reports it as the opinion of
<cb/>
Empedocles, that he ascribes the cause of it to the
weight of the air, which by its pressure forces itself
into the lungs; and much in the same way are the sentiments
of Asclepiades expressed by Plutarch, who represents
him as saying, among other things, that the
external air, by its weight, forcibly opened its way into
the breast. But nevertheless it is certain, however unreasonable
it may seem, that Aristotle's followers departed
in this instance from their master, by asserting
the contrary for many ages together. Indeed many of
the phenomena arising from this property, have been
remarked from the highest antiquity. Many centuries
since, it was known that by sucking the air from an
open pipe, having its extremity immersed in water,
this fluid rises above its level, and occupies the place
of the air. In consequence of such observations, sucking
pumps were contrived, and various other hydraulic
machines; as Heron's syphons, described in his Spiritalia
or Pneumatics, and the watering pots known in
Aristotle's time under the name of <hi rend="italics">clepsydr&#xE6;,</hi> which
alternately stop or run as the singer closes or opens their
upper orifice. Indeed the reason assigned, by philosophers
many ages after, for this phenomenon, was a pretended
horror that nature conceives for a vacuum,
which, rather then endure it, makes a body ascend contrary
to the powerful solicitation of its gravity. Even
Galileo, with all his sagacity, could not for some time
hit upon any thing more satisfactory; for he only
assigned a limit to this dread of vacuity: having observed
that sucking pumps would not raise water
higher than 16 brasses, or 34 English feet, he limited
this abhorring force of nature, to one that was equivalent
to the weight of a column of water 34 feet high,
on the same base as the void space. Consequently he
pointed out a way of making a vacuum, by means of
a hollow cylinder, whose piston is charged with a weight
sufficient to detach it from the close bottom turned
upwards: this effort he called the measure of the force
of vacuity, and made use of it for explaining the cohesion
of the parts of bodies.</p><p>Galileo however was well apprised of the weight of
the air as a body: in his Dialogues he shews two ways
of demonstrating it, by weighing it in bottles: the
transition was easy from one discovery to another: yet
still Galileo's knowledge of the matter was imperfect,
that is, as to the particular instance of the suspension
of a fluid above its level, by the pressure of the external
air.</p><p>At length Torricelli fell upon the lucky guess, that
the counterpoise which keeps fluids above their level,
when nothing presses upon their internal surface, is the
mass of air resting upon the external one. He discovered
it in the following manner: In the year 1643
this disciple of Galileo, on occasion of executing an
experiment on the vacuum formed in pumps, above
the column of water, when it exceeds 34 feet, thought
of using some heavier fluid, such as quicksilver. He
conceived that whatever might be the cause by which
a column of water of 34 feet high is sustained above
its level, the same force would sustain a column of any
other fluid, which weighed as much as that column of
water, on the same base; whence he concluded that
quicksilver, being about 14 times as heavy as water,
would not be sustained higher than 29 or 30 inches.
<pb n="50"/><cb/>
He therefore took a glass tube of several feet in length,
sealed it hermetieally at one end, and filled it with quicksilver;
then inverting it, and holding it upright, by
pressing his finger against the lower or open orifice, he
immersed that end in a vessel of quicksilver; then removing
his singer, and suffering the sluid to run out,
the event verisied his conjecture; the quicksilver, faithful
to the laws of hydrostatics, descended till the column
of it was about 30 inches high above the surface of that
in the vessel below. And hence Torricelli concluded
that it was no other than the weight of the air incumbent
on the surface of the external quicksilver, which
counterbalanced the fluid contained in the tube.</p><p>By this experiment Torricelli not only proved, what
Galileo had done before, that the air had weight, but
also that it was its weight which kept water and quicksilver
raised in pumps and tubes, and that the weight
of the whole column of it was equal to that of a like
column of quicksilver of 30 inches high, or of water
34 or 35 feet high; but he did not ascertain the weight
of any particular quantity of it, as a gallon, or a cubic
foot of it, nor its specific gravity to water, which had
been done by Galileo, though to be sure with no great
accuracy, for he only proved that water was more than
400 times heavier than air.</p><p>Torricelli's experiment became famous in a short time.
Father Mersenne, who kept up a correspondence with
most of the literati in Italy, was informed of it in 1644,
and communicated it to those of France, who presently
repeated the experiment: Messrs. Pascal and Petit made
it first, and varied it several ways; which gave occasion
to the ingenious treatise which Pascal published at 23
years of age, intitled <hi rend="italics">Experiences Nouvelles touchant la
Vuide.</hi> In this treatise indeed he makes use of the old
principle of <hi rend="italics">suga vacui;</hi> but afterwards getting some
notion of the weight of the air, he soon adopted Torricelli's
idea, and devised several experiments to consirm
it. One of these was to procure a vacuum above the
reservoir of quicksilver; in which case he found the
column sink down to the common level: but this appearing
to him not sufficiently powerful to dissipate the
prejudices of the ancient philosophy, he prevailed on
M. Perier, his brother-in-law, to execute the famous
experiment of Puy-de-Domme, who found that the
height of the quicksilver half-way up the mountain was
less, by some inches, than at the foot of it, and still less
at the top: so that it was now put out of doubt that
it was the weight of the atmosphere which counterpoised
the quicksilver.</p><p>Des Cartes too had a right notion of this effect of
the air, to sustain fluids above their level, as appears by
some of his letters about this time, and some years before;
and in one of those he lays claim to the idea of
the Puy-de-Domme experiment: After having desired
M. de Carcavi to inform him of the success of that
experiment, which public rumour had advertised him
had been made by M. Pascal himself, he adds, &#x201C;I
had reason to expect this from him, rather than
from you, because I first proposed it to him two years
since, assuring him at the same time, that although I
had not tried it, yet I could not doubt of the consequence;
but as he is a friend of M. Roberval, who
professes himself no friend to me, I suppose he is guided
<cb/>
by that gentleman's passions.&#x201D; See more of this history
under <hi rend="smallcaps">Barometer.</hi></p><p>As to the actual weight of any given portion of
common air, it seems that Galileo was the first who
determined it experimentally; and he gives two different
methods, in his Dialogues, for weighing it in bottles:
he did not however perform the experiment very
accurately, as he stated from the result that the gravity
of water was to that of air rather above 400 to 1.</p><p>A quantity of air was next weighed by Mersenne in
a very ingenious manner. His idea was to weigh a
vessel both when full of air, and when emptied of it a
to make the vacuum for this purpose, he knew no better
way than by expelling the air out of an colipile by
heating it red hot: by weighing it both when cold
and hot, he sound a certain difference; which however
was not the exact weight of that capacity of air, because
the vacuum was not perfect. But by plunging
the eolipile, when red hot, into water, just so much
water entered as was equal in bulk to the air that had
been expelled; then he took it out and weighed it
with the water, which gave the weight of the same bulk
of water; and on comparing this with the former difference,
or weight of air expelled, he found their proportion
to be as 1300 to 1. Which is as wide of the
truth as Galileo's proportion, namely 400 to 1, but the
contrary way. And it is remarkable that the mean
between the two, namely 850 to 1, is very near the
true proportion as settled by other more accurate experiments.</p><p>Mr. Boyle, by a more accurate experiment, found the
proportion to be that of 938 to 1. And Mr. Hauksbee
found it as 850 to 1, proceeding on the same principles
as Mersenne, with a three-gallon glass bottle,
but extracting the air out of it with the air pump, instead
of expelling it by fire; the height of the barometer
being at that time 29.7 inches. Also by other accurate
experiments made before the Royal Society by
Mr. Hauksbee, Dr. Halley, Mr. Cotes, and others,
the proportion was always between 800 and 900 to 1,
but rather nearer the latter, namely, being first found
as 840 to 1, then as 852 to 1, and a third time as 860
to 1; the barometer then standing at 29 3/4 inches, and
the weather warm. Mr. Cavendish determines the
ratio 800 to 1, the barometer being 29 3/4, and the thermometer
at 50&#xB0;; and Sir George Shuckburgh, by a
very accurate experiment, finds it 836 to 1, the barometer
being at that time at 29.27, and the thermometer
at 51&#xB0;. And the medium of all these is about 832 or
or 833 to 1, when reduced to the pressure of 30 inches
of the barometer, and the mean temperature 55&#xB0; of the
thermometer. Upon the whole therefore it may be
safely concluded that, when the barometer is at 30
inches, and the thermometer at the mean temperature
55&#xB0;, the density or gravity of water is to that of air, as
833 1/3 to 1, that is as 2500/3 to 1, or as 2500 to 3; and
that for any changes in the height of the barometer,
the ratio varies proportionally; and also that the density
of the air is altered by the (1/440)th part for every degree
of the thermometer above or below temperate.</p><p>This number, which is a very good medium among
them all, I have chosen with the fraction 1/37, because it
gives exactly 1 1/5 ounce for the mean weight of a cubis
<pb n="51"/><cb/>
soot of air, the weight of the cubic foot of water being
just 1000 ounces averdupois, and that of quicksilver
equal to 13600 ounces.</p><p><hi rend="italics">Air,</hi> then, having been shewn to be a heavy fluid substance,
the laws of its gravitation and pressure must be
the same as those of water and other fluids; and consequently
its pressure must be proportional to its perpendicular
altitude. Which is exactly conformable to
experiment; for on removing the Terricellian tube to
different heights, where the column of air is shorter, the
column of quicksilver which it sustains is shorter also,
and that nearly at the rate of 100 feet for 1/10 of an
inch of quicksilver. And on these principles depend the
structure and use of the barometer.</p><p>And from the same principle it likewise follows that
air, like other fluids, presses equally in all directions.
And hence it happens that soft bodies endure this pressure
without change of figure, and hard or brittle
bodies without breaking; being equally pressed on all
parts; but if the pressure be taken off, or diminished,
on one side, the effect of it is immediately perceived
on the other. See <hi rend="smallcaps">Atmosphere</hi>, for the total quantity
of effects and pressure, and the laws of different
altitudes, &amp;c.</p><p>From the weight and fluidity of the air, jointly considered,
many effects and uses of it may easily be deduced.
By the combination of these two qualities, it
closely invests the earth, with all the bodies upon it,
constringing and binding them down with a great force,
namely a pressure equal to about 15 pounds upon every
square inch. Hence, for example, it prevents the arterial
vessels of plants and animals from being too much
distended by the impetus of the circulating juices, or
by the elastic force of the air so copiously abounding
in them. For hence it happens, that on a diminution
of the pressure of the air, in the operation of cupping,
we see the parts of the body grow tumid, which causes
an alteration in the circulation of the fluids in the capillary
vessels. And the same cause hinders the fluids
from transpiring through the pores of their containing
vessels, which would otherwise cause the greatest debility,
and often destroy the animal. To the same two qualities
of the air, weight and fluidity, is owing the mixture
of bodies contiguous to one another, especially
fluids; for several liquids, as oils and salts, which readily
mix of themselves in air, will not mix at all in vacuo.
With many other natural phenomena.</p><p>3. <hi rend="italics">Elasticity.</hi> Another quality of the air, from
whence arise a multitude of effects, is its elasticity; a
quality by which it yields to the pression of any other
bodies, by contracting its volume; and dilates and expands
itself again on the removal or diminution of the
pressure. This quality is the chief distinctive property
of air, the other two being common to other fluids
also.</p><p>Of this property we have innumerable instances.
Thus, for example, a blown bladder being squeezed
in the hand, we find a sensible resistance from the included
air; and upon taking off the piessure, the compressed
parts immediately restore themselves to their
former round sigure. And on this property of elasticity
depend the structure and uses of the air-pump.</p><p>Every particle of air makes a continual effort to
dilate itself, and so it acts forcibly against all the neigh-
<cb/>
bouring particles, which also exert the like force in
return; but if their resistance happen to cease, or be
weakened, the particle immediately expands to an immense
extent. Hence it is that thin glass bubbles, or
bladders, filled with air, and placed under the receiver
of an air-pump, do, upon pumping out the air, burst
asunder by the force of the air which they contain. So
likewise a close flaccid bladder, containing only a small
quantity of air, being put under the receiver, swells as
the receiver is exhausted, and at length appears quite
full. And the same thing happens by carrying the
flaccid bladder to the top of a very high mountain.</p><p>The same experiment shews that this elastic property
of the air is very different from the elasticity of solid
bodies, and that these are dilated after a different manner
from the air. For when air ceases to be compressed,
it not only dilates, but then occupies a far greater space,
and exists under a volume immensely greater than before;
whereas solid elastic bodies only resume the figure
they had before they were compressed.</p><p>It is plain that the weight or pressure of the air does
not at all depend on its elasticity, and that it is neither
more nor less heavy than if it were not at all elastic.
But from its being elastic, it follows that it is susceptible
of a pressure, which reduces it to such a space, that
the force of its elasticity, which re-acts against the
pressing weight, is exactly equal to that weight. Now
the law os the elasticity is such, that it increases in
proportion to the density of the air, and that its density
increases in proportion to the forces or weights which
compress it. But there is a necessary equality between
action and re-action; that is, the gravity of the air,
which effects its compression, and the elasticity of it,
which gives it its tendency to expansion, are equal.</p><p>So that, the elasticity increasing or diminishing, in
the same proportion as the density increases or diminishes,
that is, as the distance between its particles decrease
or increase; it is no matter whether the air be
compressed, and retained in any space, by the weight
of the atmosphere, or by any other cause; as in either
case it must endeavour to expand with the same force.
And therefore, if such air as is near the earth be inclosed
in a vessel, so as to have no communication with
the external air, the pressure of such inclosed air will
be exactly equal to that of the whole external atmosphere.
And accordingly we find that quicksilver is
sustained to the same height, by the elastic force of
air inclosed in a glass vessel, as by the whole pressure of
the atmosphere.&#x2014;And on this principle of the condensation
and elasticity of the air, depends the structure
and use of the air-gun.</p><p>That the density of the air is always directly proportional
to the force or weight which compresses it,
was proved by Boyle and Mariotte, at least as far as
their experiments go on this head: and Mr. Mariotte
has shewn that the same rule takes place in condensed
air. However, this rule is not to be admitted as scrupulously
exact; for when air is very forcibly compressed,
so as to be reduced to (1/4)th of its ordinary bulk, the
effect does not answer precisely to the rule; for in this
case the air begins to make a greater resistance, and requires
a stronger compression, than according to the
rule. And hence it would seem, that the particles of
air cannot, by means of any possible weight or pressure,
<pb n="52"/><cb/>
how great soever, be brought into perfect contact, or
that it cannot thus be reduced to a solid mass; and consequently
that there must be a limit to which this con
densation of the air can never arrive. And the same
remark is true with regard to the rarefaction of air,
namely, that in very high degrees of rarefaction, the
elasticity is decreased rather more than in proportion to
the weight or density of the air: and hence there must
also be a limit to the rarefaction and expansion of the
air, by which it is prevented from expanding to infinity.</p><p>We know not however how to assign those limits to
the elasticity of the air, nor to destroy or alter it,
without changing the very nature of air, which is effected
by chemical processes. To what degree air is
susceptible of condensation, by compression, is not certainly
known. Mr. Boyle condensed it 13 times more
than in its natural state, by this means: others have
compressed it into (1/70)th part of its ordinary volume;
Dr. Hales made it 38 times more dense, by means of a
press; but by freezing water in a hollow cast-iron ball
or shell, he reduced it to 1838 times less space than it
naturally occupies; in which state it must have been
of more than twice the density or specific gravity of
water: And as water is not compressible, except in a
very small degree, it follows from this experiment, that
the particles of air must be of a nature very different
from those of water; since it would otherwise be impossible
to reduce air to a volume above 800 times less
than in its common state; an inference however which
militates directly against an assertion made by Dr. Halley,
from some experiments performed in London, and
others at Florence by the Academy del Cimento,
namely, that it may be safely concluded that no force
whatever is capable to reduce air into a space 800 times
less than that which it naturally occupies near the surface
of the earth.</p><p>The elasticity of the air exerts its force equally in
all directions; and when it is at liberty, and freed from
the cause which compressed it, it expands equally in all
directions, and in consequence always assumes a spherical
figure in the interstices of the fluids in which it is
lodged. This is evident in liquors placed in the receiver
of an air pump, by exhausting the air; at first there
appears a multitude of exceeding small bubbles, like
grains of fine sand, dispersed through the fluid mass,
and rising upwards; and as more air is pumped out,
they enlarge in size; but still they continue round.
Also if a plate of metal be immerged in the liquor, on
pumping, its surface will be seen covered over with
small round bubbles, composed of the air which adhered
to it, now expanding itself. And for the same
reason it is that large glass globes are always blown up
of a spherical shape, by blowing air through an iron
tube into a piece of melted glass at the end of the
pipe.</p><p>The expansion of the air, by virtue of its elastic
property, when only the compressing force is taken off,
or diminished, is found to be surprisingly great; and
yet we are far from knowing the utmost dilatation
of which it is capable. In several experiments made
by Mr. Boyle, it expanded first into 9 times its former
space; then into 31 times; then into 60, and then into
150 times. Afterwards, it was brought to dilate into
<cb/>
8000 times its first space; then into 10000, and at last
even into 13679 times its space; and this solely by its
own natural expansive force, by only removing the pressure,
but without the help of fire. And on this principle
depends the construction and use of the M<hi rend="smallcaps">ANOMETER.</hi></p><p>The elasticity of the air, under one and the same
pressure, is still farther increased by heat, and diminished
by cold, and that, by some late accurate experiments
made by Sir George Shuckburgh, at the rate of the
440th part of its volume nearly, for each degree of the
variation of heat, from that of temperate, in Fahrenheit's
thermometer.</p><p>Mr. Hauksbee observed that a portion of air inclosed
in a glass tube, when the temperature was at the
freezing point, formed a volume which was to that of
the same quantity of air in the greatest heat of summer
here in England, as 6 to 7. And it has been found by
several experiments, that air is expanded 1/3 of its natural
bulk by applying the heat of boiling water
to it.</p><p>Dr. Hales found that the air in a retort, when the
bottom of the vessel just became red hot, was dilated
into twice its former space; and that in a white, or
almost melting heat, it filled thrice its former space:
but Mr. Robins found that air was expanded, by means
of the white or fusing heat of iron, to 4 times its former
bulk.</p><p>See several ingenious experiments on the elasticity of
the air, in the Philos. Trans. for the year 1777, by Sir
George Shuckburgh and Colonel Roy.</p><p>This properly explains the common effect observed
on bringing a close flaccid bladder near the fire to
warm it; when it is presently found to swell as if more
air were blown into it. And upon this principle depends
the structure and office of the thermometer; as
also the air balloons, lately invented by Mr. Montgolfier,
for floating in the atmosphere.</p><p>M. Amontons first discovered that, with the same degree
of heat, air will expand in a degree proportioned
to its density. And on this foundation the ingenious
author has formed a discourse, to prove &#x201C;that the
spring and weight of the air, with a moderate degree
of warmth, may enable it to produce even earthquakes,
and others of the most vehement commotions of nature.&#x201D;
He computes that at the depth of the 74th
part of the earth's radius below the surface, the natural
pressure of the air would reduce to the density of
gold; and thence infers that all matter below that
depth, is probably heavier than the heaviest metal that
we know of. And hence again, as it is proved that the
more the air is compressed, the more does the same degree
of fire increase the force of its elasticity; we may
infer that a degree of heat, which in our orb can produee
only a moderate effect, may have a very violent
one in such lower orb; and that, as there are many degrees
of heat in nature, beyond that of boiling water,
it is probable there may be some whose violence, thus
assisted by the weight of the air, may be sufficiently
powerful to tear asunder the solid globe. Mem. de
l'Acad. 1703.</p><p>Many philosophers have supposed that the elastic
property of the air depends on the figure of its corpuscles,
which they take to be ramous: some maintain
<pb n="53"/><cb/>
that they are so many minute <hi rend="italics">flocculi,</hi> resembling fleeces
of wool: others conceive them rolled up like hoops,
and curled like wires, or shavings of wood, or coiled
like the springs of watches, and endeavouring to expand
themselves by virtue of their texture.</p><p>But Sir Isaac Newton (Optics, Qu. 31, &amp;c.) explains
the matter in a different way; such a contexture
of parts he thinks by no means sufficient to account
for that amazing power of elasticity observed in air,
which is capable of dilating itself into above a million
of times more space than it occupied before: but, he
observes, as it is known that all bodies have an attractive
and a repelling power; and as both these are stronger
in bodies, the denser, more compact, and solid they are;
hence it follows that when, by heat, or any other powerful
agent, the attractive force is overcome, and the
particles of the body separated so far as to be out of
the sphere of attraction; the repelling power, then
commencing, makes them recede from each other with
a strong force, proportionable to that with which they
before cohered; and thus they become permanent air.</p><p>And hence, he says, it is, that as the particles of air
are grosser, and rise from denser bodies, than those of
transient air, or vapour, true air is more ponderous
than vapour, and a moist atmosphere lighter than a dry
one.</p><p>And M. Amontons makes the elasticity of air to
arise from the fire it contains; so that by augmenting
the degree of heat, the rarefaction will be increased to
a far greater degree than by a mere spontaneous dilatation.</p><p>The elastic power of the air becomes the second
great source of the remarkable effects of this important
fluid. By this property it insinuates itself into the
pores of bodies, where, by means of this virtue of expanding,
which is so easily excited, it must put the particles
of those bodies into perpetual vibrations, and maintain
a continual motion of dilatation and contraction
in all bodies, by the incessant changes in its gravity
and density, and consequently its elasticity and expansion.</p><p>This reciprocation is observable in several instances,
particularly in plants, in which the trache&#xE6; or air-vessels
perform the office of lungs; for as the heat increases
or diminishes, the air alternately dilates and contracts,
and so by turns compresses the vessels, and eases them
again; thus promoting a circulation of their juices.
And hence it is found that no vegetation or germination
is carried on in vacuo.</p><p>It is from the same cause too, that ice is burst by the
continual action of the air contained in its bubbles.
Thus, too, glasses and other vessels are frequently cracked,
when their contained liquors are frozen; and thus
also large blocks of stone, and entire columns of marble,
sometimes split in the winter season, from some
little bubble of included air acquiring an increased
elasticity: and for the same reason it is that so few
stones will bear to be heated by a fire, without cracking
into many pieces, by the increased expansive force of
some air confined within their pores. From the same
source arise also all putrefaction and fermentation;
neither of which can be carried on in vacuo, even in
the best disposed subjects. And even respiration, and
animal-life itself, are supposed, by many authors, to be
<cb/>
conducted, in a great measure, by the same principle
of the air. And as we find such great quantities of
air generated by the solution of animal and vegetable
substances, a good deal must constantly be raised from
the dissolution of these clements in the stomach and
bowels.</p><p>In fact, all natural corruption and alteration seem
to depend on air; and even metals, particularly gold,
only seem to be durable and incorruptible, in so far as
they are impervious to air.</p><p>As to the different kinds of air, with its generation,
and the effects of different ingredients of it, &amp;c, they
are omitted here, as properly belonging to a Chemical
Dictionary, or to a General Dictionary of Arts, &amp;c.</p><p>For the resistance of the air, see <hi rend="smallcaps">Resistance.</hi></p><p><hi rend="smallcaps">Air-Gun</hi>, in <hi rend="smallcaps">Pneumatics</hi>, is a machine for propelling
bulleto with great violence, by the sole means
of condensed air.</p><p>The first account we meet with of an air-gun, is in
the <hi rend="italics">Elemens d'Artillerie</hi> of David Rivaut, who was preceptor
to Louis XIII. of France. He ascribes the
invention to one Marin, a burgher of Lisieux, who
presented one to Henry IV.</p><p>To construct a machine of this kind, it is only necessary
to take a strong vessel of any sort, into which
the air is to be thrown or condensed by means of a
syringe, or otherwise, the more the better; then a
valve is suddenly opened, which lets the air escape by
a small tube in which a bullet is placed, and which is
thus violently forced out before the air.</p><p>It is evident then that the effect is produced by
virtue of the elastic property of the air; the force of
which, as has been shewn in the last article, is directly
proportional to its condensation; and therefore the
greater quantity that can be forced into the engine, the
greater will be the effect. Now this effect will be exactly
similar to that of a gun charged with powder,
and therefore we can easily form a comparison between
them: for inflamed gun-powder is nothing more than
very condensed elastic air; so that the two forces are
exactly similar. Now it is shewn by Mr. Robins, in
his New Principles of Gunnery, that the fluid of inflamed
gun-powder, has, at the first moment, a force of
elasticity equal to about a 1000 times that of common
air; and therefore it is necessary that air should
be condensed a 1000 times more than in its natural
state, to produce the same effect as gun-powder.
But then it is to be considered, that the velocities with
which equal balls are impelled, are directly proportional
to the square roots of the forces; so that if the
air in an air-gun be condensed only 10 times, then the
velocity it will project a ball with, will be, by that rule,
(1/10)th of that arising from gun-powder; and if the air
were condensed 20 times, it would communicate a velocity
of 1/7 of that of gun-powder. But in reality the
air-gun shoots its ball with a much greater proportion
of velocity than as above, and for this reason, namely,
that as the reservoir, or magazine of condensed air, is
commonly very large in proportion to the tube which
contains the ball, its density is very little altered by expanding
through that narrow tube, and consequently
the ball is urged all the way by nearly the same uniform
force as at the first instant; whereas the elastic fluid
arising from inflamed gun-powder is but very small in
<pb n="54"/><cb/>
proportion to the tube or barrel of the gun, occupying
at first indeed but a very small portion of it next the
but-end: an dtherefore by dilating into a comparatively
large space, as it urges the ball along the barrel, its
elastic force is proportionally weakened, and it acts
always less and less on the ball in the tube. From
which cause it happens, that air condensed into a good
large machine only 10 times, will shoot its ball with a
velocity but little inferior to that given by the gunpowder.
And if the valve of communication be suddenly
shut again by a spring, after opening it to let some
air escape, then the same collection of it may serve to
impel many balls, one after another.</p><p>In all cases in which a considerable force is required,
and consequently a great condensation of air, it will be
requisite to have the condensing syringe of a small bore,
perhaps not more than half an inch in diameter: otherwise
the force to produce the compression will become
so great, that the operator cannot work the machine:
for, as the pressure against every square inch is about
15 pounds, and against every circular inch about 12
pounds, if the syringe be one inch in diameter, when
one atmosphere is injected, there will be a resistance of
12 pounds against the piston; when 2, of 24 pounds;
and when 10 are injected, there will be a force of 120
pounds to overcome; whereas 10 atmospheres act
against the half-inch piston, whose area is but 1/4 of the
former, with 1/4 of the force only, namely, 30 pounds;
and 40 at mospheres may be injected with such a syringe,
as well as ten with the larger.</p><p>There are air-guns of various constructions; an easy
and portable one is represented in Plate II, fig. 1. which is a
section lengthways through the axis, to shew the inside.
It is made ofbrass, and has two barrels; the inner barrel
D A of a small bore, from which the bullets are
shot; and a larger barrel ESCDR, on the outside of
it. In the stock of the gun there is a syringe MNPS,
whose rod M draws out to take in air; and by pushing
it in again, the piston SN drives the air before it,
through the valve PE into the cavity between the two
barrels. The ball K is put down into its place in the
small barrel, with the rammer, as in another gun.
There is another valve at SL, which, being opened by
the trigger O, permits the air to come behind the ball,
so as to drive it out with great force. If this valve be
opened and shut suddenly, one charge of condensed air
may make several discharges of bullets; because only
part of the injected air will then go out at a time, and
another bullet may be put into the place K: but if
the whole air be discharged on a single bullet, it will
impel it more forcibly. This discharge is effected by
means of a lock (fig. 2) when fixed to its place as
usual in other guns; for the trigger being pulled, the
cock will go down and drive a lever which opens the valve.</p><p>Dr. Macbride (Exper. Ess. p. 81) mentions an improvement
of the air-gun, made by Dr. Ellis; in which
the chamber sor containing the condensed air is not
in the stock, which renders the machine heavy and
unweildy, but has five or six hollow spheres belonging
to it, of about 3 inches diameter, sitted to a screw on
the lock of the gun. These spheres are contrived with
valves, to consine the air which is forced into their
cavities, so that a servant may carry them ready
charged with condensed air: and thus the gun of this
<cb/>
construction is rendered as light and portable as one of
the smallest fowling.pieces.</p><p>Fig. 3 represents one made by the late Mr. B.
Martin of London, and now by several of the mathematical
instrument and gun-makers of the metropolis;
which, for simplicity and perfection, perhaps exceeds
any other that has been contrived. A is the gun-barrel,
of the size and weight of a common fowling-piece,
with the lock, stock, and ramrod. Under the lock, at
<hi rend="italics">b,</hi> is a round steel tube, having a small moveable pin in
the inside, which is pushed out when the trigger <hi rend="italics">a</hi> is
pulled, by the springwork within the lock; to this
tube <hi rend="italics">b</hi> is serewed a hollow copper ball, perfectly airtight.
This copper ball is fully charged with condensed
air by means of a syringe, previous to its being
applied to the tube <hi rend="italics">b.</hi> Hence, if a bullet be rammed
down in the barrel, the copper ball screwed fast at <hi rend="italics">b,</hi>
and the trigger <hi rend="italics">a</hi> be pulled; then the pin in <hi rend="italics">b</hi> will
forcibly push open a valve within the copper ball, and
let out a portion of the condensed air; which air will
rush up through the aperture of the lock, and forcibly
act against the bullet, driving it to the distance of 60
or 70 yards, or farther. If the air be strongly condensed
at every discharge, only a portion of the air
escapes from the ball; therefore, by re-cocking the
piece, another discharge may be made; and this
repeated 15 or 16 times. An additional barrel is
sometimes made, and applied for the discharge of shot,
instead of the ball above described.</p><p>Sometimes the syringe is applied to the end of the
barrel C (fig. 4); the lock and trigger shut up in a brass
case <hi rend="italics">d</hi>; and the trigger pulled, or the discharge made,
by pulling the chain <hi rend="italics">b.</hi> In this contrivance there is a
round chamber for the condensed air at the end of the
spring at <hi rend="italics">e,</hi> and it has a valve acting in a similar manner
to that of the copper ball. When this instrument is
not in use, the brass case <hi rend="italics">d</hi> is made to slide off, and the
instrument then becomes a walking stick: from which
circumstance, and the barrel being made of cane, or
brass, &amp;c, it has been called the <hi rend="italics">Air-cane.</hi> The head
of the cane unscrews and takes off at <hi rend="italics">a,</hi> where the extremity
of the piston-rod in the barrel is shewn. An
iron rod is placed in a ring at the end of this, and
the air is condensed in the barrel in a manner similar to
that of the gun as above; but its force and action is
not near so strong as in the gun.</p><p><hi rend="smallcaps">Magazine Air</hi>-<hi rend="italics">Gun.</hi> This is an improvement of
the common air-gun, made by an ingenious artist, called
L. Colbe. By his contrivance, ten bullets are so lodged
in a cavity, near the place of discharge, that they may
be successively drawn into the barrel, and shot so quickly
as to be nearly of the same use as so many different
guns; the only motion required, after the air has been
injected, being that of shutting and opening the hammer,
and cocking and pulling the trigger. Fig. 3 is a
longitudinal section of this gun, as large in every part
as the gun itself; and as much of its length is shewn
as is peculiar to this construction; the rest of it being
like the ordinary air-gun. EE is part of the stock;
G is the end of the injecting syringe, with its valve H,
opening into the cavity FFFF between the barrels,
KK is the small or shooting barrel, which receives the
bullets, one at a time, from the magazine DE, being a
serpentine cavity, in which the bullets <hi rend="italics">b, b, b,</hi> &amp;c, are
<pb/><pb/><pb/><pb/><pb n="55"/><cb/>
lodged, and closed at the end D; from whence, by one
motion of the hammer, they are brought into the barrel
at I, and thence are shot out by the opening of the
valve V, which lets in the condensed air from the cavity
FFF into the channel VKI, and so along the inner
barrel KKK, whence the bullet is discharged. <hi rend="italics">s</hi> I <hi rend="italics">si</hi>
M <hi rend="italics">k</hi> is the key of a cook, having a hole through it;
which hole, in the present situation, makes part of the
barrel KK, being just of the same bore: so that the
air, which is let in at every opening of the valve V,
comes behind this cock, and taking the ball out of it,
carries it forward, and so out of the mouth of the
piece.</p><p>To bring in another bullet to succeed I, which is
done in an instant, bring the cylindrical cavity of the
key of the cock, which made part of the barrel KKK,
into the situation <hi rend="italics">ik,</hi> so that the part I may be at K;
then turning the gun upside-down, one bullet next the
cock will fall into it out of the magazine, but will go
no farther into this cylindrical cavity, than the two
little pieces <hi rend="italics">ss</hi> will permit it; by which means only
one bullet at a time will be taken in to the place I, to
be discharged again as before.</p><p>A more particular description of the several parts
may be seen in Desaguliers' Exper. Philos. vol. ii. pa.
399 et seq.</p><p><hi rend="smallcaps">Air-Pump</hi>, in <hi rend="italics">Pneumatics,</hi> is a machine for exhausting
the air out of a proper vessel, and so to make
what is commonly called a vacuum; though in reality
the air in the receiver is only rarefied to a great degree,
so as to take off the ordinary effects of the atmosphere.
So that by this machine we learn, in some measure,
what our earth would be without air; and how
much all vital, generative, nutritive, and alterative
powers depend upon it.</p><p>The principle on which the air-pump is constructed,
is the spring or elasticity of the air; as that on which
the common, or water pump is formed, is the gravity
of the same air: the one gradually exhausting the air
from a vessel by means of a piston, with a proper valve,
working in a cylindrical barrel or tube; and the other
exhausting water in a similar manner.</p><p>The air-pump has proved one of the principal
means of performing philosophical discoveries, that has
been invented by the moderns. The idea of such a
machine occurred to several persons, nearly about the
same time. But the first it seems was completed by
Otto Guericke, the celebrated consul of Magdeburg,
who exhibited his first public experiments with it, before
the emperor and the states of Germany, at the
breaking up of the imperial diet at Ratisbon, in the
year 1654. But it was not till the year 1672 that
Guericke published a description of the instrument,
with an account of his experiments, in his Experimenta
Nova Magdeburgica de Vacuo Spacia: though an account
of them had been published by Schottus in 1657,
in his Mechanica Hydraulico Pneumatica.</p><p>Dr. Hook and M. Duhamel ascribe the invention of
the air-pump to Mr. Boyle. But that great man frankly
confesses that Guericke was beforehand with him in
the execution. Some attempts, he assures us, he had
indeed made upon the same foundation, before he knew
any thing of what had been done abroad: but the information
he afterwards received from the account given
<cb/>
by Schottus, enabled him, with the assistance of Dr.
Hook, aster two or three unsuccessful trials, to bring
his design to maturity. The product of their labours
was a new air-pump, much more easy; convenient, and
manageable, than the German one. And hence, or
rather from the great variety of experiments to which
this illustrious author applied the machine, it was afterwards
called <hi rend="italics">Machina Boyliana,</hi> and the vacuum produced
by it, <hi rend="italics">Vacuum Boylianum.</hi></p><p><hi rend="italics">Structure of the Air-Pump.</hi> Most of the air-pumps
that were first made, consisted of only one barrel, or
hollow cylinder of brass, with a valve at the bottom,
opening inwards; and a moveable embolus or piston,
having likewise a valve opening upwards, and so exactly
fitted to the barrel, that when it is drawn up
from the bottom, by means of an indented iron rod or
rack, and a handle turning a small indented wheel,
playing in the teeth of that rod, all the air will be
drawn up from the cavity of the barrel: there is also
a small pipe opening into the bottom of the barrel, by
means of which it communicates with any proper
vessel to be exhausted of air, which is called a receiver,
from its office in receiving the subjects upon which experiments
are to be made in vacuo: the whole being
fixed in a convenient frame of wood-work, where the
end of the pipe turns up into a horizontal plate, upon
which the receiver is placed, just over that end of the
pipe.</p><p>The other parts of the machine, being only accidental
circumstances, chiefly respecting conveniency,
have been diversified and improved from time to time,
according to the address and several views of the
makers. That of Otto Guericke was very rude and
inconvenient, requiring the labour of two strong men,
for more than two hours, to extract the air from a glass,
which was also placed under water; and yet allowed of
no change of subjects for experiments.</p><p>Mr. Boyle, from time to time, removed several of
these inconveniences, and lessened others: but still the
working of his pump, which had but one barrel, was
laborious, by reason of the pressure of the atmosphere,
a great part of which was to be removed at every lift
of the piston, when the exhaustion was nearly completed.
Various improvements were successively made
in the machine by the philosophers about that time,
and foon after, who cultivated this new and important
branch of pneumatics; as Papin, Mersenne, Mariotte,
and others; but still they laboured under a difficulty of
working them, from the circumstance of the single
barrel, till Papin, in his farther improvements of the
air-pump, removed that inconvenience, by the use of a
second barrel and piston, contrived to rise, as the other
fell, and to sall as that rose; by which, and the great
improvements made by Mr. Hauksbee, the pressure of
the atmosphere on the descending piston, always nearly
balanced that of the ascending one; so that the winch,
which worked them up and down, was easily moved by
a very gentle force with one hand; and besides, the
exhaustion was hereby made in less than half the time.</p><p>Some of the Germans, and others likewise, made
improvements in the air-pump, and contrived it to
perform the counter office of a condenser, in order to
examine the properties of the air depending on its condensation.
<pb n="56"/><cb/></p><p>Mr. Boyle contrived a mercurial gauge or index to the
air-pump, which is described in his first and second
Physico-Mechanical Continuations, for measuring the
degrees of the air's rarefaction in the receiver. This
gauge is similar to the barometer, being a long glass
tube, having its lower end immersed in an open bason
of quicksilver, but its other end, which was open also,
communicating with the receiver: which being exhausted,
this tube is equally exhausted of air at the
same time, and the external air presses the quicksilver
up into the tube, to a height proportioned to the degree
of exhaustion.</p><p>Mr. Vream, an ingenious pneumatic operator, made
an improvement in Hauksbee's air-pump, by reducing
the alternate up-and-down motion of the hand and
winch to a circular one. In his method, the winch is
turned quite round, and yet the pistons are alternately
raised and depressed: by which the trouble of shifting
the hand backwards and forwards, as well as the loss
of time, and the shaking of the pump, are prevented.</p><p>The air-pump, thus improved, is represented in plate
III. fig. 1; where <hi rend="italics">oo</hi> is the receiver to be exhausted,
ground truly level at the bottom, set over a hole in the
plate, from which descends the bent pipe <hi rend="italics">hh</hi> to the
cistern <hi rend="italics">dd,</hi> with which the two barrels <hi rend="italics">aa</hi> communicate,
in which the pistons are worked by a toothed wheel,
by turning the handle <hi rend="italics">bb</hi>; by which the racks <hi rend="italics">cc,</hi>
with the pistons, are worked alternately up and down.
<hi rend="italics">ll</hi> is the gauge tube, immersed in a bason of quicksilver
<hi rend="italics">m</hi> at bottom, and communicating with the receiver at
top; from which however it may be occasionally disengaged,
by turning a cock. And <hi rend="italics">n</hi> is another cock,
by turning of which, the air is again let in to the exhausted
receiver; into which it is heard to rush with a
considerable hissing noise.</p><p>Notwithstanding the great excellency of Mr. Hauksbee's
air-pump, it was still subject to inconveniences,
from which it was in a great measure relieved by some
contrivances of Mr. Smeaton, which are described at
large in the Philos. Trans. for the year 1752. The
principal improvements suggested by Mr. Smeaton, relate
to the gauge, the valves of the piston, and the
piston going closer down to the bottom of the barrel;
for his pump has only one. By the last of these, the
air was extracted more perfectly at each stroke. By
the second, he remedied an inconvenience arising from
the valve hole of the piston being too wide properly to
support the bladder valve which covered it: instead of
the usual circular orisice, Mr. Smeaton perforated the
piston with seven small and equal hexagonal holes, one
in the centre, and the other six around, forming together
the appearance of a transverse section of a honeycomb;
the bars or divisions between which, served to
support the pressure of the air on the valve. His gage
consists of a bulb of glass, of a pear-like shape, and
capable of holding about half a pound of quicksilver:
it is open at the lower end, the other terminating in
a tube hermetically sealed; and it has annexed to it a
seale, divided into parts of about 1/10 of an inch, and
answering to the 1000th part of the whole capacity.
During the exhaustion of the receiver, the gage is
suspended in it by a wire; but when the pump has been
worked as much as necessary, the gage is pushed down,
till the open end be immersed in a bason of quicksilver
<cb/>
placed underneath. The air is then let into the receiver
again, and the quicksilver driven by it from the bason,
up into the gauge, till the air remaining in it become
of the same density as the air without; and as the air
always takes the highest place, the tube being uppermost,
the expansion will be determined by the number
of divisions occupied by the air at the top. This airpump
is made to act also as a condensing engine, as
some German machines had done before, by the very
simple apparatus of turning a cock.</p><p>By means of this gauge, Mr. Smeaton judged that
his machine was incomparably better than any former
ones, as it seemed to rarefy the air in the receiver 1000,
or even 2000 times, while the best of the former construction
only rarefied about 140 times: and so the
case has since been always understood, an implicit considence
being placed in Mr. Smeaton's accuracy, till
the fallacy was accidentally detected in the manner related
at large by Mr. Nairne in the Philos. Trans. for
the year 1777. This accurate and ingenious artist
wanting to make trial of Mr. Smeaton's pear-gauge,
executed an air-pump of his improved construction, in
the best manner possible; which, in various experiments
made with it, appeared, by the pear-gauge, to rarefy
the air to an amazing degree indeed, being at times
from 4000 to 10000, or 50000, or even 100000 times
rarefied. But upon measuring the same expansion by
the usual long and short tube gauges, which both accurately
agreed together, he found that these never
shewed a rarefaction of more than 600 times: widely
different from the same as measured by the pear or internal
gauge, by experiments often repeated. &#x2018;Finding,
says Mr. Nairne, still this disagreement between the
pear-gauge and the other gauges, I tried a variety of
experiments; but none of them appeared to me satisfactory,
till one day in April 1776, shewing an experiment
with one of these pumps to the honourable Henry
Cavendish, Mr. Smeaton, and several other gentlemen
of the Royal Society, when the two gauges differed
some thousand times from one another, Mr. Cavendish
accounted for it in the following manner. &#x201C;It appeared,
he said, from some experiments of his father's,
Lord Cavendish, that water, whenever the pressure of
the atmosphere on it is diminished to a certain degree,
is immediately turned into vapour, and is as immediately
turned back again into water on reftoring the pressure.
This degree of pressure is different according to the
heat of the water: when the heat is 72&#xB0; of Fahrenhest's
scale, it turns into vapour as soon as the pressure is
no greater than that of three quarters of an inch of
quicksilver, or about 1-40th of the usual pressure of
the atmosphere; but when the heat is only 41&#xB0;, the
pressure must be reduced to that of a quarter of an inch
of quicksilver before the water turns into vapour. It
is true, that water exposed to the open air, will evaporate
at any heat, and with any pressure of the atmosphere;
but that evaporation is intirely owing to the
action of the air upon it; whereas the evaporation
here spoken of, is performed without any assistance
from the air. Hence it follows, that when the receiver
is exhausted to the above-mentioned degree, the
moisture adhering to the different parts of the machine
will turn into vapour, and supply the place of the air,
which is continually drawn away by the working of
<pb n="57"/><cb/>
the pump; so that the fluid in the pear-gauge, as well
as that in the receiver, will consist in a good measure of
vapour. Now letting the air into the receiver, all the
vapour within the pear-gauge will be reduced to water,
and only the real air will remain uncondensed; consequently
the pear-gauge shews only how much real air
is left in the receiver, and not how much the pressure
or spring of the included fluid is diminished; whereas
the common gauges shew how much the pressure of the
included fluid is diminished, and that equally, whether
it consist of air or of vapour.&#x201D; Mr. Cavendish having
explained so satisfactorily the cause of the disagreement
between the two gauges, Mr. Nairne considered that,
if he were to avoid moisture as much as possible, the
two gauges should nearly agree. And in fact they were
found so to do, each shewing a rarefaction of about 600,
when all moisture was perfectly cleared away from the
pump, and the plate and the edges of the receiver were
secured by a cement instead of setting it upon a soaked
leather, as in the usual way. But by future experiments,
Mr. Nairne found that the same excellent machine would
not exhaust more than 50 or 60 times, when the receiver
was set upon leather soaked in water, the heat of the
room being about 57&#xB0;. And from the whole, Mr.
Nairne concludes that the air-pump of Otto Guericke,
and those contrived by Mr. Gratorix, and Dr. Hook,
and the improved one by Mr. Papin, both used by Mr.
Boyle, as also Hauksbee's, s'Gravesande's, Muschenbroeck's,
and those of all who have used water in the
barrels of their pumps, could never have exhausted to
more than between 40 and 50, if the heat of the place
was about 57; and although Mr. Smeaton, with his
pump, where no water was in the barrel, but where
leather soaked in a mixture of water and spirit of wine
was used on the pump-plate, to set the receiver upon,
may have exhausted all but a thousandth, or even a tenthousandth
part of the common air, according to the
testimony of his pear-gauge; yet so much vapour must
have arisen from the wet leather, that the contents of
the receiver could never be less than a 70th or 80th
part of the density of the atmosphere. But when nothing
of moisture is used about this machine, it will,
when in its greatest perfection, rarefy its contents of air
about 600 times.</p><p>It is evident that by means of these two gauges we
can ascertain the several quantities of vapour and permanent
air which make up the contents of the receiver,
after the exhaustion is made as perfect as can be; for
the usual external gauge determines the whole contents,
made up of the vapour and air, whilst the pear-gauge
shews the quantity of real permanent air; consequently
the difference is the quantity of vapour.</p><p>The principal cause which prevents this pump from
exhausting beyond the limit above-mentioned, is the
weakened elasticity of the air within the receiver, which,
decreasing in proportion as the quantity of the air
within is diminished, becomes at last incapable of lifting
up the valve of communication between the receiver
and the barrel; and consequently no more air can then
pass from the former to the latter.</p><p>Several ingenious persons have used their endeavours
to remove this imperfection in the best air-pumps.
Amongst these it seems that one Mr. Haas has succeeded
tolerably well; having, by means of a contri-
<cb/>
vance to open the communication valve in the bottom
of the barrel, made his machine so perfect, that when
every thing is in the greatest perfection, it rarefies the
contents of the receiver as far as 1000 times, even when
measured by the exterior gauge. The description of
this machine, and an account of some experiments
performed with it, are given by Mr. Cavallo in the Philos.
Trans. for the year 1783.</p><p>But the imperfections it seems have more recently
been removed by an ingenious contrivance of Mr.
Cuthbertson, a mathematical instrument maker at Amsterdam,
now of London, whose air-pump has neither
cocks nor valves, and is so constructed, that what supplies
their place has the advantages of both, without
the inconveniences of either. He has also made improvements
in the gauges, by means of which he determines
the height of the mercury in the tube, by
which the degree of exhaustion is indicated, to the
hundredth part of an inch. And to obviate the inconvenience
of the elastic vapour arising from the wet
leather, upon which the receiver is placed, for common
experiments, he recommends the use of leather
dressed with allum, and soaked in hog's lard, which he
found to yield very little of this vapour; but when
the utmost degree of exhaustion is required, his advice
is, to dry the receiver well, and set it upon the plate
without any leather, only smearing its outer edges with
hog's lard, or with a mixture of three parts of hog's
lard and one of oil. But the use of the leather has
long been laid aside by our English instrument-makers,
a circumstance which probably had not come to Mr.
Cuthbertson's knowledge. An account of this instrument,
and of some experiments performed with it,
was published at Amsterdam in the year 1787; from
which experiments it appears that, by a coincidence
of the several gauges, a rarefaction of 1200 times
was shewn; but when the atmosphere was very dry,
the exhaustion has been so complete, that the gauges
have shewn the air in the receiver to be rarefied above
2400 times.</p><p>There are made also by different persons, portable,
or small air-pumps, of various constructions, to set
upon a table, to perform experiments with. In these,
the gauge is varied according to the fancy of the maker,
but commonly it consists of a bent glass tube, like
a syphon, open only at one end. The gauge is placed
under a small receiver communicating, by a pipe, with
the principal pipe leading from the general receiver to
the barrels. The close end of the gauge, of 3 or 4
inches long, before the exhaustion, has the quicksilver
forced close up to the top by the pressure of the air on
the open end; but when the exhaustion is considerably
advanced, it begins to descend, and then the difference
of the heights of the quicksilver in the two legs, compared
with the height in the barometrical tube, determines
the degree of exhaustion: so if the difference
between the two be one inch, when the barometer
stands at 30, the air is rarefied 30 times; but if the
difference be only half an inch, the rarefaction is 60
times, and so on. See Plate 111. fig. 2.</p><p><hi rend="italics">The Use of the Air-Pump.</hi> In whatever manner or
form this machine be made, the use and operation of it
are always the same. The handle, which works the
piston, is moved up and down in the barrel, by which
<pb n="58"/><cb/>
means a barrel of the contained air is drawn out at every
stroke of the piston, in the following manner: by pushing
the piston down to the bottom of the barrel, where
the air is prevented from escaping downwards, by its
elasticity it opens the valve of the piston, and escapes
upwards above it into the open air; then raising the
piston up, the external atmosphere shuts down its valve,
and a vacuum would be made below it, but for the air
in the receiver, pipe, &amp;c, which now raises the valve
in the bottom of the barrel, and rushes in and fills it
again, till the whole air in the receiver and barrel be
of one uniform density, but less than it was before the
stroke, in proportion as the sum of all the capacities
of the receiver, pipe, and barrel together, is to the same
sum wanting the barrel. And thus is the air in the
receiver diminished at each stroke of the piston, by the
quantity of the barrel or cylinder full, and therefore
always in the same proportion: so that by thus repeating
the operation again and again, the air is rarefied to
any proposed degree, or till it has not elasticity enough
to open the valve of the piston or of the barrel, after
which the exhaustion cannot be any farther carried on:
the gauge, in comparison with the barometer, shewing
at any time what the degree of exhaustion is, according
to the particular nature and construction of it.</p><p>But, supposing no vapour from moisture, &amp;c, to rise
in the receiver, the degree of exhaustion, after any
number of strokes of the piston, may be determined
by knowing the respective capacities of the barrel and
the receiver, including the pipe, &amp;c. For as we have
seen above that every stroke diminishes the density in a
constant proportion, namely as much as the whole content
exceeds that of the cylinder or barrel; and consequently
the sum of as many diminutions as there are
strokes of the piston, will shew the whole diminution
by all the strokes. So, if the capacity of the barrel be
equal to that of the receiver, in which the communication
pipe is always to be included; then, the barrel
being half the sum of the whole contents, half the air
will be drawn out at one stroke; and consequently the
remaining half, being dilated through the whole or first
capacity, will be of only half the density of the first:
in like manner, after the second stroke, the density of
the remaining contents will be only half of that after
the sirst stroke, that is only 1/4 of the original density:
continuing this operation, it follows that the density of
the remaining air will be 1/8 after 3 strokes of the piston,
1/16 after 4 strokes, 1/32 after 5 strokes, and so on, according
to the powers of the ratio 1/2; that is, such power
of the ratio as is denoted by the number of the strokes.
In like manner, if the barrel be 1/3 of the whole contents,
that is, the receiver double of the barrel, or 2/3 of
the whole contents; then the ratio of diminution of
density being 2/3, the density of the contents, after
any number of strokes of the piston, will be denoted
by such power of 2/3 whose exponent is that number;
namely, the density will be 2/3 after one stroke, (2/3)<hi rend="sup">2</hi> or
4/9 after two strokes, (2/3)<hi rend="sup">3</hi> or 8/27 after 3 strokes, and in
general it will be (2/3)<hi rend="sup">n</hi> after <hi rend="italics">n</hi> strokes: the original
density of the air being 1. Hence then, universally,
if <hi rend="italics">s</hi> denote the sum of the contents of the receiver
and barrel, and <hi rend="italics">r</hi> that of the receiver only without
the barrel, and <hi rend="italics">n</hi> any number of strokes of the
piston; then, the original density of the air being 1,
<cb/>
the density after <hi rend="italics">n</hi> strokes will be (<hi rend="italics">r/s</hi>)<hi rend="sup">n</hi> o <hi rend="italics">r</hi><hi rend="sup">n</hi>/<hi rend="italics">s</hi><hi rend="sup">n</hi>, namely
the <hi rend="italics">n</hi> power of the ratio <hi rend="italics">r/s</hi>. So, for example, if the
capacity of the receiver be equal to 4 times that of the
barrel; then their sum <hi rend="italics">s</hi> is 5, and <hi rend="italics">r</hi> is 4; and the
density of the contents after 30 strokes, will be (4/5)<hi rend="sup">30</hi>,
or the 30th power of 4/5, which is 1/808 nearly; so that
the air in the receiver is raresied 808 times.</p><p>See also the Memoires de l'Acad. Royale des Sciences
for the years 1693 and 1705.</p><p>From the same formula, namely (<hi rend="italics">r/s</hi>)<hi rend="sup">n</hi> = <hi rend="italics">d</hi> the density,
we easily derive a rule for finding the number of
strokes of the piston, necessary to rarefy the air any
number of times, or to reduce it to a given density <hi rend="italics">d,</hi>
that of the natural air being 1. For since ((<hi rend="italics">r/s</hi>)<hi rend="sup">n</hi> = <hi rend="italics">d,</hi>
by taking the logarithm of this equation, it is <hi rend="italics">n</hi> X log.
<hi rend="italics">r/s</hi> = log. of <hi rend="italics">d;</hi> and hence ;
that is, divide the log. of the proposed density by the
log. of the ratio of the receiver to the sum of the receiver
and barrel together, and the quotient will shew
the number of strokes of the piston requisite to produce
the degree of exhaustion required. So, for example, if
the receiver be equal to 5 times the barrel, and it be
proposed to find how many strokes of the piston will
rarefy the air 100 times; then <hi rend="italics">r</hi> = 5, <hi rend="italics">s</hi> = 6, <hi rend="italics">d</hi> = 1/100,
whose log. is - 2, and <hi rend="italics">r/s</hi>=5/6, whose log. is - .07918;
therefore 2/.07918 = 25 1/4 nearly, which is the number
of strokes required.</p><p>And, farther, the same formula reduced, would give
us the proportion between the receiver and barrel, when
the air is rarefied to any degree by an assigned number
of strokes of the piston. For since  the density,
therefore, extracting the <hi rend="italics">n</hi> root of both sides, it is
: that is, the <hi rend="italics">n</hi> root of the density is equal
to the ratio of the receiver to the sum of the receiver
and barrel. So, if the density <hi rend="italics">d</hi> be 1/128, and the number
of strokes <hi rend="italics">n</hi> = 7; then the 7th root of 1/128 is 1/2;
which shews that the receiver is equal to half the receiver
and barrel together, or that the capacity of the
barrel is just equal to that of the receiver.</p><p>Some of the principal effects and phenomena of the
air-pump, are the following: That, in the exhausted
receiver, heavy and light bodies fall equally swift; so, a
guinea and feather fall from the top of a tall receiver
to the bottom exactly together. That most animals
die in a minute or two: but however, That vipers and
frogs, though they swell much, live an hour or two; and
after being seemingly quite dead, come to life again in
the open air: That snails survive about ten hours;
efts, or slow-worms, two or three days; and leeches
five or six. That oysters live for 24 hours. That the
heart of an eel taken out of the body, continues to
<pb n="59"/><cb/>
beat for good part of an hour, and that more briskly
than in the air. That warm blood, milk, gall, &amp;c, undergo
a considerable intumescence and ebullition. That
a mouse or other animal may be brought, by degrees, to
survive longer in a rarefied air, than naturally it does.
That air may retain its usual pressure, after it is become
unfit for respiration. That the eggs of silk-worms
hatch in vacuo. That vegetation stops. That fire extinguishes;
the flame of a candle usually going out in
one minute; and a charcoal in about five minutes.
That red-hot iron, however, seems not to be affected;
and yet sulphur or gun-powder are not lighted by it,
but only fused. That a match, after lying seemingly
extinct a long time, revives again on re-admitting the
air. That a flint and steel strike sparks of fire as copiously,
and in all directions, as in air. That magnets,
and magnetic needles, act the same as in air. That the
smoke of an extinguished luminary gradually settles to
the bottom in a darkish body, leaving the upper part of
the receiver clear and transparent; and that on inclining
the vessel sometimes to one side, and sometimes to
another, the fume preserves its surface horizontal, after
the nature of other fluids. That heat may be produced
by attrition. That camphire will not take fire;
and that gun-powder, though some of the grains of a
heap of it be kindled by a burning glass, will not give
fire to the contiguous grains. That glow-worms lose
their light in proportion as the air is exhausted, and at
length become totally obscure; but on re-admitting
the air, they presently recover it all. That a bell, on
being struck, is not heard to ring, or very faintly.
That water freezes. But that a syphon will not run.
That electricity appears like the aurora borealis. With
multitudes of other curious and important particulars,
to be met with in the numerous writings on this machine,
namely, besides the Philos. Transactions of most
academies and societies, in the writings of Torricelli,
Pascal, Mersenne, Guericke, Schottus, Boyle, Hook,
Duhamel, Mariotte, Hauksbee, Hales, Muschenbroeck,
Gravesande, Desaguliers, Franklin, Cotes, Helsham,
and a great many other authors.</p><p><hi rend="smallcaps">Air-Vessel</hi>, in <hi rend="italics">Hydraulics,</hi> is a vessel of air within
some water engines, which being compressed, by forcing
in a considerable quantity of water, by its uniform
spring, forces it out at the pipe in a constant uninterrupted
stream, to a great height.</p><p>Air-vessel too, in the improved fire engines, is a metallic
cylinder, placed between the two forcing pumps,
by the action of whose pistons the water is forced into
this vessel, through two pipes, with valves; then the
air, previously contained in it, is compressed by the
water, in proportion to the quantity admitted, and
this air, by its spring, forces the water through a pipe
by a constant and equal stream; whereas in the common
squirting engine, the stream is discontinued between
the several strokes.</p><p>AIRY <hi rend="smallcaps">Triplicity</hi>, in <hi rend="italics">Astrology,</hi> the signs of
Gemini, Libra, and Aquarius.</p></div1><div1 part="N" n="AJUTAGE" org="uniform" sample="complete" type="entry"><head>AJUTAGE</head><p>, or <hi rend="smallcaps">Adjutage</hi>, in <hi rend="italics">Hydraulics,</hi> part of
the apparatus of a <hi rend="italics">jet d'eau,</hi> or artisicial fountain; being
a kind of tube fitted to the aperture or mouth of
the cistern, or the pipe; through which the water is
to be played in any direction, and in any shape or
figure.
<cb/></p><p>It is chiefly the diversity in the ajutage, that makes
the different kinds of fountains. So that, by having
several ajutages, to be applied occasionally, one fountain
is made to have the effect of many.</p><p>Mariotte, Gravesande, and Desaguliers have written
pretty fully on the nature of ajutages, or spouts for jets
d'eau, and especially the former. He affirms, from experiment,
that an even polished round hole, made in
the thin end of a pipe, gives a higher jet than either
a cylindrical or a conical ajutage; but that, of these
two latter however, the conical is better than the cylindrical
figure. See his Traite du Mouvement des
Eaux, part 4.</p><p>The quantity of water discharged by ajutages of
equal area, but of different figures, is the same. And
for like figures, but of different sizes, the quantity
discharged, is directly proportional to the area of the
ajutage, or to the square of its diameter, or of any side
or other linear dimension: so, an ajutage of a double
diameter, or side, will discharge 4 times the quantity
of water; of a triple diameter, 9 times the quantity;
and so on; supposing them at an equal depth below the
surface or head of water. But if the ajutage be at
different depths below the head, then the celerity with
which the water issues, and consequently the quantity
of it run out in any given-time, is directly proportional
to the square-root of the altitude of the head, or depth
of the hole: so at 4 times the depth, the celerity and
quantity is double; at 9 times the depth, triple; and
so on.</p><p>It has been found that jets do not rise quite so high
as the head of water; owing chiefly to the resistance
of the air against it, and the pressure of the upper parts
of the jet upon the lower: and for this reason it is,
that if the direction of the ajutage be turned a very
little from the perpendicular, it is found to spout rather
higher than when the jet is exactly upright.</p><p>It is sound by experiment too, that the jet is higher
or lower, according to the size of the ajutage: that a
circular hole of about an inch and a quarter in diameter,
jets highest; and that the farther from that size,
the worse. Experience also shews that the pipe leading
to the ajutage, should be much larger than it; and if
the pipe be along one, that it should be wider the farther
it is from the ajutage.</p><p>For the other circumstances relating to jets and the
issuing of water under various circumstances, see E<hi rend="smallcaps">XHAUSTION,
Flux, Fountain, Jet d'Eau</hi>, &amp;c, to
which they more properly belong.</p></div1><div1 part="N" n="ALBATEGNI" org="uniform" sample="complete" type="entry"><head>ALBATEGNI</head><p>, an Arabic prince of Batan in
Mesopotamia, who was a celebrated astronomer, about
the year of Christ 880, as appears by his observations.
He is also called <hi rend="italics">Muhammed ben Geber Albatani, Mahomet
the son of Geber,</hi> and <hi rend="italics">Muhamedes Aractensis.</hi> He
made astronomical observations at Antioch, and at
Racah or Aracta, a town os Chaldea, which some
authors call a town of Syria or of Mesopotamia. He
is highly spoken of by Dr. Halley, as a <hi rend="italics">vir admirandi
acuminis, ac in administrandis observationibus exercitatissimus.</hi></p><p>Finding that the tables of Ptolomy were imperfect,
he computed new ones, which were long used as the
best among the Arabs: these were adapted to the meridian
of Aracta or Racah. Albategni composed in
<pb n="60"/><cb/>
Arabic a work under the title of <hi rend="italics">The Science of the
Stars,</hi> comprising all parts of astronomy, according to
his own observations and those of Ptolomy. This
work, translated into Latin by Plato of Tibur, was
published at Nuremberg in 1537, with some additions
and demonstrations of Regiomontanus; and the same
was reprinted at Bologna in 1645, with this author's
notes. Dr. Halley detected many faults in these editions:
Philos. Trans. for 1693, N&#xB0; 204.</p><p>In this work, Albategni gives the motion of the
sun's apogee since Ptolomy's time, as well as the motion
of the stars, which he makes 1 degree in 70 years.
He made the longitude of the sirst star of Aries to be
18&#xB0; 2&#x2032;; and the obliquity of the ecliptic 23&#xB0; 35&#x2032;.
And upon Albategni's observations were founded the
Alphonsine tables of the moon's motions; as is observed
by Nic. Muler, in the Tab. Frisic&#xE6;, pa. 248.</p><p>ALBERTUS <hi rend="smallcaps">Magnus</hi>, a very learned man in the
13th century, who, among a multitude of books,
wrote several upon the various mathematical sciences,
as Arithmetic, Geometry, Perspective or Optics, Music,
Astrology and Astronomy, particularly under the titles,
<hi rend="italics">de sph&#xE6;ra, de astris, de astronomia, item speculum astronomicum.</hi></p><p>Albertus Magnus was born at Lawingen on the Danube,
in Suabia, in 1205, or according to some in
1193; and he died at a great age, at Cologn, November
15, 1280. Vossius and other authors speak of him
as a great genius, and deeply skilled in all the learning
of the age. His writings were so numerous, that they
make 21 volumes in folio, in the Lyons edition of 1615.
He has passed also for the author of some writings relating
to midwifery, &amp;c, under the title of <hi rend="italics">De natura
rerum,</hi> and <hi rend="italics">De secretis mulierum,</hi> in which there are
many phrases and expressions unavoidable on such a
subject, which gave great offence, and raised a clamour
against him as the supposed author, and inconsistent
with his character, being a Dominican friar, and sometime
bishop of Ratisbon; which dignity however he
soon resigned, through his love for solitude, to enter
again into the monastic life. But the advocates of
Albert assert, that he was not the author of either of
these two works. It must be acknowledged however,
that there are, in his Comment upon the Master of
Sentences, some questions concerning the practice of
conjugal duty, in which he has used some words rather
too gross for chaste and delicate ears: but they allege
what he himself used to say in his own vindication,
that he came to the knowledge of so many monstrous
things at confession, that it was impossible to avoid
touching upon such questions. Albert was certainly a
man of a most curious and inquisitive turn of mind,
which gave rise to other accusations against him; such
as, that he laboured to find out the philosopher's stone;
that he was a magician; and that he made a machine
in the shape of a man, which was an oracle to him,
and explained all the difficulties he proposed: the common
cant accusations of those times of ignorance and
superstition. But having great knowledge in the mathematics
and mechanics, by his skill in these sciences
he probably formed a head, with springs capable of articulate
sounds; like the machines of Boetius and others.</p><p>John Matth&#xE6;us de Luna, in his treatise De Rerum
Inventor&lt;*&gt;bus, has attributed the invention of fire-arms
<cb/>
to Albert; but in this he is refuted by Naude, in his
Apologie des grands hommes.</p></div1><div1 part="N" n="ALBUMAZAR" org="uniform" sample="complete" type="entry"><head>ALBUMAZAR</head><p>, otherwise called <hi rend="smallcaps">Abuassar</hi>, and
<hi rend="smallcaps">Japhar</hi>, was a celebrated Arabian philosopher and
astrologer, of the 9th or 10th century, or according to
some authors much earlier. Blancanus, Vossius, &amp;c,
speak of him as one of the most learned astronomers of
his time, or astrologers, which was then the same thing.
He wrote a work <hi rend="italics">De Magnis Conjunctionibus Annorum
Revolutionibus, ac eorum Perfectionibus,</hi> printed at Venice
in 1515, at the expence of Melchior Sessa, a work
chiefly astrological.</p><p>He wrote also <hi rend="italics">Introductio in Astronomiam,</hi> printed in
the year 1489. And it is reported that he observed a
comet in his time, above the orb of Venus.</p></div1><div1 part="N" n="ALCOHOL" org="uniform" sample="complete" type="entry"><head>ALCOHOL</head><p>, in the Arabian Astrology, is when a
heavy slow-moving planet receives another lighter one
within its orb, so as to come in conjunction with it.</p></div1><div1 part="N" n="ALDEBARAN" org="uniform" sample="complete" type="entry"><head>ALDEBARAN</head><p>, an Arabian name of a fixed star,
of the first magnitude, just in the eye of the sign or
constellation Taurus, or the bull, and hence it is popularly
called the bull's eye. For the beginning of the
year 1800, its
<table><row role="data"><cell cols="1" rows="1" role="data">Right Ascension is</cell><cell cols="1" rows="1" role="data">&#x2014;</cell><cell cols="1" rows="1" role="data">66&#xB0;</cell><cell cols="1" rows="1" role="data">6&#x2032;</cell><cell cols="1" rows="1" role="data">51&#x2033;</cell><cell cols="1" rows="1" role="data">.10</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Annual variation in AR</cell><cell cols="1" rows="1" role="data">&#xA0;0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">.31</cell></row><row role="data"><cell cols="1" rows="1" role="data">Declination</cell><cell cols="1" rows="1" role="data">&#x2014;</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">.00 N.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">And Annual variat. in Decl.</cell><cell cols="1" rows="1" role="data">&#xA0;0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">&#xA0;8</cell><cell cols="1" rows="1" role="data">.30</cell></row></table></p></div1><div1 part="N" n="ALDERAIMIN" org="uniform" sample="complete" type="entry"><head>ALDERAIMIN</head><p>, a star of the third magnitude in
the right shoulder of the constellation Cepheus.</p></div1><div1 part="N" n="ALDHAFERA" org="uniform" sample="complete" type="entry"><head>ALDHAFERA</head><p>, or Aldhaphra, in the Arabian
Astronomy, denotes a fixed star of the third magnitude,
in the mane of the sign or constellation Leo, the lion.</p></div1><div1 part="N" n="ALEMBERT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ALEMBERT</surname> (<foreName full="yes"><hi rend="smallcaps">John le Rond</hi></foreName> D')</persName></head><p>, an eminent
French mathematician and philosopher, and one of the
brightest ornaments of the 18th century. He was
perpetual secretary to the French Academy of Sciences,
and a member of most of the philosophical
academies and societies of Europe.</p><p>D'Alembert was born at Paris, the 16th of November
1717. He derived the name of John le Rond
from that of the church near which, after his birth, he
was exposed as a foundling. But his father, informed
of this circumstance, listening to the voice of nature and
duty, took measures for the proper education of his
child, and for his future subsistence in a state of ease
and independence. His mother, it is said, was a lady of
of rank, the celebrated Mademoiselle Tencin, sister to
cardinal Tencin, archbishop of Lyons.</p><p>He received his first education among the Jansenists,
in the College of the Four Nations, where he gave
early signs of genius and capacity. In the first year
of his philosophical studies, he composed a Commentary
on the Epistle of St. Paul to the Romans. The Jansenists
considered this production as an omen, that
portended to the party of Port-Royal a restoration
to some part of their former splendor, and hoped to
find one day in d'Alembert a second Pascal. To render
this resemblance more complete, they engaged
their pupil in the study of the mathematics; but they
soon perceived that his growing attachment to this
science was likely to disappoint the hopes they had
formed with respect to his future destination: they
therefore endeavoured to divert him from this line;
but their endeavours were fruitless.
<pb n="61"/><cb/></p><p>On his quitting the college, finding himself alone,
and unconnected in the world, he sought an asylum in
the house of his nurse. He hoped that his fortune,
though not ample, would enlarge the subsistence, and
better the condition of her family, which was the only
one that he could consider as his own. It was here
therefore that he fixed his residence, resolving to apply
himself entirely to the study of geometry.&#x2014;And here
he lived, during the space of 40 years, with the greatest
fimplicity, discovering the augmentation of his means
only by increasing displays of his beneficence, concealing
his growing reputation and celebrity from these
honest people, and making their plain and uncouth
manners the subject of good-natured pleasantry and
philosophical observation. His good nurse perceived
his ardent activity; heard him mentioned as the writer
of many books; but never took it into her head that
he was a great man, and rather beheld him with a kind
of compassion. &#x201C;You will never, said she to him one
day, be any thing but a philosopher&#x2014;and what is a
philosopher?&#x2014;a fool, who toils and plagues himself
all his life, that people may talk of him when he is
dead.&#x201D;</p><p>As d'Alembert's fortune did not far exceed the demands
of necessity, his friends advised him to think of
some profession that might enable him to increase it.
He accordingly turned his views to the law, and took
his degrees in that faculty: but soon after, abandoning
this line, he applied himself to the study of medicine.
Geometry however was always drawing him back to
his former pursuits; so that after many ineffectual
struggles to resist its attractions, he renounced all views
of a lucrative profession, and gave himself up entirely
to mathematics and poverty.</p><p>In the year 1741 he was admitted a member of the
Academy of Sciences; for which distinguished literary
promotion, at so early an age (24), he had prepared
the way by correcting the errors of a celebrated work
(The <hi rend="italics">Analyse Demontr&#xE9;e</hi> of Reyneau), which was
esteemed classical in France in the line of analytics. He
afterwards set himself to examine, with close attention
and assiduity, what must be the motion and path of a
body, which passes from one fluid into another denser
fluid, in a direction oblique to the surface between the
two fluids. Every one knows the phenomenon which
happens in this case, and amuses children, under the
denomination of <hi rend="italics">Ducks and Drakes;</hi> but it was d'Alembert
who first explained it in a satisfactory and philosophical
manner.</p><p>Two years after his election to a place in the academy,
he published his <hi rend="italics">Treatise on Dynamics.</hi> The new
principle developed in this treatise, consisted in establishing
an equality, at each instant, between the
changes that the motion of a body has undergone, and
the forces or powers which have been employed to produce
them: or, to express the same thing otherwise, in
separating into <hi rend="italics">two parts</hi> the action of the moving
powers, and considering the <hi rend="italics">one</hi> as producing alone the
motion of the body, in the second instant, and the
<hi rend="italics">other</hi> as employed to destroy that which it had in the
first.</p><p>So early as the year 1744, d'Alembert had applied
this principle to the theory of the equilibrium, and the
motion of fluids: and all the problems before resolved
<cb/>
in physics, became in some measure its corollaries.
The discovery of this new principle was followed by
that of a new calculus, the first essays of which were
published in a <hi rend="italics">Discourse on the General Theory of the
Winds,</hi> to which the prize-medal was adjudged by the
Academy of Berlin in the year 1746, which proved a
new and brilliant addition to the fame of d'Alembert.
This new calculus of <hi rend="italics">Partial Differences</hi> he applied, the
year following, to the problem of vibrating chords, the
resolution of which, as well as the theory of the oscillations
of the air and the propagation of sound, had
been but imperfectly given by the mathematicians who
preceded him; and these were his masters or his
rivals.</p><p>In the year 1749 he furnished a method of applying
his principle to the motion of any body of a given
figure. He also resolved the problem of the precession
of the equinoxes; determining its quantity, and
explaining the phenomenon of the nutation of the terrestrial
axis discovered by Dr. Bradley.</p><p>In 1752, d'Alembert published a treatise on the
<hi rend="italics">Resistance of Fluids.</hi> to which he gave the modest title
of an <hi rend="italics">Essay;</hi> though it contains a multitude of original
ideas and new observations. About the same time
he published, in the Memoirs of the Academy of Berlin,
<hi rend="italics">Researches concerning the Integral Calculus,</hi> which is
greatly indebted to him for the rapid progress it has
made in the present century.</p><p>While the studies of d'Alembert were confined to
mere mathematics, he was little known or celebrated
in his native country. His connections were limited to
a small society of select friends. But his cheerful conversation,
his smart and lively sallies, a happy knack at
telling a story, a singular mixture of malice of speech
with goodness of heart, and of delicacy of wit with
simplicity of manners, rendering him a pleasing and
interesting companion, his company began to be much
sought after in the fashionable circles. His reputation
at length made its way to the throne, and rendered him
the object of royal attention and beneficence. The
consequence was a pension from government, which he
owed to the friendship of count d'Argenson.</p><p>But the tranquillity of d'Alembert was abated when
his same grew more extensive, and when it was known
beyond the circle of his friends, that a fine and enlightened
taste for literature and philosophy accompanied
his mathematical genius. Our author's eulogist
ascribes to envy, detraction, &amp;c, all the opposition and
censure that d'Alembert met with on account of the
famous Encyclop&#xE9;die, or Dictionary of Arts and
Sciences, in conjunction with Diderot. None surely
will refuse the well-deserved tribute of applause to the
eminent displays of genius, judgment, and true literary
taste, with which d'Alembert has enriched that great
work. Among others, the Preliminary Discourse he
has prefixed to it, concerning the rise, progress, connections,
and affinities of all the branches of human
knowledge, is perhaps one of the most capital productions
the philosophy of the age can boast of.</p><p>Some time after this, d'Alembert published his <hi rend="italics">Philosophical,
Historical,</hi> and <hi rend="italics">Philological Miscellanies.</hi>
These were followed by the <hi rend="italics">Memoirs of Christina
queen of Sweden;</hi> in which d'Alembert shewed that he
was acquainted with the natural rights of mankind,
<pb n="62"/><cb/>
and was bold enough to assert them. His <hi rend="italics">Essay on the
Intercourse of Men of Letters with Persons high in Rank
and Ofsice,</hi> wounded the former to the quick, as it exposed
to the eyes of the public the ignominy of those
servile chains, which they feared to shake off, or were
proud to wear. A lady of the court hearing one day
the author accused of having exaggerated the despotism
of the great, and the submission they require, answered
slyly, &#x201C;If he had consulted me, I would have told him
still more of the matter.&#x201D;</p><p>D'Alembert gave elegant specimens of his literary
abilities in his translations of some select pieces of
Tacitus. But these occupations did not divert him
from his mathematical studies: for about the same time
he enriched the Encyclop&#xE9;die with a multitude of excellent
articles in that line, and composed his <hi rend="italics">Researches
on several Important Points of the System of the World,</hi>
in which he carried to a higher degree of perfection
the solution of the problem concerning the perturbations
of the planets, that had several years before been
presented to the Academy.</p><p>In 1759 he published his <hi rend="italics">Elements of Philosophy:</hi> a
work much extolled as remarkable for its precision and
perspicuity.</p><p>The resentment that was kindled (and the disputes
that followed it) by the article <hi rend="italics">Geneva,</hi> inserted in the
Encyclop&#xE9;die, are well known. D'Alembert did not
leave this sield of controversy with flying colours. Voltaire
was an auxiliary in the contest: but as he had
no reputation to lose, in point of candour and decency;
and as he weakened the blows of his enemies, by throwing
both them and the spectators into fits of laughter,
the issue of the war gave him little uneasiness. It fell
more heavily on d'Alembert; and exposed him, even
at home, to much contradiction and opposition.</p><p>It was on this occasion that the late king of Prussia
offered him an honourable asylum at his court, and the
office of president of his academy: and the king was
not offended at d'Alembert's refusal of these distinctions,
but cultivated an intimate friendship with him
during the rest of his life. He had refused, some time
before this, a proposal made by the empress of Russia
to entrust him with the education of the Grand Duke;
&#x2014;a proposal accompanied with all the flattering offers
that could tempt a man, ambitious of titles, or desirous
of making an ample fortune: but the objects of his
ambition were tranquillity and study.</p><p>In the year 1765, he published his <hi rend="italics">Dissertation on the
Destruction of the Jesuits.</hi> This piece drew upon him
a swarm of adversaries, who only confirmed the merit
and credit of his work by their manner of attacking
it.</p><p>Beside the works already mentioned, he published
nine volumes of memoirs and treatises, under the title
of <hi rend="italics">Opuscules;</hi> in which he has resolved a multitude of
problems relating to astronomy, mathematics, and natural
philosophy; of which his panegyrist, Condorcet,
gives a particular account, more especially of those
which exhibit new subjects, or new methods of investigation.</p><p>He published also <hi rend="italics">Elements of Music</hi>; and rendered,
at length, the system of Rameau intelligible: but he did
not think the mathematical theory of the sonorous body
sufficient to account for the rules of that art.
<cb/></p><p>In the year 1772 he was chosen secretary to the
French Academy of Sciences. He formed, soon after
this preferment, the design of writing the lives of all
the deceased academicians, from 1700 to 1772; and
in the space of three years he executed this design, by
composing 70 eulogies.</p><p>D'Alembert died on the 29th of October 1783,
being nearly 66 years of age. In his moral character
there were many amiable lines of candour, modesty,
disinterestedness, and beneficence; which are described,
with a diffusive detail, in his eulogium, by Condorcet,
in the <hi rend="italics">Hist. de l'Acad. Royale des Sciences,</hi>
1783.</p><p>As it may be curious and useful to have in one
view an entire list of d'Alembert's writings, I shall here
insert a catalogue of them, from Rozier's <hi rend="italics">Nouvelle
Table des Articles contenus dans les volumes de l'Academie
Royale des Sciences de Paris,</hi> &amp;c, as follows:</p><p><hi rend="italics">Trait&#xE9; de Dynamique,</hi> in 4to, Paris, 1743. The 2d
ed. in 1758.</p><p><hi rend="italics">Trait&#xE9; de l'Equilibre et du Mouvement des Fluides.</hi>
Paris, 1744; and the 2d edition in 1770.</p><p><hi rend="italics">Reflexions sur la Cause G&#xE9;n&#xE9;rale des Vents;</hi> which
gained the prize at Berlin in 1746; and was printed at
Paris in 1747, in 4to.</p><p><hi rend="italics">Recherches sur la Pr&#xE9;cession des &#xC9;quinoxes, &amp; sur la
Nutation de l'Axe de la Terre dans le Syst&#xE8;me Newtonien.</hi>
Paris, 1749, in 4to.</p><p><hi rend="italics">Essais d'une Nouvelle Th&#xE9;orie du Mouvement des
Fluides.</hi> Paris, 1752, in 4to.</p><p><hi rend="italics">Recherches sur differens Points importans du Syst&#xE8;me
du Monde.</hi> Paris, 1754 and 1756, 3 vol. in 4to.</p><p><hi rend="italics">Elemens de Philosophie,</hi> 1759.</p><p><hi rend="italics">Opuscules Math&#xE9;matiques,</hi> ou Memoires sur diff&#xE9;rens
Sujets de G&#xE9;om&#xE9;trie, de M&#xE9;chaniques, d'Optiques,
d'Astronomie. Paris, 9 vol. in 4to; 1761 to 1773.</p><p><hi rend="italics">El&#xE9;mens de Musique,</hi> th&#xE9;orique &amp; pratique, suivant
les Principes de M. Rameau, eclair&#xE9;s, d&#xE9;velopp&#xE9;s, &amp;
simplifi&#xE9;s. 1 vol. in 8vo. &#xE0; Lyon.</p><p><hi rend="italics">De la Destruction des Jesuites,</hi> 1765.</p><p>In the Memoirs of the Academy of Paris are the
following pieces, by d'Alembert: viz,</p><p><hi rend="italics">Pr&#xE9;cis</hi> de Dynamique, 1743, Hist. 164.</p><p><hi rend="italics">Pr&#xE9;cis</hi> de l'Equilibre &amp; de Mouvement des Fluides,
1744, Hist. 55.</p><p><hi rend="italics">Methode</hi> g&#xE9;n&#xE9;rale pour d&#xE9;terminer les Orbites &amp; les
Mouvements de toutes les Plan&#xE8;tes, en ayant &#xE9;gard &#xE0;
leur action mutuelle, 1745, p. 365.</p><p><hi rend="italics">Pr&#xE9;cis</hi> des R&#xE9;flexions sur la Cause G&#xE9;n&#xE9;rale des
Vents, 1750, Hist. 41.</p><p><hi rend="italics">Pr&#xE9;cis</hi> des Recherches sur la Pr&#xE9;cession des &#xC9;quinoxes,
et sur la Nutation de l'Axe de la Terre dans
le Syst&#xE8;me Newtonien, 1750, Hist. 134.</p><p><hi rend="italics">Essai</hi> d'une Nouvelle Th&#xE9;orie sur la R&#xE9;sistance des
Fluides, 1752, Hist. 116.</p><p><hi rend="italics">Pr&#xE9;cis</hi> des Essais d'une Nouvelle Th&#xE9;orie de la R&#xE9;sistance
des Fluides, 1753, Hist. 289.</p><p><hi rend="italics">Pr&#xE9;cis</hi> des Recherches sur les differens Points importans
du Syst&#xE8;me du Monde, 1754, Hist. 125.</p><p><hi rend="italics">Recherches</hi> sur la Pr&#xE9;cession des Equinoxes, &amp; sur la
Nutation de l'Axe de la Terre, dans l'Hypothese de la
Dissimilitude des M&#xE9;ridiens, 1754, p. 413, Hist.
116.</p><p><hi rend="italics">Reponse</hi> &#xE0; un Article du M&#xE9;moire de M. l'Abb&#xE9; de
<pb n="63"/><cb/>
la Caille, su&lt;*&gt; la Th&#xE9;orie du Soleil, 1757, p. 145,
Hist. 118.</p><p><hi rend="italics">Addition</hi> &#xE0; ce M&#xE9;moire, 1757, p. 567, Hist. 118.</p><p><hi rend="italics">Pr&#xE9;cis</hi> des Opuscules Math&#xE9;matiques, 1761, Hist. 86.</p><p><hi rend="italics">Pr&#xE9;cis</hi> du troisi&#xE8;me volume des Opuscules Math&#xE9;matiques,
1764, Hist. 92.</p><p><hi rend="italics">Nouvelles</hi> Recherches sur les Verres Optiques, pour
servir de suite &#xE0; la th&#xE9;orie qui en &#xE0; &#xE9;t&#xE9; donn&#xE9;e dans le
volume 3<hi rend="sup">e</hi> des Opuscules Math&#xE9;matiques. Premier
M&#xE9;moire, 1764, p. 75, Hist. 175.</p><p><hi rend="italics">Nouvelles</hi> Recherches sur les Verres Optiques, pour
fervir de suite &#xE0; la th&#xE9;orie qui en a &#xE9;t&#xE9; donn&#xE9;e dans le
troisi&#xE8;me volume des Opuscules Math&#xE9;matiques. Second
M&#xE9;moire, 1765, p. 53.</p><p><hi rend="italics">Observations</hi> sur les Lunettes Achromatiques, 1765,
p. 53, Hist. 119.</p><p><hi rend="italics">Suite</hi> des Recherches sur les Verres Optiques.
Troisi&#xE8;me M&#xE9;moire, 1767, p. 43, Hist. 153.</p><p><hi rend="italics">Recherches</hi> sur le Calcul Int&#xE9;gral, 1767, p. 573.</p><p><hi rend="italics">Accident</hi> arriv&#xE9; par l'Explosion d'une Meule d'Emouleur,
1768, Hist. 31.</p><p><hi rend="italics">Pr&#xE9;cis</hi> des Opuscules de Math&#xE9;matiques, 4<hi rend="sup">e</hi> &amp; 5<hi rend="sup">e</hi>
volumes. Leur Analyse, 1768, Hist. 83.</p><p><hi rend="italics">Recherches</hi> sur les Mouvemens de l'Axe d'une Planete
quelconque dans l'hypothese de la Dissimilitude des M&#xE9;ridienes,
1768 p. 1, Hist. 95.</p><p><hi rend="italics">Suite</hi> des Recherches sur les Mouvemens, &amp;c, 1768
p. 332, Hist. 95.</p><p><hi rend="italics">Recherches</hi> sur le Calcul Int&#xE9;gral, 1769, p. 73.</p><p><hi rend="italics">M&#xE9;moire</hi> sur les Principes de la Mech. 1769, p. 278.</p><p>And in the Memoirs of the Academy of Berlin, are
the following pieces, by our author: viz,</p><p><hi rend="italics">Recherches</hi> sur le Calcul Int&#xE9;gral, premiere partie,
1746.</p><p><hi rend="italics">Solution</hi> de quelques problemes d'astronomie, 1747.</p><p><hi rend="italics">Recherches</hi> sur la cour be que forme une Corde Tendue,
mise en Vibration, 1747.</p><p><hi rend="italics">Suite</hi> des recherches sur le Calcul Int&#xE9;gral, 1748.</p><p><hi rend="italics">Lettre</hi> &#xE0; M. de Maupertuis, 1749.</p><p><hi rend="italics">Addition</hi> aux recherches sur la courbe que forme une
Corde Tendue mise en Vibration, 1750.</p><p><hi rend="italics">Addition</hi> aux recherches sur le Calcul Int&#xE9;gral, 1750.</p><p><hi rend="italics">Lettre</hi> &#xE0; M. le professeur Formey, 1755.</p><p><hi rend="italics">Extr.</hi> de differ. lettres &#xE0; M. de la Grange, 1763.</p><p><hi rend="italics">Sur</hi> les Tautochrones, 1765.</p><p><hi rend="italics">Extr.</hi> de differ. lettres &#xE0; M. de la Grange, 1769.</p><p>Also in the Memoirs of Turin are,</p><p><hi rend="italics">Differentes</hi> Lettres &#xE0; M. de la Grange, en 1764 &amp;
1765, tom. 3 of these Memoris.</p><p><hi rend="italics">Recherches</hi> sur differens sujets de Math. t. 4.</p></div1><div1 part="N" n="ALFECCA" org="uniform" sample="complete" type="entry"><head>ALFECCA</head><p>, or Alfeta, a name given to the star
commonly called Lucida Coron&#xE6;.</p></div1><div1 part="N" n="ALFRAGAN" org="uniform" sample="complete" type="entry"><head>ALFRAGAN</head><p>, <hi rend="smallcaps">Alfergani</hi>, or <hi rend="smallcaps">Fargani</hi>, a celebrated
Arabic astronomer, who flourished about the
year 800. He was so called from the place of his nativity,
Fergan, in Sogdiana, now called Maracanda, or
Samarcand, anciently a part of Bactria. He is also
called Ahmed (or Muhammed) ben-Cothair, or Katir.
He wrote the Elements of Astronomy, in 30 chapters
or sections. In this work the author chiefly follows
Ptolomy, using the same hypotheses, and the same terms,
and frequently citing him.</p><p>There are three Latin translations of Alfragan's
<cb/>
work. The first was made in the 12th century, by
Joannes Hispalensis; and was published at Ferrara in
1493, and at Nuremberg in 1537, with a preface by
Melancthon. The second was by James Christman,
from the Hebrew version of James Antoli, and appeared
at Frankfort in 1590. Christman added to the first
chapter of the work an ample commentary, in which he
compares together the calendars of the Romans, the
Egyptians, the Arabians, the Persians, the Syrians, and
the Hebrews, and shews the correspondence of their
years.</p><p>The third and best translation was made by Golius,
professor of mathematics and Oriental languages at
Leyden: this work, which came out in 1669, after
the death of Golius, is accompanied with the Arabic
text, and many learned notes upon the first nine chapters;
for this author was not spared to carry them
farther.</p></div1><div1 part="N" n="ALGAROTI" org="uniform" sample="complete" type="entry"><head>ALGAROTI</head><p>, commonly called <hi rend="italics">Count Algaroti,</hi> a
celebrated Italian of the present century, well skilled in
Architecture and the Newtonian philosophy, &amp;c. Algaroti
was born at Padua, but in what year has not
been mentioned. Led by curiosity, as well as a desire
of improvement, he travelled early into foreign countries;
and was very young when he arrived in France in 1736.
It was here that he composed his <hi rend="italics">Newtonian Philosophy
for the Ladies,</hi> as Fontenelle had done his Cartesian
Astronomy, in the work intitled <hi rend="italics">The Plurality of Worlds.</hi>
He was much noticed by the king of Prussia, who
conferred on him many marks of his esteem. He died
at Pisa the 23d of May, 1764, and gave orders for his
own mausoleum, with this inscription upon it; <hi rend="italics">Hic
jacet Algarotus, sed non omnis.</hi> He was esteemed to be
well skilled in painting, sculpture, and architecture.
His works, which are numerous, and upon a variety of
subjects, abound with vivacity, elegance, and wit: a
collection of them has lately been made, and printed
at Leghorn; but that for which he is chiefly intitled
to a place in this work is his <hi rend="italics">Newtonian Philosophy for
the Ladies,</hi> a sprightly, ingenious, and popular work.</p></div1><div1 part="N" n="ALGEBRA" org="uniform" sample="complete" type="entry"><head>ALGEBRA</head><p>, a general method of resolving mathematical
problems by means of equations. Or, it is
a method of performing the calculations of all sorts of
quantities by means of general signs or characters. At
first, numbers and things were expressed by their names
at full length; but afterwards these were abridged,
and the initials of the words used instead of them; and,
as the art advanced farther, the letters of the alphabet
came to be employed as general representations of all
sorts of quantities; and other marks were gradually
introduced, to express all sorts of operations and combinations;
so as to entitle it to different appellations&#x2014;
universal arithmetic, and literal arithmetic, and the
arithmetic of signs.</p><p>The etymology of the name, <hi rend="italics">Algebra,</hi> is given in
various ways. It is pretty certain, however, that the
word is Arabian, and that from those people we had
the name, as well as the art itself, as is testisied by
Lucas le Burgo, the first European author whose treatise
was printed on this art, and who also refers to former
authors and masters, from whose writings he had
learned it. The Arabic name he gives it, is <hi rend="italics">Alghebra
e Almucabala,</hi> which is explained to signify the art of
<pb n="64"/><cb/>
<hi rend="italics">restitution and comparison,</hi> or <hi rend="italics">opposition and comparison,</hi> or
<hi rend="italics">resolution and equation,</hi> all which agree well enough with
the nature of this art. Some however derive it from
various other arabic words; as from Geber, a celebrated
philosopher, chemist, and mathematician, to whom
also they ascribe the invention of this science: some
likewise derive it from the word Geber, which with the
particle <hi rend="italics">al,</hi> makes Algeber, which is purely Arabic,
and signifies the reduction of broken numbers or fractions
to integers.</p><p>But Peter Ramus, in the beginning of his Algebra,
says &#x201C;the name Algebra is Syriac, signifying the art
and doctrine of an excellent man. For <hi rend="italics">Geber,</hi> in Syriac,
is a name applied to men, and is sometimes a term of honour,
as master or doctor among us. That there was a
certain learned mathematician, who sent his Algebra,
written in the Syriac language, to Alexander the Great,
and he named it <hi rend="italics">Almucabala,</hi> that is, the book of dark or
mysterious things, which others would rather call the
doctrine of Algebra. And to this day the same book
is in great estimation among the learned in the oriental
nations, and by the Indians who cultivate this art it is
called <hi rend="italics">Aljabra,</hi> and <hi rend="italics">Alboret;</hi> though the name of the
author himself is not known.&#x201D; But Ramus gives no
authority for this singular paragraph. It has however
on various occasions been distinguished by other names.
Lucas Paciolus, or de Burgo, in Italy, called it <hi rend="italics">l'Arte
Magiore: ditta dal vulgo la Regola de la Cosa over Alghebra
e Almucabala;</hi> calling it <hi rend="italics">l'Arte Magiore,</hi> or the
greater art, to distinguish it from common arithmetic,
which is called <hi rend="italics">l'Arte Minore,</hi> or the lesser art. It seems
too that it had been long and commonly known in his
country by the name <hi rend="italics">Regola de la Cosa,</hi> or <hi rend="italics">Rule of the
Thing;</hi> from whence came our rule of coss, cosic numbers,
and such like terms. Some of his countrymen
followed his denomination of the art; but other Italian
and Latin writers called it <hi rend="italics">Regula rei &amp; census,</hi> the rule
of the thing and the product, or the root and the
square, as the unknown quantity in their equations
commonly ascended no higher than the square or second
power. From this Italian word <hi rend="italics">census,</hi> pronounced
<hi rend="italics">chensus,</hi> came the barbarous word <hi rend="italics">zenzus,</hi> used by the
Germans and others, for quadratics; with the several zenzic
or square roots. And hence [scruple], [dram], <figure/>, which
are derived from the letters <hi rend="italics">r, z, c,</hi> the initials of <hi rend="italics">res,
zenzus, cubus,</hi> or root, square, cube, came to be the
signs or characters of these words: like as &#x211E; and &#x221A;,
derived from the letters <hi rend="italics">R, r,</hi> became the signs of radicality.</p><p>Later authors, and other nations, used some the one
of those names, and some another. It was also called
<hi rend="italics">Specious Arithmetic</hi> by Vieta, on account of the species,
or letters of the alphabet, which he brought into general
use; and by Newton it was called <hi rend="italics">Universal
Arithmetic,</hi> from the manner in which it performs all
arithmetical operations by general symbols, or indeterminate
quantities.</p><p>Some authors define algebra to be <hi rend="italics">the art of resolving
mathematical problems:</hi> but this is the idea of analysis,
or the analytic art in general, rather than of algebra,
which is only one particular species of it.</p><p>Indeed algebra properly consists of two parts: first,
the method of calculating magnitudes or quantities, as
represented by letters or other characters: and secondly
<cb/>
the manner of applying these calculations in the solution
of problems.</p><p>In algebra, as applied to the resolution of problems,
the first business is to translate the problem out of the
common into the algebraic language, by expressing all
the conditions and quantities, both known and unknown,
by their proper characters, arranged in an
equation, or several equations if necessary, and treating
the unknown quantity, whether it be number, or
line, or any other thing, in the same way as if it were
a known one: this forms the composition. Then the
resolution, or analytic part, is the disentangling the unknown
quantity from the several others with which it
is connected, so as to retain it alone on one side of the
equation, while all the other, or known, quantities,
are collected on the other side, and so giving the value
of the unknown one. And as this disentangling of the
quantity sought, is performed by the converse of the
operations by which it is connected with the others,
taking them always backwards in the contrary order,
it hence becomes a species of the analytic art, and is
called the modern analysis, in contradistinction to the
ancient analysis, which chiefly respected geometry, and
its applications.</p><p>There have arisen great controversies and sharp disputes
among authors, concerning the history of the
progress and improvements of Algebra; arifing partly
from the partiality and prejudices which are natural to
all nations, and partly from the want of a closer examination
of the works of the older authors on this subject.
From these causes it has happened, that the improvements
made by the writers of one nation, have
been ascribed to those of another; and the discoveries of
an earlier author, to some one of much later date. Add to
this also, that the peculiar methods of many authors
have been described so little in detail, that our information
derived from such histories, is but very imperfect,
and amounting only to some general and vague ideas of
the true state of the arts. To remedy this inconvenience
therefore, and to reform this article, I have taken
the pains carefully to read over in succession all the older
authors on this subject, which I have been able to
meet with, and to write down distinctly a particular account
and description of their several compositions, as
to their contents, notation, improvements, and peculiarities;
from the comparison of all which, I have acquired
an idea more precise and accurate than it was possible
to obtain from other histories, and in a great many
instances very different from them. The full detail of
these descriptions would employ a volume of itself,
and would be far too extensive for this place: I must
therefore limit this article to a very brief abridgment of
my notes, remarking only the most material circumstances
in each author; from which a general idea of
the chain of improvements may be perceived, from the
first rude beginnings, down to the more perfect state;
from which it will appear that the discoveries and improvements
made by any one single author, are scarcely
ever either very great or numerous; but that, on the
contrary, the improvements are almost always very slow
and gradual, from former writers, successively made,
not by great leaps, and after long intervals of time,
but by gradations which, viewed in succession, become
almost imperceptible.
<pb n="65"/><cb/></p><p>As to the origin of the analytic art, of which Algebra
is a species, it is doubtless as old as any science in the
world, being the natural method by which the mind investigates
truths, causes, and theories, from their observed
effects and properties. Accordingly, traces of
it are observable in the works of the earliest philosophers
and mathematicians, the subject of whose enquiries
most of any require the aid of such an art. And
this process constituted their Analytics. Of that part
of analytics however which is properly called Algebra,
the oldest treatise which has come down to us, is that
of Diophantus of Alexandria, who flourished about the
year 350 after Christ, and who wrote, in the Greek language,
13 books of Algebra or Arithmetic, as mentioned
by himself at the end of his address to Dionysius, though
only 6 of them have hitherto been printed; and an imperfect
book on multangular numbers, namely in a
Latin translation only, by Xilander, in the year 1575,
and afterwards in 1621 and 1670 in Greek and Latin
by Gaspar Bachet. These books however do not contain
a treatise on the elementary parts of Algebra, but
only collections of difficult questions relating to square
and cube numbers, and other curious properties of
numbers, with their solutions. And Diophantus only
prefaces the books by an address to one Dionysius,
for whose use it was probably written, in which he
just mentions certain precognita, as it were to prepare
him for the problems themselves. In these remarks he
shews the names and generation of the powers, the
square, cube, 4th, 5th, 6th, &amp;c, which he calls dynamis,
cubus, dynamodinamis, dynamocubus, cubocubus,
according to the sum of the indices of the powers;
and he marks these powers with the initials thus <foreign xml:lang="greek">d<hi rend="sup">n_</hi>, k<hi rend="sup">n_</hi>,
dd<hi rend="sup">n_</hi>, dk<hi rend="sup">n_</hi>, kk<hi rend="sup">n_</hi></foreign>, &amp;c: the unknown quantity he calls simply
<foreign xml:lang="greek">ariqmos</foreign>, <hi rend="italics">numerus,</hi> the number; and in the solutions he
commonly marks it by the final thus <foreign xml:lang="greek">s_</foreign>; also he denotes
the monades, or indefinite unit, by <foreign xml:lang="greek">m<hi rend="sup">o_</hi></foreign>. Diophantus there
remarks on the multiplication and division of simple species
together, shewing what powers or species they produce;
declares that minus (<foreign xml:lang="greek">leiyis</foreign>) multiplied by minus
produces plus (<foreign xml:lang="greek">nparcin</foreign>); but that minus multiplied by
plus, produces minus; and that the mark used for minus
is &lt;*&gt; namely the <foreign xml:lang="greek">y</foreign> inverted and curtailed, but he
uses no mark for plus, but a word or conjunction copulative.
As to the operations, viz. of addition, subtraction,
multiplication, and division of compound
species, or those connected by plus and minus, Diophantus
does not teach, but supposes his reader to
know them. He then remarks on the preparation
or simplifying of the equations that are derived from
the questions, which we call reduction of equations,
by collecting like quantities together, adding quantities
that are minus, and subtracting such as are
plus, called by the moderns Transposition, so as to bring
the equation to simple terms, and then depressing it to
a lower degree by equal division when the powers of the
unknown quantity are in every term: which preparation,
or reduction of the complex equation, being now
made, or reduced to what we call a final equation, Diophantus
goes no farther, but barely says what the root
or <hi rend="italics">res ignota</hi> is, without giving any rules for finding it,
or for the resolution of equations; thereby intimating
that such rules were to be found in some other work,
done either by himself or others. Of the body of the
<cb/>
work, <hi rend="italics">Lib.</hi> 1 contains 43 questions, concerning one,
two, three, or four unknown numbers, having certain
relations to each other, viz. concerning their sums, differences,
ratios, products, squares, sums and differences
of squares, &amp;c, &amp;c; but none of them concerning
either square or cubic numbers. <hi rend="italics">Lib.</hi> 2 contains 36
questions. The first five questions are concerning two
numbers, though only one condition is given in each
question; but he supplies another by assuming the numbers
in a given ratio, viz, as 2 to 1. The 6th and 7th
contain each two conditions: then in the 8th question
he first comes to trcat of square numbers, which is this,
to divide a given square number into two other squares;
and the 9th is the same, but performed in a different
way: the rest, to the end, are, almost all, about one,
two, or three squares. <hi rend="italics">Lib.</hi> 3 contains 24 questions
concerning squares, chiefly including three or four numbers.
<hi rend="italics">Lib.</hi> 4 begins with cubes; the first of which is
this, to divide a given number into two cubes whose
sides shall have a given sum: here he has occasion to
cube the two binomials 5+<hi rend="italics">n</hi> and 5-<hi rend="italics">n;</hi> the manner of
doing which shews that he knew the composition of the
cube of a binomial; and many other places manifest the
same thing. Only part of the questions in this book
are concerning cubes; the rest are relating to squares.
Two or three questions in this book have general solutions,
and the theorems deduced are general, and for any
numbers indefinitely; but all the other questions, in all
the four books, find only particular numbers. <hi rend="italics">Lib.</hi> 5 is
also concerning square and cube numbers, but of a more
difficult kind, beginning with some that relate to numbers
in geometrical progression. <hi rend="italics">Lib.</hi> 6 contains 26 propositions,
concerning right-angled triangles; such as to
make their sides, areas, perimeters, &amp;c, &amp;c, squares
or cubes, or rational, &amp;c. In some parts of this book
it appears, that he was acquainted with the composition
of the 4th power of the binomial root, as he sets down all
the terms of it; and, from his great skill in such matters,
it seems probable that he was acquainted with the composition
of other higher powers, and with other parts of Algebra,
besides what are here treated of. At the end is part
of a book, in 10 propositions, concerning arithmetical progressions,
and multangular or polygonal numbers. Diophantus
once mentions a compound quadratic equation;
but the resolution of his questions is by simple equations,
and by means of only one unknown letter or character,
which he chooses so ingeniously, that all the
other unknown quantities in the question are easily expressed
by it, and the final equation reduced to the simplest
form which it seems the question can admit of.
Sometimes he substitutes for a number sought immediately,
and then expresses the other numbers or conditions
by it: at other times he substitutes for the sum or
difference, &amp;c, and thence derives the rest, so as always
to obtain the expressions in the simplest form. Thus,
if the sum of two numbers be given, he substitutes for
their difference; and if the difference be given, he substitutes
for their sum: and in both cases he has the two
numbers easily expressed by adding and subtracting the
half sum and half difference; and so in other cases he
uses other similar ingenious notations. In short, the
chief excellence in this collection of questions, which
seems to be only a set of exercises to some rules which
had been given elsewhere, is the neat mode of substitution
or notation; which being once made, the reduc-
<pb n="66"/><cb/>
tion to the final equation is easy and evident: and there
he leaves the solution, only mentioning that the root or
<foreign xml:lang="greek">ariqmos</foreign> is so much. Upon the whole, this work is treated
in a very able and masterly manner, manifesting the utmost
address and knowledge in the solutions, and forcing
a persuasion that the author was deeply skilled in
the science of Algebra, to some of the most abstruse
parts of which these questions or exercises relate. However,
as he contrives his assumptions and notations so
as to reduce all his conditions to a simple equation, or
at least a simple quadratic, it does not appear what his
knowledge was in the resolution of compound or affected
equations.</p><p>But although Diophantus was the first author on Algebra
that we now know of, it was not from him, but from
the Moors or Arabians that we received the knowledge
of Algebra in Europe, as well as that of most other
sciences. And it is matter of dispute who were the
first inventors of it; some ascribing the invention to
the Greeks, while others say that the Arabians had it
from the Persians, and these from the Indians, as well
as the arithmetical method of computing by ten characters,
or digits; but the Arabians themselves say it
was invented amongst them by one <hi rend="italics">Mahomet ben Musa,</hi>
or son of Moses, who it seems flourished about the 8th
or 9th century. It is more probable, however, that
Mahomet was not the inventor, but only a person well
skilled in the art; and it is farther probable, that the
Arabians drew their first knowledge of it from Diophantus
or other Greek writers, as they did that of
Geometry and other sciences, which they improved and
translated into their own language; and from them it
was that we received these sciences, before the Greek
authors were known to us, after the Moors settled in
Spain, and after the Europeans began to hold communications
with them, and that our countrymen began
to travel amongst them to learn the sciences. And according
to the testimony of Abulpharagius, the Arithmetic
of Diophantus was translated in Arabic by Mahomet
ben-yahya Ba&lt;*&gt;iani. But whoever were the
inventors and first cultivators of Algebra, it is certain
that the Europeans first received the knowledge, as
well as the name, from the Arabians or Moors, in consequence
of the close intercourse which subsisted between
them for several centuries. And it appears that
the art was pretty generally known, and much cultivated,
at least in Italy, if not in other parts of Europe
also, long before the invention of printing, as many
writers upon the art are still extant in the libraries of
manuscripts; and the first authors, presently after the
invention of printing, speak of many former writers
on this subject, from whom they learned the art.</p><p>It was chiefly among the Italians that this art was
first cultivated in Europe. And the first author whose
works we have in print, was Lucas Paciolus, or Lucas
de Burgo, a Cordelier, or Minorite Friar. He wrote
several treatises of Arithmetic, Algebra, and Geometry,
which were printed in the years 1470, 1476, 1481,
1487, and in 1494 his principal work, intitled <hi rend="italics">Summa
de Arithmetica, Geometria, Proportioni, et Proportionalita,</hi>
is a very masterly and complete treatise on those sciences,
as they then stood. In this work he mentions various
former writers, as Euclid, St. Augustine, Sacrobosco
or Halifax, Boetius, Prodocimo, Giordano,
<cb/>
Biagio da Parma, and Leonardus Pisanus, from whom
he learned those sciences. The order of the work is,
1st Arithmetic, 2d Algebra, and 3d Geometry. Of
the Arithmetic the contents, and the order of them,
are nearly as follow. First, of numbers figurate, odd
and even, perfect, prime and composite, and many
others. Then of Common Arithmetic in 7 parts, namely
numeration or notation, addition, subtraction, multiplication,
division, progression, and extraction of roots.
Before him, he says, duplation and mediation, or doubling
and halving, were accounted two rules in Arithmetic;
but that he omits them, as being included in multiplication
and division. He ascribes the present notation
and method of Arithmetic to the Arabs; and says that
according to some the word <hi rend="italics">Abaco</hi> is a corruption of
<hi rend="italics">Modo Arabico,</hi> but that according to others it was from
a Greek word. All those primary operations he both performs
and demonstrates in various ways, many of which
are not in use at present, proving them not only by
what is called casting out the nines, but also by casting
out the sevens, and otherwise. In the extraction of
roots he uses the initial &#x211E; for a root; and when the
roots can be extracted, he calls them discrete or rational;
otherwise surd, or indiscrete, or irrational. The
square root is extracted much the same way as at present,
namely, dividing always the last remainder by
double the root found; and so he continues the surd
roots continually nearer and nearer in vulgar fractions.
Thus, for the root of 6, he firsts finds the nearest whole
number 2, and the remainder 2 also; then 2/4 or 1/2 is the
first correction, and 2 1/2 the second root: its square is 6 1/4,
therefore 1/4 divided by 5, or 1/20 is the next correction,
and 2 1/2 minus 1/20, or 2 9/20 is the 3d root: its square is
6 1/400, therefore 1/400 divided by 4 9/10, or 1/1960, is the 3d
correction, which gives 2 881/1960 for the 4th root, whose
square exceeds 6 by only 1/3841600: and so on continually:
and this process he calls approximation. He
observes that fractions, which he sets down the same
way as we do at present, are extracted, by taking the
root of the denominator, and of the denominated, for
so he calls the numerator: and when mixed numbers
occur, he directs to reduce the whole to a fraction, and
then extract the roots of its two terms as above: as if
it be 12 1/4; this he reduces to 49/4, and then the roots give
7/2 or 3 1/2: in like manner he finds that 4 1/2 is the root of
20 1/4; 5 1/2 the root of 30 1/4; &#x201C;and so on (he adds) in
infinitum;&#x201D; which shews that he knew how to form the
series of squares by addition. He then extracts the
cube root, by a rule much the same as that which is
used at present; from which it appears that he was
well acquainted with the co-efficients of the binomial
cubed, namely 1, 3, 3, 1; and he directs how the
operation may be continued &#x201C;in infinitum&#x201D; in fractions,
like as in the square root. After this, he describes
geometrical methods for extracting the square and cube
roots instrumentally: he then treats professedly of vulgar
fractions, their reductions, addition, subtraction, and
other operations, much the fame as at present: then of
the rule-of-three, gain-and-loss, and other rules used by
merchants.</p><p>Paciolus next enters on the algebraical part of this
work, which he calls &#x201C;<hi rend="italics">L'Arte Magiore; ditta dal vulgo
la Regola de la Cosa, over Alghebra e Ahnucabala:</hi>&#x201D; which
last name he explains by <hi rend="italics">restauratio &amp; oppositio,</hi> and
<pb n="67"/><cb/>
assigns as a reason for the first name, because it treats of
things above the common affairs in business, which
make the <hi rend="italics">Arte Minore.</hi> Here he ascribes the invention
of Algebra to the Arabians, and denominates the series
of powers, with their marks or abbreviations, as <hi rend="italics">n</hi>&#xB0;,
or <hi rend="italics">numero,</hi> the absolute or known number; <hi rend="italics">co.</hi> or <hi rend="italics">cosa,</hi>
the thing or 1st power of the unknown quantity; <hi rend="italics">ce.</hi>
or <hi rend="italics">censo,</hi> the product or square; <hi rend="italics">cu.</hi> or <hi rend="italics">cubo,</hi> the cube,
or 3d power; <hi rend="italics">ce. ce.</hi> or <hi rend="italics">censo de censo,</hi> the square-squared,
or 4th power; <hi rend="italics">p</hi>&#xB0;. <hi rend="italics">r</hi>&#xB0;. or <hi rend="italics">primo relato,</hi> or 5th power;
<hi rend="italics">ce. cu.</hi> or <hi rend="italics">censo de cubo,</hi> the square of the cube, or 6th
power; and so on, compounding the names or indices
according to the multiplication of the numbers 2, 3, &amp;c,
and not according to their sum or addition, as used by
Diophantus. He describes also the other characters
made use of in this part, which are for the most part no
more than the initials or other abbreviations of the words
themselves; as &#x211E; for <hi rend="italics">radici,</hi> the root; &#x211E;. &#x211E;. <hi rend="italics">radici de
radici,</hi> the root of the root; &#x211E; u. <hi rend="italics">radici universale,</hi> or
<hi rend="italics">radici legata,</hi> or <hi rend="italics">radici unita;</hi> &#x211E; <hi rend="italics">cu. radici cuba;</hi> and
&#x2015;q[dram]<hi rend="sup">&lt;*&gt;</hi> <hi rend="italics">quantita,</hi> quantity; <hi rend="italics">p</hi> for <hi rend="italics">piu</hi> or plus, and <hi rend="italics">m</hi> for
<hi rend="italics">meno</hi> or minus; and he remarks that the necessity and
use of these two last characters are for connecting, by
addition or subtraction, different powers together; as
3 <hi rend="italics">co.</hi> p. 4 <hi rend="italics">ce.</hi> m. 5 <hi rend="italics">cu.</hi> p. 2 <hi rend="italics">ce. ce.</hi> m. 6 <hi rend="italics">n</hi><hi rend="sup">i</hi>. that is, 3
<hi rend="italics">cosa</hi> piu 4 <hi rend="italics">censa</hi> meno 5 <hi rend="italics">cubo</hi> piu 2 <hi rend="italics">censa-censa</hi> meno
6 <hi rend="italics">numeri,</hi> or, as we now write the same thing, 3<hi rend="italics">x</hi>+4<hi rend="italics">x</hi><hi rend="sup">2</hi>
- 5<hi rend="italics">x</hi><hi rend="sup">3</hi> + 2<hi rend="italics">x</hi><hi rend="sup">4</hi> - 6. He first treats very fully of proportions
and proportionalities, both arithmetical and
geometrical, accompanied with a large collection of
questions concerning numbers in continued proportion,
resolved by a kind of Algebra. He then treats of <hi rend="italics">el Cataym,</hi>
which he says, according to some, is an Arabic or
Phenician word, and signifies the Double Rule of False
Position: but he here treats of both single and double
position, as we do at present, dividing the <hi rend="italics">el Cataym</hi> into
single and double. He gives also a geometrical demonstration
of both the cases of the errors in the double
rule, namely when the errors are both plus or both
minus, and when the one error is plus and the other
minus; and adds a large collection of questions, as usual.
He then goes through the common operations of Algebra,
with all the variety of signs, as to plus and minus;
proving that, in multiplication and division, like signs
give plus, and unlike signs give minus. He next treats
of different roots <hi rend="italics">in infinitum,</hi> and the extraction of
roots; giving also a copious treatise on radicals or surds,
as to their addition, subtraction, multiplication and
division, and that both in square roots and cube roots,
and in the two together, much the same as at present.
He makes here a digression concerning the 15 lines in
the 10th book of Euclid, treating them as surd numbers,
and teaching the extraction of the roots of the same,
or of compound surds or binomials, such as of 23 <hi rend="italics">p</hi> &#x211E;
448, or of &#x211E; 18 <hi rend="italics">p</hi> &#x211E; 10; and gives this rule, among
several others, namely: Divide the first term of the binomial
into two such parts that their product may be 1/4
of the number in the second term; them the roots of
those two parts, connected by their proper sign <hi rend="italics">p</hi> or <hi rend="italics">m,</hi>
is the root of the binomial; as in this 23 <hi rend="italics">p</hi> &#x211E; 448, the
two parts of 23 are 7 and 16, whose product, 112, is 1/4
of 448, therefore their roots give 4 <hi rend="italics">p</hi> &#x211E; 7 for the root
&#x211E; u. 23 <hi rend="italics">p</hi> &#x211E; 448. He next treats of equations both
simple and quadratic, or simple and compound, as he
<cb/>
calls it; and this latter he performs by completing the
square, and then extracting the root, just as we do at
present. He also resolves equations of the simple 4th
power, and of the 4th combined with the 2d power,
which he treats the same way as quadratics; expressing
his rules in a kind of bad verse, and giving geometrical
demonstrations of all the cases. He uses both the roots
or values of the unknown quantity, in that case of the
quadratics which has two positive roots; but he takes
no notice of the negative roots in the other two cases.
But as to any other compound equations, such as the
cube and any other power, or the 4th and 1st, or 4th
and 3d, &amp;c, he gives them up as impossible, or at least
says that no general rule has yet been found for them,
any more, he adds, than for the quadrature of the circle.
&#x2014;The remainder of this part is employed on rules in
trade and merchandise, such as Fellowship, Barter, Exchange,
Interest, Composition or Alligation, with
various other cases in trade. And in the third part of
the work, he treats of Geometry, both theoretical and
practical.</p><p>From this account of Lucas de Burgo's book, we
may perceive what was the state of Algebra about the
year 1500, in Europe; and probably it was much the
same in Africa and Asia, from whence the Europeans
had it. It appears that their knowledge extended only
to quadratic equations, of which they used only the
positive roots; that they used only one unknown
quantity; that they had no marks or signs for either
quantities or operations, excepting only some few abbreviations
of the words or names themselves; and
that the art was only employed in resolving certain numeral
problems. So that either the Africans had not
carried Algebra beyond quadratic equations, or else the
Europeans had not learned the whole of the art, as it
was then known to the former. And indeed it is not
improbable but this might be the case: for whether the
art was brought to us by an European, who, travelling
in Africa, there learned it; or whether it was brought
to us by an African; in either case we might receive
the art only in an imperfect state, and perhaps far
short of the degree of perfection to which it had been
carried by their best authors. And this suspicion is
rendered rather probable by the circumstance of an
Arabic manuscript, said to be on cubic equations, deposited
in the Library of the university of Leyden by
the celebrated Warner, bearing a title which in Latin
signifies <hi rend="italics">Omar Ben Ibrahim al'Ghajam&#xE6;i Algebra cubicarum
&#xE6;quationum, sive de problematum solidorum resolutione;</hi> and
of which book I am in some hopes of procuring either
a copy or a translation, by means of my worthy friend
Dr. Damen, the learned Professor of Mathematics in that
university, and by that means to throw some light on
this doubtful subject.</p><p>Since this was written, death has prematurely put an
end to the useful labours of this ingenious and worthy
successor of Gravesande.</p><p>After the publication of the books of Lucas de Burgo,
the science of Algebra became more generally known,
and improved, especially by many persons in Italy; and
about this time, or soon after, namely about the year
1505, the first rule was there found out by Scipio
Ferreus, for resolving one case of a compound cubic
equation. But this science, as well as other branches of
<pb n="68"/><cb/>
Mathematics, was most of all cultivated and improved
there by Hieronymus Cardan of Bononia, a very learned
man, whose arithmetical writings were the next that
appeared in print, namely in the year 1539, in 9 books,
in the Latin language, at Milan, where he practised
physic, and read public lectures on Mathematics; and
in the year 1545 came out a 10th book, containing the
whole doctrine of cubic equations, which had been in
part revealed to him about the time of the publication
of his first 9 books. And as it is only this 10th book
which contains the new discoveries in Algebra, I shall
here confine myself to it alone, as it will also afford
sufficient occasion to speak of his manner of treating
Algebra in general. This book is divided into 40 chapters,
in which the whole science of cubic equations is
most amply and ably treated. Chap. 1 treats of the
nature, number and properties of the roots of equations,
and particularly of single equations that have double
roots. He begins with a few remarks on the invention
and name of the art: calls it <hi rend="italics">Ars Magna,</hi> or <hi rend="italics">Cosa,</hi> or
<hi rend="italics">Rules of Algebra,</hi> after Lucas de Burgo and others:
says it was invented by Mahomet, the son of one Moses
an Arabian, as is testified by Leonardus Pisanus; and
that he left four rules or cases, which perhaps only included
quadratic equations: that afterwards three derivatives
were added by an unknown author, though
some think by Lucas Paciolus; and after that again
three other derivatives, for the cube and 6th power, by
another unknown author; all which were resolved like
quadratics: that then Scipio Ferreus, Professor of
Mathematics at Bononia, about 1505, found out the
rule for the case <hi rend="italics">cubum &amp; rerum numero &#xE6;qualium,</hi> or, as
we now write it, , which he speaks of as a
thing admirable: that the same thing was next afterwards
found out, in 1535, by Tartalea, who revealed
it to him, Cardan, after the most earnest intreaties:
that, finally, by himself and his quondam pupil
Lewis Ferrari, the cases are greatly augmented and
extended, namely, by all that is not here expressly
ascribed to others; and that all the demonstrations
of the rules are his own, except only three adopted
from Mahomet for the quadratics, and two of Ferrari
for cubics.</p><p>He then delivers some remarks, shewing that all
square numbers have two roots, the one positive, and
the other negative, or, as he calls them, <hi rend="italics">vera &amp; ficta,</hi>
true and fictitious or false; so the <hi rend="italics">&#xE6;stimatio rei,</hi> or root,
of 9, is either 3 or - 3; of 16 it is 4 or - 4; the 4th
root of 81 is 3 or - 3; and so on for all even <hi rend="italics">denominations</hi>
or powers. And the same is remarked on compound
cases of even powers that are added together;
as if , then the &#xE6;stimatio <hi rend="italics">x</hi> is=2 or-2;
but that the form  has four answers or
roots, in real numbers, two plus and two minus, viz. 2 or
- 2, and &#x221A; 3 or - &#x221A; 3; while the case 
has no real roots; and the case  has two,
namely 2 and - 2: and in like manner for other even
powers. So that he includes both the positive and negative
roots; but rejects what we now call imaginary ones. I
here express the cases in our modern notation, for brevity
sake, as he commonly expresses the terms by words at
full length, calling the ablolute or known term the <hi rend="italics">numero,</hi>
the 1st power the <hi rend="italics">res,</hi> the 2d the <hi rend="italics">quadratum,</hi> the
3d the <hi rend="italics">cubum,</hi> and so on, using no mark for the unknown
<cb/>
quantity, and only the initials <hi rend="italics">p</hi> and <hi rend="italics">m</hi> for plus and
minus, and &#x211E; for radix or root. The <hi rend="italics">res</hi> he sometimes
calls <hi rend="italics">positio,</hi> and <hi rend="italics">quantitas ignota;</hi> and in stating or
setting down his equations, he, as well as Lucas de
Burgo before him, sets down the terms on that side
where they will be plus, and not minus.</p><p>On the other hand, he remarks that the odd denominations,
or powers, have only one &#xE6;stimatio, or root,
and that true or positive, but none sictitious or negative,
and for this reason, that no negative number raised
to an odd power, will give a positive number; so of
2<hi rend="italics">x</hi>=16, the root is 8 only; and if 2<hi rend="italics">x</hi><hi rend="sup">3</hi>=16, the root
is 2 only: and if there be ever so many odd denominations,
added together, equal to a number, there will
be only one &#xE6;stimatio or root; as if ,
the only root is 2. But that when the signs of some
of the terms are different as to plus and minus, they
may have more roots; and he shews certain relations of
the co-efficients, when they have two or more roots:
so the equation  has two &#xE6;stimatios,
the one true or 2, and the other fictitious or - 4, which
he observes is the same as the true &#xE6;stimatio of the case
, having only the sign of the absolute
number changed from the former, the 3d root 2 being
the same as the first, which therefore he does not count.
He next shews what are the relations of the co-efficients
when a cubic equation has three roots, of which two are
true, and the 3d fictitious, which is always equal to the
sum of the other two, and also equal to the true root of
the same equation with the sign of the absolute number
changed: thus, in the equation , the
two true roots are 3 and &#x221A;5 1/4 - 1 1/2, and the fictitious
one is - &#x221A; 5 1/4 - 1 1/2, which last is the same as the true
root of , viz. &#x221A; 5 1/4 + 1 1/2; and he here
infers generally that the fictitious &#xE6;stimatio of the
case ,
always answers to the true root of .
Cardan also shews what the relation of the co-efficients
is, when the case has no true roots, but only one fictitious
root, which is the same as the true root of the reciprocal
case, formed by changing the sign of the absolute number.
Thus, the case  has no true root,
and only one false root, viz. - 3, which is the same as
the true root of : and he shews in general,
that changing the sign of the absolute number in such
cases as want the 2d term, or changing the signs of the
even terms when it is not wanting, changes the signs
of all the three roots, which he also illustrates by many
examples; thus, the roots of , are
+ &#x221A;40 - 4, and - 3, and - &#x221A;40 - 4; and the
roots of , are - &#x221A;40 + 4, and + 3,
and + &#x221A;40 + 4.</p><p>And he further observes, that the sum of the three
roots, or the difference between the true and sictitious
roots, is equal to 11, the co-efficient of the 2d term.
He also shews how certain cubic cases have one, or two,
or three roots, according to circumstances: that the
case  has sometimes four roots, and sometimes
none at all, that is, no real ones: that the case
 may have three true &#xE6;quatios, or
positive roots, but no fictitious or negative ones; and
for this reason, that the odd powers of minus being
minus, and the even powers plus, the two terms <hi rend="italics">x</hi><hi rend="sup">3</hi>+<hi rend="italics">bx</hi>
would be negative, and equal to a positive sum <hi rend="italics">ax</hi><hi rend="sup">2</hi> + <hi rend="italics">c,</hi>
<pb n="69"/><cb/>
which is absurd: and farther, that the case  has three roots, one plus and two minus,
which are the same, with the signs changed, as the
roots of the case . He also shews
the relation of the co-efficients when the equation has
only one real root, in a variety of cases: but that the
case  has always one negative root,
and either two positive roots, or none at all; the number
of roots failing by pairs, or the impossible roots, as
we now call them, being always in pairs. Of all these
circumstances Cardan gives a great many particular
examples in numeral co-efficients, and subjoins geometrical
demonstrations of the properties here enumerated;
such as, that the two corresponding or reciprocal
cases have the same root or roots, but with different
signs or affections; and how many true or positive roots
each case has.</p><p>Upon the whole, it appears from this short chapter,
that Cardan had discovered most of the principal properties
of the roots of equations, and could point out the
number and nature of the roots, partly from the signs
of the terms, and partly from the magnitude and relations
of the co-efficients. He shews in effect, that
when the case has all its roots, or when none are impossible,
the number of its positive roots is the same as
the number of changes in the signs of the terms, when
they are all brought to one side: that the co-efficient of
the 2d term is equal to the sum of all the roots positive
and negative collected together, and conseqnently that
when the 2d term is wanting, the positive roots are
equal to the negative ones: and that the signs of all the
roots are changed, by changing only the signs of the
even terms: with many other remarks concerning the
nature of equations.</p><p>In chap. 2, Cardan enumerates all the cases of compound
equations of the 2d and 3d order, namely, 3
quadratics, and 19 cubics; with 44 derivatives of these
two, that is, of the same kind, with higher denominations.</p><p>In chap. 3 are treated the roots of simple cases, or
simple equations, or at least that will reduce to such,
having only two terms, the one equal to the other.
He directs to depress the denominations equally, as
much as they will, according to the height of the least;
then divide by the number or co-efficient of the greatest;
and lastly extract the root on both sides. So if 20<hi rend="italics">x</hi><hi rend="sup">3</hi>
= 180<hi rend="italics">x</hi><hi rend="sup">5</hi>, then 20 = 180<hi rend="italics">x</hi><hi rend="sup">2</hi>, and 1/9=<hi rend="italics">x</hi><hi rend="sup">2</hi>, and <hi rend="italics">x</hi> = 1/3.</p><p>Chap. 4 treats of both general and particular roots,
and contains various definitions and observations concerning
them. It is here shewn that the several cases
of quadratics and cubics have their roots of the following
forms or kinds, namely that the case
where the three parts &#x221A;<hi rend="sup">3</hi>16, 2, &#x221A;<hi rend="sup">3</hi>4, are in continual
proportion.</p><p>Chap. 5 treats of the &#xE6;stimatio of the lowest degree
of compound cases, that is, affected quadratic equations;
giving the rule for each of the three cases, which con-
<cb/>
sists in completing the square, &amp;c, as at present, and
which it seems was the method given by the Arabians;
and proving them by geometrical demonstrations from
Eucl. I. 43, and II. 4 and 5, in which he makes
some improvement of the demonstrations of Mahomet.
And hence it appears that the work of this Arabian
author was in being, and well known in Cardan's time.</p><p>Chap. 6, on the methods of finding new rules, contains
some curious speculations concerning the squares and
cubes of binomial and residual quantities, and the proportions
of the terms of which they consist, shewn from
geometrical demonstrations, with many curious remarks
and properties, forming a foundation of principles for
investigating the rules for cubic equations.</p><p>Chap. 7 is on the transmutation of equations, shewing
how to change them from one form to another, by
taking away certain terms out of them; as , to , &amp;c. The rules are demonstrated
geometrically; and a table is added, of the forms into
which any given cases will reduce; which transformations
are extended to equations of the 4th and 5th
order. And hence it appears that Cardan knew how
to take away any term out of an equation.</p><p>Chap. 8 shews generally how to find the root of any
such equation as this , where <hi rend="italics">m</hi> and <hi rend="italics">n</hi>
are any exponents whatever, but <hi rend="italics">n</hi> the greater; and the
rule is, to separate or divide the co-efficient <hi rend="italics">a</hi> into two
such parts <hi rend="italics">z</hi> and <hi rend="italics">a</hi> - <hi rend="italics">z,</hi> as that the absolute number <hi rend="italics">b</hi>
shall be equal to &#x2015;(<hi rend="italics">a</hi> - <hi rend="italics">z</hi>).<hi rend="italics">z</hi><hi rend="sup">m/(n-m)</hi>, the product of the
one part <hi rend="italics">a</hi> - <hi rend="italics">z,</hi> and the <hi rend="italics">m</hi>/(<hi rend="italics">n</hi> - <hi rend="italics">m</hi>) power of the other
part: then the root <hi rend="italics">x</hi> is = <hi rend="italics">z</hi><hi rend="sup">1/(n-m)</hi>. The rule is general
for quadratics, cubics, and all the higher powers; and
could not have been formed without the knowledge of
the composition of the terms from the roots of the
equation.</p><p>Chap. 9 and 10 contain the resolution of various
questions producing equations not higher than quadratics.</p><p>Chap. 11 is of the case or form .
Cardan now comes to the actual resolution of the first
case of cubic equations. He begins with relating a
short history of the invention of it, observing that it was
first found out, about 30 years before, by Scipio Ferreus
of Bononia, and by him taught to Antonio Maria
Florido of Venice, who having a contest afterwards
with Nicolas Tartalea of Brescia, it gave occasion to
Tartalea to find it out himself, who after great entreaties
taught it to Cardan, but suppressed the demonstration.
By help of the rule alone, however, Cardan of himself
discovered the source or geometrical investigation, which
he gives here at large, from Eucl. II. 4. In this process
he makes use of the Greek letters <foreign xml:lang="greek">a, &lt;*&gt;, g, d</foreign>, &amp;c, to
denote certain indefinite numbers or quantities, to render
the investigation general; which may be considered
as the first instance of such literal notation in Algebra.
He then gives the rule in words at length, which comes
to this,
;
illustrating it in a variety of examples; in the resolution
<pb n="70"/><cb/>
of which, he extracts the cubic roots of such of the
binomials as will admit of it, by some rule which he
had for that purpose; such as , which
.</p><p>Chap. 12, of the case . This he
treats exactly as the last, and finds the rule
;
which he illustrates by many examples, as usual. But
when <hi rend="italics">b</hi><hi rend="sup">3</hi> exceeds <hi rend="italics">c</hi><hi rend="sup">2</hi>, which has since been called the
irreducible case, he refers to another following book,
called <hi rend="italics">Aliza,</hi> for other rules of solution, to overcome
this difficulty, about which he took insinite pains.</p><p>Chap. 13, of the case . This case, by
a geometrical process, he reduces to the case in the
last chapter: thus, find the &#xE6;stimatio <hi rend="italics">y</hi> of the case
, having the same co-efficients as the given
case ; then is ,
giving two roots. He shews also how to find the
second root, when the first is known, independent of
the foregoing case. From this relation of these two
cases he deduces several corollaries, one of which is,
that the &#xE6;stimatio or root of the case , is
equal to the sum of the roots of the case .
As in the example , whose &#xE6;stimatio is
&#x221A;(9 1/4 + 1 1/2), which is equal to the sum of 3 and &#x221A;(9 1/4
- 1 1/2), the two roots of the case .</p><p>In chapters 14, 15, and 16, he treats of the three
cases which contain the 2d and 3d powers, but wanting
the first power, according to all the varieties of the
signs; which he performs by exterminating the 2d
term, or that which contains the 2d power of the unknown
quantity <hi rend="italics">x,</hi> by substituting <hi rend="italics">y</hi> &#xB1; 1/3 the co-efficient
of that term for <hi rend="italics">x,</hi> and so reducing these cases to one
of the former. In these chapters Cardan sometimes
also gives other rules; thus, for the case ,
find first the &#xE6;stimatio <hi rend="italics">y</hi> of the case ,
then is : also for the case ,
first find the two roots of , then is
<hi rend="italics">x</hi> = (&#x221A;<hi rend="sup">3</hi>4<hi rend="italics">c</hi><hi rend="sup">2</hi>)/<hi rend="italics">y</hi> the two values of <hi rend="italics">x</hi> according to the two
values of <hi rend="italics">y.</hi> He here also gives another rule, by which
a second &#xE6;stimatio or root is found, when the first is
known, namely, if <hi rend="italics">e</hi> be the first estimatio or value of <hi rend="italics">x</hi>
in the case , then is the other value of
.</p><p>In chapters 17, 18, 19, 20, 21, 22, 23, Cardan
treats of the cases in which all the four terms of the
equation are present; and this he always effects by
taking away the 2d term out of the equation, and so
reducing it to one of the foregoing cases which want
that term, giving always geometrical investigations, and
adding a great many examples of every case of the
equations.</p><p>Chap. 24, of the 44 derivative cases; which are only
higher powers of the forms of quadratics and cubics.</p><p>Chap. 25, of imperfect and special cases; containing
many particular examples when the co-efficients have
certain relations amongst them, with easy rules for
<cb/>
finding the roots; also 8 other rules for the irreducible
case .</p><p>Chap. 26, in like manner, contains easy rules for biquadratics,
when the co-efficients have certain special
relations.</p><p>Then the following chapters, from chap. 27 to chap.
38, contain a great number of questions and applications
of various kinds, the titles of which are these:
<hi rend="italics">De transitu capituli specialis in capitulum speciale; De
operationibus radicum pronicarum seu mixtarum &amp; Allellarum;
De regula modi; De regula Aurea; De regula
Magna,</hi> or the method of finding out solutions to certain
questions; <hi rend="italics">De regula &#xE6;qualis positionis,</hi> being a
method of substituting for the half sum and half difference
of two quantities, instead of the quantities
themselves; <hi rend="italics">De regula in&#xE6;qualiter ponendi, seu proportionis;
De regula medii; De regula aggregati; De regula
liber&#xE6; positionis; De regula falsum ponendi,</hi> in which
some quantities come out negative; <hi rend="italics">Quomodo excidant
partes &amp; denominationes multiplicando.</hi> Among the foregoing
collection of questions, which are chiefly about
numbers, there are some geometrical ones, being the application
of Algebra to Geometry, such as, In a rightangled
triangle, given the sum of each leg and the adjacent
segment of the hypotenuse, made by a perpendicular
from the right angle, to determine the area &amp;c; with
other such geometrical questions, resolved algebraically.</p><p>Chap. 39, <hi rend="italics">De regula qua pluribus positionibus invenimus
ignotam quantitatem</hi>; which is employed on biquadratie
equations. After some examples of his own, Cardan
gives a rule of Lewis Ferrari's, for resolving all biquadratics,
namely by means of a cubic equation, which
Ferrari investigated at his request, and which Cardan
here demonstrates, and applies in all its cases. The
method is very general, and consists in forming three
squares, thus: first, complete one side of the equation up
to a square, by adding or subtracting some multiples or
parts of some of its own terms on both sides, which it
is always easy to do: 2d, supposing now the three terms
of this square to be but one quantity, viz, the first
term of another square to which this same side is to
be completed, by annexing the square of a new and
assumed indeterminate quantity, with double the product
of the roots of both; which evidently forms
the square of a binomial, consisting of the assumed
indeterminate quantity and the root of the first square:
3d, the other side of the equation is then made
to become the square of a binomial also, by supposing
the product of its ist and 3d terms to be equal to the
square of half its 2d term; for it consists of only three
terms, or three different denominations of the original
unknown quantity: then this equality will determine
the value of the assumed indeterminate quantity, by
means of a cubic equation, and from it, that of the
original ignota, by the equal roots of the 2d and 3d
squares. Here we have a notable example of the use
of assuming a new indeterminate quantity to introduce
into an equation, long before Des Cartes was born, who
made use of a like assumption for a similar purpose.
And this method is very general, and is here applied to
all forms of biquadratics, either having all their terms,
or wanting some of them. To illustrate this rule I shall
here set down the process of one of his examples,
which is this, . Now first sub-
<pb n="71"/><cb/>
tract 2<hi rend="italics">x</hi><hi rend="sup">2</hi> + 4<hi rend="italics">x</hi> + 7 from both sides, then the first becomes
a square, viz, . Next assume the indeterminate <hi rend="italics">y,</hi> and
subtract 2<hi rend="italics">y</hi> (<hi rend="italics">x</hi><hi rend="sup">2</hi> - 1) - <hi rend="italics">y</hi><hi rend="sup">2</hi> from both sides, making
the first side again a square, viz, . Of this latter side, make the product of the 1st
and 3d terms equal to the square of half the 2d term,
that is, , which reduces to
; the positive roots of which are
<hi rend="italics">y</hi> = 2 or &#x221A;15; and hence, using 2 for <hi rend="italics">y,</hi> the equation
of equal squares becomes , the roots of which give ; and hence ; the
two positive roots of which are &#x221A;(3 + 1) and &#x221A;(5 - 1),
which are two of the values of <hi rend="italics">x</hi> in the given equation
. The other roots he leaves to be
tried by the reader.</p><p>The 40th, or last, chap. is entitled, Of modes of
general supposition relating to this art; with some
rules of an unusual kind; and &#xE6;stimatios or roots of a
nature different from the foregoing ones. Some of
these are as follow: If , and , and
<hi rend="italics">x : y :: c : d</hi>; then is .</p><p>Secondly, if ,
and ,
then is <hi rend="italics">x</hi> + <hi rend="italics">a : y</hi> - <hi rend="italics">a :: y</hi><hi rend="sup">2</hi> : <hi rend="italics">x</hi><hi rend="sup">2</hi>.</p><p>Thirdly, when , the square will be
taken away, by putting ; and then the
equation becomes .</p><p>Cardan adds some other remarks concerning the
solutions of certain cases and questions, all evincing the
accuracy of his skill, and the extent of his practice;
and then he concludes the book with a remark concerning
a certain transformation of equations, which
quite astonishes us to find that the same person who,
through the whole work, has shewn such a profound
and critical skill in the nature of equations, and the solution
of problems, should yet be ignorant of one of the
most obvious transmutations attending them, namely
increasing or diminishing the roots in any proportion.
Cardan having observed that the form  may
be changed into another similar one, viz, ,
of which the co-efficient of the term <hi rend="italics">y</hi> is the quotient
arising from the co-efficient of <hi rend="italics">x</hi> divided by the absolute
number of the first equation: and that the absolute number
of the 2d equation is the root of the quotient of 1
divided by the said absolute number of the first; he then
adds, that finding the &#xE6;stimatio or root of the one
equation from that of the other is very difficult, <hi rend="italics">valde
difficilis.</hi></p><p>It is matter of wonder that Cardan, among so many
transmutations, should never think of substituting instead
of <hi rend="italics">x</hi> in such equations, another positio or root,
greater or less than the former in any indefinite proportion,
that is, multiplied or divided by a given number;
for this would have led him immediately to the same
transformation as he makes above, and that by a way
which would have shewn the constant proportion be-
<cb/>
tween the two roots. Thus, instead of <hi rend="italics">x</hi> in the given
form , substitute <hi rend="italics">dy,</hi>
and it becomes ; and this divided
by <hi rend="italics">d</hi><hi rend="sup">3</hi> becomes ; and here if <hi rend="italics">d</hi>
be taken = &#x221A;<hi rend="italics">c,</hi> it becomes ; which is
the transformation in question, and in which it is evident
that <hi rend="italics">x</hi> is = <hi rend="italics">y&#x221A;c,</hi> and <hi rend="italics">y</hi> = <hi rend="italics">x/&#x221A;c.</hi> Instead of this, Cardan
gives the following strange way of finding the one
root <hi rend="italics">x</hi> from the other <hi rend="italics">y,</hi> when this latter is by any
means known; viz, Multiply the first given equation
by <hi rend="italics">y</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> + 1, then add <hi rend="italics">x</hi><hi rend="sup">2</hi>/4<hi rend="italics">y</hi><hi rend="sup">2</hi> to both sides, and lastly extract
the roots of both, which can always be done, as
they will always be both of them squares; and the
roots will give the value of <hi rend="italics">x</hi> by a quadratic equation.</p><p>Thus,  multiplied by <hi rend="italics">y</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> + 1 gives
;
and
theroots are ;
and this 2d side of the equation he says will always have
a root also. It is indeed true that it will have an exact
root; but the reason of it is not obvious, which is, because
<hi rend="italics">y</hi> is the root of the equation .
Cardan has not shewn the reason why this happens;
but I apprehend he made it out in this manner, viz,
similar to the way in which he forms the last square in
the case of biquadratic equations, namely, by making
the product of the 1st and 3d terms equal to the square
of half the 2d term: thus, in the present case, it is
, which reduces to  the equation in question. Therefore taking <hi rend="italics">y</hi>
the root of the equation , and substituting
its value in the quantity , this will become a complete square.
<hi rend="center"><hi rend="italics">Of</hi> Cardan's <hi rend="italics">Libellus de Aliza Regula.</hi></hi></p><p>Subjoined to the above Treatise on cubic equations,
is this <hi rend="italics">Libellus de Aliza regula,</hi> or the algebraic logistics,
in which the author treats of some of the abstruser
parts of Arithmetic and Algebra, especially cubic-equations,
with many more attempts on the irreducible case
. This book is divided into 60 chapters;
<pb n="72"/><cb/>
but I shall only set down the titles of some few of them,
whose contents require more particular notice.</p><p>Chap. 4. <hi rend="italics">De modo redigendi quantitates omnes, qu&#xE6; dicuntur
latera prima ex decimo Euclidis in compendium.</hi> He
treats here of all Euclid's irrational lines, as surd numbers,
and persorms various operations with them.</p><p>Chap. 5. <hi rend="italics">De consideratione binomiorum &amp; recisorum,
&amp;c; ubi de &#xE6;stimatione capitulorum.</hi> Contains various
operations of multiplying compound numbers and surds.</p><p>Chap. 6. <hi rend="italics">De operationibus</hi> p: &amp; m: (i. e. + and -)
<hi rend="italics">secundum communem usum.</hi> Here it is shewn that, in multiplication
and division, <hi rend="italics">plus</hi> always gives the same signs, and
<hi rend="italics">minus</hi> gives the contrary signs. So also in addition, every
quantity retains its own sign; but in subtraction they
change the signs. That the &#x221A; +, or the square root of
plus, is +; but the &#x221A; -, or the square root of minus, is
nothing as to common use: (but of this below.) That
&#x221A;<hi rend="sup">3</hi> - is -; as &#x221A; - 8 is - 2. That a residual, composed of
+ and - may have a root also composed of + and -: So
&#x221A;(5-&#x221A;24) is = &#x221A;3-&#x221A;2. The rules for the signs in
multiplication and division are illustrated by this example;
to divide 8 by 2 + &#x221A;6 or &#x221A;6 + 2. Take the
two corresponding residuals 2 - &#x221A;6 and &#x221A;6 - 2, and
by these multiply both the divisor and dividend; then
the products are + and - respectively, and the quotients
still both alike. Thus,
<table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Divid.</cell><cell cols="1" rows="1" rend="align=center" role="data">Divis.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">|</cell><cell cols="1" rows="1" rend="align=right" role="data">Divid.</cell><cell cols="1" rows="1" rend="align=center" role="data">Divis.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">&#x221A;6 + 2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">2 + &#x221A;6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">&#x221A;6 - 2</cell><cell cols="1" rows="1" rend="align=center" role="data">&#x221A;6 - 2</cell><cell cols="1" rows="1" rend="align=right" role="data">2 - &#x221A;6</cell><cell cols="1" rows="1" rend="align=center" role="data">2 - &#x221A;6</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">&#x221A;384 - 16 divide + 4</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">16 - &#x221A;384 div. - 2</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Quot. &#x221A;96-8.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Quot. &#x221A;96 - 8.</cell></row></table>
And this method of performing division of compound
surds, was fully taught before him, by Lucas de Burgo,
namely, reducing the compound divisor to a simple
quantity, by multiplying by the corresponding quantity,
having the sign changed.</p><p>In chap. 11 and 18, and elsewhere, Cardan makes a
general notation of <hi rend="italics">a, b, c, d, e, f,</hi> for any indefinite
quantities, and treats of them in a general way.</p><p>Cap. 2. <hi rend="italics">De contemplatione</hi> p: &amp; m: (or + and -),
<hi rend="italics">&amp; quod</hi> m: <hi rend="italics">in</hi> m: <hi rend="italics">facit</hi> p: <hi rend="italics">&amp; de causis horum juxta
veritatem.</hi> Cardan here demonstrates geometrically
that, in multiplication and division, like signs give plus,
and unlike signs give minus. And he illustrates this
numerically, by squaring the quantity 8, or 6 + 2, or 10
- 2, which must all produce the same thing, namely 64.</p><p>Among many of the chapters which treat of the irreducible
case , there is a peculiar kind of
way given in chap. 31, which is entitled <hi rend="italics">De &#xE6;stimatione
generali</hi>  <hi rend="italics">solida vocata, &amp; operationibus ejus;</hi> in
which he shews how to approximate to the root of that
case, in a manner similar to approximating the square
root and cube root of a number. The rule he uses for
this purpose, is the 3d in chap. 25 of the last book, and it
is this: Divide <hi rend="italics">b</hi> into two parts, such that the sum of
the products of each, multiplied by the square of the
other, may be equal to (1/2)<hi rend="italics">c</hi>; then the sum of the roots of
these parts is the &#xE6;stimatio or value
<figure/>
of <hi rend="italics">x</hi> required. So, of this equation
; the two parts are 9
and 1, and their roots 3 and 1,
and their sum 4=<hi rend="italics">x,</hi> as in the margin.
<cb/>
Again, take . Here he invents a new notation
to express the root or <hi rend="italics">radix,</hi> which he calls <hi rend="italics">solida,</hi>
viz, <hi rend="italics">x</hi>=&#x221A; <hi rend="italics">solida</hi> 6 <hi rend="italics">in</hi> 1/2, that is, the roots of the two
parts of 6, so that each part multiplied by the root of
the other, the two products may be 1/2 or (1/2)<hi rend="italics">c.</hi> Then to
free this from fractions, and make the operation easier,
multiply that root by some number as suppose 4, that is
the square part 6 by the square of 4, and the solid part 1/2
by the cube of 4; then <hi rend="italics">x</hi>=1/4&#x221A; <hi rend="italics">solida</hi> 96 <hi rend="italics">in</hi> 32. Now, by
a few trials, it is found that the parts are
nearly 95 8/9 and 1/9, which give too much,
or 95 9/10 and 1/10, which give too little,
and thereof 95 17/19 and 2/19 are still nearer. Divide both
by 4<hi rend="sup">2</hi> or 16, then 5 151/152 and 1/152 are the quot. And the
sum of their roots, or  is nearly
the value of the root <hi rend="italics">x.</hi></p><p><hi rend="italics">Cap.</hi> 42. <hi rend="italics">De duplici &#xE6;quatione comparanda in capitulo
cubi &amp; numeri &#xE6;qualium rebus.</hi> Treats of the two positive
roots of that case, neglecting the negative one;
and shewing, not only that that case has two such roots,
but that the same number may be the common root of
innumerable equations.</p><p><hi rend="italics">Cap.</hi> 57. <hi rend="italics">Detractatione &#xE6;stimationis generalis capituli</hi> . Cardan here again resumes the consideration
of the irreducible case, making ingenious observations
upon it, but still without obtaining the root by
a general rule. In this place also, as well as elsewhere,
he shews how to form an equation in this case, that
shall have a given binomial root, as suppose &#x221A;<hi rend="italics">m</hi> + <hi rend="italics">n,</hi>
where the equation will be , having &#x221A;<hi rend="italics">m</hi> + <hi rend="italics">n</hi> for one root, namely the
positive root. From which it appears that he was well
acquainted with the composition of cubic equations from
given roots.</p><p><hi rend="italics">Cap.</hi> 59. <hi rend="italics">De ordine &amp; exemplis in binomiis secudo
&amp; quinto.</hi> Contains a great many numeral forms of
the same irreducible case , with their roots;
from which are derived these following cases, with many
curious remarks. When</p><p><hi rend="italics">Cap.</hi> 60. <hi rend="italics">Demonstratio generalis capituli cubi &#xE6;qualis rebus
&amp; numero.</hi> This demonstration of the irreducible case
is geometrical, like all the rest. Some more ingenious
remarks are again added, as if he reluctantly finished
the book without perfectly overcoming the difficulty
of the irreducible case. Cardan here also uses the letters
<hi rend="italics">a</hi> and <hi rend="italics">b</hi> for any two indefinite numbers, in order to
shew the form and manner of the arithmetical operations:
thus <hi rend="italics">a/b</hi> is the fraction for their quotient, also
&#x221A;<hi rend="italics">a/b</hi> or &#x221A;<hi rend="italics">a/&#x221A;b</hi> the square root of that quotient, and
&#x221A;<hi rend="sup">3</hi><hi rend="italics">a/b</hi> or &#x221A;<hi rend="sup">3</hi><hi rend="italics">a</hi>/&#x221A;<hi rend="sup">3</hi><hi rend="italics">b</hi> the cube root of it, &amp;c.
<pb n="73"/><cb/></p><p>Having considered the chief contents of Cardan's algebra,
it will now be proper to sum them up, and set
down a list of the improvements made by him, as collected
from his writings:</p><p>And 1st, Tartalea having only communicated to him
the rules for resolving these three cases of cubic equations,
viz,
having all their terms, or wanting any of them, and having
all possible varieties of signs; demonstrating all these
rules geometrically; and treating very fully of almost all
sorts of transformations of equations, in a manner heretofore
unknown.</p><p>2nd, It appears that he was well acquainted with all
the roots of equations that are real, both positive and
negative; or, as he calls them, true and fictitious;
and that he made use of them both occasionally. He
also shewed, that the even roots of positive quantities, are
either positive or negative; that the odd roots of negative
quantities, are real and negative; but that the even roots
of them are impossible, or nothing as to common use.
He was also acquainted with,</p><p>3d, The number and nature of the roots of an equation,
and that partly from the signs of the terms, and
partly from the magnitude and relation of the coefficients.
He also knew,</p><p>4th, That the number of positive roots is equal to the
number of changes of the signs of the terms.</p><p>5th, That the coefficient of the second term of the
equation, is the difference between the positive and negative
roots.</p><p>6th, That when the second term is wanting, the sum
of the negative roots is equal to the sum of the positive
roots.</p><p>7th, How to compose equations that shall have given
roots.</p><p>8th, That, changing the signs of the even terms,
changes the signs of all the roots.</p><p>9th, That the number of roots failed in pairs; or
what we now call impossible roots were always in pairs.</p><p>10th, To change the equation from one form to another,
by taking away any term out of it.</p><p>11th, To increase or diminish the roots by a given
quantity. It appears also,</p><p>12th, That he had a rule for extracting the cube
root of such binomials as admit of extraction.</p><p>13th, That he often used the literal notation <hi rend="italics">a, b, c, d,</hi> &amp;c.</p><p>14th, That he gave a rule for biquadratic equations,
suiting all their cases; and that, in the investigation
of that rule, he made use of an assumed indeterminate
quantity, and afterwards found its value by the arbitrary
assumption of a relation between the terms.</p><p>15th, That he applied Algebra to the resolution of
geometrical problems. And</p><p>16th, That he was well acquainted with the difficulty
of what is called the irreducible case, viz, ,
upon which he spent a great deal of time, in attempting
to overcome it. And though he did not fully succeed
in this case, any more than other persons have done
since, he nevertheless made many ingenious observations
about it, laying down rules for many particular
<cb/>
forms of it, and shewing how to approximate very
nearly to the root in all cases whatever.
<hi rend="center"><hi rend="smallcaps">OF TARTALEA.</hi></hi></p><p>Nicholas Tartalea, or Tartaglia, of Brescia, was
contemporary with Cardan, and was probably older
than he was, but I do not know of any book of Algebra
published by him till the year 1546, the year after
the date of Cardan's work on Cubic Equations, when
he printed his <hi rend="italics">Quesiti &amp; Inventioni diverse,</hi> at Venice,
where he resided as a public lecturer on mathematics.
This work is dedicated to our king Henry the VIIIth
of England, and consists of 9 books, containing answers
to various questions which had been proposed to him
at different times, concerning mechanics, statics, hydrostatics,
&amp;c.; but it is only the 9th, or last, that
we shall have occasion to take notice of in this place, as
it contains all those questions which relate to arithmetic
and algebra. These are all set down in chronological
order, forming a pretty collection of questions and solutions
on those subjects, with a short account of the occasion
of each of them. Among these, the correspondence
between him and Cardan forms a remarkable part, as
we have here the history of the invention of the rules
for cubic equations, which he communicated to Cardan.
under the promise, and indeed oath, to keep them secret,
on the 25th of March 1539. But, notwithstanding his
oath, finding that Cardan published them in 1545, as
above related, it seems Tartalea published the correspondence
between them in revenge for his breach of
faith; and it elsewhere appears, that many other sharp
bickerings passed between them on the same account,
which only ended with the death of Tartalea, in the
year 1557. It seems it was a common practice among
the mathematicians, and others, of that time, to send
to each other nice and difficult questions, as trials of
skill, and to this cause it is that we owe the principal
questions and discoveries in this collection, as well as
many of the best discoveries of other authors. The
collection now before us contains questions and solutions,
with their dates, in a regular order, from the
year 1521, and ending in .1541, in 42 dialogues, the
last of which is with an English gentleman, namely,
Mr. Richard Wentworth, who it seems was no mean
mathematician, and who learned some algebra, &amp;c, of
Tartalea, while he resided at Venice. The questions
at first are mostly very easy ones in arithmetic, but
gradually become more difficult, and exercising simple
and quadratic equations, with complex calculations of
radical quantities: all shewing that he was well skilled
in the art of Algebra as it then stood, and that he
was very ingenious in applying it to the solutions of
questions. Tartalea made no alteration in the notation
or forms of expression used by Lucas de Burgo,
calling the first power of the unknown quantity, in his
language, <hi rend="italics">cosa,</hi> the second power <hi rend="italics">censa,</hi> the third <hi rend="italics">cubo,</hi>
&amp;c, and writing the names of all the operations in words
at length, without using any contractions, except the
initial R for root or radicality. So that the only thing
remarkable in this collection, is the discovery of the rules
for cubic equations, with the curious circumstances attending
the same.
<pb n="74"/><cb/></p><p>The first two of these were discovered by Tartalea
in the year 1530, namely for the two cases ,
and , as appears by Quest. 14 and 25 of
this collection, on occasion of a question then proposed
to him by one Zuanni de Tonini da Coi or Colle,
John Hill, who kept a school at Brescia. And from
the 25th letter we learn, that he discovered the rules
for the other two cases , and ,
on the 12th and 13th of February 1535, at Venice,
where he had come to reside the year before. And
the occasion of it was this: There was then at Venice
one Antonio Maria Fiore or Florido, who, by his own
account, had received from his preceptor Scipio Farreo,
about thirty years before, a general rule for resolving the
case . Being a captious man, and presuming
on this discovery, Florido used to brave his contemporaries,
and by his insults provoked Tartalea to enter into
a wager with him, that each should propose to the
other thirty different questions; and that he who soonest
resolved those of his adversary, should win from him
as many treats for himself and friends. These questions
were to be proposed on a certain day at some weeks
distance; and Tartalea made such good use of his time,
that eight days before the time appointed for delivering
the propositions, he discovered the rules both for
the case , and the case . He therefore
proposed several of his questions so as to fall either
on this latter case, or on the cases of the cube and square,
expecting that his adversary would propose his in the
former. And what he suspected fell out accordingly;
the consequence of which was, that on the day of meeting
Tartalea resolved all his adversary's questions in the
space of two hours, without receiving one answer from
Florido in return; to whom, however, Tartalea generously
remitted the forfeit of the thirty treats won of him.</p><p>Question 31 first brings us acquainted with the correspondence
between Tartalea and Cardan. This correspondence
is very curious, and would well deserve to
be given at full length in their own words, if it were
not too long for this place. I may enlarge farther upon
it under the article <hi rend="italics">Cubic Equations</hi>; but must here
be content with a brief abstract only. Cardan was then
a respectable physician, and lecturer in mathematics at
Milan; and having nearly finished the printing of a
large work on Arithmetic, Algebra, and Geometry, and
having heard of Tartalea's discoveries in cubic equations,
he was very desirous of drawing those rules from
him, that he might add them to his book before it
was finished. For this purpose he first applied to Tartalea,
by means of a third person, a bookseller, whom he
sent to him, in the beginning of the year 1539, with
many flattering compliments, and offers of his services
and friendship, &amp;c, accompanied with some critical
questions for him to resolve, according to the custom of
the times. Tartalea however refused to disclose his
rules to any one, as the knowledge of them gained him
great reputation among all people, and gave him a great
advantage over his competitors for fame, who were
commonly afraid of him on account of those very rules.
He only sent Cardan therefore, at his request, a copy
of the thirty questions which had been proposed to him
in the contest with Florido. Not to be rebuffed so easily,
Cardan next applied, in the most urgent manner,
<cb/>
by letter to Tartalea; which however procured from
him only the solution of some other questions proposed
by Cardan, with a few of the questions that had been
proposed to Florido, but none of their solutions. Finding
he could not thus prevail, with all his fair promises,
Cardan then fell upon another scheme. There was at
Milan a certain Marquis dal Valsto, a great patron of
Cardan, and, it was said, of learned men in general.
Cardan conceived the idea of making use of the influence
of this nobleman to draw Tartalea to Milan, hoping
that then, by personal intreaties, he should succeed in
drawing the long-concealed rules from him. Accordingly
he wrote a second letter to Tartalea, much in the
same strain with the former, strongly inviting him to
come and spend a few days in his house at Milan, and
representing that, having often commended him in the
highest terms to the marquis, this nobleman desired
much to see him; for which reason Cardan advised
him, as a friend, to come to visit them at Milan, as it
might be greatly to his interest, the marquis being very
liberal and bountiful; and he besides gave Tartalea to
understand, that it might be dangerous to offend such a
man by refusing to come, who might, in that case, take
offence, and do him some injury. This man&#x153;uvre had
the desired effect: Tartalea on this occasion laments to
himself in these words, &#x201C;By this I am reduced to a
great dilemma; for if I go not to Milan, the marquis
may take it amiss, and some evil may befal me on that
account; I shall therefore go, although very unwillingly.&#x201D;
When he arrived at Milan however, the
marquis was gone to Vigeveno, and Tartalea was prevailed
on to stay three days with Cardan, in expectation
of the marquis returning, at the end of which he set out
from Milan, with a letter from Cardan, to go to Vigeveno
to that nobleman. While Tartalea was at Milan
the three days, Cardan plied him by all possible means
to draw from him the rules for the cubic equations; and
at length, just as Tartalea was about to depart from
Milan, on the 25th of March 1539, he was overcome
by the most solemn protestations of secrecy that could
be made. Cardan says, &#x201C;I shall swear to you on the
holy evangelists, and by the honour of a gentleman, not
only never to publish your inventions, if you reveal them
to me; but I also promise to you, and pledge my faith
as a true christian, to note them down in cyphers, so
that after my death no other person may be able to understand
them.&#x201D; To this Tartalea replies, &#x201C;If I refuse
to give credit to these assurances, I should deservedly be
accounted utterly void of belief. But as I intend to
ride to Vigeveno, to see his excellency the marquis, as I
have been here now these three days, and am weary of
waiting so long; whenever I return therefore, I promise
to shew you the whole.&#x201D; Cardan answers, &#x201C;Since
you determine at any rate to go to Vigeveno, to the marquis,
I shall give you a letter for his excellency, that
he may know who you are. But now before you depart,
I intreat you to shew me the rule for the equations,
as you have promised.&#x201D; &#x201C;I am content,&#x201D; says
Tartalea: &#x201C;But you must know, that to be able on all
occasions to remember such operations, I have brought
the rule into rhyme; for if I had not used that precaution,
I should often have forgot it; and although my
rhymes are not very good, I do not value that, as it is
<pb n="75"/><cb/>
sufficient that they serve to bring the rule to mind as
often as I repeat them. I shall here write the rule with
my own hand, that you may be sure I give you the
discovery exactly.&#x201D; These rude verses contain, in rather
dark and enigmatical language, the rule for these three
cases, viz.
that their difference in the first case, and their sums in
the 2d and 3d, may be equal to <hi rend="italics">c</hi> the absolute number,
and their product equal to the cube of 1/3 of <hi rend="italics">b</hi> the
coefficient of the less power; then the difference of
their cube roots will be equal to <hi rend="italics">x</hi> in the first case, and
the sum of their cube roots equal to <hi rend="italics">x</hi> in the 2d and 3d
cases: that is, taking  in the 1st case, or 
in the 2d and 3d, and ; then 
in the first case, and  in the other two. At
parting, T, fails not again to remind C. of his obligation:
&#x201C;Now your excellency will remember not to break your
promised faith, for if unhappily you should insert these
rules either in the work you are now printing, or in any
other, although you should even give them under my
name, and as of my invention, I promise and swear
that I shall immediately print another work that will not
be very pleasing to you.&#x201D; &#x201C;Doubt not, says C. but that
I shall observe what I have promised: Go, and rest secure
as to that point: and give this letter of mine to the
marquis.&#x201D; It should seem however that T. was much
displeased at having suffered himself to be worried as it
were out of his rules, for as soon as he quitted Milan,
instead of going to wait upon the marquis, he turned
his horse's head, and rode straight home to Venice, saying
to himself, &#x201C;By my faith I shall not go to Vigeveno,
but shall return to Venice, come of it what will.&#x201D;</p><p>After T's departure it seems C. applied himself immediately
to resolving some examples in the cubic equations
by the new rules, but not succeeding in them, for
indeed he had mistaken the words, as it was very easy to
do in such bad verses, having mistaken ((1/3)<hi rend="italics">b</hi>)<hi rend="sup">3</hi> for (1/3)<hi rend="italics">b</hi><hi rend="sup">3</hi>,
or the cube of 1/3 of the coefficient, for 1/3 of the
cube of the coefficient; accordingly we find him writing
to T. in fourteen days after the above, blaming him
much for his abrupt departure without seeing the marquis,
who was so liberal a prince he said, and requesting
T. to resolve him the example . This T.
did to his satisfaction, rightly guessing at the nature of
his mistake; and concludes his answer with these emphatical
words, &#x201C;Remember your promise.&#x201D; On the 12th
of May following C. returns him a letter of thanks, together
with a copy of his book, saying, &#x201C;As to my
work, just finished, to remove your suspicion, I send
you a copy, but unbound, as it is yet too fresh to be
beaten. But as to the doubt you express lest I may
print your inventions, my faith which I gave you with
an oath should satisfy you; for as to the finishing of
my book, that could be no security, as I could always
add to it whenever I please. But on account of the
dignity of the thing, I excuse you for not relying on
that which you ought to have done, namely on the faith
of a gentleman, instead of the finishing of a book,
which might at any time be enlarged by the addition of
new chapters; and there are besides a thousand other
ways. But the security consists in this, that there is no
<cb/>
greater treachery than to break one's faith, and to ag&lt;*&gt;
grieve those who have given us pleasure. And when you
shall try me, you will find whether I be your friend or
not, and whether I shall make an ungrateful return for
your friendship, and the satisfaction you have given
me.&#x201D;</p><p>It was within less than two months after this, however,
that T. received the alarming news of Cardan's shewing
some symptoms of breaking the faith he had so lately
pledged to him; this was in a letter from a quondam
pupil of his, in which he writes, &#x201C;A friend of mine
at Milan has written to me, that Dr. Cardano is composing
another algebraical work, concerning some latelydiscovered
rules; hence I imagine they may be those
same rules which you told me you had taught him; so
that I fear he will deceive you.&#x201D; To which T. replies,
&#x201C;I am heartily grieved at the news you inform me of,
concerning Dr. Cardano of Milan; for if it be true,
they can be no other rules but those I gave him; and
therefore the proverb truly says, &#x2018;That which you wish
not to be known, tell to nobody.&#x2019; Pray endeavour to
learn more of this matter, and inform me of it.&#x201D;</p><p>Tartalea, after this, kept on the reserve with Cardan,
not answering several letters he sent him, till one written
on the 4th of August the same year, 1539, complaining
greatly of T's neglect of him, and farther requesting
his assistance to clear up the difficulty of the irreducible
case , which C. had thus early been embarrassed
with: he says that when ((1/3)<hi rend="italics">b</hi>)<hi rend="sup">3</hi> exceeds ((1/2)<hi rend="italics">c</hi>)<hi rend="sup">2</hi>, the
rule cannot be applied to the equation in hand, because
of the square root of the negative quantities. On this
occasion T. turns the tables on C. and plays his own
game back upon him; for being aware of the above
difficulty, and unable to overcome it himself, he wanted
to try if C. could be encouraged to accomplish it, by
pretending that the case might be done, though in another
way. He says thus to himself, &#x201C;I have a good
mind to give no answer to this letter, no more than to
the other two. However I will answer it, if it be but
to let him know what I have been told of him. And as
I perceive that a suspicion has arisen concerning the difficulty
or obstacle in the rule for the case .
I have a mind to try if he can alter the data in hand, so
as to remove the said obstacle, and to change the rule into
another form, although I believe indeed that it cannot
be done; however there is no harm in trying.&#x201D;&#x2014;&#x201C;M.
Hieronime, I have received your letter, in which you
write that you understand the rule for the case , but that when ((1/3)<hi rend="italics">b</hi>)<hi rend="sup">3</hi> exceeds ((1/2)<hi rend="italics">c</hi>)<hi rend="sup">2</hi>, you cannot resolve
the equation by following the rule, and therefore
you request me to give you the solution of this equation
. To which I reply, that you have
not used a good method in that case, and that your
whole process is intirely false. And as to resolving you
the equation you have sent, I must say that I am very
sorry that I have already given you so much as I have
done, for I have been informed, by a credible person,
that you are about to publish another Algebraical work,
and that you have been boasting through Milan of having
discovered some new rules in algebra. But, take notice,
that if you break your faith with me, I shall certainly
keep my word with you, nay, I even assure you to do
more than I promised.&#x201D; In Cardan's answer to this he
says, &#x201C;You have been misinformed as to my intention to
<pb n="76"/><cb/>
publish more on Algebra. But I suppose you have heard
something about my work <hi rend="italics">de mysteriis &#xE6;ternitatis,</hi> which
you take for some Algebra I intend to publish. As to
your repenting of having given me your rules, I am not
to be moved from the faith I promised you for any thing
you say.&#x201D; To this, and many other things contained
in the same letter, T. returned no answer, being still
suspicious of Cardan's intentions, and declining any
more correspondence with him. This however did not
discourage C. for we find him writing again to T. on
the 5th of January, 1540, to clear up another difficulty
which had occurred in this business, namely to extract
the cube root of the binomials, of which the two parts
of the rule always consisted, and for which purpose it
seems C. had not yet found out a rule. On this occasion
he informs T. that his quondam competitor Zuanne Colle
had come to Milan, where, in some contests between
them, Colle gave Cardan to understand that he had found
out the rules for the two cases , and , and farther that he had discovered a general rule for
extracting the cube roots of all such binomials as can be
extracted; and that, in particular, the cube root of &#x221A;(108
+ 10) is &#x221A;(3 + 1), and that of &#x221A;(108 - 10) is &#x221A;(3 - 1),
and consequently that . He then earnestly entreats
T. to try to find out the rule, and the solution of certain
other questions which had been proposed to him by Colle.
By this letter T. is still more confirmed in his resolution
of silence; so that, without returning any answer,
he only sets down among his own memorandums some
curious remarks on the contents of the letter, and then
concludes to himself, &#x201C;Wherefore I do not choose to
answer him again, as I have no more affection for him
than for M. Zuanne, and therefore I shall leave the matter
between them.&#x201D; Among those remarks he sets down a
rule for extracting the cube root of such binomials as can
be extracted, and that is done from either member of
the binomial alone, thus: Take either term of the binomial,
and divide it into two such parts that one of
them may be a complete cube, and the other part exactly
divisible by 3; then the cube root of the said cubic
part will be one term of the required root, and the square
root of the quotient arising from the division of 1/3 of the
2d part by the cube root of the first, will be the other
member of the root sought. This rule will be better
understood in characters thus: let <hi rend="italics">m</hi> be one member of
the given binomial, whose cube root is sought, and let it
be divided into the two parts <hi rend="italics">a</hi><hi rend="sup">3</hi> and 3<hi rend="italics">b,</hi> so that <hi rend="italics">a</hi><hi rend="sup">3</hi> + 3<hi rend="italics">b</hi>
be = <hi rend="italics">m</hi>; then is <hi rend="italics">a</hi> + &#x221A;<hi rend="italics">b/a</hi> the cube root required, if it
have one. Thus in the quantity &#x221A;(108+10), taking
the term 10 for <hi rend="italics">m,</hi> then 10 divides into 1 and 9, where
<hi rend="italics">a</hi><hi rend="sup">3</hi>=1 or <hi rend="italics">a</hi>=1, and 3<hi rend="italics">b</hi>=9 or <hi rend="italics">b</hi>=3: therefore <hi rend="italics">a</hi> + &#x221A;<hi rend="italics">b/a</hi>
becomes 1 + &#x221A;3 for the cube root of &#x221A;(108 + 10). And
taking the other member &#x221A;108, this divides into the
two equal parts &#x221A;27 and &#x221A;27, making <hi rend="italics">a</hi><hi rend="sup">3</hi> = &#x221A;27, and
3<hi rend="italics">b</hi>=&#x221A;27; hence <hi rend="italics">a</hi>=&#x221A;3, and <hi rend="italics">b</hi>=&#x221A;3 also; consequently
<hi rend="italics">a</hi> + &#x221A;<hi rend="italics">b/a</hi> is = &#x221A;3 + &#x221A;3/3 or &#x221A;(3 + 1) for the cube root
of the binomial sought, the same as before. &#x201C;And thus,
he adds, we may know whether any proposed binomial
<cb/>
or residual be a cube or a noncube; for if it be a cube,
the same two terms for the root must arise from both
the given terms separately; and if the two terms of the
root cannot thus be brought to agree both ways, such
binomial or residual will not be a cube.&#x201D; And thus ends
the correspondence between them, at least for this time.
But it seems they had still more violent disputes when
C. in violation of his faith so often pledged to the contrary,
published his work on cubic equations 4 years
afterwards, viz, in the year 1545, of which we have
before given an account, which disputes, it is said, continued
till the death of Tartalea in the year 1557.</p><p>The last article in the volume contains a dialogue on
some other forms of the cubic equations, in the year
1541, between T. and a Mr. Richard Wentworth,
an English gentleman, who it seems had resided some
time at Venice, on some public service from England,
as T. in the dedication of the volume to Henry VIII.
king of England, makes mention of him as &#x201C;a gentleman
of his sacred majesty.&#x201D; Mr. Wentworth had
learned some mathematics of T. and being about to
depart for England, requests T. to shew him his newly
discovered rules for cubic equations, as a farewell-lesson;
and it is worth while to note a few particulars in this
conference, as they shew pretty nicely the limited
knowledge of T. at that time, as to the nature and
roots of such equations. T. had before, it seems,
shewed Mr. W. the rules for the cases of the 3d and 1st
powers, and now the latter desires him to do the same
as to the three cases in which the 3d and 2d powers only
are concerned. On this T. professes great gratitude
to Mr. W. for many obligations, but desires to be excused
from giving him the rules for these, because he
says he intends soon to compose a new work on Arithmetic,
Geometry, and Algebra, which he intends to
dedicate to him, and in which he means to insert all
his new discoveries. On Mr. W. urging him further,
T. gives him the roots of some equations of that kind,
as for instance:
but not the rules for finding them.</p><p>In the course of the conversation T. tells him that
&#x201C;all such equations admit of two different answers,
and perhaps more; and hence it follows that they have,
or admit of, two different rules, and perhaps more, the
one more difficult than the other.&#x201D; And on Mr. W.
expressing his wonder at this circumstance of a plurality
of roots, T. replies, &#x201C;It is however very true, though
hardly to be believed, and indeed if experience had not
confirmed it, I should scarcely have believed it myself.&#x201D;
He then commits a strange blunder in an example which
he takes to illustrate this by, namely the equation
, which, he says, it is evident has the
number 2 for one of its roots; and yet, he adds,
&#x201C;whoever shall resolve the same equation by my rule,
will find the value of <hi rend="italics">x</hi> to be &#x221A;<hi rend="sup">3</hi>(7 + &#x221A;50) +
&#x221A;<hi rend="sup">3</hi>(7 - &#x221A;50), which is proved to be a true root by sub-
<pb n="77"/><cb/>
stituting it in the equation for <hi rend="italics">x.</hi> And therefore,
continues he, it is manifest that the case <hi rend="italics">x</hi><hi rend="sup">3</hi> + <hi rend="italics">bx</hi> = <hi rend="italics">c</hi>
admits of two rules, namely, one (as in the above example)
which ought to give the value of <hi rend="italics">x</hi> rational,
viz 2, and the other is my rule, which gives the value
of <hi rend="italics">x</hi> irrational, as appears above; and there is reason
to think that there may be such a rule as will give the
value of <hi rend="italics">x</hi> = 2, although our ancestors may not have
found it out.&#x201D;&#x2014;&#x2014;&#x201C;And these two different answers
will be found not only in every equation of this form
, when the value of <hi rend="italics">x</hi> happens to be rational,
as in the example  above, but the
same will also happen in all the other five forms of
cubic equations: and therefore there is reason to think
that they also admit of two different rules; and by
certain circumstances attending some of them, I am
almost certain that they admit of more than two rules,
as, God willing, I shall soon demonstrate.&#x201D; Now all this
discourse shews a strange mixture of knowledge and ignorance:
it is very probable that he had met with some
equations which admit of a plurality of roots; indeed it
was hardly possible for him to avoid it; but it seems he
had no suspicion what the number of roots might be,
nor that his reasoning in this instance was founded on
an error of his own, mistaking the root , of the equation , for
a different root from the number or root 2, when in
reality it is the very same, as he might easily have found,
if he had extracted the cube roots of the binomials by
the rule which he himself had just given above for
that purpose: for by that rule he would have found
, and , and therefore their sum is 2 = <hi rend="italics">x,</hi> the same root
as the other, which T. thought had been different.
And besides this root 2, the equation in hand, , admits of no other real roots. Nor does any
equation of the same form, , admit of
more than one real root.</p><p>It seems also they had not yet discovered that all
cases belong to the rules and forms for quadratic equations,
which have only two powers in them, in which
the exponent of the one is just double of the exponent
of the other, as ; but some particular
cases only of this sort they had as yet ventured
to refer to quadratics, as the case . But
in the conclusion of this dialogue T. informs W. of
another case of this sort which he had accomplished, as
a notable discovery, in these words: &#x201C;I well remember,
says he, that in the year 1536, on the night of
St. Martin, which was on a Saturday, meditating
in bed when I could not sleep, I discovered the general
rule for the case , and also for the other
two, its accompanying cases, in the same night.&#x201D; And
then he directs that they are to be resolved like quadratics,
by completing the square, &amp;c. And in these
resolutions it is remarkable that he uses only the positive
roots, without taking any notice of the negative ones.</p><p>Tartalea also published at Venice, in 1556, &amp;c, a
very large work, in folio, on Arithmetic, Geometry,
and Algebra. This is a very complete and curious
work upon the first two branches; but that of Algebra
is carried no farther than quadratic equations,
<cb/>
called <hi rend="italics">book the first,</hi> with which the work terminates.
It is evidently incomplete, owing to the death of the
author, which happened before this latter part of the
work was printed, as appears by the dates, and by the
prefaces. It appears also, from several parts of this
work, that the author had many severe conflicts with
Cardan and his friend Lewis Ferrari: and particularly,
there was a public trial of skill between them, in the
year 1547; in which it would seem that Tartalea had
greatly the advantage, his questions mostly remaining
unanswered by his antagonists.
<hi rend="center"><hi rend="smallcaps">OF MICHAEL STIFELIUS.</hi></hi></p><p>After the foregoing analysis of the works of the first
algebraic writers in Italy, it will now be proper to consider
those of their contemporaries in Germany; where,
excepting for the discoveries in cubic equations, the
art was in a more advanced state, and of a form approaching
nearer to that of our modern Algebra; the
state and circumstances indeed being so different, that
one would almost be led to suppose they had derived
their knowledge of it from a different origin.</p><p>Here Stifelius and Scheubelius were writers of the
same time with Cardan and Tartalea, and even before
their discoveries, or publication, concerning the rules
for cubic equations, Stifelius's <hi rend="italics">Arithmetica Integra</hi> was
published at Norimberg in 1544, being the year before
Cardan's work on cubic equations, and is an excellent
treatise, both on Arithmetic and Algebra. The work is
divided into three books, and is prefaced with an Introduction
by the famous Melanchthon. The first book
contains a complete and ample Treatise on Arithmetic,
the second an Exposition of the 10th book of Euclid's
Elements, and the third a Treatise of Algebra, and it
is therefore properly the part with which we are at
present concerned. In the dedication of this part, he
ascribes the invention of Algebra to Geber, an Arabic
Astronomer; and mentions besides, the authors Campanus,
Christ. Rudolph, and Adam Ris, Risen, or
Gigas, whose rules and examples he has chiefly given.
In other parts of the book he speaks, and makes use
also, of the works of Bretius, Campanus, Cardan
(i. e. his Arithmetic published in 1539, before the work
on cubic equations appeared), de Cusa, Euclid, Jordan,
Milichius, Schonerus, and Stapulensis.</p><p><hi rend="italics">Chap.</hi> 1. <hi rend="italics">Of the Rule of Algebra, and its parts.</hi> Stifelius
here describes the notation and marks of powers,
or denominations as he calls them, which marks for the
several powers are thus:
<table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1st,</cell><cell cols="1" rows="1" rend="align=right" role="data">2d,</cell><cell cols="1" rows="1" rend="align=right" role="data">3d,</cell><cell cols="1" rows="1" rend="align=right" role="data">4th,</cell><cell cols="1" rows="1" rend="align=center" role="data">5th,</cell><cell cols="1" rows="1" rend="align=right" role="data">6th,</cell><cell cols="1" rows="1" role="data">&amp;c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><figure/>,</cell><cell cols="1" rows="1" rend="align=right" role="data">[dram],</cell><cell cols="1" rows="1" rend="align=right" role="data"><figure/>,</cell><cell cols="1" rows="1" rend="align=right" role="data">[dram][dram],</cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">&#x222B;s,</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">[dram] <figure/>,</cell><cell cols="1" rows="1" role="data">&amp;c.</cell></row></table>
which are formed from the initials of the barbarous way
in which the Germans pronounced and wrote the Latin
and Italic names of the powers, namely, res or cosa, zensus,
cubo, zensi-zensus, sursolid, zenfi-cubo, &amp;c. And
the coss or first power <figure/>, he calls the radix or root,
which is the first time that we meet with this word in
the printed authors. He also here uses the signs or
characters, + and -, for addition and subtraction, and
the first of any that I know of: for in Italy they used
none of these characters for a long time after. He has
no mark however for equality, but makes use of the
word itself.
<pb n="78"/><cb/></p><p><hi rend="italics">Chap.</hi> 2. Of the Parts of the Rule of Geber or Algebra:
teaching the various reductions by addition,
subtraction, multiplication, division, involution, and
evolution, &amp;c.</p><p><hi rend="italics">Chap.</hi> 3. Of the Algorithm of Cossic Numbers:
teaching the usual operations of addition, subtraction,
multiplication, division, involution, and extraction of
roots, much the same as they are at present. Single
terms, or powers, he calls simple quantities; but such
as 1[dram] + 1<figure/> a composite or compound, and 2<figure/> - 8 a
defective one. In multiplication and division, he proves
that like signs give +, and unlike signs -. He shews that
the powers 1, <figure/>, [dram], <figure/>, &amp;c, form a geometrical progression
from unity; and that the natural series of numbers
0, 1, 2, 3, &amp;c, from 0, are the exponents of the cossic
powers; and he, for the first time, expressly calls them
exponents: thus,
<table><row role="data"><cell cols="1" rows="1" role="data">Exponents,</cell><cell cols="1" rows="1" role="data">0,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">2,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" rend="align=center" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">5,</cell><cell cols="1" rows="1" rend="align=center" role="data">6,</cell><cell cols="1" rows="1" role="data">&amp;c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Powers,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data">[dram],</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" rend="align=center" role="data">[dram] [dram],</cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">&#x222B;s,</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">[dram] <figure/>,</cell><cell cols="1" rows="1" role="data">&amp;c.</cell></row></table>
And he shews the use of the exponents, in multiplication,
division, powers, and roots, as we do at present;
viz, adding the exponents in multiplication, and subtracting
them in division, &amp;c. And these operations he
demonstrates from the nature of arithmetical and geometrical
progressions. It is remarkable that these compound
denominations of the powers are formed from the
simple ones according to the <hi rend="italics">products</hi> of the exponents,
while those of Diophantus are formed according to the
<hi rend="italics">sums</hi> of them; thus the 6th power here is [dram] <figure/> or quadrato-cubi,
but with Diophantus it is cubo-cubi; and
so of others. Which is presumptive evidence that the
Europeans had not taken their Algebra immediately
from him, independent of other proofs.</p><p><hi rend="italics">Chap.</hi> 4. Of the extraction of the roots of cossic numbers.
He here treats of quadratic equations, which he
resolves by completing the square, from Euclid II. 4 &amp;c.
Also quadratics of the higher orders, shewing how to
resolve them in all cases, whatever the height may be,
provided the exponents be but in arithmetical progression,
as
<hi rend="brace"><note anchored="true" place="unspecified">&amp;c; where it is plain that he always
counts 0 for the exponent of the unknown
quantity in the absolute term.</note>
2, 1, 0
4, 2, 0
6, 3, 0
8, 4, 0</hi></p><p><hi rend="italics">Chap.</hi> 5. Of irrational cossic numbers, and of surd
or negative numbers. In this treatise of radicals, or irrationals,
he first uses the character &#x221A; to denote a root,
and sets after it the mark of the power whose root is intended;
as &#x221A;[dram] 20 for the square root of 20, and &#x221A; <figure/> 20
for the cube root of the same, and so on. He treats
here also of negative numbers, or what he calls surd or
fictitious, or numbers less than 0. On which he takes
occasion to observe, that when a geometrical progression
is continued downwards below 1, then the exponents of
the terms, or the arithmetical progression, will go below
0 into negative numhers, and will yet be the true exponents
of the former; as in these,
<table rend="border"><row role="data"><cell cols="1" rows="1" role="data">Expon.</cell><cell cols="1" rows="1" rend="align=right" role="data">-3</cell><cell cols="1" rows="1" rend="align=right" role="data">-2</cell><cell cols="1" rows="1" rend="align=right" role="data">-1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pow.</cell><cell cols="1" rows="1" rend="align=right" role="data">1/8</cell><cell cols="1" rows="1" rend="align=right" role="data">1/4</cell><cell cols="1" rows="1" rend="align=right" role="data">1/2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell></row></table>
And he gives examples to shew that these negative
exponents perform their office the same as the positive
ones, in all the opesations.
<cb/></p><p><hi rend="italics">Chap.</hi> 6. Of the perfection of the Rule of Algebra,
and of Secondary Roots. In the reduction of equations
he uses a more general rule than those who had preceded
him, who detailed the rule in a multitude of
cases; instead of which, he directs to multiply or divide
the two sides equally, to transpose the terms with
+ or -, and lastly to extract such root as may be denoted
by the exponent of the highest power.</p><p>As to secondary roots, Cardan treated of a 2d <hi rend="italics">ignota</hi>
or unknown, which he called <hi rend="italics">quantitas,</hi> and denoted it
by the initial <hi rend="italics">q,</hi> to distinguish it from the first. But
here Stifelius, for distinction sake, and to prevent one
root from being mistaken for others, assigns literal
marks to all of them, as A, B, C, D, &amp;c, and then
performs all the usual operations with them, joining
them together as we do now, except that he subjoins the
initial of the power, instead of its numeral exponent:
thus, 3A into 9B makes 27AB,<lb/>
3[dram] into 4B makes 12[dram]B,<lb/>
2<figure/> into 4A[dram] makes 8<figure/> A[dram],<lb/>
1A squared makes 1A[dram],<lb/>
6 into 3C makes 18C,<lb/>
2A[dram] into 5A <figure/> makes 10A<hi rend="italics">&#x222B;s</hi>, &amp;c, &amp;c.<lb/>
8<figure/>A[dram] divided by 4 <figure/> makes 2A[dram], &amp;c.<lb/>
The square root of 25A[dram] is 5A, &amp;c.<lb/>
Also 2A added to 2<figure/> makes 2<figure/> + 2A,<lb/>
and 2A subtr. from 2<figure/> makes 2<figure/> - 2A.<lb/>
And he shews how to use the same, in questions concerning
several unknown numbers; where he puts a
different character for each of them, as<figure/>, A, B, C, &amp;c;
he then makes out, from the conditions of the question,
as many equations as there are characters; from these
he finds the value of each letter, in terms of some one of
the rest; and so, expelling them all but that one, reduces
the whole to a final equation, as we do at present.</p><p>The remainder of the book is employed with the
solutions of a great number of questions to exercise all
the rules and methods; some of which are geometrical
ones.</p><p>From this account of the state of Algebra in Stifelius,
it appears that the improvements made by himself, or
other Germans, beyond those of the Italians, as contained
in Cardan's book of 1539, were as follow:</p><p>1st. He introduced the characters +, -, &#x221A;, for
plus, minus, and root, or <hi rend="italics">radix,</hi> as he calls it.</p><p>2d. The initials <figure/>, [dram], <figure/>, &amp;c. for the powers.</p><p>3d. He treated all the higher orders of quadratics by
the same general rule.</p><p>4th. He introduced the numeral exponents of the
powers, -3, -2, -1, 0, 1, 2, 3, &amp;c, both positive and
negative, so far as integral numbers, but not fractional
ones; calling them by the name <hi rend="italics">exponens,</hi> exponent:
and he taught the general uses of the exponents, in the
several operations of powers, as we now use them, or
the logarithms.</p><p>5th. And lastly, he used the general literal notation
A, B, C, D, &amp;c, for so many different unknown or
general quantities.
<hi rend="center"><hi rend="smallcaps">OF SCHEUBELIUS.</hi></hi></p><p>John Scheubelius published several books upon Arithmetic
and Algebra. The one now before me, is intitled
<hi rend="italics">Algebr&#xE6; Compendiosa Facilisque Descriptio, qu&#xE2; depro-</hi>
<pb n="79"/><cb/>
<hi rend="italics">muntur magna Arithmetices miracula. Authore Johanne
Scheubelio Mathematicarum Professore in Academia Tubingensi.
Parisiis</hi> 1552. But at the end of the book it
is dated 1551. The work is most beautifully printed,
and is a very clear and succinct treatise; and both in
the form and matter much resembles a modern printed
book. He says that the writers ascribe this art to Diophantus,
which is the first time that I find this Greek
author mentioned by the modern algebraists: he farther
observes, that the Latins call it <hi rend="italics">Regula Rei &amp; Census,</hi> the
rule of the thing and the square (or of the 1st and 2d
power); and the Arabs, Algebra. His characters and
operations are much the same as those of Stifelius, using
the signs and characters +, -, &#x221A;, and the powers <figure/>,
<figure/>, [dram], <figure/>, &amp;c, where the character <figure/> is used for 1 or unity,
or a number, or the o power; prefixing also the numeral
coefficients; thus 44&#x222B;[dram] + 11[dram] + 31<figure/> - 53<figure/>.
He uses also the exponents 0, 1, 2, 3, &amp;c, of the powers,
the same way as Stifelius, before him. He performs the
algebraical calculations, first in integers, and then in
fractions, much the same as we do at present. Then of
equations, which he says may be of infinite degrees,
though he treats only of two, namely the first and second
orders, or what we call simple and quadratic equations,
in the usual way, taking however only the positive roots
of these; and adverting to all the higher orders of
quadratics, namely, <hi rend="italics">x</hi><hi rend="sup">4</hi>, <hi rend="italics">ax</hi><hi rend="sup">2</hi>, <hi rend="italics">b</hi>;
<hi rend="center"><hi rend="italics">x</hi><hi rend="sup">6</hi>, <hi rend="italics">ax</hi><hi rend="sup">3</hi>, <hi rend="italics">b</hi>;</hi>
<hi rend="center"><hi rend="italics">x</hi><hi rend="sup">8</hi>, <hi rend="italics">ax</hi><hi rend="sup">4</hi>, <hi rend="italics">b</hi>; &amp;c.</hi></p><p>Next follows a tract on surds, both simple and compound,
quadratic, cubic, binomial, and residual. Here
he first marks the notation, observing that the root is
either denoted by the initial of the word, or, after some
authors, by the mark &#x221A;:, viz. the sq. root &#x221A;:, the
cube root w&#x221A;:, and the 4th root, or root of the root
thus v&#x221A;:, which latter method he mostly uses. He
then gives the Arithmetic of surds, in multiplication,
division, addition, and subtraction. In these last two
rules he squares the sum or difference of the surds, and
then sets the root to the whole compound, which he
calls <hi rend="italics">radix collecti,</hi> what Cardan calls <hi rend="italics">radix universalis.</hi>
Thus &#x221A;12 &#xB1; &#x221A;20 is ra. col. 32 &#xB1; &#x221A;960. But when
the terms will reduce to a common surd, he then unites
them into one number; as &#x221A;27 + &#x221A;12 is equal &#x221A;75.
Also of cubic surds, and 4th roots. In binomial and
residual surds, he remarks the different kinds of them
which answer to the several irrational lines in the 10th
book of Euclid's elements; and then gives this general
rule for extracting the root of any binomial or residual
<hi rend="italics">a</hi> &#xB1; <hi rend="italics">b,</hi> where one or both parts are surds, and <hi rend="italics">a</hi> the
greater quantity, namely, that the square root of it is
; which he illustrates
by many examples. This rule will only succeed
however, so as to come out in simple terms, in certain
cases, namely, either when <hi rend="italics">a</hi><hi rend="sup">2</hi>-<hi rend="italics">b</hi><hi rend="sup">2</hi> is a square, or when
<hi rend="italics">a</hi> and &#x221A;(<hi rend="italics">a</hi><hi rend="sup">2</hi> - <hi rend="italics">b</hi><hi rend="sup">2</hi>) will reduce to a common surd, and
unite: in all other cases the root is in two compound
surds, instead of one. He gives also another rule, which
comes however to the same thing as the former, though
by the words of them they seem to be different.
<cb/></p><p>Scheubelius wrote much about the time of Cardan and
Stifelius. And as he takes no notice of cubic equations,
it is probable he had neither seen nor heard any thing
about them; which might very well happen, the one
living in Italy, and the other in Germany. And, besides,
I know not if this be the first edition of Scheubel's book:
it is rather likely it is not, as it is printed at Paris, and
he himself was professor of mathematics at Tubingen in
Germany.
<hi rend="center"><hi rend="smallcaps">ROBERT RECORDE.</hi></hi></p><p>The first part of his Arithmetic was published in 1552;
and the second part in 1557, under the title of, &#x201C;The
Whetstone of Witte, which is the seconde parte of
Arithmetike: containing the Extraction of Rootes:
The Cossike Practise, with the Rule of Equation: and
the Workes of Surde Nombers.&#x201D; The work is in
dialogue between the master and scholar; and is nearly
after the manner of the Germans, Stifelius and Scheubelius,
but especially the latter, whom he often quotes,
and takes examples from. The chief parts of the work
are, 1st. The properties of abstract and figurate numbers.
2nd. The extraction of the square and cube roots,
much the same as at present. Here, when the number is
not an exact power, but having some remainder over,
he either continues the root into decimals as far as he
pleases, by adding to the remainders always periods of
cyphers; or else makes a vulgar fraction for the remaining
part of the root, by taking the remainder for the numerator,
and double the root for the denominator, in the
square root; but in the cube root he takes for the nominator
either the triple square of the root, which is Cardan's
rule, or the triple square and triple root, with one more,
which is Scheubel's rule. 3d. Of Algebra, or &#x201C;Cossike
Nombers.&#x201D; He uses the notation of powers with their
exponents the same as Stifel, with all the operations in
simple and compound quantities, or integers and fractions.
And he gives also many examples of extracting
the roots of compound algebraic quantities, even when
the roots are from two to six terms, in imitation of
the same process in numbers, just as we do at present;
which is the first instance of this kind that I have observed.
As of this quantity:
25[dram] <figure/> + 80&#x222B;[dram]<hi rend="sup">Square</hi> - 26[dram][dram] - 63[dram] (5<figure/> +<hi rend="sup">Root.</hi> 8[dram] - 9<figure/>.</p><p>4th. &#x201C;The Rule of Equation, commonly called Algeber's
Rule.&#x201D; He here, first of any, introduces the character
=, for brevity sake. His words are, &#x201C;And to
avoide the tediouse repetition of these woordes: is
equalle to: I will sette as I doe often in woorke use, a
paire of paralleles, or gemowe lines of one lengthe,
thus:=, bicause noe 2 thynges can be moare equalle.&#x201D;
He gives the rules for simple and quadratic equations,
with many examples. He gives also some examples in
higher compound equations, with a root for each of
them, but gives no rule how to find it. 5th. &#x201C;Of Surde
Nombers.&#x201D; This is a very ample treatise on surds, both
simple and compound, and surds of various degrees, as
square, cubic, and biquadratic, marking the roots in
Scheubel's manner, thus: &#x221A;, w&#x221A;, v&#x221A;. He here uses
the names bimedial, binomial, and residual; but says
they have been used by others before him, though this
is the first place where I have observed the two latter.&#x2014;
Hence it appears that the things which chiefly are new
in this author, are these three, viz.
<pb n="80"/><cb/></p><p>1. The extraction of the roots of compound algebraic
quantities.</p><p>2. The use of the terms binomial and residual.</p><p>3. The use of the sign of equality, or =.
<hi rend="center"><hi rend="smallcaps">OF PELETARIUS.</hi></hi></p><p>The first edition of this author's algebra was printed
in 4to at Paris, in 1558, under this title, <hi rend="italics">Jacobi Peletarii
Cenomani, de occulta parte Numerorum, quam Algebram
vocant. Lib. duo.</hi></p><p>In the preface he speaks of the supposed authors of
Algebra, namely Geber, Mahomet the son of Moses,
an Arabian, and Diophantus. But he thinks the art
older, and mentions some of his contemporary writers,
or a very little before him, as Cardan, Stifel, Scheubel,
Chr. Januarius; and a little earlier again, Lucas Paciolus
of Florence, and Stephen Villafrancus a Gaul.</p><p>Of the two books, into which the work is divided,
the first is on rational, and the second on irrational or
surd quantities; each being divided into many chapters.
It will be sufficient to mention only the principal
articles.</p><p>He calls the series of powers <hi rend="italics">numeri creati,</hi> or derived
numbers, or also radicals, because they are all raised from
one root or <hi rend="italics">radix.</hi> He names them thus, radix, quadratus
cubus, quadrato-quadratus, or biquadratus, supersolidus,
quadrato-cubus, &amp;c; and marks them thus &#x211E;, <hi rend="italics">q, <figure/>, qq,
&#x222B;s, q<figure/>, b&#x222B;s, &amp;c.</hi> Of these he gives the following series
in numbers, having the common ratio 2, with their
marks set over them, and the exponents set over these
again, in an arithmctical series, beginning at 0, thus:
<table><row role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">&#x211E;</cell><cell cols="1" rows="1" role="data"><hi rend="italics">q</hi></cell><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data"><hi rend="italics">qq</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">q</hi><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">b&#x222B;s</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">qqq</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">32</cell><cell cols="1" rows="1" rend="align=center" role="data">64</cell><cell cols="1" rows="1" rend="align=center" role="data">128</cell><cell cols="1" rows="1" rend="align=center" role="data">256</cell><cell cols="1" rows="1" role="data">&amp;c.</cell></row></table>
And he shews the use of the exponents, the same as Stifelius
and Scheubelius; like whom also he prefixes coefficients
to quantities of all kinds, as also the radical &#x221A;.
But he does not follow them in the use of the signs +
and -, but employs the initials <hi rend="italics">p</hi> and <hi rend="italics">m</hi> for the same
purpose. After the operations of addition, &amp;c, he performs
involution and cvolution also much the same way
as at present: thus, in powers, raise the coefficient to
the power required, and multiply the exponent, or
sign, as he calls it, by 2, or 3, or 4, &amp;c, for the 2nd,
3d, 4th, &amp;c, power; and the reverse for extraction:
and hence he observes, if the number or coefficient will
not exactly extract, or the sign do not exactly divide, the
quantity is a surd.</p><p>After the operations of compound quantities, and
fractions, and reduction of equations, namely, simple and
quadratic equations, as usual, in chap. 16, <hi rend="italics">De Inveniendis
generatim Radicibus Denominatorum,</hi> he gives a method of
finding the roots of equations among the divisors of the
absolute number, when the root is rational, whether it
be integral or fractional; for then, he observes, the root
always lies hid in that number, and is some one of its
divisors. This is exemplified in several instances, both
of quadratic and cubic equations, and both for integral
and fractional roots. And he here observes, that he knows
not of any person who has yet given general rules for
the solution of cubic equations; which shews that when
he wrote this book, either Cardan's last book was not
published, or else it had not yet come to his knowledge.</p><p><hi rend="italics">Chap.</hi> 17 contains, in a few words, directions for
<cb/>
bringing questions to equations, and for reducing these.
He here observes, that some authors call the unknown
number <hi rend="italics">res,</hi> and others the <hi rend="italics">positio</hi>; but that he calls it <hi rend="italics">radix,</hi>
or root, and marks it thus &#x211E;: hence the term, root of an
equation. But it was before called radix by Stifelius.</p><p><hi rend="italics">Chap.</hi> 21 <hi rend="italics">&amp; seq.</hi> treat of secondary roots, or a plurality
of roots, denoted by A, B, C, &amp;c, after Stifelius.</p><p>The 2d book contains the like operations in surds, or
irrational numbers, and is a very complete work on this
subject indeed. He treats first of simple or single surds,
then of binomial surds, and lastly of trinomial surds. He
gives here the same rule for extracting the root of a
binomial and residual as Scheubelius, viz, . Individing by a
binomial or residual, he proceeds as all others before him
had done, namely, reducing the divisor to a simple quantity,
by multiplying it by the same two terms with the
sign of one of them changed, that is by the binomial if
it be a residual, or by the same residual if it be a binomial;
and multiplying the dividend by the same thing:
thus .
And, in imitation of this method, in division by trinomial
surds, he directs to reduce the trinomial divisor
first to a binomial or residual, by multiplying it by the
same trinomial with the sign of one term changed, and
then to reduce this binomial or residual to a simple
nomial as above; observing to multiply the dividend by
the same quantities as the divisor. Thus, if the divisor
be 4 + &#x221A;2 - &#x221A;3; multiplying this by 4 + &#x221A;2 +
&#x221A;3, the product is 15 + 8&#x221A;2; then this binomial
multiplied by the residual 15 - 8&#x221A;2, gives 225
- 128 or 97 for the simple divisor: and the dividend,
whatever it is, must also be multiplied by the two 4 +
&#x221A;2 + &#x221A;3 and 15 - 8&#x221A;2. Or in general, if the divisor
be <hi rend="italics">a</hi> + &#x221A;<hi rend="italics">b</hi> - &#x221A;<hi rend="italics">c</hi>; multiply
it by <hi rend="italics">a</hi> + &#x221A;<hi rend="italics">b</hi> + &#x221A;<hi rend="italics">c,</hi> which
gives ;
then multiply this by <hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi> - <hi rend="italics">c</hi> - 2<hi rend="italics">a&#x221A;b,</hi>
and it gives (<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi> - <hi rend="italics">c</hi>)<hi rend="sup">2</hi> - 4<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b,</hi> which
will be rational, and will all collect into one single term.
But Tartalea must have been in possession of some such
rule as this, as one of the questions he proposed to Florido
was of this nature, namely to find such a quantity
as multiplied by a given trinomial surd, shall make it
rational: and it appears, from what is done above,
that, the given trinomial being , the
answer will be .</p><p><hi rend="italics">Chap.</hi> 24 shews the composition of the cube of a binomial
or residual, and thence remarks on the root of
the case or equation 1 <figure/> <hi rend="italics">p</hi> 3&#x211E; eqnal to 10, which he
seems to know something about, though he had not Cardan's
rules.</p><p><hi rend="italics">Chap.</hi> 30, which is the last, treats of certain precepts
relating to square and cubic numbers, with a table of
such squares and cubes for all numbers to 140; also
shewing how to compute them both, by adding always
their differences.
<pb n="81"/><cb/></p><p>He then concludes with remarking that there are
many curious properties of these numbers, one of which
is this, that the sum of any number of the cubes, taken
from the beginning, always makes a square number, the
root of which is the sum of the roots of the cubes;
so that the series of squares so formed, have for their
roots &#x2014; 1, 3, 6, 10, 15, 21, &amp;c.
whose diff. are the natural n<hi rend="sup">os</hi> 1, 2, 3, 4, 5, 6, &amp;c.
Namely, , &amp;c.
Or in general, .</p><p>This work of Peletarius is a very ingenious and masterly
composition, treating in an able manner of the
several parts of the subject then known, excepting the
cubic equations. But his real discoveries, or improvements,
may be reduced to these three, viz.</p><p>1st. That the root of an eqnation, is one of the divisors
of the absolute term.</p><p>2d. He taught how to reduce trinomials to simple
terms, by multiplying them by compound factors.</p><p>3d. He taught eurious precepts and properties concerning
square and cube numbers, and the method of
constructing a series of each by addition only, namely
by adding successively their several orders of differences.
<hi rend="center"><hi rend="smallcaps">RAMUS.</hi></hi></p><p>Peter Ramus wrote his arithmetic and algebra about
the year 1560. His notation of the powers is thus, <hi rend="italics">l,
q, c, bq,</hi> being the initials of latus, quadratus, cubus,
biquadratus. He treats only of simple and quadratic
equations. And the only thing remarkable in his work,
is the first article, on the names and invention of Algebra,
which we have noticed at the beginning of this
history.
<hi rend="center"><hi rend="smallcaps">BOMBELLI.</hi></hi></p><p>Raphael Bombelli's Algebra was published at Bologna
in the year 1579, in the Italian language. It seems however
it was written some time before, as the dedication
is dated 1572. In a short, but neat, introduction, he
first adverts, in a few words, to the great excellence and
usefulness of arithmetic and algebra. He then laments
that it had hitherto been treated in so imperfect and irregular
a way; and declares it is his intention to remedy
all defects, and to make the science and practice of it
as easy and perfect as may be. And for this purpose he
first resolved to procure and study all the former authors.
He then mentions several of these, with a short
history or character of them; as Mahomet the son of
Moses, an Arabian; Leonard Pisano; Lucas de Burgo,
the first printed author in Europe; Oroncius; Seribelius;
Boglione Francesi; Stifelius in Germany; a certain
Spaniard, doubtless meaning Nunez or Nonius;
and lastly Cardan, Ferrari, and Tartalea; with some
others since, whose names he omits. He then adds a
curious paragraph concerning Diophantus: he says that
some years since there had been found, in the Vatican
library, a Greek work on this art, composed by a certain
Diophantus, of Alexandria, a Greek author, who
lived in the time of Antoninus Pius; which work having
been shewn to him by Mr. Antonio Maria Pazzi Reggiano,
public lecturer on mathematics at Rome; and
sinding it to be a good work, these two formed the re-
<cb/>
solution of giving it to the world, and he says that they
had already translated five books, of the six which were
then extant, being as yet hindered by other avocations
from completing the work. He then adds the following
strange circumstance, viz. <hi rend="italics">that they had found that in
the said work the Indian authors are often cited; by which
they learned that this science was known among the
Indians before the Arabians had it:</hi> a paragraph the
more remarkable as I have never understood that asty
other person could ever find, in Diophantus, any reference
to Indian writers: and I have examined his work
with some attention, for that purpose.</p><p>Bombelli's work is divided into three books. In the
first, are laid down the definitions and operations of
powers and roots, with various sorts of radicals, simple
and compound, binomial, residual, &amp;c; mostly aster
the rules and manner of former writers, excepting in
some few instances, which I shall here take notice
of. And first of his rule for the cube root of binomials
or residuals, which for the sake of brevity, may be
expressed in modern notation as follows: let &#x221A;<hi rend="italics">b</hi> + <hi rend="italics">a</hi>
be the binomial, the term &#x221A;<hi rend="italics">b</hi> being greater than
<hi rend="italics">a</hi>; then the rule for the cube root of &#x221A;(<hi rend="italics">b</hi> + <hi rend="italics">a</hi>) comes
to this, .
Which is a rule that can be of little or no use; for, in
the first place,  is the same as
&#x2015;(P + Q)<hi rend="sup">2</hi>; and , the original quantity first proposed. The
next thing remarkable in this 1st book, is his method
for the square roots of negative quantities, and his rule
for the cube roots of such imaginary binomials as arise
from the irreducible case in cubic equations. His words,
translated, are these: &#x201C;I have found another sort of cubic
root, very different from the former, which arises
from the case of the cube equal to the first power and a
number, when the cube of the 1/3d part of the (coef. of the)
1st power, is greater than the square of half the absolute
number, which sort of square root hath in its algorism,
names and operations different from the others; for in that
case, the excess cannot be called either plus or minus; I
therefore call it <hi rend="italics">plus of minus</hi> when it is to be added, and
<hi rend="italics">minus of minus</hi> when it is to be subtracted.&#x201D; He then
gives a set of rules for the signs when such roots are multiplied,
and illustrates them by a great many examples.
His rule for the cube roots of such binomials, viz. such
as <hi rend="italics">a</hi> + &#x221A;- <hi rend="italics">b,</hi> is this: First sind &#x221A;<hi rend="sup">3</hi>(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi>); then, by
trials search out a number <hi rend="italics">c,</hi> and a sq. root &#x221A;<hi rend="italics">d,</hi> such, that
the sum of their squares <hi rend="italics">c</hi><hi rend="sup">2</hi> + <hi rend="italics">d</hi> may be 
<pb n="82"/><cb/>
and also ; then shall  sought. Thus, to extract the cube root
of 2 + &#x221A; - 121: here ; then
taking <hi rend="italics">c</hi> = 2, and <hi rend="italics">d</hi> = 1, it is ,
and , as it ought; and
therefore 2+&#x221A;-1 is = the cube root of 2+&#x221A;-121,
as required.</p><p>The notation in this book, is the initial <hi rend="italics">R</hi> for root,
with <hi rend="italics">q</hi> or <hi rend="italics">c</hi> &amp;c after it, for quadrate or cubic, &amp;c root.
Also <hi rend="italics">p</hi> for <hi rend="italics">plus,</hi> and <hi rend="italics">m</hi> for <hi rend="italics">minus.</hi></p><p>In the 2d book, Bombelli treats of the algorism with
unknown quantities, and the resolution of equations.
He first gives the definitions and characters of the unknown
quantity and its powers, in which he deviates
from the former authors, but professes to imitate Diophantus.
He calls the unknown quantity <hi rend="italics">tanto,</hi> and
marks it thus <figure/>,
Its square or 2d power <hi rend="italics">potenza,</hi> <figure/>,
Its cube <hi rend="italics">cubo,</hi> <figure/>,
and the higher names are compounded of these, and
marked <figure/>, &amp;c, so that he denotes all the
powers by their exponents set over the common character
<figure/>. And all these powers he calls by the general
name <hi rend="italics">dignita,</hi> dignity. He then performs all the algorism
of these powers, by means of their exponents, as
we do at present, viz, adding them in multiplication,
subtracting in division, multiplying them by the index
in involution, and dividing by the same in evolution.</p><p>In equations he goes regularly through all the cases,
and varieties of the signs and terms; first all the simple
or single powers, and then all the compound cases; demonstrating
the rules geometrically, and illustrating
them by many examples.</p><p>In compound quadratics, he gives two rules: the first
is by freeing the <hi rend="italics">potenza</hi> or square from its coefficient
by division, and then completing the square, &amp;c, in the
usual way: and the 2d rule, when the first term has
its coefficient, may be thus expressed; if ,
then . He takes only the positive
root or roots; and in the case , which has
two, he observes that the nature of the problem must
shew which of the two is the proper one.</p><p>In the cubic equations, he gives the rules and transformations,
&amp;c, after the manner of Cardan; remarking
that some of the cases have only one root, but others two
or three, of which some are true, and others false or negative.
And in one place he says that by means of the
case  he <hi rend="italics">trisects</hi> or <hi rend="italics">divides an angle into three
equal parts.</hi></p><p>When he arrives at biquadratic equations, and particularly
to this case <hi rend="italics">x</hi><hi rend="sup">4</hi> + <hi rend="italics">ax</hi> - <hi rend="italics">b,</hi> he says, &#x201C;Since I
have seen Diophantus's work, I have always been of
opinion that his chief intention was to come to this
equation, because I observe he labours at finding always
square numbers, and such, that adding some number to
them, may make squares; and I believe that the six
books, which are lost, may treat of this equation, &amp;c.&#x201D;
&#x2014;&#x201C;But Lewis Ferrari,&#x201D; he adds, &#x201C;of this city, also
laboured in this way, and found out a rule for such
cases, which was a very fine invention, and therefore I
<cb/>
shall here treat of it the best I can.&#x201D; This he accordingly
does, in all the cases of biquadratics, both with
respect to the number of terms in the equation, and the
signs of the terms, except I think this most general case
only ; fully applying Ferrari's
method in all cases. Which concludes the 2d book.</p><p>The 3d book consists only of the resolution of near
300 practical questions, as exercises in all the rules and
equations, some of which are taken from Diophantus
and other authors.</p><p>Upon the whole it appears that this is a plain, explicit,
and very orderly treatise on algebra, in which are
very well explained the rules and methods of former
writers. But Bombelli does not produce much of improvement
or invention of his own, except his notation,
which varies from others, and is by means of one general
character, with the numeral indices of Stifelius. He
also first remarks that angles are trisected by a cubie
equation. But I know not how to account for his assertion,
that Diophantus often cites the Indian authors;
which I think must be a mistake in Bombelli.
<hi rend="center"><hi rend="smallcaps">CLAVIUS.</hi></hi></p><p>Christopher Clavius wrote his Algebra about the year
1580, though it was not published till 1608, at Orleans.
He mostly follows Stifelius and Scheubelius in his notation
and method, &amp;c, having scarcely any variations
from them; nor does he treat of cubic equations. He
mentions the names given to the art, and the opinions
about its origin, in which he inclines to ascribe it to
Diophantus, from what Diophantus says in his preface
to Dyonisius.
<hi rend="center"><hi rend="smallcaps">STEVINUS.</hi></hi></p><p>The Arithmetic of Simon Stevin of Bruges, was
published in 1585, and his Algebra a little afterwards.
They were also printed in an edition of his works at
Leyden in 1634, with some notes and additions of
Albert Girard, who it seems died the year before, this
edition being published for the benefit of Girrard's
widow and children. The Algebra is an ingenious and
original work. He denotes the <hi rend="italics">res,</hi> or unknown quantity,
in a way of his own, namely by a small circle &#x25CB;,
within which he places the numeral exponent of the
power, as &#x25CB;0, &#x25CB;1, &#x25CB;2, &#x25CB;3, &amp;c, which are the 0, 1, 2, 3,
&amp;c power of the quantity &#x25CB;; where &#x25CB;0, or the 0
power, is the beginning of quantity, or arithmetical
unit. He also extends this notation to roots or fractional
exponents, and even to radical ones.</p><p>Thus &#x25CB;1/2, &#x25CB;1/3, &#x25CB;1/4, &amp;c, are the sq. root, cube root, 4th
root, &amp;c;</p><p>and &#x25CB;4/3 is the cube root of the square;</p><p>and &#x25CB;3/2 is the sq. root of the cube. And so of others.</p><p>The first three powers, &#x25CB;1, &#x25CB;2, &#x25CB;3, he also calls
<hi rend="italics">coste</hi> (side), <hi rend="italics">quarre</hi> (square), <hi rend="italics">cube</hi> (cube); and the first of
them, &#x25CB;1, the prime quantity, which he observes is also
<hi rend="italics">metaphorically</hi> called the racine or root, (the mark of
which is also &#x221A;), because it represents the root or
origin from whence all other quantities spring or arise,
called the <hi rend="italics">potences</hi> or powers of it. He condemns the
terms sursolids, and numbers absurd, irrational, irregular,
inexplicable, or surd, and shews that all numbers are
denoted the same way, and are all equally proper ex-
<pb n="83"/><cb/>
pressions of some length or magnitude, or some power
of the same root. He also rejects all the compound
expressions of square-squared, cube-squared, cube-cubed,
&amp;c, and shews that it is best to name them all by their exponents,
as the 1st, 2d, 3d, 4th, 5th, 6th, &amp;c power or
quantity in the series. And on his extension of the new
notation he justly observes that what was before obscure,
laborious, and tiresome, will by these marks be clear,
easy, and pleasant. He also makes the notation of algebraic
quantities more general in their coefficients, including
in them not only integers, as 3&#x25CB;1, but also fractions
and radicals, as (3/4)&#x25CB;2, and &#x221A;2&#x25CB;3, &amp;c. He has various
other peculiarities in his notations; all shewing an
original and inventive mind. A quantity of several
terms, <hi rend="italics">he</hi> calls a multinomial, and also binomial, trinomial,
&amp;c, according to the number of the terms.
He uses the signs + and -, and sometimes: for equality;
also X for division of fractions, or to multiply
crosswise thus, 5/7X2/3 : 15/14.</p><p>He teaches the generation of powers
<figure/>
by means of the annexed table of
numbers, which are the coefficients
of all the terms except the first and
last. And he makes use of the same
numbers also for extracting all roots
whatever: both which things had
first been done by Stifelius. In extracting the roots of
non-quadrate or non-cubic numbers, he has the same
approximations as at present, viz, either to continue the
extraction indefinitely in decimals, by adding periods
of ciphers, or by making a fraction of the remainder
in this manner, viz,  nearly, and
 nearly; where <hi rend="italics">n</hi> is the
nearest exact root of N; which is Peletarius's rule, and
which differs from Tartalea's rule, as this wants the 1
in the denominator. And in like manner he goes on
to the roots of higher powers.</p><p>He then treats of equations, and their inventors,
which according to him are thus:
<hi rend="brace"><note anchored="true" place="unspecified">Mahomet, son of Moses, an Arabian,
invented these</note>
&#x25CB;1 egale <hi rend="italics">&#xE0;</hi> &#x25CB;0,
its derivatives,
&#x25CB;2 egale <hi rend="italics">&#xE0;</hi> &#x25CB;1, &#x25CB;0,</hi>
And some unknown author, the derivatives of this.
<hi rend="brace"><note anchored="true" place="unspecified">Some unknown author invented
these</note>
&#x25CB;3 egale <hi rend="italics">&#xE0;</hi> &#x25CB;1 &#x25CB;0,
&#x25CB;3 egale <hi rend="italics">&#xE0;</hi> &#x25CB;2 &#x25CB;0,</hi></p><p>But afterwards he mentions Ferreus, Tartalea, Cardan,
&amp;c, as being also concerned in the invetion of them.</p><p>Lewis Ferrari invented &#x25CB;4 egale <hi rend="italics">&#xE0;</hi> &#x25CB;3 &#x25CB;2 &#x25CB;1 &#x25CB;0.</p><p>He says also that Diophantus once resolves the case
&#x25CB;2 egale <hi rend="italics">&#xE0;</hi> &#x25CB;1 &#x398;. In his reduction of equations,
which is full and masterly, he always puts the highest
power on one side alone, equal to all the other terms,
set in their order, on the other side, whether they be
+ or -. And he demonstrates all the rules both arithmetically
and geometrically. In cubics, he gives up
the irreducible case, as hopeless: but says that Bombelli
resolves it by <hi rend="italics">plus of minus,</hi> and <hi rend="italics">minus of minus</hi>;
thus, if , then , that is, . He resolves biquadratics by
<cb/>
means of cubics and quadratics. In quadratics, he
takes both the two roots, but looks for no more than
two in cubics or biquadratics. He gives also a general
method of approaching indefinitely near, in decimals, to
the root of any equation whatever: but it is very laborious,
being little more than trying all numbers, one
after another, finding thus the 1st figure, then the 2d,
then the 3d, &amp;c, among these ten characters 0, 1, 2, 3,
4, 5, 6, 7, 8, 9. And finally he applies the rules in
the resolution of a great many practical questions.</p><p>Although a general air of originality and improvement
runs through the whole of Stevinus's work, yet
his more remarkable or peculiar inventions, may be reduced
to these few following: viz,</p><p>1st. He invented not only a new character for the
unknown quantity, but greatly improved the notation
of powers, by numeral indices, sirst given by Stifelius
as to integral exponents; which Stevinus extended to
fractional and all other sorts of exponents, thereby denoting
all sorts of roots the same way as powers, by numeral
exponents. A circumstance hitherto thought to
be of much later invention.</p><p>2d. He improved and extended the use and notation
of coefficients, including in them fractions and radicals,
and all sorts of numbers in general.</p><p>3d. A quantity of several terms, he called generally a
multinomial; and he denoted all nomials whatever
by particular names expressing the number of their
terms, binomial, trinomial, quadrinomial, &amp;c.</p><p>4th. A numeral resolution of all equations whatever by
one general method.</p><p>Besides which, he hints at some unknown author as
the first inventor of the rules for cubic equations; by
whom may probably be intended the author of the
Arabic manuscript treatise on cubic equations, given to
the library at Leyden by the celebrated Warner.
<hi rend="center"><hi rend="smallcaps">VIETA.</hi></hi></p><p>Most of Vieta's algebraical works were written
about or a little before, the year 1600, but some of
them were not published till after his death, which
happened in the year 1603. And his whole mathematical
works were collected together by Francis Schooten,
and elegantly printed in a folio volume in 1646. Of
these, the algebraical parts are as follow:</p><p>1. Isagoge in Artem Analyticam.</p><p>2. Ad Logisticen Speciosam Not&#xE6; priores.</p><p>3. Zeteticorum libri quinque.</p><p>4. De &#xC6;quationum Recognitione, &amp; Emendatione.</p><p>5. De Numeros&#xE2; Potestatum ad Exegesin Resolutione.</p><p>Of all these I shall give a very minute account, especially
in such parts as contain any discoveries, as we
here meet with more improvements and inventions in
the nature of equations, than in almost any former
author. And first of the <hi rend="italics">Isagoge</hi> or Introduction to
the Analytic Art. In this short introduction Vieta
lays down certain pr&#xE6;cognita in this art, as definitions,
axioms, notations, common precepts or operations
of addition, subtraction, multiplication, and division,
with rules for questions, &amp;c. From which we find,
1st. That the names of his powers are latus, quadratum,
cubus, quadrato-quadratum, quadrato-cubus, cubocubus,
&amp;c; in which he follows the method of Diophantus,
and not that derived from the Arabians.
<pb n="84"/><cb/>
2d. That he calls powers pure or adfected, and first
here uses the terms coefficient, affirmative, negative,
specious logistics or calculations, homogeneum comparationis,
or the absolute known term of an equation,
homogeneum adfectionis, or the 2d or other term which
makes the equation adfected, &amp;c. 3d. That he uses
the capital letters to denote the known as well as unknown
quantities, to render his rules and calculations
general, namely, the vowels A, E, I, O, U, Y for the
unknown quantities, and the consonants B, C, D, &amp;c,
for the known ones. 4th. That he uses the sign +
between two terms for addition; &#x2014; for subtraction,
placing the grcater before the less; and when it is not
known which term is the greater, he places = between
them for the difference, as we now use &lt;01&gt;; thus A =
B is the same as A &lt;01&gt; B; that he expresses division
by placing the terms like a fraction, as at present;
though he was not first in this. But that he uses no
characters for multiplication or equality, but writes the
words themselves, as well as the names of all the powers,
as he uses no exponents, which causes much trouble
and prolixity in the progress of his work; and the numeral
coefficients set after the literal quantities, have a
disagreeable effect.</p><p>II. <hi rend="italics">Ad Logisticen Speciosam, Not&#xE6; Priores.</hi> These
consist of various theorems concerning sums, differences,
products, powers, proportionals, &amp;c, with the genesis
of powers from binomial and residual roots, and certain
properties of rational right-angled triangles.</p><p>III. <hi rend="italics">Zeteticorum libri quinque.</hi> The zetetics or questions
in these 5 books are chiefly from Diophantus,
but resolved more generally by literal arithmetic. And
in these questions are also investigated rules for the resolution
of quadratic and cubic equations. In these
also Vieta first uses a line drawn over compound quantities,
as a vinculum.</p><p>IV. <hi rend="italics">De &#xC6;quationum Recognitione, &amp; Emendatione.</hi>
These two books, which contain Vieta's chief improvements
in Algebra, were not published till the year 1615,
by Alexander Anderson, a learned and ingenious
Scotchman, with various corrections and additions.
The 1st of these two books consists of 20 chapters.
In the first six chapters, rules are drawn from the
zetetics for the resolution of quadratic and cubic equations.
These rules are by means of certain quantities
in continued proportion, but in the resolution they
come to the same thing as Cardan's rules. In the
cubics, Vieta sometimes changes the negative roots into
affirmative, as Cardan had done, but he finds only the
affirmative roots. And he here refers the irreducible
cafe to angular sections for a solution, a method which
had been mentioned by Bombelli.</p><p><hi rend="italics">Chap.</hi> 7 treats of the general method of transforming
equations, which is done either by changing the root
in various ways, namely by substituting another instead
of it which is either increased or diminished, or multiplied
or divided by fome known number, or raised or
depressed in some known proportion; or by retaining
the same root, and equally multiplying all the terms.
Which sorts of transformation, it is evident, are intended
to make the equation become simpler, or more
convenient for solution. And all or most of these reductions
and transformations were also practised by Cardan.</p><p><hi rend="italics">Chap.</hi> 8 shews what purposes are answered by the
<cb/>
foregoing transformations; such as taking away some
of the terms out of an equation, and particularly the
2d term, which is done by increasing or diminishing the
root by the coefficient of the 2d term divided by the
index of the first: by which means also the affected
quadratic is reduced to a simple one. And various
other effects are produced.</p><p><hi rend="italics">Chap.</hi> 9 shews how to deduce compound quadratic
equations from pure ones, which is done by increasing
or diminishing the root by a given quantity, being
one application of the foregoing reductions.</p><p><hi rend="italics">Chap.</hi> 10, the reduction of cubic equations affected
with the 1st power, to such as are affected with the
2d power; by the same means.</p><p>In <hi rend="italics">chap.</hi> 11, by the same means also, the 2d term is
restored to such cubic equations as want it.</p><p>In <hi rend="italics">chap.</hi> 12, quadratic and cubic equations are raised
to higher degrees by substituting for the root, the square
or cube of another root divided by a given quantity.</p><p>In <hi rend="italics">chap.</hi> 13 affected biquadratic equations are deduced
from affected quadratics in this manner, when expressed
in the modern notation:
If , then shall .
For since , therefore ,
and its square is
: but ,
therefore
, or .
And in like manner for the biquadratic affected with its
other terms. And in a similar manner also, in chap. 14,
affected cubic equations are deduced from the affected
quadratics.</p><p>In <hi rend="italics">chap.</hi> 15 it is shewn that the quadratic  has two values of the root A, or has ambiguous
roots, as he calls them; and also that the cubics, biquadratics,
&amp;c, which are raised or deduced from that quadratic,
have also double roots.</p><p>Having, in the foregoing chapters, shewn how the
coefficients of equations of the 3d and 4th degree are
formed from those of the 2d degree, of the same root;
and that certain quadratics, and others raised from them,
have double roots; then in the 16th chap. Vieta shews
what relation those two roots bear to the coefficients of
the two lowest terms of an equation consisting of only
three terms. Thus,
And so on for the same terms with their signs variously
changed.
<pb n="85"/><cb/></p><p><hi rend="italics">Chap.</hi> 17 contains several theorems concerning quantities
in continued geometrical progression. Which are
preparatory to what follows, concerning the double
roots of equations, the nature of which he expounds by
means of such properties of proportional quantities.</p><p><hi rend="italics">Chap.</hi> 18, <hi rend="italics">&#xC6;quationum ancipitum constitutiva</hi>; treating
of the nature of the double roots of equations. Thus,
if <hi rend="italics">a, b, c, d,</hi> &amp;c, be quantities in continual progression;
then, 1st, of equations affected with the first power,
If ; then , Z=<hi rend="italics">ab,</hi> &amp; A=<hi rend="italics">a</hi> or <hi rend="italics">b.</hi>
If ; then , , &amp; A =<hi rend="italics">a</hi> or <hi rend="italics">c.</hi>
And in general, if BA-A<hi rend="sup">n+1</hi>; then , , and A=<hi rend="italics">a</hi>
or <hi rend="italics">k</hi> the first or last term. Where the number of terms<hi rend="italics">a</hi><hi rend="sup">n</hi>,
<hi rend="italics">b</hi><hi rend="sup">n</hi>, &amp;c, in B is <hi rend="italics">n</hi> + 1, and the number of terms in Z is <hi rend="italics">n.</hi></p><p>2d, For equations containing only the highest two
powers.
If ; then ,
the sum of all except the last, or sum of all except the
first; where the number of terms in B is <hi rend="italics">n</hi>+1, and the
number of terms in Z is <hi rend="italics">n.</hi></p><p>3d. Of equations affected by the intermediate powers.
If .</p><p>4th. Of the remaining cases.
If ;
then ,
and ,
and .
If ;
then ,
and ;
and .</p><p><hi rend="italics">Chap.</hi> 19. <hi rend="italics">&#xC6;qualitatum contradicentium constitutiva.</hi>
Of the relation of equations of like terms, but the sign
of one term different; containing these 5 theorems, viz,
<cb/></p><p><hi rend="italics">Chap.</hi> 20. <hi rend="italics">&#xC6;qualitatum inversarum constitutiva.</hi>
Containing these six theorems, viz,</p><p><hi rend="italics">Chap.</hi> 21. <hi rend="italics">Alia rursus &#xE6;qualitatem inversarum constitutiva.</hi>
In these two theorems:</p><p>Next follows the 2d of the pieces published by Alexander
Anderson, namely,</p><p><hi rend="italics">De Emendatione &#xC6;quationum,</hi> in 14 chapters.</p><p><hi rend="italics">Chap.</hi> 1. Of preparing equations for their resolution
in numbers, by taking away the 2d term; by which
affected quadratics are reduced to pure ones, and cubia
equations affected with the 2d term are reduced to such
as are affected with the 3d only. Several examples
of both sorts of equations are given. He here too
remarks upon the method of taking away any other term
out of an equation, when the highest power is combined
with that other term only; and this Vieta effects
by means of the coefficients, or, as he calls them, the
unci&#xE6; of the power of a binomial. All which was
also performed by Cardan for the same purpose.</p><p><hi rend="italics">Chap.</hi> 2. <hi rend="italics">De transmutatione <foreign xml:lang="greek">*prw_ton&#x2014;e)\katon</foreign>, qu&#xE6; remedium
est adversus vitium negationis.</hi> Concerning the
transformations by changing the given root A for
another root E, which is equal to the homogeneum
<pb n="86"/><cb/>
comparationis divided by the first root A; by which
means negative terms are changed to affirmative, and
radicals are taken out of the equation when they are
contained in the homogeneum comparationis.</p><p><hi rend="italics">Chap.</hi> 3, <hi rend="italics">De Anastrophe,</hi> shewing the relation between
the roots of correlate equations; from whence, having
given the root of the one equation, that of the other
becomes known; and it consists of these following 8
theorems, mostly deduced from the last 4 chapters of
the foregoing <hi rend="italics">recognitio &#xE6;quationum.</hi></p><p><hi rend="italics">Chap.</hi> 4, <hi rend="italics">De Isom&#xE6;ria, adversus vitium fractionis.</hi>
To take away fractions out of an equation. Thus,
if . Put A = E/D; then .</p><p><hi rend="italics">Chap.</hi> 5, <hi rend="italics">De Symmetrica Climactismo adversus vitium
asymmetri&#xE6;.</hi> To take away radicals or surds out of
equations, by squaring &amp;c the other side of the equation.</p><p><hi rend="italics">Chap.</hi> 6. To reduce biquadratic equations by means
of cubics and quadratics, by methods which are small
variations from those of Ferrari and Cardan.</p><p><hi rend="italics">Chap.</hi> 7. The resolution of cubic equations by rules
which are the same with Cardan's.</p><p><hi rend="italics">Chap.</hi> 8. <hi rend="italics">De Canonica &#xE6;quationum transmutatione, ut
coefficientes subgraduales sint qu&#xE6; pr&#xE6;scribuntur.</hi> To
transmute the equation so that the coefficient of the
lower term, or power, may be any given number, he
changes the root in the given proportion, thus: Let
A be the root of the equation given, E that of the
transmuted equation, B the given coefficient, and X
the required one; then take A = BE/X, which substitute
in the given equation, and it is done.&#x2014;He commonly
changes it so, that X may be 1; which he
does, that the numeral root of the equation may be
the easier found; and this he here performs by trials,
by taking the nearest root of the highest power alone;
and if that does not turn out to be the root of the
whole equation, he concludes that it has no rational
root.</p><p><hi rend="italics">Chap.</hi> 9. To reduce certain peculiar forms of cubics
to quadratics, or to simpler forms, much the same as
Cardan had done. Thus,
<cb/>
1. If ; then is .
2. If ; then is .
3. If ; then is A=2B.
4. If ; then is A=B.
5. If ; then is A=B.
6. If ; then is A=D.
7. If ; then is A=B or=D.
8. If ; then is 
9. If ; then is 
10. If ; then is .
11. If ; then is .</p><p><hi rend="italics">Chap.</hi> 10. <hi rend="italics">Similium reductionum continuatio.</hi> Being
some more similar theorems, when the equation is affected
with all the powers of the unknown quantity A.</p><p><hi rend="italics">Chap.</hi> 11, 12, 13 relate also to certain peculiar forms
of equations, in which the root is one of the terms of
a certain series of continued proportionals.</p><p><hi rend="italics">Chap.</hi> 14, which is the last in this tract, contains, in
four theorems, the general relation between the roots of
an equation and the coefficients of its terms, when all
its roots are positive. Namely,
1. If ; then is A=B or D.
2. If ; then is A=B or D or G.
3. If ; then is A=B or D or G or H.
4. If ; then is A=B or D or G or
H or K.</p><p>And from these last 4 theorems it appears that Vieta
was acquainted with the composition of these equations,
that is, when all their roots are positive, for he never
adverts to negative roots; and from other parts of the
work it appears that he was not aware that the same
properties will obtain in all sorts of roots whatever.
But it is not certain in what manner he obtained
these theorems, as he has not given any account of
the investigations, though that was usually his way
on other occasions; but he here contents himself with
barely announcing the theorems as above, and for this
strange reason, that he might at length bring his work
to a conclusion.</p><p>To this piece is added, by Alexander Anderson,
an Appendix, containing the construction of the cubic
equations by the trisection of an angle, and a demonstration
of the property referred to by Vieta for this
purpose.</p><p><hi rend="italics">De Numerosa Potestatum Purarum Resolutione.</hi> Vieta
here gives some examples of extracting the roots of
pure powers, in the way that had been long before
practised, by pointing the number into periods of
figures according to the index of the root to be extracted,
and then proceeding from one period to another,
in the usual way.
<pb n="87"/><cb/></p><p><hi rend="italics">De Numerosa Potestatum adsectarum Resolutione.</hi> And
here, in close imitation of the above method for the
roots of pure powers, Vieta extracts those of adfected
ones; or finding the roots of affected equations, placing
always the homogeneum comparationis, or absolute
term, on one side, and all the terms affected with the
unknown quantity, and their proper signs, on the
other side. The method is very laborious, and is but
little more than what was before done by Stevinus on
this subject, depending not a little upon trials. The
examples he uses are such as have either one or two
roots, and indeed such as are affected commonly with
only two powers of the unknown quantity, and which
therefore admit only of those two varieties as to the
number of roots, namely according as the higher of
the two powers is affirmative or negative, the homogeneum
comparationis, on the other side of the equation,
being always affirmative; and he remarks this
general rule, if the higher power be negative, the
equation has two roots; otherwise, only one; that is,
affirmative roots; for as to negative and imaginary ones,
Vieta knew nothing about them, or at least he takes
no notice of them. By the foregoing extraction,
Vieta finds both the greater and less root of the two
that are contained in the equation, and either of them
that he pleases; having first, for this purpose, laid
down some observations concerning the limits within
which the two roots are contained. Also, having
found one of the roots, he shews how the other root
may be found by means of another equation, which
is a degree lower than the given one; though not by
depressing the given equation, by dividing it as is now
done; but from the nature of proportionals, and the
theorems relating to equations, as given in the former
tracts, he finds the terms of another equation, different
from that last mentioned, from the root &amp;c of which,
the 2d root of the original equation may be obtained.</p><p>In the course of this work, Vieta makes also some
observations on equations that are ambiguous, or have
three roots; namely, that the equation , or as we write it  is ambiguous,
when the 2d term is negative, and the 3d
term affirmative, and when 1/3 of the square of 6 the
coefficient of the 2d term, exceeds 11, the coefficient
of the 3d term, and has then three roots. Or in general,
if , and (1/3)<hi rend="italics">a</hi><hi rend="sup">2</hi>&gt;<hi rend="italics">b,</hi> the equation
is ambiguous, and has three roots. He shews also,
from the relation of the coefficients, how to sind whether
the roots are in arithmetical progression or not,
and how far the middle root differs from the extremes,
by means of a cubic equation of this form .
In all or most of which remarks he was preceded by
Cardan.&#x2014;Vieta also remarks that the case , has three roots by the same rule, viz, 2, 2, 5,
but that two of them are equal. And farther, that
when (1/3)<hi rend="italics">a</hi><hi rend="sup">2</hi> is=<hi rend="italics">b,</hi> then all the three roots are equal, as
in the case , the three roots of
which are 2, 2, 2. But when (1/3)<hi rend="italics">a</hi><hi rend="sup">2</hi> is less than <hi rend="italics">b,</hi> the
case is not ambiguous, having but one root. And
when <hi rend="italics">ab</hi>=<hi rend="italics">c,</hi> then <hi rend="italics">a</hi>=<hi rend="italics">x</hi> is one root itself.</p><p>Many curious notes are added at the end, with remarks
on the method of finding the approximate roots,
when they are not rational, which is done in two
ways, in imitation of the same thing in the extraction
<cb/>
of pure powers, viz, the one by forming a fraction
of the remainder after all the figures of the homogeneum
comparationis are exhausted; the other by increasing
the root of the equation in a 10 fold, or 100
fold, &amp;c, proportion, and then dividing the root which
results by 10, or 100, &amp;c: and this is a decimal approximation.
AndVieta observes that the roots will be
increased 10 or 100 fold, &amp;c, by adding the corresponding
number of ciphers to the coefficient of the
2d term, double that number to the 3d, triple the
same number to the 4th, and so on. So if the equation
were
,
then  will have its root 10 fold,
and  will have it 100 fold.</p><p>Besides the foregoing algebraical works, Vieta gave
various constructions of equations by means of circles
and right lines, and angular sections, which may be
considered as an algebraical tract, or a method of exhibiting
the roots of certain equations having all their
roots affirmative, and by means of which he resolved
the celebrated equation of 45 powers, proposed to all
the world by Adrianus Romanus.</p><p>Having now delivered a particular analysis of Vieta's
algebraical writings, it will be proper, as with other
authors, to collect into one view the particulars of his
more remarkable peculiarities, inventions, and improvements.</p><p>And first it may be observed, that his writings
shew great originality of genius and invention, and
that he made alterations and improvements in most
parts of algebra; though in other parts and respects
his method is inferior to some of his predecessors;
as, for instance, where he neglects to avail himself of
the negative roots of Cardan; the numeral exponents
of Stifelius, instead of which he uses the names of the
powers themselves; or the fractional exponents of Stevinus;
or the commodious way of presixing the coefficient
before the quantity or factor; and such like
circumstances; the want of which gives his Algebra
the appearance of an age much earlier than its ownBut
his real inventions of things before not known,
may be reduced to the following particulars.</p><p>1st. Vieta introduced the general use of the letters
of the alphabet to denote indefinite given quantities;
which had only been done on some particular occasions
before his time. But the general use of letters for
the unknown quantities was before pretty common
with Stifelius and his successors. Vieta uses the vowels
A, E, I, O, U, Y for the unknown quantities, and the
consonants B, C, D, &amp;c, for known ones.</p><p>2d. He invented, and introduced many expressions
or terms, several of which are in use to this day:
such as coefficient, affirmative and negative, pure and
adsected or affected, unci&#xE6;, homogeneum adfectionis,
homogeneum comparationis, the line or vinculum over
compound quantities thus &#x2015;(A + B). And his method of
setting down his equations, is to place the homogeneum
comparationis, or absolute known term, on the righthand
side alone, and on the other side all the terms
which contain the unknown quantity, with their proper
signs.</p><p>3d. In most of the rules and reductions for cubic and
<pb n="88"/><cb/>
other equations, he made some improvements, and variations
in the modes.</p><p>4th. He shewed how to change the root of an equation
in a given proportion.</p><p>5. He derived or raised the cubic and biquadratic,
&amp;c equations, from quadratics; but not by composition
in Harriot's way, but by squaring and otherwise
multiplying certain parts of the quadratic. And as
some quadratic equations have two roots, therefore the
cubics and others raised from them, have also the same
two roots, and no more. And hence he comes to
know what relation these two roots bear to the coefficients
of the two lowest terms of cubic and other
equations, when they have only 3 terms, namely, by
comparing them with similar equations so raised from
quadratics. And, on the contrary, what the roots are,
in terms of such coefficients.</p><p>6. He made some observations on the limits of the
two roots of certain equations.</p><p>7. He stated the general relation between the roots
of certain equations and the coefficients of its terms,
when the terms are alternately plus and minus, and
none of them are wanting, or the roots all positive.</p><p>8. He extracted the roots of affected equations, by a
method of approximation similar to that for pure powers.</p><p>9. He gave the construction of certain equations,
and exhibited their roots by means of angular sections;
before adverted to by Bombelli.
<hi rend="center"><hi rend="smallcaps">OF ALBERT GIRARD.</hi></hi></p><p>Albert Girard was an ingenious Dutch or Flemish
mathematician, who died about the year 1633. He
published an edition of Stevinus's Arithmetic in 1625,
augmented with many notes; and the year after his
death was published by his widow, an edition of the
whole works of Stevinus, in the same manner, which
Girard had left ready for the press. But the work
which entitles him to a particular notice in this history,
is his &#x201C;<hi rend="italics">Invention Nouvelle en l' Algebre, tant pour la sobution
des equations, que pour recognoistre le nombre des solutions
qu'elles re&#xE7;oivent, avec plusieurs choses qui sont necessaires
a la perfection de ceste divine science</hi>;&#x201D; which
was printed at Amsterdam 1629, in small quarto in
63 pages, viz, 49 pages on Arithmetic and Algebra,
and the rest on the measure of the supersicies of
spherical triangles and polygons, by him then lately
discovered.</p><p>In this work Girard first premises a short tract on
Arithmetic; in the notation of which he has something
peculiar, viz, dividing the numbers into the
ranks of millions, billions, trillions, &amp;c.</p><p>He next delivers the common rules of Algebra, both
in integers, fractions and radicals; with the notation
of the quantities and signs. In this part he uses
sometimes the letters A, B, C, &amp;c, after the manner
of Vieta, but more commonly the characters of
Stevinus, viz, &#x25CB;0, &#x25CB;1, &#x25CB;2, &#x25CB;3, &amp;c, for the powers of the
unknown quantity, with their roots &#x25CB;5/2, &#x25CB;1/2, &#x25CB;1/4, &#x25CB;2/3, &#x25CB;&lt;*&gt;/4,
&amp;c, used by Stevinus; and sometimes the more usual
marks of the roots as, &#x221A; or &#x221A;<hi rend="sup">2</hi>, &#x221A;<hi rend="sup">3</hi>, &#x221A;<hi rend="sup">4</hi>, &amp;c; presixing
the coefficients, as 6&#x25CB;2, or 3&#x221A;<hi rend="sup">5</hi>32, or 2&#x25CB;1/2. In the
signs he follows his predecessors so far as to have + for
plus, - or &#xF7; for minus, = for general or indefinite
difference, A + B for the sum, A - B or A = B for
<cb/>
the difference, AB the product, and A/B for the quotient
of A and B. He uses the parentheses ( ) for the
vinculum or bond of compound quantities, as is now
commonly practised on the continent; as A(AB+Bq),
or &#x221A;<hi rend="sup">3</hi> (A cub. - 3AqB); and he introduces the new
characters ff for <hi rend="italics">greater than,</hi> and &#xA7; for <hi rend="italics">less than</hi>; but
he uses no character for equality, only the word itself.</p><p>Girard gives a new rule for extracting the cube root
of binomials, which however is in a good measure tentative,
and which he explains thus: To extract the cube
root of 72 + &#x221A;5120.
<table><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">The squares of the terms</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">5184</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">5120</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=right" role="data">their difference</cell><cell cols="1" rows="1" rend="align=right" role="data">64,</cell><cell cols="1" rows="1" role="data">and its</cell></row></table>
cube root 4. Which shews that the difference between
the squares of the terms required is 4; and the rational
part 72 being the greater, the greater term of the root
will be rational also; and farther, that the greater terms
of the power and root are commensurable, as also the
two less terms. Then having made a table as in the
<table><row role="data"><cell cols="1" rows="1" role="data">2 + &#x221A;0</cell></row><row role="data"><cell cols="1" rows="1" role="data">3 + &#x221A;5</cell></row><row role="data"><cell cols="1" rows="1" role="data">4 + &#x221A;12</cell></row><row role="data"><cell cols="1" rows="1" role="data">5 + &#x221A;21</cell></row></table>
margin, where the square of the rational
term always exceeds that of the other, by the
number 4 above mentioned, one of these binomials
must be the cubic root sought, if the
given quantity have such a root, and it must
be one of these four forms, for it is known
to be carried far enough by observing that the cube
root of 72 is less than 5, and the cube root of 5120
less than 21; indeed, this being the case, the last binomial
is excluded, as evidently too great; and the first
is excluded because one of its terms is 0; therefore
the root must be either 3+&#x221A;5 or 4+&#x221A;12. And to
know whether of these two it must be, try which of
them has its two terms exact divisors of the corresponding
terms of the given quantity; then it is found that
3 and 4 are both divisors of 72, but that only 5, and
not 12, is a divisor of 5120; therefore 3+&#x221A;5 is the
root sought, which upon trial is found to answer. It
is remarkable here that Girard uses 4+&#x221A;20 instead of
4+&#x221A;12, and 5+&#x221A;29 instead of 5+&#x221A;20, contrary to
his own rule.</p><p>Girard then gives distinct and plain rules for bringing
questions to equations, and for the reduction of those
equations to their simplest form, for solution, by the usual
modes, and also by the way called by Vieta <hi rend="italics">Isomeria,</hi>
multiplying the terms of the equation by the terms of
a geometrical progression, by which means the roots are
altered in the proportion of 1 to the ratio of the progression.
He then treats of the methods of finding
the roots of the several sorts of equations, quadratic,
cubic, &amp;c; and adds remarks on the proper number of
conditions or equations for limiting questions. The
quadratics are resolved by completing the square, and
both the positive and negative roots are taken; and he
observes that sometimes the equation is impossible, as 2&lt;*&gt;
equ. 6&#x25CB;1 - 25, whose roots, he adds, are 3 + &#x221A;-16
and 3 - &#x221A;-16.</p><p>The cubic equations he resolves by Cardan's rule,
except the irreducible case, which he the first of any
resolves by a table of sines; the other cases also he resolves
by tables of sines and tangents; and adds geometrical
constructions by means of the hyperbola or the
<pb n="89"/><cb/>
trisection of angles. He next adds a particular mode
of resolving all sorts of equations, that have rational
roots, upon the principle of the roots being divisors of
the last or absolute term, as before mentioned by Peletarius;
and then gives the method of approximating
to other roots that are not rational, much the same way
as Stevinus.</p><p>Having found one root of an equation, by any of the
former methods, by means of it he depresses the equation
one degree lower, then finds another root, and so on
till they are all found; for he shews that every algebraic
equation admits of as many solutions or roots, as there
are units in the index of the highest power, which
roots may be either positive or negative, or imaginary,
or, as he calls them, greater than nothing, or less than
nothing, or involved; so the roots of the equation 1&#x25CB;3
equ. 7&#x25CB;1 - 6, are 2, 1, and - 3; and the roots of the
equation 1&#x25CB;4 equ. 4&#x25CB;1 - 3 are
<table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">-1 + &#x221A;-2,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">-1 - &#x221A;-2.</cell></row></table></p><p>In depressing an equation to lower degrees, he does
not use the method of resolution of Harriot, but that
which is derived from the general relation of the roots
and coefficients of the terms, which he here fully and
universally states, viz, that the coefficient of the 2d term
is equal to the sum of all the roots; that of the 3d
term equal to the sum of all the products of the roots,
taken two by two; that of the 4th term, the sum of
the products, taken three by three; and so on, to the
last or absolute term, which is the continual product of
all the roots; a property which was before stated by
Vieta, as to the equations that have all their roots positive;
and here extended by Girard to all sorts of roots
whatever: but how either Vieta or he came by this property,
no where appears that I know of. From this
general property, among other deductions, Girard shews
how to find the sums of the powers of the roots of an
equation; thus, let A, B, C, D, &amp;c, be the 1st, 2d, 3d,
4th, &amp;c, coefficient, after the first term, or the sums of
the products taken one by one, two by two, three by
three, &amp;c; then, in all sorts of equations,
<table><row role="data"><cell cols="1" rows="1" role="data">A</cell><cell cols="1" rows="1" rend="rowspan=4" role="data"><hi rend="size(7)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=4" role="data">will be the
sum of the</cell><cell cols="1" rows="1" rend="rowspan=4" role="data"><hi rend="size(7)">{</hi></cell><cell cols="1" rows="1" role="data">roots,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aq-2B</cell><cell cols="1" rows="1" role="data">squares,</cell></row><row role="data"><cell cols="1" rows="1" role="data">A cub.-3AB+3C</cell><cell cols="1" rows="1" role="data">cubes,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aqq-4AqB+4AC+2Bq-4D</cell><cell cols="1" rows="1" role="data">biquadrates.</cell></row></table></p><p>Girard next explains the use of negative roots in
Geometry, shewing that they represent lines only
drawn in a direction contrary to those representing the
positive roots; and he remarks that this is a thing
hitherto unknown. He then terminates the Algebra
by some questions having two or more unknown quantities;
and subjoins to the whole a tract on the mensuration
of the surfaces of spherical triangles and polygons,
by him lately discovered.</p><p>From the foregoing account it appears that,</p><p>1st, He was the first person who understood the general
doctrine of the formation of the coefficients of the powers,
from the sums of their roots, and their products, &amp;c.</p><p>2d, He was the first who understood the use of negative
roots in the solution of geometrical problems.</p><p>3d, He was the first who spoke of the imaginary roots,
and understood that every equation might have as many
roots real and imaginary, and no more, as there are
<cb/>
units in the index of the highest power. And he was
the first who gave the whimsical name of <hi rend="italics">quantities less
than nothing</hi> to the negative. And,</p><p>4th, He was the first who discovered the rules for
summing the powers of the roots of any equation.
<hi rend="center"><hi rend="smallcaps">OF HARRIOT.</hi></hi></p><p>Thomas Harriot, a celebrated astronomer, philosopher,
and mathematician, flourished about the year 1610,
about which time it is probable he wrote his Algebra, as
he was then, and had been for many years before,
celebrated for his mathematical and astronomical labours.
In that year he made observations on the spots
in the sun, and on Jupiter's satellites, the same year also
in which Galileo first observed them: he left many
other curious astronomical observations, and amongst
them, some on the remarkable comets of the years 1607
and 1618. His Algebra was left behind him unpublished,
as well as those other papers, at his death, which happened
in the year 1621, being then 60 years of age, and
but six years after the first publication of the principal
parts of Vieta's Algebra by Alexander Anderson; so
that it is probable that Harriot's Algebra was written
before this time, and indeed that he had never seen these
pieces. Harriot's Algebra was published by his friend
Walter Warner, in the year 1631: and it would doubtless
be highly grateful to the learned in these sciences,
if his other curious algebraical and astronomical works
were published from his original papers in the possession
of the Earl of Egremont, to whom they have descended
from Henry Percy, the Earl of Northumberland, that
noble M&#xE6;cenas of his day. The book is in folio, and
intitled <hi rend="italics">Artis Analytic&#xE6; Praxis, ad &#xC6;quationes Algebraicas
nova, expedita, &amp; generali methodo, resolvendas</hi>;
a work in all parts of it shewing marks of great genius
and originality, and is the first instance of the modern
form of Algebra in which it has ever since appeared.
It is prefaced by 18 definitions, which are these: 1st,
Logistica Speciosa; 2d, Equation; 3d, Synthesis;
4, Analysis; 5, Composition and Resolution; 6, Forming
an Equation; 7, Reduction of an Equation; 8, Verification;
9, Numerosa &amp; Speciosa; 10, Excogitata;
11, Resolution; 12, Roots; 13 and 14, The kinds and
generation of equations by multiplication, from binomial
roots or factors, called original equations,
<table><row role="data"><cell cols="1" rows="1" role="data">as</cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">=<hi rend="italics">aa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">ba</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> - <hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">ca</hi> - <hi rend="italics">bc,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">=<hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">+<hi rend="italics">baa</hi>+<hi rend="italics">bca</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+<hi rend="italics">caa</hi>-<hi rend="italics">bda</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> - <hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-<hi rend="italics">daa</hi>-<hi rend="italics">cda</hi>-<hi rend="italics">bcd,</hi></cell></row></table>
where he puts <hi rend="italics">a</hi> for the unknown quantity, and the
small consonants, <hi rend="italics">b, c, d,</hi> &amp;c, for its literal values or
roots; 15, The first form of canonical equations, which
are derived from the above originals, by transposing
the homogeneum, or absolute term,
<table><row role="data"><cell cols="1" rows="1" role="data">thus <hi rend="italics">aa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">ba</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">ca</hi></cell><cell cols="1" rows="1" role="data">= + <hi rend="italics">bc,</hi> &amp;c;</cell></row></table>
16, The secondary canonicals, formed from the primary
by expelling the 2d term,
<table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">thus <hi rend="italics">aa</hi></cell><cell cols="1" rows="1" role="data">= + <hi rend="italics">bb,</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or <hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">- <hi rend="italics">bba</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">bca</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">cca</hi> =</cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">bbc</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ <hi rend="italics">bcc</hi>;</cell></row></table>
<pb n="90"/><cb/>
17, That these are called canonicals, because they are
adapted to canons or rules for finding the numeral roots,
&amp;c. 18, Reciprocal equations, in which the homogeneum
is the product of the coefficients of the other
terms, and the first term, or highest power of the
root, is equal to the product of the powers in the other
terms, as .</p><p>After these definitions, the work is divided into two
principal parts; 1st, of various generations, reductions,
and preparations of equations for their resolution in the
2d part. The former is divided into 6 sections as follows.</p><p><hi rend="italics">Sect.</hi> 1. <hi rend="italics">Logistices Specios&#xE6;,</hi> exemplified in the 4 operations
of addition, subtraction, multiplication, and division;
as also the reduction of algebraic fractions, and
the ordinary reduction of irregular equations to the
form proper for the resolution of them, namely, so that
all the unknown terms be on one side of the equation,
and the known term on the other, the powers in the
terms ranged in order, the greatest first, and the first or
highest power made positive, and freed from its coefficient;
as ,
<hi rend="center">or .</hi>
In this part he explains some unusual characters which
he introduces, namely
<hi rend="center">= for equality, as <hi rend="italics">a</hi> = <hi rend="italics">b.</hi></hi>
<hi rend="center">&gt; for majority, as <hi rend="italics">a</hi> &gt; <hi rend="italics">b,</hi></hi>
<hi rend="center">&lt; for minority, as <hi rend="italics">a</hi> &lt; <hi rend="italics">b</hi></hi>;
but the first had been before introduced by Robert
Recorde.</p><p><hi rend="italics">Sect.</hi> 2. The generation of original equations from
binomial factors or roots, and the deducing of canonicals
from the originals. He supposes that every equation
has as many roots as dimensions in its highest
power; then supposing the values of the unknown
letter <hi rend="italics">a</hi> in any equation to be <hi rend="italics">b, c, d, f,</hi> &amp;c, that is <hi rend="italics">a</hi>=<hi rend="italics">b,</hi>
and <hi rend="italics">a</hi>=<hi rend="italics">c,</hi> and <hi rend="italics">a</hi>=<hi rend="italics">d,</hi> &amp;c; by transposition, or equal
subtraction, these become , and , and
, &amp;c, or the same letters with contrary signs,
for negative values or roots; then two of these binomial
factors multiplied together, gives a quadratic equation,
three of them a cubic, four of them a biquadratic, and
so on, with all the terms on one side of the equation,
and 0 on the other side, since, every binomial factor
being = 0, the continual product of all of them must
also be = 0. Thus,
<table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">= <hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">baa</hi> + <hi rend="italics">bca</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ <hi rend="italics">caa</hi> - <hi rend="italics">bda</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> - <hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">daa</hi> - <hi rend="italics">cda</hi>-<hi rend="italics">bcd</hi> = 0</cell></row></table>
an original equation,
and
<table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">baa</hi> + <hi rend="italics">bca</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ <hi rend="italics">caa</hi> - <hi rend="italics">bda</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">daa</hi> - <hi rend="italics">cda</hi></cell><cell cols="1" rows="1" role="data">= + <hi rend="italics">bcd</hi></cell></row></table>
its canonical, deduced from it. And these operations
are carried through all the cases of the 2d, 3d and 4th
powers, as to the varieties of the signs + and -, and
the proportions of the roots as to equal and unequal,
with the reciprocals, &amp;c. From which are made evident,
at one glance of the eye, all the relations and
properties between the roots of equations, and the coefficients
of the terms.</p><p><hi rend="italics">Sect.</hi> 3. <hi rend="italics">&#xC6;quationum canonicarum secundariarum a
primariis reductio per gradus alicujus parodici sublatiouem</hi>
<cb/>
<hi rend="italics">radice supposititia invariata manente.</hi> Containing a great
many examples of preparing equations by taking away
the 2d, 3d, or any other of the intermediate terms,
which is done by making the positive coefficients in
that term, equal to the negative ones, by which means
the whole term vanishes, or becomes equal to nothing.</p><p>They are extended as far as equations of the 5th
degree; and at the end are collected, and placed in
regular order, all the secondary canonicals, so reduced,
so that by the uniform law which is visible through
them all, the series may be continued to the higher degrees
as far as we please.</p><p><hi rend="italics">Sect.</hi> 4. <hi rend="italics">&#xC6;quationum canonicarum tam primariarum,
quam secundariarum, radicum designatio.</hi> A great many
literal equations are here set down, and their roots assigned
from the form of the equation, that is all their
positive roots; for their negative roots are not noticed
here; and it is every where proved that they cannot
have any more positive roots than these, and consequently
the rest are negative. That those are roots, he
proves by substituting them instead of the unknown
letter <hi rend="italics">a</hi> in the equation, when they make all the terms
on one side come to the same thing as the homogeneum
on the other side.</p><p><hi rend="italics">Sect.</hi> 5, <hi rend="italics">In qua &#xE6;quationum communium per canonicarum
&#xE6;quipollentiam, radicum numerus determinatur.</hi> On the
number of the roots of common equations, that is the
positive roots. This Harriot determines by comparing
them with the like cases found among his canonical
forms, which two equations, having the same number of
terms with the same signs, and the relations of the coefficients
and homogeneum correspondent, he calls equipollents.
And whatever was the number of positive
roots used in the composition of the canonical, the
same, he infers, is the number in the proposed common
equation. It is remarkable that in all the examples
here used, the number of positive roots is just equal to
the number of the changes in the signs from + to and
from - to +, which is a circumstance, though not
here expressly mentioned, that could not escape the
observation, or the eye, of any one, much less of so
clear and comprehensive a sight as that of Harriot.
In this section are contained many ingenious disquisitions
concerning the limits and magnitudes of quantities,
with several curious lemmas laid down to demonstrate
the propositions by, which lemmas are themselves
demonstrated in a pure mathematical way, from
the magnitudes themselves, independent of geometrical
sigures; such as, 1, If a quantity be divided into any
two unequal parts, the square of half the line will be
greater than the product of the two unequal parts.
2, In three continued proportionals, the sum of the
extremes is greater than double the mean. 3, In four
continued proportionals, the sum of the extremes is
greater than the sum of the two means. 4, In any
two quantities, one-fourth the square of the sum of the
cubes, is greater than the cube of the product of the
two quantities. 5, Of any two quantities <hi rend="italics">q</hi> and <hi rend="italics">r,</hi> then
(1/27)(<hi rend="italics">qq</hi> + <hi rend="italics">qr</hi> + <hi rend="italics">rr</hi>)<hi rend="sup">3</hi> &gt; 1/4 (<hi rend="italics">qqr</hi>+<hi rend="italics">qrr</hi>)<hi rend="sup">2</hi>. 6, If any quantity
be divided into three unequal parts, the square of
1/3 of the whole quantity is greater than 1/3 of the sum of the
three products made of the three unequal parts. 7, Also
the cube of the 1/3 part of the whole, is greater than the
solid or continual product of the three unequal parts.
<pb n="91"/><cb/></p><p><hi rend="italics">Sect.</hi> 6. <hi rend="italics">&#xC6;quationum communium reductio per gradus
alicujus parodici exclusionem &amp; radicis supposititi&#xE6; mutationeni.</hi>
Here are a great many examples of reducing
and transforming equations of the 2d, 3d, and 4th degrees;
chiefly either by multiplying the roots of equations
in any proportion, as was done by Vieta, or increasing
or diminishing the root by a given quantity,
after the manner of Cardan. The former of these reductions
is performed by multiplying the terms of the
equation by the corresponding terms of a geometrical
progression, the 1st term being 1, and the 2d term the
quantity by which the root is to be multiplied. And
the other reduction, or transforming to another root,
which may be greater or less than the given root by a
given quantity, is performed commonly by substituting
<hi rend="italics">e</hi> + or - <hi rend="italics">b</hi> for the given root <hi rend="italics">a,</hi> by which the equation
is reduced to a simpler form. Other modes of substitution
are also used; one of which is this, viz, substituting
(<hi rend="italics">ee</hi> &#xB1; <hi rend="italics">bb</hi>)/<hi rend="italics">e</hi> or <hi rend="italics">e</hi> &#xB1; <hi rend="italics">bb/e</hi> for the root <hi rend="italics">a</hi> in the given equation
 by which it reduces to this quadratic
form , from whence Cardan's
forms are immediately deduced; namely , and therefore ; where he denotes
the cube or 3d root thus &#x221A;3), but without any vinculum
over the compound quantities.</p><p>In this section, Harriot makes various remarks as they
occur: thus he remarks, and demonstrates, that <hi rend="italics">eee</hi> -
3.<hi rend="italics">bbe</hi> = -<hi rend="italics">ccc</hi>
-2.<hi rend="italics">bbb</hi> is an impossible equation, or has no
affirmative root. He remarks also that the three cases
of the equation <hi rend="italics">aaa</hi> - 3.<hi rend="italics">bba</hi> = + 2.<hi rend="italics">ccc</hi> are similar to
the three conic sections; namely to the hyperbola when
<hi rend="italics">c</hi> &gt; <hi rend="italics">b,</hi> to the parabola when <hi rend="italics">c</hi> = <hi rend="italics">b,</hi> or to the ellipsis
when <hi rend="italics">c</hi> &lt; <hi rend="italics">b,</hi> and for which reason this case is not generally
resoluble in species.</p><p>Having thus shewn how to simplify equations, and
prepare them for solution, Harriot enters next upon the
second part of his work, being the
<hi rend="center"><hi rend="italics">Exegetice Numerosa,</hi></hi></p><p>or the numeral resolution of all sorts of equations by a
general method, which is exemplified in a great number
of equations, both simple and affected as far as the 5th
power inclusive; and they are commonly prepared, by
the foregoing parts, by freeing them from their 2d term,
&amp;c. These extractions are explained and performed in
a way different from that of Vieta; and the examples
are first in perfect or terminate roots, and afterwards for
irrational or interminate ones, to which Harriot approximates
by adding always periods of ciphers to the
given number or resolvend, as far as necessary in decimals,
which are continued and set down as such, but
with their proper denominator 10, or 100, or 1000, &amp;c.</p><p>He then concludes the work with
<hi rend="center"><hi rend="italics">Canones Directorii,</hi></hi></p><p>which form a collection of the cases or theorems for
making the foregoing numeral extractions, ready arranged
for use, under the various forms of equations,
with the factors necessary to form the several resolvends
and subtrahends.
<cb/></p><p>And from a review of the whole work, it appears
that Harriot's inventions, peculiarities, and improvements
in algebra, may be comprehended in the following
particulars.</p><p>1st. He introduced the uniform use of the small letters
<hi rend="italics">a, b, c, d,</hi> &amp;c, viz, the vowels <hi rend="italics">a, e,</hi> &amp;c for unknown
quantities, and the consonants <hi rend="italics">b, c, d, f,</hi> &amp;c for the
known ones; which he joins together like the letters of
a word, to represent the multiplication or product of
any number of these literal quantities, and prefixing the
numeral coefficient as we do at present, except only
separated by a point, thus 5.<hi rend="italics">bbc.</hi> For a root he set the
index of the root after the mark &#x221A;; as &#x221A;3) for the
cube root. He also introduced the characters &gt; and &lt;
for greater and less; and in the reduction of equations,
he arranged the operations in separate steps or lines, setting
the explanations in the margin on the left hand,
for each line. By which, and other means, he may be
considered as the introducer of the modern state of Algebra,
which quite changed its form under his hands.</p><p>2d. He shewed the universal generation of all the
compound or affected equations, by the continual multiplication
of so many simple ones, or binomial roots;
thereby plainly exhibiting to the eye the whole circumstances
of the nature, mystery and number of the roots
of equations; with the composition and relations of the
coefficients of the terms; and from which many of the
most important properties have since been deduced.</p><p>3d. He greatly improved the numeral exegesis, or
extraction of the roots of all equations, by clear and explicit
rules and methods, drawn from the foregoing
generation or composition of affected equations of all
degrees.
<hi rend="center"><hi rend="smallcaps">OF OUGHTRED'S CLAVIS.</hi></hi></p><p>Oughtred was contemporary with Harriot, but lived
a long time after him. His Clavis was first published in
1631, the same year in which Harriot's Algebra was
published by his friend Warner. In this work, Oughtred
chiefly follows Vieta, in the notation by the capitals
A, B, C, D, &amp;c, in the designation of products,
powers, and roots, though with some few variations.
His work may be comprehended under the following
particulars.</p><p>1. <hi rend="italics">Notation.</hi> This extends to both Algebra and
Arithmetic, vulgar and decimal. The Algebra chiefly
after the manner of Vieta, as abovesaid. And he separates
the decimals from the integers thus, 21&lt;03&gt;56, which
is the first time I have observed such a separation, and
the decimals set down without their denominator.</p><p>2. The common rules or operations of Arithmetic
and Algebra. In algebraic multiplication, he either
joins the letters together like a word, or connects them
by the mark X, which is the first introduction of this
character of multiplication: thus A X A or AA or
A<hi rend="italics">q.</hi> But omitting the vinculum over compound factors,
used by Vieta. He introduces here many neat
and useful contractions in multiplication and division of
decimals: as that common one of inverting the multiplier,
to have fewer decimals, and abridge the work;
that of omitting always one figure at a time, of the
divisor, for the same purpose; dividing by the component
factors of a number instead of the number itself; as
4 and 6 for 24; and many other neat contractions. He
<pb n="92"/><cb/>
states his proportions thus 7.9 :: 28.36, and denotes
continued proportion thus &lt;04&gt;; which is the first time I
have observed these characters.</p><p>3. Invents and describes various symbolical marks or
abbreviations, which are not now used.</p><p>4. <hi rend="italics">The genesis and analysis of powers.</hi> Denotes powers
like Vieta, and also roots, thus &#x221A;<hi rend="italics">q</hi>6, &#x221A;<hi rend="italics">c</hi>20, &#x221A;<hi rend="italics">qq</hi>24,
&amp;c; and much in his manner too performs the numeral
extraction of roots. He here gives a table of the powers
of the binomial A + E as far as the 10th power,
with all their terms and coefficients, or unci&#xE6; as he calls
them, after Vieta.</p><p>5. <hi rend="italics">Equations.</hi> He here gives express and particular
directions for the several sorts of reductions, according
as the form of the equation may require. And he uses
the letter <hi rend="italics">u</hi> after &#x221A;, for universal, instead of the vinculum
of Vieta. And observes that the signs of all the
terms of the powers of A + E are positive, but those
of A - E are alternately positive and negative.</p><p>6. Next follow many properties of triangles and
other geometrical sigures; and the first instance of applying
Algebra to Geometry, so as to investigate new
geometrical properties; and after the algebraical resolution
of each problem, he commonly deduces and gives
a geometrical construction adapted to it. He gives also
a good tract on angular sections.</p><p>7. The work concludes with the numeral resolution
of affected equations, in which he follows the manner of
Vieta, but he is more explicit.
<hi rend="center"><hi rend="smallcaps">OF DESCARTES.</hi></hi></p><p>Descartes's Geometry was first published in 1637,
being six years after the publication of Harriot's Algebra.
That work was rather an application of Algebra
to Geometry, than the science either of Algebra or Geometry
itself, purely and properly so called. And yet
he made improvements in both. We must observe
however, that all the properties of equations, &amp;c,
which he sets down, are not to be considered as even
meant by himself for new inventions or discoveries;
but as statements and enumerations of properties, before
known and taught by other authors, which he is about
to make some use or application of, and for which reason
it is that he mentions those properties.</p><p>Descartes's Geometry consists of three books. The
sirst of these is, <hi rend="italics">De Problematibus, qu&#xE6; construi possunt,
adhibendo tantum rectas lineas &amp; circulos.</hi> He here accommodates
or performs arithmetical operations by
Geometry, supposing some line to represent unity, and
then, by means of proportionals, shewing how to multiply,
divide, and extract roots by lines. He next describes
the notation he uses, but not because it is a new
one, for it is the same as had been used by former authors,
viz, <hi rend="italics">a</hi> + <hi rend="italics">b</hi> for the addition of <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> also <hi rend="italics">a</hi> - <hi rend="italics">b</hi>
for their subtraction, <hi rend="italics">ab</hi> multiplication, <hi rend="italics">a/b</hi> division, <hi rend="italics">aa</hi>
or <hi rend="italics">a</hi><hi rend="sup">2</hi> the square of <hi rend="italics">a, a</hi><hi rend="sup">3</hi> its cube, &amp;c: also &#x221A;(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi>)
for the square root of <hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi>, and 
for the cube root, &amp;c. He then observes, after Stifelius,
that there must be as many equations as there are
unknown lines or quantities; and that they must be
reduced all to one final equation, by exterminating all
the unknown letters except one; when the final equation
will appear like these,
<cb/>
Where he uses <figure/> for = or equality, setting the highest
term or power alone on one side of the equation, and all
the other terms on the other side, with their proper
signs.</p><p>Descartes next defines plane problems, namely, such
as can be resolved by right lines and circles, described
on a plane superficies; and then the final equation rises
only to the 2d power of the unknown letter. He then
constructs such equations, viz. quadratics, by the circle,
thus finding geometrically the root or roots, that is, the
positive ones. But when the lines, by which the roots
are determined, neither cut nor touch, he observes that
the equation has then no possible root, or that the problem
is impossible. He then concludes this book with
the algebraical solution of the celebrated problem, before
treated of by the antients, namely, to sind a point,
or the locus of all the points, from whence a line being
drawn to meet any number of given lines in given angles,
the product of the segments of some of them shall
have a given ratio to that of the rest.</p><p><hi rend="italics">Lib.</hi> 2. <hi rend="italics">De Natura Linearum Curvarum.</hi> This is
a good algebraical treatise on curve lines in general, and
the first of the kind that has been produced by the moderns.
Here the nature of the curve is expressed by an
equation containing two unknown or variable lines, and
others that are known or constant, as <hi rend="italics">y</hi><hi rend="sup">z</hi> <figure/> <hi rend="italics">cy</hi> - <hi rend="italics">cxy/b</hi>
+ <hi rend="italics">ay</hi> - <hi rend="italics">ac.</hi> But, not relating to pure Algebra, the particulars
will be most properly placed under the article of
curve lines, and other terms relating to them. Only one
discovery, among many ingenious applications of Algebra
to Geometry, may here be particularly noticed, as it
may be considered as the first step towards the arithmetic
of infinites; and that is the method of tangents, here
given, or, which comes to the same thing, of drawing a
line perpendicular to a curve at any point, which is an
ingenious application of the general form of an equation,
generated in Harriot's way, that has two equal roots, to
the equation of the curve. Of which a particular account
will be given at the article <hi rend="smallcaps">Tangents.</hi></p><p><hi rend="italics">Lib.</hi> 3. <hi rend="italics">De Constructione Problematum Solidorum, et
Solida excedentium.</hi> Descartes begins this book with remarks
on the nature and roots of equations, observing
that they have as many roots as dimensions, which he
shews, after Harriot, by multiplying a certain number
of simple binomial equations together, as <hi rend="italics">x</hi> - 2 <figure/> 0,
and <hi rend="italics">x</hi> - 3 <figure/> 0, and <hi rend="italics">x</hi> - 4 <figure/> 0, producing <hi rend="italics">x</hi><hi rend="sup">3</hi> - 9<hi rend="italics">xx</hi>
+ 26<hi rend="italics">x</hi> - 24 <figure/> 0. He here remarks that equations
may sometimes have their roots <hi rend="italics">false,</hi> or what we call
negative, which he opposes to those that are positive, or
as he calls them <hi rend="italics">true,</hi> as Cardan had done before. As a
natural deduction from the generation or composition of
equations, by multiplication, he infers their resolution,
or depression, or decomposition, namely, dividing them
by the binomial factors which were multiplied to produce
the equation: and he observes that by this operation
it is known that this divisor is one of the binomial
roots, and that there can be no more roots than dimensions,
or than those which form with the unknown letter
<pb n="93"/><cb/>
<hi rend="italics">x,</hi> binomials that will exactly divide the equation, as
Harriot had shewn before. Descartes adverts to several
other properties, mostly known before, which he has
occasion to make use of in the progress of his work;
such as, that equations may have as many true roots as
the terms have changes of the signs + and -, and as
many false ones as successions of the same signs: which
number and nature of the roots had before been partly
shewn by Cardan and Vieta, from the relation of the
coefficients, and their signs, and more fully by Harriot
in his 5th section. And hence Descartes infers the method
of changing the true roots to false, and the false
to true, namely by changing the signs of the even terms
only, as Cardan had taught before. Descartes then adverts
to other reductions and transmutations which had
been taught by Cardan, Vieta, and Harriot, such as,
To increase or diminish the roots by any quantity; To
take away the 2d term: To alter the roots in any proportion,
and thence to free the equation from fractions
and radicals.</p><p>Descartes next remarks that the roots of equations,
whether true or false, may be either real or imaginary;
as in the equation <hi rend="italics">x</hi><hi rend="sup">3</hi> - 6<hi rend="italics">xx</hi> + 13<hi rend="italics">x</hi> - 10 <figure/> 0, which
has only one real root, namely 2. The imaginary roots
were first noticed by Albert Girard, as before mentioned.
He then treats of the depression of a cubic equation to
a quadratic, or plane problem, that it may be constructed
by the circle, by dividing it by some one of the binomial
factors, which, in Harriot's way, compose the
equation. Peletarius having shewn that the simple root
is one of the divisors of the known term of the equation,
and Harriot that that term is the continual product
of all the roots, Descartes therefore tries all the simple
divisors of that term, till he finds one of them which,
connected with the unknown letter <hi rend="italics">x,</hi> by + or -, will
exactly divide the equation. And the process is the
same for higher powers than the cube. But when a
divisor cannot be thus found, for depressing a biquadratic
equation to a cubic, he gives another rule, which is
a new one, for dissolving it into two quadratics, by
means of a cubic equation, in this manner:
Let the given biqu. be + <hi rend="italics">x</hi><hi rend="sup">4</hi>* . <hi rend="italics">pxx. qx. r</hi> <figure/> 0;
where the sign of (1/2)<hi rend="italics">p</hi> in the two quadratics must be the
same as the sign of <hi rend="italics">p</hi> in the given equation, and in
the 1st quadratic the sign of <hi rend="italics">q</hi>/2<hi rend="italics">y</hi> must be the same as the
the sign of <hi rend="italics">q,</hi> but in the 2d quadratic the contrary.
Then if there be found the root <hi rend="italics">yy</hi> of this cubic equation
<hi rend="italics">y</hi><hi rend="sup">6</hi> . 2<hi rend="italics">py</hi><hi rend="sup">4</hi> + (+<hi rend="italics">pp</hi>)/(.4<hi rend="italics">r</hi>)<hi rend="italics">yy</hi>-<hi rend="italics">qq</hi> <figure/> 0,
where the sign of 2<hi rend="italics">p</hi> is the same as of <hi rend="italics">p</hi> in the given biquadratic,
but the sign of 4<hi rend="italics">r</hi> contrary to that of <hi rend="italics">r</hi> in
the same: Then the value of <hi rend="italics">y,</hi> hence deduced, being
substituted for it in the two quadratic equations, and
their two pairs of roots taken, they will be the four roots
of the proposed biquadratic. And thus also, he hints,
may equations of the 6th power be reduced to those of
the 5th, and those of the 8th power to those of the 7th,
<cb/>
and so on. Descartes does not give the investigation of
this rule; but it has evidently been done, by assuming
indeterminate quantities, after the manner of Ferrari
and Cardan, as coefficients of the terms of the two quadratic
equations, and, after multiplying the two together,
determining their values by comparing the resulting
terms with those of the proposed biquadratic equation.</p><p>After these reductions, which are only mentioned for
the sake of the geometrical constructions which follow,
by simplifying and depressing the equations as much as
they will admit, Descartes then gives the construction
of solid and other higher problems, or of cubic and
higher equations, by means of parabolas and circles;
where he observes that the false roots are denoted by the
ordinates to the parabola lying on the contrary side of
the axis to the true roots. Finally, these constructions
are illustrated by various problems concerning the trisecting
of an angle, and the finding of two or four mean
proportionals; which concludes this ingenious work.</p><p>From the foregoing analysis may easily be collected
the real inventions and improvements made in algebra by
Descartes. His work, as has been observed before, is
not algebra itself, but the application of algebra to geometry,
and the algebraical doctrine of curve lines,
expressing and explaining their nature by algebraical
equations, and on the contrary, constructing and explaining
equations by means of the curve lines. What respects
the geometrical parts of this tract we shall have
occasion to advert to elsewhere; and therefore shall here
only enumerate the circumstances which belong more
peculiarly to the science of Algebra, which I shall
distinguish into the two heads of improvements and
inventions. And</p><p>1st. Of his improvements. That he might fit equations
the better for their application in the construction
of problems, Descartes mentions, as it were by-the-bye,
many things concerning the nature and reduction of
equations, without troubling himself about the first inventors
of them, stating them in his own terms and manner,
which is commonly more clear and explicit, and
osten with improvements of his own. And under this
head we sind that he chiefly followed Cardan, Vieta, and
Harriot, but especially the last, and explains some of
their rules and discoveries more distinctly, and varies
but a little in the notation, putting the first letters of
the alphabet for the known, and the latter letters for the
unknown quantities; also <hi rend="italics">x</hi><hi rend="sup">3</hi> for <hi rend="italics">aaa,</hi> &amp;c; and <figure/> for =.
But Herigone used the numeral exponents in the same
manner two years before. Descartes explained or improved
most parts of the reductions of equations, in
their various transmutations, the number and nature of
their roots, true and false, real and what he calls imaginary,
called involved by Girard; and the depression
of equations to lower degrees.</p><p>2d. As to his inventions and discoveries in algebra,
they may be comprehended in these particulars, namely,
the application of algebra to the geometry of curve lines,
the constructing equations of the higher orders, and
a rule for resolving biquadratic equations by means of
a cubic and two quadratics.</p><p>Having now traced the science of Algebra from its
origin and rude state, down to its modern and more
polished form, in which it has ever since continued,
with very little variation; having analysed all or most
<pb n="94"/><cb/>
of the principal authors, in a chronological order, and
deduced the inventions and improvements made by each
of them; from this time the authors both become too
numerous, and their improvements too inconsiderable,
to merit a detail in the same minute and circumstantial
way: and besides, these will be better explained in a
particular manner under the word or article to which
each of them severally belongs. It may therefore now
suffice to enumerate, or announce only in a cursory manner,
the chief improvements and authors on algebra
down to the present time.</p><p>After the publication of the Geometry of Descartes,
a great many other ingenious men followed the same
course, applying themselves to algebra and the new
geometry, to the mutual improvement of them both;
which was done chiefly by reasoning on the nature and
forms of equations, as generated and composed by
Harriot. Before proceeding upon these however, it is
but proper to take notice here of Fermat, a learned
and ingenious mathematician, who was contemporary
and a competitor of Descartes for his brightest
discoveries, which he was in possession of before the
geometry of Descartes appeared. Namely, the application
of algebra to curve lines, which he expressed by
an algebraical equation, and by them constructing equations
of the 3d and 4th orders; also a method of tangents,
and a method de maximis et minimis, which
approach very near to the method of Fluxions or Increments,
which they strikingly resemble both in the manner
of treating the problems, and in the algebraic notation
and process. The particulars of which, see under their
proper heads. Besides these, Fermat was deeply learned
in the Diophantine problems, and the best edition of
Diophantus's Arithmetic, is that which contains the
notes of Fermat on that ingenious work.</p><p>But to return to the successors of Descartes. His
geometry having been published in Holland, several
learned and ingenious mathematicians of that country,
presently applied themselves to cultivate and improve it;
as Schooten, Hudde, Van-Heuraet, De Witte, Slusius,
Huygens, &amp;c; besides M. de Beaune, and perhaps
some others in France.</p><p>Francis Schooten, professor of mathematics in the
university of Leyden, was one of the first cultivators of
the new geometry. He translated Descartes's Geometry
out of French into Latin, and published it in 1649,
with his commentary upon it, as also Brief Notes of M.
de Beaune; both of them containing many ingenious
and useful things. And in 1659 he gave a new edition of
the same in two volumes, with the addition of several
other ingenious pieces: as two posthumous tracts of
de Beaune, the one on the nature and constitution, the
other on the limits of equations, shewing how to assign
the limits between which are contained the greatest and
least roots of equations, extended and completed by
Erasmus Bartholine: two letters of M. Hudde on the
reduction of equations, and on the maxima and minima
of quantities, containing many ingenious rules;
among which are some concerning the drawing of tangents,
and on the equal roots of equations, which he
determines by multiplying the terms of the equation by
the terms of any arithmetical progression, &lt;*&gt; being one
of the terms, the equation is commonly depressed one
degree lower: also a tract of Van Heuraet on the rectifi-
<cb/>
cation of curve lines; the elements of curves by De Witte;
Schooten's principles of universal mathematics, or introduction
to Descartes's geometry, which had before been
published by itself in 1651; and to the end of the work is
added a posthumous piece of Schooten's (for he died
while the 2d vol. was printing) intitled <hi rend="italics">Tractatus de
concinnandis demonstrationibus geometricis ex calculo algebraico.</hi>
Schooten also published, in 1657, <hi rend="italics">Exercitationes
Mathematic&#xE6;,</hi> in which are contained many curious
algebraical and analytical pieces, amongst others of a
geometrical nature.</p><p>An elaborate commentary on Descartes's Geometry
was also published by F. Rabuel, a Jesuit; and James
Bernoulli, enriched with notes, an edition of the same,
printed at Basil in 169&#x2014;.</p><p>The celebrated Huygens also, among his great discoveries,
very much cultivated the algebraical analysis:
and he is often cited by Schooten, who relates divers inventions
of his, while he was his pupil.</p><p><hi rend="italics">Slusius,</hi> a canon of Liege, published in 1659, <hi rend="italics">Mesolabum,
seu de&#xE6; medi&#xE6; propor. per circulum &amp; ellips. vel hyperb.
infinitis modis exhibit&#xE6;;</hi> by which, any solid problem
may be constructed by infinite different ways.
And in 1668 he gave a second edition of the same, with
the addition of the analysis, and a miscellaneous collection
of curious and important problems, relating to spirals,
centres of gravity, maxima and minima, points of
inflexion, and some Diophantine problems; all shewing
him deeply skilled in Algebra and Geometry.</p><p>There have been a great number of other writers and
improvers of Algebra, of which it may suffice slightly
to mention the chief part, as in the following catalogue.</p><p>Peter Nonius, or Nunez, a Spaniard, wrote about the
time of Cardan, or soon after.</p><p>In 1619 several pieces of Van Collen, or Ceulen,
were translated out of Dutch into Latin, and published
at Leyden by W. Snell; among which are contained a
particular treatise on surds, and his proportion of the
circumference of a circle, to its diameter.</p><p>In 1621 Bachet published, in Greek and Latin, an
edition of Diophantus, with many notes. And another
edition of the same was published in 1670, with additions
by Fermat.</p><p>In 1624 Bachet's <hi rend="italics">Problemes Plaisans et Delectables,</hi>
being curious problems in mathematical recreations.</p><p>In 1634 Herigone published, at Paris, the first course
of mathematics, in 5 vols. 8vo; in the 2d of which is
contained a good treatise on Algebra; in which he uses
the notation by small letters, introduced by the Algebra
of Harriot, which was published three years before,
though the rest of it does not resemble that work, and
one would suspect that Herigone had not seen it. The
whole of this piece bears evident marks of originality
and ingenuity. Besides + for <hi rend="italics">plus,</hi> he uses &lt;01&gt; for
<hi rend="italics">minus,</hi> and | for <hi rend="italics">equality,</hi> with several other usful abbrevations
and marks of his own. In the notation of powers,
he does not repeat the letters like Harriot, but subjoins
the numeral exponents, to the letter, as Descartes did
two years afterwards. And Herigone uses the same numeral
exponents for roots, as &#x221A;3 for the cube root.</p><p>In 1635 Cavalerius published his <hi rend="italics">Indivisibles;</hi> which
proved a new &#xE6;ra in analytics, and gave rise to other
new modes of computation in analytics.
<pb n="95"/><cb/></p><p>About 1640, et seq. Roberval made several notable
improvements in analytics, which are published
in the early volumes of the Memoirs of the Academy
of Sciences; as, 1. A tract on the composition of motion,
and a method of tangents. 2, <hi rend="italics">De recognitione
&#xE6;quationum.</hi> 3, <hi rend="italics">De geometrica planarum &amp; cubicarum
&#xE6;quationum resolutione.</hi> 4, A treatise on indivisibles, &amp;c.</p><p>In 1643 De Billy published <hi rend="italics">Nova Geometri&#xE6; Clavis
Algebra.</hi> And in 1670 <hi rend="italics">Diophantus Redivivus.</hi> He was an
author particularly well skilled in Diophantine problems.</p><p>In 1644 Renaldine published, in 4to, <hi rend="italics">Opus Algebraicum,</hi>
both ancient and modern, with mathematical resolution
and composition. And in 1665, in folio, the
same, greatly enlarged, or rather a new work, which is
very heavy and tedious. In this work Renaldine uses the
parentheses (<hi rend="italics">a</hi><hi rend="sup">2</hi>+<hi rend="italics">b</hi><hi rend="sup">2</hi>) as a vinculum, instead of the line
over, as &#x2015;(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi>).</p><p>In 1655 was published Wallis's <hi rend="italics">Arithmetica Infinitorum,</hi>
being a new method of reasoning on quantities, or a
great improvement on the Indivisibles of Cavalerius, and
which in a great measure led the way to infinite series,
the binomial theorem, and the method of fluxions. Wallis
here treats ingeniously of quadratures and many other
problems, and gives the sirst expression for the quadrature
of the circle by an insinite series. Another series
is here added for the same purpose, by the Lord
Brouncker.</p><p>In 1659 was published <hi rend="italics">Algebra Rhonii Germanice;</hi>
which was in 1668, translated into English by Mr. Thomas
Brancker, with additions and alterations by Dr.
John Pell.</p><p>In 1661 was published in Dutch, a neat piece of Algebra
by Mr. Kinckhuysen; which Sir I. Newton,
while he was professor of mathematics at Cambridge,
made use of and improved, and he meant to republish it,
with the addition of his method of fluxions and infinite
series; but he was prevented by the accidental burning of
some of his papers.</p><p>In 1665 or 1666 Sir Isaac Newton made several of
his brightest discoveries, though they were not published
till afterwards: such as the binomial theorem; the
method of fluxions and infinite series; the quadrature,
rectification, &amp;c of curves; to find the roots of all
sorts of equations, both numeral and literal, in infinite
converging series; the reversion of series, &amp;c. Of
each of which a particular account may be seen in their
proper articles.</p><p>In 1666 M. Frenicle gave several curious tracts concerning
combinations, magic squares, triangular numbers,
&amp;c; which were printed in the early volumes of
Memoirs of the Academy of Sciences.</p><p>In 1668 Thomas Brancker published a translation of
Rhonius's Algebra, with many additions by Dr. John
Pell, who used a peculiar method of registering the
steps in any algebraical process, by means of marks and
abbreviations in a small column drawn down the margin,
by which each line, or step, is clearly explained,
as was before done by Harriot in words at length.</p><p>In 1668 Mercator published his <hi rend="italics">Logarithmotechnia,</hi>
or method of constructing logarithms; in which he
gives the quadrature of the hyperbola, by means of
an infinite series of algebraical terms, found by dividing
a simple algebraic quantity by a compound one, and
for the first time that this operation was given to the
<cb/>
public, though Newton had before that expanded all sorte
of compound algebraical quantities into infinite series.</p><p>In the same year was published James Gregory's <hi rend="italics">Exercitationes
Geometric&#xE6;,</hi> containing, among other things,
a demonstration of Mercator's quadrature of the hyperbola,
by the same series.</p><p>And in the same year was published, in the Philosophical
Transactions, Lord Brouncker's quadrature of
the hyperbola by another infinite series of simple rational
terms, which he had been in possession of since the year
1657, when it was announced to the public by Dr.
Wallis. Lord Brouncker's series for the quadrature of
the circle, had been published by Wallis in his Arithmetic
of Infinites.</p><p>In 1669 Dr. Isaac Barrow published his Optical and
Geometrical Lectures, abounding with profound researches
on the dimensions and properties of curve lines;
but particularly to be noticed here for his method of
tangents, by a mode of calculation similar to that of
Fluxions, or Increments, from which these differ but
little, except in the notation.</p><p>In 1673 was published, in 2 vols. folio, <hi rend="italics">Elements of Algebra,</hi>
by John Kersey; a very ample and complete work,
in which Diophantus's problems are fully explained.</p><p>In 1675 were published Nouveaux Elemens des Mathematiques,
par J. Prestet, pr&#xEA;tre: a prolix and tedious
work, which he presumptuously dedicated to God Almighty.</p><p>About 1677 Leibnitz discovered his <hi rend="italics">Methodus Differrentialis,</hi>
or else made a variation in Newton's Fluxions,
or an extension of Barrow's method, for it is not certain
which. He gave the first instance of it in the Leipsic
Acts for the year 1684. He also improved infinite series,
and gave a simple one for the quadrature os the circle,
in the same acts for 1682.</p><p>In 1682 Ismael Bulliald published, in folio, his <hi rend="italics">Opus
Novum ad Arithmeticam Infinitorum,</hi> being a large amplification
of Wallis's Arithmetic of Insinites.</p><p>In 1683 Tschirnausen gave a memoir, in the Leipsic
Acts, concerning the extraction of the roots of all equations
in a general way; in which he promised too much,
as the method did not succeed.</p><p>In 1684 came out, in English and Latin, 4to, Thomas
Baker's <hi rend="italics">Geometrical Key, or Gate of Equations Unlock'd;</hi>
being an improvement of Descartes's construction
of all equations under the 5th degree, by means
of a circle and only one and the same parabola for all
equations, using any diameter instead of the axis of the
parabola.</p><p>In 1685 was published, in folio, Wallis's <hi rend="italics">Treatise of
Algebra, both Historical and Practical,</hi> with the addition
of several other pieces; shewing the origin, progress,
and advancement of that science, from time to time. It
cannot be denied that, in this work, Wallis has shewn
too much partiality to the Algebra of Harriot. Yet,
on the other hand, it is as true, that M. de Gua, in his
account of it, in the Memoirs of the Academy of Sciences
for 1741, has run at least as far into the same
extreme on the contrary side, with respect to the discoveries
of Vieta; and both these I believe from the
same cause, namely, the want of examining the works
of all former writers on Algebra, and specifying their
several discoveries; as has been done in the course of
this article.
<pb n="96"/><cb/></p><p>In 1687 Dr. Halley gave, in the Philos. Trans. the
construction of cubic and biquadratic equations, by a
parabola and circle; with improvements on what had
been done by Descartes, Baker, &amp;c. Also, in the
same Transactions, a memoir on the number of the roots
of equations, with their limits and signs.</p><p>In 1690 was published, in 4to, by M. Rolle, <hi rend="italics">Trait&#xE9;
d' Alg&#xE9;bre;</hi> in 1699 <hi rend="italics">Une Methode pour la Resolution des
Problemes indetermin&#xE9;s;</hi> and in 1704 <hi rend="italics">Memoires sur Pinverse
des tangents;</hi> and other pieces.</p><p>In 1690 Joseph Raphson published <hi rend="italics">Analysis &#xC6;quationum
Universalis;</hi> being a general method of approximating
to the roots of equations in numbers. And in
1715 he published the <hi rend="italics">History of Fluxious,</hi> both in
English and Latin.</p><p>In 1690 was also published, in 4 vols 4to, Dechale's
<hi rend="italics">Cursus seu mundus mathematicus;</hi> in which is a piece of
algebra, of a very old-fashioned sort, considering the
time when it was written.</p><p>About 1692, and at different times afterwards, De
Lagny published many pieces on the resolution of equations
in numbers, with many theorems and rules for
that purpose.</p><p>In 1693 was published, in a neat little volume, <hi rend="italics">Synopsis
Algebraica, opus posthumum Johannis Alexandri.</hi></p><p>In 1694, Dr. Halley gave, in the Philos. Trans. an
ingenious tract on the numeral extraction of all roots,
without any previous reduction. And this tract is also
added to some editions of Newton's Universal Arithmetic.</p><p>In 1695 Mr. John Ward, of Chester, published, in
8vo, <hi rend="italics">A Compendium of Algebra,</hi> containing plain, easy,
and concise rules, with examples in an easy and clear
way. And in 1706 he published the first edition of his
<hi rend="italics">Young Mathematician's Guide,</hi> or a plain and easy introduction
to the mathematics: a book which is still
in great request, especially with beginners, and which
has been ever since the ordinary introduction of the greatest
part of the mathematicians of this country.</p><p>In 1696 the Marquis de l'H&#xF2;pital published his <hi rend="italics">Analyse
des insiniment petits.</hi> And gave several papers to the
Leipsic Acts and the Memoires of the Academy of Sciences.
He left behind him also an ingenious treatise,
which was published in 1707, intitled <hi rend="italics">Trait&#xE9; analytique
des Sections Coniques, et de la construction des lieux geometriques.</hi></p><p>In 1697, and several other years, Mr. Ab. Demoivre
gave various papers, in the Philos. Trans. containing
improvements in Algebra: viz. in 1697, A method of
raising an infinite multinomial to any power, or extracting
any root of the same. In 1698, The extraction of
the root of an infinite equation. In 1707, Analytical
solution of certain equations of the 3d, 5th, 7th, &amp;c degree.
In 1722, Of algebraic fractions and recurring
series. In 1738, The reduction of radicals into more
simple forms. Also in 1730, he published <hi rend="italics">Miscellanea
analytica de seriebus &amp; quadraturis,</hi> containing great improvements
in series, &amp;c.</p><p>In 169, Mr. Richard Sault published, in 4to, <hi rend="italics">A
New Treatise of Algebra, apply'd to numeral questions and
geometry. With a converging series for all manner of
Adfected Equations.</hi> The series here alluded to, is Mr.
Raphson's method of approximation, which had been
lately published.
<cb/></p><p>In 1699 Hyac. Christopher published at Naples, in
4to, <hi rend="italics">De constructione &#xE6;quationum.</hi></p><p>In 1702 was published Ozanam's Algebra; which is
chiefly remarkable for the Diophantine analysis. He had
published his mathematical dictionary in 1691, and in
1693 his course of mathematics, in 5 vols 8vo, containing
also a piece of algebra.</p><p>In 1704, Dr. John Harris published his <hi rend="italics">Lexicon
Technicum,</hi> the first dictionary of arts and sciences: a
very plain and useful book, especially in the mathematical
articles. And in 1705 a neat little piece on algebra
and fluxions.</p><p>In 1705 M. Guisn&#xE9;e published, in 4to, his <hi rend="italics">Application
de l'algebre a la geometrie:</hi> a useful book.</p><p>In 1706 Mr. William Jones published his <hi rend="italics">Synopsis
Palmariorum Matheseos,</hi> or a new introduction to the
mathematics: a very useful compendium in the mathematical
sciences. And in 1711 he published, in 4to, a
collection of Sir Isaac Newton's papers, intitled <hi rend="italics">Analysis
per quantitatum series, fluxiones, ac differentias: cum
enumeratione linearum tertii ordinis.</hi></p><p>In 1707 was published by Mr. Whiston, the first edition
of Sir Isaac Newton's <hi rend="italics">Arithmetica Universalis: sive de compositione
et resolutione arithmetica liber:</hi> and many editions
have been published since. This work was the text book
used by our great author in his lectures, while he was
professor of mathematics in the university of Cambridge.
And although it was never intended for publication, it
contains many and great improvements in analytics; particularly
in the nature and transmutation of equations;
the limits of the roots of equations; the number of
impossible roots; the invention of divisors, both surd
and rational; the resolution of problems, arithmetical
and geometrical; the linear construction of equations;
approximating to the roots of all equations, &amp;c. To
the later editions of the book is commonly subjoined
Dr. Halley's method of finding the roots of equations.
As the principal parts of this work are not adapted to
the circumstance os beginners, there have been published
commentaries upon it by several persons, as s'Gravesande,
Castilion, Wilder, &amp;c.</p><p>In 1708 M. Reyneau published his <hi rend="italics">Analyse Demontr&#xE9;e,</hi>
in 2 vols 4to. And in 1714 <hi rend="italics">La Science du Calcul, &amp;c.</hi></p><p>In 1709 was published an English translation of Alexander's
algebra. With an ingenious appendix by Humphry
Ditton.</p><p>In 1715 Dr. Brooke Taylor published his <hi rend="italics">Methodus
Incrementorum:</hi> an ingenious and learned work. And
in the Philos. Trans. for 1718, An improvement of the
method of approximating to the roots of equations in
numbers.</p><p>In 1717 M. Nicole gave, in the memoirs of the academy
of sciences, a tract on the calculation of finite
differences. And in several following years, he gave
various other tracts on the same subject, and on the resolution
of equations of the 3d degree, and particularly
on the irreducible case in cubic equations.</p><p>Also in 1717 was published a treatise on Algebra by
Philip Ronayne.</p><p>Also in 1717 Mr. James Sterling published <hi rend="italics">Line&#xE6; tertii
Ordinis</hi>; an ingenious work, containing good improvements
in analytics. Also in 1730 <hi rend="italics">Methodus Differentialis:
sive tractatus de summatione et interpolatione serierum
insinitarum:</hi> with great improvements on infinite series.
<pb n="97"/><cb/></p><p>In 1726 and 1729 Maclaurin gave, in the Philos.
Trans. tracts on the imaginary roots of equations.
And afterwards was published, from his posthumous
papers, his treatise on Algebra, with its application
to curve lines.</p><p>In 1727 came out s'Gravesande's Algebra, with a
specimen of a commentary on Newton's universal
arithmetic.</p><p>In 1728 Mr. Campbell gave, in the Philos. Trans.
an ingenious paper on the number of impossible roots of
equations.</p><p>In 1732 was published Wolsius's Algebra, in his
course of mathematics, in 5 vols. 4to.</p><p>In 1735 Mr. John Kirkby published his arithmetic
and algebra. And in 1748 his doctrine of ultimators.</p><p>In 1740 were published Mr. Thomas Simpson's Essays;
in 1743 his Dissertations, and in 1757 his Tracts;
in all which are contained several improvements in series
and other parts of Algebra. As also in his algebra, first
printed in 1745, and in his Select Exercises, in 1752.</p><p>Also in 1740 was published professor Saunderson's
Elements of Algebra, in 2 vols. 4to.</p><p>In 1741 M. de la Caille published <hi rend="italics">le&#xE7;ons de mathematiques;
ou elemens d'algebre &amp; de geometrie.</hi></p><p>Also in 1741, in the memoirs of the academy of sciences,
were given two articles by M. de Gua, on the
number of positive, negative, and imaginary roots of
equations. With an historical account of the improvements
in Algebra; in which he severely censures Wallis
for his partiality; a circumstance in which he himself
is not less faulty.</p><p>In 1746 M. Clairaut published his Elemens d'algebre,
in which are contained several improvements, especially
on the irreducible case in cubic equations. He has also
several good papers on different parts of analytics, in
the memoirs of the academy of sciences.</p><p>In 1747 M. Fontaine gave, in the memoirs of the
academy of sciences, a paper on the resolution of equations.
Besides some analytical papers in the memoirs of
other years.</p><p>In 1761 M. Castillion published, in 2 vols 4to, Newton's
universal arithmetic, with a large commentary.</p><p>In 1763 Mr. Emerson published his Increments. In
1764 his Algebra, &amp;c.</p><p>In 1764 Mr. Landen published his Residual Analysis.
In 1765 his Mathematical Lucubrations. And in 1780
his Mathematical Memoirs. All containing good improvements
in infinite series, &amp;c.</p><p>In 1770 was published, in the German language, Elements
of Algebra by M. Euler. And in 1774 a French
translation of the same. The memoirs of the Berlin and
Petersburgh academies also abound with various improvements
on series and other branches of analysis by
this great man.</p><p>In 1775 was published at Bologna, in 2 vols 4to,
<hi rend="italics">Compendio d'Analisi di</hi> Girolamo Saladini.</p><p>Besides the soregoing, there have been many other
authors who have given treatises on Algebra, or who
have made improvements on series and other parts of
Algebra; as Schonerus, Coignet, Salignac, Laloubere,
Hemischius, Degraave, Mescher, Henischins, Roberval,
the Bernoullis, Malbranche, Agnesi, Wells, Dodson,
Manfredi, Regnault, Rowning, Maseres, Waring, Lorg-
<cb/>
na, de la Grange, de la Place, Bertrand, Kuhnius'
Hales, and many others.</p><div2 part="N" n="Algebra" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Algebra</hi></head><p>, <hi rend="italics">numeral,</hi> is that which is chiefly concerned
in the solution of numeral problems, and in which
all the given quantities are expressed by numbers only.
As used by the more early authors, Diophantus, Paciolus,
Stifelius, &amp;c.</p></div2><div2 part="N" n="Algebra" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Algebra</hi></head><p>, <hi rend="italics">specious,</hi> or <hi rend="italics">literal,</hi> is that commonly used
by the moderns, in which all the quantities, both known
and unknown, are represented or expressed by species or
general characters, as the letters of the alphabet, &amp;c;
in consequence of which general designation, all the
conclusions become universal theorems for performing
every operation of the like kind. There are specimens
of this method from Cardan and others about his time,
but it was more generally employed and introduced by
Vieta.</p></div2><div2 part="N" n="Algebraical" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Algebraical</hi></head><p>, something relating to <hi rend="italics">algebra.</hi></p><p>Thus we say <hi rend="italics">algebraical</hi> solutions, curves, characters
or symbols, &amp;c.</p><p><hi rend="smallcaps">Algebraical</hi> <hi rend="italics">Curve,</hi> is a curve in which the general
relation between the abscisses and ordinates can be
expressed by a common algebraical equation.</p><p>These are also called <hi rend="italics">geometrical lines</hi> or <hi rend="italics">curves,</hi> in contradistinction
to <hi rend="italics">mechanical</hi> or <hi rend="italics">transcendental</hi> ones.</p></div2></div1><div1 part="N" n="ALGEBRAIST" org="uniform" sample="complete" type="entry"><head>ALGEBRAIST</head><p>, a person skilled in <hi rend="italics">algebra.</hi></p></div1><div1 part="N" n="ALGENEB" org="uniform" sample="complete" type="entry"><head>ALGENEB</head><p>, or <hi rend="smallcaps">Algenib</hi>, a fixed star of the second
magnitude, on the right side of Perseus.</p></div1><div1 part="N" n="ALGOL" org="uniform" sample="complete" type="entry"><head>ALGOL</head><p>, or <hi rend="italics">Medusa's Head,</hi> a fixed star of the third
magnitude, in the constellation Perseus.</p></div1><div1 part="N" n="ALGORAB" org="uniform" sample="complete" type="entry"><head>ALGORAB</head><p>, a fixed star of the third magnitude, in
the right wing of the constellation Corvus.</p></div1><div1 part="N" n="ALGORISM" org="uniform" sample="complete" type="entry"><head>ALGORISM</head><p>, or <hi rend="smallcaps">Algorithm</hi>, is similar to logistics,
signifying the art of computing in any particular
way, or about some particular subject; or the common
rules of computing in any art. As the <hi rend="italics">algorithm</hi> of numbers,
of algebra, of integers, of fractions, of surds, &amp;c;
meaning the common rules for performing the operations
of arithmetic, or algebra, or fractions, &amp;c.</p></div1><div1 part="N" n="ALHAZEN" org="uniform" sample="complete" type="entry"><head>ALHAZEN</head><p>, <hi rend="smallcaps">Allacen</hi>, or <hi rend="smallcaps">Abdilazum</hi>, was a
learned Arabian, who lived in Spain about the year
1100, according to his editor Risner, and Weidler.
He wrote upon Astrology; and his work upon Optics
was printed, in Latin, at Basil, in 1572, under
the title of <hi rend="italics">Optic&#xE6; Thesaurus,</hi> by Risner. Alhazen
was the first who shewed the importance of refractions
in astronomy, so little known to the ancients. He is also
the first author who has treated on the twilight, upon
which he wrote a work, in which he also speaks of the
height of the clouds.</p></div1><div1 part="N" n="ALIDADE" org="uniform" sample="complete" type="entry"><head>ALIDADE</head><p>, an Arabic name for the label, index,
or ruler, which is moveable about the centre of an
astrolabe, quadrant, &amp;c, and carrying the sights or telescope,
and by which are shewn the degrees cut off
the limb or arch of the instrument.</p><p>ALIQUANT <hi rend="italics">part,</hi> is that part which will not exactly
measure or divide the whole, without leaving some
remainder. Or the <hi rend="italics">aliquant</hi> part is such, as being taken
or repeated any number of times, does not make up the
whole exactly, but is either greater or less than it.
Thus 4 is an aliquant part of 10; for 4 twice taken
makes 8 which is less than 10, and three times taken
makes 12 which is greater than 10.
<pb n="98"/><cb/></p><p>ALIQUOT <hi rend="italics">part,</hi> is such a part of any whole, as
will exactly measure it without any remainder. Or the
<hi rend="italics">aliquot</hi> part is such, as being taken or repeated a certain
number of times, exactly makes up, or is equal to the
whole. So 1 is an aliquot part of 6, or of any other whole
number; 2 is also an aliquot part of 6, being contained
just 3 times in 6; and 3 is also an aliquot part of 6, being
contained just 2 times: so that all the aliquot parts
of 6 are 1, 2, 3.</p><p>All the <hi rend="italics">aliquot</hi> parts of any number may be thus found:
Divide the given number by its least divisor; then divide
the quotient also by its least divisor; and so on always
dividing the last quotient by its least divisor, till
the quotient 1 is obtained; and all the divisors, thus
taken, are the prime aliquot parts of the given number.
Then multiply continually together these prime divisors,
viz. every two of them, every three of them, every four
of them, &amp;c; and the products will be the other or compound
<hi rend="italics">aliquot</hi> parts of the given number. So if the
<hi rend="italics">aliquot</hi> parts of 60 be required; first divide it by 2, and
the quotient is 30: then 30 divided by 2 also, gives 15,
and 15 divided by 3 gives 5, and 5 divided by 5 gives 1:
so that all the prime divisors or aliquot parts are 1, 2, 2,
3, 5. Then the compound ones, by multiplying every
two, are 4, 6, 10, 15; and every three 10, 20, 30. So
that all the <hi rend="italics">aliquot</hi> parts of the given number 60, are 1,
2, 3, 4, 5, 6, 10, 12, 15, 20, 30.&#x2014;In like manner it
will be found that all the <hi rend="italics">aliquot</hi> parts of 360 are 1, 2,
3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40,
45, 60, 72, 180.</p><p>ALLEN (<hi rend="smallcaps">Thomas</hi>) a celebrated mathematician of
the 16th century. He was born at Uttoxeter in Staffordshire,
in 1542; was admitted a scholar of Trinity
college, Oxford, in 1561; where he took his degree of
master of arts in 1567. In 1570 he quitted his college
and fellowship, and retired to Glocester hall, where he
studied very closely, and became famous for his knowledge
in antiquity, philosophy and mathematics. He
received an invitation from Henry earl of Northumberland,
a great friend and patron of the mathematicians,
and he spent some time at the earl's house; where he became
acquainted with those celebrated mathematicians
Thomas Harriot, John Dee, Walter Warner, and Nathaniel
Torporley. Robert earl of Leicester, too, had
a great esteem for Allen, and would have conferred a
bishopric upon him; but his love for solitude and retirement
made him decline the offer. His great skill in
the mathematics gave occasion to the ignorant and vulgar
to look upon him as a magician or conjurer. Allen
was very curious and indefatigable in collecting scattered
manuscripts relating to history, antiquity, astronomy,
philosophy, and mathematics: which collections have
been quoted by several learned authors, and mentioned
as in the Bibliotheca Alleniana. He published in Latin
the second and third books of Ptolemy, <hi rend="italics">Concerning the
Judgment of the Stars,</hi> or, as it is usually called, of the
<hi rend="italics">quadripartite construction,</hi> with an exposition. He wrote
also notes on many of Lilly's books, and some on John
Bale's work, <hi rend="italics">De scriptoribus Maj. Britanni&#xE6;.</hi> He died
at Glocester hall in 1632, being 90 years of age.</p><p>Mr. Burton, the author of his funeral oration, calls
him the very soul and sun of all the mathematicians of his
age. And Selden mentions him as a person of the most
extensive learning and consummate judgment, the bright-
<cb/>
est ornament of the university of Oxford. Also Camden
says he was skilled in most of the best arts and sciences.
A. Wood has also transcribed part of his character from
a manuscript in the library of Trinity college, in these
words: &#x201C;He studied polite literature with great application;
he was strictly tenacious of academic discipline,
always highly esteemed both by foreigners and those of
the university, and by all of the highest stations in the
church of England, and the university of Oxford. He
was a sagacious observer, an agreeable companion, &amp;c.&#x201D;</p></div1><div1 part="N" n="ALLIGATION" org="uniform" sample="complete" type="entry"><head>ALLIGATION</head><p>, one of the rules in arithmetic, by
which are resolved questions which relate to the compounding
or mixing together of divers simples or ingredients,
being so called from <hi rend="italics">alligare,</hi> to tie or connect
together, probably from certain vincula, or crooked ligatures,
commonly used to connect or bind the numbers
together.</p><p>It is probable that this rule came to us from the
Moorish or Arabic writers, as we find it, with all the
other rules of arithmetic, in <hi rend="italics">Lucas de Burgo,</hi> and the
other early authors in Europe.</p><p><hi rend="italics">Alligation</hi> is of two kinds, <hi rend="italics">medial</hi> and <hi rend="italics">alternate.</hi></p><p><hi rend="smallcaps">Alligation</hi> <hi rend="italics">medial</hi> is the method of finding the
rate or quality of the composition, from having given
the rates and quantities of the simples or ingredients.</p><p>The rule of operation is this: multiply each quantity
by its rate, and add all the products together; then
divide the sum of the products by the sum of the quantities,
or whole compound, and the quotient will be the
rate sought.</p><p><hi rend="italics">For example,</hi> Suppose it were required to mix together
6 gallons of wine, worth 5s. a gallon; 8 gallons, worth
6s. the gallon; and 4 gallons, worth 8s. the gallon; and
to find the worth or value, per gallon, of the whole
mixture.
<table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Gal.</cell><cell cols="1" rows="1" rend="align=center" role="data">s.</cell><cell cols="1" rows="1" rend="align=center" role="data">products.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Here</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">mult. by 5 gives</cell><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">&quot; by 6 &quot;</cell><cell cols="1" rows="1" rend="align=center" role="data">48</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">&quot; by 8 &quot;</cell><cell cols="1" rows="1" rend="align=center" role="data">32</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">whole comp.</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">110</cell><cell cols="1" rows="1" role="data">sum of prod.</cell></row></table>
<table><row role="data"><cell cols="1" rows="1" role="data">Then 18)</cell><cell cols="1" rows="1" role="data">110 (6 2/18 or (6 1/9)s, is the rate sought.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">108</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">&#xA0;&#xA0;&#xA0;2</cell></row></table></p><p><hi rend="smallcaps">Alligation</hi> <hi rend="italics">alternate</hi> is the method of finding the
quantities of ingredients or simples, necessary to form a
compound of a given rate.</p><p>The <hi rend="italics">rule</hi> of operation is this: 1st, Place the given
rates of the simples in a column, under each other;
noting which rates are less, and which are greater than
the proposed compound. 2d, Connect or link with a
crooked line, each rate which is less than the proposed
compound rate, with one or any number of those which
are greater than the same; and every greater rate with
one or any number of the less ones. 3d, Take the difference
between the given compound rate and that of each
simple rate, and set this difference opposite every rate
with which that one is linked. 4th, Then if only one
difference stand opposite any rate, it will be the quantity
belonging to that rate; but when there are several
differences to any one, take their sum for its quantity.</p><p><hi rend="italics">For example,</hi> Suppose it were required to mix together
gold of various degrees of fineness, viz. of 19,
<pb n="99"/><cb/>
of 21, and of 23 caracts fine, so that the mixture shall
be of 20 caracts fine. Hence,
<table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Rates</cell><cell cols="1" rows="1" rend="align=center" role="data">Diffs.</cell><cell cols="1" rows="1" rend="align=center" role="data">Sum of Diffs.</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=3" role="data">Comp. rate 20</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(7)">{</hi></cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(7)">{</hi></cell><cell cols="1" rows="1" role="data">1 of 21 caracts sine,</cell></row><row role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1+3</cell><cell cols="1" rows="1" role="data">4 of 19 caracts sine,</cell></row><row role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1 of 23 caracts sine.</cell></row></table>
That is, there must be an equal quantity of 21 and 23
caracts fine, and 4 times as much of 19 caracts fine.</p><p>Various limitations, both of the compound and the
ingredients, may be conceived; and in such cases, the
differences are to be altered proportionally.</p><p>Questions of this sort are however commonly best and
easiest resolved by common Algebra, of which they form
a species of indeterminate problems, as they admit of
many, or an indesinite number of answers.</p><p>There is recorded a remarkable instance of a discovery
made by Archimedes, both by <hi rend="italics">alligation</hi> and specific
gravity at the same time, namely, concerning the
crown of Hiero, king of Syracuse. The king had
ordered a crown to be made of pure gold, but when
brought to him, a suspicion arose that it was mixed with
alloy of either silver or copper, and the king recommended
it to Archimedes to discover the cheat without
defacing the crown. Archimedes, after long thinking
on the matter, without lighting on the means of doing
it, being one day in the bath, and observing how his
body raised the water higher, conceived the idea that
different metals of the same weight would occupy different
spaces, and so raise or expel different quantities
of water. Upon which he procured two other masses,
each of the same weight with the crown, the one of
pure gold, and the other of alloy; then immersing
them all three, separately, in water, and observing the
space each occupied, by the quantity it raised the water,
he from thence computed the quantities of gold and
alloy contained in the crown.</p></div1><div1 part="N" n="ALLIOTH" org="uniform" sample="complete" type="entry"><head>ALLIOTH</head><p>, a star in the tail of the great bear.
The word in Arabic denotes a horse; and they gave
this name to each of the three stars, in the tail of the
great bear, as they are placed like three horses, thus
arranged for the purpose of drawing the waggon commonly
called Charles's wain, represented by the four
stars on the body of the same constellation.</p><p>ALMACANTAR. See <hi rend="smallcaps">Almucantar.</hi></p></div1><div1 part="N" n="ALMAGEST" org="uniform" sample="complete" type="entry"><head>ALMAGEST</head><p>, the name of a celebrated book
composed by Ptolemy; being a collection of a great
number of the observations and problems of the ancients,
relating to geometry and astronomy, but especially the
latter. And being the sirst work of this kind which
has come down to us, and containing a catalogue of the
fixed stars, with their places, beside numerous records
and observations of eclipses, the motions of the planets,
&amp;c, this work will ever be held dear and valuable to the
cultivators of astronomy.</p><p>In the original Greek it is called <foreign xml:lang="greek">suntacis m&lt;*&gt;gish</foreign>, the
<hi rend="italics">great composition</hi> or <hi rend="italics">collection.</hi> And to the word <foreign xml:lang="greek">megish</foreign>,
<hi rend="italics">megiste,</hi> the Arabians joined the particle <hi rend="italics">al,</hi> and thence
called it <hi rend="italics">Almaghesti,</hi> or, as we call it, from them, the
Almagest.</p><p>Ptolemy was born about the year of Christ 69, and
died in 147, and wrote this work, consisting of 13 books,
at Alexandria in Egypt, where the Arabians found it
on the capture of that kingdom. It was by them
<cb/>
translated out of Greek, into Arabic, by order of the
caliph Almaimon, about the year 827; and sirst into
Latin about 1230, by favour of the emperor Frederic II.
The Greek text however was not known in Europe till
about the beginning of the 15th century, when it was
brought from Constantinople, then taken by the Turks,
by <hi rend="italics">George,</hi> a monk of Trabezond, who translated it into
Latin, which translation has several times been published.</p><p>Riccioli, an Italian jesuit, also published, in 1651, a
body of Astronomy, which, in imitation of Ptolemy, he
called <hi rend="italics">Almagestum Novum,</hi> the <hi rend="italics">New Almagest</hi>; being a
large collection of ancient and modern observations and
discoveries, in the science of Astronomy.</p></div1><div1 part="N" n="ALMAMON" org="uniform" sample="complete" type="entry"><head>ALMAMON</head><p>, caliph of Bagdat, a philosopher and
astronomer in the beginning of the 9th century, he
having ascended the throne in the year 814. He was
son of Harun Al-Rashid, and grand son of Almansor.
His name is otherwise written <hi rend="italics">Mamon, Almaon, Almamun,
Alamoun,</hi> or <hi rend="italics">Al-Maimon.</hi> Having been educated
with great care and with a love for the liberal sciences,
he applied himself to cultivate and encourage them in his
own country. For this purpose he requested the Greek
emperors to supply him with such books on philosophy
as they had among them; and he collected skilful interpreters
to translate them into the Arabic language.
He also encouraged his subjects to study them; frequenting
the meetings of the learned, and assisting at
their exercises and deliberations. He caused Ptolemy's
Almagest to be translated in 827, by Isaac Ben-honain,
and Thabet Ben-korah, according to Herbelot, but according
to others by Sergius, and Alhazen, the son of
Joseph. In his reign, and doubtless by his encouragement,
an astronomer of Bagdat, named Habash, composed
three sets of astronomical tables.</p><p>Almamon himself made many astronomical observations,
and determined the obliquity of the ecliptic to be
then 23&#xB0; 35&#x2032; (or 23&#xB0; 33&#x2032; in some manuscripts), but
Vossius says 23&#xB0; 51&#x2032; or 23&#xB0; 34&#x2032;. He also caused skilful
observers to procure proper instruments to be made,
and to exercise themselves in astronomical observations;
which they did accordingly at Shemasi in the province
os Bagdat, and upon Mount Casius near Damas.</p><p>Under the auspices of Mamon also a degree of the
meridian was measured on the plains of Sinjar or Sindgiar
(or according to some Fingar), upon the borders of
the Red Sea; by which the degree was found to contain
56 2/3 miles, of 4000 coudees each, the coudee being
a foot and a half: but it is not known what foot is here
meant, whether the Roman, the Alexandrian, or some
other. Riccioli makes this measure of the degree
amount to 81 ancient Roman miles, which value answers
to 62046 French toises; a quantity more than the true
value of the degree by almost one-third.</p><p>Finally, Mamon revived the sciences in the East to
such a degree, that many learned men were found, not
only in his own time, but after him, in a country where
the study of the sciences had been long forgotten. This
learned king died near Tarsus in Cilicia, by having eaten
too freely of some dates, on his return from a military
expedition, in the year 833.</p></div1><div1 part="N" n="ALMANAC" org="uniform" sample="complete" type="entry"><head>ALMANAC</head><p>, a calendar or table, in which are set
down and marked the days and feasts of the year, the
common ecclesiastical notes, the course and phases of
<pb n="100"/><cb/>
the moon, &amp;c, sor each month: and answers to the
<hi rend="italics">fasti</hi> of the ancient Romans.</p><p>The etymology of the word is much controverted
among grammarians.&#x2014;Some derive it from the Arabic,
viz, from the particle <hi rend="italics">al,</hi> and <hi rend="italics">manah,</hi> to count. While
Scaliger, and others, derive it from the same <hi rend="italics">al,</hi> and the
Greek <foreign xml:lang="greek">manakos</foreign>, the course of the months. But Golius
controverts these opinions, and ascribes the word to
another origin, though he still makes it of Arabic extract,
which it more evidently is. He says that, in the East
it is the custom for the people, at the beginning of the
year, to make presents to their princes; and that, among
the rest, the astrologers present them with their almanacs,
or ephemerides, for the year ensuing; whence
these came to be called <hi rend="italics">almanha,</hi> that is, new-year's
gifts. But this derivation seems rather strained and
improbable; for, by the same rule, the gifts or productions
of other artists, or classes of men, might also be
called almanacs. There are other guesses at the etymology;
and Verstegan writes the word <hi rend="italics">almonac,</hi> and
makes it of German original. Our ancestors, he observes,
used to carve the courses of the moon, for the
whole year, upon a square piece of wood, which they
called <hi rend="italics">al-monaght,</hi> which is as much as to say, in old
English or Saxon, <hi rend="italics">all-moon-heed.</hi></p><p>Almanacs are of various kinds and composition, some
books, others sheets, &amp;c, some annual, others perpetual.
The essential part is the calendar of months, weeks, and
days; the motions, changes, and phases of the moon;
with the rising and setting of the sun and moon. To
these are commonly added various matters, astronomical,
astrological, chronological, meteorological, and even
political, rural, medical, &amp;c; as also eclipses, solar ingresses,
aspects, and configurations of the heavenly bodies,
lunations, heliocentric and geocentric motions of the
planets, prognostications of the weather, and predictions
of other events, the tides, twilight, equation of time, &amp;c.</p><p>Till about the 4th century, almanacs bore the marks
of heatheniim only; from thence to the 7th century,
they were a mixture of heathenism and christianity;
and ever since they have been altogether christian: but
at all times, astrological and other predictions have been
considered as an essential part, and still are so to this day
with several of them, notwithstanding that most people
<hi rend="italics">assect</hi> to disbelieve in such predictions.</p><p><hi rend="italics">Nautical</hi> <hi rend="smallcaps">Almanac</hi>, and <hi rend="italics">Astronomical Ephemeris,</hi> is a
kind of national almanac, chiefly for nautical purposes,
which was begun in the year 1767 under the direction
of the Board of Longitude, on the recommendation of
the present worthy Astronomer Royal, who has the
immediate conducting of it. It is still published by anticipation
for several years before hand, for the convenience
of ships going out upon long voyages, for
which it is highly useful, and was found eminently so in
the course of the late voyages round the world for
making discoveries. Besides most things essential to
general use, that are to be found in other almanacs, it
contains many new and important particulars; more
especially, the distances of the moon from the sun and
fixed stars, which are computed for the meridian of the
Royal Observatory of Greenwich, and set down to every
three hours of time, expressly designed for computing
the longitude at sea, by comparing these with the like
distances observed there.
<cb/></p></div1><div1 part="N" n="ALMANAR" org="uniform" sample="complete" type="entry"><head>ALMANAR</head><p>, in the Arabian astrology, denotes the
pre-eminence or prevalence of one planet over another.</p></div1><div1 part="N" n="ALMUCANTARS" org="uniform" sample="complete" type="entry"><head>ALMUCANTARS</head><p>, <hi rend="smallcaps">Almacantars</hi>, or A<hi rend="smallcaps">LMICANTARS</hi>,
from the Arabic <hi rend="italics">almocantharat,</hi> are circles
parallel to the horizon, conceived to pass through every
degree of the meridian; serving to shew the height of
the sun, moon, or stars, &amp;c; and are the same as the
parallels of altitude.</p><p><hi rend="smallcaps">Almucantar</hi>-<hi rend="italics">Staff,</hi> was an instrument formerly
used at sea to observe the sun's amplitude at rising or
setting, and thence to determine the variation of the
compass, &amp;c. The instrument had an arch of 15 degrees,
made of some smooth wood.</p><p>ALPHONSINE <hi rend="italics">Tables,</hi> are astronomical tables
compiled by order of Alphonsus, king of Castile. In
the compiling of these it is thought that prince himself
assisted. See <hi rend="italics">Astronomical tables.</hi></p><p>ALPHONSUS the 10th, king of Leon and Castile,
who has been surnamed The Wise, on account of his attachment
to literature, and is now more celebrated for
having been an astronomer than a king. He was born
in 1203; succeeded his father Ferdinand the 3d, in
1252; and died in 1284, consequently at the age of 81.</p><p>The affairs of the reign of Alphonsus were very
extraordinary and unfortunate for him. But we shall
here only consider him in that part of his character, on
account of which he has a place in this work, namely,
as an astronomer and man of letters. He understood
astronomy, philosophy, and history, as if he had been
only a man of letters; and composed books upon the
motions of the heavens, and on the history of Spain,
which are highly commended. &#x201C;What can be more
surprising,&#x201D; says Mariana, &#x201C;than that a prince, educated
in a camp, and handling- arms from his childhood,
should have such a knowledge of the stars, of philosophy,
and the transactions of the world, as men of
leisure can scarcely acquire in their retirements?
There are extant some books of Alphonsus on the
motions of the stars, and the history of Spain, written
with great skill and incredible care.&#x201D; In his astronomical
pursuits he discovered that the tables of Ptolemy were
full of errors; and thence he conceived the first of any
the resolution of correcting them. For this purpose,
about the year 1240, and during the life of his father,
he assembled at Toledo the most skilful astronomers of
his time, Christians, Moors, or Jews, when a plan was
formed for constructing new tables. This task was accomplished
about 1252, the first year of his reign; the
tables being drawn up chiefly by the skill and pains of
Rabbi Isaac Hazan a learned Jew, and the work called
the <hi rend="italics">Alphonsine Tables,</hi> in honour of the prince, who was
at vast expences concerning them. He fixed the epoch
of the tables to the 30th of May 1252, being the day
of his accession to the throne. They were printed for
the first time in 1483, at Venice, by Radtolt, who excelled
in printing at that time; an edition extremely
rare: there are others of 1492, 1521, 1545, &amp;c.
(Weidler, p. 280).</p><p>We must not omit a memorable saying of Alphonfus,
which has been recorded for its boldness and pretended
impiety; namely, &#x201C;that if he had been of God's privy
council when he made the world, he could have advised
him better.&#x201D; Mariana however says only in general,
that Alphonsus was so bold as to blame the works of
<pb n="101"/><cb/>
Providence, and the construction of our bodies; and he
says that this story concerning him rested only upon a
vulgar tradition. The Jesuit's words are curious:
&#x201C;Emanuel, the uncle of Sanchez (the son of Alphonsus),
in his own name, and in the name of other nobles,
deprived Alphonsus of his kingdom by a public sentence:
which that prince merited, for daring severely and boldly
to censure the works of divine Providence, and the construction
of the human body, as tradition says he did.
Heaven most justly punished the folly of his tongue.&#x201D;
Though the silence of such an historian as Mariana, in
regard to Ptolemy's system, ought to be of some weight,
yet we cannot think it improbable, that if Alphonsus
did pass so bold a censure on any part of the univerfe,
it was on the celestial sphere, and meant to glance upon
the contrivers and supporters of that system. For, besides
that he studied nothing more, it is certain that at
that time astronomers explained the motions of the
heavens by intricate and confused hypotheses, which did
no honour to God, nor anywise answered the idea of
an able workman. So that, from considering the multitude
of spheres composing the system of Ptolemy, and
those numerous eccentric cycles and epicycles with
which it is embarrassed, if we suppose Alphonsus to have
said, &#x201C;That if God had asked his advice when he made
the world, he would have given him better council,&#x201D;
the boldness and impiety of the censure will be greatly
diminished.</p></div1><div1 part="N" n="ALSTED" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ALSTED</surname> (<foreName full="yes"><hi rend="smallcaps">John-Henry</hi></foreName>)</persName></head><p>, a German protestant
divine, and one of the most indefatigable writers of the
17th century. He was some time professor of philosophy
and divinity at Herborn in the county of Nassau:
from thence he went into Transilvania, to be professor
at Alba Julia; where he continued till his death, which
happened in 1638, being then 50 years of age. He
applied himself chiefly to compose methods, and to reduce
the several branches of arts and sciences into systems.
His <hi rend="italics">Encyclop&#xE6;dia</hi> has been much esteemed even by
Roman Catholics; it was printed at Lyons, and sold
very well throughout all France. Vossius mentions the
Encyclop&#xE6;dia in general, but speaks of his treatise of
<hi rend="italics">Arithmetic</hi> more particularly, and allows the author to
have been a man of great reading and universal erudition.
His <hi rend="italics">Thesaurus Chronologicus</hi> is by some esteemed one
of his best works, and has gone through several editions,
though others speak of it with contempt. In his
<hi rend="italics">Triumphus Biblicus</hi> Alsted endeavours to prove that
the materials and principles of all the arts and sciences
may be found in the scriptures; but he gained very few
to his opinion. John Himmelius wrote a piece against
his <hi rend="italics">Theologia Polemica,</hi> which was one of Alsted's best
performances. It seems he was a millenarian, having
published, in 1672, a treatise <hi rend="italics">De Mille Annis,</hi> in which
he asserts that the faithful shall reign with Jesus Christ
upon earth a thousand years; after which will be the
general resurrection, and the last judgment; and he pretended
that this reign would commence in the year 1694.</p><p>ALTERNATE <hi rend="italics">angles,</hi> are the internal
angles, A and B, or <hi rend="italics">a</hi> and <hi rend="italics">b,</hi>
<figure/>
made by a line cutting two parallel
lines, and lying on opposite sides of
the cutting line. It is the property
of these angles to be always equal to
each other, namely the angle A = the
<cb/>
angle B, and the angle <hi rend="italics">a</hi> = the angle <hi rend="italics">b.</hi> And the exterior
alternate angles are also equal.</p><p><hi rend="smallcaps">Alternate</hi> <hi rend="italics">Ratio</hi> or <hi rend="italics">Proportion,</hi> is the ratio of the
one antecedent to the other, or of one consequent to
the other, in any proportion, in which the quantities
are of the same kind. So if A : B :: C : D,
then alternately, or by alternation A : C :: B : D.</p></div1><div1 part="N" n="ALTERNATION" org="uniform" sample="complete" type="entry"><head>ALTERNATION</head><p>, or <hi rend="italics">Permutation,</hi> of quantities
or things, is the varying or changing the order or position
of them.</p><p>As suppose two things <hi rend="italics">a</hi> and <hi rend="italics">b</hi>; these may be placed
either thus <hi rend="italics">ab</hi> or <hi rend="italics">ba</hi> that is two ways, or 1 X 2. If
there be three things, <hi rend="italics">a, b, c,</hi> then the 3d thing <hi rend="italics">c,</hi> may
be placed three different ways with respect to each of
the two positions <hi rend="italics">ab</hi> and <hi rend="italics">ba</hi> of the other two things,
it may stand either before them, or between them, or
after them both, that is, it may stand either 1st, 2d, or
3d; and therefore with three things there will be three
times as many changes as with two, that is 1X2X3 or
six changes with three things. Again, if there be four
things <hi rend="italics">a, b, c, d</hi>; then the fourth thing <hi rend="italics">d</hi> may be
placed in four different ways with respect to each of the
six positions of the other three; for it may be set either
1st or 2d or 3d or 4th in the order of each position;
consequently from four things there will be four times
as many alternations as there are from three things;
and therefore 1 X 2 X 3 X 4 = 24 is the number of
changes with four things. And so on, always multiplying
the last found number of alternations by the next
number of things; or to find the number of changes for
any number of things, as <hi rend="italics">n,</hi> multiply the series of natural
numbers 1, 2, 3, 4, 5, &amp;c, to <hi rend="italics">n,</hi> continually together,
and the last product will be the number of alternations
sought; so 1X2X3X4X5 - - - - <hi rend="italics">n</hi> is the number of
changes in <hi rend="italics">n</hi> things.</p><p>So if, for example, it were required to find how many
changes may be rung on 12 bells; it would be
1 X 2 X 3 X 4 X 5 X 6X7X8X9X10X11X12=
479001600, the number of changes. Now supposing
there might be rung 10 changes in one minute, that is
10X12 or 120 strokes in a minute, or 2 strokes in each
second of time; then, according to this rate, it would
take upwards of 91 years to ring over all these changes
on the 12 bells only. Also, if but two more bells were
added, making 14 bells; then, at the same rate of ringing,
it would require about 16575 years to ring all the changes
on 14 bells but once over. And if the number of bells
were 24, it would require more than 117000000000000000,
years to ring all the different changes upon them!</p></div1><div1 part="N" n="ALTIMETRY" org="uniform" sample="complete" type="entry"><head>ALTIMETRY</head><p>, <hi rend="smallcaps">Altimetria</hi>, the art of taking
or measuring altitudes or heights, whether accessible or
inaccessible. Or</p><p>Altimetria is the part of practical Geometry which
respects the theory and practice of measuring both
heights and depths, and both in respect of perpendicular
and oblique lines.</p></div1><div1 part="N" n="ALTING" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ALTING</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, was born at Heidelberg in
1618. He travelled into England in 1640, where he
was ordained by the learned Dr. Prideaux, bishop of
Worcester. He afterwards succeeded Gomarus in the
professorship of Groninghen. He died in 697; and
recommended the edition of his works to Menso Alting
(author of Notitia German. Infer. Antiqu&#xE6;); but
they were published in 5 vols folio, with his life, by
<pb n="102"/><cb/>
Bekker of Amsterdam. They contain various analytical,
exegetical, practical, problematical, and philosophical
tracts, which shew his great industry and knowledge.</p></div1><div1 part="N" n="ALTITUDE" org="uniform" sample="complete" type="entry"><head>ALTITUDE</head><p>, in Geometry is the third dimension of
body, considered with respect to its elevation above the
ground: and is otherwise called its height when measured
from bottom to top, or its depth when measured
from top to bottom.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of a figure,</hi> is the distance of its vertex
from the base, or the length of a perpendicular let fall
from its vertex to the base. The altitudes of sigures
are useful in computing their areas or solidities.</p><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">Height</hi> of any point of a terrestrial
object, is the perpendicular let fall from that point to the
plane of the horizon. <hi rend="italics">Altitudes</hi> are distinguished into
<hi rend="italics">accessible</hi> and <hi rend="italics">inaccessible.</hi></p><p><hi rend="italics">Accessible</hi> <hi rend="smallcaps">Altitude</hi> of an object, is that whose base
there is access to, to measure the nearest distance to it
on the ground, from any place.</p><p><hi rend="italics">Inaccessible</hi> <hi rend="smallcaps">Altitude</hi>, of an object, is that whose
base there is not free access to, by which a distance
may be measured to it, by reason of some impediment,
such as water, wood, or the like.</p><p><hi rend="italics">To measure</hi> or <hi rend="italics">take</hi> <hi rend="smallcaps">Altitudes.</hi> If an altitude
cannot be measured by stretching a string from top to
bottom, which is the direct and most accurate way, then
some indirect way is used, by actually measuring some
other line or distance which may serve as a basis, in conjunction
with some angles, or other proportional lines,
either to compute, or geometrically determine, the altitude
of the object sought.</p><p>There are various ways of measuring altitudes, or
depths, by means of different instruments, and by shadows
or reflected images, on optical principles. There are
also various ways of computing the altitude in numbers,
from the measurements taken as above, either by geometrical
construction, or trigonometrical calculation, or
by simple numeral computation from the property of
parallel lines, &amp;c.</p><p>The instrumcnts mostly used in measuring altitudes,
are the quadrant, theodolite, geometrical square, line
of shadows, &amp;c; the descriptions of each of which may
be seen under their respective names.</p><p><hi rend="italics">To measure an Accessible Altitude Geometrically.</hi> Thus,
suppose the height of the accessible tower AB be required.
First, by means of two rode, the one longer
than the other: plant the longer upright at C; then
move the shorter back from it, till by trials you find such
a place, D, that the eye placed at the top of it at E, may
see the top of the other, F, and the top of the object B
straight in a line: next measure the distances DA or
EG and DC or EH, also HF the difference between the
heights of the rods: then, by similar triangles, as EH :
EG :: HF : the 4th proportional GB; to which add
AG or DE, and the sum will be the whole altitude AB
sought.
<figure/>
<cb/></p><p>Or, with one rod CF only: plant it at such a place C,
that the eye at the ground, or near it, at I, may see the
tops F and B in a right line: then, having measured IC,
IA, CF, the 4th proportional to these will be the altitude
AB sought.</p><p>Or thus, by means of Shadows. Plant a rod <hi rend="italics">ab</hi> at
<hi rend="italics">a,</hi> and measure its shadow <hi rend="italics">ac,</hi> as also the shadow AC of
the object AB; then the 4th proportional to <hi rend="italics">ac, ab,</hi> AC
will be the altitude AB sought.
<figure/></p><p>Or thus, by means of Optical Reflection. Place a
vessel of water, or a mirror or other reflecting smface,
horizontal at C; and move off from it to such a distance,
D, that the eye E may see the image of the top of the
object in the mirror at C: then, by similar figures, CD :
DE :: CA : AB the altitude sought.
<figure/></p><p>Or thus, by the Geometrical Square. At any place,
C, fix the stand, and turn the square about the centre of
motion, D, till the eye there see the top of the object
through the sights or telescope on the side DE of the
quadrant, and note the number of divisions cut off the
other side by the plumb line EG: then as EF : FG
:: DH : HB; to which add AH or CD, for the
whole height AB.
<figure/></p><p><hi rend="italics">To measure an Accessible Altitude Trigonometrically.</hi> At
any convenient station, C, with a quadrant, theodolite,
or other graduated instrument, observe the angle of
elevation ACB above the horizontal line AC; and measure
the distance AC. Then, A being a right angle, it
will be, as radius is to the tangent of the angle A, so is
AC to AB sought.</p><p>If AC be not horizontal, but an inclined plane; then
the angle above it must be observed at two stations C and
D in a right line, and the distances AC, CD both measured.
Then, from the angle C take the angle D, and
there remains the angle CBD; hence in the triangle
BCD, are given the angles and the side DC, to sind the
side CB; and then in the triangle ABC, are given the
<pb n="103"/><cb/>
two sides CA and CB, with the included angle C, to
find the third side AB.
<figure/></p><p>Or thus, measure only the distance AC, and the angles
A and C: then, in the triangle ABC, are given all
the angles and the side AC, to find the side AB.</p><p><hi rend="italics">To measure an Inaccessible Altitude,</hi> as a hill, cloud,
or other object. This is commonly done, by observing
the angle of its altitude at two stations, and measuring
the distance between them. Thus, for the height
AB of a hill, measure the distance CD at the foot of it,
and observe the quantity of the two angles C and D.
Then, from the angle C taking the angle D, leaves the
augle CBD; hence
As sine [angle]CBD: sine [angle]D :: CD : CB; and
As rad.: sine [angle]ACB :: CB : AB the altitude.
<figure/></p><p>And for a balloon, or cloud, or other moveable object
C, let two observers at A and B, in a plane with C, take
at the same time the angles A and B, and measure the
distance between them AB; then calculate the altitude
CD exactly as in the last example.
<figure/></p><p>To find the height of an object, by knowing the utmost
distance at which its top can be just seen in the horizon.
As suppose the top H of a tower FH can be
just seen from E when the distance EF is 25 miles, supposing
the circumference of the earth to be 25000 miles,
or the radius 3979 miles or 21009120 feet. First, as
25000 : 25 :: 360&#xB0; : 21&#x2032; 36&#x2033; equal to the angle G;
then as radius : sec. [angle]G :: EG : GH, which will be
found to be 21009536 feet; from which take EG or
GF, and there remains 416 feet, for FH the height of
the tower sought.&#x2014; Or rather thus, as 10000000 radius:
198=sec. [angle]G&#x2014;radius :: 21009120=EG : 416 =
FH, as before.</p><p>Or the same may be found easier thus: The horizon
dips nearly 8 inches or 2/3 of a foot at the distance of 1
mile, and according to the square of the distance for
<cb/>
other distances; therefore as 1<hi rend="sup">2</hi> or 1 : 25<hi rend="sup">2</hi> or 625 :: 2/3 :
2/3 of 625 or 416 feet, the same as before.</p><p>There is a very easy method of taking great terrestrial
altitudes, such as mountains &amp;c, by means of the difference
between the heights of the barometer observed at
the bottom and top of the same. Which see under the
article <hi rend="smallcaps">Barometer.</hi></p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the Eye,</hi> in <hi rend="italics">Perspective,</hi> is a right line
let fall from the eye, perpendicular to the geometrical
plane.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi>, in <hi rend="italics">Astronomy</hi></head><p>, is the arch of a vertical
circle, measuring the height of the sun, moon, star, or
other celestial object, above the horizon.</p><p>This altitude may be either <hi rend="italics">true</hi> or <hi rend="italics">apparent.</hi> The
apparent altitude is that which appears by sensible observations
made at any place on the surface of the earth.
And the true altitude is that which results by correcting
the apparent, on account of refraction and parallax.</p><p>The quantity of the refraction is different at different
altitudes; and the quantity of the parallax is different
according to the distance of the different luminaries: in
the fixed stars this is too small to be observed; in the sun
it is but about 8 3/4 seconds; but in the moon it is about
52 minutes.</p><p>Altitudes are observed by a quadrant, or sextant,
or by the shadow of a gnomon or high pole, and by
various other ways, as may be seen in most books of
astronomy.</p><p><hi rend="italics">Meridian</hi> <hi rend="smallcaps">Altitude</hi>, is an arch of the meridian intercepted
between any point in it and the horizon. So
if HO be the horizon, and HEZO the meridian; then
the arch HE, or the angle HCE, is the meridian altitude
of an object in the meridian at the point E.
<figure/></p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">elevation, of the Pole,</hi> is the angle
OCP, or arch OP of the meridian, intercepted between
the horizon and pole P.</p><p>This is equal to the latitude of the place; and it may
be found by observing the meridian altitude of the pole
star, when it is both above and below the pole, and
taking half the sum, when corrected on account of
refraction. Or the same may be found by the declination
and meridian altitude of the sun.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">elevation,</hi> of the <hi rend="italics">equator,</hi> is the angle
HCE, or arch HE of the meridian, between the horizon
and the equator at E; and it is equal to ZP the colatitude
of the place.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the Tropics,</hi> the same as what is otherwise
called the <hi rend="italics">solstitial altitude</hi> of the sun, or his meridian
altitude when in the solstitial points.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, <hi rend="italics">or height, of the horizon,</hi> or of stars &amp;c
seen in it, is the quantity by which it is raised by refraction.</p><p><hi rend="italics">Refraction of</hi> <hi rend="smallcaps">Altitude</hi>, is an arch of a vertical circle,
by which the true altitude of the moon, or a star,
or other object, is increased by means of the refraction;
<pb n="104"/><cb/>
and is different at different altitudes, being nothing in
the zenith, and greatest at the horizon, where it is about
33&#x2032;.</p><p><hi rend="italics">Parallax of</hi> <hi rend="smallcaps">Altitude</hi>, is an arch of a vertical circle,
by which the true altitude, observed at the centre of the
earth, exceeds that which is observed on the surface;
or the difference between the angles <figure/> LM and <figure/> IK of
altitude there; and is equal to the angle I <figure/> L formed at
the moon or other body, and subtended by the radius
IL of the earth.</p><p>It is evident that this angle is less, as the luminary is
farther distant from the earth; and also less, for any
one luminary, as it is higher above the horizon; being
greatest there, and nothing in the zenith.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the Nonagesimal,</hi> is the altitude of the
90th degree of the ecliptic, counted upon it from where
it cuts the horizon, or of the middle or highest point of
it which is above the horizon, at any time; and is equal
to the angle made by the ecliptic and horizon where they
intersect at that time.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the cone of the earth's or moon's shadow,</hi>
the height of the shadow of the body, made by the sun,
and measured from the centre of the body. To find it,
say, As the tangent of the angle of the sun's apparent
semidiameter is to radius, so is 1 to a 4th proportional,
which will be the height of the shadow, in semidiameters
of the body.</p><p>So, the greatest height of the earth's shadow, is
217.8 semidiameters of the earth, when the sun is at
his greatest distance, or his semidiameter subtends an
angle of about 15&#x2032; 47&#x2033;; and the height of the same is
210.7 semidiameters of the earth, when the sun is nearest
the earth, or when his semidiameter is about 16&#x2032; 19&#x2033;:
And proportionally between these limits for the intermediate
distances or semidiameters of the sun.</p><p>The altitudes of the shadows of the earth and moon,
are nearly as 11 to 3, the proportion of their diameters.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">exaltation,</hi> in astrology, denotes the
second of the five essential dignities, which the planets
acquire by virtue of the signs they are found in.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of motion,</hi> is a term used by Dr. Wallis,
for the measure of any motion, estimated in the line of
direction of the moving force.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, in speaking of fluids, is more frequently
expressed by the term <hi rend="italics">depth.</hi> The pressure of fluids,
in every direction, is in proportion to their altitude or
depth.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the mercury,</hi> in the <hi rend="italics">barometer</hi> and <hi rend="italics">thermometer,</hi>
is marked by degrees, or equal divisions, placed
by the side of the tube of those instruments.</p><p>The altitude of the barometer, or of the mercury in
its tube, at London, is usually comprised between the
limits of 28 and 31 inches; and the mean height, for
every day in several years, is nearly 29.87 inches.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the pyramids in Egypt,</hi> was measured
so long since as the time of Thales, which he effected
by means of their shadow, and that of a pole set upright
beside them, making the altitudes of the pole and pyramid
proportional to the lengths of their shadows. Plutarch
has given an account of the manner of this operation,
which is one of the first geometrical observations
we have an exact account of.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, <hi rend="italics">circles of, parallels of, quadrant of, &amp;c.</hi>
See the respective words.
<cb/></p><p><hi rend="italics">Equal</hi> <hi rend="smallcaps">Altitude</hi> <hi rend="italics">Instrument,</hi> is an instrument used to
observe a celestial object, when it has the same or an
equal altitude, on both sides of the meridian, or before
and after it passes the meridian: an instrument very
useful in adjusting clocks &amp;c, and for comparing equal
and apparent time.</p></div2></div1><div1 part="N" n="AMBIENT" org="uniform" sample="complete" type="entry"><head>AMBIENT</head><p>, encompassing round about; as the bodies
which are placed about any other body, are called ambient
bodies, and sometimcs <hi rend="italics">circum-ambient</hi> bodies; and
the whole mass of the air or atmosphere, because it encompasses
all things on the face of the earth, is called
the <hi rend="italics">ambient</hi> air.</p><p>AMBIGENAL <hi rend="italics">Hyperbola,</hi> a name given by Newton,
in his <hi rend="italics">Enumeratio linearum tertii ordinis,</hi> to one of the triple
hyperbolas EGF of the second order, having one
of its infinite legs, as EG, falling within the angle
ACD, formed by the asymptotes AC, CD, and the
other leg GF falling without that angle.
<figure/></p></div1><div1 part="N" n="AMBIT" org="uniform" sample="complete" type="entry"><head>AMBIT</head><p>, <hi rend="italics">of a figure,</hi> in Geometry, is the perimeter,
or line, or sum of the lines, by which the figure
is bounded.</p></div1><div1 part="N" n="AMBLIGON" org="uniform" sample="complete" type="entry"><head>AMBLIGON</head><p>, or <hi rend="smallcaps">Ambligonal</hi>, in Geometry, signisies
obtuse-angular, as a triangle which has one of its
angles obtuse, or consisting of more than 90 degrees.</p><p>AMICABLE <hi rend="italics">numbers,</hi> denote pairs of numbers, of
which each of them is mutually equal to the sum of all
the aliquot parts of the other. So the first or least pair
of amicable numbers are 220 and 284; all the aliquot
parts of which, with their sums, are as follow, viz,
of 220, they are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55,
110, their sum &#x2014; &#x2014; 284;
of 284, they are 1, 2, 4, 71, 142, and their sum is 220.</p><p>The 2d pair of amicable numbers are 17296 and
18416, which have also the same property as above.</p><p>And the 3d pair of amicable numbers are 9363584
and 9437056.</p><p>These three pairs of amicable numbers were found
out by F. Schooten, sect. 9 of his <hi rend="italics">Exercitationes Mathematic&#xE6;,</hi>
who I believe first gave the name of <hi rend="italics">amicable</hi> to
such numbers, though such properties of numbers it
seems had before been treated of by Rudolphus, Descartes,
and others.</p><p>To find the sirst pair, Schooten puts 4<hi rend="italics">x</hi> and 4<hi rend="italics">yz,</hi> or
<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> and <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">yz</hi> for the two numbers where <hi rend="italics">a</hi> = 2; then
making each of these equal to the sum of the aliquot
parts of the other, gives two equations, from which are
found the values of <hi rend="italics">x</hi> and <hi rend="italics">z,</hi> and consequently, assuming
a proper value for <hi rend="italics">y,</hi> the two amicable numbers themselves
4<hi rend="italics">x</hi> and 4<hi rend="italics">yz.</hi></p><p>In like manner for the other pairs of such numbers;
in which he finds it necessary to assume 16<hi rend="italics">x</hi> and 16<hi rend="italics">yz</hi> or
<hi rend="italics">a</hi><hi rend="sup">4</hi><hi rend="italics">x</hi> and <hi rend="italics">a</hi><hi rend="sup">4</hi><hi rend="italics">yz</hi> for the 2d pair, and 128<hi rend="italics">x</hi> and 128<hi rend="italics">yz</hi> or
<hi rend="italics">a</hi><hi rend="sup">7</hi><hi rend="italics">x</hi> and <hi rend="italics">a</hi><hi rend="sup">7</hi><hi rend="italics">yz</hi> for the 3d pair.
<pb n="105"/><cb/></p><p>Schooten then gives this practical rule, from Descartes,
for finding amicable numbers, viz, Assume the
number 2, or some power of the number 2, such that
if unity or 1 be subtracted from each of these three following
quantities, viz;
from 3 times the assumed number,<lb/>
also from 6 times the assumed number,<lb/>
and from 18 times the square of the assumed number,<lb/>
the three remainders may be all prime numbers; then
the last prime number being multiplied by double the assumed
number, the product will be one of the amicable
numbers sought, and the sum of its aliquot parts will be
the other.</p><p>That is, if <hi rend="italics">a</hi> be put = the number 2, and <hi rend="italics">n</hi> some
integer number, such that 3<hi rend="italics">a</hi><hi rend="sup">n</hi>-1, and 6<hi rend="italics">a</hi><hi rend="sup">n</hi>-1, and
18<hi rend="italics">a</hi><hi rend="sup">2n</hi>-1 be all three prime numbers; then is &#x2015;(18<hi rend="italics">a</hi><hi rend="sup">2n</hi>-1)
X2<hi rend="italics">a</hi><hi rend="sup">n</hi> one of the amicable numbers; and the sum of its
aliquot parts is the other.</p></div1><div1 part="N" n="AMONTONS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">AMONTONS</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an ingenious French
experimental philosopher, was born in Normandy the
31st of August 1663. While at the grammar school,
he by sickness contracted a deafness that almost excluded
him from the conversation of mankind. In this situation
he applied himself to the study of geometry and
mechanics; with which he was so delighted that it is
said he refused to try any remedy for his disorder, either
because he deemed it incurable, or because it increased
his attention to his studies. Among other objects of his
study, were the arts of drawing, of land-surveying, and
of building; and shortly after he acquired some knowledge
of those more sublime laws by which the universe
is regulated. He studied with great care the nature of
barometers and thermometers; and wrote his treatise of
<hi rend="italics">Observations and Experiments concerning a new Hour-glass,
and concerning Barometers, Thermometers, and Hygroscopes;</hi>
as also some pieces in the Journal des Savans. In 1687,
he presented a new hygroscope to the Academy of Sciences,
which was much approved. He found out a
method of conveying intelligence to a great distance in
a short space of time: this was by making signals from
one person to another, placed at as great distances from
each other as they could see the signals by means of telescopes.
When the Royal Academy was new regulated
in 1699, Amontons was chosen a member of it, as an
eleve under the third Astronomer; and he read there
his <hi rend="italics">New Theory of Friction,</hi> in which he happily cleared
up an important object in mechanics. In fact he had a
particular genius for making experiments: his notions
were just and delicate: and he knew how to prevent the
inconveniences of his new inventions, and had a wonderful
skill in executing them. He died of an inflammation
in his bowels, the 11th of October 1705, being
only 42 years of age.</p><p>The eloge of Amontons may be seen in the volume of
the Memoirs of the Academy of Sciences for the year
1705, Hist. pa. 150. And his pieces contained in
the different volumes of that work, which are pretty
numerous, and upon various subjects, as the air, action
of fire, barometers, thermometers, hygrometers, friction,
machines, heat, cold, rarefactions, pumps, &amp;c, may be
seen in the volumes for the years 1696, 1699, 1702,
1703, 1704, and 1705.</p></div1><div1 part="N" n="AMPHISCII" org="uniform" sample="complete" type="entry"><head>AMPHISCII</head><p>, or <hi rend="smallcaps">Amphiscians</hi>, are the people
who inhabit the torrid zone; which are so called, because
<cb/>
they have their shadow at noon turned sometimes one
way, and sometimes another, namely, at one time of
the year towards the north, and at the other towards the
south.</p></div1><div1 part="N" n="AMPLITUDE" org="uniform" sample="complete" type="entry"><head>AMPLITUDE</head><p>, in <hi rend="italics">gunnery,</hi> the range of the projectile,
or the right line upon the ground subtending the
curvilinear path in which it moves.</p><div2 part="N" n="Amplitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Amplitude</hi>, in <hi rend="italics">astronomy</hi></head><p>, is an arch of the horizon,
intercepted between the true east or west point, and the
centre of the sun or a star at its rising or setting: so
that the amplitude is of two kinds; <hi rend="italics">ortive</hi> or eastern, and
<hi rend="italics">occiduous</hi> or western. Each of these amplitudes is also
either northern or southern, according as the point of
rising or setting is in the northern or southern part of the
horizon: and the complement of the amplitude, or the
arch of distance of the point of rising or setting, from
the north or south point of the horizon, is the azimuth.</p><p>The amplitude is of use in navigation, to find the variation
of the compass or magnetic needle. And the
rule to find it is this: As the cosine of the latitude is to
radius, so is the sine of the sun's or star's declination,
to the sine of the amplitude. So in the latitude of London,
viz, 51&#xB0; 31&#x2032;, when the sun's declination is 23&#xB0; 28&#x2032;; then
<table><row role="data"><cell cols="1" rows="1" role="data">cos. 51&#xB0; 31&#x2032; the lat.</cell><cell cols="1" rows="1" rend="align=right" role="data">-9.7939907</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 23 28 the decl.</cell><cell cols="1" rows="1" rend="align=right" role="data">+9.6001181</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 39 47 the ampl.</cell><cell cols="1" rows="1" rend="align=right" role="data">9.8061274</cell></row></table>
That is, the sun then rises or sets 39&#xB0; 47&#x2032; from the east
or west point, to the north or south according as the
declination is north or south.</p><p><hi rend="italics">Magnetical</hi> <hi rend="smallcaps">Amplitude</hi>, is an arch of the horizon,
contained between the sun or star, at the rising or setting,
and the magnetical east or west point of the horizon,
pointed out by the magnetical compass, or the
amplitude or azimuth compass. And the difference between
this magnetical amplitude, so observed, and the
true amplitude, as computed in the last article, is the
variation of the compass.</p><p>So if, for instance, the magnetical amplitude be observed,
by the compass, to be 61&#xB0; 47&#x2032;, at the time when
<table><row role="data"><cell cols="1" rows="1" role="data">it is computed to be</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">47,</cell></row><row role="data"><cell cols="1" rows="1" role="data">then the difference</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">0 is the variation west.</cell></row></table></p></div2></div1><div1 part="N" n="ANABIBAZON" org="uniform" sample="complete" type="entry"><head>ANABIBAZON</head><p>, a name sometimes given to the
dragon's tail, or northern node of the moon.</p></div1><div1 part="N" n="ANACAMPTICS" org="uniform" sample="complete" type="entry"><head>ANACAMPTICS</head><p>, or the science of the reflections
of sounds, frequently used in reference to echoes, which
are said to be sounds produced <hi rend="italics">anacamptically,</hi> or by reflection.
And in this sense it was used by the ancients
for that part of optics which is otherwise called Catoptrics.</p></div1><div1 part="N" n="ANACHRONISM" org="uniform" sample="complete" type="entry"><head>ANACHRONISM</head><p>, in Chronology, an error in
computation of time, by which an event is placed earlier
than it really happened. Such is that of Virgil, who
makes Dido to reign at Carthage in the time of &#xC6;neas,
though, in reality, she did not arrive in Africa till 300
years after the taking of Troy.</p><p>An error on the other side, by which a fact is placed
later, or lower than it should be, is called a <hi rend="italics">parachronism.</hi>
But in common use, this distinction, though proper, is
not attended to; and the word <hi rend="italics">anachronism</hi> is used indifferently
for the mistake on both sides.</p></div1><div1 part="N" n="ANACLASTICS" org="uniform" sample="complete" type="entry"><head>ANACLASTICS</head><p>, <hi rend="italics">or</hi> <hi rend="smallcaps">Anaclatics</hi>, an ancient name
for that part of Optics which considers refracted light;
<pb n="106"/><cb/>
being the same as what is more usually called <hi rend="italics">dioptrics.</hi>
See the Compendium of <hi rend="italics">Ambrosius Rhodius, lib.</hi> 3. <hi rend="italics">Optic&#xE6;,
pa.</hi> 384 <hi rend="italics">&amp; seq.</hi></p><p><hi rend="smallcaps">Anaclastic</hi> <hi rend="italics">Curves,</hi> a name given by M. de Mairan
to certain apparent curves formed at the bottom of a
vessel full of water, to an eye placed in the air; or the
vault of the heavens, seen by refraction through the atmosphere.</p><p>M. de Mairan determines these curves by a principle
not admitted by all authors; but Dr. Barrow, at the end
of his Optics, determines the same curves by other principles.</p></div1><div1 part="N" n="ANALEMMA" org="uniform" sample="complete" type="entry"><head>ANALEMMA</head><p>, a planisphere, or projection of the
sphere, orthographically made on the plane of the
meridian, by perpendiculars from every point of that
plane, the eye supposed to be at an infinite distance,
and in the east or west point of the horizon. In this
projection, the solstitial colure, and all its parallels, are
projected into concentric circles, equal to the real circles
in the sphere; and all circles whose planes pass
through the eye, as the horizon and its parallels, are projected
into right lines equal to their diameters; but all
oblique circles are projected into ellipses, having the diameter
of the circle for the transverse axis.</p><p>This instrument, having the furniture drawn on a
plate of wood or brass, with an horizon fitted to it,
is used for resolving many astronomical problems; as
the time of the sun's rising and setting, the length
and hour of the day, &amp;c. It is also useful in dialling,
for laying down the signs of the zodiac, with
the lengths of days, and other matters of furniture, upon
dials.</p><p>The oldest treatise we have on the analemma, was
written by Ptolemy, which was printed at Rome in
1562, with a commentary by F. Commandine. Pappus
also treated of the same. Since that time, many other
authors have treated very well of the analemma; as
Aguilonius, Taquet, Dechales, Witty, &amp;c.</p></div1><div1 part="N" n="ANALOGY" org="uniform" sample="complete" type="entry"><head>ANALOGY</head><p>, the same as proportion, or equality,
or similitude of ratios. Which see.</p></div1><div1 part="N" n="ANALYSIS" org="uniform" sample="complete" type="entry"><head>ANALYSIS</head><p>, is, generally, the resolution of any
thing into its component parts, to discover the thing or
the composition. And in mathematics it is properly the
method of resolving problems, by reducing them to
equations. <hi rend="italics">Analysis</hi> may be distinguished into the <hi rend="italics">ancient</hi>
and the <hi rend="italics">modern.</hi></p><p>The <hi rend="italics">ancient analysis,</hi> as described by Pappus, is the method
of proceeding from the thing sought as <hi rend="italics">taken</hi> for
granted, through its consequences, to something that is
<hi rend="italics">really</hi> granted or known; in which sense it is the reverse
of synthesis or composition, in which we lay that down
<hi rend="italics">first</hi> which was the <hi rend="italics">last</hi> step of the analysis, and tracing
the steps of the analysis back, making that antecedent
here which was consequent there, till we arrive at the
thing sought, which was taken or assumed as granted in
the first step of the analysis. This chiefly respected
geometrical enquiries.</p><p>The principal authors on the ancient analysis, as recounted
by Pappus, in the 7th book of his <hi rend="italics">Mathematical
Collections,</hi> are Euclid in his <hi rend="italics">Data, Porismata, &amp; de Locis
ad Superficiem;</hi> Apollonius <hi rend="italics">de Sectione Rationis, de Sectione
Spatii, de Taclionibus, de Inclinationibus, de Locis Planis,
&amp; de Sectionibus Conicis;</hi> Arist&#xE6;us, <hi rend="italics">de Locis Solidis;</hi> and
Eratosthenes, <hi rend="italics">de Mediis Proportionalibus;</hi> from which
<cb/>
Pappus gives many examples in the same book. T
these authors we may add Pappus himself. The same
sort of analysis has also been well cultivated by many of
the moderns; as Fermat, Viviani, Getaldus, Snellius,
Huygens, Simson, Stewart, Lawson, &amp;c, and more
especially Hugo d'Omerique, in his <hi rend="italics">Analysis Geometrica,</hi>
in which he has endeavoured to restore the Analysis of
the ancients. And, on this head, Dr. Pemberton tells
us &#x201C;that Sir Ifaac Newton used to censure himself for
not following the ancients more closely than he did; and
spoke with regret of his mistake, at the beginning of
his mathematical studies, in applying himself to the
works of Descartes, and other algebraical writers, before
he had considered the Elements of Euclid with that attention
so excellent a writer deserves: that he highly
approved the laudable attempt of Hugo d'Omerique to
restore the ancient analysis.&#x201D;</p><p>In the application of the ancient analysis in geometrical
problems, every thing cannot be brought within strict
rules; nor any invariable directions given, by which we
may succeed in all cases; but some previous preparation
is necessary, a kind of mental contrivance and construction,
to form a connexion between the <hi rend="italics">data</hi> and <hi rend="italics">qu&#xE6;sita,</hi>
which must be left to every one's fancy to find out; being
various, according to the various nature of the problems
proposed: Right lines must be drawn in particular directions,
or of particular magnitudes; bisecting perhaps
a given angle, or perpendicular to a given line; or perhaps
tangents must be drawn to a given curve, from a
given point; or circles described from a given centre,
with a given radius, or touching given lines, or other
given circles; or such-like other operations. Whoever
is conversant with the works of Archimedes, Apollonius,
or Pappus, well knows that they founded their analysis
upon some such previous operations; and the great skill
of the analyst consists in discovering the most proper affections
on which to found his analysis: for the same
problem may often be effected in many different ways:
of which it may be proper to give here an example or
two. Let there be taken, for instance, this problem,
which is the 155th prop. of the 7th book of Pappus.</p><p>From the extremities of the base A, B, of a given
segment of a circle, it is required to draw two lines AC,
BC, meeting at a point C in the circumference, so that
they shall have a given ratio to each other, suppose that
of F to G.</p><p>The solution of this problem, as given by Pappus, is
thus.
<hi rend="center"><hi rend="smallcaps">Analysis.</hi></hi></p><p>Suppose the thing done, and that the point C is found:
then suppose CD is drawn a tangent to the circle at C,
and meeting the line AB produced in the point D. Now
by the hypothesis AC : BC :: F : G, and also AC<hi rend="sup">2</hi> :
BC<hi rend="sup">2</hi> :: DA : DB, as may be thus proved.
<figure/></p><p>Since DC touches the circle, and BC cuts it, the angle
BCD is equal to BAC by Euc. iii. 32; also the angle
<pb n="107"/><cb/>
D is common to both the triangles DCA, DCB; these
are therefore similar, and so, by vi 4, DA : DC :: DC :
DB, and hence DA<hi rend="sup">2</hi> : DC<hi rend="sup">2</hi> :: DA : DB by cor. vi
20. But also, by vi 4, DA : AC :: DC : CB, and by
permutation DA : DC :: AC : BC, or DA<hi rend="sup">2</hi> : DC<hi rend="sup">2</hi> ::
AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi>; and hence, by equality, AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi> :: DA :
DB.</p><p>But the ratio of AC<hi rend="sup">2</hi> to BC<hi rend="sup">2</hi> is given by prop. LVII
of Simson's edition of the <hi rend="italics">Data,</hi> because the ratio of
AC to BC is given, and consequently that of DA to DB
is given. Now since the ratio of DA to DB is given,
therefore also, by Data vi, that of DA to AB, and
hence, by Data ii, DA is given in magnitude.</p><p>And here the analysis properly ends. For it having
been shewn that DA is given, or that a point D may be
found in AB produced, such, the a tangent being drawn
from it to the circumserence, the point of contact will
be the point sought; we may now begin the composition,
or synthetical demonstration; which must be done
by finding the point D, or laying down the line AD,
which, it was affirmed, was given, in the last step of the
analysis.
<hi rend="center"><hi rend="smallcaps">Synthesis.</hi></hi></p><p><hi rend="italics">Construction.</hi> Make as F<hi rend="sup">3</hi> : G<hi rend="sup">2</hi> :: AD : DB, (which
may be done, since AB is given, by making it as F<hi rend="sup">2</hi>G<hi rend="sup">2</hi> : G<hi rend="sup">2</hi> :: AB : DB, and then by composition it will be
as F<hi rend="sup">2</hi> : G<hi rend="sup">2</hi> :: AD : DB); and then from the point D,
thus found, draw a tangent to the circle, and from the
point of contact C drawing CA and CB, the thing is
done.</p><p><hi rend="italics">Demonstration.</hi> Since, by the constr. F<hi rend="sup">2</hi> : G<hi rend="sup">2</hi> ::
AD : DB, and also AD : DB :: AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi>, which
has been already demonstrated in the analysis, and might
be here proved in the same manner. Therefore F<hi rend="sup">2</hi> :
G<hi rend="sup">2</hi> :: AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi>, and consequently F : G :: AC :
BC. <hi rend="italics">Q.E.D.</hi></p><p>Here we see an instance of the method of <hi rend="italics">resolution</hi>
and <hi rend="italics">composition,</hi> as it was practised by the ancients, the
solution here given being that of Pappus himself. But as
the method of referring and reducing every thing to the
<hi rend="italics">Data,</hi> and constantly quoting the same, may appear now
to be tedious and troublesome: and indeed it is unnecessary
to those who have already made themselves masters
of the substance of that valuable book of Euclid,
and have by practice and experience acquired a facility
of reasoning in such matters: I shall therefore now
shew how we may abate something of the rigour and
strict from of the ancient method of solution, without
diminishing any part of its admirable elegance and perspicuity.
And this may be done by the instance of another
solution, of the many more which might be given,
of the same problem, as follows.
<hi rend="center"><hi rend="smallcaps">Analysis.</hi></hi></p><p>Let us again suppose that the thing is done, viz AC :
BC :: F : G, and let there be drawn BH making the
angle ABH equal to the angle ACB, and meeting AC
produced in H. Then, the angle A being also common,
the two triangles ABC and ABH are equiangular,
and therefore, by vi 4, AC : BC :: AB : BH, in a
given ratio; and, AB being given, therefore BH is
given in position and magnitude.
<cb/>
<hi rend="center"><hi rend="smallcaps">Synthesis.</hi></hi></p><p><hi rend="italics">Construction.</hi> Draw BH making the angle ABH
equal to that which may be contained in the given segment,
and take AB to BH in the given ratio of F to G.
Draw ACH, and BC.</p><p><hi rend="italics">Demonstration.</hi> The triangles ABC, ABH are equiangular,
therefore, vi 4, AC : CB :: AB : BH, which
is the given ratio by construction.</p><p><hi rend="italics">Modern</hi> <hi rend="smallcaps">Analysis</hi>, consists chiefly of algebra, arithmetic
of insinites, insinite series, increments, fluxions,
&amp;c; of each of which a particular account may be seen
under their respective articles.</p><p>These form a kind of arithmetical and symbolical analysis,
depending partly on modes of arithmetical computation,
partly on rules peculiar to the symbols made
use of, and partly on rules drawn from the nature and
species of the quantities they represent, or from the
modes of their existence or generation.</p><p>The modern analysis is a general instrument by which
the sinest inventions and the greatest improvements have
been made in mathematics and philosophy, for near two
centuries past. It furnishes the most perfect examples of
the manner in which the art of reasoning should be employed;
it gives to the mind a wonderful skill for discovering
things unknown, by means of a small number
that are given; and by employing short and easy symbols
for expressing ideas, it presents to the understanding
things which otherwise would seem to lie above its sphere.
By this means geometrical demonstrations may be greatly
abridged: a long train of arguments, in which the mind
cannot, without the greatest effort of attention, discover
the connection of ideas, is converted into visible symbols;
and the various operations which they require, are
simply effected by the combination of those symbols.
And, what is still more extraordinary, by this artifice,
a great number of truths are often expressed in one line
only: instead of which, by following the ordinary way
of explanation and demonstration, the same truths would
occupy whole pages or volumes. And thus, by the bare
contemplation of one line of calculation, we may undersland
in a short time whole sciences, which otherwise
could hardly be comprehended in several years.</p><p>It is true that Newton, who best knew all the advantages
of analysis in geometry and other sciences, laments,
in several parts of his works, that the study of the ancient
geometry is abandoned or neglected. And indeed
the method employed by the ancients in their geometrical
writings, is commonly regarded as more rigorous,
than that of the modern analysis: and though it
be greatly inferior to that of the moderns, in point of
dispatch and facility of invention; it is nevertheless
highly useful in strengthening the mind, improving the
reasoning faculties, and in accustoming the young mathematician
to a pure, clear, and accurate mode of investigation
and demonstration, though by a long and laboured
process, which he would with difficulty have submitted
to if his taste had before been vitiated, as it were, by
the more piquant sweets of the modern analysis. And
it is principally on this that the complaints of Newton
are founded, who feared lest by the too early and frequent
use of the modern analysis, the science of geometry
should lose that rigour and purity which characterise its
investigations, and the mind become debilitated by the
<pb n="108"/><cb/>
facility of our analysis. This great man was therefore
well founded, in recommending, to a certain extent, the
study of the ancient geometricians: for, their demonstrations
being more difficult, give more exercise to the
mind, accustom it to a closer application, give it a
greater scope, and habituate it to patience and resolution,
so necessary for making discoveries. But this is the
only or principal advantage from it; for if we should
look no farther than the method of the ancients, it is
probable that, even with the best genius, we should
have made but few or small discoveries, in comparison
of those obtained by means of the modern analysis.
And even with regard to the advantage given to investigations
made in the manner of the ancients, namely
of being more rigorous, it may perhaps be doubted
whether this pretension be well founded. For to instance
in those of Newton himself, although his demonstrations
be managed in the manner of the ancients;
yet at the same time it is evident that he investigates
his theorems by a method different from that employed
in the demonstrations, which are commonly analytical
calculations, disguised by substituting the name of lines
for their algebraical value: and though it be true that
his demonstrations are rigorous, it is no less so
that they would be the same when translated and delivered
in algebraic language; and what difference can it
make in this respect, whether we call a line AB, or
denote it by the algebraic character <hi rend="italics">a?</hi> Indeed this last
designation has this peculiarity, that when all the lines
are denoted by algebraic characters, many operations
can be performed upon them, without thinking of the
lines or the figure. And this circumstance proves of no
small advantage: the mind is relieved, and spared as
much as possible, that its whole force may be employed
in overcoming the natural difficulty of the problem
alone.</p><p>Upon the whole therefore the state of the comparison
seems to be this; That the method of the ancients is
fittest to begin our studies with, to form the mind and
to establish proper habits; and that of the moderns to
succeed, by extending our views beyond the present
limits, and enabling us to make new discoveries and
improvements.</p><p><hi rend="italics">Analysis</hi> is divided, with respect to its object, into that
of <hi rend="italics">finites,</hi> and that of <hi rend="italics">infinites.</hi></p><p><hi rend="italics">Analysis of finite quantities,</hi> is what is otherwise called
<hi rend="italics">algebra,</hi> or <hi rend="italics">specious arithmetic.</hi></p><p><hi rend="italics">Analysis of infinites,</hi> called also the <hi rend="italics">new analysis,</hi> is
that which is concerned in calculating the relations of
quantities which are considered as infinite, or infinitely
little; one of its chief branches being the <hi rend="italics">method of
fluxions,</hi> or the <hi rend="italics">differential calculus.</hi> And the great advantage
of the modern mathematicians over the ancients,
arises chiefly from the use of this modern analysis.</p><p><hi rend="smallcaps">Analysis</hi> <hi rend="italics">of powers,</hi> is the same as resolving them
into their roots, and is otherwise called <hi rend="italics">evolution.</hi></p><p><hi rend="smallcaps">Analysis</hi> <hi rend="italics">of curve lines,</hi> shews their constitution,
nature and properties, their points of inflexion, station,
retrogradation, variation, &amp;c.</p></div1><div1 part="N" n="ANALYST" org="uniform" sample="complete" type="entry"><head>ANALYST</head><p>, a person who analyses something, or
makes use of the analytical method. In mathematics,
it is a person skilled in algebra, or in the mathematical
analysis in general.
<cb/></p><div2 part="N" n="Analyst" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Analyst</hi></head><p>, the title of an ingenious, though sophistical
book, written by the celebrated Dr. Berkeley,
against the doctrine of fluxions.</p></div2></div1><div1 part="N" n="ANALYTIC" org="uniform" sample="complete" type="entry"><head>ANALYTIC</head><p>, or <hi rend="smallcaps">Analytical</hi>, something belonging
to, or partaking of, the nature of analysis; or performed
by the method of analysis.</p><p>Thus we say <hi rend="italics">analytical</hi> demonstration, <hi rend="italics">analytical</hi> enquiry,
<hi rend="italics">analytical</hi> table or scheme, <hi rend="italics">analytical</hi> method, &amp;c.
The <hi rend="italics">analytical</hi> stands opposed to the <hi rend="italics">synthetical,</hi> or that
which proceeds by the way of <hi rend="italics">synthesis.</hi></p></div1><div1 part="N" n="ANALYTICS" org="uniform" sample="complete" type="entry"><head>ANALYTICS</head><p>, the science, or doctrine, and use
of analysis.</p></div1><div1 part="N" n="ANAMORPHOSIS" org="uniform" sample="complete" type="entry"><head>ANAMORPHOSIS</head><p>, in perspective and painting,
a monstrous projection; or a representation of some
image, either on a plane or curve surface, deformed
or distorted; but which in a certain point of view shall
appear regular, and drawn in just proportion.</p><p><hi rend="italics">To construct an Anamorphosis, or monstrous projection, on
a plane.</hi>&#x2014;Draw the square ABCD (fig. 1), of any
size at pleasure, and divide it by crossing lines into a
number of areol&#xE6; or smaller squares: and then in this
square, or reticle, called also the <hi rend="italics">cratioular prototype,</hi>
draw the regular image which is to be distorted.&#x2014;Or,
about any image, proposed to be distorted, draw a reticle
of small squares.
<figure/></p><p>Then draw the line <hi rend="italics">ab</hi> (fig. 2.) equal to AB, dividing
it into the same number of equal parts, as the
side of the prototype AB; and on its middle point
E erect the perpendicular EV, and also VS perpendicular
to EV, making EV so much the longer, and VS
so much the shorter, as it is intended the image shall be
more distorted. From each of the points of division draw
right lines to the point V, and draw the right line <hi rend="italics">a</hi>S.
Lastly through the points <hi rend="italics">c, e, f, g,</hi> &amp;c, draw lines parallel
to <hi rend="italics">ab:</hi> So shall<hi rend="italics">abcd</hi> be the space upon which the monstrous
projection is to be drawn; and is called the <hi rend="italics">craticular ectype.</hi></p><p>Then, in every areola, or small trapezium, of the
space <hi rend="italics">abcd,</hi> draw what appears contained in the corresponding
areola of the original space ABCD: so shall
there be produced a deformed image in the spacc <hi rend="italics">abcd,</hi>
which yet will appear in just proportion to an eye distant
<pb n="109"/><cb/>
from it the length of EV, and raised above it by a
height equal to VS.</p><p>It will be amusing to contrive it so, that the deformed
image may not represent a mere chaos, but some certain
figure: thus, a river with soldiers, waggons, and other
objects on the side of it, have been so drawn and distorted,
that when viewed by an eye at S, it appeared
like the face of a satyr.</p><p>An image may also be distorted mechanically, by
perforating through in several places with a fine pin;
then, placing it against a candle or lamp, observe where
the rays, which pass through these small holes, fall on
any plane or curve superficies; for they will give the
correspondent points of the image deformed, and by
means of which the deformation may be completed.</p><p><hi rend="italics">To draw an Anamorphosis upon the convex surface of
a cone.</hi> It appears from the construction above, that we
have only to make a craticular ectype upon the surface
of the cone, which may appear equal to the craticular
prototype, to an eye placed at a proper height above
the vertex of the cone. Hence,</p><p>Let the base, or circumference, ABCD, of the cone
(fig. 3) be divided by radii into any number of equal
parts; and let some one radius be likewise divided into
equal parts; then through each point of division draw
concentric circles: so shall the craticular prototype be
formed.
<figure/></p><p>With double the diameter AB, as a radius, describe
the quadrant EFG (fig. 4) so as the arch EG be
equal to the whole periphery; then this quadrant, being
plied or bent round, will form the superficies of a cone,
whose base is the circle.</p><p>Next divide the arch EG into the same number of
equal parts as the craticular prototype is divided into;
and draw radii from all the points of division. Produce
GF to I, so that FI be equal to FG; and from the centre
I, with the radius IF, describe the quadrant FKH; and
draw the right line IE. Then divide the arch KF into
the same number of equal parts as the radius of the
eraticular prototype is divided into; and from the centre
I draw radii through all the points of division, meeting
EF in 1, 2, 3, &amp;c. Lastly, from the centre F, with
the radii F1, F2, F3, &amp;c, describe concentric circles.
So will the craticular ectype be formed, whose areolas
will appear equal to each other.</p><p>Hence, what is delineated in every areola of the craticular
prototype, being transferred into the areolas of
the craticular ectype, the images will be distorted or
deformed; and yet they will appear in just proportion
<cb/>
to an eye elevated above the vertex at a height equal to
the height of the cone itself.</p><p>If the chords of the quadrants be drawn in the craticular
prototype, and chords of each of the 4th parts
in the craticular ectype, every thing else remaining the
same, there will be obtained the craticular ectype in a
quadrangular pyramid.</p><p>And hence it will be easy to deform an image, in
any other pyramid, whose base is any regular polygon.</p><p>Because the illusion is more perfect when the eye, by
the contiguous objects, cannot estimate the distance of
the parts of the deformed image, it is therefore proper
to view it through a small hole.</p><p>Anamorphoses, or monstrous images, may also be
made to appear in their natural shape and just proportions,
by means of mirrors of certain shapes, from which
those images are reflected again; and then they are said
to be reformed.</p><p>For farther particulars, see Wolfius's <hi rend="italics">Catoptrics and
Dioptrics,</hi> and some other optical authois.</p></div1><div1 part="N" n="ANAPHORA" org="uniform" sample="complete" type="entry"><head>ANAPHORA</head><p>, in <hi rend="italics">Astrology,</hi> the second house, or
that part of the heavens which is 30 degrees from the
horoscope.</p><p>The term <hi rend="italics">anaphora</hi> is also sometimes applied promiscuously
to some of the succeeding houses, as the 5th,
the 8th, and the 11th. In this sense <hi rend="italics">anaphora</hi> is the
same as <hi rend="italics">epanaphora,</hi> and stands opposed to <hi rend="italics">cataphora.</hi></p><p>ANASTROUS <hi rend="italics">signs,</hi> in <hi rend="italics">Astronomy,</hi> a name given to
the <hi rend="italics">duodecatemoria,</hi> or the 12 portions of the ecliptic,
which the signs possessed anciently, but have since deserted
by the precession of the equinox.</p></div1><div1 part="N" n="ANAXAGORAS" org="uniform" sample="complete" type="entry"><head>ANAXAGORAS</head><p>, one of the most celebrated philosophers
among the ancients. He was born at Clazomene
in Ionia, about the 70th Olympiad. He was a
disciple of Anaximenes; and he gave up his patrimony,
to be more at leisure for the study of philosophy, giving
lectures in that science at Athens. Being persecuted
in this place, and at last banished from it, he opened a
school at Lampsacum, where he was greatly honoured
during his life, and still more after his death, statues
having been erected to his memory. It is said he made
some predictions relative to the phenomena of nature,
as earthquakes &amp;c, upon which he wrote some treatises.
His principal tenets may be reduced to the following:&#x2014;
All things were in the beginning confusedly mixed together,
without order and without motion. The principle
of things is at the same time one and multiplex,
which had the name of <hi rend="italics">hom&#xE6;meries,</hi> or similar particles,
deprived of life. But there is beside this, from all
eternity, another principle, an infinite and incorporeal
spirit, who gave motion to these particles; in virtue of
which, such as are homogeneal united, and such as were
heterogeneal separated according to their different kinds.
All things being thus put into motion by the spirit, and
every thing being united to such as are similar, those
that had a circular motion produced heavenly bodies,
the lighter particles ascending, while those that were
heavier descended. The rocks of the earth, being drawn
up by the whirling force of the air, took fire, and became
stars, beneath which the sun and moon took their stations.
&#x2014;It was said he also wrote upon the <hi rend="italics">Quadrature of the
Circle;</hi> the treatise upon which, Plutarch says, he
composed during his imprisonment at Athens.
<pb n="110"/><cb/></p></div1><div1 part="N" n="ANAXIMANDER" org="uniform" sample="complete" type="entry"><head>ANAXIMANDER</head><p>, a very celebrated Greek philosopher,
was born at Miletus in the 42d olympiad; for,
according to Apollodorus, he was 64 years of age in the
2d year of the 58th olympiad. He was one of the first
who publicly taught philosophy, and wrote upon philosophical
subjects. He was the kinsman, companion,
and disciple of Thales. He wrote also upon the sphere
and geometry, &amp;c. And he carried his researches into
nature very far, for the time in which he lived. It is
said that he discovered the obliquity of the zodiac; that
he first published a geographical table; that he invented
the gnomon, and set up the first sun-dial in an open
place at Laced&#xE6;mon. He taught, that infinity of
things was the principal and universal element; that
this infinite always preserved its unity, but that its parts
underwent changes; that all things came from it; and
that all were about to return to it. By this obscure and
indeterminate principle he probably meant the chaos of
other philosophers. He held that the worlds are insinite;
that the stars are composed of air and sire, which are
carried about in their spheres, and that these spheres are
gods; and that the earth is placed in the midst of the
universe, as in a common centre. Farther, that insinite
worlds were the produce of infinity; and that corruption
proceeded from separation.</p></div1><div1 part="N" n="ANAXIMENES" org="uniform" sample="complete" type="entry"><head>ANAXIMENES</head><p>, an eminent Greek philosopher,
born at Miletus, the friend, scholar, and successor of
Anaximander. He diffused some degree of light upon the
obscurity of his master's system. He made the first principle
of things to consist in the air, which he considered as
infinite or immense, and to which he ascribed a perpetual
motion; that this air was the same as spirit or God,
since the divine power resided in it, and agitated it.
The stars were as fiery nails in the heavens; the sun a
flat plate of fire; the earth an extended flat surface, &amp;c.</p></div1><div1 part="N" n="ANDERSON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ANDERSON</surname> (<foreName full="yes"><hi rend="smallcaps">Alexander</hi></foreName>)</persName></head><p>, one of the brightest
ornaments of the mathematical world, who flourished
about 200 years ago. He was born at Aberdeen in
Scotland, it would seem towards the latter part of the
16th century, as he was professor of mathematics at
Paris in the early part of the 17th, where he published
several ingenious works in geometry and algebra, both
of his own, and of his friend Vieta's. Thus he published
his &#x201C;Supplementum Apollonii Redivivi; (of Ghetaldus)
sive analysis problematis hactenus desiderati ad
Apollonii Perg&#xE6;i doctrinam <foreign xml:lang="greek">weri neusewn</foreign>, a Marino Ghetaldo
Patritio Ragusino hujusque, non ita pridem restitutam.
In qua exhibetur mechanice &#xE6;qualitatum tertii
gradus sive solidarum, in quibus magnitudo omnino data,
&#xE6;quatur homogene&#xE6; sub altero tantum co&#xEB;fficiente
ignoto. Huic subnexa est variorum problematum practice.&#x201D;
Paris, 1612, in 4to.</p><p>&#x201C;<foreign xml:lang="greek">*aitiologia</foreign>: Pro Zetetico Apolloniani problematis
a se jam pridem edito in supplemento Apollonii Redivivi.
Ad clarissimum &amp; ornatissimum virum Marinum
Ghetaldum Patritium Ragusinum. In qua ad ea quae
obiter mihi perstrinxit Ghetaldus respondetur, &amp; analytices
clarius detegitur.&#x201D; Paris, 1615, in 4to.
<hi rend="center">He published also,</hi></p><p>&#x201C;Francisci Viet&#xE6; Fontenacensis de Aequationum
Recognitione &amp; Emendatione Tractatus duo.&#x201D; Paris,
1615, in 4to; with a Dedication, Prefaee, and an
Appendix, by Anderson.
<cb/></p><p>And Vieta's Angulares Sectiones, with the Demonstrations
by Anderson.</p><p>Alexander was cousin german to a Mr. David Anderson,
of Finshaugh, a gentleman who also possessed a singular
turn for mathematical and mechanical knowledge.
This mathematical genius was hereditary in the family
of the Andersons, and from them it seems to have been
transmitted to their descendants of the name of Gregory
in the same country: the daughter of the said David
Anderson having been the mother of the celebrated
mathematician James Gregory, and who herself first
instructed her son James in the elements of the Mathematics,
upon her observing in him, while yet a child, a
strong propensity to those sciences.</p><p>The time either of the birth or death of our author
Alexander, has not come to my knowledge.</p></div1><div1 part="N" n="ANDROGYNOUS" org="uniform" sample="complete" type="entry"><head>ANDROGYNOUS</head><p>, an appellation given, by astrologers,
to such of the planets as are sometimes hot, and
sometimes cold; as mercury, which is accounted hot and
dry when near the sun, and cold and moist when near the
moon.</p></div1><div1 part="N" n="ANDROMEDA" org="uniform" sample="complete" type="entry"><head>ANDROMEDA</head><p>, in <hi rend="italics">Astronomy,</hi> a constellation of
the northern hemisphere, representing the sigure of a
woman almost naked, her seet at a distance from each
other, and her arms extended and chained; being one
of the original 48 asterisms, or sigures under which the
ancients comprehended the stars, as derived to us from
the Greeks, who probably had them from the Egyptians
or Indians, and who, it is suspected, altered their
names, and accompanied them with fabulous stories
of their own. According to them, Cepheus, the father
of Andromeda, was obliged to give her up to be devoured
by a monster, to preserve his kingdom from the
plague; but that she was delivered by Perseus, who slew
the monster, and espoused her. And the family were
all translated by Minerva to heaven, the mother being
the constellation Cassiopeia.</p><p>She is sometimes called, in Latin, <hi rend="italics">Persea, Mulier catenata,
Virgo devota,</hi> &amp;c. The Arabians, whose religion
did not permit them to draw the figure of the
human body on any occasion whatever, have changed
this constellation into the figure of a sea-calf. Schickard
has changed the name for that of the scripture name
<hi rend="italics">Abigail.</hi> And Schiller has also changed the figure of
the constellation, for that of a sepulchre, and calls it
the <hi rend="italics">Holy Sepulchre.</hi></p><p>This constellation contains about 27 stars that are
visible to the naked eye; of which the principal are, <foreign xml:lang="greek">a</foreign>
<hi rend="italics">Andromeda's head;</hi> <foreign xml:lang="greek">b</foreign> in the girdle, and called <hi rend="italics">mirach</hi> or
<hi rend="italics">mizar;</hi> <foreign xml:lang="greek">g</foreign> on the south foot, and named <hi rend="italics">alamak,</hi> and
sometimes <hi rend="italics">alhames.</hi></p><p>The number of stars placed in this constellation by
the catalogue of Ptolemy is 23, by that of Tycho Brahe
also 23, by that of Hevelius 47, and by that of Flamsteed
66.</p></div1><div1 part="N" n="ANEMOMETER" org="uniform" sample="complete" type="entry"><head>ANEMOMETER</head><p>, an instrument for measuring the
force of the wind.</p><p>An instrument of this sort, it seems, was first invented
by Wolfius in the year 1708, and first published
in his <hi rend="italics">Areometry</hi> in 1709, also in the <hi rend="italics">Acta Eruditorum</hi>
of the same year; afterwards in his <hi rend="italics">Mathematical Dictionary,</hi>
and in his <hi rend="italics">Elem. Matheseos.</hi> He says he tried
the goodness of it, and observes that the internal struc-
<pb n="111"/><cb/>
ture may be preserved, so as to measure the force of running
water, or that of men or horses when they draw
or pull. The machine consists of sails, A, B, C, like
those of a wind-mill, against which the wind blows, and
by turning them about, raises an arm K with a weight
L upon it, to different angles of elevation, shewn by
the index M, according to the force of the wind. (Plate
III. fig. 3)</p><p>In the Philos. Trans. another anemometer is described,
in which the wind being supposed to blow
directly against a flat side, or board, which moves along
the graduated arch of a quadrant, the number of degrees
it advances shews the comparative forceof the wind.</p><p>In the same Transactions, for the year 1766, Mr.
Alex. Brice describes a method, successfully practised
by him, of measuring the velocity of the wind, by
means of that of the shadow of clouds passing over a
plane upon the earth.</p><p>Also in the same Transactions, for the year 1775,
Dr. Lind gives a description of a very ingenious portable
<hi rend="italics">Wind-Gauge,</hi> by which the force of the wind is
easily measured; a brief description of the principal
parts of which here follows. This simple instrument
consists of two glass tubes, AB, CD, (Plate III. fig. 4.)
which should not be less than 8 or 9 inches long, the
bore of each being about 4/10 of an inch diameter, and
connected together by a small bent glass tube <hi rend="italics">ab,</hi> only
of about 1/10 of an inch diameter, to check the undulations
of the water caused by a sudden gust of wind.
On the upper end of the leg AB is fitted a thin metal
tube, which is bent perpendicularly outwards, and
having its mouth open to receive the wind blowing
horizoutally into it. The two tubes, or rather the two
branches of the tube, are connected to a steel spindle KL
by slips of brass near the top and bottom, by the sockets
of which at <hi rend="italics">e</hi> and <hi rend="italics">f</hi> the whole instrument turns easily
about the spindle, which is fixed into a block by a
screw in its bottom, by the wind blowing in at the
orifice at F. When the instrument is used, a quantity
of water is poured in, till the tubes are about half full;
then exposing the instrument to the wind, by blowing in
at the orifice F, it forces the water down lower in the
tube AB, and raises it so much higher in the other tube;
and the distance between the surfaces of the water in
the two tubes, estimated by a scale of inches and parts
HI, placed by the sides of the tubes, will be the height
of a column of water whose weight is equal to the force
or momentum of the wind blowing or striking against an
equal base. And as a cubic foot of water weighs 1000
ounces, or 62 1/2 pounds, the 12th part of which is 5 5/24
or 5 1/5 pounds nearly, therefore for every inch the surface
of the water is raised, the force of the wind will be equal
to so many times 5 1/5 pounds on a square foot. Thus,
suppose the water stand 3 inches higher in the one tube,
than in the other; then 3 times 5 1/5 or 15 3/5 pounds is
equal to the pressure or force of the wind on the surface
of a foot square.</p><p>This instrument of Dr. Lind's, measures only the
force or momentum of the wind, but not its velocity.
However the velocity of the wind may be deduced from
its force so obtained, by help of some experiments performed
by me at the Royal Military Academy, in the
years 1786, 1787, and 1788; from which experiments
it appears that a plane sursace of a square foot suffers a
<cb/>
resistance of 12 ounces from the wind, when blowing
with a velocity of 20 feet per second; and that the sorce
is nearly as the square of the velocity. Hence then,
taking the force of 15 2/5 pounds, above found for the
force of the wind when it sustains 3 inches of water, and
taking the square roots of the forces, it will be, as &#x221A;12
: &#x221A;15 3/5 :: 20 : 22 4/5 the 4th proportional, that is a
velocity of 22 4/5 feet per second, or 15 1/2 miles per hour,
is the rate or velocity at which the wind blows, when it
raises the water 3 inches higher in the one tube than
the other. And farther, as the said height is as the
force, and the force as the square of the velocity, we
shall have the force and velocity, corresponding to several
heights of the water in the one tube, above that
in the other, as in the following table.
<hi rend="center"><hi rend="italics">Table of the corresponding height of water, force on a
square foot, and velocity of wind.</hi></hi>
<table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Height of
water.</cell><cell cols="1" rows="1" rend="align=center" role="data">Force of
wind.</cell><cell cols="1" rows="1" rend="align=center" role="data">Velocity of
wind per hour.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Inches.</cell><cell cols="1" rows="1" rend="align=center" role="data">Pounds.</cell><cell cols="1" rows="1" rend="align=center" role="data">Miles.</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;0 1/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4.5</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;0 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;1</cell><cell cols="1" rows="1" role="data">&#xA0;5.2</cell><cell cols="1" rows="1" rend="align=right" role="data">9.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;2</cell><cell cols="1" rows="1" role="data">10.4</cell><cell cols="1" rows="1" rend="align=right" role="data">12.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;3</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" rend="align=right" role="data">15.5</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;4</cell><cell cols="1" rows="1" role="data">20.8</cell><cell cols="1" rows="1" rend="align=right" role="data">18.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;5</cell><cell cols="1" rows="1" role="data">26.0</cell><cell cols="1" rows="1" rend="align=right" role="data">20.1</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;6</cell><cell cols="1" rows="1" role="data">31.25</cell><cell cols="1" rows="1" rend="align=right" role="data">22.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;7</cell><cell cols="1" rows="1" role="data">36.5</cell><cell cols="1" rows="1" rend="align=right" role="data">23.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;8</cell><cell cols="1" rows="1" role="data">41.7</cell><cell cols="1" rows="1" rend="align=right" role="data">25.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">&#xA0;9</cell><cell cols="1" rows="1" role="data">46.9</cell><cell cols="1" rows="1" rend="align=right" role="data">27.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">52.1</cell><cell cols="1" rows="1" rend="align=right" role="data">28.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">57.3</cell><cell cols="1" rows="1" rend="align=right" role="data">29.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">62.5</cell><cell cols="1" rows="1" rend="align=right" role="data">31.0</cell></row></table></p><p>In one instance Dr.
Lind found that the
force of the wind was
such as to be equal 34 9/10
pounds, on a square
foot; and this by proportion,
in the following
table, will be found
to answer to a velocity
of 23 1/4 miles per hour.</p><p>Mr. Leutmann improved upon Wolfius's anemometer,
by placing the sails horizontal, instead of vertical, which
are easier to move, and turn what way soever the wind
blows.</p><p>Mr. Benjamin Martin also (Plate III. fig. 5) improved
upon the same. He made the axis like the
fusee of a watch, having a cord winding upon it, with
two weights at the ends which make always a balance
to the force of the wind on the sails. See his Philos.
Britan.</p><p>And M. D'Ons-en-Bray invented a new anemometer,
which of itself expresses on paper, not only the several
winds that have blown during the space of 24 hours,
and at what hour each began and ended, but also the
different strength or velocity of each. See Mem. Acad.
Scienc. an. 1734. See also the article <hi rend="italics">Wind-Gauge.</hi></p></div1><div1 part="N" n="ANEMOSCOPE" org="uniform" sample="complete" type="entry"><head>ANEMOSCOPE</head><p>, is sometimes used to denote a
machine invented to foretell the changes of the wind,
or weather; and sometimes for an instrument shewing
by an index what the present direction of the wind is.
Of this latter sort, it seems, was that used by the ancients,
and described by Vitruvius; and we have many
of them at present in large or public buildings, where
an index withinside a room or hall, points to the name
of the quarter from whence the wind blows without;
which is simply effected by connecting an index to the
lower end of the spindle of a weather-cock.
<pb n="112"/><cb/></p><p>It has been observed that hygroscopes made of catgut,
or such like, prove very good anemoscopes; seldom
failing, by the turning of the index, to foretell the
shifting of the wind. See accounts of two different
anemoscopes; one by Mr. Pickering, vol. 43 Philos.
Trans. the other by Mr. B. Martin, vol. 2 of his Philos.
Britan.</p><p>Otto Gueric also gave the title anemoscope to a
machine invented by him to foretell the change of the
weather, as to rain and fair. It consisted of the small
wooden figure of a man, which rose and fell in a glass
tube, as the atmosphere was more or less heavy. Which
was only an application of the common barometer, as
shewn by M. Couriers in the <hi rend="italics">Acta Eruditorum</hi> for 1684.</p></div1><div1 part="N" n="ANGLE" org="uniform" sample="complete" type="entry"><head>ANGLE</head><p>, <hi rend="italics">Angulus,</hi> in <hi rend="italics">Geometry,</hi> the opening or mutual
inclination of two lines, or two planes, or three
planes, meeting in a point called the vertex or angular
point. Such as the angle formed by, or between,
the two lines AB and AC, at the vertex or angular
point A.&#x2014;Also the two lines AB and AC, are called
the <hi rend="italics">legs</hi> or the <hi rend="italics">sides</hi> of the angle.
<figure/></p><p>Angles are sometimes denoted, or named, by the
single letter placed at the angular point, as the angle A;
and sometimes by three letters, placing always that of
the vertex in the middle. The former method is
used when only one angle has the same vertex; and the
latter method it is necessary to use when several angles
have the same vertex, to distinguish them from one
another.</p><p>The measure of an angle, by which its quantity or
magnitude is expressed, is an arch, as DE described
from the centre A, with any radius at pleasure, and
contained between its legs AB and AC.&#x2014;Hence
angles are compared and distinguished by the ratio of
the arcs which subtend them, to the whole circumference
of the circle; or by the number of degrees
contained in the arc DE by which they are measured,
to 360, the number of degrees in the whole circumference
of the circle. And thus an angle is said to be
of so many degrees, viz, as are contained in the arc DE.</p><p>Hence it matters not, with what radius the arc is
described, by which an angle is measured, when great
or small, as AD, or A<hi rend="italics">d,</hi> or any other: for the arcs
DE, <hi rend="italics">de,</hi> being similar, have the same ratio as their
respective radii or circumferences, and therefore they
contain the same number of degrees.&#x2014;Hence it follows,
that the quantity or magnitude of the angle remains
still the same, though the legs be ever so much increased
or diminished.&#x2014;And thus, in similar figures,
the like or corresponding angles are equal.</p><p>The <hi rend="italics">taking</hi> or <hi rend="italics">measuring</hi> of <hi rend="italics">angles,</hi> is an operation of
great use and extent in surveying, navigation, geography,
astronomy, &amp;c. And the instruments chiefly
used for this purpose, are quadrants, sextants, octants,
<cb/>
theodolites, circumferentors, &amp;c. Mr. Hadley invented
an excellent instrument for taking the <hi rend="italics">larger</hi> sort of
angles, where much accuracy is required, or where
the motion of the object, or any circumstance causing
an unsteadiness in the common instruments, renders the
observations difficult, or uncertain. And Mr. Dollond
contrived an instrument for measuring <hi rend="italics">small</hi> angles.
See <hi rend="italics">Hadley's Quadrant, Micrometer,</hi> and the Philos.
Trans. Numbers 420, 425, and vol. 48.
<hi rend="center"><hi rend="italics">To measure the Quantity of an Angle.</hi></hi></p><p>1. <hi rend="italics">On paper.</hi> Apply the centre of a protractor to
the vertex A of the angle, so that the radius may coincide
with one leg, as AB; then the degree on the
arch that is cut by the other leg AC, will give the
measure of the angle required.</p><p>Or thus, by a line of chords. Take off the chord of
60 with a pair of compasses; and with that radius, from
the centre A, describe an arc as DE. Then take this
arc DE between the compasses, and apply the extent to
the scale of chords, which will give the degrees in the
angle as before.</p><p>M. De Lagny gave, in several memoirs of the Royal
Academy of Sciences, a new method of measuring
angles, which he called <hi rend="italics">Goniometry.</hi> The method consists
in measuring, with a pair of compasses, the are
which subtends the proposed angle, not by applying its
extent to a pre-constructed scale, like chords, but in the
following manner: From the angular point as a centre,
with a pretty large radius, describe a circle, producing
one leg of the angle backwards to cut off a semicircle;
then search out what part of the semicircle the arc is
which measures the given angle, in this manner; viz,
take the extent of this arc with a very fine pair of compasses,
and apply it several times to the arc of the semicircle,
to find how often it is contained, with a small
part remaining over; in the same manner take the extent
of this small part, and apply it to the first arc, to
find how often it is contained in it; and what remains
this 2d time, apply in like manner to the first remainder;
then the 3d remainder apply to the 2d, and so on,
always counting how often the last remainder is contained
in the next foregoing, till nothing remain, or till
the remainder is insensible, and too small to be measured:
Then, beginning at the last, and returning backwards,
make a series of fractions of which the numerators are
always 1, and the denominators are the number of times
each remainder is contained in its next remainder, with
the fractional part more, as derived from the following
remainder; then the last fraction, thus obtained, will
shew what part the given angle is of 180&#xB0; or the semicircle;
and being turned into degrees &amp;c, will be the
measure of the angle, and nearer, it is asserted, than it
can be obtained by any other means; whether it be
measuring, or calculating by trigonometrical tables.&#x2014;
Thus, if it be required to measure the angle GFH:
With a large radius describe the semicircle GHI,
meeting the leg FG produced in I; then take the extent
of the arc GH in the compasses, and applying it
from G upon the semicircle, suppose it contains 4 times
to the point 4, and the part 4 I over; take 4 I and
apply it from H to 1, so that HG contains 4 I once,
and 1 G over; also apply this remainder to the former
4 I, and it contains 5 times, from 4 to 5, and 5 I over;
<pb n="113"/><cb/>
and lastly the remainder 5 I is just two times contained
in the former remainder 1 G or 12, without
any remaindor. Here then, the series of quotients, or
numbers of times contained, are 4, 1, 5, 2; therefore,
beginning at the last, the first fraction is 1/2, or the last
remainder is half the preceding one; and the 2d fraction
is 1/(5 1/2) or 2/11; the 3d is 1/(1 2/11) or 11/13; and the fourth is
1/(4 11/13) or 13/63; that is, the arc GH is 13/63 of a semicircle,
or the angle GFH is 13/63 of two right angles, or of 180&#xB0;,
which is equivalent to 37 1/7 degrees, or 37&#xB0; 8&#x2032; 34&#x2033; 2/7.</p><p>2. <hi rend="italics">On the ground.</hi> Place a surveying instrument with
its centre over the angular point to be measured, turning
the instrument about till 0, the beginning of its arch,
fall in the line or direction of one leg of the angle;
then turn the index about to the direction of the other
leg, and it will cut off from the arch the degrees answering
to the given angle.</p><p>To plot or lay down any given angle, either on paper
or on the ground, may be performed in the same manner;
and the method is farther explained under the
articles <hi rend="smallcaps">Plotting</hi> and <hi rend="smallcaps">Protracting</hi>, and under the
names of the several instruments.</p><p><hi rend="italics">To bisect a given angle,</hi> as suppose the angle LKM.
From the centre K, with any radius, describe the arc
LM; then with the centres L and M, describe two
arcs intersecting in N; and draw the line KN, which
will bisect the given angle LKM, dividing it into the
two equal angles LKN, MKN.
<figure/></p><p><hi rend="italics">To trisect an angle,</hi> see <hi rend="smallcaps">Trisection.</hi></p><p>Pappus, in his Mathematical Collections, book 4,
treats of angular sections, but particularly and more
largely, of trisections. He also treats of any section in
general, in the 36th and following propositions.</p><p><hi rend="smallcaps">Angles</hi> are of various kinds and denominations.
With regard to the form of their legs, they are divided
into <hi rend="italics">rectilinear, curvilinear,</hi> and <hi rend="italics">mixed.</hi></p><p><hi rend="italics">Rectilinear,</hi> or <hi rend="italics">right-lined</hi> <hi rend="smallcaps">Angle</hi>, is that whose legs
are both right lines; as the foregoing angle CAB.</p><p><hi rend="italics">Curvilinear</hi> <hi rend="smallcaps">Angle</hi>, is that whose legs are both of
them curves.</p><p><hi rend="italics">Mixt,</hi> or <hi rend="italics">mixtilinear</hi> <hi rend="smallcaps">Angle</hi>, is that of which one
leg is a right line, and the other a curve.</p><p>With regard to their <hi rend="italics">magnitude,</hi> angles are again divided
into <hi rend="italics">right</hi> and <hi rend="italics">oblique, acute</hi> and <hi rend="italics">obtuse.</hi></p><p><hi rend="italics">Right</hi> <hi rend="smallcaps">Angle</hi>, is that which is formed by one line
perpendicular to another; or that which is subtended
by a quadrant of a circle. As the angle BAC.&#x2014;Therefore
the measure of a right angle is a quadrant of a
circle, or 90&#xB0;; and consequently all right angles are
equal to each other.</p><p><hi rend="italics">Oblique</hi> <hi rend="smallcaps">Angle</hi>, is a common name for any angle that
is not a right one; and it is either <hi rend="italics">acute</hi> or <hi rend="italics">obtuse.</hi>
<cb/></p><p><hi rend="italics">Acute</hi> <hi rend="smallcaps">Angle</hi>, is that which is less than a righ&lt;*&gt;
angle, or less than 90 degrees; as the angle BAD
And</p><p><hi rend="italics">Obtuse</hi> <hi rend="smallcaps">Angle</hi>, is greater than a right angle, or whos&lt;*&gt;
measure exceeds 90 degrees; as the angle BAE.</p><p>With regard to their situation in respect of eac&lt;*&gt;
other, angles are distinguished into <hi rend="italics">contiguous, adjacen&lt;*&gt;
vertical, opposite,</hi> and <hi rend="italics">alternate.</hi></p><p><hi rend="italics">Contiguous</hi> <hi rend="smallcaps">Angles</hi>, are such as have the same vertex&lt;*&gt;
and one leg common to both. As the angles BAD
CAD, which have AD common.</p><p><hi rend="italics">Adjacent</hi> <hi rend="smallcaps">Angles</hi>, are those of which a leg of th&lt;*&gt;
one produced forms a leg of the other: as the angle&lt;*&gt;
GFH and IFH, which have the legs IF and FG in &lt;*&gt;
straight line.&#x2014;Hence adjacent angles are supplement&lt;*&gt;
to each other, making together 180 degrees. An&lt;*&gt;
therefore if one of these be given, the other will b&lt;*&gt;
known by subtracting the given one from 180 degrees.
Which property is useful in surveying, to find the
quantity of an inaccessible angle; viz, measure its adjacent
accessible one, and subtract this from 180 degrees.
<figure/></p><p><hi rend="italics">Vertical</hi> or <hi rend="italics">opposite</hi> <hi rend="smallcaps">Angles</hi>, are such as have their
legs mutually continuations of each other; as the two
angles <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> or <hi rend="italics">c</hi> and <hi rend="italics">d.</hi>&#x2014;The property of these is,
that the vertical or opposite angles are always equal to
each other, viz, [angle] <hi rend="italics">a</hi> = [angle] <hi rend="italics">b,</hi> and [angle] <hi rend="italics">c</hi> = [angle] <hi rend="italics">d.</hi> And
hence the quantity of an inaccessible angle of a field,
&amp;c, may be found, by measuring its accessible opposite
angle.</p><p><hi rend="italics">Alternate</hi> <hi rend="smallcaps">Angles</hi>, are those made on the opposite
sides of a line cutting two parallel lines; so, the angles
<hi rend="italics">e</hi> and <hi rend="italics">f,</hi> or <hi rend="italics">g</hi> and <hi rend="italics">h,</hi> are alternates. And these are always
equal to each other; viz, the [angle] <hi rend="italics">c</hi> = [angle] <hi rend="italics">f,</hi> or [angle] <hi rend="italics">g</hi>
= [angle] <hi rend="italics">h.</hi>
<figure/></p><p><hi rend="italics">External</hi> <hi rend="smallcaps">Angles</hi>, are the angles of a figure made
without it, by producing its sides outwards; as the
angles <hi rend="italics">i, k, l, m.</hi> All the external angles of any rightlined
figure, taken together, are equal to 4 right
angles; and the external angle of a triangle is equal to
both the internal opposite ones taken together; also
any external angle of a trapezium inseribed in a circle,
is equal to the internal opposite angle.</p><p><hi rend="italics">Internal</hi> <hi rend="smallcaps">Angles</hi>, are the angles within any figure,
made by the sides of it; as the angles <hi rend="italics">n, o, p, q.</hi>&#x2014;In
any right-lined figure, an internal angle as <hi rend="italics">n,</hi> and its adjacent
external angle <hi rend="italics">k,</hi> together make two right angles,
or 180 degrees; and all the internal angles <hi rend="italics">n, o, p, q,</hi>
<pb n="114"/><cb/>
taken together, make twice as many right angles, wanting
4 right angles; also any two opposite internal
angles of a trapezium inscribed in a circle, taken together,
make two right angles, or 180 degrees.</p><p><hi rend="italics">Homologous,</hi> or <hi rend="italics">like</hi> <hi rend="smallcaps">Angles</hi>, are such angles in two
figures, as retain the same order from the first, in both
figures.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">out of the centre,</hi> as G, is one whose vertex
is not in the centre of the circle.&#x2014;And its measure is
half the sum (<hi rend="italics">a</hi>+<hi rend="italics">b</hi>)/2 of the arcs intercepted by its legs
when it is within the circle, or half the difference (<hi rend="italics">a</hi>-<hi rend="italics">b</hi>)/2
when it is without.
<figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">at the centre,</hi> is an angle whose vertex is in
the centre; as the angle AFC, formed by two radii AF,
FC, and measured by the arc ADC.&#x2014;An angle at the
centre, as AFC, is always double of the angle ABC
at the circumference, standing upon the same arc ADC;
and all angles at the centre are equal that stand upon the
same or equal arcs: also all angles at the centre, are
proportional to the arcs they stand upon; and so also
are all angles at the circumference.
<figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">at the circumference,</hi> is an angle whose vertex
is somewhere in the circumference of a circle; as
the angle ABC.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">in a segment,</hi> is an angle whose legs meet the
extremities of the base of the segment, and its vertex
is anywhere in its arch; as the angle B is in the segment
ABC, or standing upon the supplemental segment
ADC; and is comprehended between two chords
AB and BC.&#x2014;An angle at the circumference is measured
by half the arc ADC upon which it stands; and
all the angles ABC, AEC, in the same segment, are
equal to each other.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">in a semicircle</hi> is an angle at the circumference
contained in a semicircle, or standing upon a
semicircle, or on a diameter.&#x2014;An angle in a semicircle,
is always a right angle; in a greater segment, the
angle is less, and in a less segment the angle is greater
than a right angle.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of a segment,</hi> is that made by a chord with a
tangent, at the point of contact. So IHK is the
<cb/>
angle of the less segment IMH, and IHL, the angle of
the greater segment INH.&#x2014;And the measure of each
of these angles, is half the alternate or supplemental
segment, or equal to the angle in it; viz, the [angle] IHK
= [angle] INH, and the [angle] IHL = [angle] IMH.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of a semicircle,</hi> is the angle which the diameter
of a circle makes with the circumference. And
Euclid demonstrates that this is less than a right angle,
but greater than any acute angle.
<figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of contact,</hi> is that made by a curve line and
a tangent to it, at the point of contact; as the angle
IHK. It is proved by Euclid, that the angle of contact
between a right line and a circle, is less than any
right-lined angle whatever; though it does not therefore
follow that it is of no magnitude or quantity.
This has been the subject of great disputes amongst
geometricians, in which Peletarius, Clavius, Taquet,
Wallis, &amp;c, bore a considerable share; Peletarius and
Wallis contending that it is no angle at all, against
Clavius, who rightly asserts that it is not absolutely
nothing in itself, but only of no magnitude in comparison
with a right-lined angle, being a quantity of a different
kind or nature; like as a line in respect to a surface,
or a surface in respect to a solid, &amp;c. And since
his time, it has been proved by Sir I. Newton, and
others, that angles of contact can be compared to each
other, though not to right-lined angles, and what are
the proportions which they bear to each other. Thus,
the circular angles of contact IHK, IHL, are to each
other in the reciprocal subduplicate ratio of the diameters
HM, HN. And hence the circular angle of contact
may be divided, by describing intermediate circles, into
any number of parts, and in any proportion. And if,
instead of circles, the curves be parabolas, and the
point of contact H the common vertex of their axes;
the angles of contact would then be reciprocally in the
subduplicate ratio of their parameters. But in such
elliptical and hyperbolical angles of contact, these will
be reciprocally in the subduplicate of the ratio compounded
of the ratios of the parameters, and the
transverse axes. Moreover, if TOQ be a common parabola,
to the axis OP, and tangent VOW, and whose
equation is , or <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">2</hi>, where <hi rend="italics">x</hi> is the absciss
OP, and <hi rend="italics">y</hi> the ordinate PQ, the parameter being 1&lt;*&gt;
and if OR, OS, &amp;c, be other parabolas to the same
axis, tangent, and parameter, their ordinate <hi rend="italics">y</hi> being PR,
or PS, &amp;c, and their equations <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">4</hi>, <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">5</hi>,
&amp;c: then the series of angles of contact will be in succession
infinitely greater than each other, viz, the angle
of contact WOQ infinitely greater than WOR, and
this infinitely greater than WOS, and so on infinitely.</p><p>And farther, between the angles of contact of any
two of this kind, may other angles of contact be found
<hi rend="italics">ad infinitum,</hi> which shall infinitely exceed each other,
<pb n="115"/><cb/>
and yet the greatest of them be infinitely less than the
smallest right-lined angle. So also <hi rend="italics">x</hi><hi rend="sup">2</hi>=<hi rend="italics">y</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">3</hi>=<hi rend="italics">y</hi><hi rend="sup">4</hi>,
<hi rend="italics">x</hi><hi rend="sup">4</hi>=<hi rend="italics">y</hi><hi rend="sup">5</hi>, &amp;c, denote a series of curves, of which every
succeeding one makes an angle with its tangent, infinitely
greater than the preceding one; and the least
of these, viz, that whose equation is <hi rend="italics">x</hi><hi rend="sup">2</hi>=<hi rend="italics">y</hi><hi rend="sup">3</hi>, or the
semicubical parabola, is infinitely greater than any circular
angle of contact.</p><p><hi rend="smallcaps">Angles</hi> are again divided into <hi rend="italics">plane, spherical,</hi> and <hi rend="italics">solid.</hi></p><p><hi rend="italics">Plane</hi> <hi rend="smallcaps">Angles</hi>, are all those above treated of; which
are defined by the inclination of two lines in a plane,
meeting in a point.</p><p><hi rend="italics">Spherical</hi> <hi rend="smallcaps">Angle</hi>, is an angle formed on the surface
of a sphere by the intersection of two great circles; or,
it is the inclination of the planes of the two great circles.</p><p>The measure of a spherical angle, is the arc of a great
circle of the sphere, intercepted between the two planes
which form the angle, and which cuts the said planes at
right angles. For their properties, &amp;c, see <hi rend="smallcaps">Sphere,
Spherical</hi>, and <hi rend="smallcaps">Spherical Trigonometry.</hi></p><p><hi rend="italics">Solid</hi> <hi rend="smallcaps">Angle</hi>, is the mutual inclination of more than
two planes, or plane angles, meeting in a point, and
not contained in the same plane; like the angles or
corners of solid bodies. For their measure, properties,
&amp;c, see <hi rend="smallcaps">Solid</hi> <hi rend="italics">Angle.</hi></p><p><hi rend="italics">Angles</hi> of other less usual kinds and denominations,
are also to be found in some books of Geometry. As,</p><p><hi rend="italics">Horned</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus cornutus,</hi> that which is
made by a right line, whether a tangent or secant, with
the circumserence of a circle.</p><p><hi rend="italics">Lunular</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus lunularis,</hi> is that which is
formed by the intersection of two curve lines, the one
concave, and the other convex.</p><p><hi rend="italics">Cissoid</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus cissoides,</hi> the inner angle made
by two spherical convex lines intersecting each other.</p><p><hi rend="italics">Sistroid</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus sistroides,</hi> is that which is in
form of a sistrum.</p><p><hi rend="italics">Pelecoid</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus pelecoides,</hi> is that in form of
a hatchet.</p><div2 part="N" n="Angle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angle</hi></head><p>, in <hi rend="italics">Trigonometry.</hi> See <hi rend="smallcaps">Triangle</hi>, T<hi rend="smallcaps">RICONOMETRY,
Sine, Tangent</hi>, <hi rend="italics">&amp;c.</hi></p></div2><div2 part="N" n="Angle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angle</hi>, in <hi rend="italics">Mechanics.&#x2014;Angle of Direction</hi></head><p>, is that
which is comprehended between the lines of direction
of two conspiring forces.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of Elevation,</hi> is that which is comprehended
between the line of direction and any plane upon which
the projection is made, whether horizontal or oblique.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of Incidence,</hi> is that made by the line of direction
of an impinging body, at the point of impact.
As the angle ABC.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of Reflection,</hi> is that made by the line of direction
of the reflected body, at the point of impact.
As the angle DBE.</p><p>Instead of the angles of incidence and reflection being
estimated from the plane on which the body impinges,
sometimes the complements of these are understood,
viz, as estimated from a perpendicular to the reflecting
plane; as the two angles ABF and DBF.
<figure/>
<cb/></p><p><hi rend="smallcaps">Angle</hi> in <hi rend="italics">Optics.&#x2014;Visual</hi> or <hi rend="italics">Optic Angle,</hi> is the
angle included between the two rays drawn from the
two extreme points of an object to the centre of the
pupil of the eye: as the angle HGI. The apparent
magnitude of objects is greater or less, according to the
angle under which they appear.&#x2014;Objects seen under
the same or an equal angle, always appear equal&#x2014;.The
least <hi rend="italics">visible</hi> angle, or least angle under which a body can
be seen, according to Dr. Hook, is one minute; but
Dr. Jurin shews, that at the time of his debate with
Hevelius on this subject, the latter could probably discover
a single star under so small an angle as 20&#x2033;. But
bodies are visible under smaller angles as they are more
bright or luminous. Dr. Jurin states the grounds of
this controversy, and discusses the question at large, in
his Essay upon distinct and indistinct Vision, published
in Smith's Optics, pa. 148, <hi rend="italics">&amp; seq.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the interval,</hi> of two places, is the angle
subtended by two lines directed from the eye to those
places.</p><p><hi rend="smallcaps">Angle</hi> of <hi rend="italics">incidence,</hi> or <hi rend="italics">reflection,</hi> or <hi rend="italics">refraction,</hi> &amp;c.
See the respective words <hi rend="smallcaps">Incidence, Reflection,
Refraction</hi>, &amp;c.</p><p><hi rend="smallcaps">Angle</hi> in <hi rend="italics">Astronomy.&#x2014;Angle of Commutation.</hi> See
<hi rend="smallcaps">Commutation.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of elongation,</hi> or <hi rend="italics">Angle at the Earth.</hi> See
<hi rend="smallcaps">Elongation.</hi></p><p><hi rend="italics">Parallactic</hi> <hi rend="smallcaps">Angle</hi>, or the <hi rend="italics">parallax,</hi> is the angle
made at the centre of a star, the sun, &amp;c, by two lines
drawn, the one to the centre of the earth, and the other
to its surface. See <hi rend="smallcaps">Parallactic</hi>, and <hi rend="smallcaps">Parallai.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the position of the sun, of the sun's apparent
semi-diameter,</hi> &amp;c. See the respective words.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">at the sun,</hi> is the angle under which the
distance of a planet from the ecliptic, is seen from the
sun.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the East.</hi> See <hi rend="smallcaps">Nonagesimal.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of obliquity,</hi> of the ecliptic, or the angle of
inclination of the axis of the earth, to the axis of the
ecliptic, is now nearly 23&#xB0; 28&#x2032;. See <hi rend="smallcaps">Obliquity</hi>, and
<hi rend="smallcaps">Ecliptic.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of longitude,</hi> is the angle which the circle of
a star's longitude makes with the meridian, at the pole
of the ecliptic.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of right ascension,</hi> is the angle which the
circle of a star's right ascension makes with the meridian
at the pole of the equator.</p><p><hi rend="smallcaps">Angle</hi> in <hi rend="italics">Navigation.</hi> <hi rend="smallcaps">Angle</hi> <hi rend="italics">of the rhumb,</hi> or
<hi rend="italics">loxodromic angle.</hi> See <hi rend="smallcaps">Rhumb</hi> and <hi rend="smallcaps">Loxodromic.</hi></p></div2><div2 part="N" n="Angles" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angles</hi>, in <hi rend="italics">Fortification</hi></head><p>, are understood of those
formed by the several lines used in fortifying, or making
a place defensible.</p><p>These are of two sorts; <hi rend="italics">real</hi> and <hi rend="italics">imaginary.&#x2014;Real
angles</hi> are those which actually exist and appear in the
works. Such as the <hi rend="italics">flanked angle,</hi> the <hi rend="italics">angle of the epaule,
angle of the flank,</hi> and the <hi rend="italics">re-entering angle of the counterscarp.</hi>
Imaginary, or <hi rend="italics">occult angles,</hi> are those which
are only subservient to the construction, and which
exist no more after the fortification is drawn. Such as
the <hi rend="italics">angle of the centre, angle of the polygan, flanking angle,
sallant angle of the counterscarp,</hi> &amp;c.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of,</hi> or <hi rend="italics">at, the centre,</hi> is the angle formed at the
centre of the polygon, by two radii drawn from the
centre to two adjacent angles, and subtended by a side
<pb n="116"/><cb/>
of it, as the angle ACB. This is found by dividing
360 degrees by the number of sides in the regular polygon.
<figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the Polygon,</hi> is the angle intercepted between
two sides of the polygon; as DAB, or ABE.
This is the supplement of the angle at the centre, and
is therefore found by subtracting the angle C from
380 degrees.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the Triangle,</hi> is half the angle of the polygon;
as CAB or CBA; and is therefore half the
supplement of the angle C at the centre.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the Bastion,</hi> is the angle FAG made by
the two faces of the bastion. And is otherwise called
the flanked angle.</p><p><hi rend="italics">Diminisbed</hi> <hi rend="smallcaps">Angle</hi>, is the angle BAG made by the
meeting of the exterior side of the polygon with the
face AG of the bastion.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the curtin,</hi> or <hi rend="italics">of the flank,</hi> is the angle
GHI made between the curtin and the flank.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the epaule,</hi> or <hi rend="italics">shoulder,</hi> is the angle AGH
made by the flank and the face of the bastion.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the tenaille,</hi> or <hi rend="italics">exterior flanking angle,</hi> is
the angle AKB made by the two rasant lines of defence,
or the faces of two bastions produced.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the counterscarp,</hi> is the angle made by the
two sides of the counterscarp, meeting before the middle
of the curtin.</p></div2><div2 part="N" n="Angle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angle</hi></head><p>, <hi rend="italics">flanking inward,</hi> is the angle made by the
flanking line with the curtin.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">forming the flank,</hi> is that consisting of one
flank and one demigorge.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">forming the face,</hi> is that composed of one
flank and one face.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the moat,</hi> is that made before the curtin,
where it is intersected.</p><p><hi rend="italics">Re-entering,</hi> or <hi rend="italics">re-entrant</hi> <hi rend="smallcaps">Angle</hi>, is that whose
vertex is turned inwards, towards the place; as H
or I.</p><p><hi rend="italics">Saliant,</hi> or <hi rend="italics">sortant</hi> <hi rend="smallcaps">Angle</hi>, is that turned outwards,
advancing its point towards the field; as A or G.</p><p><hi rend="italics">Dead</hi> <hi rend="smallcaps">Angle</hi>, is a re-entering angle, which is not
flanked or defended.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of a wall,</hi> in <hi rend="italics">Architecture,</hi> is the point or
corner where the two sides or faces of a wall meet.</p></div2><div2 part="N" n="Angles" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angles</hi>, in <hi rend="italics">Astrology</hi></head><p>, denote certain houses of a
figure, or scheme of the heavens. So the horoscope of
the first house, is termed the <hi rend="italics">angle of the east.</hi></p><p>ANGUINEAL <hi rend="italics">Hyperbola,</hi> a name given by Sir
<cb/>
I. Newton to four of his curves of the second order,
viz, species 33, 34, 35, 36, expressed by the equation
; being hyperbolas
of a serpentine figure. See <hi rend="smallcaps">Curves.</hi></p></div2></div1><div1 part="N" n="ANGULAR" org="uniform" sample="complete" type="entry"><head>ANGULAR</head><p>, something relating to, or that hath
angles.</p><p>At a distance, angular bodies appear round; the angles
and small inequalities disappearing at a much less
distance than the bulk of the body.</p><p>ANGULAR <hi rend="italics">Motion,</hi> is the motion of a body which
moves circularly about a point; or the variation in the
angle described by a line, or radius, connecting a body
with the centre about which it moves.&#x2014;Thus, a pendulum
has an angular motion about its centre of motion;
and the planets have an angular motion about
the sun.&#x2014;Two moveable points M
and O, of which the one describes
<figure/>
the arc MN, and the other the arc
OP, in the same time, have an equal,
or the same angularmotion, although
the real motion of the point O be
much greater than that of the point
M, viz, as the arc OP is greater than
the arc MN. The angular motions of revolving bodies,
as of the planets about the sun, are reciproeally proportional
to their periodic times. And they are also, as
their real or absolute motions directly, and as their radii
of motion inverfely.</p><p><hi rend="smallcaps">Angular</hi> <hi rend="italics">motion</hi> is also a kind of compound motion
composed of a circular and a rectilinear motion; like
the wheel of a coach, or other vehicle.</p><p>ANIMATED <hi rend="italics">needle,</hi> a needle touched with a magnet
or load-stone.</p></div1><div1 part="N" n="ANNUAL" org="uniform" sample="complete" type="entry"><head>ANNUAL</head><p>, in <hi rend="italics">Astronomy,</hi> something that returns
every year, or which terminates with the year.</p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">motion of the earth.</hi> See <hi rend="smallcaps">Earth.</hi></p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">argument of langitude.</hi> See <hi rend="smallcaps">Argument.</hi></p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">epacts.</hi> See <hi rend="smallcaps">Epact.</hi></p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">equation</hi> of the mean motion of the sun and
moon, and of the moon's apogee and nodes. See E<hi rend="smallcaps">QUATION.</hi></p></div1><div1 part="N" n="ANNUITIES" org="uniform" sample="complete" type="entry"><head>ANNUITIES</head><p>, a term for any periodical income,
arising from money lent, or from houses, lands, salaries,
pensions, &amp;c; payable from time to time; either annually,
or at other intervals of time.</p><p>Annuities may be divided into such as are <hi rend="italics">certain,</hi>
and such as depend on some <hi rend="italics">contingency,</hi> as the continuance
of a life, &amp;c.</p><p>Annuities are also divided into annuities in <hi rend="italics">possession,</hi>
and annuities in <hi rend="italics">reversion;</hi> the former meaning such as
have commenced; and the latter such as will not commence
till some particular event has happened, or till
some given period of time has elapsed.</p><p><hi rend="italics">Annuities</hi> may be farther considered as payable either
<hi rend="italics">yearly,</hi> or <hi rend="italics">half yearly,</hi> or <hi rend="italics">quarterly,</hi> &amp;c.</p><p>The <hi rend="italics">present value</hi> of an annuity, is that sum, which,
being improved at interest, will be sufficient to pay the
annuity.</p><p>The <hi rend="italics">present value</hi> of an <hi rend="italics">annuity certain,</hi> payable yearly,
is calculated in the following manner.&#x2014;Let the annuity
be 1, and let <hi rend="italics">r</hi> denote the amount of 1<hi rend="italics">l.</hi> for a year, or
1<hi rend="italics">l.</hi> increased by its interest for one year. Then, 1 being
the present value of the sum <hi rend="italics">r,</hi> and having to find the
present value of the sum 1, it will be, by proportion
<pb n="117"/><cb/>
thus, <hi rend="italics">r</hi> : 1 :: 1 : 1/<hi rend="italics">r</hi> the present value of 1<hi rend="italics">l.</hi> due a
year hence. In like manner 1/<hi rend="italics">r</hi><hi rend="sup">2</hi> will be the present value
of 1<hi rend="italics">l.</hi> due 2 years hence; for <hi rend="italics">r</hi> : 1 :: 1/<hi rend="italics">r</hi> : 1/<hi rend="italics">r</hi><hi rend="sup">2</hi>. In like
manner 1/<hi rend="italics">r</hi><hi rend="sup">3</hi>, 1/<hi rend="italics">r</hi><hi rend="sup">4</hi>, 1/<hi rend="italics">r</hi><hi rend="sup">5</hi>, &amp;c, will be the present value of
1<hi rend="italics">l.</hi> due at the end of 3, 4, 5, &amp;c, years respectively;
and in general, 1/<hi rend="italics">r</hi><hi rend="sup">n</hi> will be the value of 1<hi rend="italics">l.</hi> to be received
after the expiration of <hi rend="italics">n</hi> years. Consequently the sum
of all these, or (1/<hi rend="italics">r</hi>)+(1/<hi rend="italics">r</hi><hi rend="sup">2</hi>)+(1/<hi rend="italics">r</hi><hi rend="sup">3</hi>)+(1/<hi rend="italics">r</hi><hi rend="sup">4</hi>)+ &amp;c, contined to
<hi rend="italics">n</hi> terms, will be the present value of all the <hi rend="italics">n</hi> years annuities.
And the value of the perpetuity, is the sum of
the series continued <hi rend="italics">ad infinitum.</hi></p><p>But this series, it is evident, is a geometrical progression,
whose first term and common ratio are each 1/<hi rend="italics">r,</hi>
and the number of its terms <hi rend="italics">n</hi>; and therefore the sum <hi rend="italics">s</hi>
of all the terms, or the present value of all the annual
payments, will be .</p><p>When the annuity is a perpetuity, it is plain that
the last term 1/<hi rend="italics">r</hi><hi rend="sup">n</hi> vanishes, and therefore (1/(<hi rend="italics">r</hi>-1))X(1/<hi rend="italics">r</hi><hi rend="sup">n</hi>) also
vanishes; and consequently the expression becomes
barely <hi rend="italics">s</hi>=1/(<hi rend="italics">r</hi>-1); that is, any annuity divided by its
interest for one year, is the value of the perpetuity.
So, if the rate of interest be 5 per cent; then (100/5)=20
is the value of the perpetuity at 5 per cent. Also 100/4
=25 is the value of the perpetuity at 4 per cent. And
100/3 = 33 1/3 is the value of the perpetuity at 3 per cent.
interest. And so on.</p><p>If the annuity is not to be entered on immediately,
but after a certain number of years, as <hi rend="italics">m</hi> years; then
the present value of the reversion is equal to the difference
between two present values, the one for the first
term of <hi rend="italics">m</hi> years, and the other for the end of the last
term <hi rend="italics">n</hi>: that is, equal to the difference between
.</p><p>Annuities certain differ in value, as they are made
payable <hi rend="italics">yearly, half-yearly,</hi> or <hi rend="italics">quarterly.</hi> And by proceeding
as above, using the interest or amount of a
half year, or a quarter, as those for the whole year were
<cb/>
used, the following set of theorems will arise; where &lt;*&gt;
denotes, as before, the amount of 1<hi rend="italics">l.</hi> and its interest for
a year, and <hi rend="italics">n</hi> the number of years, during which, any
annuity is to be paid; also P denotes the perpetnity
1/(<hi rend="italics">r</hi>-1), Y denotes (1/(<hi rend="italics">r</hi>-1))-(1/(<hi rend="italics">r</hi>-1))X(1/<hi rend="italics">r</hi><hi rend="sup">n</hi>) the value of
the annuity supposed payable yearly, H the value of the
same when it is payable half-yearly, and Q the value
when payable quarterly; or universally, M the value
when it is payable every <hi rend="italics">m</hi> part of a year.</p><p><hi rend="smallcaps">Theor.</hi> 1. .</p><p><hi rend="smallcaps">Theor.</hi> 2. .</p><p><hi rend="smallcaps">Theor.</hi> 3. .</p><p><hi rend="smallcaps">Theor.</hi> 4. 
<hi rend="center"><hi rend="italics">Example</hi> 1.</hi></p><p>Let the rate of interest be 4. per cent, and the term
5 years; and consequently <hi rend="italics">r</hi> = 1.04, <hi rend="italics">n</hi> = 5, P = 25;
also let <hi rend="italics">m</hi> = 12, or the interest payable monthly in
theorem 4: then the present value of such annuity of
1<hi rend="italics">l.</hi> a year, for 5 years, according as it is supposed payable
1<hi rend="italics">l.</hi> yearly, or (1/2)<hi rend="italics">l.</hi> every half year, or (1/4)<hi rend="italics">l.</hi> every quarter,
or (1/12)<hi rend="italics">l.</hi> every month or (1/12)th part of a year, will
be as follows:</p><p><hi rend="smallcaps">Example</hi> 2. Supposing the annuity to continue 25
years, the rate of interest and every thing else being as
before; then the values of the annuities for 25 years will
be</p><p><hi rend="smallcaps">Example</hi> 3. And if the term be 50 years, the values
will be</p><p><hi rend="smallcaps">Example</hi> 4. Also if the term be 100 years, the
values will be</p><p>Hence the difference in the value by making periods
of payments smaller, for any given term of years, is the
more as the intervals are smaller, or the periods more
frequent. The same difference is also variable, both as
the rate of interest varies, and also as the whole term
of years <hi rend="italics">n</hi> varies; and, for any given rate of interest, it
<pb n="118"/><cb/>
is evident that the difference, for any periods <hi rend="italics">m</hi> of payments,
first increases from nothing as the term <hi rend="italics">n</hi> increases,
when <hi rend="italics">n</hi> is 0, to some certain finite term or value
of <hi rend="italics">n,</hi> when the difference D is the greatest or a maximum;
and that afterwards, as <hi rend="italics">n</hi> increases more, that
difference will continually decrease to nothing again, and
vanish when <hi rend="italics">n</hi> is infinite: also the term or value of <hi rend="italics">n,</hi>
for the maximum of the difference, will be different according
to the periods of payment, or value of <hi rend="italics">m.</hi> And
the general value of <hi rend="italics">n,</hi> when the difference is a maximum
between the yearly payments and the payments of <hi rend="italics">m</hi>
times in a year, is expressed by this formula, viz,
, where <hi rend="italics">l.</hi> denotes the logarithm
of the quantity following it. Hence, taking
the different values of <hi rend="italics">m,</hi> viz, 2 for half years, 4 for
quarters, 12 for monthly payments, &amp;c, and substituting
in the general formula, the term or value of <hi rend="italics">n</hi> for each
case, when the difference in the present worths of the
annuities, will be as follows, reckoning interest at 4 per
cent, viz,
 for half-yearly payments,
 for quarterly payments,
 for monthly payments.</p><p><hi rend="italics">Annuities</hi> may also be considered as in arrears, or as
forborn, for any number of years; in which case each
payment is to be considered as a sum put out to interest
for the remainder of the term after the time it
becomes due. And as 1<hi rend="italics">l.</hi> due at the end of 1 year,
amounts to <hi rend="italics">r</hi> at the end of another year, and to <hi rend="italics">r</hi><hi rend="sup">2</hi> at
the end of the 3d year, and to <hi rend="italics">r</hi><hi rend="sup">3</hi> at the end of the 4th
year, and so on; therefore by adding always the last
year's annuity, or 1, to the amounts of all the former
years, the sum of all the annuities and their interests,
will be the sum of the following geometrical series,
1 + <hi rend="italics">r</hi> + <hi rend="italics">r</hi><hi rend="sup">2</hi> + <hi rend="italics">r</hi><hi rend="sup">3</hi> + <hi rend="italics">r</hi><hi rend="sup">4</hi> to <hi rend="italics">r</hi><hi rend="sup">n-x</hi>, continued
till the last term be <hi rend="italics">r</hi><hi rend="sup">n-x</hi>, or till the number of
terms be <hi rend="italics">n,</hi> the number of years the annuity is forborn.
But the sum of this geometrical progression is (<hi rend="italics">r</hi><hi rend="sup">n</hi>-1)/(<hi rend="italics">r</hi>-1),
<cb/>
which therefore is the amount of 1<hi rend="italics">l.</hi> annuity forborn for
<hi rend="italics">n</hi> years. And this quantity being multiplied by any
other annuity <hi rend="italics">a,</hi> instead of 1, will produce the amount
for that other annuity.</p><p>But the amounts of annuities, or their present values,
are easiest found by the two following tables of numbers
for the annuity of 1<hi rend="italics">l.</hi> ready computed from the foregoing
principles.
<table rend="border"><head><hi rend="smallcaps">Table</hi> I.</head><head><hi rend="italics">The Amount of an Annuity of</hi> 1<hi rend="italics">l. at Comp. Interest.</hi></head><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Yrs.</cell><cell cols="1" rows="1" rend="align=center" role="data">at 3 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">3 1/2 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 1/2 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">5 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">6 per
cent.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">2.03000</cell><cell cols="1" rows="1" rend="align=right" role="data">2.03500</cell><cell cols="1" rows="1" rend="align=right" role="data">2.04000</cell><cell cols="1" rows="1" rend="align=right" role="data">2.04500</cell><cell cols="1" rows="1" rend="align=right" role="data">2.05000</cell><cell cols="1" rows="1" rend="align=right" 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rend="align=right" role="data">75.40126</cell><cell cols="1" rows="1" rend="align=right" role="data">84.55028</cell><cell cols="1" rows="1" rend="align=right" role="data">95.02552</cell><cell cols="1" rows="1" rend="align=right" role="data">107.03032</cell><cell cols="1" rows="1" rend="align=right" role="data">120.79977</cell><cell cols="1" rows="1" rend="align=right" role="data">154.76197</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">78.66330</cell><cell cols="1" rows="1" rend="align=right" role="data">88.50954</cell><cell cols="1" rows="1" rend="align=right" role="data">99.82654</cell><cell cols="1" rows="1" rend="align=right" role="data">112.84669</cell><cell cols="1" rows="1" rend="align=right" role="data">127.83976</cell><cell cols="1" rows="1" rend="align=right" role="data">165.04768</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">82.02320</cell><cell cols="1" rows="1" rend="align=right" role="data">92.60737</cell><cell cols="1" rows="1" rend="align=right" role="data">104.81960</cell><cell cols="1" rows="1" rend="align=right" role="data">118.92479</cell><cell cols="1" rows="1" rend="align=right" role="data">135.23175</cell><cell cols="1" rows="1" rend="align=right" role="data">175.95054</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">85.48389</cell><cell cols="1" rows="1" rend="align=right" role="data">96.84863</cell><cell cols="1" rows="1" rend="align=right" role="data">110.01238</cell><cell cols="1" rows="1" rend="align=right" role="data">125.27640</cell><cell cols="1" rows="1" rend="align=right" role="data">142.99334</cell><cell cols="1" rows="1" rend="align=right" role="data">187.50758</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">89.04841</cell><cell cols="1" rows="1" rend="align=right" role="data">101.23833</cell><cell cols="1" rows="1" rend="align=right" role="data">115.41288</cell><cell cols="1" rows="1" rend="align=right" role="data">131.91384</cell><cell cols="1" rows="1" rend="align=right" role="data">151.14301</cell><cell cols="1" rows="1" rend="align=right" role="data">199.75803</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">92.71986</cell><cell cols="1" rows="1" rend="align=right" role="data">105.78167</cell><cell cols="1" rows="1" rend="align=right" role="data">121.02939</cell><cell cols="1" rows="1" rend="align=right" role="data">138.84997</cell><cell cols="1" rows="1" rend="align=right" role="data">159.70016</cell><cell cols="1" rows="1" rend="align=right" role="data">212.74351</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">96.50146</cell><cell cols="1" rows="1" rend="align=right" role="data">110.48403</cell><cell cols="1" rows="1" rend="align=right" role="data">126.87057</cell><cell cols="1" rows="1" rend="align=right" role="data">146.09821</cell><cell cols="1" rows="1" rend="align=right" role="data">168.68516</cell><cell cols="1" rows="1" rend="align=right" role="data">226.50812</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">100.39650</cell><cell cols="1" rows="1" rend="align=right" role="data">115.35097</cell><cell cols="1" rows="1" rend="align=right" role="data">132.94539</cell><cell cols="1" rows="1" rend="align=right" role="data">153.67263</cell><cell cols="1" rows="1" rend="align=right" role="data">178.11942</cell><cell cols="1" rows="1" rend="align=right" role="data">241.09861</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">104.40840</cell><cell cols="1" rows="1" rend="align=right" role="data">120.38826</cell><cell cols="1" rows="1" rend="align=right" role="data">139.26321</cell><cell cols="1" rows="1" rend="align=right" role="data">161.58790</cell><cell cols="1" rows="1" rend="align=right" role="data">188.02539</cell><cell cols="1" rows="1" rend="align=right" role="data">256.56453</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">108.54065</cell><cell cols="1" rows="1" rend="align=right" role="data">125.60185</cell><cell cols="1" rows="1" rend="align=right" role="data">145.83373</cell><cell cols="1" rows="1" rend="align=right" role="data">169.85936</cell><cell cols="1" rows="1" rend="align=right" role="data">198.42666</cell><cell cols="1" rows="1" rend="align=right" role="data">272.95840</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">112.79687</cell><cell cols="1" rows="1" rend="align=right" role="data">130.99791</cell><cell cols="1" rows="1" rend="align=right" role="data">152.66708</cell><cell cols="1" rows="1" rend="align=right" role="data">178.50303</cell><cell cols="1" rows="1" rend="align=right" role="data">209.34800</cell><cell cols="1" rows="1" rend="align=right" role="data">290.33590</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" rend="align=right" role="data">117.18077</cell><cell cols="1" rows="1" rend="align=right" role="data">136.58284</cell><cell cols="1" rows="1" rend="align=right" role="data">159.77377</cell><cell cols="1" rows="1" rend="align=right" role="data">187.53566</cell><cell cols="1" rows="1" rend="align=right" role="data">220.81540</cell><cell cols="1" rows="1" rend="align=right" role="data">308.75606</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">121.69620</cell><cell cols="1" rows="1" rend="align=right" role="data">142.36324</cell><cell cols="1" rows="1" rend="align=right" role="data">167.16472</cell><cell cols="1" rows="1" rend="align=right" role="data">196.97477</cell><cell cols="1" rows="1" rend="align=right" role="data">232.85617</cell><cell cols="1" rows="1" rend="align=right" role="data">328.28142</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">126.34708</cell><cell cols="1" rows="1" rend="align=right" role="data">148.34595</cell><cell cols="1" rows="1" rend="align=right" role="data">174.85131</cell><cell cols="1" rows="1" rend="align=right" role="data">206.83863</cell><cell cols="1" rows="1" rend="align=right" role="data">245.49897</cell><cell cols="1" rows="1" rend="align=right" role="data">348.97831</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">131.13750</cell><cell cols="1" rows="1" rend="align=right" role="data">154.53806</cell><cell cols="1" rows="1" rend="align=right" role="data">182.84536</cell><cell cols="1" rows="1" rend="align=right" role="data">217.14637</cell><cell cols="1" rows="1" rend="align=right" role="data">258.77392</cell><cell cols="1" rows="1" rend="align=right" role="data">370.91701</cell></row></table>
<pb n="119"/><cb/>
<table rend="border"><head><hi rend="smallcaps">Table</hi> II.</head><head><hi rend="italics">The present Value of an Annuity of</hi> 1<hi rend="italics">l.</hi></head><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Yrs.</cell><cell cols="1" rows="1" rend="align=center" role="data">at 3 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">3 1/2 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 1/2 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">5 per
cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">6 per
cent.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0.97087</cell><cell cols="1" rows="1" rend="align=right" role="data">0.96618</cell><cell cols="1" rows="1" rend="align=right" role="data">0.96154</cell><cell cols="1" rows="1" rend="align=right" role="data">0.95694</cell><cell cols="1" rows="1" rend="align=right" role="data">0.95238</cell><cell cols="1" rows="1" rend="align=right" role="data">0.94340</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">1.91347</cell><cell cols="1" rows="1" rend="align=right" role="data">1.89969</cell><cell cols="1" rows="1" rend="align=right" role="data">1.88610</cell><cell cols="1" rows="1" rend="align=right" role="data">1.87267</cell><cell cols="1" rows="1" rend="align=right" role="data">1.85941</cell><cell cols="1" rows="1" rend="align=right" role="data">1.83339</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">2.82861</cell><cell cols="1" rows="1" rend="align=right" role="data">2.80164</cell><cell cols="1" rows="1" rend="align=right" role="data">2.77509</cell><cell cols="1" rows="1" rend="align=right" role="data">2.74896</cell><cell cols="1" rows="1" rend="align=right" role="data">2.72325</cell><cell cols="1" rows="1" rend="align=right" role="data">2.67301</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">3.71710</cell><cell cols="1" rows="1" rend="align=right" role="data">3.67308</cell><cell cols="1" rows="1" rend="align=right" role="data">3.62990</cell><cell cols="1" rows="1" rend="align=right" role="data">3.58753</cell><cell cols="1" rows="1" rend="align=right" role="data">3.54595</cell><cell cols="1" rows="1" rend="align=right" role="data">3.46511</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">4.57971</cell><cell cols="1" rows="1" rend="align=right" role="data">4.51505</cell><cell cols="1" rows="1" rend="align=right" role="data">4.45182</cell><cell cols="1" rows="1" rend="align=right" role="data">4.38998</cell><cell cols="1" rows="1" rend="align=right" role="data">4.32948</cell><cell cols="1" rows="1" rend="align=right" role="data">4.21236</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5.41719</cell><cell cols="1" rows="1" rend="align=right" role="data">5.32855</cell><cell cols="1" rows="1" rend="align=right" role="data">5.24214</cell><cell cols="1" rows="1" rend="align=right" role="data">5.15787</cell><cell cols="1" rows="1" rend="align=right" role="data">5.07569</cell><cell cols="1" rows="1" rend="align=right" role="data">4.91732</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">6.23028</cell><cell cols="1" rows="1" rend="align=right" role="data">6.11454</cell><cell cols="1" rows="1" rend="align=right" role="data">6.00205</cell><cell cols="1" rows="1" rend="align=right" role="data">5.89270</cell><cell cols="1" rows="1" rend="align=right" role="data">5.78637</cell><cell cols="1" rows="1" rend="align=right" role="data">5.58238</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">7.01969</cell><cell cols="1" rows="1" rend="align=right" role="data">6.87396</cell><cell cols="1" rows="1" rend="align=right" role="data">6.73274</cell><cell cols="1" rows="1" rend="align=right" role="data">6.59589</cell><cell cols="1" rows="1" rend="align=right" role="data">6.46321</cell><cell cols="1" rows="1" rend="align=right" role="data">6.20979</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">7.78611</cell><cell cols="1" rows="1" rend="align=right" role="data">7.60769</cell><cell cols="1" rows="1" rend="align=right" role="data">7.43533</cell><cell cols="1" rows="1" rend="align=right" role="data">7.26879</cell><cell cols="1" rows="1" rend="align=right" role="data">7.10782</cell><cell cols="1" rows="1" rend="align=right" role="data">6.80169</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">8.53020</cell><cell cols="1" rows="1" rend="align=right" role="data">8.31661</cell><cell cols="1" rows="1" rend="align=right" role="data">8.11090</cell><cell cols="1" rows="1" rend="align=right" role="data">7.91272</cell><cell cols="1" rows="1" rend="align=right" role="data">7.72173</cell><cell cols="1" rows="1" rend="align=right" role="data">7.36009</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">9.25262</cell><cell cols="1" rows="1" rend="align=right" role="data">9.00155</cell><cell cols="1" rows="1" rend="align=right" role="data">8.76048</cell><cell cols="1" rows="1" rend="align=right" role="data">8.52892</cell><cell cols="1" rows="1" rend="align=right" role="data">8.30541</cell><cell cols="1" rows="1" rend="align=right" role="data">7.88687</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">9.95400</cell><cell cols="1" rows="1" rend="align=right" role="data">9.66333</cell><cell cols="1" rows="1" rend="align=right" role="data">9.38507</cell><cell cols="1" rows="1" rend="align=right" role="data">9.11858</cell><cell cols="1" rows="1" rend="align=right" role="data">8.86325</cell><cell cols="1" rows="1" rend="align=right" role="data">8.38384</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">10.63496</cell><cell cols="1" rows="1" rend="align=right" role="data">10.30274</cell><cell cols="1" rows="1" rend="align=right" role="data">9.98565</cell><cell cols="1" rows="1" rend="align=right" role="data">9.68285</cell><cell cols="1" rows="1" rend="align=right" role="data">9.39357</cell><cell cols="1" rows="1" rend="align=right" role="data">8.85268</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">11.29607</cell><cell cols="1" rows="1" rend="align=right" role="data">10.92052</cell><cell cols="1" rows="1" rend="align=right" role="data">10.56312</cell><cell cols="1" rows="1" rend="align=right" role="data">10.22283</cell><cell cols="1" rows="1" rend="align=right" role="data">9.89864</cell><cell cols="1" rows="1" rend="align=right" role="data">9.29498</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">11.93794</cell><cell cols="1" rows="1" rend="align=right" role="data">11.51741</cell><cell cols="1" rows="1" rend="align=right" role="data">11.11839</cell><cell cols="1" rows="1" rend="align=right" role="data">10.73955</cell><cell cols="1" rows="1" rend="align=right" role="data">10.37966</cell><cell cols="1" rows="1" rend="align=right" role="data">9.71225</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">12.56110</cell><cell cols="1" rows="1" rend="align=right" role="data">12.09412</cell><cell cols="1" rows="1" rend="align=right" role="data">11.65230</cell><cell cols="1" rows="1" rend="align=right" role="data">11.23402</cell><cell cols="1" rows="1" rend="align=right" role="data">10.83777</cell><cell cols="1" rows="1" rend="align=right" role="data">10.10590</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">13.16612</cell><cell cols="1" rows="1" rend="align=right" role="data">12.65132</cell><cell cols="1" rows="1" rend="align=right" role="data">12.16567</cell><cell cols="1" rows="1" rend="align=right" role="data">11.70719</cell><cell cols="1" rows="1" rend="align=right" role="data">11.27407</cell><cell cols="1" rows="1" rend="align=right" role="data">10.47726</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">13.75351</cell><cell cols="1" rows="1" rend="align=right" role="data">13.18968</cell><cell cols="1" rows="1" rend="align=right" role="data">12.65930</cell><cell cols="1" rows="1" rend="align=right" role="data">12.15999</cell><cell cols="1" rows="1" rend="align=right" role="data">11.68959</cell><cell cols="1" rows="1" rend="align=right" role="data">10.82760</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">14.32380</cell><cell cols="1" rows="1" rend="align=right" role="data">13.70984</cell><cell cols="1" rows="1" rend="align=right" role="data">13.13394</cell><cell cols="1" rows="1" rend="align=right" role="data">12.59329</cell><cell cols="1" rows="1" rend="align=right" role="data">12.08532</cell><cell cols="1" rows="1" rend="align=right" role="data">11.15812</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">14.87747</cell><cell cols="1" rows="1" rend="align=right" role="data">14.21240</cell><cell cols="1" rows="1" rend="align=right" role="data">13.59033</cell><cell cols="1" rows="1" rend="align=right" role="data">13.00794</cell><cell cols="1" rows="1" rend="align=right" role="data">12.46221</cell><cell cols="1" rows="1" rend="align=right" role="data">11.46992</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">15.41502</cell><cell cols="1" rows="1" rend="align=right" role="data">14.69797</cell><cell cols="1" rows="1" rend="align=right" role="data">14.02916</cell><cell cols="1" rows="1" rend="align=right" role="data">13.40472</cell><cell cols="1" rows="1" rend="align=right" role="data">12.82115</cell><cell cols="1" rows="1" rend="align=right" role="data">11.76408</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">15.93692</cell><cell cols="1" rows="1" rend="align=right" role="data">15.16712</cell><cell cols="1" rows="1" rend="align=right" role="data">14.45112</cell><cell cols="1" rows="1" rend="align=right" 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