自然哲学的数学原理-35

to the motion lost,was inches.The number of the oscillations in water.The number of the oscillations in air.16. li . 3 . 7 . lH.12f.13j85i . 287 . 535SEC. VI.] OF NATURAL PHILOSOPHY. 319In the experiments of the 4th column there were equal motions lost in535 oscillations made in the air, and If in water. The oscillations in theair were indeed a little swifter than those in the water. But if the oscillations in the water were accelerated in such a ratio that the motions ofthe pendulums might be equally swift in both mediums, there would bestill the same number 1 j of oscillations in the water, and by these thesame quantity of motion would be lost as before ; because the resistance i>increased, and the square of the time diminished in the same duplicate ratio. The pendulums, therefore, being of equal velocities, there were equalmotions lost in 535 oscillations in the air, and 1} in the water; and therefore the resistance of the pendulum in the water is to its resistance in theair as 535 to 1 }. This is the proportion of the whole resistances in thecase of the 4th column.Now let AV + CV 2represent the difference of the arcs described in thedescent and subsequent ascent by the globe moving in air with the greatestvelocity V ; and since the greatest velocity is in the case of the 4th columnto the greatest velocity in the case of the 1st column as 1 to 8 ; and thatdifference of the arcs in the case of the 4th column to the difference in the2 16case of the 1st column as ^ to7, or as 86J to 4280 ; put in thesecases 1 and 8 for the velocities, and 85 1 and 4280 for the differences ofthe arcs, and A + C will be S5|, and 8A -f 640 == 4280 or A + SC= 535 ; and then by reducing these equations, there will come out TC =449^ and C = 64T。 and A = 21f ; and therefore the resistance, which isas TVAV + fCV 2, will become as 13 T6TV + 48/^Y 2. Therefore in thecase of the 4th column, where the velocity was 1, the whole resistance is toits part proportional to the square of the velocity as 13 T6T + 48/F or61 }f to 48/e ; and therefore the resistance of the pendulum in water is tothat part of the resistance in air, which is proportional to the square of thevelocity, and which in swift motions is the only part that deserves consideration, as 61}^ to 4S/g and 535 to 1} conjunctly, that is, as 571 to 1.If the whole thread of the pendulum oscillating in the water had been immersed, its resistance would have been still greater ; so that the resistanceof the pendulum oscillating in the water, that is, that part which is proportional to the square of the velocity, and which only needs to be considered in swift bodies, is to the resistance of the same whole pendulum, oscillating in air with the same velocity, as about 850 to 1, that is as, the density of water to the density of air, nearly.In this calculation we ought also to have taken in that part of the resistance of the pendulum in the water which was as the square of the velocity ; but I found (which will perhaps seem strange) that the resistancein the water was augmented in more than a duplicate ratio of the velocity.In searching after the cause, I thought upon this, that the vessel was toe320 THE MATHEMATICAL PRINCIPLES [BOOK II.narrow for the magnitude of the pendulous globe, and by its narrownessobstructed the motion of the water as it yielded to the oscillating globe.For when I immersed a pendulous globe, whose diameter was one inch only,the resistance was augmented nearly in a duplicate ratio of the velocity,I tried this by making a pendulum of two globes, of which the lesser andlower oscillated in the water, and the greater and higher was fastened tothe thread just above the water, and, by oscillating in the air, assisted themotion of the pendulum, and continued it longer. The experiments madeby this contrivance proved according to the following table.Arc descr. in first descent . .16.8. 4.Arc descr. in last ascent . . 12 . 6 . 3 . li . J . | . T3FDif. of arcs, proport. to 1 . pi imotion lost$T r T*Number of oscillations... 3f . 6j . 12^. 211 . 34 . 53 . 62)In comparing the resistances of the mediums with each other, I alsocaused iron pendulums to oscillate in quicksilver. The length of the ironwire was about 3 feet, and the diameter of the pendulous globe about i ofan inch. To the wire, just above the quicksilver, there was fixed anotherleaden globe of a bigness sufficient to continue the motion of the pendulumfor some time. Then a vessel, that would hold about 3 pounds of quicksilver, was filled by turns with quicksilver and common water, that, bymaking the pendulum oscillate successively in these two different fluids, Imight find the proportion of their resistances ; and the resistance of thequicksilver proved to be to the resistance of water as about 13 or 14 to 1;that is. as the density of quicksilver to the density of water. When I madeuse of a pendulous globe something bigger, as of one whose diameter wasabout ^ or | of an inch, the resistance of the quicksilver proved to be tothe resistance of the water as about 12 or 10 to 1. But the former experiment is more to be relied on, because in the latter the vessel was too narrow in proportion to the magnitude of the immersed globe; for the vesselought to have been enlarged together with the globe. I intended to haverepeated these experiments with larger vessels, and in melted metals, andother liquors both cold and hot ; but I had not leisure to try all: and besides, from what is already described, it appears sufficiently that the resistance of bodies moving swiftly is nearly proportional to the densities ofthe fluids in which they move. I do not say accurately ; for more tenacious fluids, of equal density, will undoubtedly resist more than those thatare more liquid ; as cold oil more than warm, warm oil more than rainwater, and water more than spirit of wine. But in liquors, which are sensibly fluid enough, as in air, in salt and fresh water, in spirit of wine, ofturpentine, and salts, in oil cleared of its fseces by distillation and warmed,in oil of vitriol, and in mercury, and melted metals, and any other suchlike, that are fluid enough to retaia for some time the motion impressedSEC. VI.J OF NATURAL PHILOSOPHY. 321upon them by the agitation of the vessel, and which being poured out areeasily resolved into drops, I doubt not but the rule already laid down maybe accurate enough, especially if the experiments be made with largerpendulous bodies and more swiftly moved.Lastly, since it is the opinion of some that there is a certain ^etherealmedium extremely rare and subtile, which freely pervades the pores of allbodies ; and from such a medium, so pervading the pores of bodies, some resistance must needs arise; in order to try whether the resistance, which wreexperience in bodies in motion, be made upon their outward superficies only,or whether their internal parts meet with any considerable resistance upontheir superficies, I thought of the following experiment I suspended around deal box by a thread 11 feet long, on a steel hook, by means of a ringof the s-ime metal, so as to make a pendulum of the aforesaid length. Thehook had a sharp hollowredge on its upper part, so that the upper arc ofthe ring pressing on the edge might move the more freely ; and the threadwas fastened to the lower arc of the ring. The pendulum being thus prepared, I drew it aside from the perpendicular to the distance of about 6feet, and that in a plane perpendicular to the edge of the hook, lest thering, while the pendulum oscillated, should slide to and fro on the edge ofthe hook : for the point of suspension, in which the ring touches the hook,ought to remain immovable. I therefore accurately noted the place towhich the pendulum was brought, and letting it go, I marked three otherplaces, to which it returned at the end of the 1st, 2d, and 3d oscillation.This I often repeated, that I might find those places as accurately as possible. Then I filled the box with lead and other heavy metals that werenear at hand. But, first, I weighed the box when empty, and that pnrt ofthe thread that went round it, and half the remaining part, extended between the hook and the suspended box ; for the thread so extended alwaysacts upon the pendulum, when drawn aside from the perpendicular, with halfits weight. To this weight I added the weight of the air contained in thebox And this whole weight was about -fj of the weight of the box whenfilled wr ith the metals. Then because the box when full of the metals, by extending the thread with its weight, increased the length of the pendulum,f shortened the thread so as to make the length of the pendulum, when oscillating, the same as before. Then drawing aside the pendulum to theplace first marked, and letting it go, I reckoned about 77 oscillations beforethe box returned to the second mark, and as many afterwards before it cameto the third mark, and as many after that before it came to the fourthxnark. From whence I conclude that the whole resistance of the box, whenfull, had not a greater proportion to the resistance of the box, when empty,than 78 to 77. For if their resistances were equal, the box, when full, byreason of its vis insita, which was 78 times greater than the vis tfuritoofthe same when empty, ought to have continued its oscillating motion so21322 THE MATHEMATICAL PRINCIPLES| BOOK II.much the longer, and therefore to have returned to those marks at the endof 78 oscillations. But it returned to them at the end of 77 oscillations.Let, therefore, A represent the resistance of the box upon its externalsuperficies, and B the resistance of the empty box on its internal superficies ;and if the resistances to the internal parts of bodies equally swift be as thematter, or the number of particles that are resisted, then 78B will be theresistance made to the internal parts of the box, when full; and thereforethe whole resistance A + B of the empty box will be to the whole resistance A + 7SB of the full box as 77 to 78, and, by division, A + B to 77Bas 77 to 1; and thence A + B to B as 77 X 77 to 1, and, by divisionagain, A to B as 5928 to 1. Therefore the resistance of the empty box inits internal parts will be above 5000 times less than the resistance on itsexternal superficies. This reasoning depends upon the supposition that thegreater resistance of the full box arises not from any other latent cause,but only from the action of some subtile fluid upon the included metal.This experiment is related by memory, the paper being lost in which Ihad described it;so that I have been obliged to omit some fractional parts,which are slipt out of my memory ; and I have no leisure to try it again.The first time I made it, the hook being weak, the full box was retardedsooner. The cause I found to be, that the hook was not strong enough tobear the weight of the box : so that, as it oscillated to and fro, the hookwas bent sometimes this and sometimes that way. I therefore procured ahook of sufficient strength, so that the point of suspension might remainunmoved, and then all things happened as is above described.SEC. VI I.] OF NATURAL PHILOSOPHY. 323SECTION VII.Of the, motion offluids, and the resistance made to projected bodies.PROPOSITION XXXII. THEOREM XXVI.Suppose two similar systems of bodies consisting of an equal number ofparticles, and let the correspondent particles be similar and proportional, each in, one system to each in the other, and have a like situation among themselves, and the same given ratio of density to eachother ; and let them begin to move anwng themselves in proportionaltimes, and with like motions (that is, those in one system among oneanother, and those in the other among one another). And if the particles that are in the same system do not touch otte another, except irthe moments of reflexion ; nor attract, nor repel each other, except withthat are as the diameters of the correspondent particles inversely, and the squares of the velocities directly ; I say, that theparticles of those systems will continue to move among themselves witItlike motions and in proportional times.Like bodies in like situations are said to be moved among themselveswith like motions and in proportional times, when their situations at theend of those times are always found alike in respect of each other ; as suppose we compare the particles in one system with the correspondent particles in the other. Hence the times will be proportional, in which similarand proportional parts of similar figures will be described by correspondentparticles. Therefore if we suppose two systems of this kind; the correspondent particles, by reason of the similitude of the motions at theirbeginning, will continue to be moved with like motions, so long as theymove without meeting one another ;for if they are acted on by no forces,they will go on uniformly in right lines, by the 1st Law. But if they doagitate one another with some certain forces, and those forces are as thediameters of the correspondent particles inversely and the squares of thevelocities directly, then, because the particles are in like situations, andtheir forces are proportional, the whole forces with which correspondentparticles are agitated, and which are compounded of each of the agitatingforces (by Corol. 2 of the Laws), will have like directions, and have thesame effect as if they respected centres placed alike among the particles ;and those whole forces will be to each other as the several forces whichcompose them, that is, as the diameters of the correspondent particles inversely, and the squares of the velocities directly : and therefore will cans**3^4 THE MATHEMATICAL PRINCIPLES [BOOK 11.correspondent particles to continue to describe like figures. These thingswill be so (by Cor. 1 and S, Prop. IV.; Book 1), if those centres are at restbut if they are moved, yet by reason of the similitude of the translations,their situations among the particles of the system will remain similar , sothat the changes introduced into the figures described by the particles willstill be similar. So that the motions of correspondent and similar particles will continue similar till their first meeting with each other; andthence will arise similar collisions, and similar reflexions; which will againbeget similar motions of the particles among themselves (by what was justnow shown), till they mutually fall upon one another again, and so on adinfinitum.COR. 1. Hence if any two bodies, which are similar and in like situationsto the correspondent particles of the systems, begin to move amongst themin like manner and in proportional times, and their magnitudes and densities be to each other as the magnitudes and densities of the correspondingparticles, these bodies will continue to be moved in like manner and inproportional times: for the case of the greater parts of both systems and ofthe particles is the very same.COR. 2. And if all the similar and similarly situated parts of both systems be at rest among themselves ; and two of them, which are greater thanthe rest, and mutually correspondent in both systems, begin to move inlines alike posited, with any similar motion whatsoever, they will excitesimilar motions in the rest of the parts of the systems, and will continueto move among those parts in like manner and in proportional times ; andwill therefore describe spaces proportional to their diameters.PROPOSITION XXXIII. THEOREM XXVII.The same things biting supposed, I say, that the greater parts of thesystems are resisted in a ratio compounded of the duplicate ratio oftheir velocities, and the duplicate ratio of their diameters, and Ihe simple ratio of the density of the parts of the systems.For the resistance arises partly from the centripetal or centrifugal, forceswith which the particles of the system mutually act on each other, partlyfrom the collisions and reflexions of the particles and the greater parts.The resistances of the first kind are to each other as the whole motiveforces from which they arise, that is, as the whole accelerative forces andthe quantities of matter in corresponding parts ; that is (by the supposition), as the squares of the velocities directly, and the distances of thecorresponding particles inversely, and the quantities of matter in the correspondent parts directly : and therefore since the distances of the particles in one system are to the correspondent distances of the particles of the;ther S3 the diameter of one particle or part in *he former system to theSEC. VII.] OF NATURAL PHILOSOPHY. C>2"diameter of the correspondent particle or part in the other, and since thequantities of matter are as the densities of the parts and the cubes of thediameters ; the resistances arc to each other as the squares of the velocitiesand the squares of the diameters and the densities of the parts of the systems. Q.E.D. The resistances of the latter sort are as the number ofsorrespondent reflexions and the forces of those reflexions conjunctly ; butthe number of the reflexions are to each other as the velocities of the corresponding parts directly and the spaces between their reflexions inversely.And the forces of the reflexions are as the velocities and the magnitudesand the densities of the corresponding parts conjunctly ; that is, as the velocities and the cubes of the diameters and the densities of the parts. And,joining all these ratios, the resistances of the corresponding parts are toeach other as the squares of the velocities and the squares of the diametersand the densities of the parts conjunctly. Q.E.T).COR. 1. Therefore if those systems are two elastic fluids, like our air,and their parts are at rest among themselves ; and two similar bodies proportional in magnitude and density to the parts of the fluids, and similarlygituated among those parts, be any how projected in the direction of linessimilarly posited ; and the accelerative forces with which the particles ofthe fluids mutually act upon each other are as the diameters of the bodiesprojected inversely and the squares of their velocities directly ; those bodieswill excite similar motions in the fluids in proportional times, and will describe similar spaces and proportional to their diameters.COR. 2. Therefore in the same fluid a projected body that moves swiftlymeets with a resistance that is, in the duplicate ratio of its velocity, nearly.For if the forces with which distant particles act mutually upon oneanother should be augmented in the duplicate ratio of the velocity, theprojected body would be resisted in the same duplicate ratio accurately ;and therefore in a medium, whose parts when at a distance do not act mutually with any force on one another, the resistance is in the duplicate ratio of the velocity accurately. Let there be, therefore, three mediums A,B, C, consisting of similar and equal parts regularly disposed at equaldistances. Let the parts of the mediums A and B recede from each otherwith forces that are among themselves as T and V ; and let the parts ofthe medium C be entirely destitute of any such forces. And if four equalbodies D, E, P7 G, move in these mediums, the two first D and E in thetwo first A and B, and the other two P and G in the third C ; and if thevelocity of the body D be to the velocity of the body E, and the velocityof the body P to the velocity of the body G, in the subduplicate ratio ofthe force T to the force V ; the resistance of the body D to the resistanceof the body E, and the resistance of the body P to the resistance of thebody G, will be in the duplicate ratio of the velocities ; and therefore theresistance of the body D will be to the resistance of the body P as the re326 THE MATHEMATICAL PRINCIPLES [BOOK IIsistance of the body E to the resistance of the body G. Let the bodies 1)and F be equally swift, as also the bodies E and G ; and, augmenting thevelocities of the^bodies D arid F in any ratio, and diminishing the forcesof the particles of the medium B in the duplicate of the same ratio, themedium B will approach to the form and condition of the medium C atpleasure ; and therefore the resistances of the equal and equally swiftbodies E and G in these mediums will perpetually approach to equalityso that their difference will at last become less than any given. Therefore since the resistances of the bodies D and F are to each other as theresistances of the bodies E and G, those will also in like manner approachto the ratio of equality. Therefore the bodies 1) and F, when they movewith very great swiftness, meet with resistances very nearly equal; andtherefore since the resistance of the body F is in a duplicate ratio of thevelocity, the resistance of the body D will be nearly in the same ratio.Con. 3. The resistance of a body moving very swift in an elastic fluidis almost the same as if the parts of the fluid were destitute of their centrifugal forces, and did not fly from each other; if so be that the elasticity of the fluid arise from the centrifugal forces of the particles, and thevelocity be so great as not to allow the particles time enough to act.COR. 4. Therefore, since the resistances of similar and equally swiftbodies, in a medium whose distant parts do not fly from each other, are asthe squares of the diameters, the resistances made to bodies moving withvery great and equal velocities in an elastic fluid will be as the squares ofthe diameters, nearly.COR. 5. And since similar, equal, and equally swift bodies, movingthrough mediums of the same density, whose particles do not fly from eachother mutually, will strike against an equal quantity of matter in equaltimes, whether the particles of which the medium consists be more andsmaller, or fewer and greater, and therefore impress on that matter an equalquantity of motion, and in return (by the 3d Law of Motion) suffer anequal re-action from the same, that is, are equally resisted ;it is manifest,also, that in elastic fluids of the same density, when the bodies move withextreme swiftness, their resistances are nearly equal, whether the fluidsconsist of gross parts, or of parts ever so subtile. For the resistance ofprojectiles moving with exceedingly great celerities is not much diminishedby the subtilty of the medium.COR. G. All these things are so in fluids whose elastic force takes its risefrom the centrifugal forces of the particles. But if that force arise fromsome other cause, as from the expansion of the particles after the mannerof wool, or the boughs of trees, or any other cause, by which the particlesare hindered from moving freely among themselves, the resistance, byreason of the lesser fluidity of the medium, will be greater than in theCorollaries above.SEC. VII. OF NATURAL PHILOSOPHY. 32?KL, POPROPOSITION XXXIV. THEOREM XXV1I1.If iu a rare medium, consisting of equal particles freely disposed atequal distances from each other, a globe and a cylinder described onequal diameters move with equal velocities in the. direction of the axisof the cylinder, the resistance of the globe ivill be but half so great anthat of the cylinder.For since the action of the medium upon the body is the same (byCor. 5 of the Laws) whether the bodymove in a quiescent medium, orwhether the particles of the mediumimpinge with the same velocity uponthe quiescent body, let us considerthe body as if it were quiescent, andsee with what force it would be impelledby the moving medium. Let, therefore, ABKI represent a sphericalbody described from the centre C with the semi-diameter CA, and let theparticles of the medium impinge with a given velocity upon that sphericalbody in the directions of right lines parallel to AC : and let FB be one ofthose right lines. In FB take LB equal to the semi-diameter CB, anddraw BI) touching the sphere in B. Upon KG and BD let fall the perpendiculars BE, LD ; and the force with which a particle of the medium,impinging on the globe obliquely in the direction FB, would strike theglobe in B, will be to the force with which the same particle, meeting thecylinder ONGQ, described about the globe with the axis ACI, would strikeit perpendicularly in b, as LD to LB, or BE to BC. Again ; the efficacyof this force to move the globe, according to the direction of its incidenceFB or AC, is to the efficacy of the same to move the globe, according tothe direction of its determination, that is, in the direction of the right lineBC in which it impels the globe directly, as BE to BC. And, joining