Descartes, Rene Mechanics 1665 London Thomas Salusbury en desca_mecha_01_en_1665

DISCOURSES OF THE MECHANICKS: A MANVSCRIPT of Mon&longs;ieur Des-Cartes.

The Explication.

Of Engines, by help of which we may rai&longs;e a very great weight with &longs;mall &longs;trength.

The Invention of all the&longs;e Engines de­pends 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 Weight which it cau­&longs;eth to be rai&longs;ed by a le&longs;&longs;er, are the Pulley, the Inclined Plane, the 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, Trochlea.

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;up­po&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 Cal­culation, 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 rai&longs;eth it, and cannot be e&longs;timated &longs;ave wthin a &longs;mall matter.

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 Pul­leys 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: be­cau&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 great­e&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.

The Inclined PLANE.

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;ide­rable.

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, which likewi&longs;e have for their Center that of the Earth, for that it is not any way &longs;en&longs;ible.

The WEDGE, Cuneus.

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 Pro­portion 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.

The CRANE, or the CAPSTEN, Axis in Peritrochio.

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 Cir­cumference 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 Diame­ters: 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 al&longs;o in&longs;tead of the Chord that rolleth about the Cylinder C, place 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.

The SCREW, Cochlea.

When once the Force of the Cap&longs;ten and of the In­clined 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 v. gr. ten turns to advance forwards the length of a foot in the 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 Remo­ving, 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 v. gr. this Wheel have thirty Coggs, whil&longs;t the Screw maketh one entire turn, it &longs;hall not cau&longs;e the Wheel to make more than the thirtieth part of a turn, and if the Weight be fa&longs;tned to a Chord that rowling about the Axis of this Wheel &longs;hall rai&longs;e it but one foot in the time that the Wheel makes one entire revolution, 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 Weight of that bigne&longs;s with this In&longs;trument, which is called the Perpetual Screw, as would require 300 men with­out 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 Weight, but by the Force or Matter of the In­&longs;trument: which difficulty is more &longs;en&longs;ible in it than in tho&longs;e afore­going, fora&longs;much as it hath greater Force.

The LEAVER, Vectis.

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 diffi­cult 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;cri­beth the whole Semicircle A B C D E, and acteth accord­ing to the Line A B C D E, al­though that the Weight de&longs;cri­beth 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 ra­ther 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 un­der&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 equi­di&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 the Line A B) is much le&longs;&longs;er than the Line S O, which &longs;heweth 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 ope­rates 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;e­quently 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 be­twixt 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 Contin­gent 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 Paral­lels: 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 Ba­lance, and &longs;hew, That then when it is mo&longs;t exact, and for in&longs;tance, &longs;uppo&longs;ing it's Centre at O by which it is &longs;u&longs;tained to be no more 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;i­ne&longs;&longs;e than this little which I have now writ thereof, for then they would not be in danger of decei­ving them&longs;elves in their Computation, as they frequently do in &longs;uppo&longs;ing other Principles.

FINIS.

A LETTER OF Mon&longs;ieur Des-Cartes TO THE REVEREND FATHER MARIN MERSENNE.

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 Sept.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 (eadem numero)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 this place &longs;peak of all &longs;uch as I &longs;u&longs;pect, which for the mo&longs;t part 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 Ve­locity, &longs;o that, for Example, in the Leaver, or (which is the &longs;ame) the Ba­llance 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 Equilibrium, in&longs;tead of &longs;aying, that that which cau&longs;eth this Equilibrium is, that if the Weight C did &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 quick­ly, 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 dou­ble 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, I have thought good to omit this Con&longs;ideration, and in this manner 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 un­der&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 ex­plaineth 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 be­ing 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 con­clude no more but this, that by this application the one of the Di­men&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, be­cometh 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, with­out 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 Di­men&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 double, which is repre&longs;ented by the Line FH; and &longs;o the Force 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 Superfi­cies, 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 ac­cording 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 Ma­chine, 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 Ex­amples 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 oc­ca&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;po­ken 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 Rectangle F H de&longs;cribeth whil&longs;t it draweth the Weight D along 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 Rectan­gle 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 to­wards A, one may ima­gine its Motion to be compo&longs;ed of two others, of which the one carrieth it from B R to­wards 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, &c. of the Plane do not 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 ex­plain the nature of the Spiral Line that repre&longs;enteth the Plane equally enclined, which hath many qualities that render it &longs;uffi­ciently 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,