GALILEUS,
HIS
MECHANICKS:
OF THE BENEFIT DERIVED
FROM THE SCIENCE OF MECHANICKS,
AND FROM ITS INSTRUMENTS.
I judged it extreamly nece&longs;&longs;ary, before our
de&longs;cending to the Speculation of Mecha
nick In&longs;truments, to con&longs;ider how I might,
as it were, &longs;et before your eyes in a gene
ral Di&longs;cour&longs;e, the many benefits that are
derived from the &longs;aid In&longs;truments: and
this I have thought my &longs;elf the more ob
liged to do, for that (if I am not mi&longs;taken)
I have &longs;een the generality of
ans deceive them&longs;elves in going about to apply Machines to many
operations of their own nature impo&longs;&longs;ible; by the &longs;ucce&longs;&longs;e where
of they have been di&longs;appointed, and others likewi&longs;e fru&longs;trate of
the hope which they had conceived upon the promi&longs;e of tho&longs;e pre
&longs;umptuous undertakers: of which mi&longs;takes I think I have found
the principall cau&longs;e to be the belief and con&longs;tant opinion the&longs;e
to move and rai&longs;e great weights; (in a certain manner with their
Machines cozening nature, who&longs;e In&longs;tinct, yea mo&longs;t po&longs;itive con
&longs;titution it is, that no Re&longs;i&longs;tance can be overcome, but by a Force
more potent then it:) which conjecture how fal&longs;e it is, I hope by
the en&longs;uing true and nece&longs;&longs;ary Demon&longs;trations to evince.
In the mean time, &longs;ince I have hinted, that the benefit and help
derived from Machines is not, to be able with le&longs;&longs;e Force, by help
of the Machine to move tho&longs;e weights, which, without it, could
not be moved by the &longs;ame Force: it would not be be&longs;ides the
purpo&longs;e to declare what the Commodities be which are derived to
us from &longs;uch like faculties, for if no profit were to be hoped for,
all endeavours employed in the acqui&longs;t thereof will be but lo&longs;t
labour.
Proceeding therefore according to the nature of the&longs;e Studies,
let us fir&longs;t propo&longs;e four things to be con&longs;idered. Fir&longs;t, the weight
to be transferred from place to place; and &longs;econdly, the Force
and Power which &longs;hould move it; thirdly, the Di&longs;tance between
the one and the other Term of the Motion; Fourthly, the Time
in which that mutation is to be made: which Time becometh the
&longs;ame thing with the Dexterity, and Velocity of the Motion; we
determining that Motion to be more &longs;wift then another, which in
le&longs;&longs;e Time pa&longs;&longs;eth an equal Di&longs;tance.
Now, any determinate Re&longs;i&longs;tance and limited Force what&longs;oever
being a&longs;&longs;igned, and any Di&longs;tance given, there is no doubt to be
made, but that the given Force may carry the given Weight to the
determinate Di&longs;tance; for, although the Force were extream
&longs;mall, yet, by dividing the Weight into many &longs;mall parts, none
of which remain &longs;uperiour to the Force, and by transferring them
one by one, it &longs;hall at la&longs;t have carried the whole Weight to the
a&longs;&longs;igned Term: and yet one cannot at the end of the Work with
Rea&longs;on &longs;ay, that that great Weight hath been moved, and tran&longs;
ported by a Force le&longs;&longs;e then it &longs;elf, howbeit indeed it was done
by a Force, that many times reiterated that Motion, and that
Space, which &longs;hall have been mea&longs;ured but only once by the whole
Weight. From whence it appears, that the Velocity of the Force
hath been as many times Superiour to the Re&longs;i&longs;tance of the weight,
as the &longs;aid Weight was &longs;uperiour to the Force; for that in the
&longs;ame Time that the moving Force hath many times mea&longs;ured the
intervall between the Terms of the Motion, the &longs;aid Moveable
happens to have pa&longs;t it onely once: nor therefore ought we to
affirm a great Re&longs;i&longs;tance to have been overcome by a &longs;mall Force,
contrary to the con&longs;titution of Nature. Then onely may we &longs;ay
the Natural Con&longs;titution is overcome, when the le&longs;&longs;er Force tran&longs;
fers the greater Re&longs;i&longs;tance, with a Velocity of Motion like to that
impo&longs;&longs;ible to be done with any Machine imaginable. But becau&longs;e
it may &longs;ometimes come to pa&longs;&longs;e, that having but little Force, it is
required to move a great Weight all at once, without dividing it
in pieces, on this occa&longs;ion it will be necei&longs;ary to have recour&longs;e to
the Machine, by means whereof the propo&longs;ed Weight may be
transferred to the a&longs;&longs;igned Space by the Force given. But yet
this doth not hinder, but that the &longs;ame Force is to move, mea&longs;uring
that &longs;ame Space, or another equall to it, as many &longs;everall times as
it is exceeded by the &longs;aid Weight. So that in the end of the a
ction we &longs;hall &longs;ind that we have received from the Machine no
other benefit tnen only that of tran&longs;porting the &longs;aid Weight with
the given Force to the Term given, all at once. Which Weight,
being divided into parts, would without any Machine have been
carried by the &longs;ame Force, in the &longs;ame Time, through the &longs;ame
Intervall. And this ought to pa&longs;&longs;e for one of the benefits taken
from the Mechanicks: for indeed it frequently happens, that be
ing &longs;canted in Force but not Time, we are put upon moving great
Weights unitedly or in gro&longs;&longs;e: but he that &longs;hould hope, and at
tempt to do the &longs;ame by the help of Machines without increa&longs;e of
Tardity in the Moveable, would certainly be deceived, and would
declare his ignorance of the u&longs;e of Mechanick In&longs;truments, and
the rea&longs;on of their effects.
Another benefit is drawn from the In&longs;truments, which depend
eth on the place wherein the operation is to be made: for all In
&longs;truments cannot be made u&longs;e of in all places with equall conve
nience. And &longs;o we &longs;ee (to explain our &longs;elves by an example) that
for drawing of Water out of a Well, we make u&longs;e of onely a
Rope and a Bucket fitted to receive and hold Water, wherewith
we draw up a determinate quantity of Water, in a certain Time,
with our limited &longs;trength: and he that &longs;hould think he could with
a Machine of what&longs;oever Force, with the &longs;ame &longs;trength, and in
the &longs;ame Time, take up a great quantity of Water, is in a gro&longs;&longs;e
Errour. And he &longs;hall find him&longs;elf &longs;o much the more deceived,
the more he &longs;hall vary and multiply his Inventions: Yet never
thele&longs;&longs;e we &longs;ee Water drawn up with other Engines, as with a Pump
that drinks up Water in the Hold of Ships; where you mu&longs;t note
that the Pump was not imployed in tho&longs;e Offices, for that it draws
up more Water in the &longs;ame Time, and with the &longs;ame &longs;trength
then that which a bare Bucket would do, but becau&longs;e in that place
the u&longs;e of the Bucket or any &longs;uch like Ve&longs;&longs;el could not effect what
is de&longs;ired, namely to keep the Hold of the Ship quite dry from e
very little quantity of Water; which the Bucket cannot do, for
that it cannot dimerge and dive, where there is not a con&longs;iderable
depth of Water. And thus we &longs;ee the Holds of Ships by the
ly be drawn up, which the ordinary u&longs;e of the Bucket would not
effect, which ri&longs;eth and de&longs;cends with its Rope perpendicu
larly.
The third is a greater benefit, haply, then all the re&longs;t that are
derived from Mechanick In&longs;truments, and re&longs;pects the a&longs;&longs;i&longs;tance
which is borrowed of &longs;ome Force exanimate, as of the &longs;tream of a
River, or el&longs;e animate, but of le&longs;&longs;e expence by far, then that which
would be nece&longs;&longs;ary for maintaining humane &longs;trength: as when to
turn Mills, we make u&longs;e of the Current of a River, or the &longs;trength
of a Hor&longs;e, to effect that, which would require the &longs;trength of five
or fix Men. And this we may al&longs;o advantage our &longs;elves in rai&longs;ing
Water, or making other violent Motions, which mu&longs;t have been
done by Men, if there were no other helps; becau&longs;e with one &longs;ole
Ve&longs;&longs;el we may take Water, and rai&longs;e, and empty it where occa&longs;ion
requires; but becau&longs;e the Hor&longs;e, or &longs;uch other Mover wanteth
Rea&longs;on, and tho&longs;e In&longs;truments which are requi&longs;ite for holding and
emptying the Ve&longs;&longs;el in due time, returning again to fill it, and one
ly is endued with Force, therefore it's nece&longs;&longs;ary that the Mecha
nitian &longs;upply the naturall defect of that Mover, furni&longs;hing it with
&longs;uch devices and inventions, that with the &longs;ole application of it's
Force the defired effect may follow. And therein is very great
advantage, not becau&longs;e that a Wheel or other Machine can enable
one to tran&longs;port the &longs;ame Weight with le&longs;&longs;e Force, and greater
Dexterity, or a greater Space than an equall Force, without tho&longs;e
In&longs;truments, but having Judgment and proper Organs, could have
done; but becau&longs;e that the &longs;tream of a River co&longs;teth little or
nothing, and the charge of keeping of an Hor&longs;e or other Bea&longs;t,
who&longs;e &longs;trength is greater then that of eight, or it may be more
Men, is far le&longs;&longs;e then what &longs;o many Men would be kept
for.
The&longs;e then are the benefits that may be derived from Mecha
nick In&longs;truments, and not tho&longs;e which ignorant Engineers dream
of, to their own di&longs;grace, and the abu&longs;e of &longs;o many Princes,
whil&longs;t they undertake impo&longs;&longs;ible enterprizes; of which, both
by the little which hath been hinted, and by the much which
&longs;hall be demon&longs;trated in the Progre&longs;&longs;e of this Treati&longs;e, we &longs;hall
come to a&longs;&longs;ure our &longs;elves, if we attentively heed that which &longs;hall
be &longs;poken.
DEFINITIONS.
That which in all Demon&longs;trative Sciences is nece&longs;&longs;ary to be
ob&longs;erved, we ought al&longs;o to follow in this Di&longs;cour&longs;e, that is;
to propound the Definitions of the proper Terms of this
Art, and the primary Suppo&longs;itions, from which, as from &longs;eeds full
of fecundity, may of con&longs;equence &longs;pring and re&longs;ult the cau&longs;es,
and true Demon&longs;trations, of the Nature of all the Mechanick
Engines which are u&longs;ed, for the mo&longs;t part about the Motions of
Grave Matters, therefore we will determine, fir&longs;t, what is
VITIE.
We call
naturally downwards, which is found in &longs;olid Bodies, cau&longs;ed by
the greater or le&longs;&longs;e quantity of matter, whereof they are con&longs;ti
tuted.
much by the Gravity of the moveable, as by the di&longs;po&longs;ure which
divers Grave Bodies have in relation to one another; by means of
whichMoment, we oft &longs;ee a Body le&longs;s Grave counterpoi&longs;e another
of greater Gravity: as in the Stiliard, a great Weight is rai&longs;ed by
a very &longs;mall counterpoi&longs;e, not through exce&longs;s of Gravity, but
through the remotene&longs;&longs;e from the point whereby the Beam is up
held, which conjoyned to the Gravity of the le&longs;&longs;er weight adds
thereunto Moment, and
Moment of the other greater Gravity may be exceeded.
MENT
of Gravity, Po&longs;ition, and the like, whereby that propenfion may
be occa&longs;ioned
The
in every Grave Body, about which con&longs;i&longs;t parts of equall Moment:
&longs;o that, imagining &longs;ome Grave Body to be &longs;u&longs;pended and &longs;u&longs;tain
ed by the &longs;aid point, the parts on the right hand will Equilibrate
tho&longs;e on the left, the Anteriour, the Po&longs;teriour, and tho&longs;e above
tho&longs;e below; &longs;o that be it in any what&longs;oever fite, and po&longs;ition,
provided it be &longs;u&longs;pended by the &longs;aid
&longs;till: and this is that point which would gladly unite with the
univer&longs;all Center of Grave Bodies, namely withthat of the Earth,
if it might thorow &longs;ome free From
whence we take the&longs;e Suppo&longs;itions.
SUPPOSITIONS.
Any Grave Body, (as to what belongeth to it's proper ver
tue) moveth downwards, &longs;o that the Center of it's Gravity
never &longs;trayeth out of that Right Line which is produced
from the &longs;aid Center placed in the fir&longs;t Term of the Motion unto
the univer&longs;al Center of Grave Bodies. Which is a Suppo&longs;ition
very manife&longs;t, becau&longs;e that &longs;ingle Center being obliged to endea
vour to unite with the common Center, it's nece&longs;&longs;ary, unle&longs;&longs;e &longs;ome
impediment intervene, that it go &longs;eeking it by the &longs;horte&longs;t Line,
which is the Right alone: And from hence may we &longs;econdarily
&longs;uppo&longs;e
Every Grave Body putteth the greate&longs;t &longs;tre&longs;&longs;e, and weigheth
mo&longs;t on the Center of it's Gravity, and to it, as to its proper &longs;eat,
all
cour&longs;e.
We la&longs;tly &longs;uppo&longs;e the Center of the Gravity of two Bodies e
qually Grave to be in the mid&longs;t of that Right Line which conjoyns
the &longs;aid two Centers; or that two equall weights, &longs;u&longs;pended in
equall di&longs;tence, &longs;hall have the point of
Center, or meeting of tho&longs;e equal Di&longs;tances. As for Example,
the Di&longs;tance C E being equall to the Di&longs;tance E D, and there be
ing by them two equall weights &longs;u&longs;pended, A and B, we &longs;uppo&longs;e
the point of
greater rea&longs;on for inclining to
one, then to the other part. But
here is to be noted, that the Di
&longs;tances ought to be mea&longs;ured
with Perpendicular Lines, which
from the point of Su&longs;pen&longs;ion E,
fall on the Right Lines, that from
the Center of the Gravity of the
Weights A and B, are drawn to
the common Center of things
Grave; and therefore if the Di&longs;tance E D were tran&longs;ported into
E F, the weight B would not counterpoi&longs;e the weight A, becau&longs;e
drawing from the Centers of Gravity two Right Lines to the Cen
ter of the Earth, we &longs;hall &longs;ee that which cometh from the Center
of the Weight I, to be nearer to the Center E, then the other
produced from the Center of the weight A. Therefore our &longs;aying
that equal Weights are &longs;u&longs;pended by [or at] equal Di&longs;tances, is
to be under&longs;tood to be meant when as the Right Lines that go from
their Centers & to &longs;eek out the common Center of Gravity, &longs;hall be
equidi&longs;ta nt from that Right Line, which is produced from the &longs;aid
the &longs;ame Center of the Earrh.
The&longs;e things determined and &longs;uppo&longs;ed, we come to the explica
tion of a Principle, the mo&longs;t common and materiall of the greater
part of Mechanick In&longs;truments: demon&longs;trating, that unequall
Weights weigh equally when &longs;u&longs;pended by [or at] unequal Di&longs;tan
ces, which have contrary proportion to that which tho&longs;e weights
are found to have, See the Demon&longs;tration in the beginning of the
&longs;econd Dialogue of Local-Motions.
Now being that Weights unequall come to acquire equall
Moment, by being alternately &longs;u&longs;pended at Di&longs;tances that
have the &longs;ame proportion with them; I think it not fit to
over pa&longs;&longs;e with &longs;ilence another congruicy and probability, which
may confirm the &longs;ame truth; for let the Ballance A B, be con&longs;ide
red, as it is divided into unequal parts in the point C, and let the
Weights be of the &longs;ame propor
tion that is between the Di&longs;tan
ces B C, and C A, alternately
&longs;u&longs;pended by the points A, and
B: It is already manife&longs;t, that
the one will counterpoi&longs;e the
other, and con&longs;equently, that
were there added to one of them
a very &longs;mall Moment of Gravity, it would preponderate, rai&longs;ing
the other, &longs;o that an in&longs;en&longs;ible Weight put to the Grave B, the
Ballance would move and de&longs;cend from the point B towards E,
and the other extream A would a&longs;cend into D, and in regard that
to weigh down B, every &longs;mall Gravity is &longs;ufficient, therefore not
keeping any accompt of this in&longs;en&longs;ible Moment, we will put no
difference between one Weights Now, let us con&longs;ider the Motion which the
Weight B makes, de&longs;cending into E, and that which the other
A makes in a&longs;cending into D, we &longs;hall without doubt find the
Space B E to be &longs;o much greater than the Space A D, as the Di
&longs;tance B C is greater than C A, forming in the Center C two an
gles D C A, and E C B, equall as being at the Cock, and con&longs;e
quently two Circumferences A D and B E alike; and to have the
&longs;ame proportion to one another, as have the Semidiameters B C,
and C A, by which they are de&longs;cribed: &longs;o that then the Velocity
of the Motion of the de&longs;cending Grave B cometh to be &longs;o much
Superiour to the Velocity of the other a&longs;cending Moveable A, as
the Gravity of this exceeds the Gravity of that; and it not being
ly, unle&longs;&longs;e the other Weight B do move to E &longs;wiftly, it will not
be &longs;trange, or incon&longs;i&longs;tent with the Order of Nature, that the
Velocity of the Motion of the Grave B, do compen&longs;ate the greater
Re&longs;i&longs;tance of the Weight A, &longs;o long as it moveth &longs;lowly to D,
and the other de&longs;cendeth &longs;wiftly to E, and &longs;o on the contrary,
the Weight A being placed in the point D, and the other B in
the point E, it will not be unrea&longs;onable that that falling lea&longs;urely
to A, &longs;hould be able to rai&longs;e the other ha&longs;tily to B, recovering by
its Gravity what it had lo&longs;t by it's Tardity of Motion. And by
this Di&longs;cour&longs;e we may come to know how the Velocity of the
Motion is able to encrea&longs;e Moment in the Moveable, according to
that &longs;ame proportion by which the &longs;aid Velocity of the Motion is
augmented.
There is al&longs;o another thing, before we proceed any farther, to
be confidered; and this is touching the Di&longs;tances, whereat, or
wherein Weights do hang: for it much imports how we are to
under&longs;tand Di&longs;tances equall, and unequall; and, in &longs;um, in what
manner they ought to be mea
&longs;ured: for that A B being the
Right Line, and two equall
Weights being &longs;u&longs;pended at
the very ends thereof, the point
C being taken in the mid&longs;t of
the &longs;aid Line, there &longs;hall be an
And the rea&longs;on is for that the
Di&longs;tance C B is equal to C A. But if elevating the Line C B, moving it about the point C, it
&longs;hall be transferred into CD, &longs;o that the Ballance &longs;tand according
to the two Lines A C, and C D, the two equall Weights hanging
at the Terms A and D, &longs;hall no longer weigh equally on that
point C, becau&longs;e the di&longs;tance of the Weight placed in D, is made
le&longs;&longs;e then it was when it hanged in B. For if we confider the Lines,
along [or by] which the &longs;aid Graves make their Impul&longs;e, and
would de&longs;cend, in ca&longs;e they were freely moved, there is no doubt
but that they would make or de&longs;cribe the Lines A G, D F, B H:
Therefore the Weight hanging on the point D, maketh it's Moment
and
B, it made
nearer to the Fulciment C, then is the Line B H Therefore we
are to under&longs;tand that the Weights hanging on the points A and D,
are not equi-di&longs;tant from the point C, as they be when they are
con&longs;tituted according to their Right Line A C B: And la&longs;tly,
we are to take notice, that the Di&longs;tance is to be mea&longs;ured by
hang, and would move, if &longs;o be they were permitted to de&longs;cend
freely.
Of the BALLANCE and LEAVER.
Having under&longs;tood by certain Demon&longs;tration, one of the
fir&longs;t Principles, from which, as from a plenti&longs;ul Fountain,
many of the Mechanical In&longs;truments are derived, we may
take occa&longs;ion without any difficulty to come to the knowledge of
the nature of them: and fir&longs;t &longs;peaking of the Stiliard, an In&longs;tru
ment of mo&longs;t ordinary u&longs;e, with which divers Merchandizes are
weighed, &longs;u&longs;taining them, though very heavy, with a very &longs;mall
counterpoi&longs;e, which is com
monly called the Roman or
Plummet, we &longs;hall prove that
there is no more to be done in
&longs;uch an operation, but to re
duce into act and practice
what hath been above contemplated. For if we propo&longs;e the Bal
lance A B, who&longs;e Fulciment or Lanquet is in the point C, by
which, at the &longs;mall Di&longs;tance C A, hangeth the heavy Weight D,
and if along the other greater C B, (which we call the Needle of
the Stiliard) we &longs;hould &longs;uppo&longs;e the Roman F, though of but little
weight in compari&longs;on of the Grave Body D to be &longs;lipped to and
fro, it &longs;hall be pof&longs;ible to place it &longs;o remotely from the Lanquet C,
that the &longs;ame proportion may be found between the two Weights
D and F, as is between the Di&longs;tances F C, and C A: and then &longs;hall
an
alternately proportional to them.
Nor is this In&longs;trument different from that other called
and vulgarly the ^{*} Leaver, wherewith great Weights are moved
by &longs;mall Force; the application of which is according to the Fi
gure prefixed; wherein the Leaver
is repre&longs;ented by the Bar of wood
or other &longs;olid matter,
the heavy Weight to be rai&longs;ed be
A, and let the &longs;teadfa&longs;t &longs;upport
or Fulciment on which the Leaver
re&longs;ts and moves be &longs;uppo&longs;ed to be
E, and putting one end of the
Leaver under the Weight A, as
may be &longs;een in the point C, en
crea&longs;ing the Weight or Force at the other end D, it will be able
to lift up the Weight A, though not much, whenever the Force in
A, in the point C: as the Di&longs;tance
whereby it's clear, that the nearer the Fulciment E &longs;hall approach
to the Term B, encrea&longs;ing the proportion of the Di&longs;tance D C to
the Di&longs;tance C
is to rai&longs;e the Weight A. And here it is to be noted, which I &longs;hall
al&longs;o in its place remember you of, that the benefit drawn from all
Mechanical In&longs;truments, is not that which the vulgar Mechanitians
do per&longs;wade us, to wit, &longs;uch, that there by Nature is overcome, and
in a certain manner deluded, a &longs;mall Force over-powring a very
great Re&longs;i&longs;tance with help of the Leaver; for we &longs;hall demon&longs;trate,
that without the help of the length of the Leaver, the &longs;ame Force,
in the &longs;ame Time, &longs;hall work the &longs;ame effect. For taking the &longs;ame
Leaver B C D, who&longs;e re&longs;t or Fulci
ment is in C, let the Di&longs;tance C D
be &longs;uppo&longs;ed, for example, to be
in quintuple proportion to the
Di&longs;tance C
be moved till it come to I C G: In
the Time that the Force &longs;hall have
pa&longs;&longs;ed the Space D I, the Weight
&longs;hall have been moved from B
to G: and becau&longs;e the Di&longs;tance
D C, was &longs;uppo&longs;ed quintuple to the other C B, it is manife&longs;t from
the things demon&longs;trated, that the Weight placed in B may be five
times greater then the moving Force &longs;uppo&longs;ed to be in D: but now,
if on the contrary, we take notice of the ^{*} Way pa&longs;&longs;ed by
the Force from D unto I, whil&longs;t the Weight is moved from B unto
G, we &longs;hall find likewi&longs;e the Way D I, to be quintuple to the Space
B G. Moreover if we take the Di&longs;tance C L, equal to the Di&longs;tance
C B, and place the &longs;ame Force that was in D, in the point L, and
in the point B the fifth part onely of the Weight that was put there
at fir&longs;t, there is no que&longs;tion, but that the Force in L being now
equal to this Weight in B, and the Di&longs;tances L C and C B being
equall, the &longs;aid Force &longs;hall be able, being moved along the Space LM
to transfer the Weight equall to it &longs;elf, thorow the other equall
Space B G: which five times reiterating this &longs;ame action, &longs;hall tran&longs;
port all the parts of the &longs;aid Weight to the &longs;ame Term G: But
the repeating of the Space L M, is certainly nothing more nor le&longs;&longs;e
then the onely once mea&longs;uring the Space D I, quintuple to the
&longs;aid L M. Therefore the transferring of the Weight from B to G,
requireth no le&longs;&longs;e Force, nor le&longs;&longs;e Time, nor a &longs;horter Way if it
wee placed in D, than it would need if the &longs;ame were applied
in L: And, in &longs;hort, the benefit that is derived from the length of
the Leaver C D, is no other, &longs;ave the enabling us to move that
Force, in the &longs;ame Time, with an equall Motion, &longs;ave onely in
pieces, without the help of the Leaver.
called a Crow,
if of wood, a Bar
or Hand-&longs;pike.
The In&longs;truments which we are now about to declare, have
immediate dependence upon the Leaver, nay, are no other
but a perpetual Vectis or Leaver. For if we &longs;hall &longs;uppo&longs;e the
Leaver B A C to be &longs;u&longs;tained in
the point A, and the Weight G to
hang at the point B, the Force be
ing placed in C; It is manife&longs;t,
that transferring the Leaver unto
the points D A E, the Weight G
doth alter according to the Di
&longs;tance B D, but cannot much far
ther continue to rai&longs;e it, &longs;o that
if it were required to elevate it yet
higher, it would be nece&longs;&longs;ary to
&longs;tay it by &longs;ome other Fulciment
in this Po&longs;ition, and to remit or return the Leaver to its former Po
&longs;ition B A C, and &longs;u&longs;pending the Weight anew thereat, to rai&longs;e it
once again to the like height B D; and in this manner repeating
the work, many times one &longs;hall come with an interrupted Motion
to effect the drawing up of the Weight, which for many re&longs;pects
will not prove very beneficial: whereupon this difficulty hath bin
thought on, and remedied, by finding out a way how to unite to
gether almo&longs;t infinite Leavers, perpetuating the operation without
any interruption; and this hath been done by framing a Wheel
about the Center A, according to the Semidiameter A C, and an
Axis or Nave, about the &longs;ame Center, of which let the Line A B
be the Semidiameter; and all this of very tough wood, or of other
&longs;trong and &longs;olid matter, afterwards &longs;u&longs;taining the whole Machine
upon a Gudgeon or Pin of Iron planted in the point A, which
pa&longs;&longs;eth quite thorow, where it is held fa&longs;t by two fixed Fulciments,
and the Rope D B G, at which the weight G hangeth, being be-laid
or wound about the Axis or Barrell, and applying another Rope
about the greater Wheel, at which let the other Grave I be hang
ed: It is manife&longs;t, that the length C A having to the other A B
the &longs;elf-&longs;ame proportion that the Weight G hath to the Weight I,
it may &longs;u&longs;tain the Grave G, and with any little Moment more &longs;hall
move it: and becau&longs;e the Axis turning round together with the
Wheel, the Ropes that &longs;u&longs;tain the Weights are alwaies pendent and
contingent with the extream Circumferences of that Wheel and
in re&longs;pect of the Di&longs;tances B A and A C, the Motion &longs;hall be
perpetuated, the Weight I de&longs;cending, and forcing the other G
to a&longs;cend. Where we are to ob&longs;erve the nece&longs;&longs;ity of be-laying
or winding the Rope about the Wheel, that &longs;o the Weight I may
hang according to the Line that is tangent to the &longs;aid Wheel: for
if one &longs;hould &longs;u&longs;pend the &longs;aid Weight, &longs;o as that it did hang by the
point F, cutting the &longs;aid Wheel, as is &longs;een along the Line F N M,
the Motion would cea&longs;e, the Moment of the Weight M being di
mini&longs;hed; which would weigh no more then if it did hang by the
point N: becau&longs;e the Di&longs;tance of its Su&longs;pen&longs;ion from the Center
A, cometh to be determined by the Line A N, which falleth per
pendicularly upon the Rope F M, and is no longer terminated by
the Semidiameter of the Wheel A F, which falleth at unequall
Angles upon the &longs;aid Line F M. A violence therefore being offered
in the Circumference of the Wheel by a Grave and Exanimate
Body that hath no other
nece&longs;&longs;ary that it be &longs;u&longs;tained by a Line that is contingent with
the Wheel, and not by one that cutteth it. But if in the &longs;ame
Circumference an Animate Force were employed, that had a Mo
ment or Faculty of making an
be effected in any whatever place of the &longs;aid Circumference. And
thus being placed in F, it would draw up the Weight by turning
the Wheel about, pulling not according to the Line F M down
wards, but &longs;ide-waies according to the Contingent Line F L, which
maketh a Right Angle, with that which is drawn from the Center
A unto the point of Contact F: &longs;o, that if in this manner one do
mea&longs;ure the Di&longs;tance from the Center A to the Force placed in
F, according to the Line A F perpendicular to F L, along which
the
u&longs;e of the ordinary Leaver. And we mu&longs;t note, that the &longs;ame
would be po&longs;&longs;ible to be done likewi&longs;e with an Exanimate Force,
in ca&longs;e that a way were found out to cau&longs;e that its Moment might
make Impul&longs;e in the point F, drawing according to the Contingent
Line F L: which would be done by adjoyning beneath the Line F L
a turning Pulley, making the Rope wound about the Wheel to
pa&longs;&longs;e along upon it, as it is &longs;een to do by the Line F L X, &longs;u&longs;pending
at the end thereof the Weight X equall to the other I, which ex
erci&longs;ing its Force according to the Line F L, &longs;hall alwaies keep a
Di&longs;tance from the Center A equall unto the Semidiameter of the
Wheel. And from what hath been declared we will gather for a
Conclu&longs;ion, That in this In&longs;trument the Force hath alwaies the
&longs;ame proportion to the Weight, as the Semidiameter of the Axis
or Barrell hath to the Semidiameter of the Wheel.
From the In&longs;trument la&longs;t de&longs;cribed, the other In&longs;trument which
we call the Crane is not much different, as to form, nay, differeth
nothing, &longs;ave in the way of applying or employing it: For that the
Cap&longs;ten moveth and is con&longs;tituted perpendicular to the Horizon,
and the Crane worketh with its Moment parallel to the &longs;ame Ho
rizon. For if upon the Circle D A E we &longs;uppo&longs;e an Axis to be
placed Column-wi&longs;e, turning about the Center B, and about which
the Rope D H, fa&longs;tened to the Weight that is to be drawn, is be
laid, and if the Bar F E B D be let into the &longs;aid Axis [
tace B
Animal apt to draw, be applyed at its end F, which moving round,
pa&longs;&longs;eth along the Circumference F G C, the Crane &longs;hall be framed
and fini&longs;hed, &longs;o that by carrying round the Bar F B D, the Barrell
or Axis E A D &longs;hall turn about, and the Rope which is twined a
bout it, &longs;hall con&longs;train the Weight H to go forward: And becau&longs;e
the point of the Fulciment about which the Motion is made, is the
point B, and the Moment keeps at a Di&longs;tance from it according to
the Line B F, and the Re&longs;i&longs;tor at the Di&longs;tance B D, the Leaver
F B D is formed, by vertue of which the Force acquireth Moment
equall to the Re&longs;i&longs;tance, if &longs;o be, that it be in proportion to it, as
the Line B D is to B F, that is, as the Semidiameter of the Axis to
the Semidiameter of the Circle, along who&longs;e Circumference the
Force moveth. And both in this, and in the other In&longs;trument we
are to ob&longs;erve that which hath been frequently mentioned, that is,
That the benefit which is derived from the&longs;e Machines, is not that
which the generality of the Vulgar promi&longs;e them&longs;elves from the
Mechanicks; namely, that being too hard for Nature, its po&longs;&longs;ible
&longs;mall Force, in regard, that we &longs;hall manife&longs;tly prove that the &longs;ame
Force placed in F, might in the &longs;ame Time conveigh the &longs;ame
Weight, with the &longs;ame Motion, unto the &longs;ame Di&longs;tance, without
any Machine at all: For &longs;uppo&longs;ing, for example, that the Re&longs;i&longs;tance
of the Grave H be ten times greater than the Force placed in F, it
will be requi&longs;ite for the mo
ving of the &longs;aid Re&longs;i&longs;tance,
that the Line F B be decuple
to B D; and con&longs;equently,
that the Circumference of the
Circle F G C be al&longs;o decuple
to the Circumference E A D:
and becau&longs;e when the Force
&longs;hall be moved once along the
whole Circumference of the
Circle F G C, the Barrel EAD,
about which the Rope is be-laid which draweth the Weight, &longs;hall
likewi&longs;e have given one onely turn; it is manife&longs;t, that the Weight
H &longs;hall not have been moved more than the tenth part of that way
which the Mover &longs;hall have gone. If therefore the Force that is to
move a Re&longs;i&longs;tance that is greater than it &longs;elf, for &longs;uch an a&longs;&longs;igned
Space by help of this Machine, mu&longs;t of nece&longs;&longs;ity move ten times as
far, there is no doubt, but that dividing that Weight into ten parts,
each of them &longs;hall be equall to the Force, and con&longs;equently, might
have been tran&longs;ported one at a Time, as great a Space as that
which it &longs;elf did move, &longs;o that making ten journeys, each equal to
the Circumference E A D, it &longs;hall not have gone any farther than
if it did move but once alone about the Circumference F G C;
and &longs;hall have conveighed the &longs;ame Weight H to the &longs;ame Di
&longs;tance. The benefit therefore that is to be derived from the&longs;e
Machines is, that they carry all the Weight together, but not with
le&longs;&longs;e Labour, or with greater Expedition, or a greater Way than
the &longs;ame Force might have done conveying it by parcels.
Of PULLIES.
The In&longs;truments, who&longs;e Natures are reducible unto the Bal
lance, as to their Principle and Foundation, and others little
differing from them, have been already de&longs;cribed; now for
the under&longs;tanding of that which we have to &longs;ay touching Pullies,
it is requi&longs;ite, that we con&longs;ider in the fir&longs;t place another way to u&longs;e
the Leaver, which will conduce much towards the inve&longs;tigation of
the Force of Pullies, and towards the under&longs;tanding of other Me
chanical Effects. The u&longs;e of the Leaver above declared &longs;uppo&longs;ed
the Fulciment placed in &longs;ome point between the extreams: but we
may make u&longs;e of the Leaver another way, yet, placing, as we &longs;ee,
the Fulciment in the extream A, the Force in the other extream C,
and &longs;uppo&longs;ing the Weight D to hang by &longs;ome point in the mid&longs;t,
as here we &longs;ee by the point B, in
this example it's manife&longs;t, that if
the Weight did hang at a point
Equi-di&longs;tant from the two ex
treams A and C, as at the point F,
the labour of &longs;u&longs;taining it would
be equally divided betwixt the
two points A and C, &longs;o that half
the Weight would be felt by the
Force C, the other half being &longs;u
&longs;tained by the Fulciment A: but if the Grave Body &longs;hall be hanged
at another place, as at B, we &longs;hall &longs;hew that the Force in C is &longs;uffi
cient to &longs;u&longs;tain the Weight in B, as it hath the &longs;ame proportion
to it, that the Di&longs;tance, A B hath to the Di&longs;tance A C. For De
mon&longs;tration of which, let us imagine the Line B A to be continued
right out unto G, and let the Di&longs;tance B A be equall to A G, and
let the Weight hanging at G, be &longs;uppo&longs;ed equall to D: It is ma
nife&longs;t, that by rea&longs;on of the equality of the Weights D and E, and
of the Di&longs;tances G A and A B, the Moment of the Weight E
&longs;hall equalize the Moment of the Weight D, and is &longs;ufficient to
&longs;u&longs;tain it: Therefore whatever Force &longs;hall have Moment equall to
that of the Weight E, and that &longs;hall be able to &longs;u&longs;tain it, &longs;hall be
&longs;ufficient likewi&longs;e to &longs;u&longs;tain the Weight D: But for &longs;u&longs;taining the
Weight E, let there be placed in the point C &longs;uch a Force, who&longs;e
Moment hath that proportion to the Weight E, that the Di&longs;tance
G A hath to the Di&longs;tance A C, it &longs;hall be &longs;ufficient to &longs;u&longs;tain it:
Therefore the &longs;ame Force &longs;hall likewi&longs;e be able to &longs;u&longs;tain the
Weight D, who&longs;e Moment is equall to the of E: But look what
Proportion the Line G A hath to the Line A C; and A B al&longs;o hath
the &longs;ame to the &longs;aid A C, G A having been &longs;uppo&longs;ed equall to A B:
And becau&longs;e the Weights E and D are equall, each of them &longs;hall
have the &longs;ame proportion to the Force placed in C: Therefore the
Force in C is concluded to equall the Moment of the Weight D,
as often as it hath unto it the &longs;ame proportion that the Di&longs;tance B A
hath to the Di&longs;tance C A. And by moving the Weight, with the
Leaver u&longs;ed in this manner, it is gathered in this al&longs;o, as well as in
the other In&longs;truments, that what is gained in Force is lo&longs;t in Velo
city: for the Force C rai&longs;ing the Leaver, and transferring it to A I,
the Weight is moved the Space B H, which is as much le&longs;&longs;er than
the Space C I pa&longs;&longs;ed by the Force, as the Di&longs;tance A B is le&longs;&longs;er
Weight.
The&longs;e Principles being declared, we will pa&longs;&longs;e to the Contem
plation of Pullies, the compo&longs;ition and &longs;tructure of which, together
with their u&longs;e, &longs;hall be de&longs;cribed by us. And fir&longs;t let us &longs;uppo&longs;e the
^{*} Little Pulley A B C, made of Mettall or hard Wood, voluble a
bout it's Axis which pa&longs;&longs;eth thorow it's Center D, and about this
Pulley let the Rope E A B C be put,
at one end of whichlet the Weight E
hang, and at the other let us &longs;uppo&longs;e
the Force F. I &longs;ay, that the Weight
being &longs;u&longs;tained by a Force equall to
it &longs;elf in the upper Nut or Pulley
A B C, bringeth &longs;ome benefit, as the
moving or &longs;u&longs;taining of the &longs;aid
Weight with the Force placed in F:
For if we &longs;hall under&longs;tand, that from
the Center D, which is the place of the Fulciment, two Lines be
drawn out as far as the Circumference of the Pulley in the points
A and C, in which the pendent Cords touch the Circumference, we
&longs;hall have a Ballance of equal Arms which determine the Di&longs;tance
of the two Su&longs;pen&longs;ions from the Center and Fulciment D: Where
upon it is manife&longs;t, that the Weight hanging at A cannot be &longs;u&longs;tain
ed by a le&longs;&longs;er Weight hanging at G, but by one equal to it; &longs;uch
is the nature of equal Weights hanging at equal Di&longs;tances. And
although in moving downwards, the Force F cometh to turn about
the Pulley A B C, yet there followeth no alteration of the Alti
tude or Re&longs;pect, that the Weight and Force have unto the two
Di&longs;tances A D and D C, nay, the Pulley encompa&longs;&longs;ed becometh a
Ballance equal to A C, but perpetuall. Whence we may learn,
how childi&longs;hly
the &longs;mall Pulley A B C bigger, one might draw up the Weight with
a le&longs;&longs;er Force; he con&longs;idering that upon the enlargement of the
&longs;aid Pulley, the Di&longs;tance D C encrea&longs;ed, but not con&longs;idering that
there was as great an encrea&longs;e of the other Di&longs;tance of the Weight,
that is, the other Semidiameter D A. The benefit therefore that may
be drawn from the In&longs;trument above &longs;aid, is nothing at all as to the
diminution of the labour: and if any one &longs;hould ask how it hap
pens, that on many occa&longs;ions of rai&longs;ing Weights, this means is made
u&longs;e of to help the Axis, as we &longs;ee, for example, in drawing up the
Water of Wells; it is an&longs;wered, that that is done, becau&longs;e that
by this means the manner of employing the Force is found more
commodious: for being to pull downwards, the proper Gravity of
our Arms and other parts help us, whereas if we were to draw
the fame Weight upwards with a meer Rope, by the &longs;ole &longs;trength
Armes, be&longs;ides the extern Weight, we are to lift up the Weight of
our own Armes, in which greater pains is required. Conclude we,
therefore, that this upper Pulley doth not bring any Facility to the
Force &longs;imply con&longs;idered, but onely to the manner of applying it:
but if we &longs;hall make u&longs;e of the like Machine
in another manner, as we are now about to
declare; we may rai&longs;e the Weight with di
minution of Forces: For let the Pulley
B D C be voluble about the Center E placed
in it's Frame B L C, at which hang the
Grave G; and let the Rope A B D C F
pa&longs;&longs;e about the Pulley; of which let the end
A be fa&longs;tned to &longs;ome fixed &longs;tay, and in the
other F let the Force be placed; which
moving to wards H &longs;hall rai&longs;e the Machine
B L C, and con&longs;equently the Weight G:
and in this operation I &longs;ay, that the Force in
F is the half of the Weight &longs;u&longs;tained by it. For the &longs;aid Weight being kept to Rights by the two ^{*} Ropes A B
and F C, it is manife&longs;t, that the Labour is equally &longs;hared betwixt
the Force F and the Fulciment A: and more &longs;ubtilly examining the
nature of this In&longs;trument, if we but continue forth the Diameter
B E C, we &longs;hall &longs;ee a Leaver to be made, at the mid&longs;t of which, that
is at the point E, the Grave doth hang, and the Fulciment cometh
to be at the end B, and the Force in the Term C: whereupon, by
what hath been above demon&longs;trated, the Force &longs;hall have the &longs;ame
proportion to the Weight, that the Di&longs;tance E B hath to the Di
&longs;tance; Therefore it &longs;hall be the half of the &longs;aid Weight: And
becau&longs;e the Force ri&longs;ing towards A, the Pulley turneth round,
therefore that Re&longs;pect or Con&longs;titution which the Fulciment B and
Center E, on which the Weight and Term C, in which the Force
is employed do depend, &longs;hall not change all the while; but yet in
the Circuinduction the Terms B and C happen to vary in number,
but not in vertue, others and others continually &longs;ucceeding in their
place, whereby the Leaver B C cometh to be perpetuated. And
here (as hath been done in the other In&longs;truments, and &longs;hall be in
tho&longs;e that follow) we will not pa&longs;&longs;e without con&longs;idering how that
the journey that the Force maketh, is double to the Moment of the
Weight. For in ca&longs;e the Weight &longs;hall be moved &longs;o far, till that
the Line B C come to arrive with it's points B and C, at the points
A and F, it is nece&longs;&longs;ary that the two equal Ropes be di&longs;tended in
one &longs;ole Line F H, and con&longs;equently, when the Weight &longs;hall have
a&longs;cended along the Intervall B A, the Force &longs;hall have been moved
twice as far, that is, from Then con&longs;idering that the
to exanimate Movers, as being for the mo&longs;t part Grave Bodies, is al
together impo&longs;&longs;ible, or at lea&longs;t more laborious,
than the making of the &longs;ame
wards: Therefore to help this inconvenience,
a Remedy hath been found by adjoyning an
other Nut or Pulley above, as in the adjacent
been made to pa&longs;s about the upper Pulley
upheld by the Hook L, &longs;o that the Rope pa&longs;&longs;ing
to H, and thither transferring the
&longs;hall be able to move the Weight X by pulling
downwards, but not that it may be le&longs;&longs;er than
it was in E:
D G of the upper Pulley, do alwaies continue
equal; nor doth that upper Pulley (as hath
been &longs;hewn above) come to produce any di
minution in the Labour. Moreover it having been nece&longs;&longs;ary by
the addition of the upper Pulley to introduce the Appendix B, by
which it is &longs;u&longs;tained, it will prove of &longs;ome benefit to us to rai&longs;e
the other A, to which one end of the Rope was fa&longs;tned, transferring
it to a Ring annexed to the lower part of the
Pulley, as we &longs;ee it done in M. Now finally, this Machine com
pounded of upper and lower Pullies, is that which the Greeks call
a Nut.
the &longs;ame Rope.
chlea.
We have hitherto explained, how by help of Pullies one may
double the
&longs;ible, we &longs;hew the way how to encrea&longs;e it according to any Multi
plicity. And fir&longs;t we will &longs;peak of the Multiplicity according to
the even numbers, and then the odde: To &longs;hew how we may mul
tiply the
following Speculation as the Soul of all that followeth.
Take two Leavers, A B, C D, with the
treams A and C; and at the middles
of each of them let the Grave G hang,
&longs;u&longs;tained by two
ment placed in B and D. I &longs;ay, that
the Moment of each of them will
equal the Moment of the fourth part
of the Weight G.
ces B and D bearing equally, it is
manife&longs;t, that the
contra&longs;ted with more then one half of the Weight G: But if the
the &longs;aid
proportion which the Di&longs;tance
Which is &longs;ubduple proportion: Therefore the Moment D is &longs;ub
duple to the Moment of half of the Weight G &longs;u&longs;tained by it:
Wherefore it followeth, that it is the fourth part of the Moment
of the whole Weight. And in the &longs;ame manner the &longs;ame thing is
demon&longs;trated, of the Moment
Weight G being &longs;u&longs;tained by the four points, A,
them &longs;hould feel an equall part of the Labour.
Let us come now to apply this Con&longs;ideration to Pullies, and let
the Weight X be &longs;uppo&longs;ed to hang at the two Pullies A B and D E
entwining about them, and about the uppermo&longs;t Pulley G H, the
Rope, as we &longs;ee, I D E H G A B, &longs;u&longs;taining the whole Machine in
the point K. Now I &longs;ay, that placing the Force in L, it &longs;hall be able
to &longs;u&longs;tain the Weight X, if &longs;o be, it be equal to the fourth part of
it. For if we do imagine the two Diameters D E and A B, and the
Weights hanging at the middle points F and C, we &longs;hall have two
Leavers like to tho&longs;e before de&longs;cribed, the Fulciments of which an
&longs;wer to the points D and A. Whereupon the Force placed in B,
or if you will, in L, &longs;hall be able to &longs;u
&longs;tain the Weight X, being the fourth
part of it: And if we adde another Pul
ley above the other two, making the
Rope or Cord to pa&longs;s along L M N, trans
ferring the Force L into N, it &longs;hall be
able to bear the &longs;ame Weight gravitating
downwards, the upper Pulley neither aug
menting or dimini&longs;hing the Force, as hath
been declared. And we will likewi&longs;e
note, that to make the: Weight a&longs;cend the
four Ropes B L, E H, D I, and A G
ought to pa&longs;s, whereupon the Mover will
be to begin, as much as tho&longs;e Ropes are
long; and yet neverthele&longs;s the Weight
&longs;hall move but only as much as the length
of one of them: So that we may &longs;ay by
way of adverti&longs;ement, and for confirma
tion of what hatn been many times &longs;po
ken, namely, that look with what proportion the Labour of the
Mover is dimini&longs;hed, the length of the Way, on the contrary, is
encrea&longs;ed with the &longs;ame proportion
of the &longs;ame Rope
rilla
Shiver, Rundle,
or &longs;mall Wheel
of a Pulley, tran
&longs;lated by we
&longs;ometimes Pul
ley, &longs;ometimes
Nut or Girill.
But if we would encrea&longs;e the Force in &longs;excuple proportion, it
will be requi&longs;ite that we adjoyn another ^{*} &longs;mall Pulley or Gyrill
to the inferiour Pulley which that you may the better under&longs;tand Suppo&longs;e, there
fore, that A B, C D, and E F are three Leavers; and that on the
middle points of them G, H, and I the Weight K doth hang in
common, &longs;o that every one of them &longs;hall &longs;u&longs;tain the third part of
it: And becau&longs;e the Power in
B, &longs;u&longs;taining with the Leaver
B A thependent Weight in G,
hapneth to be the half of the
&longs;aid Weight, and it hath been
already &longs;aid, that it &longs;u&longs;taineth
the third part of the Weight
K: Therefore the Moment of
the Force B is equal to half of
the third part of the Weight K; that is, to the &longs;ixth part of it:
And the &longs;ame &longs;hall be demon&longs;trated of the other Forces D and F:
From whence we may ea&longs;ily gather, that putting three Gyrils or
Rundles into the inferiour Pulley, and two or three into the upper
mo&longs;t, we may multiply the Force accor
ding to our ^{*}
encrea&longs;e it according to any other even
Number, the Gyrils of the Pulley below
mu&longs;t be multiplyed according to the half
of that Number, according to which the
Force is to be multiplyed, circumpo&longs;ing
the Rope about the Pulleys, &longs;o as that one
of the ends be fa&longs;tned to the upper Pul
ley, and let the Force be in the other; as
in this Figure adjoyning may manife&longs;tly
be gathered.
proportion.
Now pa&longs;&longs;ing to the Declaration of the
manner how to multiply the Force ac
cording to the odd Numbers, and begin
ning at the triple proportion: fir&longs;t, let us
propo&longs;e the pre&longs;ent Contemplation, as
that, on the under&longs;tanding of which the
knowledge of all the Work in hand
doth depend. Let therefore the Leaver
be A B, its Fulciment A, and from the
middle of it, that is, at the point C let
the Grave D be hanged; and let it be &longs;u
&longs;tained by two equal Forces; and let one of them be applied to the
point C, and the other to the term B. I &longs;ay, that each of tho&longs;e Powers
have Moment equal to the third part of the Weight D. For the
Force in C &longs;u&longs;taineth a Weight equal to it &longs;elf, being placed in the
&longs;ame Line in which the Weight D doth hang & Gravitate: But the
Di&longs;tance from the Fulciment A, that is, the Line B A being dou
ble to the Di&longs;tance A C at which the Grave hangeth: But becau&longs;e
the two Forces in B and C are &longs;uppo&longs;ed to be equal to each other:
Therefore the part of the Weight D, which is &longs;u&longs;tained by the
Force in B, is double to the part &longs;u&longs;tained by the Force in C. If
therefore, of the Grave D two parts be made, the one double to
the remainder, the greater is &longs;u&longs;tained by the Force in B, and the
le&longs;&longs;er by the Force in C: But this le&longs;&longs;er is the third part of the
Weight D: Therefore the Moment of the Force in C is equal to
the Moment of the third part of the Weight D; to which, of
con&longs;equence, the Force B &longs;hall be equal, we having &longs;uppo&longs;ed it
equal to the other Force C: Wherefore our intention is manifell,
which we were to demon&longs;trate, how that each of the two Powers
C and B is equal to the third part of the Weight D. Which be
ing demon&longs;trated, we will pa&longs;s forwards to the Pulleys, and will
de&longs;cribe the inferiour Gyrils of A C B, voluble about the Center
G, and the Weight H hanging thereat, we will draw the other up
per one E F, winding about them both the Rope D F E A C B I,
of which let the end D be fa&longs;tned to the inferiour Pulley, and to
the other I let the Force be applyed:
Which, I &longs;ay, &longs;u&longs;taining or moving the
Weight H, &longs;hall feele no more than the
third part of the Gravity of the &longs;ame. For
con&longs;idering the contrivance of this Ma
chine, we &longs;hall find that the Diameter A B
&longs;upplieth the place of a Leaver, in who&longs;e
term B the Force I is applied, and in the
other A the
dle G the Grave H is hanged, and another
the Weight is fa&longs;tned to the ^{*} three Ropes
I B,
&longs;u&longs;tain the Weight. Now, by what hath
already been contemplated, the two
D and B being applied, one, to the mid&longs;t of the Leaver A B, and
the other to the extream term B, it is manife&longs;t, that each of them
holdeth no more but the third part of the Weight H: Therefore
the Power I, having a Moment equal to the third part of the
Weight H, &longs;hall be able to &longs;u&longs;tain and move it: but yet the Way
of the
pa&longs;s; the &longs;aid Force being to di&longs;tend it &longs;elf according to the
Length of the three Ropes I B,
mea&longs;ureth the Way of the Weight H.
of one Rope.
Among&longs;t the re&longs;t of Mechanick In&longs;truments for &longs;undry u&longs;es
found out by the Wit of Man, the Screw doth, in my opi
nion, both for Invention and for Utility, hold the fir&longs;t
place, as that which is appo&longs;itely accommodated, and &longs;o contrived
not only to move, but al&longs;o to &longs;tay and pre&longs;s with very great Force,
that taking up but little room, it worketh tho&longs;e effects which other
In&longs;truments cannot, unle&longs;s they were reduced to a great Machine. The Screw therefore being of mo&longs;t ingenious and commodious
contrivance, we ought de&longs;ervedly to be at &longs;ome pains in explaining,
with all the plainne&longs;s that is po&longs;&longs;ible, the Original and Nature of
it. The which that we may do, we will begin at a Speculation,
which, though at fir&longs;t blu&longs;h it may appear &longs;omewhat remote from
the con&longs;ideration of this In&longs;trument, yet is the
tion thereof.
No doubt, but that Natures operation in the Motions of Grave
Bodies is &longs;uch, that any whatever Body that hath a Gravity in it
hath a propen&longs;ion of moving, being at liberty, towards the Cen
ter, and that not only ^{*} by the Right Line perpendicularly, but al
&longs;o (when it cannot do otherwi&longs;e) by any other Line, which ha
ving &longs;ome inclination towards the Center goeth more and more
aba&longs;ing. And thus we &longs;ee the Water not only to fall downwards
along the Perpendicular from &longs;ome eminent place, but al&longs;o to run
about the Surface of the Earth along Lines though very little en
clined; as we &longs;ee in the Cour&longs;e of Rivers, the Waters of which, if &longs;o
be that the Bed have any the lea&longs;t declivity, go freely declining
downwards. Which very effect, like as it is di&longs;cerned in all Fluid
Bodies, would appear al&longs;o in hard Bodies, if &longs;o be, that their Fi
gure and other Accidental and Extern Impediments did not hinder
it. So that we, having a Superficies very well &longs;moothed and poli
&longs;hed, as for in&longs;tance, that of a Looking-gla&longs;s, and a Ball exactly
rotund and &longs;leek, either of Marble, or of Gla&longs;s, or of any other
Matter apt to be poli&longs;hed, this being placed upon that Superficies
&longs;hall trundle along, in ca&longs;e that this have any, though very &longs;mall,
inclination; and &longs;hall lie &longs;till only upon that Superficies which is
exactly levelled and parallel to the Plane of the Horizon: as is
that, for example, of a Lake or &longs;tanding Water being frozen, up
on which the &longs;aid Spherical Body would &longs;tand &longs;till, but in a con
dition of being moved by every &longs;mall Force. For we having &longs;up
po&longs;ed that if that Plane did incline but an hairs breadth only, the
&longs;aid Ball would move along it &longs;pontaneou&longs;ly towards the part de
clining, and on the oppo&longs;ite would have a Re&longs;i&longs;tance, nay, would
not be able without &longs;ome Violence to move towards the part
Superficies which is exactly equilibrated, the &longs;aid Ball remaineth in
different and dubious between Motion and Re&longs;t, &longs;o that every &longs;mall
Force is &longs;ufficient to move it, as on the contrary, every &longs;mall Re&longs;i
&longs;tance, and no greater than that of the meer Air that environs it, is
able to hold it &longs;till.
From whence we may take this Conclu&longs;ion for indubitable, That
Crave Bodies, all Extern and Adventitious Impediments being re
moved, may be moved along the Plane of the Horizon by any ne
ver &longs;o &longs;mall Force: but when the &longs;ame Grave is to be thrown along
an A&longs;cending Plane, then, it beginning to &longs;trive again&longs;t that a&longs;cent,
having an inclination to the contrary Motion, there &longs;hall be requi
red greater Violence, and &longs;till greater the more Elevation that &longs;ame
Plane &longs;hall have. As for example, the Moveable G, being po&longs;ited
upon the Line A B parallel to the Horizon, it &longs;hall, as hath been
&longs;aid, be indifferent on it either to Motion or Re&longs;t, &longs;o that it may
be moved by a very &longs;mall Force: But if we &longs;hall have the Planes
Elevated, they &longs;hall not be driven along without Violence; which
Violence will be required to be
greater to move it along the Line
A D, than along A C; and &longs;till
greater along A E than along A D:
The which hapneth, becau&longs;e it hath
greater
wards along A E than along A D,
and along A D than along A C. So
that we may likewi&longs;e conclude
Grave Bodies to have greater Re&longs;i&longs;tance upon Planes differently
Elevared, to their being moved along the &longs;ame, according as one
&longs;hall be more or le&longs;s elevated than the other; and, in fine, that the
greate&longs;t Re&longs;i&longs;tance of the &longs;ame Grave to its being rai&longs;ed is in the
Perpendicular A F. But it will be nece&longs;&longs;ary to declare exactly what
proportion the Force mu&longs;t have to the Weight, that it may be able
to carry it along &longs;everal elevated Planes, before we proceed any
farther, to the end that we may perfectly under&longs;tand all that which
remains to be &longs;poken.
Letting, therefore, Perpendiculars fall from the points C, D,
and E unto the Horizontal Line A B, which let be C H, D I, and
E K: it &longs;hall be demon&longs;trated that the &longs;ame Weight &longs;hall be mo
ved along the Plane A C with le&longs;&longs;er Force than along the Perpendi
cular A F, (where it is rai&longs;ed by a Force equal to it &longs;elf) accor
ding to the proportion by which the Perpendicular C H is le&longs;s than
A C: and that along the Plane A D, the Force hath the &longs;ame pro
portion to the Weight, that the Perpendicular I D hath to D A:
and, la&longs;tly, that in the Plane A E the
veth the proportion of E K and E A.
The pre&longs;ent Speculation hath been attempted by
andrinusMathemat.
right, he hath not hit the mark, and was over&longs;een in the A&longs;&longs;umpti
on that he maketh, where he &longs;uppo&longs;eth that the Weight ought to
be moved along the Horizontal Line by a
fal&longs;e: there needing no &longs;en&longs;ible
Impediments, which in the Theory are not regarded) to move the
given Weight along the Horizon, &longs;o that he goeth about in vain
afterwards to &longs;eek with what
elevated Plane. It will be therefore better, the
the Weight upwards perpendicularly, (which equalizeth the Gra
vity of that Weight which is to be moved) being given, to
&longs;eek the
we will endeavour to do in a Method different from that of
Let us therefore &longs;uppo&longs;e the Circle A I C, and in it the Diame
ter A B C, and the Center B, and two Weights of equal Moment
in the extreams B and C; &longs;o that the Line A C being a Leaver,
or Ballance moveable about the Center B, the Weight C &longs;hall
come to be &longs;u&longs;tained by the Weight A. But if we &longs;hall imagine
the Arm of the Ballance B C to be inclined downwards according
to the Line B F, but yet in &longs;uch a manner that the two Lines
and
the Moment of the Weight C &longs;hall not be equal to the Moment
of the Weight
&longs;tance of the point
of Direction, which goeth accord
ing to B I, from the
to the Center of the Earth, is dimi
ni&longs;hed: But if from the point
erect a Perpendicular unto B C, as is
Line K
Di&longs;tance K B is dimini&longs;hed by the
Di&longs;tance B
by the Moment of the
the
&longs;till dimini&longs;h and &longs;hall be as if it did hang at the Di&longs;tance
cording to the
a
Di&longs;tance B See therefore that
the
downwards along the Circumference C
its Moment and
B L: But the con&longs;idering that this Grave de&longs;cending, and &longs;u&longs;tained
by the Semidiameters B F and B L is one while le&longs;s, and another
while more con&longs;trained to pa&longs;s along the Circumference C F L, is
no other, than if we &longs;hould imagine the &longs;ame Circumference
C F L I to be a Super&longs;icies &longs;o curved, and put under the &longs;ame
Moveable: &longs;o that bearing it &longs;elf thereon it were con&longs;trained to
de&longs;cend along thereby; for if in the one and other manner the
Moveable de&longs;cribeth the &longs;ame Cour&longs;e or Way, it will nothing im
port whether, if &longs;u&longs;pended at the Center B, it is &longs;u&longs;tained by the
Semidiameter of the Circle, or el&longs;e, whether that Fulciment being
taken away, it proceed along the Circumference C F L I: So that
we may confidently affirm, that the Grave de&longs;cending downwards
from the point C along the Circumference C F L I, its Moment
of De&longs;cent in the point C is total and entire, becau&longs;e it is not in
any part &longs;u&longs;tained by the Circumference: And there is not in that
fir&longs;t point C, any indi&longs;po&longs;ition to Motion different from that, which
being at liberty, it would make along the Perpendicular and Con
tingent Line D C E: But if the Moveable &longs;hall be placed in the
point F, then its Gravity is in part &longs;u&longs;tained, and its Moment of
De&longs;cent is dimini&longs;hed by the Circular Path or Way that is placed
under it, in that proportion wherewith the
by
&longs;uch its Motion, it be as if it were in the Plane elevated according
to the Contingent
Circumference in the point F differeth not from the inclination of
the Contingent
the Contact. And in the &longs;ame manner we &longs;hall find the Moment
of the &longs;aid Moveable to dimini&longs;h in the point
is dimini&longs;hed by B C; &longs;o that in the Plane contingent to the Circle
in the point
Moment of De&longs;cent dimini&longs;heth in the Moveable with the &longs;ame
proportion. If therefore ^{*} upon the Plane HG the Moment of the
Moveable be dimini&longs;hed by the total
Perpendicular D C E, according to the proportion of the
to the
K B F and K F H the &longs;ame proportion betwixt the
F H, as betwixt the &longs;aid K B and
proportion of the entire and ab&longs;olute Moment, that the Moveable
hath in the Perpendicular to the Horizon to that which it hath up
on the Inclined Plane H F, hath the &longs;ame proportion that the
Inclined Plane hath to the Perpendicular which &longs;hall fall from it
unto the Horizon. So that pa&longs;&longs;ing to a more di&longs;tinct Figure, &longs;uch
as this here pre&longs;ent, the Moment of De&longs;cending which the Move
wherewith it gravitates in the Perpendicular to the Horizon C P the
&longs;ame proportion that the &longs;aid Line P C hath to C A. And if thus it
be, it is manife&longs;t, that like as the Force that &longs;u&longs;tai
neth the Weight in the Perpendiculation P C ought
to be equal to the &longs;ame, &longs;o for &longs;u&longs;taining it in the
inclined Plane C A, it will &longs;uffice that it be &longs;o much
le&longs;&longs;er, by how much the &longs;aid Perpendicular C P wan
teth of the Line C A: and becau&longs;e, as &longs;ometimes we
&longs;ce, it &longs;ufficeth, that the Force for moving of the
Weight do in&longs;en&longs;ibly &longs;uperate that which &longs;u&longs;taineth it, therefore
we will infer this univer&longs;al Propo&longs;ition, [That upon an Elevated
Plane the Force hath to the Weight the &longs;ame proportion, as the
Perpendicular let fall from the Plane unto the Horizon hath to the
Length of the &longs;aid Plane.]
Returning now to our fir&longs;t Intention, which was to inve&longs;tigate
the Nature of the Screw, we will con&longs;ider the Triangle A B C, of
which the Line A B is Horizontal, B C perpendicular to the &longs;aid
Horizon, and A C a Plane elevated; upon which the Moveable D
&longs;hall be drawn by a Force &longs;o much le&longs;s than it, by how much the
Line B C is &longs;horter than C A: But to elevate or rai&longs;e the &longs;aid
Weight along the &longs;aid Plane A C, is as much as if the Triangle
C A B &longs;tanding &longs;till, the Weight
D be moved towards C, which is
the &longs;ame, as if the &longs;ame Weight
never removing from the Perpen
dicular A E, the Triangle did
pre&longs;s forwards towards H. For if
it were in the Site F H G, the
Moveable would be found to
have mounted the height A I.
Now, in fine, the primary Form and E&longs;&longs;ence of the Screw is no
thing el&longs;e but &longs;uch a Triangle A C B, which being forced for
wards, &longs;hall work it &longs;elf under the Grave Body to be rai&longs;ed, and
lifteth it up, as we &longs;ay, by the ^{*} head and &longs;houlders. And this was
its fir&longs;t Original: For its fir&longs;t Inventor (whoever he was) con&longs;i
dering how that the Triangle A B C going forwards rai&longs;eth the
Weight D, he might have framed an In&longs;trument like to the &longs;aid
Triangle, of a very &longs;olid Matter, which being thru&longs;t forwards did
rai&longs;e up the propo&longs;ed Weight: But afterwards con&longs;idering better,
how that that &longs;ame Machine might be reduced into a much le&longs;&longs;er
and more commodious Form, taking the &longs;ame Triangle he twined
and wound it about the Cylinder A B C D in &longs;uch a fa&longs;hion, that
the height of the &longs;aid Triangle, that is the Line C B, did make the
Height of the Cylinder, and the A&longs;cending Plane did beget upon
which we vulgarly call the Wale of the Screw, which was produ
ced by the Line A C. And in this manner is the In&longs;trument made,
which is by the Greeks called
winding about
cometh to work
and in&longs;inu
ate with its
Wales under
the Weight, and
with facility rai
&longs;eth it. And we
having demon
&longs;trated, That up
on [
the elevated Plane the Force hath the &longs;ame proportion to the
Weight, that the perpendicular Altitude of the &longs;aid Plane hath to
its Length; &longs;o, &longs;uppo&longs;ing that the Force in the Screw A B C D is
multiplied according to the proportion by which the Length of the
whole Wale exceedeth the Altitude C B, from hence we come
to know that making the Screw with its Helix's more thick or clo&longs;e
together, it becometh &longs;o much the more forceable, as being begot
by a Plane le&longs;s elevated, and who&longs;e Length regards its own Per
pendicular Altitude with greater proportion. But we will not
omit to adverti&longs;e you, that de&longs;iring to find the Force of a propo
&longs;ed Screw, it will not be needful that we mea&longs;ure the Length of
all its Wales, and the Altitude of the whole Cylinder, but it
will be enough if we &longs;hall but examine how many times the Di
&longs;tance betwixt two &longs;ingle and Contiguous terms do enter into one
&longs;ole Turn of the &longs;ame Wale, as for example, how many times
the Di&longs;tance AF is contained in the Length of the Turn AEF:
For this is the &longs;ame proportion that the Altitude CB hath to all
the Wale.
&longs;ignfieth to lift
on high by force
tine
Screw winding
like the Shell of
a Snail.
If all that be under&longs;tood which we have hitherto &longs;poken touch
ing the Nature of this In&longs;trument, I do not doubt in the lea&longs;t but
that all the other circum&longs;tances may without difficulty be compre
hended: as for in&longs;tance, that in&longs;teed of making the Weight to
mount upon the Screw if one accommodates its Nut with
the Helix incavated or made hollow, into which the Male Screw
that is the Wale entring, & then being turned round it rai&longs;eth and
lifteth up the Nut or Male Screw together with the Weight which
was hanged thereat. La&longs;tly, we are not to pa&longs;s over that Con&longs;idera
tion with &longs;ilence which at the beginning hath been &longs;aid to be nece&longs;
&longs;ary for us to have in all Mechanick In&longs;truments, to wit, That
what is gained in Force by their a&longs;&longs;i&longs;tance, is lo&longs;t again in Time,
to &longs;ome &longs;o true and manife&longs;t in the pre&longs;ent Contemplation; nay,
rather it &longs;eems, that in this ca&longs;e the Force is multiplied without the
Movers moving a longer way than the Moveable: In regard, that
if we &longs;hall in the Triangle A B C &longs;uppo&longs;e the Line A B to be the
Plane of the Horizon, A C the elevated Plane, who&longs;e Altitude is
mea&longs;ured by the Perpendicular C B, a Moveable placed upon the
Plane A C, and the Cord E D
applyed in
Gravity of the Weight E the
&longs;ame proportion that the Line
B C hath to C A; by what
hath been demon&longs;trated, the
Weight
downwards, drawing the
Moveable E along the eleva
ted Plane; nor &longs;hall the Move
able E mea&longs;ure a greater Space
when it &longs;hall have pa&longs;&longs;ed the
whole Line A
de&longs;cent downwards. But here yet it mu&longs;t be adverti&longs;ed, that al
though the Moveable E &longs;hall have pa&longs;&longs;ed the whole Line A C, in
the &longs;ame Time that the other Grave
like Space, neverthele&longs;s the Grave E &longs;hall not have retired from the
common Center of things Grave more than the Space of the Per
pendicular
be aba&longs;ed a Space equal to the whole Line A
Bodies make no Re&longs;i&longs;tance to Tran&longs;ver&longs;al Motions, but only &longs;o
far as they happen to recede from the
fore the Moveable E in all the Motion A
than the length of the Line
pendicularly the quantity of all the Line A
de&longs;ervedly affirm that Way of the
proportion to the
the Weight E to the Weight
to con&longs;ider by [
cially in exanimate Grave Bodies, the Moments of which have their
total Vigour, and entire Re&longs;i&longs;tance in the
the Horizon; and in the others tran&longs;ver&longs;ally Elevated and Inclined
they feel the more or le&longs;s Vigour,
or le&longs;s tho&longs;e Inclinations approach unto the Perpendicular Inclina
tion.
I Do not think it &longs;it in this place to pa&longs;s over with Silence the
Invention of
is not only marvellous, but miraculous: for we &longs;hall find that
the Water a&longs;cendeth in the Screw continually de&longs;cending; and in
a given Time, with a given Force doth rai&longs;e an un&longs;peakable quan
tity therof. But before we proceed any farther, let us declare the u&longs;e
of the Screw in making Water to ri&longs;e: And in the en&longs;uing Figure,
let us con&longs;ider the Line I L O P Q
R S H being wrapped or twined
about the Collumn M I K H,
which Line you are to &longs;uppo&longs;e to
be a Chanel thorow which the
Water may run: If we &longs;hall put
the end I into the Water, making
the Screw to &longs;tand leaning, &longs;o as
the point L may be lower than
the fir&longs;t I, as the Diagram &longs;hew
eth, and &longs;hall turn it round about
on the two Axes, T and V, the Water &longs;hall run thorow the Cha
nel, till that in the end it &longs;hall di&longs;charge &longs;orth at the mouth H. Now I &longs;ay, that the Water, in its conveyance from the point I to
the point H, doth go all the way de&longs;cending, although the point H
be higher than the point I. Which that it is &longs;o, we will declare
in this manner. We will de&longs;cribe the Triangle A C B, which is
that of which the Screw H I is generated, in &longs;uch &longs;ort that the
Chanel of the Screw is repre&longs;ented by the Line A C, who&longs;e
A&longs;cent and Elevation is determined by the Angle C A B; that is
to &longs;ay, if &longs;o be, that that Angle &longs;hall be the third or fourth part of a
Right Angle, then the Elevation of the Chanel A C &longs;hall be ac
cording to 1/3, or 1/4 of a Right Angle. And it is manife&longs;t; that the
Ri&longs;e of that &longs;ame Chanel A C will be taken away deba&longs;ing the
point C as far as to B: for then the Chanel A C &longs;hall have no
Elevation. And deba&longs;ing the point C a little below B, the Water
will naturally run along the Chanel A C downwards from the
point A towards C. Let us therefore conclude, that the Angle A
being 1/3 of a Right Angle, the Chanel A C &longs;hall no longer have any
Ri&longs;e, deba&longs;ing it on the part
The&longs;e things under&longs;tood, let us infold the Triangle about the
Column, and let us make the Screw B A E F G, &c. which if it
&longs;hall be placed at Right Angles with the end B in the Water, turn
ing it about, it &longs;hall not this way draw up the Water, the Chanel
about the Column being elevated, as may be &longs;een by the part B A.But although the Column &longs;tand erect at Right-Angles, yet for all
that, the Ri&longs;e along the Screw, folded about the Column, is not of
a greater Elevation than of 1/3 of a Right Angle, it being generated
by the Elevation of the Chanel A C: Therefore if we incline the
Column but 1/3 of the
&longs;aid Right Angle, and
a little more, as we &longs;ee
I K H M, there is a
Tran&longs;ition and Moti
on along the Chanel
I L: Therefore the
Water from the point
I to the point L &longs;hall
move de&longs;cending, and
the Screw being turned
about, the other parts
of it &longs;hall &longs;ucce&longs;&longs;ively
di&longs;po&longs;e or pre&longs;ent
them&longs;elves to the Wa
ter in the &longs;ame Po&longs;ition as the part I L: Whereupon the Water
&longs;hall go &longs;ucce&longs;&longs;ively de&longs;cending, and in the end &longs;hall be found to
be a&longs;cended from the point I to the point H. Which how admira
ble a thing it is, I leave &longs;uch to judge who &longs;hall perfectly have un
der&longs;tood it. And by what hath been &longs;aid, we come to know, That
the Screw for rai&longs;ing of Water ought to be inclined a little more
than the quantity of the Angle of the Triangle by which the &longs;aid
Screw is de&longs;cribed.
HAMMER, MALLET, or BEETLE.
The Inve&longs;tigation of the cau&longs;e of the Force of the&longs;e Percuti
ents is nece&longs;&longs;ary for many Rea&longs;ons: and fir&longs;t, becau&longs;e that
there appeareth in it much more matter of admiration than
is ob&longs;erved in any other Mechanick In&longs;trument what&longs;oever. For
&longs;triking with the Hammer upon a Nail, which is to be driven into
a very tough Po&longs;t, or with the Beetle upon a Stake that is to pene
trate into very &longs;tiffe ground, we &longs;ee, that by the &longs;ole vertue of the
blow of the Percutient both the one and the other is thru&longs;t for
wards: &longs;o that without that, only laying the Beetle upon the
Nail or Stake it will not move then, nay, more, although you
&longs;hould lay upon them a Weight very much heavier than the &longs;aid
Beetle. An effect truly admirable, and &longs;o much the more worthy
of Contemplation, in that, as I conceive, none of tho&longs;e who have
which we may take for a certain Sign and Argument of the Ob&longs;cu
rity and difficulty of this For
who would reduce the cau&longs;e of this admirable Effect unto the
length of the
made to &longs;ee their mi&longs;take in the effect of tho&longs;e In&longs;truments, which
having no Handle, yet percu&longs;s, either in falling from on high
downwards, or by being thrown with Velocity &longs;idewaies. There
fore it is requi&longs;ite, that we have recour&longs;e to &longs;ome other Principle, if
we would find out the truth of this bu&longs;ine&longs;s; the cau&longs;e of which,
although it be of its own nature &longs;omewhat ob&longs;cure, and of diffi
cult con&longs;ideration, yet neverthele&longs;s we will attempt with the grea
te&longs;t per&longs;picuity po&longs;&longs;ible to render it clear and obvious, &longs;hewing, for
a clo&longs;e of all, that the Principle and Original of this Effect is deri
ved from no other Fountain than this, from which the rea&longs;ons of all
other Mechanick Effects do proceed: and this we will do, by &longs;etting
before your eyes that very thing which is &longs;een to befall in every
other Mechanick Operation,
and the Space by which the Motion is made, do go alternately
with &longs;uch proportion operating, and with &longs;uch a rate an&longs;wering to
each other, that a Re&longs;i&longs;tance, equal to the Force, &longs;hall be moved by
the &longs;aid Force along an equal Space, with Velocity equal to that
with which it is moved. Likewi&longs;e, That a Force that is le&longs;s by half
than a Re&longs;i&longs;tance &longs;hall be able to move it, &longs;o that it be moved
with double Velocity, or, if you will, for a Di&longs;tance twice as great
as that which the moved Re&longs;i&longs;tance &longs;hall pa&longs;s: and, in a word, it
hath been &longs;een in all the other In&longs;truments, that any, never &longs;o great,
Re&longs;i&longs;tance may be moved by every &longs;mall Force given, provided,
that the Space, along which the Re&longs;i&longs;tance &longs;hall move, have the
&longs;ame proportion that is found to be betwixt the &longs;aid great Re&longs;i
&longs;tance and the Force: and that this is according to the nece&longs;&longs;ary
Order and Con&longs;titution of Nature: So that inverting the Di&longs;cour&longs;e,
and Arguing the contrary way, what wonder &longs;hall it be, if that
Power that &longs;hall move a &longs;mall Re&longs;i&longs;tance a great way, &longs;hall carry
one an hundred times bigger an hundredth part of that Di&longs;tance? Certainly none at all: nay, it would be ab&longs;urd, yea, impo&longs;&longs;ible
that it &longs;hould be otherwi&longs;e. Let us therefore con&longs;ider, what the
Re&longs;i&longs;tance of the Beetle unto Motion may be in that point where
it is to &longs;trike, and how far, if it do not &longs;trike, it would be carryed
by the received Force beyond that point: and again, what Re&longs;i
&longs;tance to Motion there is in him who &longs;triketh, and how much by
that &longs;ame Percu&longs;&longs;ion he is moved: and, having found that this
great Re&longs;i&longs;tance goeth forwards by a percu&longs;&longs;ion &longs;o much le&longs;s than
the Beetle driven by the
by how much that &longs;ame great Re&longs;i&longs;tance is greater than that of
in the lea&longs;t exceed the terms of Natural Con&longs;titutions, and of
what hath been &longs;poken. Let us, for better under&longs;tanding, give an
example thereof in particular Terms. There is a Beetle, which ha
ving four degrees of Re&longs;i&longs;tance, is moved by &longs;uch a Force, that
being freed from it in that term where it maketh the Percu&longs;&longs;ion, it
would, meeting with no &longs;top, go ten Paces beyond it, and in that
term a great po&longs;t being oppo&longs;ed to it, who&longs;e Re&longs;i&longs;tance to Moti
on is as four thou&longs;and, that is, a thou&longs;and times greater than that of
the Beetle, (but yet is not immoveable) &longs;o that it without mea
&longs;ure or proportion exceeds the Re&longs;i&longs;tance of the Beetle, yet the
Percu&longs;&longs;ion being made on it, it &longs;hall be driven forwards, though in
deed no more but the thou&longs;andth part of the ten Paces which the
Beetle &longs;hall be moved: and thus in an inverted method, changing
that which hath been &longs;poken touching the other Mechanical Effects,
we may inve&longs;tigate the rea&longs;on of the Force of the Percutient. I
know that here ari&longs;e difficulties and objections unto &longs;ome, which
they will not ea&longs;ily be removed from, but we will freely remit them
to the ^{*} Problems Mechanical, which we &longs;hall adjoyn in the end of
this Di&longs;cour&longs;e.
blems he here
promi&longs;eth were
never yet ex
tant.
THE
BALLANCE
OF
In which, in immitation of
Problem of the Crown, he &longs;heweth how to
find the proportion of the Alloy of
Mixt-Metals; and how to make
the &longs;aid In&longs;trument.
As it is well known, by &longs;uch who take the pains to read
old Authors, that
the Gold&longs;mith in the Crown of ^{*}
hitherto unknown what method this Great Philo&longs;o
pher ob&longs;erved in that Di&longs;covery: for the opinion, that he did per
form it by putting the Crown into the Water, having fir&longs;t put in
to it &longs;uch another Ma&longs;s of pure Gold, and another of Silver &longs;eve
rally, and that from the differences in their making the Water
more or le&longs;s ri&longs;e and run over, he came to know the Mixture or
Alloy of the Gold with the Silver, of which that Crown was
compounded; &longs;eems a thing (if I may &longs;peak it) very gro&longs;s, and
far from exactne&longs;s. And it will &longs;eem &longs;o much the more dull to
&longs;uch who have read and under&longs;tood the exqui&longs;ite Inventions of &longs;o
Divine a Man among&longs;t the Memorials that are extant of him; by
which it is very manife&longs;t that all other Wits are inferiour to that
of
that
Water, &longs;ome Writer of tho&longs;e Times committed the memory there
of to Po&longs;terity, and that this per&longs;on, that he might add &longs;omething
to that little which he had heard by common Fame, did relate that
hath been by the generality of men believed.
and Kin&longs;man to
that Great Ma
thematician.
Marcel.
But in regard I know, that that method is altogether fallacious,
and falls &longs;hort of that exactne&longs;s which is required in Mathematical
Matters, I have often thought in what manner, by help of the
Water, one might exactly find the Mixture of two Metals, and
in the end, after I had diligently peru&longs;ed that which
demon&longs;trateth in his Books
exqui&longs;itely re&longs;olveth our Que&longs;tion; which Rule I believe to be
the &longs;ame that
u&longs;e that is to be made of the Water, the exactne&longs;s of the Work
dependeth al&longs;o upon certain Demon&longs;trations found by the &longs;aid
The way is by help of a Ballance, who&longs;e Con&longs;truction and U&longs;e
&longs;hall be &longs;hewn by and by, after we &longs;hall have declared what is
nece&longs;&longs;ary for the knowledge thereof. You mu&longs;t know there
fore, that the Solid Bodies that &longs;ink in the Water weigh &longs;o much
le&longs;s in the Water than in the Air, as a Ma&longs;s of Water equal to
the &longs;aid Solid doth weigh in the Air: which hath been demon
&longs;trated by
mediate, becau&longs;e I would not be over long, laying it a&longs;ide, I &longs;hall
declare the &longs;ame another way. Let us con&longs;ider, therefore, that
putting into the Water g.
of Water it would have no weight at all: For the Water moveth
neither upwards, nor downwards in the Water: It remains,
therefore, that the Ma&longs;s of Gold weigheth in the Water only &longs;o
much as the Gravity of the Gold exceeds the Gravity of the Wa
ter. And the like is to be under&longs;tood of other Metals.
And be
cau&longs;e the Metals are different from each other in Gravity, their
Gravity in the Water &longs;hall dimini&longs;h according to &longs;everal proporti
ons. As for example: Let us &longs;uppo&longs;e that Gold weigheth twenty
times more than Water, it is manife&longs;t by that which hath been
&longs;poken, that the Gold will weigh le&longs;s in the Water than in the
Air by a twentieth part of its whole weight. Now, let us &longs;uppo&longs;e
that Silver, as being le&longs;s Grave than Gold, weigheth 12 times more
than Water: this then, being weighed in the Water, &longs;hall di
mini&longs;h in Gravity the twelfth part of its whole weight. Therefore
the Gravity of Gold in the Water decrea&longs;eth le&longs;s than that of
Silver; for that dimini&longs;heth a twentieth part, and this a twelfth. If therefore in an exqui&longs;ite Ballance we &longs;hall hang a Metal at the
one Arm, and at the other a Counterpoi&longs;e that weigheth equally
with the &longs;aid Metal in the Water, leaving the Counterpoi&longs;e in the
Air, to the end that it may equivalate and compen&longs;ate the Me
tal, it will be nece&longs;&longs;ary to hang it nearer the Perpendicular or
Cook. As for example, Let the Ballance be A B, its Perpendicu
lar C, and let a
Ma&longs;s of &longs;ome
Metal be &longs;u
&longs;pended at B,
counterpoi&longs;edby
the Weight D: putting the Weight B into the Water, the
Weight D in A would weigh more: therefore that they may
Perpendicular C, as gr.
&longs;tance C A &longs;hall contain A E, &longs;o many times &longs;hall the Metal
weigh more than the Water. Let us therefore &longs;uppo&longs;e that the
Weight in B be Gold, and that weighed in the Water it with
draws the Counterpoi&longs;e D into E; and then doing the &longs;ame with
pure Silver, let us &longs;uppo&longs;e that its Counterpoi&longs;e, when afterwards
it is weighed in the Water, returneth to F: which point &longs;hall be
nearer to the point C, as Experience &longs;heweth, becau&longs;e the Silver
is le&longs;s grave than the Gold: And the Di&longs;tance that is between
A and F &longs;hall have the &longs;ame Difference with the Di&longs;tance A E,
that the Gravity of the Gold hath with that of the Silver. But if
we have a Mixture of Gold and Silver, it is clear, that by rea&longs;on it
participates of Silver, it &longs;hall weigh le&longs;s than the pure Gold, and
by rea&longs;on it participates of Gold, it &longs;hall weigh more than the
pure Silver: and therefore being weighed in the Air, and de&longs;iring
that the &longs;ame Counterpoi&longs;e &longs;hould counterpoi&longs;e it, when that
Mixture &longs;hall be put into the Water it will be nece&longs;&longs;ary to draw
the &longs;aid Counterpoi&longs;e more towards the Perpendicular C, than the
point E is, which is the term of the Gold; and more from C
than F is, which is the term of the pure Silver; Therefore it &longs;hall
fall between the points E and F: And the proportion into which
the Di&longs;tance EF &longs;hall be divided, &longs;hall exactly give the proportion
of the two Metals which compound that Mixture. As for exam
ple: Let us &longs;uppo&longs;e the Mixture of Gold and Silver to be in B,
counterpoi&longs;ed in
the Air by D,
which Counter
poi&longs;e when the
Compound Me
tal is put into the Water returneth into G: I &longs;ay now, that the
Gold and the Silver which compound this Mixture are to one ano
ther in the &longs;ame proportion, as the Di&longs;tance F G is to the Di&longs;tance
G E. But you mu&longs;t know that the Di&longs;tance G F terminated in
the mark of the Silver, &longs;hall denote unto us the quantity of the
Gold, and the Di&longs;tance G E, terminated in the mark of the Gold,
&longs;hall &longs;hew us the quantity of the Silver: in&longs;omuch that if F G
&longs;hall prove double to G E, then that Mixture &longs;hall be two parts
Gold, and one part Silver: and in the &longs;ame method proceeding in
the examination of other Mixtures, one &longs;hall exactly find the
quantity of the &longs;imple Metals.
To compo&longs;e the Ballance, therefore, take a Rod at lea&longs;t a yard
long, (and the longer it is, the exacter the In&longs;trument &longs;hall be)
and divide it in the mid&longs;t, where place the Perpendicular: then
adju&longs;t the Arms that they may &longs;tand in
note the terms to which the Counterpoi&longs;es of &longs;imple Metals return
when they &longs;hall be weighed in the Water: taking care to weigh the
pure&longs;t Metals that can be found. This being done, it remaineth
that we find out a way, how we may with facility di&longs;cover the
proportion, according to which, the Di&longs;tances between the terms
of the &longs;imple and pure Metals are divided by the Marks of the
Mixt Metals: Which &longs;hall be effected in this manner.
We are to have two very &longs;mall Wires drawn thorow the &longs;ame
drawing-Iron, one of Steel, the other of Bra&longs;s, and above the
terms of the &longs;imple Metals we mu&longs;t wind the Steel Wyer; as for
example: above the point E, the term of the pure Gold, we are
to wind the Steel Wyer, and under it the other Bra&longs;s Wyre, and
having made ten folds of the Steel Wyer, we mu&longs;t make ten
more with that of Bra&longs;s, and thus we are to continue to do with
ten of Steel, and ten of Bra&longs;s, until that the whole Space be
tween the points E and F, the terms of the pure Metals, be full;
cau&longs;ing tho&longs;e two terms to be alwaies vi&longs;ible and per&longs;picuous:
and thus the Di&longs;tance E F &longs;hall be divided into many equal parts,
and numbred by ten and ten. And if at any time we would know
the proportion that is between F G and G E, we mu&longs;t count the
Wyers F G, and the Wyers G E: and finding the Wyers F G
to be, for example, 40, and the Wyers G E, 21: we will &longs;ay that
there is in the mixt Metal 40 parts of Gold, and 21 of Silver. But
here you mu&longs;t note, that there is &longs;ome difficulty in the counting,
for tho&longs;e Wyers being very &longs;mall, as it is requi&longs;ite for exactne&longs;s
&longs;ake, it is not po&longs;&longs;ible with the eye to tell them, becau&longs;e the
&longs;malne&longs;s of the Spaces dazleth & confoundeth the Sight. Therefore
to number them with facility, take a Bodkin as &longs;harp as a Needle
and &longs;et it into an handle, or a very fine pointed Pen-knife, with
which we may ea&longs;ily run over all the &longs;aid Wyers, and this way
partly by help of hearing, partly by the impediments the hand
&longs;hall feel at every Wyer, tho&longs;e Wyers &longs;hall be counted;
the number of which, as I &longs;aid before, &longs;hall give us the exact
quantity of the &longs;unple Metals, of which the Mixt-Metal is com
pounded: taking notice that the Simple an&longs;wer alternately to the
Di&longs;tances. As for example, in a Mixture of Gold and Silver,
the Wyers that &longs;hall be towards the term of Gold &longs;hall &longs;hew us
the quantity of the Silver: And the &longs;ame is to be under&longs;tood of
other Metals.
Annotations of
lance of
Fir&longs;t, I conceive that the difficulty of Numbring the Wyres
is removed by wrapping about the Ballance ten of Steel,
and then ten of Bra&longs;s, which being divided by tens, there
only remains that tenth part to be numbred, in which the term
of the Mixt Metal falleth. For although
Author of this Invention, makes mention of two Wyres, one of
Steel, the other of Bra&longs;s, yet he doth not &longs;ay, that we are to
take ^{*} ten of the one, and ten of the other: which it may be
hapneth by the negligence of him that hath tran&longs;cribed it; al
though I mu&longs;t confe&longs;s that the Copy which came to my hands was
of his own writing.
expre&longs;ly in this
Copy which I fol
low, but might
omit it in the Co
py which came to
the hands of
tovani.
Secondly, it is &longs;uppo&longs;ed in this Problem that the Compo&longs;ition
of two Metals do retain the &longs;ame proportion of Ma&longs;s in the
Mixture as the two Simple Metals, of which it is compounded,
had at fir&longs;t. I mean, that the Simple Metals retain and keep in
the Compo&longs;ition (after that they are incorporated and commix
ed) the &longs;ame proportion in Ma&longs;s that the Simple Metals had
when they were &longs;eparated: Which in the Ca&longs;e of
leo,
deny, nor particularly confe&longs;s. But if one would, for example,
unite 101 pounds of Copper with 21 pounds of Tin, to make
thereof 120 pounds of Bell-Metal, (I abate two pounds,
&longs;uppo&longs;ed to be wa&longs;ted in the Melting) I do think that 120
pounds of Compound Metal will have a le&longs;s Bulk than the 100
pounds of pure Copper, and the 20 pounds of Tin unmixt, that
is, before they were incorporated and melted into one Ma&longs;s, and
that the Compo&longs;ition is more grave
per, and the &longs;ingle Bra&longs;s: and in the Ca&longs;e of
Compo&longs;ition of Gold and Silver is &longs;uppo&longs;ed to be lighter
than the pure Gold, and heavier Of
which it would be ea&longs;ie to make &longs;ome &longs;uch like experiment, melt
ing together, gr.
and ob&longs;erving whether tho&longs;e 15 pounds, or whatever the Mixture
maketh, do give the difference betwixt the weight in the Water
to the weight in the Air, in the proportion that the 15 pounds of
the two Metals di&longs;-united gave before: I do not &longs;ay, the &longs;ame diffe
rence, becau&longs;e I pre &longs;uppo&longs;e that they will wa&longs;te in melting down,
and that the Compound will be le&longs;s than 15 pounds, therefore I
&longs;ay in proportion.
Thirdly, He doth al&longs;o &longs;uppo&longs;e, that one ought to take the
weight as the Mixture, although he doth not &longs;ay &longs;o; which may
be collected in that he marketh the ballance only betwixt the
Terms of the Gold and the Silver, which is the cau&longs;e of the great
facility in re&longs;olving the Problem by only counting the
Wyers.
One might take the pure Gold, and pure Silver of the &longs;ame
weight, in re&longs;pect of one another, but yet different from the
weight of the Mixture, that is, either more or le&longs;s grave than the
Mixt Metal: and being equal in weight to one another they
might &longs;hew the proportion of the Ma&longs;s of the Gold to that of the
Silver; but yet with this difference, that the more grave will &longs;hew
the &longs;aid proportion more exactly than the &longs;mall and le&longs;s grave. But the Simple and pure Metals not being of the &longs;ame weight as
the Compound, it will be nece&longs;&longs;ary, having found the proportion
of the Ma&longs;s of the Gold to that of the Silver; to find by numbers
proportionally the exact quantity of each of the two Metals com
pounding the Mixture.
A man may likewi&longs;e u&longs;e the quantity of the &longs;imple Metals ac
cording to nece&longs;&longs;ity and convenience, although of different
Weights, both as to each other, and to the Mixture, provided that
each of them be pure in its kind: but then we mu&longs;t after
wards by numbers find the proportion of the Ma&longs;&longs;es of the two
Simple ones of equal weight (which is &longs;oon done, taking them of
equal weight as was &longs;aid before) and then according to this pro
portion to find, by means of the Weight, and of the Ma&longs;s of the
Compound Metal, the di&longs;tinct quantity of each of the two Sim
ple ones that make the Compo&longs;ition: of each of which Ca&longs;es
examples might be given. But to conclude, if the pure Gold,
and pure Silver, and the Mixt Metal &longs;hould be of equal Ma&longs;s,
they would be unequal in Weight, and it would not need to
weigh them in the Water, for being of equal Bulk, the differen
ces of their Weights in the Air and in the Water would be al&longs;o
equal: for the difference of the weight of any Body in the Air
to its weight in the Water, is alwaies equal to the Weight of &longs;o
much Water as equalleth the &longs;ame Body in Ma&longs;s, by
his fifth Propo&longs;ition,
And la&longs;t of all, the Simple and pure Metals may have the &longs;ame
proportion in Gravity, mutually or reciprocally, as their Bodies
have in Bulk: In which ca&longs;e, as well the Ma&longs;s, found by help of
the weight in Water, or by any other meanes, as their Weight in
the Air &longs;hall &longs;hew the proportion of their Specifical Gravities; as
their Weights in the Water do when their Weights in the Air
are equal; but yet alternately weighed: that is to &longs;ay, the Spe
cifical Gravity of the Gold &longs;hall have &longs;uch proportion to the
the Ma&longs;s of the Gold; that is, as the difference betwixt the
Weight in Water and Weight in Air of the Silver, hath to the
difference betwixt the Weight in Water and Weight in Air of
the Gold.
With this &longs;ame Ballance one may with facility mea&longs;ure the
Ma&longs;s or Magnitude of any Body, in any manner what&longs;oever Irre
gular in manner following, namely:
We will have at hand a Solid Body of a &longs;ub&longs;tance more grave
of Wood, or other matter more light
it may be made heavier by fa&longs;tning unto it Lead, or &longs;ome other
thing that makes it &longs;ink in the Water, and let us take &longs;ome
known Mea&longs;ure, and with it mea&longs;ure the Irregular Solid; as for
in&longs;tance, the Roman Palm, the Geometrical Foot, or any other
known mea&longs;ure, or part of the &longs;ame, as the half Foot, the quar
ter of a Foot, or any &longs;uch like part known; then let it be weighed
in the Air, and &longs;uppo&longs;e that it weigh 10 pounds; let the &longs;ame
Mea&longs;ure be weighed in the Air, and &longs;uppo&longs;e that it weigh 8
pounds: and &longs;ub&longs;tract 8 pounds, the Weight in the Water, from
10 pounds, the Weight in the Air, and there remaineth 2 pounds
for the Weight of a Body of Water equal in Magnitude to the
Mea&longs;ure known. Now, if we would mea&longs;ure a Statue of Mar
ble, let it be weighed fir&longs;t in the Air, and then in the Water, and
&longs;ub&longs;tract the Weight in the
the remainder &longs;hall be the weight of &longs;o much
the Statue in Ma&longs;s; which being divided by the difference betwixt
the
the Quotient will give how many times the Statue containeth the
&longs;ame given Mea&longs;ure. As for example; if the Statue in Air weigh
100 pounds, and in the
&longs;tracted from 100 there re&longs;teth 20 pounds for the
much But becau&longs;e the
difference betwixt the
equal in Magnitude to the Mea&longs;ure known, was &longs;uppo&longs;ed to be
2 pounds; divide 18 pounds by two pounds, and the Quotient
is 9, for the number of times that the propo&longs;ed Statue containeth
the given Mea&longs;ure. The &longs;ame Method may be ob&longs;erved, if it
were required, to mea&longs;ure a Statue, or other Ma&longs;s of any kind of
Metal: only it mu&longs;t be adverti&longs;ed, that all the holes mu&longs;t be
&longs;topt, that the
but he that de&longs;ireth only the Solid content of the Metal of the
&longs;aid Statue mu&longs;t open the holes, and with Tunnels fill the whole
cavity of the Statue with And if the Statue were of a
Sub&longs;tance lighter
poi&longs;e, that maketh it &longs;ink in the
Counterpoi&longs;e, as above, and to &longs;ub&longs;tract its mea&longs;ure from the
Compound Body, and there will remain the Mea&longs;ure of the
Statue of And la&longs;tly, to make u&longs;e of the &longs;aid Ballance,
in&longs;tead of &longs;eeking the numbers of the pounds of the Differences
of the
to be mea&longs;ured in
being very &longs;mall will give the
Mea&longs;ure exactly.