自然哲学的数学原理-9

arc EAF, and (withdrawing the bodyB) let it go from thence, and after one oscillation suppose it to return tothe point V : then RV will be the retardation arising from the resistanceof the air. Of this RV let ST be a fourth part, situated in the middle.to wit, so as RS and TV may be equal, and RS may be to ST as 3 to 2then will ST represent very nearly the retardation during the descentfrom S to A. Restore the body B to its place: and, supjx sing the bodyA to be let fall from the point S, the velocity thereof in the place of reflexion A, without sensible error, will be the same as if it had descendedm vacit.o from the point T. Upon which account this velocity may berepresented by the chord of the arc TA. For it is a proposition wellknown to geometers, that the velocity of a pendulous body in the loAvestpoint is as the chord of the arc which it has described in its descent. AftciOF NATUltAL PHILOSOPHY. 9 Ireflexion, suppose the body A comes to the place s, and the body B to theplace k. Withdraw the body B, and find the place v, from which if thebody A, being let go, should after one oscillation return to the place r, stmay be a fourth part of rv. so placed in the middle thereof as to leave isequal to tv, and let the chord of the arc tA represent the velocity whichthe body A had in the place A immediately after reflexion. For t will bethe true and correct place to which the body A should have ascended, ifthe resistance of the air had been taken off. In the e way we are tocorrect the place k to which the body B ascends, by finding the place I towhich it should have ascended in vacuo. And thus everything may besubjected to experiment, in the same manner as if we were really placedin vacuo. These things being done, we are to take the product (if I mayso say) of the body A, by the chord of the arc TA (which represents itsvelocity), that we may have its motion in the place A immediately beforereflexion ; and then by the chord of the arc /A, that we may have its motion in the place A immediately after reflexion. And so we are to takethe product of the body B by the chord of the arc B/, that we may havethe motion of the same immediately after reflexion. And in like manner,when two bodies are let go together from different places, we are to findthe motion of each, as well before as after reflexion; and then we maycompare the motions between themselves, and collect the effects of the reflexion. Thus trying the thing with pendulums of ten feet, in unequalas well as equal bodies, and making the bodies to concur after a descentthrough large spaces, as of 8, 12, or 16 feet, I found always, without anerror of 3 inches, that when the bodies concurred together directly, equalchanges towards the contrary parts were produced in their motions, and,of consequence, that the action and reaction were always equal. As if thebody A impinged upon the body B at rest with 9 parts of motion, andlosing 7, proceeded after reflexion with 2, the body B was carried backwards with those 7 parts. If the bodies concurred with contrary motions,A with twelve parts of motion, and B with six, then if A receded with J4,B receded with 8 ;to wit, with a deduction of 14 parts of motion oneach side. For from the motion of A subducting twelve parts, nothingwill remain ; but subducting 2 parts more, a motion will be generated of2 parts towards the contrary way ; and so, from the motion of the bodyB of 6 parts, subducting 14 parts, a motion is generated of 8 parts towardsthe contrary way. But if the bodies were made both to move towards thesame way, A, the swifter, with 14 parts of motion, B, the slower, with 5,and after reflexion A went on with 5, B likewise went on with 14 parts ;9 parts being transferred from A to B. And so in other cases. By thecongress and collision of bodies, the quantity of motion, collected from thesum of the motions directed towards the same way, or from the difference,of those that were directed towards contrary ways, was never changed.For the error of an inch or two in measures may be easily ascribed to tht92 THE MATHEMATICAL PRINCIPLESdifficulty of executing everything with accuracy. It was not easy to letgo the two pendulums so exactly together that the bodies should impingeone upon the other in the lowermost place AB ; nor to mark the places s,and ky to which the bodies ascended after congress. Nay, and some errors,too, might have happened from the unequal density of the parts of the pendulous bodies themselves, and from the irregularity of the texture proceeding from other causes.But to prevent an objection that may perhaps be alledged against therule, for the proof of which this experiment was made, as if this rule didsuppose that the bodies were either absolutely hard, or at least perfectlyelastic (whereas no such bodies are to be found in nature), 1 must add. thatthe experiments we have been describing, by no means depending uponthat quality of hardness, do succeed as well in soft as in hard bodies. Forif the rule is to be tried in bodies not perfectly hard, we are only to diminish the reflexion in such a certain proportion as the quantity of theelastic force requires. By the theory of Wren and Huygens, bodies absolutely hard return one from another with the same velocity with whichthey meet. But this may be affirmed with more certainty of bodies perfectly elastic. In bodies imperfectly elastic the velocity of the return is tobe diminished together with the elastic force ; because that force (exceptwhen the parts of bodies are bruised by their congress, or suffer some suchextension as happens under the strokes of a hammer) is (as far as I can perceive) certain and determined, and makes the bodies to return one fromthe other with a relative velocity, which is in a given ratio to that relativevelocity with which they met. This I tried in balls of wool, made uptightly, and strongly compressed. For, first, by letting go the pendulousbodies, and measuring their reflexion, I determined the quantity of theirelastic force; and then, according to this force, estimated the reflexionsthat ought to happen in other cases of congress. And with this computation other experiments made afterwards did accordingly agree ; the ballsalways receding one from the other with a relative velocity, which was tothe relative velocity with which they met as about 5 to 9. Balls of steelreturned with almost the same velocity : those of cork with a velocity some-^thing less; but in balls of glass the proportion was as about 15 to 16.And thus the third Law, so far as it regards percussions and reflexions, isproved by a theory exactly agreeing with experience.In attractions, I briefly demonstrate the thing after this manner. Suppose an obstacle is interposed to hinder the congress of any two bodies A.B, mutually attracting one the other : then if either body, as A, is moreattracted towards the other body B, than that other body B is towards thefirst body A, the obstacle will be more strongly urged by the pressure ofthe body A than by the pressure of the body B, and therefore will notremain in equilibrio : but the stronger pressure will prevail, and will makethe system of the two bodies, together with the obstacle, to move directlyOF NATURAL PHILOSOPHY. 93towards the parts on which B lies;arid in free spaces, to go forward ininfmitiim with a motion perpetually accelerated ; which is absurd andcontrary to the first Law. For, by the first Law, the system ought to persevere in its state of rest, or of moving uniformly forward in a right line :and therefore the bodies must equally press the obstacle, and be equallyattracted one by the other. I made the experiment on the loadstone andiron. If these, placed apart in proper vessels, are made to float by oneanother in standing water, neither of them will propel the other ; but,by being equally attracted, they will sustain each other s pressure, and restat last in an equilibrium.So the gravitation betwixt the earth and its parts is mutual. Let theearth FI be cut by any plane EG into two parts EGFand EGI, and their weights one towards the otherwill be mutually equal. For if by another planeHK, parallel to the former EG, the greater partFJEGI is cut into two parts EGKH and HKI.whereof HKI is equal to the part EFG, first cutoft,it is evident that the middle part EGKH, willhave no propension by its proper weight towards either side, but will hangas it were, and rest in an equilibrium betwixt both. But the one extremepart HKI will with its whole weight bear upon and press the middle parttowards the other extreme part EGF : and therefore the force with whichEGI, the sum of the parts HKI and EGKH, tends towards the third partEGF, is equal to the weight of the part HKI, that is, to the weight ofthe third part EGF. And therefore the weights of the two parts EGIand EGF, one towards the other, are equal, as I was to prove. And indeed if those weights were not equal, the whole earth floating in the nonresistingaether would give way to the greater weight, and, retiring fromit, would be carried off in infinitum.And as those bodies are equipollent in the congress and reflexion, whosevelocities are reciprocally as their innate forces, so in the use of mechanicinstruments those agents are equipollent, and mutually sustain each thecontrary pressure of the other, whose velocities, estimated according to thedetermination of the forces, are reciprocally as the forces.So those weights are of equal force to move the arms of a balance;which during the play of the balance are reciprocally as their velocitiesupw ards and downwards ; that is, if the ascent or descent is direct, thoseweights are of equal force, which are reciprocally as the distances of thepoints at which they are suspended from the axis oi the balance : but ifthey are turned aside by the interposition of oblique planes, or other obstacles, and made to ascend or descend obliquely, those bodies will beequipollent, wThich are reciprocally as the heights of their ascent and descent taken according to the perpendicular ; and that on account of thedetermination of gravity downwards.94 THE MATHEMATICAL PRINCIPLESAnd in like manner in the pully, or in a combination of pullies, theforce of a hand drawing the rope directly, which is to the weight, whethelascending directly or obliquely, as the velocity of the perpendicular ascentof the weight to the velocity of the hand that draws the rope, will sustainthe weight.In clocks and such like instruments, made up from a combination ofwheels, the contrary forces that promote and impede the motion of thewheels, if they are reciprocally as the velocities of the parts of the wheelon which they are impressed, will mutually sustain the one the other.The force of the screw to press a body is to the force of the hand thatturns the handles by which it is moved as the circular velocity of thehandle in that part where it is impelled by the hand is to the progressivevelocity of the screw towards the pressed body.The forces by which the wedge presses or drives the two parts of thewood it cleaves are to the force of the mallet upon the wedge as the propressof the wedge in the direction of the force impressed upon it by themallet is to the velocity with which the parts of the wood yield to thewedge, in the direction of lines perpendicular to the sides of the wedge.And the like account is to be given of all machines.The power and use of machines consist only in this, that by diminishingthe velocity we may augment the force, and the contrary : from whencein all sorts of proper machines, we have the solution of this problem ; 7move a given weight with a given power, or with a given force to overcome any other given resistance. For if machines are so contrived that thevelocities of the agent and resistant are reciprocally as their forces, theagent will just sustain the resistant, but with a greater disparity of velocity will overcome it. So that if the disparity of velocities is so greatas to overcome all that resistance which commonly arises either from theattrition of contiguous bodies as they slide by one another, or from thecohesion of continuous bodies that are to be separated, or from the weightsof bodies to be raised, the excess of the force remaining, after all those resistances are overcome, will produce an acceleration of motion proportionalthereto, as well in the parts of {he machine as in the resisting body. Butto treat of mechanics is not my present business. I was only willing toshow by those examples the great extent and certainty of the third Law otmotion. For if we estimate the action of the agent from its force andvelocity conjunctly, and likewise the reaction of the impediment conjuncthfrom the velocities of its several parts, and from the forces of resistancearising from the attrition, cohesion, weight, and acceleration of those parts,the action and reaction YL the use of all sorts of machines will b" foundalways equal to one another. And so far as the action is propagated bythe intervening instruments, and at last impressed upon tic resistingbody, the ultimate determination of the action will be always contrary tothe determination of the reaction.OF NATURAL PHILOSOPHY 95BOOK I.OF THE MOTION OF BODIES.SECTION I.Of the method offirst and last ratios of quantities, by the help wJicreojwe demonstrate the propositions that follow.LEMMA I.Quantities, and the ratios of quantities, which in anyfinite time convergecontinually to equality, and before the end of that time approach nearerthe one to the other than by any given difference, become ultimatelyequal.If you deny it, suppose them to be ultimately unequal, and let D betheir ultimate difference. Therefore they cannot approach nearer toequality than by that given difference D ; which is against the supposition,LEMMA II.If in any figure AacE, terminated by the right (flines A a. AE, and the curve acE, there be inscribed any number of parallelograms Ab, Be,Cd, fyc., comprehended under equal bases AB,BC, CD, ^c., and the sides, Bb, Cc, Dd, ^c.,parallel to one side Aa of the figure ; and theparallelograms aKbl, bLcm, cMdn, *c., are completed. Then if the breadth of those parallelo- 。grams be supposed to be diminisJied, and their X BF C D |;number to be augmented in infinitum : / say, that :he ultimate ratioswhich the inscribed fignre AKbLcMdD, the tin nmscribed figureAalbmcndoE, and enrvilijiear figure AabcdE, will have to one another,are ratios of equality.For the difference of the inscribed and circumscribed figures is the sumof the parallelograms K7, Lw, M//. Do. that is (from the equality of alltheir bases), the rectangle under one of their bases K6 and the sum of theiraltitudes Aa, that is, the rectangle ABla. But this rectangle, becauseMa96 THE MATHEMATICAL PRINCIPLES [BOOK 1its breadth AB is supposed diminished in infinitum, becomes less thanany given space. And therefore (by Lem. I) the figures inscribed andcircumscribed become ultimately equal one to the other; and much morewill the intermediate curvilinear figure be ultimately equal to either*Q.E.D.LEMMA III.The same ultimate ratios are also ratios of equality, when the breadth^AB, BC, DC, fyc., of the parallelograms are unequal, and are all diminished in infinitum.For suppose AF equal to the greatest breadth, andcomplete the parallelogram FAaf. This parallelogram will be greater than the difference of the inscribed and circumscribed figures ; but, because itsbreadth AF is diminished in infinitum, it will become less than any given rectangle. Q.E.D.COR. 1. Hence the ultimate sum of those evanescent parallelograms will in all parts coincide withthe curvilinear figure. A BF C D ECOR. 2. Much more will the rectilinear figure^comprehendcd under tnechords of the evanescent arcs ab, be, cd, (fee., ultimately coincide with tl.ccurvilinear figure.COR. 3. And also the circumscribed rectilinear figure comprehendedunder the tangents of the same arcs.COR. 4 And therefore these ultimate figures (as to their perimeters acE)are not rectilinear, but curvilinear limi s of rectilinear figures.LEMMA IV.If in two figures AacE, PprT, you inscribe (as before)two ranks of parallelograms, an equal number ineach rank, and, when their breadths are diminishedin infinitum. the ultimate ratios of the parallelogramsin one figure to those in the other, each to each respectively, are the same; I say, that those two figuresAacE, PprT, are to one another in that same ratio.For as the parallelograms in the one are severally to pthe parallelograms in the other, so (by composition) is the <sum of all in the one to the sum of all in the other : andso is the one figure to the other; because (by Lem. Ill) theformer figure to the former sum, and the latter figure to thelatter sum, are both in the ratio of equality. Q.E.D.COR. Hence if two quantities of any kind are anyhow divided into an equal number of parts, and those ASEC. I.] OF NATURAL PHILOSOPHY. 97parts, when their number is augmented, and their magnitude diminishedin infinitum, have a given ratio one to the other, the first to the first, thesecond to the second, and so on in order, the whole quantities will be one tothe other in that same given ratio. For if, in the figures of this Lemma,the parallelograms are taken one to the other in the ratio of the parts, thesum of the parts will always be as the sum of the parallelograms ; andtherefore supposing the number of the parallelograms and parts to be augmented, and their magnitudes diminished in infinitum, those sums will bein the ultimate ratio of the parallelogram in the one figure to the correspondent parallelogram in the other ;that is (by the supposition), in theultimate ratio of any part of the one quantity to the correspondent part ofthe other.LEMMA V.In similar figures, all sorts of homologous sides, whether curvilinear orrectilinear, are proportional ; and the areas are in the duplicate ratioof the homologous sides.LEMMA VI.If any arc ACB, given in position, is snb- _jtended by its chord AB, and in any pointA, in the middle of the contiinied curvature, is touched by a right line AD, produced both ways ; then if the points A Rand B approach one another and meet,I say, the angle RAT), contained between,the chord and the tangent, will be dimin- ?ished in infinitum, a/id ultimately will vanish.For if that angle does not vanish, the arc ACB will contain with thetangent AD an angle equal to a rectilinear angle ; and therefore the curvature at the point A will not be continued, which is against the supposition.LEMMA VII.The same things being supposed, I say that the ultimate ratio of the arc,chord, and tangent, any one to any other, is the ratio of equality.For while the point B approaches towards the point A, consider alwaysAB and AD as produced to the remote points b and d, and parallel to thesecant BD draw bd : and let the arc Acb be always similar to the arcACB. Then, supposing the points A and B to coincide, the angle dAbwill vanish, by the preceding Lemma; and therefore the right lines Ab,Arf (which are always finite), and the intermediate arc Acb, will coincide,and become equal among themselves. Wheref ,re, the right lines AB, AD,98 THE MATHEMATICAL PRINCIPLES [SEC. I.and the intermediate arc ACB (which are always proportional to theformer), will vanish, and ultimately acquire the ratio of equality. Q.E.D.COR. 1. Whence if through B we draw ABP parallel to the tangent, always cuttingany right line AF passing through A in F/ i-P, this line BP will be ultimately in theratio of equality with the evanescent arc ACB ; because, completing theparallelogram APBD, it is always in a ratio of equality with AD.COR. 2. And if through B and A more right lines are drawn, as BE,I5D, AF, AG, cutting the tangent AD and its parallel BP : the ultimateratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB,any one to any other, will be the ratio of equality.COR. 3. And therefore in all our reasoning about ultimate ratios, wemay freely use any one of those lines for any other.LEMMA VIII.If the right lines AR, BR, with the arc ACB, the chord AB, and thetangent AD, constitute three triangles RAB. RACB, RAD, and thepoints A and B approach and meet : I say, that the ultimate form ojthese evanescent triangles is that of similitude, and their ultimateratio that of equality.For while the point B approaches towards Athe point A, consider always AB, AD, AR,as produced to the remote points b, d, and r,and rbd as drawn parallel to RD, and letthe arc Acb be always similar to the arcACB. Then supposing the points A and Bto coincide, the angle bAd will vanish ; andtherefore the three triangles rAb, rAcb,rAd^which are always finite), will coincide, and on that account become bothsimilar and equal. And therefore the triangles RAB. RACB, RADwhich are always similar and proportional to these, will ultimately become both similar and equal among themselves. Q..E.D.COR. And hence in all reasonings about ultimate ratios, we may indifferently use any one of those triangles for any other.LEMMA IX.If a ngnt line AE. and a curve tine ABC, both given by position, cuteach other in a given angle, A ; and to that right line, in anothergiven angle, BD, CE are ordinately applied, meeting the curve in B,C : and the points B and C together approach towards and meet inthe point A : / say, that the areas of the triangles ABD, ACE, wiltultimately be one to the other in the duplicate ratio of the sides.BOOK LI OF NATURAL PHILOSOPHY.For while the points B, C, approachtowards the point A, suppose always ADto be produced to the remote points d and .e, so as Ad, Ae may be proportional toAD, AE ; and the ordinates db, ec, to bedrawn parallel to the ordinates DB andEC, and meeting AB and AC produced Din b and c. Let the curve Abe be similarto the curve A BC, and draw the right lineAg- so as to touch both curves in A, andcut the ordinates DB, EC, db ec, in F, G,J] g. Then, supposing the length Ae to remain the same, let the points Band C meet in the point A ; and the angle cAg vanishing, the curvilinearareas AW, Ace will coincide with the rectilinear areas A/rf, Age ; andtherefore (by Lem. V) will be one to the other in the duplicate ratio ofthe sides Ad, Ae. But the areas ABD, ACE are always proportional tothese areas ; and so the sides AD, AE are to these sides. And thereforethe areas ABD, ACE are ultimately one to the other in the duplicate ratioof the sides AD, AE. Q.E.D.LEMMA X.The spaces which a bodij describes by anyfinite force urging it. whetherthat force is determined and immutable, or is continually augmentedor continually diminished, are in the very beginning of the motion oneto the other in the duplicate ratio of the times.Let the times be represented by the lines AD, AE, and the velocitiesgenerated in those times by the ordinates DB, EC. The spaces describedwith these velocities will be as the areas ABD, ACE. described by thoseordinates, that is, at the very beginning of the motion (by Lem. IX), inthe duplicate ratio of the times AD, AE. Q..E.D.COR. 1. And hence one may easily infer, that the errors of bodies describing similar parts of similar figures in proportional times, are nearly