, an Arithmetician. In this sense we find the word used by William of Malmesbury, in his History de Gestis Anglorum, 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.

, in Arithmetic, an ancient instrument used by most nations for casting up accounts, or performing arithmetical calculations: it is by some derived from the Greek a<*>ac, which signifies a cupboard or beaufet, perhaps from the similarity of the form of this instrument; and by others it is derived from the Phœnician abak, 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.

We find this instrument for computation in use, under some variations, with most nations, as the Greeks, Romans, Germans, French, Chinese, &c.

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- tions were easily made. Mahudel, in Hist. Acad. R. Inscr. t. 3. p. 390.

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.

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 SHWANPAN.

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, 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.

Besides the above instruments of computation, there have been several others invented by different persons; as Napier's rods or bones, deseribed in his Rabdologia, which see under the word Napier; also the Abacus Rhabdologicus, a variation of Napier's, which is described in the first vol. of Machines et Inventions approuvées par l'Academie Royale des Sciences. 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 Palpable Arithmetic: which see.

, a weight used in Persia for weighing pearls; and is an eighth part lighter than the European carat.

, a silver coin current in Perlia, deriving its name from Schaw Abbas II. King of Persia, and is worth near eighteen pence English money.

, or Abattis, from the French abattre, 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.

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.

So , by dividing the terms continually by 2.

And , by dividing by 2, 3, and 7.

Also , by dividing by 3 and by 2.

And , by dividing by 4 ax.

And , by dividing by a+x.

, of fractions, in Arithmetic and Algebra, the reducing them to lower terms.

, in Astronomy, an apparent motion of the celestial bodies, occasioned by the progressive motion of light, and the earth's annual motion in her orbit.

This effect may be explained and familiarized by the 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

M. de Maupertuis, in his Elements of Geography, gives also a familiar and ingenious idea of the aberration, in this manner: “It is thus,” says he, “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.” 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.

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.

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 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.

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 Voyage d'Uranibourg, 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″ 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.

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 g 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.

Flamsteed having observed the pole star with his mural quadrant, in 1689 and the following years, found that its declination was 40″ 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.

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.

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 g 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″ 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.

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″ more northerly than in June, and upwards of 39″ 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.

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 g 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 g 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 g 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.

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 maxima, 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 g 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.

These considerations determined him; and by the contrivance and direction of the same ingenious person, 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 Capella, the only star of the first magnitude that came so near his zenith.

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.

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 g Draconis changed its declination above twice as much as the before-mentioned small star that was nearly op- 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 maxima 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.

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.

He considered this matter in the following manner. He imagined CA to be a ray of

, in Algebra, 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.

So the equation , by putting p = a, q = ab, and r = abc, becomes

And the equation , by putting , and , becomes .

, Abscisse, or Abscissa, is a part or segment cut off a line, terminated at some certain point, by an ordinate to a curve; as AP or BP.

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.

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.

In the common parabola, each ordinate PQ has but

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 = PQ2; 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.

When the natures or properties of curves are expressed by algebraic equations, any general absciss, as AP, is commonly denoted by the letter x, and the ordinate PQ by the letter y; 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 d is the diameter AB; parabola - px = y2 , where p is the parameter; ellipse - t2 : c2 :: tx - x2 : y2, hyperbola t2 : c2 :: tx + x2 : y2, where t is the transverse, & c the conjugate axis.

, ABSIDES. See Apsis, Apsides.

ABSOLUTE Equation, in Aftronomy, 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. See Equation, and Optic Inequality.

Absolute Number, 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 homogeneum comparationis. So, of the equation , or , the absolute number, or known term, is 36.

Absolute Gravity, Motion, Space, Time, &c. See the respective substantives.

ABSTRACT Mathematics, 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, &c.

Abstract Number, 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.

, or Absurdum, 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 propositions are proved in this way, which mode of proof is called reductio ad absurdum.

ABUNDANT Number, in Arithmetic, 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.

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.

, a member of a society called an academy, instituted for the promotion of arts, sciences, or natural knowledge in general.

, an ancient sect of philosophers, who followed the doctrine of Socrates and Plato, as to the uncertainty of knowledge, and the incomprehensibility of truth.

Academic, 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 academici, academician or academic; whereas those who did the same since the restoration of learning, have assumed the denomination of Platonists.

We usually reckon three sects of academics; though some make five. The ancient academy was that of which Plato was the chief.

Arcessilas, one of Plato's successors, introducing some alterations into the philosophy of this sect, founded what they call the second academy.

The establishment of the third, called also the new academy, is attributed to Lacydes, or rather to Carneades.

Some authors add a fourth, founded by Philo; and a fifth, by Antiochus, called the Antiochan, which tempered the ancient academy with Stoicism.

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.

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.

, the same as Academician.

, Academia, in Antiquity, 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 and philosophical conference; which gave the name to the sect of Academics.

The house took its name, Academy, 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.

The academy was farther improved by Cimon, and adorned with fountains, trees, shady walks, &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.

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 Academies.

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.

Cicero too had a villa, or country retirement, near Puzzuoli, which he called by the same name, Academia. Here he used to entertain his philosophical friends; and here it was that he composed his Academical Questions, and his books De Naturâ Deorum.

, in Architecture, 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 Abacus.

ACCELERATED Motion, 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 accessions, in equal times, either increase or decrease, then the motion is unequably or variably accelerated.

Acceleration is directly opposite to retardation, which denotes a diminution of velocity.

Acceleration 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.

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.

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.

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.

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.

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.

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.

The Cartesians account for acceleration, by reiterated impulses of their materia subtilis, acting continually on falling bodies, and propelling them downwards: a conceit equally unintelligible and absurd with the former.

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.

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.

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.

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.

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 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.

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, &c, were first discovered and taught by the great Galileo, and that nearly in the following manner.

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

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 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, &c, for the times; the spaces run through during those times, will be as the areas or rectangles CF, DG, &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, &c; so that the degree of motion remains the same to the instant that a new acceleration takes place.

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,

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.

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 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.

Hence then, from the foregoing considerations are deduced the following general laws of uniformly accelerated motions, namely,

1st. That the velocities acquired, are constantly proportional to the times; in a double time a double velocity, &c.

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, &c, the spaces will be as 1, 4, 9, 16, 25, &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.

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, &c, 1, 4, 9, 16, 25, &c, which are the differences of the squares or whole spaces 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.

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.

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 a denote the velocity acquired at the end of the first part of time; then will 2a, 3a, 4a, &c, represent the velocities at the end of the 2d, 3d, 4th, &c, part of time, because the velocities are as the times; and for the same reason 1/2a, 3/2a, 5/2a, &c, will be the velocities at the middle point of the first, second, third, &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 1/2a, 3/2a, 5/2a, &c, for the spaces passed over in the successive parts of the time; that is, the space 1/2a in the first time, 3/2a in the second, 5/2a in the third, and so on: then add these spaces successively to one another, and we obtain 1/2a, 4/2a, 9/2a, 16/2a, &c, for the whole spaces described from the beginning of the motion to the end of the first, second, third, &c, portion of time; namely 1/2a space in one time, 4/2a in 2 times, 9/2a in 3 times, and so on: and it is evident that these spaces are as the numbers 1, 4, 9, 16, &c, which are as the squares of the times.

And from this mode of demonstration, all the properties above mentioned evidently flow: such as that the whole spaces 1/2a, 4/2a, 9/2a, &c, are as the squares of the times 1, 2, 3, &c, that the separate spaces 1/2a, 3/2a, 5/2a, &c, 1, 3, 5, &c, described in the successive times, are as the odd numbers and that the velocity a acquired in any time 1, is double the space 1/2a described in the same time.

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. “Ppoposition 1.

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.

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. “Proposition II.

If a point begins to move in the direction of a straight line, and continues to move in the same di- 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.

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.

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

And hence the other laws of the spaces, before<*> mentioned, easily follow.

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 Abdfh, and AC a tangent

From the properties above-demonstrated, are derived the following practical formulas or theorems for use. Namely, if g denote the space passed over in the first second of time, by a body urged by any constant force, denoted by 1, and t denote the time or number of seconds in which the body passes over any other space s, and v the velocity acquired at the end of that time; then from the foregoing laws we have v = 2gt, and s = gt2; and from these two equations result the following general formulas:

And here, when the constant force 1, is the natural force of gravity, then the distance g 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 g will be different, in proportion as the force is more or less.

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.

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.

Accelerated Motion of Bodies on Inclined Planes. 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 Inclined Plane.

Accelerated Motion of Pendulums. See PENDULUM.

Accelerated Motion of Projectiles. See PROJECTILE.

Accelerated Motion of Compressed Bodies, in ex- panding or restoring themselves. See Dilatation, Compression, and Elasticity.

Accelerating Force, 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, &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.

But accelerating forces are sometimes variable, as well as sometimes constant; and the variation may be either increasing or decreasing.

The nature of constant and variable accelerating forces, may be illuftrated in the following manner. Let two weights W, w, be connected by a thread

, 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 ALTIMETRY, Longimetry, or Heights-and-Distances.

, Accident, Philos.

Per Accidens 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 per se, which denotes the nature and essence of a thing. Thus, fire is said to burn per se, or considered as sire, and not per accidens; but a piece of iron, though red-hot, only burns per accidens, by a quality accidental to it, and not considered as iron.

, something that partakes of the nature of an accident; or that is indifferent, or not essential to its subject.—Thus whiteness is accidental to marble, and sensible heat to iron.

Accidental Colours, so called by M. Buffon, are those which depend on the affections of the eye, in contradistinction to such as belong to light itself.

The impressions made upon the eye, by looking stedfastly on objects of a particular colour, are various 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.

The subject has also been considered by M. de la Hire, and M. Aepinus, &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.

Accidental Point, 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.

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.

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.

Accidental Dignities, and Debilities, in Astrology, 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.

, the slope or steepness of a line or plane inclined to the horizon, taken upwards; in contradistinction to declivity, which is taken downwards. So the ascent of a hill, is an acclivity: the descent of the same, a declivity.

Some writers on fortification use acclivity for talus: though more commonly the word talus is used to denote the slope, whether in ascending or descending.

, in Music, 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.

The Accompanyment is used in recitative, as well as in song; on the stage, as well as in the choir, &c.

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.

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.

Organists sometimes apply the word to several pipes which they occasionally touch to accompany the treble; as the drone, the flute, &c.

ACCOMPT. See Account.

, 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.

They divide Accords into persect and imperfect; and again into consonances and dissonances.

Accord is more commonly called Concord, which see.

Accord is also spoken of the state of an instrument, when its fixed sounds have among themselves all the justness that they ought to have.

, or Accompt, in Arithmetic, &c, a calculation or computation of the number or order of certain things; as the computation of time, &c.

There are various ways of accounting; as, by enumeration, or telling one by one; or by the rules of arithmetic, addition, subtraction, &c.

, 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.

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- 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 &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.

, in Optics, 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 Aberration and Telescope.

Achromatic Telescope, 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.

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.

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.

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.

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- 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.

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.

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.

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.

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.

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.

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. 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 TELESCOPE.

There has lately appeared in the Gentleman's Magazine (1790, pa. 890) a paper on the refracting telescope, by an author who signs Veritus, in which the invention is ascribed to another person, not heretofore mentioned; in these words: “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.

This glass has been examined by several gentlemen of eminence and scientific abilities, and found to possess the properties of the present achromatic glasses.

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.

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 “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.” 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.

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.”

, or Achronycal. See Acronychal.

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 Stentrophonic Tube.

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 Sound, Ear, Voice, and Echo.

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.

, from the Saxon æcre, or German acker, a field, of the Latin ager. 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.

The acre, in surveying, is divided into 4 roods, and the rood is 40 perches.

The French acre, arpent, is equal to 1 1/4 English acre;

The Strasburg contains about 1/2 an English acre;

The Welch acre contains about 2 English acres;

The Irish acre contains 1 ac. 2 r. 19 27/121 p. English.

Sir William Petty, in his Political Arithmetic, reckons that England contains 39 million acres: but Dr. Greve shews, in the Philos. Trans. N° 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.

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.

, or Acronycal, in Astronomy, is said of a star or planet, when it is opposite to the sun. It is from the Greek axronuxos, 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.

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.

, or Acroters, in Architecture, small pedestals, usually without bases, placed on pediments, and serving to support statues.

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.

Acroteria 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.

It is also sometimes used to denote those sharp pinacles or spiry battlements, that stand in ranges about flat buildings, with rails and balustres.

, in Mechanics or Physics, 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.

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.

It is one of the laws of nature, that action and reaction are always equal, and contrary to each other in their directions.

Action is either instantaneous or continued; that is, either by collition or perc<*>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.

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.

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.

Quantity of Action, 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 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.

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.

In the same year that Maupertuis communicated the idea of his principle, professor Euler, in the supplement to his book, intitled Methodus inveniendi lineas curvas maximi vel minimi proprietate gaudentes, 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.

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.

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.

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 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 caúse 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.

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.

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 s, with the velocity v, 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. to-s.; whence it follows, that those sines are in the direct ratio of the velocities V, v; 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.

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.

, 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, &c.

Sir Isaac Newton shews that the quantity of motion in the world must be always deereasing, in consequence of the vis inertiæ, &c. So that there is a necessity for 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.

, the virtue or faculty of acting. As the activity of an acid, a poison, &c: the activity of fire exceeds all imagination.

According to Sir Isaac Newton, bodies derive their activity from the principle of attraction.

Sphere of Activity, 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, &c.

, in Astronomy, the Arabic name of a star of the fourth magnitude, in the southern claw of Cancer, marked a by Bayer. Its longitude for 1761,

, or sharp; a term opposed to obtuse. Thus, Acute Angle, in Geometry, is that which is less than a right angle; and is measured by less than 90°, or by less than a quadrant of a circle. As the angle ABC.

Acute angled Triangle, is that whose three angles are all acute; and is otherwise called an oxygenous triangle. As the triangle DEF.

Acute-angled Cone, 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.

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

Acute-angled Section of a Cone, 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.

, in Music, one of the terms used by the Italians to express a degree or distinction of time.

Adagio denotes the slowest time except grave.

Sometimes the word is repeated, as adagio, adagio, to denote a still slower time than the former.

Adagio also signifies a slow movement, when used substantively.

, in Astrology, a name given to the moon.

, in the Hebrew Chronology, 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.

, the uniting or joining of two or more things together; or the finding of one quantity equal to two or more others taken together.

, 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.

, or Athelard, 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.

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, “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.” Wallis's Algebra, pa. 6.

, Aldhelmus, or Althelmus, 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.

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 &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.

, or Alderaimin, the Arabic name of a star of the third magnitude, in the left shoulder of Cepheus, marked a by Bayer. Its longitude for 1761, g 9° 30′ 8″ north latitude 68° 56′ 20″.

, see Affected.

, Adherence, in Physics, 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 Cohesion.

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 planes adhere firmly together, partly by the external pressure of the atmosphere, and partly by the attraction of their parts.

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.

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.

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.

, in Astronomy, a star, of the sixth magnitude, upon the garment of Andromeda, under the last star in her foot.

, whatever lies immediately by the side of another.

Adjacent Angle, in Geometry, 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.

Adjacent bodies, in Physics, are understood of those that are near, or next to, some other body.

, or rather AJUTAGE; which see.

, in Trigonometry, is used by some mathematicians, for the tangents of arcs. Vieta calls them also prosines.

ADVANCE-Fosse, in Fortification, a ditch thrown round the esplanade or glacis of a place, to prevent its being surprised by the besiegers.

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.

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, &c.

, Adventus, in the Calendar, the time immediately preceding Christmas; and was anciently employed in pious preparation for the adventus, or coming on, of the feast of the Nativity.

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.

ÆOLIPILE, Æolipile, 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.

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 Phædom. 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.

In Italy it is said that the Æolipile 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.

And some have imagined that the æolipile 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 out than up, as it is not air, but rarefied water, that is thus violently blown through the pipe.

ÆOLUS, 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.

The machine is adapted to supply the place of a 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.

ÆOLUS's Harp, or Æolian Harp, an instrument so named, from its producing an agreeable melody, merely by the action of the wind.

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 & Phonurgia.

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.

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.

ÆRA, 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.

The word is sometimes also written cra 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 a. er. a. the abbreviations of the words, annus erat Augusti, or from a. e. r. a. the initials of the words annus erat regni Augusti, because the Spaniards began their computation from the time that their country came under the dominion of Augustus. Others derive it from æs, brass, the tribute money with which Augustus taxed the world. It is also said that æra originally signified a number stamped on money to determine its current value. And that the ancients used æs or æra as an article, as we do the word item, to each particular of an account; and hence it came to stand for a sum or number itself.

ÆRA also means the way or mode of accounting time. Thus we say such a year of the Christian æra, &c.

Spanish ÆRA, otherwise called the year of Cæsar, 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 æra, the Alani, the Vandals, Suevi, &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 æra 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.

Christian ÆRA. 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 æra 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.

See an account of the other principal æras under the word Epoch.

AERIAL Perspective, is that which represents bodies diminished and weakened, in proportion to their distance from the eye.

Aërial 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.

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.

, a description of the air, or atmosphere, its limits, dimensions, properties, &c.

, the doctrine or science of the air, and its phænomena, its properties, good and bad qualities, &c. It is much the same with the soregoing word, Aerography.

, Aerometria, 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, &c.

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 Cursus Mathematicus, in five volumes in 4to.

, the pretended art of sailing through the air, or atmosphere, in a vessel, sustained as a ship in the sea.

, is properly the doctrine of the weight, pressure, and balance of the air and atmosphere.

, 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 aerostats, or aerostatic machines; and which, on account of their round and bell-like shape, are otherwise called air ballcons. Also aeronaut is the name given to the person who navigates or sloats in the air by means of such machines.

Principles of Aerostation. 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.

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.

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.

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 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.

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.

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.

History of Aerostation. 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 artisicial flying, 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.

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.

Bishop Wilkins too, who died in 1672, in several of his works, makes mention of fimilar ideas being entertained by divers persons. “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.” And again, in his Dedalus, chap. 6, “Scaliger conceives the framing of such volant automata to be very easy. Volantis columbæ machinulam, cujus autorem Archytam tradunt, vel facillime profiteri audeo. Those ancient motions were thought to be contrived by the force of some included air: So Gellius, Ita erat scilicet libramentis suspensum, & aura spiritus inclusa, atque occulta consitum, &c. 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.” From which it would seem that Bishop Wilkins had some confused notion of such a thing as a heated air-balloon.

Again F. Francisco Lana, in his Prodroma, printed in 1670, proposes the same method with that of Roger Bacon, as his own thought.

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.

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 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.

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.

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.

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, 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.

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, &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.

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, 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.

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.

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<*> 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.

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.

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, 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.

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.

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.

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.

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 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 bègun 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.

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 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.

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.

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.

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 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.

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.

The first of December was fixed on for the display of this grand experiment; and every preparation was 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, &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.

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.

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, being afraid, he cut open the balloons with a knife, and so descended.

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:—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, &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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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; 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.

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 <*> 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 manœuvres 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.

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<*>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.

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.

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 does not appear that the greatest action of their oars produced any effect in directing the course of the balloon.

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.

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.

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.

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.

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- 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.

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.

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.

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 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.

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 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.

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.

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.

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.

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.

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.

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 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, &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. “I did not mention,” says he, “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.”

“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.”

“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.”

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. For several figures of balloons, see plate 1.

Practice of Aerostation. 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.

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, &c, it shews that for those purposes a larger balloon is necessary, on 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.

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.

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 Height in miles.Density.01.2001/41.1411/21.0853/41.03110.9801 1/40.9321 1/20.8861 3/40.84220.8002 1/40.7612 1/20.7232 3/40.68730.653 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 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 Atmosphere, where the more accurate rules are given at large.

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

, or Adfected, Equation, in Algebra, 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 x, namely, x3, x2, and x.

The term, affected, is also used sometimes in algebra, when speaking of quantities that have co-efficients.

Thus in the quantity 2a, a is said to be affected with the co-efficient 2.

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.

The term adfected, or affected, I think, was introduced by Vieta.

, in Geometry, a term used by some ancient writers, signifying the same as property.

Affection. Phys. The affections of a body are certain modifications occasioned or induced by motion; in virtue of which the body is disposed after such, or such a manner.

The affections of bodies, are sometimes divided into primary and secondary.

Primary Affections, 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.

Secondary, or derivative Affections, are such as arise out of primary ones, as divisibility, continuity, contiguity, &c, which arise out of quantity; regularity, irregularity, &c, which arise out of figure, &c.

AFFIRMATIVE Quantity, or Positive Quantity, one which is to be added, or taken effectively; in contradistinction to one that is to be subtracted, or taken defectively.—The term affirmative was introduced by Vieta.

Affirmative Sign, or Positive Sign, in Algebra, the sign of addition, thus marked +, and is called plus, or more, or added to. 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 + a, 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 a + b + c denote the sum of the quantities represented by a, b and c, 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.

The early writers on Algebra used the word plus in Latin, or piu in Italian, for addition, and afterwards the initial p only, as a contraction; like as they used minus, or meno, or the initial m 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.

, in Chronology, is used for a century, being a system or a period of a hundred years.

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.

Age of the Moon, 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.—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.

, Agens, in Physics, that by which a thing is done or effected; or any thing having a power by which it acts on another, called the patient, or by its action induces some change in it.

, the sum or result of several things added together. See Sum.

, in Physics, 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.

, or Aquilon, 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 Optics, in 6 books, which was published in folio, at Antwerp, in 1613; and a treatise of Projections of the Sphere. He promised also to treat upon Catoptrics and Dioptrics, but this was prevented by his death, which happened at Seville, in the year 1617.

, in Physics, a thin, fluid, transparent, elastic, compressible, and dilatable body, which surrounds this terraqueous globe, and covers it to a considerable height.

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.

Elementary Air, or Air 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.

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.

Dr. Hook, and some others, maintain that it is the same with the ether, 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 sui generis, incorruptible, immutable, incapable of being generated, but present in all places, and in all bodies.

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 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.

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, &c.

Common, or heterogeneous Air, 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.

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.

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.

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.

3. The whole fossil kingdom must necessarily be found in the atmosphere; for all of that kind, as salts, sulphurs, stones, metals, &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.

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.

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.

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.

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.

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.

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, &c.

Mechanical Properties and Effects of Air. Of these the most considerable are its fluidity, its weight, and its elasticity.

1. Its Fluidity.—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.

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- 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.

2. Its Weight or Gravity.—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.

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 Stobæus 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 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 clepsydræ, 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.

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.

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. 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.

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.

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 Experiences Nouvelles touchant la Vuide. In this treatise indeed he makes use of the old principle of suga vacui; 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.

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, “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 by that gentleman's passions.” See more of this history under Barometer.

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.

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.

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°; 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°. 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° 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°, 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.

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 soot of air, the weight of the cubic foot of water being just 1000 ounces averdupois, and that of quicksilver equal to 13600 ounces.

Air, 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.

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 Atmosphere, for the total quantity of effects and pressure, and the laws of different altitudes, &c.

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.

3. Elasticity. 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.

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.

Every particle of air makes a continual effort to dilate itself, and so it acts forcibly against all the neigh- 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.

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.

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.

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.—And on this principle of the condensation and elasticity of the air, depends the structure and use of the air-gun.

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, 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.

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.

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.

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 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 MANOMETER.

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.

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.

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.

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.

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.

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 “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.” 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.

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 that they are so many minute flocculi, 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.

But Sir Isaac Newton (Optics, Qu. 31, &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.

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.

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.

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.

This reciprocation is observable in several instances, particularly in plants, in which the tracheæ 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.

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 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.

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.

As to the different kinds of air, with its generation, and the effects of different ingredients of it, &c, they are omitted here, as properly belonging to a Chemical Dictionary, or to a General Dictionary of Arts, &c.

For the resistance of the air, see Resistance.

Air-Gun, in Pneumatics, is a machine for propelling bulleto with great violence, by the sole means of condensed air.

The first account we meet with of an air-gun, is in the Elemens d'Artillerie 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.

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.

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 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.

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.

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.

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 construction is rendered as light and portable as one of the smallest fowling.pieces.

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 b, is a round steel tube, having a small moveable pin in the inside, which is pushed out when the trigger a is pulled, by the springwork within the lock; to this tube b 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 b. Hence, if a bullet be rammed down in the barrel, the copper ball screwed fast at b, and the trigger a be pulled; then the pin in b 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.

Sometimes the syringe is applied to the end of the barrel C (fig. 4); the lock and trigger shut up in a brass case d; and the trigger pulled, or the discharge made, by pulling the chain b. In this contrivance there is a round chamber for the condensed air at the end of the spring at e, 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 d 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, &c, it has been called the Air-cane. The head of the cane unscrews and takes off at a, 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.

Magazine Air-Gun. 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 b, b, b, &c, are 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. s I si M k 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.

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 ik, 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 ss 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.

A more particular description of the several parts may be seen in Desaguliers' Exper. Philos. vol. ii. pa. 399 et seq.

Air-Pump, in Pneumatics, 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.

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.

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.

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 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 Machina Boyliana, and the vacuum produced by it, Vacuum Boylianum.

Structure of the Air-Pump. 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.

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.

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.

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.

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.

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.

The air-pump, thus improved, is represented in plate III. fig. 1; where oo 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 hh to the cistern dd, with which the two barrels aa communicate, in which the pistons are worked by a toothed wheel, by turning the handle bb; by which the racks cc, with the pistons, are worked alternately up and down. ll is the gauge tube, immersed in a bason of quicksilver m at bottom, and communicating with the receiver at top; from which however it may be occasionally disengaged, by turning a cock. And n 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.

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 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.

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. ‘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. “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° 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°, 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 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.” 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°. 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.

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.

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.

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- 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.

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.

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.

The Use of the Air-Pump. 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 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, &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.

But, supposing no vapour from moisture, &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, &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)2 or 4/9 after two strokes, (2/3)3 or 8/27 after 3 strokes, and in general it will be (2/3)n after n strokes: the original density of the air being 1. Hence then, universally, if s denote the sum of the contents of the receiver and barrel, and r that of the receiver only without the barrel, and n any number of strokes of the piston; then, the original density of the air being 1, the density after n strokes will be (r/s)n o rn/sn, namely the n power of the ratio r/s. So, for example, if the capacity of the receiver be equal to 4 times that of the barrel; then their sum s is 5, and r is 4; and the density of the contents after 30 strokes, will be (4/5)30, or the 30th power of 4/5, which is 1/808 nearly; so that the air in the receiver is raresied 808 times.

See also the Memoires de l'Acad. Royale des Sciences for the years 1693 and 1705.

From the same formula, namely (r/s)n = d 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 d, that of the natural air being 1. For since ((r/s)n = d, by taking the logarithm of this equation, it is n X log. r/s = log. of d; 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 r = 5, s = 6, d = 1/100, whose log. is - 2, and r/s=5/6, whose log. is - .07918; therefore 2/.07918 = 25 1/4 nearly, which is the number of strokes required.

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 n root of both sides, it is : that is, the n root of the density is equal to the ratio of the receiver to the sum of the receiver and barrel. So, if the density d be 1/128, and the number of strokes n = 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.

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 beat for good part of an hour, and that more briskly than in the air. That warm blood, milk, gall, &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.

Air-Vessel, in Hydraulics, 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.

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.

AIRY Triplicity, in Astrology, the signs of Gemini, Libra, and Aquarius.

, or Adjutage, in Hydraulics, part of the apparatus of a jet d'eau, 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.

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.

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.

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.

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.

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.

For the other circumstances relating to jets and the issuing of water under various circumstances, see EXHAUSTION, Flux, Fountain, Jet d'Eau, &c, to which they more properly belong.

, 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 Muhammed ben Geber Albatani, Mahomet the son of Geber, and Muhamedes Aractensis. 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 vir admirandi acuminis, ac in administrandis observationibus exercitatissimus.

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 Arabic a work under the title of The Science of the Stars, 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° 204.

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° 2′; and the obliquity of the ecliptic 23° 35′. And upon Albategni's observations were founded the Alphonsine tables of the moon's motions; as is observed by Nic. Muler, in the Tab. Frisicæ, pa. 248.

ALBERTUS Magnus, 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, de sphæra, de astris, de astronomia, item speculum astronomicum.

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, &c, under the title of De natura rerum, and De secretis mulierum, 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.

John Matthæus de Luna, in his treatise De Rerum Inventor<*>bus, has attributed the invention of fire-arms to Albert; but in this he is refuted by Naude, in his Apologie des grands hommes.

, otherwise called Abuassar, and Japhar, was a celebrated Arabian philosopher and astrologer, of the 9th or 10th century, or according to some authors much earlier. Blancanus, Vossius, &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 De Magnis Conjunctionibus Annorum Revolutionibus, ac eorum Perfectionibus, printed at Venice in 1515, at the expence of Melchior Sessa, a work chiefly astrological.

He wrote also Introductio in Astronomiam, printed in the year 1489. And it is reported that he observed a comet in his time, above the orb of Venus.

, 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.

, 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 Right Ascension is—66°6′51″.10Annual variation in AR 0051.31Declination—16552.00 N.And Annual variat. in Decl. 00 8.30

, a star of the third magnitude in the right shoulder of the constellation Cepheus.

, 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.

, 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.

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.

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.

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.—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. “You will never, said she to him one day, be any thing but a philosopher—and what is a philosopher?—a fool, who toils and plagues himself all his life, that people may talk of him when he is dead.”

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.

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 Analyse Demontrée 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 Ducks and Drakes; but it was d'Alembert who first explained it in a satisfactory and philosophical manner.

Two years after his election to a place in the academy, he published his Treatise on Dynamics. 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 two parts the action of the moving powers, and considering the one as producing alone the motion of the body, in the second instant, and the other as employed to destroy that which it had in the first.

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 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 Discourse on the General Theory of the Winds, 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 Partial Differences 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.

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.

In 1752, d'Alembert published a treatise on the Resistance of Fluids. to which he gave the modest title of an Essay; 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, Researches concerning the Integral Calculus, which is greatly indebted to him for the rapid progress it has made in the present century.

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.

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, &c, all the opposition and censure that d'Alembert met with on account of the famous Encyclopédie, 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.

Some time after this, d'Alembert published his Philosophical, Historical, and Philological Miscellanies. These were followed by the Memoirs of Christina queen of Sweden; in which d'Alembert shewed that he was acquainted with the natural rights of mankind, and was bold enough to assert them. His Essay on the Intercourse of Men of Letters with Persons high in Rank and Ofsice, 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, “If he had consulted me, I would have told him still more of the matter.”

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 Encyclopédie with a multitude of excellent articles in that line, and composed his Researches on several Important Points of the System of the World, 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.

In 1759 he published his Elements of Philosophy: a work much extolled as remarkable for its precision and perspicuity.

The resentment that was kindled (and the disputes that followed it) by the article Geneva, inserted in the Encyclopédie, 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.

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; —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.

In the year 1765, he published his Dissertation on the Destruction of the Jesuits. 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.

Beside the works already mentioned, he published nine volumes of memoirs and treatises, under the title of Opuscules; 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.

He published also Elements of Music; 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.

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.

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 Hist. de l'Acad. Royale des Sciences, 1783.

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 Nouvelle Table des Articles contenus dans les volumes de l'Academie Royale des Sciences de Paris, &c, as follows:

Traité de Dynamique, in 4to, Paris, 1743. The 2d ed. in 1758.

Traité de l'Equilibre et du Mouvement des Fluides. Paris, 1744; and the 2d edition in 1770.

Reflexions sur la Cause Générale des Vents; which gained the prize at Berlin in 1746; and was printed at Paris in 1747, in 4to.

Recherches sur la Précession des Équinoxes, & sur la Nutation de l'Axe de la Terre dans le Système Newtonien. Paris, 1749, in 4to.

Essais d'une Nouvelle Théorie du Mouvement des Fluides. Paris, 1752, in 4to.

Recherches sur differens Points importans du Système du Monde. Paris, 1754 and 1756, 3 vol. in 4to.

Elemens de Philosophie, 1759.

Opuscules Mathématiques, ou Memoires sur différens Sujets de Géométrie, de Méchaniques, d'Optiques, d'Astronomie. Paris, 9 vol. in 4to; 1761 to 1773.

Elémens de Musique, théorique & pratique, suivant les Principes de M. Rameau, eclairés, développés, & simplifiés. 1 vol. in 8vo. à Lyon.

De la Destruction des Jesuites, 1765.

In the Memoirs of the Academy of Paris are the following pieces, by d'Alembert: viz,

Précis de Dynamique, 1743, Hist. 164.

Précis de l'Equilibre & de Mouvement des Fluides, 1744, Hist. 55.

Methode générale pour déterminer les Orbites & les Mouvements de toutes les Planètes, en ayant égard à leur action mutuelle, 1745, p. 365.

Précis des Réflexions sur la Cause Générale des Vents, 1750, Hist. 41.

Précis des Recherches sur la Précession des Équinoxes, et sur la Nutation de l'Axe de la Terre dans le Système Newtonien, 1750, Hist. 134.

Essai d'une Nouvelle Théorie sur la Résistance des Fluides, 1752, Hist. 116.

Précis des Essais d'une Nouvelle Théorie de la Résistance des Fluides, 1753, Hist. 289.

Précis des Recherches sur les differens Points importans du Système du Monde, 1754, Hist. 125.

Recherches sur la Précession des Equinoxes, & sur la Nutation de l'Axe de la Terre, dans l'Hypothese de la Dissimilitude des Méridiens, 1754, p. 413, Hist. 116.

Reponse à un Article du Mémoire de M. l'Abbé de la Caille, su<*> la Théorie du Soleil, 1757, p. 145, Hist. 118.

Addition à ce Mémoire, 1757, p. 567, Hist. 118.

Précis des Opuscules Mathématiques, 1761, Hist. 86.

Précis du troisième volume des Opuscules Mathématiques, 1764, Hist. 92.

Nouvelles Recherches sur les Verres Optiques, pour servir de suite à la théorie qui en à été donnée dans le volume 3e des Opuscules Mathématiques. Premier Mémoire, 1764, p. 75, Hist. 175.

Nouvelles Recherches sur les Verres Optiques, pour fervir de suite à la théorie qui en a été donnée dans le troisième volume des Opuscules Mathématiques. Second Mémoire, 1765, p. 53.

Observations sur les Lunettes Achromatiques, 1765, p. 53, Hist. 119.

Suite des Recherches sur les Verres Optiques. Troisième Mémoire, 1767, p. 43, Hist. 153.

Recherches sur le Calcul Intégral, 1767, p. 573.

Accident arrivé par l'Explosion d'une Meule d'Emouleur, 1768, Hist. 31.

Précis des Opuscules de Mathématiques, 4e & 5e volumes. Leur Analyse, 1768, Hist. 83.

Recherches sur les Mouvemens de l'Axe d'une Planete quelconque dans l'hypothese de la Dissimilitude des Méridienes, 1768 p. 1, Hist. 95.

Suite des Recherches sur les Mouvemens, &c, 1768 p. 332, Hist. 95.

Recherches sur le Calcul Intégral, 1769, p. 73.

Mémoire sur les Principes de la Mech. 1769, p. 278.

And in the Memoirs of the Academy of Berlin, are the following pieces, by our author: viz,

Recherches sur le Calcul Intégral, premiere partie, 1746.

Solution de quelques problemes d'astronomie, 1747.

Recherches sur la cour be que forme une Corde Tendue, mise en Vibration, 1747.

Suite des recherches sur le Calcul Intégral, 1748.

Lettre à M. de Maupertuis, 1749.

Addition aux recherches sur la courbe que forme une Corde Tendue mise en Vibration, 1750.

Addition aux recherches sur le Calcul Intégral, 1750.

Lettre à M. le professeur Formey, 1755.

Extr. de differ. lettres à M. de la Grange, 1763.

Sur les Tautochrones, 1765.

Extr. de differ. lettres à M. de la Grange, 1769.

Also in the Memoirs of Turin are,

Differentes Lettres à M. de la Grange, en 1764 & 1765, tom. 3 of these Memoris.

Recherches sur differens sujets de Math. t. 4.

, or Alfeta, a name given to the star commonly called Lucida Coronæ.

, Alfergani, or Fargani, 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.

There are three Latin translations of Alfragan's 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.

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.

, commonly called Count Algaroti, a celebrated Italian of the present century, well skilled in Architecture and the Newtonian philosophy, &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 Newtonian Philosophy for the Ladies, as Fontenelle had done his Cartesian Astronomy, in the work intitled The Plurality of Worlds. 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; Hic jacet Algarotus, sed non omnis. 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 Newtonian Philosophy for the Ladies, a sprightly, ingenious, and popular work.

, 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— universal arithmetic, and literal arithmetic, and the arithmetic of signs.

The etymology of the name, Algebra, 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 Alghebra e Almucabala, which is explained to signify the art of restitution and comparison, or opposition and comparison, or resolution and equation, 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 al, makes Algeber, which is purely Arabic, and signifies the reduction of broken numbers or fractions to integers.

But Peter Ramus, in the beginning of his Algebra, says “the name Algebra is Syriac, signifying the art and doctrine of an excellent man. For Geber, 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 Almucabala, 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 Aljabra, and Alboret; though the name of the author himself is not known.” 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 l'Arte Magiore: ditta dal vulgo la Regola de la Cosa over Alghebra e Almucabala; calling it l'Arte Magiore, or the greater art, to distinguish it from common arithmetic, which is called l'Arte Minore, or the lesser art. It seems too that it had been long and commonly known in his country by the name Regola de la Cosa, or Rule of the Thing; 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 Regula rei & census, 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 census, pronounced chensus, came the barbarous word zenzus, used by the Germans and others, for quadratics; with the several zenzic or square roots. And hence [scruple], [dram],

Later authors, and other nations, used some the one of those names, and some another. It was also called Specious Arithmetic by Vieta, on account of the species, or letters of the alphabet, which he brought into general use; and by Newton it was called Universal Arithmetic, from the manner in which it performs all arithmetical operations by general symbols, or indeterminate quantities.

Some authors define algebra to be the art of resolving mathematical problems: 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.

Indeed algebra properly consists of two parts: first, the method of calculating magnitudes or quantities, as represented by letters or other characters: and secondly the manner of applying these calculations in the solution of problems.

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.

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.

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, &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 dn_, kn_, ddn_, dkn_, kkn_, &c: the unknown quantity he calls simply ariqmos, numerus, the number; and in the solutions he commonly marks it by the final thus s_; also he denotes the monades, or indefinite unit, by mo_. Diophantus there remarks on the multiplication and division of simple species together, shewing what powers or species they produce; declares that minus (leiyis) multiplied by minus produces plus (nparcin); but that minus multiplied by plus, produces minus; and that the mark used for minus is <*> namely the y 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 res ignota 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 work, Lib. 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, &c, &c; but none of them concerning either square or cubic numbers. Lib. 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. Lib. 3 contains 24 questions concerning squares, chiefly including three or four numbers. Lib. 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+n and 5-n; 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. Lib. 5 is also concerning square and cube numbers, but of a more difficult kind, beginning with some that relate to numbers in geometrical progression. Lib. 6 contains 26 propositions, concerning right-angled triangles; such as to make their sides, areas, perimeters, &c, &c, squares or cubes, or rational, &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, &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- tion to the final equation is easy and evident: and there he leaves the solution, only mentioning that the root or ariqmos 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.

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 Mahomet ben Musa, 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<*>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.

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 Summa de Arithmetica, Geometria, Proportioni, et Proportionalita, 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, 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 Abaco is a corruption of Modo Arabico, 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 ℞ 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; “and so on (he adds) in infinitum;” 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 “in infinitum” 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.

Paciolus next enters on the algebraical part of this work, which he calls “L'Arte Magiore; ditta dal vulgo la Regola de la Cosa, over Alghebra e Ahnucabala:” which last name he explains by restauratio & oppositio, and assigns as a reason for the first name, because it treats of things above the common affairs in business, which make the Arte Minore. Here he ascribes the invention of Algebra to the Arabians, and denominates the series of powers, with their marks or abbreviations, as n°, or numero, the absolute or known number; co. or cosa, the thing or 1st power of the unknown quantity; ce. or censo, the product or square; cu. or cubo, the cube, or 3d power; ce. ce. or censo de censo, the square-squared, or 4th power; p°. r°. or primo relato, or 5th power; ce. cu. or censo de cubo, 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, &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 ℞ for radici, the root; ℞. ℞. radici de radici, the root of the root; ℞ u. radici universale, or radici legata, or radici unita; ℞ cu. radici cuba; and ―q[dram]<*> quantita, quantity; p for piu or plus, and m for meno 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 co. p. 4 ce. m. 5 cu. p. 2 ce. ce. m. 6 ni. that is, 3 cosa piu 4 censa meno 5 cubo piu 2 censa-censa meno 6 numeri, or, as we now write the same thing, 3x+4x2 - 5x3 + 2x4 - 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 el Cataym, 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 el Cataym 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 in infinitum, 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 p ℞ 448, or of ℞ 18 p ℞ 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 p or m, is the root of the binomial; as in this 23 p ℞ 448, the two parts of 23 are 7 and 16, whose product, 112, is 1/4 of 448, therefore their roots give 4 p ℞ 7 for the root ℞ u. 23 p ℞ 448. He next treats of equations both simple and quadratic, or simple and compound, as he 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, &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. —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.

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 Omar Ben Ibrahim al'Ghajamæi Algebra cubicarum æquationum, sive de problematum solidorum resolutione; 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.

Since this was written, death has prematurely put an end to the useful labours of this ingenious and worthy successor of Gravesande.

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 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 Ars Magna, or Cosa, or Rules of Algebra, 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 cubum & rerum numero æqualium, 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.

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, vera & ficta, true and fictitious or false; so the æstimatio rei, 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 denominations or powers. And the same is remarked on compound cases of even powers that are added together; as if , then the æstimatio x 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 √ 3 or - √ 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 numero, the 1st power the res, the 2d the quadratum, the 3d the cubum, and so on, using no mark for the unknown quantity, and only the initials p and m for plus and minus, and ℞ for radix or root. The res he sometimes calls positio, and quantitas ignota; 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.

On the other hand, he remarks that the odd denominations, or powers, have only one æstimatio, 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 2x=16, the root is 8 only; and if 2x3=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 æstimatio 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 æstimatios, the one true or 2, and the other fictitious or - 4, which he observes is the same as the true æstimatio 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 √5 1/4 - 1 1/2, and the fictitious one is - √ 5 1/4 - 1 1/2, which last is the same as the true root of , viz. √ 5 1/4 + 1 1/2; and he here infers generally that the fictitious æstimatio 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 + √40 - 4, and - 3, and - √40 - 4; and the roots of , are - √40 + 4, and + 3, and + √40 + 4.

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 æquatios, 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 x3+bx would be negative, and equal to a positive sum ax2 + c, 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.

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.

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.

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 20x3 = 180x5, then 20 = 180x2, and 1/9=x2, and x = 1/3.

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 √316, 2, √34, are in continual proportion.

Chap. 5 treats of the æstimatio of the lowest degree of compound cases, that is, affected quadratic equations; giving the rule for each of the three cases, which con- sists in completing the square, &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.

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.

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 , &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.

Chap. 8 shews generally how to find the root of any such equation as this , where m and n are any exponents whatever, but n the greater; and the rule is, to separate or divide the co-efficient a into two such parts z and a - z, as that the absolute number b shall be equal to ―(a - z).zm/(n-m), the product of the one part a - z, and the m/(n - m) power of the other part: then the root x is = z1/(n-m). 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.

Chap. 9 and 10 contain the resolution of various questions producing equations not higher than quadratics.

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 a, <*>, g, d, &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 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 .

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 b3 exceeds c2, which has since been called the irreducible case, he refers to another following book, called Aliza, for other rules of solution, to overcome this difficulty, about which he took insinite pains.

Chap. 13, of the case . This case, by a geometrical process, he reduces to the case in the last chapter: thus, find the æstimatio y 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 æstimatio or root of the case , is equal to the sum of the roots of the case . As in the example , whose æstimatio is √(9 1/4 + 1 1/2), which is equal to the sum of 3 and √(9 1/4 - 1 1/2), the two roots of the case .

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 x, by substituting y ± 1/3 the co-efficient of that term for x, 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 æstimatio y of the case , then is : also for the case , first find the two roots of , then is x = (√34c2)/y the two values of x according to the two values of y. He here also gives another rule, by which a second æstimatio or root is found, when the first is known, namely, if e be the first estimatio or value of x in the case , then is the other value of .

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.

Chap. 24, of the 44 derivative cases; which are only higher powers of the forms of quadratics and cubics.

Chap. 25, of imperfect and special cases; containing many particular examples when the co-efficients have certain relations amongst them, with easy rules for finding the roots; also 8 other rules for the irreducible case .

Chap. 26, in like manner, contains easy rules for biquadratics, when the co-efficients have certain special relations.

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: De transitu capituli specialis in capitulum speciale; De operationibus radicum pronicarum seu mixtarum & Allellarum; De regula modi; De regula Aurea; De regula Magna, or the method of finding out solutions to certain questions; De regula æqualis positionis, being a method of substituting for the half sum and half difference of two quantities, instead of the quantities themselves; De regula inæqualiter ponendi, seu proportionis; De regula medii; De regula aggregati; De regula liberæ positionis; De regula falsum ponendi, in which some quantities come out negative; Quomodo excidant partes & denominationes multiplicando. 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 &c; with other such geometrical questions, resolved algebraically.

Chap. 39, De regula qua pluribus positionibus invenimus ignotam quantitatem; 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- tract 2x2 + 4x + 7 from both sides, then the first becomes a square, viz, . Next assume the indeterminate y, and subtract 2y (x2 - 1) - y2 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 y = 2 or √15; and hence, using 2 for y, the equation of equal squares becomes , the roots of which give ; and hence ; the two positive roots of which are √(3 + 1) and √(5 - 1), which are two of the values of x in the given equation . The other roots he leaves to be tried by the reader.

The 40th, or last, chap. is entitled, Of modes of general supposition relating to this art; with some rules of an unusual kind; and æstimatios or roots of a nature different from the foregoing ones. Some of these are as follow: If , and , and x : y :: c : d; then is .

Secondly, if , and , then is x + a : y - a :: y2 : x2.

Thirdly, when , the square will be taken away, by putting ; and then the equation becomes .

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 y is the quotient arising from the co-efficient of x 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 æstimatio or root of the one equation from that of the other is very difficult, valde difficilis.

It is matter of wonder that Cardan, among so many transmutations, should never think of substituting instead of x 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- tween the two roots. Thus, instead of x in the given form , substitute dy, and it becomes ; and this divided by d3 becomes ; and here if d be taken = √c, it becomes ; which is the transformation in question, and in which it is evident that x is = y√c, and y = x/√c. Instead of this, Cardan gives the following strange way of finding the one root x from the other y, when this latter is by any means known; viz, Multiply the first given equation by y2x + 1, then add x2/4y2 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 x by a quadratic equation.

Thus, multiplied by y2x + 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 y 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 y the root of the equation , and substituting its value in the quantity , this will become a complete square. Of Cardan's Libellus de Aliza Regula.

Subjoined to the above Treatise on cubic equations, is this Libellus de Aliza regula, 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; but I shall only set down the titles of some few of them, whose contents require more particular notice.

Chap. 4. De modo redigendi quantitates omnes, quæ dicuntur latera prima ex decimo Euclidis in compendium. He treats here of all Euclid's irrational lines, as surd numbers, and persorms various operations with them.

Chap. 5. De consideratione binomiorum & recisorum, &c; ubi de æstimatione capitulorum. Contains various operations of multiplying compound numbers and surds.

Chap. 6. De operationibus p: & m: (i. e. + and -) secundum communem usum. Here it is shewn that, in multiplication and division, plus always gives the same signs, and minus gives the contrary signs. So also in addition, every quantity retains its own sign; but in subtraction they change the signs. That the √ +, or the square root of plus, is +; but the √ -, or the square root of minus, is nothing as to common use: (but of this below.) That √3 - is -; as √ - 8 is - 2. That a residual, composed of + and - may have a root also composed of + and -: So √(5-√24) is = √3-√2. The rules for the signs in multiplication and division are illustrated by this example; to divide 8 by 2 + √6 or √6 + 2. Take the two corresponding residuals 2 - √6 and √6 - 2, and by these multiply both the divisor and dividend; then the products are + and - respectively, and the quotients still both alike. Thus, Divid.Divis.|Divid.Divis.8√6 + 282 + √6√6 - 2√6 - 22 - √62 - √6√384 - 16 divide + 416 - √384 div. - 2Quot. √96-8.Quot. √96 - 8. 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.

In chap. 11 and 18, and elsewhere, Cardan makes a general notation of a, b, c, d, e, f, for any indefinite quantities, and treats of them in a general way.

Cap. 2. De contemplatione p: & m: (or + and -), & quod m: in m: facit p: & de causis horum juxta veritatem. 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.

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 De æstimatione generali solida vocata, & operationibus ejus; 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 b 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)c; then the sum of the roots of these parts is the æstimatio or value

, a person skilled in algebra.

, or Algenib, a fixed star of the second magnitude, on the right side of Perseus.

, or Medusa's Head, a fixed star of the third magnitude, in the constellation Perseus.

, a fixed star of the third magnitude, in the right wing of the constellation Corvus.

, or Algorithm, 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 algorithm of numbers, of algebra, of integers, of fractions, of surds, &c; meaning the common rules for performing the operations of arithmetic, or algebra, or fractions, &c.

, Allacen, or Abdilazum, 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 Opticæ Thesaurus, 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.

, an Arabic name for the label, index, or ruler, which is moveable about the centre of an astrolabe, quadrant, &c, and carrying the sights or telescope, and by which are shewn the degrees cut off the limb or arch of the instrument.

ALIQUANT part, is that part which will not exactly measure or divide the whole, without leaving some remainder. Or the aliquant 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.

ALIQUOT part, is such a part of any whole, as will exactly measure it without any remainder. Or the aliquot 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.

All the aliquot 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, &c; and the products will be the other or compound aliquot parts of the given number. So if the aliquot 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 aliquot parts of the given number 60, are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30.—In like manner it will be found that all the aliquot 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.

ALLEN (Thomas) 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, Concerning the Judgment of the Stars, or, as it is usually called, of the quadripartite construction, with an exposition. He wrote also notes on many of Lilly's books, and some on John Bale's work, De scriptoribus Maj. Britanniæ. He died at Glocester hall in 1632, being 90 years of age.

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- 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: “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, &c.”

, 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 alligare, to tie or connect together, probably from certain vincula, or crooked ligatures, commonly used to connect or bind the numbers together.

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 Lucas de Burgo, and the other early authors in Europe.

Alligation is of two kinds, medial and alternate.

Alligation medial is the method of finding the rate or quality of the composition, from having given the rates and quantities of the simples or ingredients.

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.

For example, 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. Gal.s.products.Here6mult. by 5 gives308" by 6 "484" by 8 "32whole comp.18110sum of prod. Then 18)110 (6 2/18 or (6 1/9)s, is the rate sought.108   2

Alligation alternate is the method of finding the quantities of ingredients or simples, necessary to form a compound of a given rate.

The rule 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.

For example, Suppose it were required to mix together gold of various degrees of fineness, viz. of 19, of 21, and of 23 caracts fine, so that the mixture shall be of 20 caracts fine. Hence, RatesDiffs.Sum of Diffs.Comp. rate 20{211{1 of 21 caracts sine,191+34 of 19 caracts sine,2311 of 23 caracts sine. That is, there must be an equal quantity of 21 and 23 caracts fine, and 4 times as much of 19 caracts fine.

Various limitations, both of the compound and the ingredients, may be conceived; and in such cases, the differences are to be altered proportionally.

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.

There is recorded a remarkable instance of a discovery made by Archimedes, both by alligation 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.

, 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.

ALMACANTAR. See Almucantar.

, 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, &c, this work will ever be held dear and valuable to the cultivators of astronomy.

In the original Greek it is called suntacis m<*>gish, the great composition or collection. And to the word megish, megiste, the Arabians joined the particle al, and thence called it Almaghesti, or, as we call it, from them, the Almagest.

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 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 George, a monk of Trabezond, who translated it into Latin, which translation has several times been published.

Riccioli, an Italian jesuit, also published, in 1651, a body of Astronomy, which, in imitation of Ptolemy, he called Almagestum Novum, the New Almagest; being a large collection of ancient and modern observations and discoveries, in the science of Astronomy.

, 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 Mamon, Almaon, Almamun, Alamoun, or Al-Maimon. 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.

Almamon himself made many astronomical observations, and determined the obliquity of the ecliptic to be then 23° 35′ (or 23° 33′ in some manuscripts), but Vossius says 23° 51′ or 23° 34′. 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.

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.

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.

, 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 the moon, &c, sor each month: and answers to the fasti of the ancient Romans.

The etymology of the word is much controverted among grammarians.—Some derive it from the Arabic, viz, from the particle al, and manah, to count. While Scaliger, and others, derive it from the same al, and the Greek manakos, 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 almanha, 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 almonac, 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 al-monaght, which is as much as to say, in old English or Saxon, all-moon-heed.

Almanacs are of various kinds and composition, some books, others sheets, &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, &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, &c.

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 assect to disbelieve in such predictions.

Nautical Almanac, and Astronomical Ephemeris, 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.

, in the Arabian astrology, denotes the pre-eminence or prevalence of one planet over another.

, Almacantars, or ALMICANTARS, from the Arabic almocantharat, 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, &c; and are the same as the parallels of altitude.

Almucantar-Staff, 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, &c. The instrument had an arch of 15 degrees, made of some smooth wood.

ALPHONSINE Tables, are astronomical tables compiled by order of Alphonsus, king of Castile. In the compiling of these it is thought that prince himself assisted. See Astronomical tables.

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.

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. “What can be more surprising,” says Mariana, “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.” 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 Alphonsine Tables, 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, &c. (Weidler, p. 280).

We must not omit a memorable saying of Alphonfus, which has been recorded for its boldness and pretended impiety; namely, “that if he had been of God's privy council when he made the world, he could have advised him better.” Mariana however says only in general, that Alphonsus was so bold as to blame the works of 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: “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.” 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, “That if God had asked his advice when he made the world, he would have given him better council,” the boldness and impiety of the censure will be greatly diminished.

, 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 Encyclopædia has been much esteemed even by Roman Catholics; it was printed at Lyons, and sold very well throughout all France. Vossius mentions the Encyclopædia in general, but speaks of his treatise of Arithmetic more particularly, and allows the author to have been a man of great reading and universal erudition. His Thesaurus Chronologicus is by some esteemed one of his best works, and has gone through several editions, though others speak of it with contempt. In his Triumphus Biblicus 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 Theologia Polemica, which was one of Alsted's best performances. It seems he was a millenarian, having published, in 1672, a treatise De Mille Annis, 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.

ALTERNATE angles, are the internal angles, A and B, or a and b,

, or Permutation, of quantities or things, is the varying or changing the order or position of them.

As suppose two things a and b; these may be placed either thus ab or ba that is two ways, or 1 X 2. If there be three things, a, b, c, then the 3d thing c, may be placed three different ways with respect to each of the two positions ab and ba 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 a, b, c, d; then the fourth thing d 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 n, multiply the series of natural numbers 1, 2, 3, 4, 5, &c, to n, continually together, and the last product will be the number of alternations sought; so 1X2X3X4X5 - - - - n is the number of changes in n things.

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!

, Altimetria, the art of taking or measuring altitudes or heights, whether accessible or inaccessible. Or

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.

, 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. Antiquæ); but they were published in 5 vols folio, with his life, by Bekker of Amsterdam. They contain various analytical, exegetical, practical, problematical, and philosophical tracts, which shew his great industry and knowledge.

, 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.

Altitude of a figure, 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.

, encompassing round about; as the bodies which are placed about any other body, are called ambient bodies, and sometimcs circum-ambient bodies; and the whole mass of the air or atmosphere, because it encompasses all things on the face of the earth, is called the ambient air.

AMBIGENAL Hyperbola, a name given by Newton, in his Enumeratio linearum tertii ordinis, 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.

, of a figure, in Geometry, is the perimeter, or line, or sum of the lines, by which the figure is bounded.

, or Ambligonal, in Geometry, signisies obtuse-angular, as a triangle which has one of its angles obtuse, or consisting of more than 90 degrees.

AMICABLE numbers, 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 — — 284; of 284, they are 1, 2, 4, 71, 142, and their sum is 220.

The 2d pair of amicable numbers are 17296 and 18416, which have also the same property as above.

And the 3d pair of amicable numbers are 9363584 and 9437056.

These three pairs of amicable numbers were found out by F. Schooten, sect. 9 of his Exercitationes Mathematicæ, who I believe first gave the name of amicable to such numbers, though such properties of numbers it seems had before been treated of by Rudolphus, Descartes, and others.

To find the sirst pair, Schooten puts 4x and 4yz, or a2x and a2yz for the two numbers where a = 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 x and z, and consequently, assuming a proper value for y, the two amicable numbers themselves 4x and 4yz.

In like manner for the other pairs of such numbers; in which he finds it necessary to assume 16x and 16yz or a4x and a4yz for the 2d pair, and 128x and 128yz or a7x and a7yz for the 3d pair.

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, also from 6 times the assumed number, and from 18 times the square of the assumed number, 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.

That is, if a be put = the number 2, and n some integer number, such that 3an-1, and 6an-1, and 18a2n-1 be all three prime numbers; then is ―(18a2n-1) X2an one of the amicable numbers; and the sum of its aliquot parts is the other.

, 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 Observations and Experiments concerning a new Hour-glass, and concerning Barometers, Thermometers, and Hygroscopes; 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 New Theory of Friction, 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.

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, &c, may be seen in the volumes for the years 1696, 1699, 1702, 1703, 1704, and 1705.

, or Amphiscians, are the people who inhabit the torrid zone; which are so called, because 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.

, in gunnery, the range of the projectile, or the right line upon the ground subtending the curvilinear path in which it moves.

, a name sometimes given to the dragon's tail, or northern node of the moon.

, or the science of the reflections of sounds, frequently used in reference to echoes, which are said to be sounds produced anacamptically, or by reflection. And in this sense it was used by the ancients for that part of optics which is otherwise called Catoptrics.

, 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 Æneas, though, in reality, she did not arrive in Africa till 300 years after the taking of Troy.

An error on the other side, by which a fact is placed later, or lower than it should be, is called a parachronism. But in common use, this distinction, though proper, is not attended to; and the word anachronism is used indifferently for the mistake on both sides.

, or Anaclatics, an ancient name for that part of Optics which considers refracted light; being the same as what is more usually called dioptrics. See the Compendium of Ambrosius Rhodius, lib. 3. Opticæ, pa. 384 & seq.

Anaclastic Curves, 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.

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.

, 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.

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, &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.

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, &c.

, the same as proportion, or equality, or similitude of ratios. Which see.

, 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. Analysis may be distinguished into the ancient and the modern.

The ancient analysis, as described by Pappus, is the method of proceeding from the thing sought as taken for granted, through its consequences, to something that is really granted or known; in which sense it is the reverse of synthesis or composition, in which we lay that down first which was the last 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.

The principal authors on the ancient analysis, as recounted by Pappus, in the 7th book of his Mathematical Collections, are Euclid in his Data, Porismata, & de Locis ad Superficiem; Apollonius de Sectione Rationis, de Sectione Spatii, de Taclionibus, de Inclinationibus, de Locis Planis, & de Sectionibus Conicis; Aristæus, de Locis Solidis; and Eratosthenes, de Mediis Proportionalibus; from which 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, &c, and more especially Hugo d'Omerique, in his Analysis Geometrica, in which he has endeavoured to restore the Analysis of the ancients. And, on this head, Dr. Pemberton tells us “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.”

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 data and quæsita, 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.

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.

The solution of this problem, as given by Pappus, is thus. Analysis.

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 AC2 : BC2 :: DA : DB, as may be thus proved.

Since DC touches the circle, and BC cuts it, the angle BCD is equal to BAC by Euc. iii. 32; also the angle D is common to both the triangles DCA, DCB; these are therefore similar, and so, by vi 4, DA : DC :: DC : DB, and hence DA2 : DC2 :: DA : DB by cor. vi 20. But also, by vi 4, DA : AC :: DC : CB, and by permutation DA : DC :: AC : BC, or DA2 : DC2 :: AC2 : BC2; and hence, by equality, AC2 : BC2 :: DA : DB.

But the ratio of AC2 to BC2 is given by prop. LVII of Simson's edition of the Data, 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.

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. Synthesis.

Construction. Make as F3 : G2 :: AD : DB, (which may be done, since AB is given, by making it as F2G2 : G2 :: AB : DB, and then by composition it will be as F2 : G2 :: 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.

Demonstration. Since, by the constr. F2 : G2 :: AD : DB, and also AD : DB :: AC2 : BC2, which has been already demonstrated in the analysis, and might be here proved in the same manner. Therefore F2 : G2 :: AC2 : BC2, and consequently F : G :: AC : BC. Q.E.D.

Here we see an instance of the method of resolution and composition, 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 Data, 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. Analysis.

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. Synthesis.

Construction. 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.

Demonstration. The triangles ABC, ABH are equiangular, therefore, vi 4, AC : CB :: AB : BH, which is the given ratio by construction.

Modern Analysis, consists chiefly of algebra, arithmetic of insinites, insinite series, increments, fluxions, &c; of each of which a particular account may be seen under their respective articles.

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.

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.

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 sacility 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 a? 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.

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.

Analysis is divided, with respect to its object, into that of finites, and that of infinites.

Analysis of finite quantities, is what is otherwise called algebra, or specious arithmetic.

Analysis of infinites, called also the new analysis, 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 method of fluxions, or the differential calculus. And the great advantage of the modern mathematicians over the ancients, arises chiefly from the use of this modern analysis.

Analysis of powers, is the same as resolving them into their roots, and is otherwise called evolution.

Analysis of curve lines, shews their constitution, nature and properties, their points of inflexion, station, retrogradation, variation, &c.

, 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.

, or Analytical, something belonging to, or partaking of, the nature of analysis; or performed by the method of analysis.

Thus we say analytical demonstration, analytical enquiry, analytical table or scheme, analytical method, &c. The analytical stands opposed to the synthetical, or that which proceeds by the way of synthesis.

, the science, or doctrine, and use of analysis.

, 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.

To construct an Anamorphosis, or monstrous projection, on a plane.—Draw the square ABCD (fig. 1), of any size at pleasure, and divide it by crossing lines into a number of areolæ or smaller squares: and then in this square, or reticle, called also the cratioular prototype, draw the regular image which is to be distorted.—Or, about any image, proposed to be distorted, draw a reticle of small squares.

Then draw the line ab (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 aS. Lastly through the points c, e, f, g, &c, draw lines parallel to ab: So shallabcd be the space upon which the monstrous projection is to be drawn; and is called the craticular ectype.

Then, in every areola, or small trapezium, of the space abcd, draw what appears contained in the corresponding areola of the original space ABCD: so shall there be produced a deformed image in the spacc abcd, which yet will appear in just proportion to an eye distant from it the length of EV, and raised above it by a height equal to VS.

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.

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.

To draw an Anamorphosis upon the convex surface of a cone. 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,

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.

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.

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, &c. Lastly, from the centre F, with the radii F1, F2, F3, &c, describe concentric circles. So will the craticular ectype be formed, whose areolas will appear equal to each other.

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 to an eye elevated above the vertex at a height equal to the height of the cone itself.

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.

And hence it will be easy to deform an image, in any other pyramid, whose base is any regular polygon.

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.

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.

For farther particulars, see Wolfius's Catoptrics and Dioptrics, and some other optical authois.

, in Astrology, the second house, or that part of the heavens which is 30 degrees from the horoscope.

The term anaphora is also sometimes applied promiscuously to some of the succeeding houses, as the 5th, the 8th, and the 11th. In this sense anaphora is the same as epanaphora, and stands opposed to cataphora.

ANASTROUS signs, in Astronomy, a name given to the duodecatemoria, or the 12 portions of the ecliptic, which the signs possessed anciently, but have since deserted by the precession of the equinox.

, 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 &c, upon which he wrote some treatises. His principal tenets may be reduced to the following:— 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 homæmeries, 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. —It was said he also wrote upon the Quadrature of the Circle; the treatise upon which, Plutarch says, he composed during his imprisonment at Athens.

, 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, &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 Lacedæmon. 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.

, 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, &c.

, 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 “Supplementum Apollonii Redivivi; (of Ghetaldus) sive analysis problematis hactenus desiderati ad Apollonii Pergæi doctrinam weri neusewn, a Marino Ghetaldo Patritio Ragusino hujusque, non ita pridem restitutam. In qua exhibetur mechanice æqualitatum tertii gradus sive solidarum, in quibus magnitudo omnino data, æquatur homogeneæ sub altero tantum coëfficiente ignoto. Huic subnexa est variorum problematum practice.” Paris, 1612, in 4to.

“*aitiologia: Pro Zetetico Apolloniani problematis a se jam pridem edito in supplemento Apollonii Redivivi. Ad clarissimum & ornatissimum virum Marinum Ghetaldum Patritium Ragusinum. In qua ad ea quae obiter mihi perstrinxit Ghetaldus respondetur, & analytices clarius detegitur.” Paris, 1615, in 4to. He published also,

“Francisci Vietæ Fontenacensis de Aequationum Recognitione & Emendatione Tractatus duo.” Paris, 1615, in 4to; with a Dedication, Prefaee, and an Appendix, by Anderson.

And Vieta's Angulares Sectiones, with the Demonstrations by Anderson.

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.

The time either of the birth or death of our author Alexander, has not come to my knowledge.

, 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.

, in Astronomy, 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.

She is sometimes called, in Latin, Persea, Mulier catenata, Virgo devota, &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 Abigail. And Schiller has also changed the figure of the constellation, for that of a sepulchre, and calls it the Holy Sepulchre.

This constellation contains about 27 stars that are visible to the naked eye; of which the principal are, a Andromeda's head; b in the girdle, and called mirach or mizar; g on the south foot, and named alamak, and sometimes alhames.

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.

, an instrument for measuring the force of the wind.

An instrument of this sort, it seems, was first invented by Wolfius in the year 1708, and first published in his Areometry in 1709, also in the Acta Eruditorum of the same year; afterwards in his Mathematical Dictionary, and in his Elem. Matheseos. He says he tried the goodness of it, and observes that the internal struc- 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)

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.

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.

Also in the same Transactions, for the year 1775, Dr. Lind gives a description of a very ingenious portable Wind-Gauge, 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 ab, 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 e and f 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.

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 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 √12 : √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. Table of the corresponding height of water, force on a square foot, and velocity of wind. Height of water.Force of wind.Velocity of wind per hour.Inches.Pounds.Miles. 0 1/44.5 0 1/26.4 1 5.29.0 210.412.7 315.615.5 420.818.0 526.020.1 631.2522.0 736.523.8 841.725.4 946.927.01052.128.41157.329.81262.531.0

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.

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.

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.

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 Wind-Gauge.

, 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.

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.

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 Acta Eruditorum for 1684.

, Angulus, in Geometry, 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.—Also the two lines AB and AC, are called the legs or the sides of the angle.

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.

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.—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.

Hence it matters not, with what radius the arc is described, by which an angle is measured, when great or small, as AD, or Ad, or any other: for the arcs DE, de, being similar, have the same ratio as their respective radii or circumferences, and therefore they contain the same number of degrees.—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.—And thus, in similar figures, the like or corresponding angles are equal.

The taking or measuring of angles, is an operation of great use and extent in surveying, navigation, geography, astronomy, &c. And the instruments chiefly used for this purpose, are quadrants, sextants, octants, theodolites, circumferentors, &c. Mr. Hadley invented an excellent instrument for taking the larger 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 small angles. See Hadley's Quadrant, Micrometer, and the Philos. Trans. Numbers 420, 425, and vol. 48. To measure the Quantity of an Angle.

1. On paper. 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.

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.

M. De Lagny gave, in several memoirs of the Royal Academy of Sciences, a new method of measuring angles, which he called Goniometry. 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° or the semicircle; and being turned into degrees &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.— 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; 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°, which is equivalent to 37 1/7 degrees, or 37° 8′ 34″ 2/7.

2. On the ground. 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.

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 Plotting and Protracting, and under the names of the several instruments.

To bisect a given angle, 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.

To trisect an angle, see Trisection.

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.

Angles are of various kinds and denominations. With regard to the form of their legs, they are divided into rectilinear, curvilinear, and mixed.

Rectilinear, or right-lined Angle, is that whose legs are both right lines; as the foregoing angle CAB.

Curvilinear Angle, is that whose legs are both of them curves.

Mixt, or mixtilinear Angle, is that of which one leg is a right line, and the other a curve.

With regard to their magnitude, angles are again divided into right and oblique, acute and obtuse.

Right Angle, 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.—Therefore the measure of a right angle is a quadrant of a circle, or 90°; and consequently all right angles are equal to each other.

Oblique Angle, is a common name for any angle that is not a right one; and it is either acute or obtuse.

Acute Angle, is that which is less than a righ<*> angle, or less than 90 degrees; as the angle BAD And

Obtuse Angle, is greater than a right angle, or whos<*> measure exceeds 90 degrees; as the angle BAE.

With regard to their situation in respect of eac<*> other, angles are distinguished into contiguous, adjacen<*> vertical, opposite, and alternate.

Contiguous Angles, are such as have the same vertex<*> and one leg common to both. As the angles BAD CAD, which have AD common.

Adjacent Angles, are those of which a leg of th<*> one produced forms a leg of the other: as the angle<*> GFH and IFH, which have the legs IF and FG in <*> straight line.—Hence adjacent angles are supplement<*> to each other, making together 180 degrees. An<*> therefore if one of these be given, the other will b<*> 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.

Vertical or opposite Angles, are such as have their legs mutually continuations of each other; as the two angles a and b, or c and d.—The property of these is, that the vertical or opposite angles are always equal to each other, viz, [angle] a = [angle] b, and [angle] c = [angle] d. And hence the quantity of an inaccessible angle of a field, &c, may be found, by measuring its accessible opposite angle.

Alternate Angles, are those made on the opposite sides of a line cutting two parallel lines; so, the angles e and f, or g and h, are alternates. And these are always equal to each other; viz, the [angle] c = [angle] f, or [angle] g = [angle] h.

External Angles, are the angles of a figure made without it, by producing its sides outwards; as the angles i, k, l, m. 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.

Internal Angles, are the angles within any figure, made by the sides of it; as the angles n, o, p, q.—In any right-lined figure, an internal angle as n, and its adjacent external angle k, together make two right angles, or 180 degrees; and all the internal angles n, o, p, q, 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.

Homologous, or like Angles, are such angles in two figures, as retain the same order from the first, in both figures.

Angle out of the centre, as G, is one whose vertex is not in the centre of the circle.—And its measure is half the sum (a+b)/2 of the arcs intercepted by its legs when it is within the circle, or half the difference (a-b)/2 when it is without.

Angle at the centre, 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.—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.

Angle at the circumference, is an angle whose vertex is somewhere in the circumference of a circle; as the angle ABC.

Angle in a segment, 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.—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.

Angle in a semicircle is an angle at the circumference contained in a semicircle, or standing upon a semicircle, or on a diameter.—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.

Angle of a segment, is that made by a chord with a tangent, at the point of contact. So IHK is the angle of the less segment IMH, and IHL, the angle of the greater segment INH.—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.

Angle of a semicircle, 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.

Angle of contact, 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, &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, &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 x=y2, where x is the absciss OP, and y the ordinate PQ, the parameter being 1<*> and if OR, OS, &c, be other parabolas to the same axis, tangent, and parameter, their ordinate y being PR, or PS, &c, and their equations x=y3, x=y4, x=y5, &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.

And farther, between the angles of contact of any two of this kind, may other angles of contact be found ad infinitum, which shall infinitely exceed each other, and yet the greatest of them be infinitely less than the smallest right-lined angle. So also x2=y3, x3=y4, x4=y5, &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 x2=y3, or the semicubical parabola, is infinitely greater than any circular angle of contact.

Angles are again divided into plane, spherical, and solid.

Plane Angles, are all those above treated of; which are defined by the inclination of two lines in a plane, meeting in a point.

Spherical Angle, 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.

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, &c, see Sphere, Spherical, and Spherical Trigonometry.

Solid Angle, 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, &c, see Solid Angle.

Angles of other less usual kinds and denominations, are also to be found in some books of Geometry. As,

Horned Angle, angulus cornutus, that which is made by a right line, whether a tangent or secant, with the circumserence of a circle.

Lunular Angle, angulus lunularis, is that which is formed by the intersection of two curve lines, the one concave, and the other convex.

Cissoid Angle, angulus cissoides, the inner angle made by two spherical convex lines intersecting each other.

Sistroid Angle, angulus sistroides, is that which is in form of a sistrum.

Pelecoid Angle, angulus pelecoides, is that in form of a hatchet.

, something relating to, or that hath angles.

At a distance, angular bodies appear round; the angles and small inequalities disappearing at a much less distance than the bulk of the body.

ANGULAR Motion, 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.—Thus, a pendulum has an angular motion about its centre of motion; and the planets have an angular motion about the sun.—Two moveable points M and O, of which the one describes

Angular motion is also a kind of compound motion composed of a circular and a rectilinear motion; like the wheel of a coach, or other vehicle.

ANIMATED needle, a needle touched with a magnet or load-stone.

, in Astronomy, something that returns every year, or which terminates with the year.

Annual motion of the earth. See Earth.

Annual argument of langitude. See Argument.

Annual epacts. See Epact.

Annual equation of the mean motion of the sun and moon, and of the moon's apogee and nodes. See EQUATION.

, a term for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c; payable from time to time; either annually, or at other intervals of time.

Annuities may be divided into such as are certain, and such as depend on some contingency, as the continuance of a life, &c.

Annuities are also divided into annuities in possession, and annuities in reversion; 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.

Annuities may be farther considered as payable either yearly, or half yearly, or quarterly, &c.

The present value of an annuity, is that sum, which, being improved at interest, will be sufficient to pay the annuity.

The present value of an annuity certain, payable yearly, is calculated in the following manner.—Let the annuity be 1, and let r denote the amount of 1l. for a year, or 1l. increased by its interest for one year. Then, 1 being the present value of the sum r, and having to find the present value of the sum 1, it will be, by proportion thus, r : 1 :: 1 : 1/r the present value of 1l. due a year hence. In like manner 1/r2 will be the present value of 1l. due 2 years hence; for r : 1 :: 1/r : 1/r2. In like manner 1/r3, 1/r4, 1/r5, &c, will be the present value of 1l. due at the end of 3, 4, 5, &c, years respectively; and in general, 1/rn will be the value of 1l. to be received after the expiration of n years. Consequently the sum of all these, or (1/r)+(1/r2)+(1/r3)+(1/r4)+ &c, contined to n terms, will be the present value of all the n years annuities. And the value of the perpetuity, is the sum of the series continued ad infinitum.

But this series, it is evident, is a geometrical progression, whose first term and common ratio are each 1/r, and the number of its terms n; and therefore the sum s of all the terms, or the present value of all the annual payments, will be .

When the annuity is a perpetuity, it is plain that the last term 1/rn vanishes, and therefore (1/(r-1))X(1/rn) also vanishes; and consequently the expression becomes barely s=1/(r-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.

If the annuity is not to be entered on immediately, but after a certain number of years, as m years; then the present value of the reversion is equal to the difference between two present values, the one for the first term of m years, and the other for the end of the last term n: that is, equal to the difference between .

Annuities certain differ in value, as they are made payable yearly, half-yearly, or quarterly. And by proceeding as above, using the interest or amount of a half year, or a quarter, as those for the whole year were used, the following set of theorems will arise; where <*> denotes, as before, the amount of 1l. and its interest for a year, and n the number of years, during which, any annuity is to be paid; also P denotes the perpetnity 1/(r-1), Y denotes (1/(r-1))-(1/(r-1))X(1/rn) 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 m part of a year.

Theor. 1. .

Theor. 2. .

Theor. 3. .

Theor. 4. Example 1.

Let the rate of interest be 4. per cent, and the term 5 years; and consequently r = 1.04, n = 5, P = 25; also let m = 12, or the interest payable monthly in theorem 4: then the present value of such annuity of 1l. a year, for 5 years, according as it is supposed payable 1l. yearly, or (1/2)l. every half year, or (1/4)l. every quarter, or (1/12)l. every month or (1/12)th part of a year, will be as follows:

Example 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

Example 3. And if the term be 50 years, the values will be

Example 4. Also if the term be 100 years, the values will be

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 n varies; and, for any given rate of interest, it is evident that the difference, for any periods m of payments, first increases from nothing as the term n increases, when n is 0, to some certain finite term or value of n, when the difference D is the greatest or a maximum; and that afterwards, as n increases more, that difference will continually decrease to nothing again, and vanish when n is infinite: also the term or value of n, for the maximum of the difference, will be different according to the periods of payment, or value of m. And the general value of n, when the difference is a maximum between the yearly payments and the payments of m times in a year, is expressed by this formula, viz, , where l. denotes the logarithm of the quantity following it. Hence, taking the different values of m, viz, 2 for half years, 4 for quarters, 12 for monthly payments, &c, and substituting in the general formula, the term or value of n 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.

Annuities 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 1l. due at the end of 1 year, amounts to r at the end of another year, and to r2 at the end of the 3d year, and to r3 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 + r + r2 + r3 + r4 to rn-x, continued till the last term be rn-x, or till the number of terms be n, the number of years the annuity is forborn. But the sum of this geometrical progression is (rn-1)/(r-1), which therefore is the amount of 1l. annuity forborn for n years. And this quantity being multiplied by any other annuity a, instead of 1, will produce the amount for that other annuity.

But the amounts of annuities, or their present values, are easiest found by the two following tables of numbers for the annuity of 1l. ready computed from the foregoing principles. Table I.The Amount of an Annuity of 1l. at Comp. Interest.Yrs.at 3 per cent.3 1/2 per cent.4 per cent.4 1/2 per cent.5 per cent.6 per cent.11.000001.000001.000001.000001.000001.0000022.030002.035002.040002.045002.050002.0600033.090903.106233.121603.137033.152503.1836044.183634.214944.246464.278194.310134.3746255.309145.362475.416325.470715.525635.6370966.468416.550156.632986.716896.801916.9753277.662467.779417.898298.019158.142018.3938488.892349.051699.214239.380019.549119.89747910.1591110.3685010.5828010.8021111.0265611.491321011.4638811.7313912.0061112.2882112.5778913.180791112.8078013.1419913.4863513.8411814.2067914.971641214.1920314.6019615.0258115.4640315.9171316.869941315.6177916.1130316.6268417.1599117.7129818.882141417.0863217.6769918.2919118.9321119.5986321.015071518.5989119.2956820.3235920.7840521.5785623.275971620.1568820.9710321.8245322.7193423.6574925.672531721.7615922.7050223.6975124.7417125.8403728.212881823.4144424.4996925.6454126.8550828.1323830.905651925.1168726.3571827.6712329.0635630.5390033.759992026.8703728.2796829.7780831.3714233.0659536.785592128.6764930.2694731.9692033.7831435.7192539.992732230.5367832.3289034.2479736.3033838.5052143.392292332.4528834.4604136.6178938.9370341.4304846.995832434.4264736.6665339.0826041.6892044.5020050.815582536.4592638.9498641.6459144.5652147.7271054.864512638.5530441.3131044.3117447.5706451.1134559.156382740.7096343.7590647.0842150.7113254.6691363.705772842.9309246.2906349.9675853.9933358.4025868.528112945.2188548.9108052.9662957.4230362.3227173.639803047.5754251.6226856.0849461.0070766.4388579.058193150.0026854.4294759.3283464.7523970.7607984.801683252.5027657.3345062.7014768.6662575.2988390.889783355.0778460.3412166.2095372.7562380.0637797.343163457.7301863.4531569.8579177.0302685.06696104.183753560.4620866.6740173.6522281.4966290.32031111.434783663.2759470.0076077.5983186.1639795.83632119.120873766.1742273.4578781.7022591.04134101.62814127.268123869.1594577.0288985.9703496.13820107.70955135.904213972.2342380.7249190.40915101.46442114.09502145.058464075.4012684.5502895.02552107.03032120.79977154.761974178.6633088.5095499.82654112.84669127.83976165.047684282.0232092.60737104.81960118.92479135.23175175.950544385.4838996.84863110.01238125.27640142.99334187.507584489.04841101.23833115.41288131.91384151.14301199.758034592.71986105.78167121.02939138.84997159.70016212.743514696.50146110.48403126.87057146.09821168.68516226.5081247100.39650115.35097132.94539153.67263178.11942241.0986148104.40840120.38826139.26321161.58790188.02539256.5645349108.54065125.60185145.83373169.85936198.42666272.9584050112.79687130.99791152.66708178.50303209.34800290.3359051117.18077136.58284159.77377187.53566220.81540308.7560652121.69620142.36324167.16472196.97477232.85617328.2814253126.34708148.34595174.85131206.83863245.49897348.9783154131.13750154.53806182.84536217.14637258.77392370.91701 Table II.The present Value of an Annuity of 1l.Yrs.at 3 per cent.3 1/2 per cent.4 per cent.4 1/2 per cent.5 per cent.6 per cent.10.970870.966180.961540.956940.952380.9434021.913471.899691.886101.872671.859411.8333932.828612.801642.775092.748962.723252.6730143.717103.673083.629903.587533.545953.4651154.579714.515054.451824.389984.329484.2123665.417195.328555.242145.157875.075694.9173276.230286.114546.002055.892705.786375.5823887.019696.873966.732746.595896.463216.2097997.786117.607697.435337.268797.107826.80169108.530208.316618.110907.912727.721737.36009119.252629.001558.760488.528928.305417.88687129.954009.663339.385079.118588.863258.383841310.6349610.302749.985659.682859.393578.852681411.2960710.9205210.5631210.222839.898649.294981511.9379411.5174111.1183910.7395510.379669.712251612.5611012.0941211.6523011.2340210.8377710.105901713.1661212.6513212.1656711.7071911.2740710.477261813.7535113.1896812.6593012.1599911.6895910.827601914.3238013.7098413.1339412.5932912.0853211.158122014.8774714.2124013.5903313.0079412.4622111.469922115.4150214.6979714.0291613.4047212.8211511.764082215.9369215.1671214.4511213.7844213.1630012.041582316.4436115.6204114.8568414.1477713.4885712.303382416.9355416.0583715.2469614.4954813.7986412.550362517.4131516.4815115.6220814.8282114.0939412.783362617.8768416.8903515.9827715.1466114.3751913.003172718.3270317.2853616.3295915.4513014.6430313.210532818.7641117.6670216.6630615.7428714.8981313.406162919.1884518.0357716.9837116.0218915.1410713.590723019.6004418.3920517.2920316.2888915.3724513.764833120.0004318.7362817.5884916.5443915.5928113.929093220.3887719.0688717.8735516.7888915.8026814.084043320.7657919.3902118.1476517.0228616.0025514.230233421.1318419.7006818.4112017.2467616.1929014.368143521.4872220.0006618.6646117.4610116.3741914.498253621.8322520.2904918.9082817.6660416.5468514.620993722.1672420.<*>705319.1425817.8622416.7112914.736783822.4924620.8410919.3678618.0499916.8678914.846023922.8082221.1025019.5844818.2296617.0170414.949074023.1147721.3550719.7927718.4015817.1590915.046304123.4124021.5991019.9930518.5661117.2943715.138024223.7013621.8348820.1856318.7235517.4232115.224544323.9819022.0626920.3707918.8742117.5459115.306174424.2542722.2827920.5488419.0183817.6627715.383184524.5187122.4954520.7200419.1563517.7740715.455834624.7754522.7009220.8846519.2883717.8800715.524374725.0247122.8994421.0429419.4147117.9810215.589034825.2667123.0912421.1951319.5356118.0771615.650034925.5016623.2765621.3414719.6513018.1687215.707575025.7297623.4556221.4821819.7620118.2559315.761865125.9512323.6286221.6174919.8679518.3389815.813085226.1662423.7957621.7475819.9693318.4180715.861395326.3749923.9572621.8726720.0663418.4934015.906975426.5776624.1133021.9929620.1591818.5651515.94998 The Use of Table I.

To find the Amount of an annuity forborn any number of years. Take out the amount from the 1st table, for the proposed years and rate of interest; then multiply it by the annuity in question; and the product will be its amount for the same number of years, and rate of interest.

And the converse to find the rate or time.

Exam. 1. To find how much an annuity of 50l. will amount to in 20 years at 3 1/2 per cent. compound interest.—On the line of 20 years, and in the column of 3 1/2 per cent, stands 28.27968, which is the amount of an annuity of 1l. for the 20 years; and therefore 28.27968 multiplied by 50, gives 1413.9841. or 1413l. 19s. 8d. for the answer.

Exam. 2. In what time will an annuity of 20l. amount to 1000l. at 4 per cent. compound interest?—Here the amount of 1000l. divided by 20l. the annuity, gives 50, the amount of 1l. annuity for the same time and rate. Then, the nearest tabular number in the column of 4 per cent. is 49.96758, which standing on the line of 28, shews that 28 years is the answer.

Exam. 3. If it be required to find at what rate of interest an annuity of 20l. will amount to 1000l. forborn for 28 years.—Here 1000 divided by 20 gives 50 as before. Then looking along the line of 28 years, for the nearest to this number 50, I find 49.96758 in the column of 4 per cent. which is therefore the rate of interest required. The Use of Table II.

Exam. 1. To find the present value of an annuity of 50l. which is to continue 20 years, at 3 1/2 per cent.— By the table, the present value of 1l. for the same rate and time, is 14.21240; therefore 14.2124 X 50 = 710.62l. or 710l. 128. 4d. is the present value sought.

Exam. 2. To find the present value of an annuity of 20l. to commence 10 years hence, and then to continue for 40 years, or to terminate 50 years hence, at 4 per cent. interest.—In such cases as this, it is plain we have to sind the difference between the present values of two equal annuities, for the two given times; which therefore will be effected by subtracting the tabular value of the one term from that of the other, and multiplying by the annuity. Thus, tabular value for 50 years21.48218tabular value for 10 years8.11090the difference13.37128mult. by20gives267.4256or2671. 8s. 6d.the answer.

The foregoing observations, rules, and tables, contain all that is important in the doctrine of annuities certain. And for farther information, reference may be had to arithmetical writings, particularly Malcolm's Arithmetic, page 595; Simpson's Algebra, sect. 16; Dodson's Mathematical Repository, page 298, &c; Jones's Synopsis, ch. 10; Philos. Trans. vol. lxvi, page 109.

For what relates to the doctrine of annuities on lives, see Assurance, Complement, Expectation, Life Annuities, Reversions, &c.

, in Architecture, are small square members, in the Doric capital, placed under the quarter round.

Annulet is also used for a narrow flat moulding, common to other parts of a column, as well as to the capital; and so called, because it encompasses the column round. In which sense annulet is frequently used for aguette, or little astragal.

, a species of Voluta. See also Ring.

ANOMALISTICAL Year, in Astronomy, called <*>lso periodical year, is the space of time in which the <*>arth, or a planet, passes through its orbit. The ano- <*>nalistical, or common year, is somewhat longer than <*>he tropical year; by reason of the precession of the <*>quinox.

And the apses of all the planets have a like progresive motion; by which it happens that a longer time is <*>ecessary to anive at the aplielion, which has advanced <*> little, than to arrive at the same fixed star. For eximple, the tropical revolution of the sun, with respect to the equinox, is365da5h48m45s;but the sidereal, or return to the same star,3656 911;and the anomalistic revolution is36561520, because the sun's apogee advances each year 65″ 1/2 with respect to the equinoxes, and the sun cannot arrive at the apogee till he has passed over the 65″ 1/2 more than the revolution of the year answering to the equinoxes.

To find the anomalistic revolution, say, As the whole secular motion of a planet minus the motion of its aphelion, is to 100 years or 3155760000 seconds, so is 360°, to the duration of the anomalistic revolution.

, is something irregular, or that deviates from the ordinary rule and method of other things of the same kind.

, in Astronomy, is an irregularity in the motion of a planet, by which it deviates from the aphelion or apogee; or it is the angular distance of the planet from the aphelion or apogee; that is, the angle formed by the line of the apses, and another line drawn through the planet.

Kepler distinguishes three kinds of anomaly; mean, eccentric, and true.

Mean or Simple Anomaly, in the ancient astronomy, is the distance of a planet's mean place from the apogee. Which Ptolomy calls the angle of the mean motion.

But in the modern astronomy, in which a planet P is considered as describing an ellipse APB about the sun S, placed in one focus, it is the time in which the planet moves from its aphelion A, to the mean place or point of its orbit P.

Hence, as the elliptic area ASP is proportional to the time in which the planet describes the arc AP, that area may represent the mean anomaly.—Or, if PD be drawn perpendicular to the transverse axis AB, and meet the circle in D described on the same axis; then the mean anomaly may also be represented by the cir- cular trilineal ASD, which is always proportional to the elliptic one ASP, as is proved in my Mensuration, pr. 3, page 296, second edition.—Or, drawing SG perpendicular to the radius DC produced; then the mean anomaly is alse proportional to SG + the circular arc AD, as is demonstrated by Keil in his Lect. Astron.—Hence, taking DH = SG, the arc AH, or angle ACH will be the mean anomaly in practice, as expressed in degrees of a circle, the number of those degrees being to 360°, as the elliptic trilineal area ASP, is to the whole area of the ellipse; the degrees of mean anomaly, being those in the arc AH, or angle ACH.

Eccentric Anomaly, or of the centre, in the modern astronomy, is the arc AD of the circle ADB intercepted between the apsis A and the point D determined by the perpendicular DPE to the line of the apses, drawn through the place P of the planet. Or it is the angle ACD at the centre of the circle.—Hence the eccentric anomaly is to the mean anomaly, as AD to AD + SG, or as AD to AH, or as the angle ACD to the angle ACH.

True or Equated Anomaly, is the angle ASP at the sun, which the planet's distance AP from the aphelion, appears under; or the angle formed by the radius vector or line SP drawn from the sun to the planet, with the line of the apses.

The true anomaly being given, it is easy from thence to find the mean anomaly. For the angle ASP, which is the true anomaly, being given, the point P in the ellipse is given, and thence the proportion of the area ASP to the whole ellipse, or of the mean anomaly to 360 degrees. And for this purpose, the following easy rules for practice are deduced from the properties of the ellipse, by M. de la Caille in his Elements of Astronomy, and M. de la Lande, art. 1240 &c of his astronomy: 1st, As the square root of SB the perihelion distance, is to the square root of SA the aphelion distance, so is the tangent of half the true anomaly ASP, to the tangent of half the eccentric anomaly ACD. 2nd, The difference DH or SG between the eccentric and mean anomaly, is equal to the product of the eccentricity CS, by the sine of SCG the eccentric anomaly just found. And in this case, it is proper to express the eccentricity in seconds of a degree, which will be found by this proportion, as the mean distance 1: the eccentricity :: 206264.8 seconds, or 57° 17′ 44″.8, in the arch whose length is equal to the radius, to the seconds in the are which is equal to the eccentricity CS; which being multiplied by the sine of the eccentric anomaly, to radius 1, as above, gives the seconds in SG, or in the are DH, being the difference between the mean and eccentric anomalies. 3d, To find the radius vector SP, or distance of the planet from the sun, say either, as the sine of the true anomaly is to the sine of the eccentric anomaly, so is half the less axis of the orbit, to the radius vector SP; or as the sine of half the true anomaly is to the sine of half the eccentric anomaly, so is the square root of the perihelion distance SB, to the square root of the radius vector or planet's distance SP.

But the mean anomaly being given, it is not so easy to find the true anomaly, at least by a direct process<*> Kepler, who first proposed this problem, could not find a direct way of resolving it, and therefore made use of an indirect one, by the rule of false position, as may be seen page 695 of Kepler's Epitom. Astron. Copernic. See also §628 Wolfius Elem. Astron. Now the easiest method of performing this operation, would be to work first for the eccentric anomaly, viz, assume it nearly, and from it so assumed compute what would be its mean anomaly by the rule above given, and find the difference between this result and the mean anomaly given; then assume another eccentric anomaly, and proceed in the same way with it, finding another computed mean anomaly, and its difference from the given one; and treating these differences as in the rule of position for a nearer value of the eccentric anomaly: repeating the operation till the result comes out exact. Then, from the eccentric anomaly, thus found, compute the true anomaly by the 1st rule above laid down.

Of this problem, Dr. Wallis first gave the geometrical solution by means of the protracted cycloid; and Sir Isaac Newton did the same at prop. 31 lib. 1 Princip. But these methods being unsit for the purpose of the practical astronomer, various series for approximation have been given, viz, several by Sir Isaac Newton in his Fragmenta Epistolarum, page 26, as also in the Schol. to the prop. above-mentioned, which is his best, being not only fit for the planets, but also for the comets, whose orbits are very eccentric. Dr. Gregory, in his Astron. lib. 3, has also given the solution by a series, as well as M. Reyneau, in his Analyse Demontrée, page 713, &c. And a better still for converging is given by Keil in his Prælect. Astron. page 375; he says, if the are AH be the mean anomaly, calling its sine e, consine f, the eccentricity g, also putting z = ge, and ; then the eccentric anomaly AD will be , supposing r = 57.29578 degrees; of which the first term rz/a is sufficient for all the planets, even for Mars itself, where the error will not exceed the 200th part of a degree; and in the orbit of the earth, the error is less than the 10000th part of a degree.

Dr. Seth Ward, in his Astronomia Geometrica, takes the angle AFP at the other focus, where the sun is not, for the mean anomaly, and thence gives an elegant solution. But this method is not sufficiently accurate when the orbit is very eccentric, as in that of the planet Mars, as is shewn by Bullialdus, in his defence of the Philolaic. Astron. against Dr. Ward. However, when Newton's correction is made, as in the Schol. abovementioned, and the problem resolved according to Ward's hypothesis, Sir Isaac affirms that, even in the orbit of Mars, there will scarce ever be an error of more than one second.

ANSÆ, Anses, in Astronomy, those seemingly prominent parts of the ring of the planet Saturn, discovered in its opening, and appearing like handles to the body of the planet; srom which appearance the name ansæ is taken.

, in Astronomy, a small star, of the 5th or 6th magnitude, in the milky-way, between the eagle and swan, first brought into order by Hevelius.

ANTARCTIC pole, denotes the southern pole, or southern end of the earth's axis.—The stars near the antarctic pole never appear above our horizon in these latitudes.

ANTARCTIC circle, is a small circle parallel to the equator, at the distance of 23° 28′ from the antarctic or south pole.—At one time of the year the sun never rises above the horizon of any part within this circle; and at other times he never sets.

, in Astronomy, the scorpion's heart; a fixed star of the first magnitude, in the constellation Scorpio.

ANTECANIS is used by some astronomers, to denote the constellation otherwise called canis minor, or the star procyon. It is so called, as preceding, or being the forerunner of the canis major, and rising a little before it.

, of a ratio, denotes the first of the two terms of the ratio, or that term which is compared with the other. Thus, if the ratio be 2 to 3. or a to b; then 2 or a is the antecedent.

ANTECEDENTAL Method, is a branch of general geometrical proportion, or universal comparison, and is derived from an examination of the Antecedents of ratios, having given consequents, and a given standard of comparison, in the various degrees of augmentation and diminution, which they undergo by composition and decomposition. This is a method invented by Mr. James Glenie, and published by him in 1793; a method which he says he always used instead of the fluxional and differential methods, and which is totally unconnected with the ideas of motion and time. See the author's treatise above-mentioned, and also his Doctrine of Universal Comparison, or General Proportion, 1789, upon which it is founded.

, a term used by astronomers when a planet &c moves westward, or contrary to the order of the signs aries, taurus, &c.—Like as when it moves eastward, or according to the order of the signs aries, taurus, &c, it is then said to move in consequentia.

, or Antoeci, in Geography, the inhabitants of the earth which occupy the same semicircle of the same meridian, but equally distant from the equator, the one north and the other south; as Peloponnesus and the Cape of Good Hope.

These have their noon, or midnight, or any other hour at the same time; but their seasons are contrary, being spring to the one, when it is autumn with the other; and summer with the one, when it is winter with the other; also the length of the day to the one, is equal to the length of night to the other.

, in Architecture, are small pilastres placed at the corners of buildings.

, in Architecture, figures of men, beasts, &c, placed as ornaments to buildings.

, in Architecture, a porch before a door; also that part of a temple, which is called the outer temple, and lies between the body of the temple and the portico.

, the complement of the logarithm of a sine, tangent, secant, &c, to that of the radius. This is sound by beginning at the lest hand, subtracting each sigure from 9, and the last figure from 10.

, in Astronomy, a part of the constellation aquila, or the eagle.

ANTIOCHIAN Sect, or Academy, a name given to the sifth academy or branch of academics. It took its name from being founded by Antiochus, a philosopher contemporary with Cicero; and it succeeded the Philonian academy. Though Antiochus was really a stoic, and only nominally an academic.

Antiochian epocha, a method of computing time from the proclamation of liberty granted to the city of Antioch, about the time of the battle of Pharsalia.

, in Geometry, are those lines which make equal angles with two other lines, but contrary ways; that is, calling the former pair the first and 2d lines, and the latter pair the 3d and 4th lines, if the angle made by the 1st and 3d liues be equal to the angle made by the 2d and 4th, and contrariwise the angle made by the 1st and 4th equal to the angle made by the 2d and 3d; then each pair of lines are antiparallels with respect to each other, viz, the first and 2d, and the 3d and 4th. So, if AB and AC be any two lines, and FC and FE be two others, cutting themso, that the angle B is equal to the angle E, and the angle C is equal to the angle D; then BC and DE are antiparallels with respect to AB and AC; also these latter are antiparallels with regard to the two former.—See also Subcontrary.

It is a property of these lines, that each pair cuts the other into proportional segments, taking them alternately, viz AB : AC :: AE : AD :: DB : EC, and FE : FC :: FB : FD :: DE : BC.

, in Geography, are the inhabitants of two places on the earth which lie diametrically opposite to each other, or that walk feet to feet; that is, if a line be continued down from our feet, quite through the centre of the earth, till it arrive at the surface on the other side, it will fall on the feet of our Antipodes, and vice versa.——Antipodes are 180 degrees distant from each other every way on the surface of the globe; they have equal latitudes, the one north and the other south, but they differ by 180 degrees of longitude: they have therefore the same climates or degrees of heat and cold, with the same seasons and length of days and nights; but all of these at contrary times, it being day to the one, when it is night to the other, summer to the one when it is winter to the other, &c: they have also the same horizon, the one being as far distant on the one side, as the other on the other side, and therefore when the sun, &c, rises to the one, it sets to the other. The Antipodes to London are a part a little south of New Zealand.

It has been said that Plato first started the notion of Antipodes, and gave them the name; which is likely enough, as he conceived that the earth was of a glo- bular figure. But there have been many disputes upon this point, and the fathers of the church have greatly opposed it, especially Lactantius and Augustine, who laughed at it, and were greatly perplexed to think how men and trees should hang pendulous in the air with their feet uppermost, as he thought they must do, in the other hemisphere.

, or ANTISCII, in Geography, are people who dwell in the opposite hemispheres of the earth, as to north and south, and whose shadows at noon fall in contrary directions. This term is more general than antæci, with which it is often confounded. The Antiscians stand contradistinguished from Periscians.

Antiscii is also used sometimes, among Astrologers, for two points of the heavens equally distant from the tropics. Thus the signs Leo and Taurus are accounted antiscii to each other.

ANTŒCI, see Antecians.

, in Geometry, is used for the space left between two lines which mutually incline towards each other, to form an angle.

, or Aphelium, in Astronomy, that point in the orbit of the earth, or a planet, in which it is at the greatest distance from the sun. Which is the point A (in the fig. to the art. Anomaly) or extremity of the transverse axis, of the elliptic orbit, farthest from the focus S, where the sun is placed; and diametrically opposite to the perihelion B, or nearer extremity of the same axis. In the Ptolemaic system, or in the supposition that the sun moves about the earth, the aphelion becomes the apogee.

The times of the aphelia of the primary planets, may be known by their apparent diameter appearing the smallest, and also by their moving slowest in a given time. Calculations and methods of finding them have been given by many astronomers, as Ricciolli, Almag. Nov. lib. 7, sect. 2 and 3; Wolfius, Elem. Astron. § 659; Dr. Halley, Philos. Trans. No. 128; Sir I. Newton, Princip. lib. 3, prop. 14; Dr. Gregory, Astron. lib. 3, prop. 14; Keil, Astron. Lect.; De la Lande, Memoires de l'Acad. 1755, 1757, 1766, and in his Astron. liv. 22; also in the writings of MM. Euler, D'Alembert, Clairaut, &c, upon attraction.

The aphelia of the planets are not fixed; for their mutual actions upon one another keep those points of their orbits in a continual motion, which is greater or less in the different planets. This motion is made in consequentia, or according to the order of the signs; and Sir I. Newton shews that it is in the sesquiplicate ratio of the distance of the planet from the sun, that is, as the square root of the cube of the distance.

The quantities of this motion, as well as the place of the aphelion for a given time, are variously given by different authors. Kepler states them, for the year 1700 as in the following table. Planets.Aphelion.Annual Motion.Mercury

 8°25′30″1′45″Venus

 32427118Mars

 051291 7Jupiter

 81040047Saturn

28 348110The Earth

 82530 By De la Hire they are given as follows, for the same year 1700. Planets.Aphelion.Annual Motion.Mercury

13° 3′40″1′39″Venus

 65610126Mars

 035251 7Jupiter

101714134Saturn

291441122 And De la Lande states them as follows, for the year 1750. Planets.Aphelion.Secular Motion.Mercury 813°33′1°57′40″Venus10 813410 0Mars 5 12815140Jupiter 6102214320Saturn 8295322320The Earth 9 83814910

Of the new planet, Herschel, or Georgium Sidus, the aphelion for 1790 was 11<*>23°29′42″, and its annual motion 50″ 3/8. See Connoissance des Temps, 1786 and 1787.

, in Chronology, denotes the eleventh month in the Bythinian year, commencing on the 25th of July in ours.

APIAN or Appian (Peter), called in German Bienewitz, a celebrated astronomer and mathematician, was born at Leisnig or Leipsick in Misnia, 1495, and made professor of mathematics at Ingolstadt, in 1524, where he died in the year 1552, at 57 years of age.

Apian wrote treatises upon many of the mathematical sciences, and greatly improved them; more especially astronomy and astrology, which in that age were much the same thing; also geometry, geography, arithmetic, &c. He particularly enriched astronomy with many instruments, and observations of eclipses, comets, &c. His principal work was the Astronomicum Cæsareum, published in folio at Ingolstadt in 1540, and which contains a number of interesting observations, with the descriptions and divisions of instruments. In this work he predicts eclipses, and constructs the figures of them in plano. In the 2d part of the work, or the Meteoroscopium Planum, he gives the description of the most accurate astronomical quadrant, and its uses. To it are added observations of five different comets, viz, in the years 1531, 1532, 1533, 1538, and 1539; where he first shews that the tails of comets are always projected in a direction from the sun.

Apian also wrote a treatise on Cosmography, or Geographical Instruction, with various mathematical instruments. This work Vossius says he published in 1524, and that Gemma Frisius republished it in 1540. But Weidler says he wrote it only in 1530, and that Gemma Frisius published it at Antwerp in 1550 and 1584, with observations of many eclipses. The truth may be, that perhaps all these editions were published.

In 1533 he made, at Norimberg, a curious instrument, which from its sigure he called Folium Populi; which, by the sun's rays, shewed the hour in all parts of the earth, and even the unequal hours of the Jews.

In 1534 he published his Inscriptiones Orbis.

In 1540, his Instrumentum Sinuum, sive Primi Mobilis, with 100 problems.

Beside these, Apian was the author of many other works: among which may be mentioned the Ephemerides from the year 1534 to 1570: Books upon Shadows: Arithmetical Centilogues: Books upon Arithmetic, with the Rule of Coss (Algebra) demonstrated: Upon Gauging: Almanacs, with Astrological directions: A book upon Conjunctions: Ptolemy, with very correct figures, drawn in a quadrangular form: Ptolemy's works in Greek: Books of Eclipses: the works of Azoph, a very ancient astrologer: the works of Gebre: the Perspective of Vitello: of Critical Days, and of the Rainbow: a new Astronomical and Geometrical Radius, with various uses of Sines and Chords: Universal Astrolabe of Numbers: Maps of the World, and of particular countries: &c, &c.

Apian left a son, who many years afterwards taught mathematics at Ingolstadt, and at Tubinga. Tycho has preserved (Progymn. p. 643) his letter to the Landgrave of Hesse, in which he gives an opinion on the new star in Cassiopeia, of the year 1572.

One of the comets observed by Apian, viz, that of 1532, had its elements nearly the same as of one observed 128 1/4 years after, viz, in 1661, by Hevelius and other Astronomers; from hence Dr. Halley judged that they were the same comet, and that therefore it might be expected to appear again in the beginning of the year 1789. But it was not found that it returned at this period, although the astronomers then looked anxiously for it; and it is doubtful whether the disappointment might be owing to its passing unobserved, or to any errors in the observations of Apian, or to its period being disturbed and greatly altered by the actions of the superior planets, &c.

, musca, the Bee, or Fly, in Astronomy, one of the southern constellations, containing 4 stars.

, in Astronomy, is the period of a planet, or the time employed in returning to the same point of the zodiac from whence it set out.

, Apogæum, in Astronomy, that point in the orbit of the sun, moon, &c, which is sarthest distant from the earth. It is at the extremity of the line of the apsides; and the point opposite to it is called the perigee, where the distance from the earth is the least.

The ancient astronomers, considering the earth as the centre of the system, chiefly regarded the apogee and perigee: but the moderns, placing the sun in the centre, change these terms for the aphelion and perihelion. —The apogee of the sun, is the same thing as the aphelion of the earth; and the perigee of the sun is the same as the perihelion of the earth.

The manner of finding the apogee of the sun or moon, is shewn by Ricciolus, Almag. Nov. lib. 3, cap. 24; by Wolfius in Elem. Astr. § 618; by Cassini, De la Hire, and many others: see also Memoires de l'Academie, the Philos. Trans. vol. 5, 47, &c.

The quantity of motion in the apogee may be found by comparing two observations of it made at a great distance of time; converting the difference into minutes, and dividing them by the number of years elapsed between the two observations; the quotient gives the annual motion of the apogee. Thus, srom an observation made by Hipparchus in the year before Christ 140, by which the sun's apogee was found 5° 30′ of

But the moon's apogee moves unequably. When she is in the syzygy with the sun, it moves forwards; but in the quadratures, backwards; and these progressions and regressions are not equable, but it goes forward slower when the moon is in the quadratures, or perhaps goes retrograde; and when the moon is in the syzygy, it goes forward the fastest of all.—See also Newton's Theory of the Moon for more upon this subject.

, a celebrated architect, under Trajan and Adrian, was born at Damascus, and flourished about the year of Christ 100. He had the direction of the stone bridge which Trajan ordered to <*>e built over the Danube in the year 104, which was asteemed the most magnifieent of all the works of that emperor. Adrian, one day as Trajan was discoursing with this architect upon the buildings he had raised at Rome, would needs give his judgment, in which he shewed that he knew nothing of the matter. Apollodorus turned upon him bluntly, and said to him, Go paint Citruls, for you are very ignorant of the subject we are talking upon. Adrian at this time boasted of his painting Citruls well. This was the first step towards the ruin of Apollodorus; a slip which he was so far from attempting to retrieve, that he even added a new offence, and that too after Adrian was advanced to the empire, upon the following occasion: Adrian sent to him the plan of a temple of Venus; and though he asked his opinion, yet to shew that he had no need of him, and that he did not mean to be directed by it, the temple was already built. Apollodorus wrote his opinion very freely, and remarked such essential faults in it, as the emperor could neither deny nor remedy. He shewed that it was neither high nor large enough; that the statues in it were disproportioned to its bulk: for, said he, if the goddesses should have a mind to rise and go out, they could not do it. This put Adrian into a great passion, and prompted him to the destruction of Apollodorus. He banished him at first; then under the pretext of certain supposed crimes, of which he had him accused, he at last put him to death.

, of Perga, a city in Pamphilia, was a celebrated geometrician who flourished in the reign of Ptolemy Euergetes, about 240 years before Christ; being about 60 years after Euclid, and 30 years later than Archimedes. He studied a long time in Alexandria under the disciples of Euclid; and afterwards he composed several curious and ingenious geometrical works, of which only his books of Conic Sections are now extant, and even these not perfect. For it appears from the author's dedicatory epistle to Eudemus, a geometrician in Pergamus, that this work consisted of 8 books; only 7 of which however have come down to us.

From the Collections of Pappus, and the Commentaries of Eutocius, it appears that Apollonius was the author of various pieces in geometry, on account of which he acquired the title of the Great Geometrician. His Conics was the principal of them. Some have thought that Apollonius appropriated the writings and discoveries of Archimedes; Heraclius, who wrote the life of Archimedes, affirms it; though Eutocius endeavours to refute him. Although it should be allowed a groundless supposition, that Archimedes was the first who wrote upon Conics, notwithstanding his treatise on Conics was greatly esteemed; yet it is highly probable that Apollonius would avail himself of the writings of that author, as well as others who had gone before him; and, upon the whole, he is allowed the honour of explaining a difficult subject better than had been done before; having made several improvements both in Archimedes's problems, and in Euclid. His work upon Conics was doubtless the most perfect of the kind among the ancients, and in some respects among the moderns also. Before Apollonius, it had been customary, as we are informed by Eutocius, for the writers on Conics to require three disserent sorts of cones to cut the three different sections from, viz, the parabola from a right angled cone, the ellipse from an acute, and the hyperbola from an obtuse cone; because they always supposed the sections made by a plane cutting the cones to be perpendicular to the side of them: but Apollonius cut his sections all from any one cone, by only varying the inclination or position of the cutting plane; an improvement that has been followed by all other authors since his time. But that Archimedes was acquainted with the same manner of cutting any cone, is sufficiently proved, against Eutocius, Pappus, and others, by Guido Ubaldus, in the beginning of his Commentary on the 2d book of Archimedes's Equiponderantes, published at Pisa in 1588.

The first four books of Apollonius's Conics only have come down to us in their original Greek language; but the next three, the 5th, 6th, and 7th, in an Arabic version; and the 8th not at all. These have been commented upon, translated, and published by various authors. Pappus, in his Mathematical Collections, has lest some account of his various works, with notes and lemmas upon them, and particularly on the Conics. And Eutocius wrote a regular elaborate commentary on the propositions of several of the books of the Conics.

The first four books were badly translated by Joan. Baptista Memmius. But a better translation of these in Latin was made by Commandine, and published at Bononia in 1566.—Vossius mentions an edition of the Conics in 1650; the 5th, 6th, and 7th books being recovered by Golius.—Claude Richard, Professor of mathematics in the imperial college of his order at Madrid, in the year 1632, explained, in his public lectures, the first four books of Apollonius, which were printed at Antwerp in 1655, in folio.—And the Grand Duke Ferdinand the 2d, and his brother Prince Leopold de Medicis, employed a professor of the Oriental languages at Rome to translate the 5th, 6th, and 7th books into Latin. These were published at Florence in 1661, by Borelli, with his own notes, who also maintains that these books are the genuine production of Apollonius, by many strong authorities, against Mydorgius and others, who suspected that these three books were not the real production of Apollonius.

As to the 8th book, some mention is made of it in a book of Golius's, where he had written that it had not been translated into Arabic; because it was wanting in the Greek copies, from whence the Arabians translated the others. But the learned Mersenne, in the preface to Apollonius's Conics, printed in his Synopsis of the Mathematics, quotes the Arabic philosopher Aben Nedin for a work of his about the year 400 of Mahomet, in which is part of that 8th book, and who asserts that all the books of Apollonius are extant in his language, and even more than are enumerated by Pappus; and Vossius says he has read the same; De Scientiis Mathematicis, pa. 55.—A neat edition of the first four books in Latin was published by Dr. Barrow, in 4to, at London in 1675.—A magnificent edition of all the 8 books, was published in folio, by Dr. Halley, at Oxford in 1710; together with the Lemmas of Pappus, and the Commentaries of Eutoeius. The first four in Greek and Latin, but the latter four in Latin only, the 8th book being restored by himself.

The other writings of Apollonius, mentioned by Pappus, are,

1. The Section of a Ratio, or Proportional Sections, two books.

2. The Section of a Space, in two books.

3. Determinate Section, in two books.

4. The Tangencies, in two books.

5. The Inclinations, in two books.

6. The Plane Loci, in two books.

The contents of all these are mentioned by Pappus, and many lemmas are delivered relative to them; but none, or very little of these books themselves have descended down to the moderns. From the account however that has been given of their contents, many restorations have been made of these works, by the modern mathematicians, as follow: viz,

Victa, Apollonius Gallus. The Tangencies. Paris, 1600, in 4to.

Snellius, Apollonius Batavus. Determinate Section. Lugd. 1601, 4to.

Snellius, Sectio Rationis & Spatii. 1607.

Ghetaldus, Apollonius Redivivus. The Inclinations. Venice, 1607, 4to.

Ghetaldus, Supplement to the Apollonius Redivivus. Tangencies. 1607.

Ghetaldus, Apollonius Redivivus, lib. 2. 1613.

Alex. Anderson, Supplem. Apol. Redivivi. Inclin. Paris, 1612, 4to.

Alex. Anderson. Pro Zetetico Apolloniani problematis a se jam pridem edito in Supplemento Apollonil Redivivi. Paris, 1615, 4to.

Schooten, Loca Plana restituta. Lug. Bat. 1656.

Fermat, Loca Plana, 2 lib. Tolos. 1679, solio.

Halley, Apol. de Sectione Rationis libri duo ex Arabico MS. Latine versi duo restituti. Oxon. 1706, 8vo.

Simson, Loca Plana, libri duo. Glasg. 1749, 4to.

Simson, Sectio Determinat. Glasg. 1776, 4to.

Horsley, Apol. Inclinat. libri duo. Oxon. 1770, 4to.

Lawson, The Tangencies, in two books. Lond. 1771, 4to.

Lawson, Determinate Section, two books. Lond. 1772, 4to.

Wales, Determinate Section, two books. Lond. 1772, 4to.

Burrow, The Inclinations. Lond. 1779, 4to.

, a learned astronomer and philosopher, was born at Apono near Padua, about the year 1250. He described the Astrolabium Planum, by which were shewn the equations of the celestial houses for any hour and minute, and sor any part of the world: it was published at Venice in 1502. He acquired the name of the Conciliator, on account of a book of his, in which he reconciies the writings of the ancient philosophers and physicians: the book was published at Venice in 1483. He resided at Padua, where, from his practising medicine, and his skill in astronomy, he fell under the suspicion of magic. He died in 1316, at 66 years of age.

, in Architecture, is a concave part or ring of a column, lying above or below the flat member; and it owes its origin to the ring by which the ends of wooden columns were hooped, to prevent them from splitting.

, the remainder or difference between two lines or quantities which are only commensurable in power. Such is the difference between 1 and √2, or the difference between the side of the square and its diagonal.

The term is used by Euclid; and a pretty full explanation of such quantities is given in the tenth book of his Elements, where he distinguishes six kinds of apotomes, and shews how to find them all geometrically.

Apotome Prima, is when the greater term is rational, and the difference of the squares of the two is a square number; as the difference 3-√5.

Apotome Secunda, is when the less number is rational, and the square root of the difference of the squares of the two terms, has to the greater term, a ratio expressible in numbers; such is √18-4, because the difference of the fquares 18 and 16 is 2, and √2 is to √18 as √1 to √9 or as 1 to 3.

Apotome Tertia, is when both the terms are irrational, and, as in the second, the square root of the difference of their squares, has to the greater term, a rational ratio: as √24-√18; for the difference of their squares 24 and 18 is 6, and √6 is to √24 as √1 to √4 or as 1 to 2.

Apotome Quarta, is when the greater term is a rational number, and the square root of the difference of the squares of the two terms, has not a rational ratio to it: as 4-√3, where the difference of the squares 16 and 3 is 13, and √13 has not a ratio in numbers to 4.

Apotome Quinta, is when the less term is a rational number, and the square root of the difference of the squares of the two, has not a rational ratio to the greater: as √6-2, where the difference of the squares 6 and 4 is 2, and √2 to √6 or √1 to √3 or 1 to √3 is not a rational ratio.

Apotome Sexta, is where both terms are irrational, and the square root of the difference of their squares has not a rational ratio to the greater: as √6-√2; where the difference of the squares 6 and 2 is 4, and √4 to √6 or 2 to √6, is not a rational ratio.

The doctrine of apotomes, in lines, as delivered by Euclid in the tenth book, is a very curious subject, and has always been much admired and cultivated by all mathematicians who have rightly understood this part of the elements; and therefore Peter Ramus has greatly exposed his judgment by censuring that book. And the first algebraical writers in Europe commonly employed a considerable portion of their works on an algebraical exposition of that book, which led them to the doctrine of surd quantities; as Lucas de Burgo, Cardan, Tartalea, Stifelius, Peletarius, &c, &c. See also Pappus, lib. 4, prop. 3, and the introduc. to lib. 7. And Dr. Wallis's Algebra, pa. 109.

, that which is visible, or evident to the eye, or the understanding.

Apparent conjunction of the planets, is when a right line, supposed to be drawn through their centres, passes through the eye of the spectator, and not through the centre of the earth.—And, in general, the apparent conjunction of any objects, is when they appear or are placed in the same right line with the eye.

Apparent Altitude, Diameter, Distance, Horizon, Magnitude, Motion, Place, Time, &c. See the respective substantives, for the quantity and measure of it.

The apparent state of things, is commonly very different from their real state, either as to distance, figure, magnitude, position, &c, &c. Thus,

Apparent Diameter, or Magnitude, as for example of the heavenly bodies, is not the real length of the diameter, but the angle which they subtend at the eye, or under which they appear. And hence, the angle, or apparent extent, diminishing with the distance of the object, a very small object, as AB, may have the same apparent diameter as a very large one FG; and indeed the objects have all the same apparent diameter, that are contained in the same angle FEG. And if these are parallel, the real magnitudes are directly proportional to their distances.

But the apparent magnitude varies not only by the distance, but also by the position of it. So, if the object CD be changed from the direct position to the oblique one Cd, its apparent magnitude would then be only the angle CEd, instead of the angle CED.

If the eye E be placed between two parallels AB, CD, these parallels will appear to converge or come nearer and nearer to each other the farther they are continued out, and at last they will appear to coincide in that point where the sight terminates, which will happen when the optic angle BED becomes equal to about one minute of a degree, the smallest angle under which an object is visible.—Also the apparent magnitudes of the same object FG or BD, seen at different distances, that is the angles FEG, BED, are in a less ratio than the reciprocal ratio of the distances, or the distance increases in a greater ratio than the angle or apparent magnitude diminishes. But when the object is very remote, or the optic angle is very small, as one degree or thereabouts, the angle then varies nearly as the distance reciprocally.

But although the optic angle be the usual or sensible measure of the apparent magnitude of an object, yet habit, and the frequent experience of looking at distant objects, by which we know that they are larger than they appear, has so far prevailed upon the imagination and judgment, as to cause this too to have some share in our estimation of apparent magnitudes; so that these will be judged to be more than in the ratio of the optic angles.

The apparent magnitude of the same object, at the same distance, is different to different persons, and different animals, and even to the same person, when viewed in different lights, all which may be occasioned by the different magnitudes of the eye, causing the optic angle to differ as that is greater or less: and since, in the same person, the more light there comes from an object, the less is the pupil of the eye, looking at that object; therefore the optic angle will also be less, and consequently the apparent magnitude of the object. Every one must have experienced the truth of this, by looking at another person in a room, and afterwards abroad in the sunshine, when he always appears smaller than in a room where the light is less. So also, objects up in the air, having more light coming from them than when they are upon the ground, or near it, may appear less in the former case than in the latter; like as the ball of the cross on the top of St. Paul's church, which is 6 feet in diameter, appears less than an object of the same diameter seen at the same distance below, near the ground. And this may be the chief reason why the sun and moon appear so much larger when seen in the horizon, where their beams are weak, then when they are raised higher, and their light is more bright and glaring.

Again, if the eye be placed in a rare medium, and view an object through a denser, as glass or water, having plane surfaces; the object will appear larger than it is: and contrariwise, smaller. And hence it is that fishes, and other objects, seen in the water, by an eye in the air, always appear larger than in the air.—In like manner, an object will appear larger when viewed through a globe of glass or water, or any convex spherical segments of these; and, on the contrary, it will appear smaller when viewed through a concave of glass or water.

Apparent Distance, is that distance which we judge an object to be from us, when seen afar off. This is commonly very different from the true distance; because we are apt to think that all very remote objects, whose parts cannot well be distinguished, and which have no other visible objects near them, are at the same distance from us; though perhaps they may bethousands or millions of miles off; as in the case of the sun and moon. The apparent distances of objects are also greatly altered by the refraction of the medium through which they are seen.

Apparent Figure, is the figure or shape which an object appears under when viewed at a distance; and is osten very different from the true figure. For a straight line, viewed at a distance, may appear but as a point; a surface, as a line; and a solid, as a surface. Also these may appear of different magnitudes, and the surface and solid of different figures, according to their situation with respect to the eye: thus, the arch of a circle may appear a straight line; a square, a trapezium, or even a triangle; a circle, an ellipsis; angular magnitudes, round; and a sphere, a circle. Also all objects have a tendency to roundness and smoothness, or appear less angular, as their distance is greater: for, as the distance is increased, the smaller angles and asperities first disappear, by subtending a less angle than one minute; after these, the next larger disappear, for the same reason; and so on continually, as the distance is more and more increased; the object seeming still more and more round and smooth. So, a triangle, or square, at a great distance, shews only as a round speck; and the edge of the moon appears round to the eye, notwithstanding the hills and valleys on her surface. And hence it is also, that near objects, as a range of lamps, or such like, seen at a great distance, appear to be contiguous, and to form one uniform continued magnitude, by the intervals between them disappearing, from the smallness of the angles subtended by them.

Apparent Motion, is either that motion which we perceive in a distant body that moves, the eye at the same time being either in motion or at rest; or that motion which an object at rest seems to have, while the eye itself only is in motion.

The motions of bodies at a great distance, though really moving equally, or passing over equal spaces in equal times, may appear to be very unequal and irregular to the eye, which can only judge of them by the mutation of the angle at the eye. And motions, to be equally visible, or appear equal, must be directly proportional to the distances of the objects moving. Again, very swift motions, as those of the luminaries, may not appear to be any motions at all, but like that of the hour hand of a clock, on account of the great distance of the objects: and this will always happen, when the space actually passed over in one second of time, is less than about the 14000th part of its distance from the eye; for the hour hand of a clock, and the stars about the earth, move at the rate of 15 seconds of a degree in one second of time, which is only the 13751 part of the radius or distance from the eye. On the other hand, it is possible for the motion of a body to be so swift, as not to appear any motion at all; as when through the whole space it describes there constantly appears a continued surface or solid as it were generated by the motion of the object, like as when any thing is whirled very swiftly round, describing a ring, &c.

Also, the more oblique the eye is to the line which a distant body moves in, the more will the apparent motion differ from the true one. So, if a body revolve with an equable motion in the circumference of the circle ABCD &c, and the eye be

If an eye move directly forwards from E to O, &c; any remote object at rest at P, will appear to move the contrary way, or from P to Q,

Apparent Place of an object, in Optics, is that in which it appears, when seen in or through glass, water, or other refracting mediums; which is commonly different from the true place. So, if an object be seen in or through glass, or water, either plane or concave, it will appear nearer to the eye than its true place; but when seen through a convex glass, it appears more remote from the eye than the real place of it.

Apparent Place of the Image of an object, in Catoptrics, is that where the image of an object made by the reflexion of a speculum appears to be; and the optical writers, from Euclid downwards, give it as a general rule that this is where the reflected rays meet the perpendicular to the speculum drawn from the object: so that if the speculum be a plane, the apparent place of the image will be at the same distance behind the speculum as the eye is besore it; if convex, it will appear behind the glass nearer to the same; but if concave, it will appear before the speculum. And yet in some cases there are some exceptions to this rule, as is shewn by Kepler in his Paralipomena in Vitellionem, prop. 18. See also Wolfius Catoptr. § 51, 188, 233, 234.

Apparent Place of a Planet, &c, in Astronomy, is that point in the sursace of the sphere of the world, where the centre of the luminary appears from the surface of the earth.

, in Astronomy, denotes a star's or other luminary's becoming visible, which before was hid. So, the heliacal rising, is rather an apparition than a proper rising.

Circle of perpetual Apparition. See Circle of perpetual apparition.

, in Perspective, is the representation or projection of a figure, body, or the like object, on the perspective plane.—The appearance of an objective right line, is always a right line. See PERSPECTIVE.—Having given the appearance of an opake body, and of a luminary, to sind the appearance of the shadow; see Shadow.

Appearance of a star or planet. See Apparition.

, Applicata, Ordinate Applicate, in Geometry, is a right line drawn to a curve, and bisected by its diameter. This is otherwise called an Ordinate, which see.

Applicate Number. See Concrete.

, the curve of equable approach. It was first proposed by Leibnitz, namely, to find a curve, down which a body descending by the force of gravity, shall make equal approaches to the horizon in equal portions of time. It has been found by Bernoulli and others, that the curve is the second cubical parabola, placed with its vertex uppermost, and which the descending body must enter with a certain determinate velocity.—Varignon rendered the question general for any law of gravity, by which a body may approach towards a given point by equal spaces in equal times. And Maupertuis also resolved the problem in the cafe of a body descending in a medium which resists as the square of the velocity. See Hist. de l' Acad. des Sciences for 1699 and 1730.

Method of Approaches, a name given by Dr. Wallis, in his Algebra, to a method of resolving certain problems relating to square numbers, &c. This is done by first assigning certain limits to the quantities required, and then approaching nearer and nearer till a coincidence is obtained.—In this sense, the method of Trial-and-error, or double rule of False Position, may be considered as a method of approaches.

, in Fortification, the several works made by the besiegers, for advancing or getting nearer to a fortress or place besieged. Such as the trenches, mines, saps, lodgments, batteries, galleries, epaulments, &c.

, a continual approach, still nearer and nearer, to a root or any quantity sought.— Methods of continual approximation for the square roots and cube roots of numbers, have been employed by algebraists and arithmeticians, from Lucas de Burgo down to the present time. And the later writers have given various approximations, not only for the roots of higher powers, or all simple equations, but for the roots of all sorts of compound equations whatever: especially Newton, Wallis, Raphson, Halley, De Lagny, &c, &c; all of them forming a kind of insinite series, either expressed or understood, converging nearer and nearer to the quantity sought, according to the nature of the process.

It is evident that if a number proposed be not a true square, then no exact square root of it can be found, explicable by rational numbers, whether integers or fractions: therefore, in such cases, we must be content with approximations, or coming continually nearer and nearer to the truth. In like manner, for the cube and other roots, when the proposed quantities are not exact cubes, or other powers.

The most easy and general method of approximation, is perhaps by the rule of Double Position, or, what is sometimes called, the Method of Trial-and-error; which method see under its own name. And among all the methods for the roots of pure powers, of which there are many, I believe the best is that which was discovered by myself, and given in the first volume of my Mathematical Tracts, in point of ease, both of execution and for remembering it. The method is this: if N denote any number, out of which is to be extracted the root whose index is denoted by r, and if n be the nearest root first taken; then shall the required root of N very nearly; or as r - 1 times the given number added to r + 1 times the nearest power, is to r + 1 times the given number added to r - 1 times the nearest power, so is the assumed root n, to the required root, very nearly. Then this last value of the root, so found, if one still nearer is wanted, is to be used for n in the same theorem, to repeat the operation with it. And so on, repeating the operation as often as necessary. Which theorem includes all the rational formulæ of Halley and De Lagny.

For example, suppose it were required to double the cube, or to find the cube root of the number 2. Here r=3; consequently , and ; and therefore the general theorem becomes for the cube root of N; or as N + 2n3 : 2N + n3 :: n : the root sought nearly. Now, in this case, N = 2, and therefore the nearest root n is 1, and its cube n3 = 1 also: hence , and ; therefore, as 4 : 5 :: 1 : 5/4 or 1 1/4=1.25 the first approximation. Again, taking r=(5/4), and consequently r3=(125/64); hence ; therefore as 378 : 381, or as 126 : 127 :: 5/4 : (635/504)= 1.259921, which is the cube root of 2, true in all the sigures. And by taking 635/504 for a new value of n, and repeating the process again, a great many more figures may be found.

Of the Roots of Equations by Approximation.— Stevinus and Vieta gave methods for sinding values, always nearer and nearer, of the roots of equations. And Oughtred and others pursued and improved the same. These however were very tedious and imperfect, and required a different process for every degree of equations. But Newton introduced, not only general methods for expressing radical quantities by approximating infinite series, but also for the roots of all sorts of compound equations whatever, which are both easy and expeditious: which will be more particularly described under each respective word or article. His method for approximating of roots, is in substance this: First take a value of the root as near as may be, by trials, either greater or less; then assuming another letter to denote the unknown difference between this and the true value, substitute into the equation the sum or difference of the approximate root and this assumed letter, instead of the unknown letter or root of the equation, which will produce a new equation having only the assumed small difference for its root or unknown letter; and, by any means, find, from this equation, a near value of this small assumed quantity. Assume then another letter for the small difference between this last value and the true one, and substitute the sum or difference of them into the last equation, by which will arise a third equation, involving the second assumed quantity; whose near value is found as before. Proceeding thus as far as we please, all the near values, connected together by their proper signs, will form a series approaching still nearer and nearer to the true value of the root of the first or proposed equation. The approximate values of the several small assumed differences, may be found in different ways: Newton's method is this: As the quan tity sought is small, its higher powers decrease more and more, and therefore neglecting them will not lead to any great error, Newton therefore neglects all the terms having in them the 2d and higher powers, leaving only the 1st power and the absolute known term; from which simple equation he always sinds the value of the assumed unknown letter nearly, in a very simple and easy manner. Halley's method of doing the samething, was to neglect all the terms above the square or 2d power, and then to sind the root of the remaining quadratic equation; which would indeed be a nearer value of the assumed letter than Newton's was, but then it is much more troublesome to perform.—Raphson has another way, which is a little varied from that of New- ton's again, which is this: having found a near value of the first assumed small quantity or difference, by this he corrects the first approximation to the root of the proposed equation; and then, assuming another letter for the next, or smaller difference, he introduces it into the original equation in the same way as before. And thus he proceeds, from one correction to another, employing always the first proposed equation to find them, instead of the fuccessive new equations used by Newton.

For example, let it be required to find the root of the equation , or :—Here the root x, it is evident, is nearly = 8; for x therefore take 8 + z, and substitute 8 + z for x in the given equation, and the terms will be thus; Hence, then, collecting all the assumed differences, with their signs, it is found that the root of the equation required, by Newton's method. The same by Raphson's way. .

Example 2. Again, taking the cubic equation the root of the equation . And in the same manner Newton performs the approximation for the roots of literal equations, that is, equations having literal coefficients; so the root of this equation

See also a memoir on this method by the Marquis de Courtivron, in the Memoires de l'Academie for 1744.

Other Methods of Approximation. Besides the foregoing general methods, other particular ways of approximating, for various purposes, have been given by many other persons.—As for example, methods of approximating, by series, to the roots of cubic equations belonging to the irreducible case, by Nicole in the same Memoirs, by M. Clairaut in his Algebra, and by myself in the Philos. Trans. for 1780. See also several parts of Simpson's works, and my Tracts vol. 1. Also the methods of infinite series by Wallis, Newton, Gregory, Mercator, &c, may be considered as approximations, in quadratures, and other branches of the mathematics, many instances of which may be seen in Wallis's Algebra, and other books:—Likewise the method of exhaustions of the ancients, by which Archimedes and others have approximated to the quadrature and rectification of the circle, &c, which was performed by continually bisecting the sides of polygons, both inscribed in a circle and circumscribed about it; by which means the sum of the sides of the like polygons approach continually nearer and nearer together, and the circumference of the circle is nearly a mean between the two sums. See also Equations.

, in Astronomy, means the actual contact of two luminaries, according to some authors; but others describe it as their near approach to each other, so as to be seen, for instance, within the same telescope.

The appulses of the planets to the fixed stars have always been very useful to astronomers, as serving to fix and determine the places of the former. The ancients, wanting an easy method of comparing the planets with the ecliptic, which is not visible, had scarce any other way of fixing their situations, but by observing their track among the fixed stars, and marking their appulses to some of those visible points. See Hist. Acad. Scienc. for 1710, pa. 417. And Philos. Trans. No. 369, where Dr. Halley has given a method of determining the places of the planets, by observing their near appulses to the fixed stars. See also Philos. Trans. No. 76, pa. 361, and Mem. Acad. Scienc. for 1708, where Flamsteed and De la Hire have given observations of the moon's appulses to the Pleiades. See also Flamsteed's Historia Cœlestis, where a multitude of observations of appulses, or small distances, of the moon and planets, from the fixed stars, are recorded. And Dr. Halley has published a map or planisphere of the starry zodiac, in which are accurately laid down all the stars to which the moon's appulse has ever been observed in any part of the world. See Philos. Trans. No. 369; or Abridg. vol. vi. pa. 170.

, the 4th month of the year according to the common computation, and the 2d from the vernal equinox.—The word is derived from Aprilis, of aperio, I open; because the earth, in this month, begins to open her bosom for the production of vegetables.—In this month the sun travels through part of the signs Aries and Taurus.

, in Gunnery, a piece of thin or sheet lead, used to cover the vent or touch-hole of a cannon.

, in Astronomy, are the two points in the orbits of planets, where they are at their greatest and least distance, from the sun or the earth. The point at the greatest distance being called the higher apsis, and that at the nearest distance the lower apsis. And the two apses are also called auges. Also the higher apsis is more particularly called the aphelion, or the apogee; and the lower apsis, the perihelion, or the perigee. The diameter which joins these two points, is called the line of the apses or of the apsides; and it paffes through the centre of the orbit of the planet, and the centre of the sun or the earth; and in the modern astronomy this line makes the longer or transverse axis of the elliptical orbit of the planet. In this line is counted the excentricity of the orbit; being the distance between the centre of the orbit and the focus, where is placed the sun or the earth.

The foregoing definitions suppose the lines of the greatest and least distances to lie in the same straight line; which is not always precisely the case; as they are sometimes out of a right line, making an angle greater or less than 180 degrees, and the difference from 180 degrees is called the motion of the line of the apses: when the angle is less than 180 degrees, the motion of the apses is said to be contrary to the order of the signs; on the other hand, when the angle exceeds 180 degrees, the motion is according to the order of the signs.

Different means have been employed to determine the motion of the apses. Dr. Keil explains, in his Astronomy, the method used by the ancients, who supposed the orbits of the planets to be perfectly circular, and the sun out of the centre. But since it has been discovered that they describe elliptical orbits, various other methods have been devised for determining it. Halley has given one, which supposes to be known only the time of the planet's revolution, or periodic time. Seth Ward has also given a determination from three different observations of a planet, in any three places of its orbit: but his method being founded on an hypothesis not strictly true, Euler has given one much more exact in vol. 7. of the Petersburgh Commentaries. See various ways explained in the Astronomy of Keil and Mounier.

Newton has also given, in the Principia, an excellent method of determining the same motion, on the supposition that the orbits of the planets differ but little from circles, which is the case nearly. That great philosopher shews, that if the sun be immoveable, and all the planets gravitate towards him in the inverse ratio of the squares of their distances, then the apses will be fixed, or their motion nothing; that is, the lines of greatest and least distance will form one right line, and the apses will be directly opposite, or at 180 degrees distance from each other. But, because of the mutual tendency of the planets towards each other, their gravitation towards the sun is not precisely in that ratio; and hence it happens, that the apses are not always exactly in a right line with the sun. And Newton has given a very elegant method of determining the motion of the apses, on the supposition that we know the force which is thereby added to the gravitation of the planet towards the sun, and that this additional force is always in that direction.

APUS or Apous, Avis Indica, in Astronomy, a constellation of the southern hemisphere, situated near the south pole, between the triangulum australe and the chameleon, and supposed to represent the bird of paradise. Also supposed to be one of the birds named Apodes, as having no feet.

The number of stars contained in this constellation, are 11 in the British Catalogue, in Bayer's Maps 12, and a still greater number in La Caille's Catalogue; the principal star being but of the 4th or 5th order of magnitude. See Cœlum Australe Stelliferum, and the Memoires de l'Acad. for 1752, pa. 569.

, in Astronomy, one of the celestial constellations, being the eleventh sign in the zodiac, reckoning from Aries, and is marked by the character

The poets feign that Aquarius was Ganymede, whom Jupiter ravished under the shape of an eagle, and carried away into Heaven to serve as a cup-bearer, instead of Hebe and Vulcan; whence the name. Others hold, that the sign was thus called, because that when it appears in the horizon, the weather commonly proves rainy.

The stars in the constellation Aquarius, are, in Ptolemy's Catalogue, 45; in Tycho's 41; in Hevelius's 47; and in Flamsteed's 108. See the article CONSTELLATION; also Catalogue.

, or Aquæduct, as much as to say ductus aquæ, a conduit of water, is a construction of stone or timber built on uneven ground, to preserve the level of water, and convey it, by a canal, from one place to another.—Some aqueducts are under ground, being conducted through hills, &c; and others are raised above ground, and supported on arches, to conduct the water over vallies, &c.

The Romans were very magnificent in their aqueducts; having some that extended a hundred miles, or more. Frontinus, a man of consular dignity, who had the direction of the aqueducts under the emperor Nerva, speaks of nine that emptied themselves through 13594 pipes, of an inch diameter. And it is observed by Vigenere, that in the space of 24 hours, Rome received from these aqueducts not less than 500000 hogsheads of water. The chief aqueducts now in being, are these: 1st, that of the Aqua Virginia, repaired by pope Paul IV; 2d, the Aqua Felice, conitructed by pope Sixtus V, and is called from the name he assumed before he was exalted to the papal throne; 3d, the Aqua Paulina, repaired by pope Paul V, in the year 1611; and 4thly, the aqueduct built by Lewis XIV, near Maintenon, to convey the river Bure to Versailles, which is perhaps the largest in the world; being 7000 fathoms long, elevated 2560 fathoms in height, and containing 242 arcades. See Philos. Trans. for 1685, No. 171; or Abridg. vol. 1. pa. 594.

AQUEOUS Humour, or the walry humour of the eye, is the first or outermost, and the rarest of the three humours of the eye. It is transparent and colourless, like water; and it fills up the space that lies between the cornea tunica, and the crystalline humour.

, the Eagle, or the Vulture as it is sometimes called, is a constellation of the northern hemisphere, usually joined with Antinous. It is one of the 48 old constellations, according to the division of which Hipparchus made his Catalogue of the Fixed Stars, and which are described by Ptolemy. The number of stars in Aquila, and those near it, now in the later-formed constellation Antinous, amount to 15 in Ptolemy's Catalogue, to 19 in Tycho's, to 42 in that of Hevelius, and to 71 in Flamsteed's. But in Aquila alone, Tycho counts only 12 stars, and Hevelius 23; the principal star being Lucida Aquila, and is between the 1st and 2d magnitude. The Greeks, as usual, relate various fables of this constellation, to make the science appear as of their own invention.

, the Altar, one of the 48 old constellations, mentioned by the ancient astronomers, and is situated in the southern hemisphere; containing only 7 stars in Ptolemy's Catalogue, and 9 in that of Flamsteed; none of which exceed the 4th magnitude.

, celebrated for his Greek poem intitled faino/mena, the Phenomena, flourished about the 127th Olympiad, or near 300 years besore Christ, while Ptolomy Philadelphus reigned in Egypt. Being educated under Dionysius Heracleotes, a Stoic philosopher, he espoused the principles of that sect, and became physician to Antigonus Gonatus, the son of Demetrius Poliorcetes, King of Macedon. The Phenomena of Aratus gives him a title to the character of an astronomer, as well as a poet. In this work he describes the nature and motion of the stars, and shews their various dispositions and relations; he describes the figures of the constellations, their situations in the sphere, the origin of the names which they bear in Greece and in Egypt, the fables which have given rise to them, the rising and setting of the stars, and he indicates the manner of knowing the constellations by their respective situations.

The poem of Aratus was commented upon and translated by many authors: of whom among the ancients were Cicero, Germanicus Cæsar, and Festus Avienus, who made Latin translations of it; a part of the former of which is still extant. Aratus must have been much esteemed by the ancients, since we sind so great a number of scholiasts and commentators upon him; among whom are Aristarchus of Samos, the Arystylli the geometricians, Apollonius, the Evæneti, Crates, Numenius the grammarian, Pyrrhus of Magnesia, Thales, Zeno, and many others, as may be seen in Vossius, p. 156. Suidas ascribes several other works to Aratus. Virgil, in his Georgies, has translated or imitated many passages from this author: Ovid speaks of him with admiration, as well as many others of the poets: And St. Paul has quoted a passage from him; which is in his speech to the Athenians (Acts xvii. 28) where he tells them that some of their own poets have said, For we are also his offspring, these words being the beginning of the 5th line of the Phenomena of Aratus.

His modern editors are as follow: Henry Stephens published his poem at Paris in 1566, in his collection of the poets, in folio. Grotius published an edition of the Phenomena at Leyden in quarto, 1600, in Greek and Latin, with the fragments of Cicero's version, and the translations of Germanicus and Avienus; all which the editor has illustrated with curious notes. Also a neat and correct edition of Aratus was published at Oxford, 1672, in 8vo. with the Scholia.

ARÆOMETER, see Areometer.

, or Arch; which see.

, in Architecture, denotes an opening in the wall of a building formed by an arch.

ARC-BOUTANT, is a kind of arched buttress, formed of a flat arch, or part of an arch, abutting against the feet or sides of another arch or vault, to support them and prevent them from bursting or giving way.

, a name by which some of the old writers call the star Arcturus; a single and very bright star of the first magnitude, between the legs of the constellation Bootes. They say Areas, the son of Calisto by Jupiter, when he was about to have killed his mother in the shape of a bear, was, together with her, snatched up into Heaven; where she was converted into the constellation of the Great Bear, near the north pole, and the youth into this single star.

, Arc, Arcus, in Geometry, a part of any curve line; as, of a circle, or ellipsis, or the like.

It is by means of circular arcs, or arches, that all angles are measured; the arc being deseribed fiom the angular point as a centre. For this purpose, every circle is supposed to be divided into 360 degrees, or equal parts; and an arch, or the angle it subtends and measures, is estimated according to the number of those degrees it contains: thus, an are, or angle, is said to be of 30 or 80, or 100 degrees.—Circular arcs are also of great use in finding of fluents.

Concentric Arcs, are such as have the same centre.

Equal Arcs, are such arcs, of the same circle, or of equal circles, as contain the same number of degrees. These have also equal chords, sines, tangents, &c.

Similar Arcs, of unequal circles, &c, are such as contain the same number of de-

Of the Length of Circular Arcs. The lengths of circular arcs, as found and expressed in various ways, may be seen in my large Treatise on Mensuration, pa. 118, & seq. 2d edition: some of which are as follow. The radius of a circle being 1; and of any arc a, if the tangent be t, the sine s, the cosine c, and the versed sine v: then the arc a will be truly expressed by several series, as follow, viz, the arc ; where d denotes the number of degrees in the given arc. Also nearly; where C is the chord of the arc, and c the chord of half the arc; whatever the radius is.

To investigate the length of the arc of any curve. Put x=the absciss, y=the ordinate, of the arc z, of any eurve whatever. Put ; then, by means of the equation of the curve, find the value of x. in terms of y., or of y. in terms of x., and substitute that value instead of it in the above expression ; hence, taking the fluents, they will give the length of the arc z, in terms of x or y.

, or Sagittarius, one of the constellations of the northern hemisphere, and one of the twelve signs of the zodiac, placed between the Scorpion and Capricorn. See Sagittarius.

, one of the most celebrated mathematicians among the ancients, who flourished about 250 years before Christ, being about 50 years later than Euclid. He was born at Syracuse in Sicily, and was related to Hiero, who was then king of that city. The mathematical genius of Archimedes set him with such distinguished excellence in the view of the world, as rendered him both the honour of his own age, and the admiration of posterity. He was indeed the prince of the ancient mathematicians, being to them what Newton is to the moderns, to whom in his genius and character he bears a very near resemblance. He was frequently lost in a kind of reverie, so as to appear hardly sensible; he would study for days and nights together, neglecting his food; and Plutarch tells us that he used to be carried to the baths by force. Many particulars of his life, and works, mathematical and mechanical, are recorded by several of the ancients, as Polybius, Livy, Plutarch, Pappus, &c. He was equally skilled in all the sciences, astronomy, geometry, mechanics, hydrostatics, optics, &c, in all of which he excelled, and made many and great inventions.

Archimedes, it is said, made a sphere of glass, of a most furprising contrivance and workmanship, exhibiting the motions of the heavenly bodies in a very pleasing manner. Claudian has an epigram upon this invention, which has been thus translated: When in a glass's narrow space confin'd, Iove saw the fabric of th' almighty mind, He smil'd, and said, Can mortals' art alone, Our heavenly labours mimic with their own? The Syracusian's brittle work contains Th' eternal law, that through all nature reigns. Fram'd by his art, see stars unnumber'd burn, And in their courses rolling orbs return: His sun through various signs describes the year; And every month his mimic moons appear. Our rival's laws his little planets bind, And rule their motions with a human mind. Salmoneus could our thunder imitate, But Archimedes can a world create.

Many wonderful stories are told of his discoveries, and of his very powerful and curious machines, &c. Hiero once admiring them, Archimedes replied, these effects are nothing, “But give me, said he, some other place to fix a machine on, and I shall move the earth.” He fell upon a curious device for discovering the deceit which had been practiced by a workman, employed by the said king Hiero to make a golden crown. Hiero, having a mind to make an offering to the gods of a golden crown, agreed for one of great value, and weighed out the gold to the artificer. After some time he brought the crown home of the full weight; but it was afterwards discovered or suspected that a part of the gold had been stolen, and the like weight of silver substituted in its stead. Hiero, being angry at this imposition, desired Archimedes to take it into consideration, how such a fraud might be certainly discovered. While engaged in the solution of this difficulty, he happened to go into the bath; where observing that a quantity of water overflowed, equal to the bulk of his body, it presently occurred to him, that Hiero's question might be answered by a like method: upon which he leaped out, and ran homeward, crying out eu(\rhka! eu(\rhka! I have found it! I have found it! He then made two masses, each of the same weight as the crown, one of gold and the other of silver: this done, he filled a vessel to the brim with water, and put the silver mass into it, upon which a quantity of water overflowed equal to the bulk of the mass; then taking the mass of silver out he filled up the vessel again, measuring the water exactly, which he put in; this shewed him what measure of water answered to a certain quantity of silver. Then he tried the gold in like manner, and sound that it caused a less quantity of water to overflow, the gold being less in bulk than the silver, though of the same weight. He then filled the vessel a third time, and putting in the crown itself, he found that it caused more water to overflow than the golden mass of the same weight, but less than the silver one; so that, finding its bulk between the two masses of gold and silver, and that in certain known proportions, he hence computed the real quantities of gold and silver in the crown, and so manifestly discovered the fraud.

Archimedes also contrived many machines for useful and beneficial purposes: among these, engines for launching large ships; screw pumps, for exhausting the water out of ships, marshes or overflowed lands, as Egypt, &c, which they would do from any depth.

But he became most famous by his curious contrivances, by which the city of Syracuse was so long defended, when besieged by the Roman consul Marcellus; showering upon the enemy sometimes long darts, and stones of vast weight and in great quantities; at other times lifting their ships up into the air, that had come near the walls, and dashing them to pieces by letting them fall down again; nor could they find their safety in removing out of the reach of his cranes and levers, for there he contrived to fire them with the rays of the sun reflected srom burning glasses.

However, notwithstanding all his art, Syracuse was at length taken by storm, and Archimedes was so very intent upon some geometrical problem, that he neither heard the noise, nor minded any thing else, till a soldier that found him tracing of lines, asked him his name, and upon his request to begone, and not disorder his figures, slew him. “What gave Marcellus the greatest concern, says Plutarch, was the unhappy fate of Archimedes, who was at that time in his museum; and his mind, as well as his eyes, so sixed and intent upon some geometrical sigures, that he neither heard the noise and hurry of the Romans, nor perceived the city to be taken. In this depth of study and contemplation, a soldier came suddenly upon him, and commanded him to follow him to Marcellus; which he refusing to do, till he had finished his problem, the soldier, in a rage, drew his sword, and ran him through.” Livy says he was slain by a soldier, not knowing who he was, while he was drawing schemes in the dust: that Marcellus was grieved at his death, and took care of his funeral; and made his name a protection and honour to those who could claim a relationship to him. His death it seems happened about the 142 or 143 Olympiad, or 210 years before the birth of Christ.

When Cicero was questor for Sicily, he discovered the tomb of Archimedes, all overgrown with bushes and brambles; which he caused to be cleared, and the place set in order. There was a sphere and cylinder cut upon it, with an inscription, but the latter part of the verses quite worn out.

Many of the works of this great man are still extant, though the greatest part of them are lost. The pieces remaining are as follow: 1. Two books on the Sphere and Cylinder.—2. The Dimension of the Circle, or proportion between the diameter and the circumference.— 3. Of Spiral lines.—4. Of Conoids and Spheroids.— 5. Of Equiponderants, or Centres of Gravity.—6. The Quadrature of the Parabola.—7. Of Bodies floating on Fluids.—8. Lemmata.—9. Of the Number of the Sand.

Among the works of Archimedes which are lost, may be reckoned the descriptions of the following inventions, which may be gathered from himself and other ancient authors. 1. His account of the method which he employed to discover the mixture of gold and silver in the crown, mentioned by Vitruvius.—2. His description of the Cochleon, or engine to draw water out of places where it is stagnated, still in use under the name of Archimedes's Screw. Athenæus, speaking of the prodigious ship built by the order of Hiero, says, that Archimedes invented the cochleon, by means of which the hold, notwithstanding its depth, could be drained by one man. And Diodorus Siculus says, that he contrived this machine to drain Egypt, and that by a wonderful mechanism it would exhaust the water from any depth.—3. The Helix, by means of which, Athenæus informs us, he launched Hiero's great ship.—4. The Trispaston, which, according to Tzetzes and Oribasius, could draw the most stupendous weights.—5. The machines, which, according to Polybius, Livy, and Plutarch, he used in the defence of Syracuse against Marcellus, consisting of Tormenta, Balistæ, Catapults, Sagittarii, Scorpions, Cranes, &c.—6. His Burning Glasses, with which he set fire to the Roman gallies.—7. His Pneumamatic and Hydrostatic engines, concerning which subjects he wrote some books, according to Tzetzes, Pappus, and Tertullian.—8. His Sphere, which exhibited the celestial motions. And probably many others.

A whole volume might be written upon the curious methods and inventions of Archimedes, that appear in his mathematical writings now extant only. He was the first who squared a curvilineal space; unless Hypocrates must be excepted on account of his lunes. In his time the conic sections were admitted into geometry, and he applied himself closely to the measuring of them, as well as other figures. Accordingly he determined the relations of spheres, spheroids, and conoids, to cylinders and cones; and the relations of parabolas to rectilineal planes whose