Galilei, Galileo, Mechanics, 1665

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Author: Galilei, Galileo
Title: Mechanics
Date: 1665

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Permanent URL: http://echo.mpiwg-berlin.mpg.de/MPIWG:Y8U1Z9QU

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Copyright: Max Planck Institute for the History of Science (unless stated otherwise)
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1
GALILEUS,
HIS
MECHANICKS:
OF THE BENEFIT DERIVED
FROM THE SCIENCE OF MECHANICKS,
AND FROM ITS INSTRUMENTS.
I judged it extreamly neceſſary, before our
deſcending to the Speculation of Mecha­
nick Inſtruments, to conſider how I might,
as it were, ſet before your eyes in a gene­
ral Diſcourſe, the many benefits that are
derived from the ſaid Inſtruments: and
this I have thought my ſelf the more ob­
liged to do, for that (if I am not miſtaken)
I have ſeen the generality of Mechaniti­
ans deceive themſelves in going about to apply Machines to many
operations of their own nature impoſſible; by the ſucceſſe where­
of they have been diſappointed, and others likewiſe fruſtrate of
the hope which they had conceived upon the promiſe of thoſe pre­
ſumptuous undertakers: of which miſtakes I think I have found
the principall cauſe to be the belief and conſtant opinion theſe
1Artificers had, and ſtill have, that they are able with a ſmall force
to move and raiſe great weights; (in a certain manner with their
Machines cozening nature, whoſe Inſtinct, yea moſt poſitive con­
ſtitution it is, that no Reſiſtance can be overcome, but by a Force
more potent then it:) which conjecture how falſe it is, I hope by
the enſuing true and neceſſary Demonſtrations to evince.
In the mean time, ſince I have hinted, that the benefit and help
derived from Machines is not, to be able with leſſe Force, by help
of the Machine to move thoſe weights, which, without it, could
not be moved by the ſame Force: it would not be beſides the
purpoſe to declare what the Commodities be which are derived to
us from ſuch like faculties, for if no profit were to be hoped for,
all endeavours employed in the acquiſt thereof will be but loſt
labour.
Proceeding therefore according to the nature of theſe Studies,
let us firſt propoſe four things to be conſidered. Firſt, the weight
to be transferred from place to place; and ſecondly, the Force
and Power which ſhould move it; thirdly, the Diſtance 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
ſame thing with the Dexterity, and Velocity of the Motion; we
determining that Motion to be more ſwift then another, which in
leſſe Time paſſeth an equal Diſtance.
Now, any determinate Reſiſtance and limited Force whatſoever
being aſſigned, and any Diſtance given, there is no doubt to be
made, but that the given Force may carry the given Weight to the
determinate Diſtance; for, although the Force were extream
ſmall, yet, by dividing the Weight into many ſmall parts, none
of which remain ſuperiour to the Force, and by transferring them
one by one, it ſhall at laſt have carried the whole Weight to the
aſſigned Term: and yet one cannot at the end of the Work with
Reaſon ſay, that that great Weight hath been moved, and tranſ­
ported by a Force leſſe then it ſelf, howbeit indeed it was done
by a Force, that many times reiterated that Motion, and that
Space, which ſhall have been meaſured 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 Reſiſtance of the weight,
as the ſaid Weight was ſuperiour to the Force; for that in the
ſame Time that the moving Force hath many times meaſured the
intervall between the Terms of the Motion, the ſaid Moveable
happens to have paſt it onely once: nor therefore ought we to
affirm a great Reſiſtance to have been overcome by a ſmall Force,
contrary to the conſtitution of Nature. Then onely may we ſay
the Natural Conſtitution is overcome, when the leſſer Force tranſ­
fers the greater Reſiſtance, with a Velocity of Motion like to that
1wherewith it ſelf doth move; which we affirm abſolutely to be
impoſſible to be done with any Machine imaginable. But becauſe
it may ſometimes come to paſſe, that having but little Force, it is
required to move a great Weight all at once, without dividing it
in pieces, on this occaſion it will be neceiſary to have recourſe to
the Machine, by means whereof the propoſed Weight may be
transferred to the aſſigned Space by the Force given. But yet
this doth not hinder, but that the ſame Force is to move, meaſuring
that ſame Space, or another equall to it, as many ſeverall times as
it is exceeded by the ſaid Weight. So that in the end of the a­
ction we ſhall ſind that we have received from the Machine no
other benefit tnen only that of tranſporting the ſaid 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 ſame Force, in the ſame Time, through the ſame
Intervall. And this ought to paſſe for one of the benefits taken
from the Mechanicks: for indeed it frequently happens, that be­
ing ſcanted in Force but not Time, we are put upon moving great
Weights unitedly or in groſſe: but he that ſhould hope, and at­
tempt to do the ſame by the help of Machines without increaſe of
Tardity in the Moveable, would certainly be deceived, and would
declare his ignorance of the uſe of Mechanick Inſtruments, and
the reaſon of their effects.
Another benefit is drawn from the Inſtruments, which depend­
eth on the place wherein the operation is to be made: for all In­
ſtruments cannot be made uſe of in all places with equall conve­
nience. And ſo we ſee (to explain our ſelves by an example) that
for drawing of Water out of a Well, we make uſe 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 ſtrength: and he that ſhould think he could with
a Machine of whatſoever Force, with the ſame ſtrength, and in
the ſame Time, take up a great quantity of Water, is in a groſſe
Errour. And he ſhall find himſelf ſo much the more deceived,
the more he ſhall vary and multiply his Inventions: Yet never­
theleſſe we ſee Water drawn up with other Engines, as with a Pump
that drinks up Water in the Hold of Ships; where you muſt note
that the Pump was not imployed in thoſe Offices, for that it draws
up more Water in the ſame Time, and with the ſame ſtrength
then that which a bare Bucket would do, but becauſe in that place
the uſe of the Bucket or any ſuch like Veſſel could not effect what
is deſired, 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 conſiderable
depth of Water. And thus we ſee the Holds of Ships by the
1ſaid Inſtrument kept dry, when Water cannot but onely oblique­
ly be drawn up, which the ordinary uſe of the Bucket would not
effect, which riſeth and deſcends with its Rope perpendicu­
larly.
The third is a greater benefit, haply, then all the reſt that are
derived from Mechanick Inſtruments, and reſpects the aſſiſtance
which is borrowed of ſome Force exanimate, as of the ſtream of a
River, or elſe animate, but of leſſe expence by far, then that which
would be neceſſary for maintaining humane ſtrength: as when to
turn Mills, we make uſe of the Current of a River, or the ſtrength
of a Horſe, to effect that, which would require the ſtrength of five
or fix Men. And this we may alſo advantage our ſelves in raiſing
Water, or making other violent Motions, which muſt have been
done by Men, if there were no other helps; becauſe with one ſole
Veſſel we may take Water, and raiſe, and empty it where occaſion
requires; but becauſe the Horſe, or ſuch other Mover wanteth
Reaſon, and thoſe Inſtruments which are requiſite for holding and
emptying the Veſſel in due time, returning again to fill it, and one­
ly is endued with Force, therefore it's neceſſary that the Mecha­
nitian ſupply the naturall defect of that Mover, furniſhing it with
ſuch devices and inventions, that with the ſole application of it's
Force the defired effect may follow. And therein is very great
advantage, not becauſe that a Wheel or other Machine can enable
one to tranſport the ſame Weight with leſſe Force, and greater
Dexterity, or a greater Space than an equall Force, without thoſe
Inſtruments, but having Judgment and proper Organs, could have
done; but becauſe that the ſtream of a River coſteth little or
nothing, and the charge of keeping of an Horſe or other Beaſt,
whoſe ſtrength is greater then that of eight, or it may be more
Men, is far leſſe then what ſo many Men would be kept
for.
Theſe then are the benefits that may be derived from Mecha­
nick Inſtruments, and not thoſe which ignorant Engineers dream
of, to their own diſgrace, and the abuſe of ſo many Princes,
whilſt they undertake impoſſible enterprizes; of which, both
by the little which hath been hinted, and by the much which
ſhall be demonſtrated in the Progreſſe of this Treatiſe, we ſhall
come to aſſure our ſelves, if we attentively heed that which ſhall
be ſpoken.
1
DEFINITIONS.
That which in all Demonſtrative Sciences is neceſſary to be
obſerved, we ought alſo to follow in this Diſcourſe, that is;
to propound the Definitions of the proper Terms of this
Art, and the primary Suppoſitions, from which, as from ſeeds full
of fecundity, may of conſequence ſpring and reſult the cauſes,
and true Demonſtrations, of the Nature of all the Mechanick
Engines which are uſed, for the moſt part about the Motions of
Grave Matters, therefore we will determine, firſt, what is GRA­
VITIE.
We call GRAVITIE then, That propenſion of moving
naturally downwards, which is found in ſolid Bodies, cauſed by
the greater or leſſe quantity of matter, whereof they are conſti­
tuted.
MOMENT is the propenſion of deſcending, cauſed not ſo
much by the Gravity of the moveable, as by the diſpoſure which
divers Grave Bodies have in relation to one another; by means of
whichMoment, we oft ſee a Body leſs Grave counterpoiſe another
of greater Gravity: as in the Stiliard, a great Weight is raiſed by
a very ſmall counterpoiſe, not through exceſs of Gravity, but
through the remoteneſſe from the point whereby the Beam is up­
held, which conjoyned to the Gravity of the leſſer weight adds
thereunto Moment, and Impetus of deſcending, wherewith the
Moment of the other greater Gravity may be exceeded. MO­
MENT then is that IMPETUS of deſcending, compounded
of Gravity, Poſition, and the like, whereby that propenfion may
be occaſioned
The CENTER of GRAVITY we define to be that point
in every Grave Body, about which conſiſt parts of equall Moment:
ſo that, imagining ſome Grave Body to be ſuſpended and ſuſtain­
ed by the ſaid point, the parts on the right hand will Equilibrate
thoſe on the left, the Anteriour, the Poſteriour, and thoſe above
thoſe below; ſo that be it in any whatſoever fite, and poſition,
provided it be ſuſpended by the ſaid CENTER, it ſhall ſtand
ſtill: and this is that point which would gladly unite with the
univerſall Center of Grave Bodies, namely withthat of the Earth,
if it might thorow ſome free Medium deſcend thither. From
whence we take theſe Suppoſitions.
1
SUPPOSITIONS.
Any Grave Body, (as to what belongeth to it's proper ver­
tue) moveth downwards, ſo that the Center of it's Gravity
never ſtrayeth out of that Right Line which is produced
from the ſaid Center placed in the firſt Term of the Motion unto
the univerſal Center of Grave Bodies. Which is a Suppoſition
very manifeſt, becauſe that ſingle Center being obliged to endea­
vour to unite with the common Center, it's neceſſary, unleſſe ſome
impediment intervene, that it go ſeeking it by the ſhorteſt Line,
which is the Right alone: And from hence may we ſecondarily
ſuppoſe
Every Grave Body putteth the greateſt ſtreſſe, and weigheth
moſt on the Center of it's Gravity, and to it, as to its proper ſeat,
all Impetus, all Ponderoſity, and, in ſome, all Moment hath re­
courſe.
We laſtly ſuppoſe the Center of the Gravity of two Bodies e­
qually Grave to be in the midſt of that Right Line which conjoyns
the ſaid two Centers; or that two equall weights, ſuſpended in
equall diſtence, ſhall have the point of Equilibrium in the common
Center, or meeting of thoſe equal Diſtances. As for Example,
the Diſtance C E being equall to the Diſtance E D, and there be­
ing by them two equall weights ſuſpended, A and B, we ſuppoſe
the point of Equilibrium to be in the point E, there being no
greater reaſon for inclining to
one, then to the other part. But
1[Figure 1]
here is to be noted, that the Di­
ſtances ought to be meaſured
with Perpendicular Lines, which
from the point of Suſpenſion 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 Diſtance E D were tranſported into
E F, the weight B would not counterpoiſe the weight A, becauſe
drawing from the Centers of Gravity two Right Lines to the Cen­
ter of the Earth, we ſhall ſee 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 ſaying
that equal Weights are ſuſpended by [or at] equal Diſtances, is
to be underſtood to be meant when as the Right Lines that go from
their Centers & to ſeek out the common Center of Gravity, ſhall be
equidiſta nt from that Right Line, which is produced from the ſaid
1Term of thoſe Diſtances, that is from the point of Suſpenſion, to
the ſame Center of the Earrh.
Theſe things determined and ſuppoſed, we come to the explica­
tion of a Principle, the moſt common and materiall of the greater
part of Mechanick Inſtruments: demonſtrating, that unequall
Weights weigh equally when ſuſpended by [or at] unequal Diſtan­
ces, which have contrary proportion to that which thoſe weights
are found to have, See the Demonſtration in the beginning of the
ſecond Dialogue of Local-Motions.
Some Adveriiſements about what hath been ſaid.
Now being that Weights unequall come to acquire equall
Moment, by being alternately ſuſpended at Diſtances that
have the ſame proportion with them; I think it not fit to
over paſſe with ſilence another congruicy and probability, which
may confirm the ſame truth; for let the Ballance A B, be conſide­
red, as it is divided into unequal parts in the point C, and let the
Weights be of the ſame propor­
2[Figure 2]
tion that is between the Diſtan­
ces B C, and C A, alternately
ſuſpended by the points A, and
B: It is already manifeſt, that
the one will counterpoiſe the
other, and conſequently, that
were there added to one of them
a very ſmall Moment of Gravity, it would preponderate, raiſing
the other, ſo that an inſenſible Weight put to the Grave B, the
Ballance would move and deſcend from the point B towards E,
and the other extream A would aſcend into D, and in regard that
to weigh down B, every ſmall Gravity is ſufficient, therefore not
keeping any accompt of this inſenſible Moment, we will put no
difference between one Weights ſuſtaining, and one Weights
moving another. Now, let us conſider the Motion which the
Weight B makes, deſcending into E, and that which the other
A makes in aſcending into D, we ſhall without doubt find the
Space B E to be ſo much greater than the Space A D, as the Di­
ſtance 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 conſe­
quently two Circumferences A D and B E alike; and to have the
ſame proportion to one another, as have the Semidiameters B C,
and C A, by which they are deſcribed: ſo that then the Velocity
of the Motion of the deſcending Grave B cometh to be ſo much
Superiour to the Velocity of the other aſcending Moveable A, as
the Gravity of this exceeds the Gravity of that; and it not being
1poſſible that the Weight A ſhould be raiſed to D, although ſlow­
ly, unleſſe the other Weight B do move to E ſwiftly, it will not
be ſtrange, or inconſiſtent with the Order of Nature, that the
Velocity of the Motion of the Grave B, do compenſate the greater
Reſiſtance of the Weight A, ſo long as it moveth ſlowly to D,
and the other deſcendeth ſwiftly to E, and ſo on the contrary,
the Weight A being placed in the point D, and the other B in
the point E, it will not be unreaſonable that that falling leaſurely
to A, ſhould be able to raiſe the other haſtily to B, recovering by
its Gravity what it had loſt by it's Tardity of Motion. And by
this Diſcourſe we may come to know how the Velocity of the
Motion is able to encreaſe Moment in the Moveable, according to
that ſame proportion by which the ſaid Velocity of the Motion is
augmented.
There is alſo another thing, before we proceed any farther, to
be confidered; and this is touching the Diſtances, whereat, or
wherein Weights do hang: for it much imports how we are to
underſtand Diſtances equall, and unequall; and, in ſum, in what
manner they ought to be mea­
3[Figure 3]
ſured: for that A B being the
Right Line, and two equall
Weights being ſuſpended at
the very ends thereof, the point
C being taken in the midſt of
the ſaid Line, there ſhall be an
Equilibrium upon the ſame:
And the reaſon is for that the
Diſtance C B is equal to C A.
But if elevating the Line C B, moving it about the point C, it
ſhall be transferred into CD, ſo that the Ballance ſtand according
to the two Lines A C, and C D, the two equall Weights hanging
at the Terms A and D, ſhall no longer weigh equally on that
point C, becauſe the diſtance of the Weight placed in D, is made
leſſe then it was when it hanged in B. For if we confider the Lines,
along [or by] which the ſaid Graves make their Impulſe, and
would deſcend, in caſe they were freely moved, there is no doubt
but that they would make or deſcribe the Lines A G, D F, B H:
Therefore the Weight hanging on the point D, maketh it's Moment
and Impetus according to the Line D F: but when it hanged in
B, it made Impetus in the Line B H: and becauſe the Line D F is
nearer to the Fulciment C, then is the Line B H Therefore we
are to underſtand that the Weights hanging on the points A and D,
are not equi-diſtant from the point C, as they be when they are
conſtituted according to their Right Line A C B: And laſtly,
we are to take notice, that the Diſtance is to be meaſured by
1Lines, which fall at Right Angles on thoſe whereon the Weights
hang, and would move, if ſo be they were permitted to deſcend
freely.
Of the BALLANCE and LEAVER.
Having underſtood by certain Demonſtration, one of the
firſt Principles, from which, as from a plentiſul Fountain,
many of the Mechanical Inſtruments are derived, we may
take occaſion without any difficulty to come to the knowledge of
the nature of them: and firſt ſpeaking of the Stiliard, an Inſtru­
ment of moſt ordinary uſe, with which divers Merchandizes are
weighed, ſuſtaining them, though very heavy, with a very ſmall
counterpoiſe, which is com­
monly called the Roman or
4[Figure 4]
Plummet, we ſhall prove that
there is no more to be done in
ſuch an operation, but to re­
duce into act and practice
what hath been above contemplated. For if we propoſe the Bal­
lance A B, whoſe Fulciment or Lanquet is in the point C, by
which, at the ſmall Diſtance C A, hangeth the heavy Weight D,
and if along the other greater C B, (which we call the Needle of
the Stiliard) we ſhould ſuppoſe the Roman F, though of but little
weight in compariſon of the Grave Body D to be ſlipped to and
fro, it ſhall be pofſible to place it ſo remotely from the Lanquet C,
that the ſame proportion may be found between the two Weights
D and F, as is between the Diſtances F C, and C A: and then ſhall
an Equilibrium ſucceed; unequall Weights hanging at Diſtances
alternately proportional to them.
Nor is this Inſtrument different from that other called Vectis,

and vulgarly the ^{*} Leaver, wherewith great Weights are moved
by ſmall Force; the application of which is according to the Fi­
gure prefixed; wherein the Leaver
is repreſented by the Bar of wood
or other ſolid matter, B C D, let
5[Figure 5]
the heavy Weight to be raiſed be
A, and let the ſteadfaſt ſupport
or Fulciment on which the Leaver
reſts and moves be ſuppoſed to be
E, and putting one end of the
Leaver under the Weight A, as
may be ſeen in the point C, en­
creaſing 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
1D hath the ſame proportion to the Reſiſtance made by the Weight
A, in the point C: as the Diſtance B C hath to the Diſtance C D,
whereby it's clear, that the nearer the Fulciment E ſhall approach
to the Term B, encreaſing the proportion of the Diſtance D C to
the Diſtance C B, the more may one diminiſh the Force in D which
is to raiſe the Weight A. And here it is to be noted, which I ſhall
alſo in its place remember you of, that the benefit drawn from all
Mechanical Inſtruments, is not that which the vulgar Mechanitians
do perſwade us, to wit, ſuch, that there by Nature is overcome, and
in a certain manner deluded, a ſmall Force over-powring a very
great Reſiſtance with help of the Leaver; for we ſhall demonſtrate,
that without the help of the length of the Leaver, the ſame Force,
in the ſame Time, ſhall work the ſame effect. For taking the ſame
Leaver B C D, whoſe reſt or Fulci­
ment is in C, let the Diſtance C D
6[Figure 6]
be ſuppoſed, for example, to be
in quintuple proportion to the
Diſtance C B, & the ſaid Leaver to
be moved till it come to I C G: In
the Time that the Force ſhall have
paſſed the Space D I, the Weight
ſhall have been moved from B
to G: and becauſe the Diſtance
D C, was ſuppoſed quintuple to the other C B, it is manifeſt from
the things demonſtrated, that the Weight placed in B may be five
times greater then the moving Force ſuppoſed to be in D: but now,

if on the contrary, we take notice of the ^{*} Way paſſed by
the Force from D unto I, whilſt the Weight is moved from B unto
G, we ſhall find likewiſe the Way D I, to be quintuple to the Space
B G. Moreover if we take the Diſtance C L, equal to the Diſtance
C B, and place the ſame 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 firſt, there is no queſtion, but that the Force in L being now
equal to this Weight in B, and the Diſtances L C and C B being
equall, the ſaid Force ſhall be able, being moved along the Space LM
to transfer the Weight equall to it ſelf, thorow the other equall
Space B G: which five times reiterating this ſame action, ſhall tranſ­
port all the parts of the ſaid Weight to the ſame Term G: But
the repeating of the Space L M, is certainly nothing more nor leſſe
then the onely once meaſuring the Space D I, quintuple to the
ſaid L M. Therefore the transferring of the Weight from B to G,
requireth no leſſe Force, nor leſſe Time, nor a ſhorter Way if it
wee placed in D, than it would need if the ſame were applied
in L: And, in ſhort, the benefit that is derived from the length of
the Leaver C D, is no other, ſave the enabling us to move that
1Body all at once, which would not have been moved by the ſame
Force, in the ſame Time, with an equall Motion, ſave onely in
pieces, without the help of the Leaver.
If of Iron, it is
called a Crow,
if of wood, a Bar
or Hand-ſpike.
Or Space.
Of the CAPSTEN and of the CRANE.
The Inſtruments 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 ſhall ſuppoſe the
Leaver B A C to be ſuſtained in
the point A, and the Weight G to
7[Figure 7]
hang at the point B, the Force be­
ing placed in C; It is manifeſt,
that transferring the Leaver unto
the points D A E, the Weight G
doth alter according to the Di­
ſtance B D, but cannot much far­
ther continue to raiſe it, ſo that
if it were required to elevate it yet
higher, it would be neceſſary to
ſtay it by ſome other Fulciment
in this Poſition, and to remit or return the Leaver to its former Po­
ſition B A C, and ſuſpending the Weight anew thereat, to raiſe it
once again to the like height B D; and in this manner repeating
the work, many times one ſhall come with an interrupted Motion
to effect the drawing up of the Weight, which for many reſpects
will not prove very beneficial: whereupon this difficulty hath bin
thought on, and remedied, by finding out a way how to unite to­
gether almoſt 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 ſame Center, of which let the Line A B
be the Semidiameter; and all this of very tough wood, or of other
ſtrong and ſolid matter, afterwards ſuſtaining the whole Machine
upon a Gudgeon or Pin of Iron planted in the point A, which
paſſeth quite thorow, where it is held faſt 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 manifeſt, that the length C A having to the other A B
the ſelf-ſame proportion that the Weight G hath to the Weight I,
it may ſuſtain the Grave G, and with any little Moment more ſhall
move it: and becauſe the Axis turning round together with the
Wheel, the Ropes that ſuſtain the Weights are alwaies pendent and
contingent with the extream Circumferences of that Wheel and
1Axis, ſo that they ſhall conſtantly maintain alike Site and Poſition
in reſpect of the Diſtances B A and A C, the Motion ſhall be
perpetuated, the Weight I deſcending, and forcing the other G
to aſcend. Where we are to obſerve the neceſſity of be-laying
or winding the Rope about the Wheel, that ſo the Weight I may
hang according to the Line that is tangent to the ſaid Wheel: for
if one ſhould ſuſpend the ſaid Weight, ſo as that it did hang by the
point F, cutting the ſaid Wheel, as is ſeen along the Line F N M,
the Motion would ceaſe, the Moment of the Weight M being di­
miniſhed; which would weigh no more then if it did hang by the
point N: becauſe the Diſtance of its Suſpenſion 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 ſaid Line F M. A violence therefore being offered
in the Circumference of the Wheel by a Grave and Exanimate
Body that hath no other Impetus then that of Deſcending, it is
neceſſary that it be ſuſtained by a Line that is contingent with
the Wheel, and not by one that cutteth it. But if in the ſame
Circumference an Animate Force were employed, that had a Mo­
ment or Faculty of making an Impulſe on all ſides, the work might
be effected in any whatever place of the ſaid 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 ſide-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: ſo, that if in this manner one do
meaſure the Diſtance from the Center A to the Force placed in
F, according to the Line A F perpendicular to F L, along which
the Impetus is made, a man ſhall not in any part have altered the
uſe of the ordinary Leaver. And we muſt note, that the ſame
would be poſſible to be done likewiſe with an Exanimate Force,
in caſe that a way were found out to cauſe that its Moment might
make Impulſe 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
paſſe along upon it, as it is ſeen to do by the Line F L X, ſuſpending
at the end thereof the Weight X equall to the other I, which ex­
erciſing its Force according to the Line F L, ſhall alwaies keep a
Diſtance from the Center A equall unto the Semidiameter of the
Wheel. And from what hath been declared we will gather for a
Concluſion, That in this Inſtrument the Force hath alwaies the
ſame proportion to the Weight, as the Semidiameter of the Axis
or Barrell hath to the Semidiameter of the Wheel.
1
From the Inſtrument laſt deſcribed, the other Inſtrument which
we call the Crane is not much different, as to form, nay, differeth
nothing, ſave in the way of applying or employing it: For that the
Capſten moveth and is conſtituted perpendicular to the Horizon,
and the Crane worketh with its Moment parallel to the ſame Ho­
8[Figure 8]
rizon. For if upon the Circle D A E we ſuppoſe an Axis to be
placed Column-wiſe, turning about the Center B, and about which
the Rope D H, faſtened to the Weight that is to be drawn, is be­
laid, and if the Bar F E B D be let into the ſaid Axis [by the Mor­
tace B] and the Force of a Man, of an Horſe, or of ſome other
Animal apt to draw, be applyed at its end F, which moving round,
paſſeth along the Circumference F G C, the Crane ſhall be framed
and finiſhed, ſo that by carrying round the Bar F B D, the Barrell
or Axis E A D ſhall turn about, and the Rope which is twined a­
bout it, ſhall conſtrain the Weight H to go forward: And becauſe
the point of the Fulciment about which the Motion is made, is the
point B, and the Moment keeps at a Diſtance from it according to
the Line B F, and the Reſiſtor at the Diſtance B D, the Leaver
F B D is formed, by vertue of which the Force acquireth Moment
equall to the Reſiſtance, if ſo 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 whoſe Circumference the
Force moveth. And both in this, and in the other Inſtrument we
are to obſerve that which hath been frequently mentioned, that is,
That the benefit which is derived from theſe Machines, is not that
which the generality of the Vulgar promiſe themſelves from the
Mechanicks; namely, that being too hard for Nature, its poſſible
1with a Machine to overcome a Reſiſtance, though great, with a
ſmall Force, in regard, that we ſhall manifeſtly prove that the ſame
Force placed in F, might in the ſame Time conveigh the ſame
Weight, with the ſame Motion, unto the ſame Diſtance, without
any Machine at all: For ſuppoſing, for example, that the Reſiſtance
of the Grave H be ten times greater than the Force placed in F, it
9[Figure 9]
will be requiſite for the mo­
ving of the ſaid Reſiſtance,
that the Line F B be decuple
to B D; and conſequently,
that the Circumference of the
Circle F G C be alſo decuple
to the Circumference E A D:
and becauſe when the Force
ſhall 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, ſhall
likewiſe have given one onely turn; it is manifeſt, that the Weight
H ſhall not have been moved more than the tenth part of that way
which the Mover ſhall have gone. If therefore the Force that is to
move a Reſiſtance that is greater than it ſelf, for ſuch an aſſigned
Space by help of this Machine, muſt of neceſſity move ten times as
far, there is no doubt, but that dividing that Weight into ten parts,
each of them ſhall be equall to the Force, and conſequently, might
have been tranſported one at a Time, as great a Space as that
which it ſelf did move, ſo that making ten journeys, each equal to
the Circumference E A D, it ſhall not have gone any farther than
if it did move but once alone about the Circumference F G C;
and ſhall have conveighed the ſame Weight H to the ſame Di­
ſtance. The benefit therefore that is to be derived from theſe
Machines is, that they carry all the Weight together, but not with
leſſe Labour, or with greater Expedition, or a greater Way than
the ſame Force might have done conveying it by parcels.
Of PULLIES.
The Inſtruments, whoſe Natures are reducible unto the Bal­
lance, as to their Principle and Foundation, and others little
differing from them, have been already deſcribed; now for
the underſtanding of that which we have to ſay touching Pullies,
it is requiſite, that we conſider in the firſt place another way to uſe
the Leaver, which will conduce much towards the inveſtigation of
the Force of Pullies, and towards the underſtanding of other Me­
chanical Effects. The uſe of the Leaver above declared ſuppoſed
1the Weight to be at one extream, and the Force at the other, and
the Fulciment placed in ſome point between the extreams: but we
may make uſe of the Leaver another way, yet, placing, as we ſee,
the Fulciment in the extream A, the Force in the other extream C,
and ſuppoſing the Weight D to hang by ſome point in the midſt,
10[Figure 10]
as here we ſee by the point B, in
this example it's manifeſt, that if
the Weight did hang at a point
Equi-diſtant from the two ex­
treams A and C, as at the point F,
the labour of ſuſtaining it would
be equally divided betwixt the
two points A and C, ſo that half
the Weight would be felt by the
Force C, the other half being ſu­
ſtained by the Fulciment A: but if the Grave Body ſhall be hanged
at another place, as at B, we ſhall ſhew that the Force in C is ſuffi­
cient to ſuſtain the Weight in B, as it hath the ſame proportion
to it, that the Diſtance, A B hath to the Diſtance A C. For De­
monſtration of which, let us imagine the Line B A to be continued
right out unto G, and let the Diſtance B A be equall to A G, and
let the Weight hanging at G, be ſuppoſed equall to D: It is ma­
nifeſt, that by reaſon of the equality of the Weights D and E, and
of the Diſtances G A and A B, the Moment of the Weight E
ſhall equalize the Moment of the Weight D, and is ſufficient to
ſuſtain it: Therefore whatever Force ſhall have Moment equall to
that of the Weight E, and that ſhall be able to ſuſtain it, ſhall be
ſufficient likewiſe to ſuſtain the Weight D: But for ſuſtaining the
Weight E, let there be placed in the point C ſuch a Force, whoſe
Moment hath that proportion to the Weight E, that the Diſtance
G A hath to the Diſtance A C, it ſhall be ſufficient to ſuſtain it:
Therefore the ſame Force ſhall likewiſe be able to ſuſtain the
Weight D, whoſe Moment is equall to the of E: But look what
Proportion the Line G A hath to the Line A C; and A B alſo hath
the ſame to the ſaid A C, G A having been ſuppoſed equall to A B:
And becauſe the Weights E and D are equall, each of them ſhall
have the ſame 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 ſame proportion that the Diſtance B A
hath to the Diſtance C A. And by moving the Weight, with the
Leaver uſed in this manner, it is gathered in this alſo, as well as in
the other Inſtruments, that what is gained in Force is loſt in Velo­
city: for the Force C raiſing the Leaver, and transferring it to A I,
the Weight is moved the Space B H, which is as much leſſer than
the Space C I paſſed by the Force, as the Diſtance A B is leſſer
1than the Diſtance A C; that is, as the Force is leſſe than the
Weight.
Theſe Principles being declared, we will paſſe to the Contem­
plation of Pullies, the compoſition and ſtructure of which, together
with their uſe, ſhall be deſcribed by us. And firſt let us ſuppoſe the

^{*} Little Pulley A B C, made of Mettall or hard Wood, voluble a­
bout it's Axis which paſſeth thorow it's Center D, and about this
11[Figure 11]
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 ſuppoſe
the Force F. I ſay, that the Weight
being ſuſtained by a Force equall to
it ſelf in the upper Nut or Pulley
A B C, bringeth ſome benefit, as the
moving or ſuſtaining of the ſaid
Weight with the Force placed in F:
For if we ſhall underſtand, 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
ſhall have a Ballance of equal Arms which determine the Diſtance
of the two Suſpenſions from the Center and Fulciment D: Where­
upon it is manifeſt, that the Weight hanging at A cannot be ſuſtain­
ed by a leſſer Weight hanging at G, but by one equal to it; ſuch
is the nature of equal Weights hanging at equal Diſtances. 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 Reſpect, that the Weight and Force have unto the two
Diſtances A D and D C, nay, the Pulley encompaſſed becometh a
Ballance equal to A C, but perpetuall. Whence we may learn,
how childiſhly Ariſtotle deceiveth himſelf, who holds, that by making
the ſmall Pulley A B C bigger, one might draw up the Weight with
a leſſer Force; he conſidering that upon the enlargement of the
ſaid Pulley, the Diſtance D C encreaſed, but not conſidering that
there was as great an encreaſe of the other Diſtance of the Weight,
that is, the other Semidiameter D A. The benefit therefore that may
be drawn from the Inſtrument above ſaid, is nothing at all as to the
diminution of the labour: and if any one ſhould ask how it hap­
pens, that on many occaſions of raiſing Weights, this means is made
uſe of to help the Axis, as we ſee, for example, in drawing up the
Water of Wells; it is anſwered, that that is done, becauſe 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 ſole ſtrength
1of the Members and Muſcles, and as we uſe to ſay, by Force of
Armes, beſides 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 ſimply conſidered, but onely to the manner of applying it:
but if we ſhall make uſe of the like Machine
12[Figure 12]
in another manner, as we are now about to
declare; we may raiſe 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
paſſe about the Pulley; of which let the end
A be faſtned to ſome fixed ſtay, and in the
other F let the Force be placed; which
moving to wards H ſhall raiſe the Machine
B L C, and conſequently the Weight G:
and in this operation I ſay, that the Force in
F is the half of the Weight ſuſtained by it.
For the ſaid Weight being kept to Rights by the two ^{*} Ropes A B

and F C, it is manifeſt, that the Labour is equally ſhared betwixt
the Force F and the Fulciment A: and more ſubtilly examining the
nature of this Inſtrument, if we but continue forth the Diameter
B E C, we ſhall ſee a Leaver to be made, at the midſt 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 demonſtrated, the Force ſhall have the ſame
proportion to the Weight, that the Diſtance E B hath to the Di­
ſtance; Therefore it ſhall be the half of the ſaid Weight: And
becauſe the Force riſing towards A, the Pulley turneth round,
therefore that Reſpect or Conſtitution which the Fulciment B and
Center E, on which the Weight and Term C, in which the Force
is employed do depend, ſhall 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 ſucceeding in their
place, whereby the Leaver B C cometh to be perpetuated. And
here (as hath been done in the other Inſtruments, and ſhall be in
thoſe that follow) we will not paſſe without conſidering how that
the journey that the Force maketh, is double to the Moment of the
Weight. For in caſe the Weight ſhall be moved ſo 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 neceſſary that the two equal Ropes be diſtended in
one ſole Line F H, and conſequently, when the Weight ſhall have
aſcended along the Intervall B A, the Force ſhall have been moved
twice as far, that is, from F unto H. Then conſidering that the
1Force in F, that it may raiſe the Weight, muſt move upwards, which
to exanimate Movers, as being for the moſt part Grave Bodies, is al­
13[Figure 13]
together impoſſible, or at leaſt more laborious,
than the making of the ſame Force down­
wards: Therefore to help this inconvenience,
a Remedy hath been found by adjoyning an­
other Nut or Pulley above, as in the adjacent
Figure is ſeen, where the Rope C E F hath
been made to paſs about the upper Pulley F G
upheld by the Hook L, ſo that the Rope paſſing
to H, and thither transferring the Force E, it
ſhall be able to move the Weight X by pulling
downwards, but not that it may be leſſer than
it was in E: For the Motions of the Force
F H, hanging at the equal Diſtances F D and
D G of the upper Pulley, do alwaies continue
equal; nor doth that upper Pulley (as hath
been ſhewn above) come to produce any di­
minution in the Labour. Moreover it having been neceſſary by
the addition of the upper Pulley to introduce the Appendix B, by
which it is ſuſtained, it will prove of ſome benefit to us to raiſe
the other A, to which one end of the Rope was faſtned, transferring
it to a Ring annexed to the lower part of the Frame of the upper
Pulley, as we ſee it done in M. Now finally, this Machine com­
pounded of upper and lower Pullies, is that which the Greeks call

Τποχίλιον.
*Called by ſome
a Nut.
* Or two ends of
the ſame Rope.
In Latine Tro­
chlea.
We have hitherto explained, how by help of Pullies one may
double the Force, it remaineth that with the greateſt brevity poſ­
ſible, we ſhew the way how to encreaſe it according to any Multi­
plicity. And firſt we will ſpeak of the Multiplicity according to
the even numbers, and then the odde: To ſhew how we may mul­
tiply the Force in a quadruple Proportion, we will propound the
following Speculation as the Soul of all that followeth.
Take two Leavers, A B, C D, with the Fulciments in the ex­
14[Figure 14]
treams A and C; and at the middles
of each of them let the Grave G hang,
ſuſtained by two Forces of equal Mo­
ment placed in B and D. I ſay, that
the Moment of each of them will
equal the Moment of the fourth part
of the Weight G. For the two For­
ces B and D bearing equally, it is
manifeſt, that the Force D hath not
contraſted with more then one half of the Weight G: But if the
Force D do by benefit of the Leaver D C ſuſtain the half of the
1Weight G hanging at F, it hath been already demonſtrated, that
the ſaid Force D hath to the Weight ſo by it ſuſtained, that ſame
proportion which the Diſtance F C hath to the Diſtance C D:
Which is ſubduple proportion: Therefore the Moment D is ſub­
duple to the Moment of half of the Weight G ſuſtained by it:
Wherefore it followeth, that it is the fourth part of the Moment
of the whole Weight. And in the ſame manner the ſame thing is
demonſtrated, of the Moment B; and it is but reaſonable, that the
Weight G being ſuſtained by the four points, A, B, C, D, each of
them ſhould feel an equall part of the Labour.
Let us come now to apply this Conſideration to Pullies, and let
the Weight X be ſuppoſed to hang at the two Pullies A B and D E
entwining about them, and about the uppermoſt Pulley G H, the
Rope, as we ſee, I D E H G A B, ſuſtaining the whole Machine in
the point K. Now I ſay, that placing the Force in L, it ſhall be able
to ſuſtain the Weight X, if ſo 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 ſhall have two
Leavers like to thoſe before deſcribed, the Fulciments of which an­
ſwer to the points D and A. Whereupon the Force placed in B,
15[Figure 15]
or if you will, in L, ſhall be able to ſu­
ſtain 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 paſs along L M N, trans­
ferring the Force L into N, it ſhall be
able to bear the ſame Weight gravitating
downwards, the upper Pulley neither aug­
menting or diminiſhing the Force, as hath
been declared. And we will likewiſe
note, that to make the: Weight aſcend the

four Ropes B L, E H, D I, and A G
ought to paſs, whereupon the Mover will
be to begin, as much as thoſe Ropes are
long; and yet nevertheleſs the Weight
ſhall move but only as much as the length
of one of them: So that we may ſay by
way of advertiſement, and for confirma­
tion of what hatn been many times ſpo­
ken, namely, that look with what proportion the Labour of the

Mover is diminiſhed, the length of the Way, on the contrary, is
encreaſed with the ſame proportion
* Or four parts
of the ſame Rope
* The word Gy­
rilla ſignifieth a
Shiver, Rundle,
or ſmall Wheel
of a Pulley, tran­
ſlated by we
ſometimes Pul­
ley, ſometimes
Nut or Girill.
But if we would encreaſe the Force in ſexcuple proportion, it
will be requiſite that we adjoyn another ^{*} ſmall Pulley or Gyrill
to the inferiour Pulley which that you may the better underſtand
1we will ſet before you the preſent Contemplation. Suppoſe, 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, ſo that every one of them ſhall ſuſtain the third part of
16[Figure 16]
it: And becauſe the Power in
B, ſuſtaining with the Leaver
B A thependent Weight in G,
hapneth to be the half of the
ſaid Weight, and it hath been
already ſaid, that it ſuſtaineth
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 ſixth part of it:
And the ſame ſhall be demonſtrated of the other Forces D and F:
From whence we may eaſily gather, that putting three Gyrils or
Rundles into the inferiour Pulley, and two or three into the upper­
17[Figure 17]
moſt, we may multiply the Force accor­

ding to our ^{*} Senarius. And if we would
encreaſe it according to any other even
Number, the Gyrils of the Pulley below
muſt be multiplyed according to the half
of that Number, according to which the
Force is to be multiplyed, circumpoſing
the Rope about the Pulleys, ſo as that one
of the ends be faſtned to the upper Pul­
ley, and let the Force be in the other; as
in this Figure adjoyning may manifeſtly
be gathered.
* Or in Sexcuple
proportion.
Now paſſing to the Declaration of the
manner how to multiply the Force ac­
cording to the odd Numbers, and begin­
18[Figure 18]
ning at the triple proportion: firſt, let us
propoſe the preſent Contemplation, as
that, on the underſtanding 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 ſu­
ſtained by two equal Forces; and let one of them be applied to the
point C, and the other to the term B. I ſay, that each of thoſe Powers
have Moment equal to the third part of the Weight D. For the
Force in C ſuſtaineth a Weight equal to it ſelf, being placed in the
ſame Line in which the Weight D doth hang & Gravitate: But the
1Force in B ſuſtaineth a part of the Weight D double to it ſelf, its
Diſtance from the Fulciment A, that is, the Line B A being dou­
ble to the Diſtance A C at which the Grave hangeth: But becauſe
the two Forces in B and C are ſuppoſed to be equal to each other:
Therefore the part of the Weight D, which is ſuſtained by the
Force in B, is double to the part ſuſtained by the Force in C. If
therefore, of the Grave D two parts be made, the one double to
the remainder, the greater is ſuſtained by the Force in B, and the
leſſer by the Force in C: But this leſſer 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
conſequence, the Force B ſhall be equal, we having ſuppoſed it
equal to the other Force C: Wherefore our intention is manifell,
which we were to demonſtrate, how that each of the two Powers
C and B is equal to the third part of the Weight D. Which be­
ing demonſtrated, we will paſs forwards to the Pulleys, and will
deſcribe 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 faſtned to the inferiour Pulley, and to
19[Figure 19]
the other I let the Force be applyed:
Which, I ſay, ſuſtaining or moving the
Weight H, ſhall feele no more than the
third part of the Gravity of the ſame. For
conſidering the contrivance of this Ma­
chine, we ſhall find that the Diameter A B
ſupplieth the place of a Leaver, in whoſe
term B the Force I is applied, and in the
other A the Fuiciment is placed, at the mid­
dle G the Grave H is hanged, and another
Force D applied at the ſame place: ſo that

the Weight is faſtned to the ^{*} three Ropes
I B, F D, and E A, which with equal Labour
ſuſtain the Weight. Now, by what hath
already been contemplated, the two Forces
D and B being applied, one, to the midſt of the Leaver A B, and
the other to the extream term B, it is manifeſt, 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, ſhall be able to ſuſtain and move it: but yet the Way
of the Force in I ſhall be triple to the Way that the Weight ſhall
paſs; the ſaid Force being to diſtend it ſelf according to the
Length of the three Ropes I B, F D, and E A, of which one alone
meaſureth the Way of the Weight H.
1
* Or three parts
of one Rope.
Of the SCREW.
Amongſt the reſt of Mechanick Inſtruments for ſundry uſes
found out by the Wit of Man, the Screw doth, in my opi­
nion, both for Invention and for Utility, hold the firſt
place, as that which is appoſitely accommodated, and ſo contrived
not only to move, but alſo to ſtay and preſs with very great Force,
that taking up but little room, it worketh thoſe effects which other
Inſtruments cannot, unleſs they were reduced to a great Machine.
The Screw therefore being of moſt ingenious and commodious
contrivance, we ought deſervedly to be at ſome pains in explaining,
with all the plainneſs that is poſſible, the Original and Nature of
it. The which that we may do, we will begin at a Speculation,
which, though at firſt bluſh it may appear ſomewhat remote from
the conſideration of this Inſtrument, yet is the Baſis and Founda­
tion thereof.
No doubt, but that Natures operation in the Motions of Grave
Bodies is ſuch, that any whatever Body that hath a Gravity in it
hath a propenſion of moving, being at liberty, towards the Cen­

ter, and that not only ^{*} by the Right Line perpendicularly, but al­
ſo (when it cannot do otherwiſe) by any other Line, which ha­
ving ſome inclination towards the Center goeth more and more
abaſing. And thus we ſee the Water not only to fall downwards
along the Perpendicular from ſome eminent place, but alſo to run
about the Surface of the Earth along Lines though very little en­
clined; as we ſee in the Courſe of Rivers, the Waters of which, if ſo
be that the Bed have any the leaſt declivity, go freely declining
downwards. Which very effect, like as it is diſcerned in all Fluid
Bodies, would appear alſo in hard Bodies, if ſo be, that their Fi­
gure and other Accidental and Extern Impediments did not hinder
it. So that we, having a Superficies very well ſmoothed and poli­
ſhed, as for inſtance, that of a Looking-glaſs, and a Ball exactly
rotund and ſleek, either of Marble, or of Glaſs, or of any other
Matter apt to be poliſhed, this being placed upon that Superficies
ſhall trundle along, in caſe that this have any, though very ſmall,
inclination; and ſhall lie ſtill 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 ſtanding Water being frozen, up­
on which the ſaid Spherical Body would ſtand ſtill, but in a con­
dition of being moved by every ſmall Force. For we having ſup­
poſed that if that Plane did incline but an hairs breadth only, the
ſaid Ball would move along it ſpontaneouſly towards the part de­
clining, and on the oppoſite would have a Reſiſtance, nay, would
not be able without ſome Violence to move towards the part
1riſing or aſcending: it of neceſſity remaineth manifeſt, that in the
Superficies which is exactly equilibrated, the ſaid Ball remaineth in­
different and dubious between Motion and Reſt, ſo that every ſmall
Force is ſufficient to move it, as on the contrary, every ſmall Reſi­
ſtance, and no greater than that of the meer Air that environs it, is
able to hold it ſtill.
* Or along.
From whence we may take this Concluſion 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 ſo ſmall Force: but when the ſame Grave is to be thrown along
an Aſcending Plane, then, it beginning to ſtrive againſt that aſcent,
having an inclination to the contrary Motion, there ſhall be requi­
red greater Violence, and ſtill greater the more Elevation that ſame
Plane ſhall have. As for example, the Moveable G, being poſited
upon the Line A B parallel to the Horizon, it ſhall, as hath been
ſaid, be indifferent on it either to Motion or Reſt, ſo that it may
be moved by a very ſmall Force: But if we ſhall have the Planes
Elevated, they ſhall not be driven along without Violence; which
20[Figure 20]
Violence will be required to be
greater to move it along the Line
A D, than along A C; and ſtill
greater along A E than along A D:
The which hapneth, becauſe it hath
greater Impetus of going down­
wards along A E than along A D,
and along A D than along A C. So
that we may likewiſe conclude
Grave Bodies to have greater Reſiſtance upon Planes differently
Elevared, to their being moved along the ſame, according as one
ſhall be more or leſs elevated than the other; and, in fine, that the
greateſt Reſiſtance of the ſame Grave to its being raiſed is in the
Perpendicular A F. But it will be neceſſary to declare exactly what
proportion the Force muſt have to the Weight, that it may be able
to carry it along ſeveral elevated Planes, before we proceed any
farther, to the end that we may perfectly underſtand all that which
remains to be ſpoken.
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 ſhall be demonſtrated that the ſame Weight ſhall be mo­
ved along the Plane A C with leſſer Force than along the Perpendi­
cular A F, (where it is raiſed by a Force equal to it ſelf) accor­
ding to the proportion by which the Perpendicular C H is leſs than
A C: and that along the Plane A D, the Force hath the ſame pro­
portion to the Weight, that the Perpendicular I D hath to D A:
and, laſtly, that in the Plane A E the Force to the Weight obſer­
veth the proportion of E K and E A.
1
The preſent Speculation hath been attempted by Pappus Alex­
andrinus in Lib. 8. de Collection. Mathemat. but, if I be in the
right, he hath not hit the mark, and was overſeen in the Aſſumpti­
on that he maketh, where he ſuppoſeth that the Weight ought to
be moved along the Horizontal Line by a Force given; which is
falſe: there needing no ſenſible Force (removing the Accidental
Impediments, which in the Theory are not regarded) to move the
given Weight along the Horizon, ſo that he goeth about in vain
afterwards to ſeek with what Force it is to be moved along the
elevated Plane. It will be therefore better, the Force that moveth
the Weight upwards perpendicularly, (which equalizeth the Gra­
vity of that Weight which is to be moved) being given, to
ſeek the Force that moveth it along the Elevated Plane: Which
we will endeavour to do in a Method different from that of
Pappus.
Let us therefore ſuppoſe 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; ſo that the Line A C being a Leaver,
or Ballance moveable about the Center B, the Weight C ſhall
come to be ſuſtained by the Weight A. But if we ſhall imagine
the Arm of the Ballance B C to be inclined downwards according
to the Line B F, but yet in ſuch a manner that the two Lines A B
and B F do continue ſolidly conjoyned in the point B, in this caſe
the Moment of the Weight C ſhall not be equal to the Moment
21[Figure 21]
of the Weight A, for that the Di­
ſtance of the point F from the Line
of Direction, which goeth accord­
ing to B I, from the Fulciment B un­
to the Center of the Earth, is dimi­
niſhed: But if from the point F we
erect a Perpendicular unto B C, as is
F K, the Moment of the Weight in
F ſhall be as if it did hang by the
Line K F, and look how much the
Diſtance K B is diminiſhed by the
Diſtance B A, ſo much is the Moment of the Weight F diminiſhed
by the Moment of the Weight A. And in this faſhion inclining
the Weight more, as for inſtance, according to B L, its Moment ſhall
ſtill diminiſh and ſhall be as if it did hang at the Diſtance B M, ac­
cording to the Line M L, in which point L it ſhall be ſuſtained by
a Weight placed in A, ſo much leſs than it ſelf, by how much the
Diſtance B A is greater than the Diſtance B M. See therefore that
the Weight placed in the extream of the Leaver B C, in inclining
downwards along the Circumference C F L I, cometh to diminiſh
its Moment and Impetus of going downwards from time to time,
1more and leſs, as it is more or leſs ſuſtained by the Lines B F and
B L: But the conſidering that this Grave deſcending, and ſuſtained
by the Semidiameters B F and B L is one while leſs, and another
while more conſtrained to paſs along the Circumference C F L, is
no other, than if we ſhould imagine the ſame Circumference
C F L I to be a Superſicies ſo curved, and put under the ſame
Moveable: ſo that bearing it ſelf thereon it were conſtrained to
deſcend along thereby; for if in the one and other manner the
Moveable deſcribeth the ſame Courſe or Way, it will nothing im­
port whether, if ſuſpended at the Center B, it is ſuſtained by the
Semidiameter of the Circle, or elſe, whether that Fulciment being
taken away, it proceed along the Circumference C F L I: So that
we may confidently affirm, that the Grave deſcending downwards
from the point C along the Circumference C F L I, its Moment
of Deſcent in the point C is total and entire, becauſe it is not in
any part ſuſtained by the Circumference: And there is not in that
firſt point C, any indiſpoſition 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 ſhall be placed in the
point F, then its Gravity is in part ſuſtained, and its Moment of
Deſcent is diminiſhed by the Circular Path or Way that is placed
under it, in that proportion wherewith the Line B K is overcome
by B C: But if when the Moveable is in F, at the firſt inſtant of
ſuch its Motion, it be as if it were in the Plane elevated according
to the Contingent Line G F H, for that reaſon the inclination of the
Circumference in the point F differeth not from the inclination of
the Contingent Line F G any more ſave the inſenſible Angle of
the Contact. And in the ſame manner we ſhall find the Moment
of the ſaid Moveable to diminiſh in the point L, as the Line BM
is diminiſhed by B C; ſo that in the Plane contingent to the Circle
in the point L, as for inſtance, according to the Line N L O, the
Moment of Deſcent diminiſheth in the Moveable with the ſame
proportion. If therefore ^{*} upon the Plane HG the Moment of the

Moveable be diminiſhed by the total Impetus which it hath in its
Perpendicular D C E, according to the proportion of the Line K B
to the Line B C, and B F, being by the Solicitude of the Triangles
K B F and K F H the ſame proportion betwixt the Lines K F and
F H, as betwixt the ſaid K B and B F, we will conclude that the
proportion of the entire and abſolute 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 ſame proportion that the
Line H F hath to the Line F K; that is, that the Length of the
Inclined Plane hath to the Perpendicular which ſhall fall from it
unto the Horizon. So that paſſing to a more diſtinct Figure, ſuch
as this here preſent, the Moment of Deſcending which the Move­
1able hath upon the inclined Plane C A hath to its total Moment
wherewith it gravitates in the Perpendicular to the Horizon C P the
ſame proportion that the ſaid Line P C hath to C A. And if thus it
be, it is manifeſt, that like as the Force that ſuſtai­
neth the Weight in the Perpendiculation P C ought
22[Figure 22]
to be equal to the ſame, ſo for ſuſtaining it in the
inclined Plane C A, it will ſuffice that it be ſo much
leſſer, by how much the ſaid Perpendicular C P wan­
teth of the Line C A: and becauſe, as ſometimes we
ſce, it ſufficeth, that the Force for moving of the
Weight do inſenſibly ſuperate that which ſuſtaineth it, therefore
we will infer this univerſal Propoſition, [That upon an Elevated
Plane the Force hath to the Weight the ſame proportion, as the
Perpendicular let fall from the Plane unto the Horizon hath to the
Length of the ſaid Plane.]
* Or along
Returning now to our firſt Intention, which was to inveſtigate
the Nature of the Screw, we will conſider the Triangle A B C, of
which the Line A B is Horizontal, B C perpendicular to the ſaid
Horizon, and A C a Plane elevated; upon which the Moveable D
ſhall be drawn by a Force ſo much leſs than it, by how much the
Line B C is ſhorter than C A: But to elevate or raiſe the ſaid
Weight along the ſaid Plane A C, is as much as if the Triangle
C A B ſtanding ſtill, the Weight
23[Figure 23]
D be moved towards C, which is
the ſame, as if the ſame Weight
never removing from the Perpen­
dicular A E, the Triangle did
preſs 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 Eſſence of the Screw is no­
thing elſe but ſuch a Triangle A C B, which being forced for­
wards, ſhall work it ſelf under the Grave Body to be raiſed, and
lifteth it up, as we ſay, by the ^{*} head and ſhoulders. And this was

its firſt Original: For its firſt Inventor (whoever he was) conſi­
dering how that the Triangle A B C going forwards raiſeth the
Weight D, he might have framed an Inſtrument like to the ſaid
Triangle, of a very ſolid Matter, which being thruſt forwards did
raiſe up the propoſed Weight: But afterwards conſidering better,
how that that ſame Machine might be reduced into a much leſſer
and more commodious Form, taking the ſame Triangle he twined
and wound it about the Cylinder A B C D in ſuch a faſhion, that
the height of the ſaid Triangle, that is the Line C B, did make the
Height of the Cylinder, and the Aſcending Plane did beget upon
1the ſaid Cylinder the Helical Line deſcribed by the Line AEFGH,
which we vulgarly call the Wale of the Screw, which was produ­
ced by the Line A C. And in this manner is the Inſtrument made,
which is by the Greeks called Κόχλος, and by us a Screw; which

winding about
cometh to work
24[Figure 24]
and inſinu­
ate with its
Wales under
the Weight, and
with facility rai­
ſeth it. And we
having demon­
ſtrated, That up­
on [or along]
the elevated Plane the Force hath the ſame proportion to the
Weight, that the perpendicular Altitude of the ſaid Plane hath to
its Length; ſo, ſuppoſing 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 cloſe
together, it becometh ſo much the more forceable, as being begot
by a Plane leſs elevated, and whoſe Length regards its own Per­
pendicular Altitude with greater proportion. But we will not
omit to advertiſe you, that deſiring to find the Force of a propo­
ſed Screw, it will not be needful that we meaſure the Length of
all its Wales, and the Altitude of the whole Cylinder, but it
will be enough if we ſhall but examine how many times the Di­
ſtance betwixt two ſingle and Contiguous terms do enter into one
ſole Turn of the ſame Wale, as for example, how many times
the Diſtance AF is contained in the Length of the Turn AEF:
For this is the ſame proportion that the Altitude CB hath to all
the Wale.
Levar in capo
ſignfieth to lift
on high by force
* Κόχλος, in La­
tine Cocblea, any
Screw winding
like the Shell of
a Snail.
If all that be underſtood which we have hitherto ſpoken touch­
ing the Nature of this Inſtrument, I do not doubt in the leaſt but
that all the other circumſtances may without difficulty be compre­
hended: as for inſtance, that inſteed 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 raiſeth and
lifteth up the Nut or Male Screw together with the Weight which
was hanged thereat. Laſtly, we are not to paſs over that Conſidera­
tion with ſilence which at the beginning hath been ſaid to be neceſ­
ſary for us to have in all Mechanick Inſtruments, to wit, That
what is gained in Force by their aſſiſtance, is loſt again in Time,
1and in the Velocity: which peradventure, might not have ſeemed
to ſome ſo true and manifeſt in the preſent Contemplation; nay,
rather it ſeems, that in this caſe the Force is multiplied without the
Movers moving a longer way than the Moveable: In regard, that
if we ſhall in the Triangle A B C ſuppoſe the Line A B to be the
Plane of the Horizon, A C the elevated Plane, whoſe Altitude is
meaſured by the Perpendicular C B, a Moveable placed upon the
Plane A C, and the Cord E D F tyed to it, and a Force or Weight
applyed in F that hath to the
Gravity of the Weight E the
25[Figure 25]
ſame proportion that the Line
B C hath to C A; by what
hath been demonſtrated, the
Weight F ſhall deſcend
downwards, drawing the
Moveable E along the eleva­
ted Plane; nor ſhall the Move­
able E meaſure a greater Space
when it ſhall have paſſed the
whole Line A C, than that which the ſaid Grave F meaſureth in its
deſcent downwards. But here yet it muſt be advertiſed, that al­
though the Moveable E ſhall have paſſed the whole Line A C, in
the ſame Time that the other Grave F ſhall have been abaſed the
like Space, nevertheleſs the Grave E ſhall not have retired from the
common Center of things Grave more than the Space of the Per­
pendicular C B. but yet the Grave F deſcending Perpendicularly ſhall
be abaſed a Space equal to the whole Line A C. And becauſe Grave
Bodies make no Reſiſtance to Tranſverſal Motions, but only ſo
far as they happen to recede from the Center of the Earth; There­
fore the Moveable E in all the Motion A C being raiſed no more
than the length of the Line CB, but the other F being abaſed per­
pendicularly the quantity of all the Line A C: Therefore we may
deſervedly affirm that Way of the Force E maintaineth the ſame
proportion to the Force F that the Line A C hath to C B; that is,
the Weight E to the Weight F. It very much importeth, therefore,
to conſider by [or along] what Lines the Motions are made, eſpe­
cially in exanimate Grave Bodies, the Moments of which have their
total Vigour, and entire Reſiſtance in the Line Perpendicular to
the Horizon; and in the others tranſverſally Elevated and Inclined
they feel the more or leſs Vigour, Impetus, or Reſiſtance, the more
or leſs thoſe Inclinations approach unto the Perpendicular Inclina­
tion.
1
Of the SCREW of ARCHIMEDES
to draw Waier.
I Do not think it ſit in this place to paſs over with Silence the
Invention of Archimedes to raiſe Wa er with the Screw, which
is not only marvellous, but miraculous: for we ſhall find that
the Water aſcendeth in the Screw continually deſcending; and in
a given Time, with a given Force doth raiſe an unſpeakable quan­
tity therof. But before we proceed any farther, let us declare the uſe
of the Screw in making Water to riſe: And in the enſuing Figure,
let us conſider the Line I L O P Q
26[Figure 26]
R S H being wrapped or twined
about the Collumn M I K H,
which Line you are to ſuppoſe to
be a Chanel thorow which the
Water may run: If we ſhall put
the end I into the Water, making
the Screw to ſtand leaning, ſo as
the point L may be lower than
the firſt I, as the Diagram ſhew­
eth, and ſhall turn it round about
on the two Axes, T and V, the Water ſhall run thorow the Cha­
nel, till that in the end it ſhall diſcharge ſorth at the mouth H.
Now I ſay, that the Water, in its conveyance from the point I to
the point H, doth go all the way deſcending, although the point H
be higher than the point I. Which that it is ſo, we will declare
in this manner. We will deſcribe the Triangle A C B, which is
that of which the Screw H I is generated, in ſuch ſort that the
Chanel of the Screw is repreſented by the Line A C, whoſe
Aſcent and Elevation is determined by the Angle C A B; that is
to ſay, if ſo be, that that Angle ſhall be the third or fourth part of a
Right Angle, then the Elevation of the Chanel A C ſhall be ac­
cording to 1/3, or 1/4 of a Right Angle. And it is manifeſt; that the
Riſe of that ſame Chanel A C will be taken away debaſing the
point C as far as to B: for then the Chanel A C ſhall have no
Elevation. And debaſing 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 ſhall no longer have any
Riſe, debaſing it on the part C for 1/3 of a Right Angle.
Theſe things underſtood, let us infold the Triangle about the
Column, and let us make the Screw B A E F G, &c. which if it
ſhall be placed at Right Angles with the end B in the Water, turn­
ing it about, it ſhall not this way draw up the Water, the Chanel
about the Column being elevated, as may be ſeen by the part B A.
1But although the Column ſtand erect at Right-Angles, yet for all
that, the Riſe 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
27[Figure 27]
ſaid Right Angle, and
a little more, as we ſee
I K H M, there is a
Tranſition and Moti­
on along the Chanel
I L: Therefore the
Water from the point
I to the point L ſhall
move deſcending, and
the Screw being turned
about, the other parts
of it ſhall ſucceſſively
diſpoſe or preſent
themſelves to the Wa­
ter in the ſame Poſition as the part I L: Whereupon the Water
ſhall go ſucceſſively deſcending, and in the end ſhall be found to
be aſcended from the point I to the point H. Which how admira­
ble a thing it is, I leave ſuch to judge who ſhall perfectly have un­
derſtood it. And by what hath been ſaid, we come to know, That
the Screw for raiſing of Water ought to be inclined a little more
than the quantity of the Angle of the Triangle by which the ſaid
Screw is deſcribed.
Of the Force of the
HAMMER, MALLET, or BEETLE.
The Inveſtigation of the cauſe of the Force of theſe Percuti­
ents is neceſſary for many Reaſons: and firſt, becauſe that
there appeareth in it much more matter of admiration than
is obſerved in any other Mechanick Inſtrument whatſoever. For
ſtriking with the Hammer upon a Nail, which is to be driven into
a very tough Poſt, or with the Beetle upon a Stake that is to pene­
trate into very ſtiffe ground, we ſee, that by the ſole vertue of the
blow of the Percutient both the one and the other is thruſt for­
wards: ſo that without that, only laying the Beetle upon the
Nail or Stake it will not move then, nay, more, although you
ſhould lay upon them a Weight very much heavier than the ſaid
Beetle. An effect truly admirable, and ſo much the more worthy
of Contemplation, in that, as I conceive, none of thoſe who have
1hitherto diſcourſed upon it, have ſaid any thing that hits the mark;
which we may take for a certain Sign and Argument of the Obſcu­
rity and difficulty of this Speculation. For Ariſtotle, or others,
who would reduce the cauſe of this admirable Effect unto the
length of the Manubrium, or Handle, may, in my judgement, be
made to ſee their miſtake in the effect of thoſe Inſtruments, which
having no Handle, yet percuſs, either in falling from on high
downwards, or by being thrown with Velocity ſidewaies. There­
fore it is requiſite, that we have recourſe to ſome other Principle, if
we would find out the truth of this buſineſs; the cauſe of which,
although it be of its own nature ſomewhat obſcure, and of diffi­
cult conſideration, yet nevertheleſs we will attempt with the grea­
teſt perſpicuity poſſible to render it clear and obvious, ſhewing, for
a cloſe of all, that the Principle and Original of this Effect is deri­
ved from no other Fountain than this, from which the reaſons of all
other Mechanick Effects do proceed: and this we will do, by ſetting
before your eyes that very thing which is ſeen to befall in every
other Mechanick Operation, ſcilicet, That the Force, the Reſiſtance,
and the Space by which the Motion is made, do go alternately
with ſuch proportion operating, and with ſuch a rate anſwering to
each other, that a Reſiſtance, equal to the Force, ſhall be moved by
the ſaid Force along an equal Space, with Velocity equal to that
with which it is moved. Likewiſe, That a Force that is leſs by half
than a Reſiſtance ſhall be able to move it, ſo that it be moved
with double Velocity, or, if you will, for a Diſtance twice as great
as that which the moved Reſiſtance ſhall paſs: and, in a word, it
hath been ſeen in all the other Inſtruments, that any, never ſo great,
Reſiſtance may be moved by every ſmall Force given, provided,
that the Space, along which the Reſiſtance ſhall move, have the
ſame proportion that is found to be betwixt the ſaid great Reſi­
ſtance and the Force: and that this is according to the neceſſary
Order and Conſtitution of Nature: So that inverting the Diſcourſe,
and Arguing the contrary way, what wonder ſhall it be, if that
Power that ſhall move a ſmall Reſiſtance a great way, ſhall carry
one an hundred times bigger an hundredth part of that Diſtance?
Certainly none at all: nay, it would be abſurd, yea, impoſſible
that it ſhould be otherwiſe. Let us therefore conſider, what the
Reſiſtance of the Beetle unto Motion may be in that point where
it is to ſtrike, and how far, if it do not ſtrike, it would be carryed
by the received Force beyond that point: and again, what Reſi­
ſtance to Motion there is in him who ſtriketh, and how much by
that ſame Percuſſion he is moved: and, having found that this
great Reſiſtance goeth forwards by a percuſſion ſo much leſs than
the Beetle driven by the Impetus of him that moveth it would do,
by how much that ſame great Reſiſtance is greater than that of
1the Beetle; we ſhall ceaſe to wonder at the Effect, which doth not
in the leaſt exceed the terms of Natural Conſtitutions, and of
what hath been ſpoken. Let us, for better underſtanding, give an
example thereof in particular Terms. There is a Beetle, which ha­
ving four degrees of Reſiſtance, is moved by ſuch a Force, that
being freed from it in that term where it maketh the Percuſſion, it
would, meeting with no ſtop, go ten Paces beyond it, and in that
term a great poſt being oppoſed to it, whoſe Reſiſtance to Moti­
on is as four thouſand, that is, a thouſand times greater than that of
the Beetle, (but yet is not immoveable) ſo that it without mea­
ſure or proportion exceeds the Reſiſtance of the Beetle, yet the
Percuſſion being made on it, it ſhall be driven forwards, though in­
deed no more but the thouſandth part of the ten Paces which the
Beetle ſhall be moved: and thus in an inverted method, changing
that which hath been ſpoken touching the other Mechanical Effects,
we may inveſtigate the reaſon of the Force of the Percutient. I
know that here ariſe difficulties and objections unto ſome, which
they will not eaſily be removed from, but we will freely remit them

to the ^{*} Problems Mechanical, which we ſhall adjoyn in the end of
this Diſcourſe.
* Theſe Pro­
blems he here
promiſeth were
never yet ex­
tant.
1
THE
BALLANCE
OF
Signeur GALILEO GALILEI;
In which, in immitation of Archimedes in the
Problem of the Crown, he ſheweth how to
find the proportion of the Alloy of
Mixt-Metals; and how to make
the ſaid Inſtrument.
As it is well known, by ſuch who take the pains to read
old Authors, that Archimedes detected the Cheat of
the Goldſmith in the Crown of ^{*} Hieron, ſo I think it

hitherto unknown what method this Great Philoſo­
pher obſerved in that Diſcovery: for the opinion, that he did per­
form it by putting the Crown into the Water, having firſt put in­

to it ſuch another Maſs of pure Gold, and another of Silver ſeve­
rally, and that from the differences in their making the Water
more or leſs riſe and run over, he came to know the Mixture or
Alloy of the Gold with the Silver, of which that Crown was
compounded; ſeems a thing (if I may ſpeak it) very groſs, and
far from exactneſs. And it will ſeem ſo much the more dull to
ſuch who have read and underſtood the exquiſite Inventions of ſo
Divine a Man amongſt the Memorials that are extant of him; by
which it is very manifeſt that all other Wits are inferiour to that
of Archimedes. Indeed I believe, that Fame divulging it abroad,
that Archimedes had diſcovered that ſame Fraud by means of the
Water, ſome Writer of thoſe Times committed the memory there­
of to Poſterity, and that this perſon, that he might add ſomething
to that little which he had heard by common Fame, did relate that
Archimedes had made uſe of the Water in that manner, as ſince
hath been by the generality of men believed.
* King of Sicily,
and Kinſman to
that Great Ma­
thematician.
Plutarch in Vit.
Marcel.
But in regard I know, that that method is altogether fallacious,
and falls ſhort of that exactneſs 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 peruſed that which Archimedes
demonſtrateth in his Books De inſidentibus aquæ, and thoſe others
1De æquiponder antium, there came into my thoughts a Rule which
exquiſitely reſolveth our Queſtion; which Rule I believe to be
the ſame that Archimedes made uſe of, ſeeing that beſides the
uſe that is to be made of the Water, the exactneſs of the Work
dependeth alſo upon certain Demonſtrations found by the ſaid
Archimedes.
The way is by help of a Ballance, whoſe Conſtruction and Uſe
ſhall be ſhewn by and by, after we ſhall have declared what is
neceſſary for the knowledge thereof. You muſt know there­
fore, that the Solid Bodies that ſink in the Water weigh ſo much
leſs in the Water than in the Air, as a Maſs of Water equal to
the ſaid Solid doth weigh in the Air: which hath been demon­
ſtrated by Archimedes. But, in regard his Demonſtration is very
mediate, becauſe I would not be over long, laying it aſide, I ſhall
declare the ſame another way. Let us conſider, therefore, that
putting into the Water v. g. a Maſs of Gold, if that Maſs were
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 Maſs of Gold weigheth in the Water only ſo
much as the Gravity of the Gold exceeds the Gravity of the Wa­
ter. And the like is to be underſtood of other Metals. And be­
cauſe the Metals are different from each other in Gravity, their
Gravity in the Water ſhall diminiſh according to ſeveral proporti­
ons. As for example: Let us ſuppoſe that Gold weigheth twenty
times more than Water, it is manifeſt by that which hath been
ſpoken, that the Gold will weigh leſs in the Water than in the
Air by a twentieth part of its whole weight. Now, let us ſuppoſe
that Silver, as being leſs Grave than Gold, weigheth 12 times more
than Water: this then, being weighed in the Water, ſhall di­
miniſh in Gravity the twelfth part of its whole weight. Therefore
the Gravity of Gold in the Water decreaſeth leſs than that of
Silver; for that diminiſheth a twentieth part, and this a twelfth.
If therefore in an exquiſite Ballance we ſhall hang a Metal at the
one Arm, and at the other a Counterpoiſe that weigheth equally
with the ſaid Metal in the Water, leaving the Counterpoiſe in the
Air, to the end that it may equivalate and compenſate the Me­
tal, it will be neceſſary to hang it nearer the Perpendicular or
Cook. As for example, Let the Ballance be A B, its Perpendicu­
28[Figure 28]
lar C, and let a
Maſs of ſome
Metal be ſu­
ſpended at B,
counterpoiſedby
the Weight D: putting the Weight B into the Water, the
Weight D in A would weigh more: therefore that they may
1weigh equally it would be neceſſary to hang it nearer to the
Perpendicular C, as v. gr. in E: and look how many times the Di­
ſtance C A ſhall contain A E, ſo many times ſhall the Metal
weigh more than the Water. Let us therefore ſuppoſe that the
Weight in B be Gold, and that weighed in the Water it with­
draws the Counterpoiſe D into E; and then doing the ſame with
pure Silver, let us ſuppoſe that its Counterpoiſe, when afterwards
it is weighed in the Water, returneth to F: which point ſhall be
nearer to the point C, as Experience ſheweth, becauſe the Silver
is leſs grave than the Gold: And the Diſtance that is between
A and F ſhall have the ſame Difference with the Diſtance 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 reaſon it
participates of Silver, it ſhall weigh leſs than the pure Gold, and
by reaſon it participates of Gold, it ſhall weigh more than the
pure Silver: and therefore being weighed in the Air, and deſiring
that the ſame Counterpoiſe ſhould counterpoiſe it, when that
Mixture ſhall be put into the Water it will be neceſſary to draw
the ſaid Counterpoiſe 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 ſhall
fall between the points E and F: And the proportion into which
the Diſtance EF ſhall be divided, ſhall exactly give the proportion
of the two Metals which compound that Mixture. As for exam­
ple: Let us ſuppoſe the Mixture of Gold and Silver to be in B,
29[Figure 29]
counterpoiſed in
the Air by D,
which Counter­
poiſe when the
Compound Me­
tal is put into the Water returneth into G: I ſay now, that the
Gold and the Silver which compound this Mixture are to one ano­
ther in the ſame proportion, as the Diſtance F G is to the Diſtance
G E. But you muſt know that the Diſtance G F terminated in
the mark of the Silver, ſhall denote unto us the quantity of the
Gold, and the Diſtance G E, terminated in the mark of the Gold,
ſhall ſhew us the quantity of the Silver: inſomuch that if F G
ſhall prove double to G E, then that Mixture ſhall be two parts
Gold, and one part Silver: and in the ſame method proceeding in
the examination of other Mixtures, one ſhall exactly find the
quantity of the ſimple Metals.
To compoſe the Ballance, therefore, take a Rod at leaſt a yard
long, (and the longer it is, the exacter the Inſtrument ſhall be)
and divide it in the midſt, where place the Perpendicular: then
adjuſt the Arms that they may ſtand in Equilibrium, by filing or
1ſhaving that leſs which weigheth moſt; and upon one of the Arms
note the terms to which the Counterpoiſes of ſimple Metals return
when they ſhall be weighed in the Water: taking care to weigh the
pureſt Metals that can be found. This being done, it remaineth
that we find out a way, how we may with facility diſcover the
proportion, according to which, the Diſtances between the terms
of the ſimple and pure Metals are divided by the Marks of the
Mixt Metals: Which ſhall be effected in this manner.
We are to have two very ſmall Wires drawn thorow the ſame
drawing-Iron, one of Steel, the other of Braſs, and above the
terms of the ſimple Metals we muſt 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 Braſs Wyre, and
having made ten folds of the Steel Wyer, we muſt make ten
more with that of Braſs, and thus we are to continue to do with
ten of Steel, and ten of Braſs, until that the whole Space be­
tween the points E and F, the terms of the pure Metals, be full;
cauſing thoſe two terms to be alwaies viſible and perſpicuous:
and thus the Diſtance E F ſhall 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 muſt 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 ſay that
there is in the mixt Metal 40 parts of Gold, and 21 of Silver. But
here you muſt note, that there is ſome difficulty in the counting,
for thoſe Wyers being very ſmall, as it is requiſite for exactneſs
ſake, it is not poſſible with the eye to tell them, becauſe the
ſmalneſs of the Spaces dazleth & confoundeth the Sight. Therefore
to number them with facility, take a Bodkin as ſharp as a Needle
and ſet it into an handle, or a very fine pointed Pen-knife, with
which we may eaſily run over all the ſaid Wyers, and this way
partly by help of hearing, partly by the impediments the hand
ſhall feel at every Wyer, thoſe Wyers ſhall be counted;
the number of which, as I ſaid before, ſhall give us the exact
quantity of the ſunple Metals, of which the Mixt-Metal is com­
pounded: taking notice that the Simple anſwer alternately to the
Diſtances. As for example, in a Mixture of Gold and Silver,
the Wyers that ſhall be towards the term of Gold ſhall ſhew us
the quantity of the Silver: And the ſame is to be underſtood of
other Metals.
1
Annotations of Dominico Mantovani upon the Bal­
lance of Signore Galileo Galilei.
Firſt, I conceive that the difficulty of Numbring the Wyres
is removed by wrapping about the Ballance ten of Steel,
and then ten of Braſs, 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 Signore Galileo, who is
Author of this Invention, makes mention of two Wyres, one of
Steel, the other of Braſs, yet he doth not ſay, 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 tranſcribed it; al­
though I muſt confeſs that the Copy which came to my hands was
of his own writing.
* Galileus ſaith it
expreſly in this
Copy which I fol­
low, but might
omit it in the Co­
py which came to
the hands of Man­
tovani.
Secondly, it is ſuppoſed in this Problem that the Compoſition
of two Metals do retain the ſame proportion of Maſs in the
Mixture as the two Simple Metals, of which it is compounded,
had at firſt. I mean, that the Simple Metals retain and keep in
the Compoſition (after that they are incorporated and commix­
ed) the ſame proportion in Maſs that the Simple Metals had
when they were ſeparated: Which in the Caſe of Signore Gali­
leo, touching the Commixtion of Gold and Silver, I do neither
deny, nor particularly confeſs. 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,
ſuppoſed to be waſted in the Melting) I do think that 120
pounds of Compound Metal will have a leſs 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 Maſs, and
that the Compoſition is more grave in Specie than the ſingle Cop­
per, and the ſingle Braſs: and in the Caſe of Signore Galileo the
Compoſition of Gold and Silver is ſuppoſed to be lighter in Specie
than the pure Gold, and heavier in Specie than the pure Silver. Of
which it would be eaſie to make ſome ſuch like experiment, melt­
ing together, v. gr. 10 pounds of Lead with 5 pounds of Tin,
and obſerving whether thoſe 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 diſ-united gave before: I do not ſay, the ſame diffe­
rence, becauſe I pre ſuppoſe that they will waſte in melting down,
and that the Compound will be leſs than 15 pounds, therefore I
ſay in proportion.
Thirdly, He doth alſo ſuppoſe, that one ought to take the
1Simple Metals, that is, the Gold and the Silver, each of the ſame
weight as the Mixture, although he doth not ſay ſo; which may
be collected in that he marketh the ballance only betwixt the
Terms of the Gold and the Silver, which is the cauſe of the great
facility in reſolving the Problem by only counting the
Wyers.
One might take the pure Gold, and pure Silver of the ſame
weight, in reſpect of one another, but yet different from the
weight of the Mixture, that is, either more or leſs grave than the
Mixt Metal: and being equal in weight to one another they
might ſhew the proportion of the Maſs of the Gold to that of the
Silver; but yet with this difference, that the more grave will ſhew
the ſaid proportion more exactly than the ſmall and leſs grave.
But the Simple and pure Metals not being of the ſame weight as
the Compound, it will be neceſſary, having found the proportion
of the Maſs 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 likewiſe uſe the quantity of the ſimple Metals ac­
cording to neceſſity 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 muſt after­
wards by numbers find the proportion of the Maſſes of the two
Simple ones of equal weight (which is ſoon done, taking them of
equal weight as was ſaid before) and then according to this pro­
portion to find, by means of the Weight, and of the Maſs of the
Compound Metal, the diſtinct quantity of each of the two Sim­
ple ones that make the Compoſition: of each of which Caſes
examples might be given. But to conclude, if the pure Gold,
and pure Silver, and the Mixt Metal ſhould be of equal Maſs,
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 alſo
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 ſo
much Water as equalleth the ſame Body in Maſs, by Archimedes
his fifth Propoſition, De ijs quæ vehuntur in aqua.
And laſt of all, the Simple and pure Metals may have the ſame
proportion in Gravity, mutually or reciprocally, as their Bodies
have in Bulk: In which caſe, as well the Maſs, found by help of
the weight in Water, or by any other meanes, as their Weight in
the Air ſhall ſhew 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 ſay, the Spe­
cifical Gravity of the Gold ſhall have ſuch proportion to the
1Specifical Gravity of the Silver, as the Maſs of the Silver hath to
the Maſs 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 ſame Ballance one may with facility meaſure the
Maſs or Magnitude of any Body, in any manner whatſoever Irre­
gular in manner following, namely:
We will have at hand a Solid Body of a ſubſtance more grave
in Specie than the Water; as for inſtance of Lead; or if it were
of Wood, or other matter more light in Specie than the Water,
it may be made heavier by faſtning unto it Lead, or ſome other
thing that makes it ſink in the Water, and let us take ſome
known Meaſure, and with it meaſure the Irregular Solid; as for
inſtance, the Roman Palm, the Geometrical Foot, or any other
known meaſure, or part of the ſame, as the half Foot, the quar­
ter of a Foot, or any ſuch like part known; then let it be weighed
in the Air, and ſuppoſe that it weigh 10 pounds; let the ſame
Meaſure be weighed in the Air, and ſuppoſe that it weigh 8
pounds: and ſubſtract 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
Meaſure known. Now, if we would meaſure a Statue of Mar­
ble, let it be weighed firſt in the Air, and then in the Water, and
ſubſtract the Weight in the Water from the Weight in the Air, and
the remainder ſhall be the weight of ſo much Water as equalleth
the Statue in Maſs; which being divided by the difference betwixt
the Weight in Water and the Weight in Air of the Meaſure known,
the Quotient will give how many times the Statue containeth the
ſame given Meaſure. As for example; if the Statue in Air weigh
100 pounds, and in the Water 80 pounds, 80 pounds being ſub­
ſtracted from 100 there reſteth 20 pounds for the Weight of ſo
much Water in Maſs as equalleth the Statue. But becauſe the
difference betwixt the Weight in Water, and the Weight in Air
equal in Magnitude to the Meaſure known, was ſuppoſed to be
2 pounds; divide 18 pounds by two pounds, and the Quotient
is 9, for the number of times that the propoſed Statue containeth
the given Meaſure. The ſame Method may be obſerved, if it
were required, to meaſure a Statue, or other Maſs of any kind of
Metal: only it muſt be advertiſed, that all the holes muſt be
ſtopt, that the Water may not enter into the Body of the Statue:
but he that deſireth only the Solid content of the Metal of the
ſaid Statue muſt open the holes, and with Tunnels fill the whole
cavity of the Statue with Water. And if the Statue were of a
Subſtance lighter in Specie than the Water; as, for example, of
1Wax, it will be requiſite to add unto the Statue ſome Counter­
poiſe, that maketh it ſink in the Water, and then to meaſure the
Counterpoiſe, as above, and to ſubſtract its meaſure from the
Compound Body, and there will remain the Meaſure of the
Statue of Wax. And laſtly, to make uſe of the ſaid Ballance,
inſtead of ſeeking the numbers of the pounds of the Differences
of the Weights of the Meaſure known, and of the Solid
to be meaſured in Water, and in Air, we may count the
Wyers of the Arm of the Ballance, which
being very ſmall will give the
Meaſure exactly.
FINIS.