DISCOURSES
OF THE
MECHANICKS:
A MANVSCRIPT of
Mon&longs;ieur Des-Cartes.
The Explication.
weight with &longs;mall &longs;trength.
The Invention of all the&longs;e Engines de
pends upon one &longs;ole Principle, which is,
That the &longs;ame Force that can lift up a
Weight, for example, of 100 pounds to
the height of one foot, can life up one of
200 pounds to the height of half a foot,
or one of 400 pounds to the height of a
fourth part of a foot, and &longs;o of the re&longs;t,
be there never &longs;o much applyed to it: and
this Principle cannot be denied if we con&longs;ider, that the Effect
ought to be proportioned to the Action that is nece&longs;&longs;ary for the
production of it; &longs;o that, if it be nece&longs;&longs;ary to employ an Action by
which we may rai&longs;e a Weight of 100 pounds to the height of two
foot, for to rai&longs;e one &longs;uch to the height of one foot only this &longs;ame
ought to weigh 200 pounds: for its the &longs;ame thing to rai&longs;e 100
pounds to the height of one foot, and again yet another 100
pounds to the height of one foot, as to rai&longs;e one of 200 pounds to
the height of one foot, and the &longs;ame, al&longs;o, as to rai&longs;e 100 pounds
to the height of two feet.
Now, the Engines which &longs;erve to make this Application of a
Force which acteth at a great Space upon a
Wedg, the Cap&longs;ten, or Wheel, the Screw, the Leaver, and &longs;ome
others, for if we will not apply or compare them one to another,
we cannot well number more, and if we will apply them we need
not in&longs;tance in &longs;o many.
The PVLLEY,
Let A B C be a Chord put about the Pulley D, to which let
the Weight E be fa&longs;tned; and fir&longs;t, &longs;uppo&longs;ing that two
men &longs;u&longs;tain or pull up equally each of them one of the
ends of the &longs;aid Chord:
it is manife&longs;t, that if the
Weight weigheth 200
pounds, each of tho&longs;e
men &longs;hal employ but the
half thereof, that is to &longs;ay,
the Force that is requi&longs;ite
for &longs;u&longs;taining or rai&longs;ing
of 100 pounds, for each
of them &longs;hal bear but the
half of it.
Afterwards, let us &longs;up
po&longs;e that A, one of the
ends of this Chord, being
made fa&longs;t to &longs;ome Nail,
the other C be again &longs;u
&longs;tained by a Man; and it
is manife&longs;t, that this Man in C, needs not (no more than before)
for the &longs;u&longs;taining the Weight E, more Force than is requi&longs;ite for
the &longs;u&longs;taining of 100 pounds: becau&longs;e the Nail at A doth the
&longs;ame Office as the Man which we &longs;uppo&longs;ed there before. In fine,
let us &longs;uppo&longs;e that this Man in C do pull the Chord to make the
Weight E to ri&longs;e, and it is manife&longs;t, that if he there employeth
the Force which is requi&longs;ite for the rai&longs;ing of 100 pounds to the
height of two feet, he &longs;hall rai&longs;e this Weight E of 200 pounds to
the height of one foot: for the Chord A B C being doubled, as it
is, it mu&longs;t be pull'd two feet by the end C, to make the Weight E
ri&longs;e as much, as if two men did draw it, the one by the end A,
and the other by the end C, each of them the length of one foot
only.
There is alwaies one thing that hinders the exactne&longs;s of the Cal
culation, that is the pondero&longs;ity of the Chord or Pulley, and the
difficulty that we meet with in making the Chord to &longs;lip, and in
bearing it: but this is very &longs;mall in compari&longs;on of that which
Moreover, it is nece&longs;&longs;ary to ob&longs;erve, that it is nothing but the
redoubling of the Chord, and not the Pulley, that cau&longs;eth this
Force: for if we fa&longs;ten yet another Pulley towards A, about
which we pa&longs;s the Chord A B C H, there will be required no le&longs;s
Force to draw H towards K, and &longs;o to lift up the Weight E, than
there was before to draw C towards G. But if to the&longs;e two Pul
leys we add yet another towards D, to which we fa&longs;ten the Weight,
and in which we make the Chord to run or &longs;lip, ju&longs;t as we did in
the fir&longs;t, then we &longs;hall need no more Force to lift up this Weight
of 200 pounds than to lift up 50 pounds without the Pulley: be
cau&longs;e that in drawing four feet of Chord we lift it up but one
foot. And &longs;o in multiplying of the Pulleys one may rai&longs;e the great
e&longs;t Weights with the lea&longs;t Forces. It is requi&longs;ite al&longs;o to ob&longs;erve,
that a little more Force is alwaies nece&longs;&longs;ary for the rai&longs;ing of a
Weight than for the &longs;u&longs;taining of it: which is the rea&longs;on why I
have &longs;poken here di&longs;tinctly of the one and of the other.
If not having more Force than &longs;ufficeth to rai&longs;e 100 pounds, one
would neverthele&longs;s rai&longs;e this Body F, that weigheth 200 pounds,
to the height of the Line B A, there needs no more but to draw
or rowl it along the Inclined Plane C A, which I &longs;uppo&longs;e to be
twice as long as the Line
A B, for by this means,
for to make it arrive at
the point A, we mu&longs;t
there employ the Force
that is nece&longs;&longs;ary for the
rai&longs;ing 100 pounds twice
as high, and the more inclined this Plane &longs;hall be made, &longs;o much
the le&longs;s Force &longs;hall there need to rai&longs;e the Weight F. But yet there
is to be rebated from this Calculation the difficulty that there is
in moving the Body F, along the Plane A C, if that Plane were
laid down upon the Line B C, all the parts of which I &longs;uppo&longs;e to
be equidi&longs;tant from the Center of the Earth.
It is true, that this impediment being &longs;o much le&longs;s as the Plane is
more united, more hard, more even, and more polite; it cannot
likewi&longs;e be e&longs;timated but by gue&longs;s, and it is not very con&longs;ide
rable.
We need not neither much to regard that the Line B C being a
part of a Circle that hath the &longs;ame Center with the Earth, the
Plane A C ought to be (though but very little) curved, and to
have the Figure of part of a Spiral, de&longs;cribed between two Circles,
is not any way &longs;en&longs;ible.
The Force of the Wedge A B C D is ea&longs;ily under&longs;tood after
that which hath been &longs;poken above of the Inclined Plane,
for the Force wherewith we &longs;trike downwards acts as if it
were to make it move according to the Line B D; and the Wood,
or other thing and Body that it cleaveth, openeth not, or the
Weight that it rai&longs;eth doth not ri&longs;e, &longs;ave only according to the
Line A C, in&longs;omuch that the Force,
wherewith one driveth or &longs;triketh this
Wedge, ought to have the &longs;ame Pro
portion to the Re&longs;i&longs;tance of this
Wood or Weight, that A C hath to
A B. Or el&longs;e again, to be exact, it
would be convenient that B D were
a part of a Circle, and A D and
C D two portions of Spirals that had the &longs;ame Center with the
Earth, and that the Wedge were of a Matter &longs;o perfectly hard
and polite, and of &longs;o &longs;mall weight, as that any little Force would
&longs;uffice to move it.
We &longs;ee al&longs;o very ea&longs;ily, that the Force wherewith the Wheel
A or Cogg B is turned, which make the Axis or Cylinder C
to move, about which a Chord is rolled, to which the
Weight D, which we would rai&longs;e, is fa&longs;tned, ought to have the
&longs;ame proportion to the &longs;aid
Weight, as the Circumference of
the Cylinder hath to the Cir
cumference of a Circle which
that Force de&longs;cribeth, or that the
Diameter of the one hath unto
the Diameter of the other; for
that the Circumferences have the
&longs;ame proportion as the Diame
ters: in&longs;omuch that the Cylinder C, having no more but one foot
in Diameter, if the Wheel AB be &longs;ix feet in its Diameter, and the
Weight D do weigh 600 pounds, it &longs;hall &longs;uffice that the Force in
B &longs;hall be capable to rai&longs;e 100 pounds, and &longs;o of others. One may
there a &longs;mall Wheel with teeth or Coggs, that may turn another
greater, and by that means multiply the power of the Force as
much as one &longs;hall plea&longs;e, without having any thing to deduct of
the &longs;ame, &longs;ave only the difficulty of moving the Machine, as in the
others.
When once the Force of the Cap&longs;ten and of the In
clined Plane is under&longs;tood, that of the Screw is ea&longs;ie
to be computed, for it is compo&longs;ed only of a Plane
much inclined, which windeth about a Cylinder: and if this Plane
be in &longs;uch manner Inclined, as that the Cylinder ought to make gr.
Screw, and that the bigne&longs;s of the Circumference of the Circle
which the Force that turneth it
about doth de&longs;cribe be of ten
feet; fora&longs;much as ten times ten
are one hundred, one Man alone
&longs;hall be able to pre&longs;s as &longs;trongly
with this In&longs;trument, or Screw, as
one hundred without it, provided
alwaies, that we rebate the Force
that is required to the turning
of it.
Now I &longs;peak here of Pre&longs;&longs;ing rather than of Rai&longs;ing, or Remo
ving, in regard that it is about this mo&longs;t commonly that the Screw
is employed, but when we would make u&longs;e of it for the rai&longs;ing of
Weights, in&longs;tead of making it to advance into a Female Screw, we
joyn or apply unto it a Wheel of many Coggs, in &longs;uch &longs;ort
made, that if gr.
maketh one entire turn, it &longs;hall not cau&longs;e the
than the thirtieth part of a turn, and if the
a Chord that rowling about the Axis of this
one foot in the time that the
and that the greatne&longs;s of the Circumference of the Circle that is
de&longs;cribed by the Force that turneth the Screw about be al&longs;o of ten
&longs;eet, by rea&longs;on that 10 times 30 make 300, one &longs;ingle Man &longs;hall be
able to rai&longs;e a
is called the Perpetual Screw, as would require 300 men with
out it.
Provided, as before, that we thence deduct the difficulty that
we meet with in turning of it, which is not properly cau&longs;ed by the
Pondero&longs;ity of the
going, fora&longs;much as it hath greater Force.
The LEAVER,
I Have deferred to &longs;peak of the Leaver until the la&longs;t, in regard
that it is of all Engines for rai&longs;ing of Weights, the mo&longs;t diffi
cult to be explained.
Let us &longs;uppo&longs;e that C H is a Leaver, in &longs;uch manner &longs;upported
at the point O, (by means of an Iron Pin that pa&longs;&longs;eth thorow it
acro&longs;s, or otherwi&longs;e) that it may turn about on this point O, its
part C de&longs;cribing the Semicircle A B C D E, and its part H the
Semicircle F G H I K; and that
the Weight which we would
rai&longs;e by help of it were in H,
and the Force in C, the Line
C O being &longs;uppo&longs;ed triple of
O H. Then let us con&longs;ider that
in the Time whil&longs;t the Force
that moveth this Leaver de&longs;cri
beth the whole Semicircle
A B C D E, and acteth accord
ing to the Line A B C D E, al
though that the Weight de&longs;cri
beth likewi&longs;e the Semicircle
F G H I K, yet it is not rai&longs;ed to
the length of this curved Line
F G H I K, but only to that of the Line F O K; in&longs;omuch that the
Proportion that the Force which moveth this Weight ought to
have to its Pondero&longs;ity ought not to be mea&longs;ured by that which is
between the two Diameters of the&longs;e Circles, or between their two
Circumferences, as it hath been &longs;aid above of the Wheel, but ra
ther by that which is betwixt the Circumference of the greater,
and the Diameter of the le&longs;&longs;er. Furthermore let us con&longs;ider, that
there is a nece&longs;&longs;ity that this Force needeth not to be &longs;o great, at
&longs;uch time as it is near to A, or near to E, for the turning of the
Leaver, as then when it is near to B, or to D; nor &longs;o great when
it is near to B or D, as then when it is near to C: of which the rea
&longs;on is, that the Weights do there mount le&longs;s: as it is ea&longs;ie to un
der&longs;tand, if having &longs;uppo&longs;ed that the Line C O H is parallel to the
Horizon, and that A O F cutteth it at Right Angles, we take the
point G equidi&longs;tant from the points F and H, and the point B equi
di&longs;tant from A and C; and that having drawn G S perpendicular
to F O, we ob&longs;erve that the Line F S (which &longs;heweth how much
the Weight mounteth in the Time that the Force operates along
how much it mounteth in the Time that the Force opperates along
the Line B C.
And to mea&longs;ure exactly what his Force ought to be in each Point
of the curved Line A B C D E, it is requi&longs;ite to know that it ope
rates there ju&longs;t in the &longs;ame manner as if it drew the Weight along
a Plane Circularly Inclined, and that the Inclination of each of the
Points of this circular Plane were to be mea&longs;ured by that of the
right Line that toucheth the Circle in this Point. As for example,
when the Force is at the Point B, for to find the proportion that it
ought to have with the pondero&longs;ity of the Weight which is at that
time at the Point G, it is nece&longs;&longs;ary to draw the Contingent Line
G M, and to account that the pondero&longs;ity of the Weight is to the
Force which is required to draw it along this Plane, and con&longs;e
quently to rai&longs;e it, according to the Circle F G H, as the Line G M
is to SM Again, for as much as B O is triple of O G, the Force
in B needs to be to the Weight in G but as the third part of the
Line SM is unto the whole Line G M. In the &longs;elf &longs;ame manner,
when the Force is at the Point D, to know how much the Weight
weigheth at I, it is nece&longs;&longs;ary to draw the Contingent Line betwixt
I and P, and the right Line I N perpendicular upon the Horizon,
and from the Point P taken at di&longs;cretion in the Line I P, provided
that it be below the Point I, you mu&longs;t draw P N parallel to the
&longs;ame Horizon, to the end you may have the proportion that is be
twixt the Line I P and the third part of the Line I N, for that which
betwixt the pondero&longs;ity of the Weight, and the Force that ought to
be at the Point D for the moving of it: and &longs;o of others. Where,
neverthele&longs;s, you mu&longs;t except the Point H, at which the Contin
gent Line being perpendicular upon the Horizon, the Weight can
be no other than triple the Force which ought to be in C for the
moving of it: in the Points F and K, at which the Contingent
Line being parallel unto the Horizon it &longs;elf, the lea&longs;t Force that
one can a&longs;&longs;ign is &longs;ufficient to move the Weight. Moreover, that you
may be perfectly exact, you mu&longs;t ob&longs;erve that the Lines S G and
P N ought to be parts of a Circle that have for their Center that
of the Earth; and GM and I P parts of Spirals drawn between two
&longs;uch Circles; and, la&longs;tly, that the right Lines S M and I N both
tending towards the Center of the Earth are not exactly Paral
lels: and furthermore, that the Point H where I &longs;uppo&longs;e the
Contingent Line to be perpendicular unto the Horizon ought
to be &longs;ome &longs;mall matter nearer to the Point F than to K, at the
which F and K the Contingent Lines are Parallels unto the &longs;aid
Horizon.
This done, we may ea&longs;ily re&longs;olve all the difficulties of the Ba
lance, and &longs;hew, That then when it is mo&longs;t exact, and for in&longs;tance,
but an indivi&longs;ible Point, like as I have &longs;uppo&longs;ed here for the Leaver,
if the Armes be declined one way or the other, that which &longs;hall be
the lowermo&longs;t ought evermore to be adjudged the heavier; &longs;o that
the Centre of Gravity is not &longs;ixed and immoveable in each &longs;everal
Body, as the Ancients have &longs;uppo&longs;ed, which no per&longs;on, that I
know of, hath hitherto ob&longs;erved.
But the&longs;e la&longs;t Con&longs;iderations are of no moment in Practice, and
it would be good for tho&longs;e who &longs;et them&longs;elves to invent new
Machines, that they knew nothing more of this bu&longs;i
ne&longs;&longs;e than this little which I have now writ thereof,
for then they would not be in danger of decei
ving them&longs;elves in their Computation,
as they frequently do in &longs;uppo&longs;ing
other Principles.
A
LETTER
OF
Mon&longs;ieur Des-Cartes
TO THE
REVEREND FATHER
I Did think to have deferred writing unto you
yet eight or fifteen dayes, to the end I might
not trouble you too often with my Letters,
but I have received yours of the fir&longs;t of
which giveth me to under&longs;tand that it is an
hard matter to admit the Principle which I
have &longs;uppo&longs;ed in my Examination of the
Geo&longs;tatick Que&longs;tion, and in regard that if it
be not true, all the re&longs;t that I have inferred from it would be yet
le&longs;&longs;e true: I would not one onely day defer &longs;ending you a more
particular Explication. It is requi&longs;ite above all things to con&longs;ider
that I did &longs;peak of the Force that &longs;erveth to rai&longs;e a Weight to &longs;ome
heighth, the which Force hath evermore two Dimen&longs;ions, and not
of that which &longs;erveth in each point to &longs;u&longs;tain it, which hath never
more than one Dimen&longs;ion, in&longs;omuch that the&longs;e two Forces differ
as much the one from the other, as a Superficies differs from a Line:
for the &longs;ame Force which a Nail ought to have for the &longs;u&longs;taining of
a Weight of 100 pound one moment of time, doth al&longs;o &longs;uffice for
to &longs;u&longs;tain it the &longs;pace of a year, provided that it do not dimini&longs;h,
but the &longs;ame Quantity of this Force which &longs;erveth to rai&longs;e the
Weight to the heighth of one foot, &longs;ufficeth not
to rai&longs;e it two feet; and it is not more manife&longs;t that two and two
make four, than it's manife&longs;t that we are to employ double as much
therein.
Now, fora&longs;much as that this is nothing but the &longs;ame thing that
I have &longs;uppo&longs;ed for a Principle, I cannot gue&longs;&longs;e on what the Scruple
&longs;hould be grounded that men make of receiving it; but I &longs;hall in
ari&longs;e onely from this, that men are before-hand over-knowing in
the Mechanicks; that is to &longs;ay, that they are pre-occupied with
Principles that others prove touching the&longs;e matters, which not being
ab&longs;olutely true, they deceive the more, the more true they &longs;eem to
be.
The fir&longs;t thing wherewith a man may be pre-occupied in this
bu&longs;ine&longs;&longs;e, is, that they many times confound the Con&longs;ideration of
Space, with that of Time, or of the Ve
locity, &longs;o that, for Example, in the
llance A B C D having &longs;uppo&longs;ed that
the Arm A B is double to B C, and the
Weight in C double to the Weight
in A, and al&longs;o that they are in
that which cau&longs;eth this
&longs;u&longs;tain, or was rai&longs;ed up by the Weight A, it did not pa&longs;&longs;e more
than half &longs;o much Space as it, they &longs;ay that it did move &longs;lower by
the half: which is a fault &longs;o much the more prejudicial, in that it is
very difficult to be known: for it is not the difference of
the Velocity that is the cau&longs;e why the&longs;e Weights are to be
one double to the other, but the difference of the Space, as
appeareth by this, that to rai&longs;e, for Example, the Weight F
with the hand unto G, it is not nece&longs;&longs;ary to employ a Force
that is preci&longs;ely double to that which one &longs;hould have
therein employed the fir&longs;t bout, to rai&longs;e it twice as quick
ly, but it is requi&longs;ite to employ therein either more or le&longs;s
than the double, according to the different proportion that
this Velocity may have unto the Cau&longs;es that re&longs;i&longs;t it.
In&longs;tead of requiring a Force ju&longs;t double for the rai&longs;ing of it with
the &longs;ame Velocity twice as high, unto H, I &longs;ay that it is ju&longs;t dou
ble in counting (as two and two make four) that one and one make
two, for it is requi&longs;ite to employ a certain quantity of this Force
to rai&longs;e the Weight from F to G, and again al&longs;o, as much more of
the &longs;ame Force to rai&longs;e it from G to H.
For if I had had a mind to have joyned the Con&longs;ideration of the
Velocity with that of the Space, it had been nece&longs;&longs;ary to have
a&longs;&longs;igned three Dimen&longs;ions to the Force, whereas I have a&longs;&longs;igned it
no more but two, on purpo&longs;e to exclude it. And if I have te&longs;tified
that there is &longs;o little of worth in any part of this &longs;mall Tract of the
Staticks, yet I de &longs;ire that men &longs;hould know, that there is more in
this alone than in all the re&longs;t: for it's impo&longs;&longs;ible to &longs;ay any thing
that is good and &longs;olid touching Velocity, without having rightly
explained what we are to under&longs;tand by Gravity, as al&longs;o the whole
Sy&longs;teme of the World. Now becau&longs;e I would not under take it,
to &longs;ingle out the&longs;e others that I could explain without it: for
though there be no Motion but hath &longs;ome Velocity, neverthele&longs;s
it is onely the Augmentations and Diminutions of this Velocity
that are con&longs;iderable. And now that &longs;peaking of the Motion of a
Body, we &longs;uppo&longs;e that it is made according to the Velocity which
is mo&longs;t naturall to it, which is the &longs;ame as if we did not con&longs;ider it
at all.
The other rea&longs;on that may have hindred men from rightly un
der&longs;tanding my Principle is, that they have thought that they could
demon&longs;trate without it &longs;ome of tho&longs;e things which I demon&longs;trate
not without it: As, for example, touching the Pulley A B C, they
have thought that it was enough to know that the Nail in A did
&longs;u&longs;tain the half of the Weight B; to conclude
that the Hand in C had need but of half &longs;o much
Force to &longs;u&longs;tain or rai&longs;e the Weight, thus wound
about the Pulley, as it would need for to &longs;u&longs;tain
or rai&longs;e it without it. But howbeit that this ex
plaineth very well, how the application of the
Force at C is made unto a Weight double to that
which it could rai&longs;e without a Pulley, and that I
my &longs;elf did make u&longs;e thereof, yet I deny that
this is &longs;imply, becau&longs;e that that the Nail A &longs;u
&longs;taineth one part of the Weight B, that the Force
in C, which &longs;u&longs;taineth it, might be le&longs;s than if it
had been &longs;o &longs;u&longs;tained. For if that had been true, the Rope C E be
ing wound about the Pulley D, the Force in E might by the &longs;ame
rea&longs;on be le&longs;s than the Force in C: for that the Nail A doth not
&longs;u&longs;tain the Weight le&longs;s than it did before, and that there is al&longs;o
another Nail that &longs;u&longs;tains it, to wit, that to wich the Pulley D is
fa&longs;tned. Thus therefore, that we may not be mi&longs;taken in this, that
the Nail A &longs;u&longs;taineth the half of the Weight B, we ought to con
clude no more but this, that by this application the one of the Di
men&longs;ions of the Force that ought to be in C
to rai&longs;e up this Weight is dimini&longs;hed the one
half; and that the other, of con&longs;equence, be
cometh double, in &longs;uch &longs;ort that if the Line
F G repre&longs;ent the Force that is required for
the &longs;u&longs;taining the Weight B in a point, with
out the help of any Machine, and the
Quadrangle G H that which is required for
the rai&longs;ing of it to the height of a foot, the
&longs;upport of the Nail A dimini&longs;heth the Di
men&longs;ion which is repre&longs;ented by the Line F G the one half, and the
redoubling of the Rope A B C maketh the other Dimen&longs;ion to
that ought to be in C for the rai&longs;ing of the Weight B to the height
of one foot is repre&longs;ented by the Quadrangle IK; and, as we know
in Geometry, that a Line being added to, or taken from a Superfi
cies, neither augmenteth, nor dimini&longs;heth it in the lea&longs;t, &longs;o the
Force where with the Nail A &longs;u&longs;tains the Weight B, having but one
&longs;ole Dimen&longs;ion, cannot cau&longs;e that the Force in C, con&longs;idered ac
cording to its two Dimen&longs;ions, ought to be le&longs;s for the rai&longs;ing in
like manner the Weight E, than for the rai&longs;ing it without any
Pulley.
The third thing which may make men imagine &longs;ome Ob&longs;curity
in my Principle is, that they, it may be, have not had regard to all
the words by which I explain it; for I do not &longs;ay &longs;imply that the
Force that can rai&longs;e a Weight of 50 pounds to the height of four
feet can rai&longs;e one of 200 pounds to the height of one foot; but I
&longs;ay that it may do it, if &longs;o be that it be applyed to it: now it is
impo&longs;&longs;ible to apply the &longs;ame thereto, but by the means of &longs;ome Ma
chine, or other Invention that &longs;hall cau&longs;e this Weight to a&longs;cend
but one, in the time whil&longs;t the Force pa&longs;&longs;eth the whole length
of four feet, and &longs;o that it do transform the Quandrangle, by
which the Force is repre&longs;ented that is required to rai&longs;e this
Weight of 400 pounds to the height of one foot into another
that is equall and like to that which repre&longs;ents the Force that is
required for to rai&longs;e a Weight of 50 pounds to the height of four
feet.
In fine, it may be that men may have thought the wor&longs;e of my
Principle, becau&longs;e they have imagined that I have alledged the Ex
amples of the Pulley, of the Inclined Plane, and of the Leaver, to
the end that I might better per&longs;warde the truth thereof, as if it had
been dubious, or el&longs;e that I had &longs;o ill di&longs;cour&longs;ed as to offer to a&longs;&longs;ume
from thence a Principle, which ought of it felf to be &longs;o clear, as not
to need any proof by things that are &longs;o difficult to comprehend as
that; it may be, they have never been well demon&longs;trated by any
man: but neither have I made u&longs;e of them, &longs;ave only with a de&longs;ign
to &longs;hew that this Principle extends it &longs;elf to all matters of which
one treateth in the Staticks: or, rather, I have made u&longs;e of this oc
ca&longs;ion for to in&longs;ert them into my Treati&longs;e, for that I conceived
that it would have been too dry and barren if I had therein &longs;po
ken of nothing el&longs;e but of this Que&longs;tion, that is of no u&longs;e, as of
that of the Geo&longs;taticks, which I purpo&longs;ed to examine.
Now one may perceive, by what hath already been &longs;aid, how
the Forces of the Leaver and Pulley are demon&longs;trated by my
Principle &longs;o well, that there only remains the Inclined Plane, of
which you &longs;hall clearly &longs;ee the Demon&longs;tration by this Figure; in
which G F repre&longs;ents the fir&longs;t Dimen&longs;ion of the Force that the
the Plane B A, by the means of a Chord parallel to this Plane, and
pa&longs;&longs;ing about the Pulley E, in &longs;uch &longs;ort, that H G, that is the height
of this Rectangle, is equal to B A, along which the Weight D is to
move, whil&longs;t it mounteth to the height of the Line C A. And N O
repre&longs;ents the fir&longs;t Dimen&longs;ion of &longs;uch another Force, that is de
&longs;cribed by the Rectan
gle N P, in the time that
it is rai&longs;ing the Weight
L to M. And I &longs;uppo&longs;e
that L M is equal to B A,
or double to C A; and
that N O is to F G, as
O P is to G H. This
done, I con&longs;ider that at
&longs;uch time as the Weight
D is moved from B to
wards A, one may ima
gine its Motion to be
compo&longs;ed of two others, of which the one carrieth it from B R to
wards C A, (to which operation there is no Force required, as all
tho&longs;e &longs;uppo&longs;e who treat of the Mechanicks) and the other rai&longs;eth
it from B C towards R A, for which alone the Force is required:
in&longs;omuch that it needs neither more nor le&longs;s Force to move it
along the Inclined Plane B A, than along the Perpendicular C A. For I &longs;uppo&longs;e that the unevenne&longs;&longs;es,
at all hinder it, like as it is alwaies &longs;uppo&longs;ed in treating of this
matter.
So then the whole Force F H is employed only about the rai&longs;ing
of D to the height of C A: and fora&longs;much as it is exactly equal to
the Force N P, that is required for the rai&longs;ing of L to the Height
of L M, double to C A, I conclude by my Principle that the
Weight D is double to the Weight L. For in regard that it is
nece&longs;&longs;ary to employ as much Force for the one as for the other,
there is as much to be rai&longs;ed in the one as in the other; and no
more knowledge is required than to count unto two for the
knowing that it is alike facile to rai&longs;e 200 pounds from C to A,
as to rai&longs;e 100 pounds from L to M: &longs;ince that L M is double
to C A.
You tell me, moreover, that I ought more particularly to ex
plain the nature of the Spiral Line that repre&longs;enteth the Plane
equally enclined, which hath many qualities that render it &longs;uffi
ciently knowable.
For if A be the Center of the Earth,
and A N B C D the Spiral Line, having
drawn the Right Lines A B, A D, and the
like, there is the &longs;ame proportion betwixt
the Curved Line A N B and the Right Line
AB, as is betwixt the Curved Line A N B C,
and the Right Line A C; or betwixt
A N B C D and A D: and &longs;o of the
re&longs;t.
And if one draw the Tangents D E, C F,
and B G, the Angles A D E, A C F, A B G, &c. &longs;hall be equal.
As for the re&longs;t I will, &c.----
Reverend Father,
Your very humble Servant