Principia Mathematica (Motte Translation, 1848) — part 24 of 45

In the experiments of the 4th column there were equal motions lost in 535 oscillations made in the air, and 1} in water. The oscillations in the air were indeed a little swifter than those in the water. But if the oscil- lations in the water were accelerated in such a ratio that the motions of the pendulums might be equally swift in both mediums, there would be still the same number 1| of oscillations in the water, and J^ these the same quantity of motion would be lost as before ; because the resistance is increased, and the square of the time diminished in the same duplicate ra- tio. The pendulums, therefore, being of equal velocities, there were equal motions lost in 535 oscillations in the air, and 1} in the water; and there- fore the resistance of the pendulum in the water is to its resistance in the air as 535 to 1 }. This is the proportion of the whole resistances in the case of the 4th column. '

Now let AV -h CV^ represent the diiference of the arcs described in the

descent and subsequent ascent by the globe moving in air with the greatest

velocity V ; and since the greatest velocity is in the case of the 4th column

to the greatest velocity in the case of the 1st column as 1 to 8 ; and that

difference of the arcs in the case of the 4th column to the difference in the

2 16 case of the 1st column as -^^ to ^^, or as 85|- to 4280 ; put in these

cases 1 and 8 for the velocities, and 85 1 and 4280 for the differences of the arcs, and A + C will be =^ 85i and 8A + 640 =- 4280 or A + 80 = 535 ; and then by reducing these equations, there will come out 70 == 4491 and 0 = 64y\ and A = 21f ; and therefore the resistance, which is as /i-AV + fOY^, will become as 13p\V + 483\Y2. Therefore in the case of the 4th column, where the velocity was 1, the whole resistance is to its part proportional to the square of the velocity as 13y\ + 48/g or 61 If to 48/6 7 ^^^ therefore the resistance of the pendulum in water is to that part of the resistance in air, which is proportional to the square of the velocity, and which in swift motions is the only part that deserves consid- eration, as 61 If to 48/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 im- mersed, its resistance would have been still greater ; so that the resistance of the pendulum oscillating in the water, that is, that part which is pro- portional to the square of the velocity, and which only needs to be consid- ered in swift bodies, is to the resistance of the same whole pendulum, oscil- lating in air with the same velocity, as about 850 to 1, that is as, the den- sity of water to the density of air, nearly.

In this calculation we ought also to have taken in that part of the re- sistance of the pendulum in the water which was as the square of the ve- locity ; but I found (which will perhaps seem strange) that the resistance in 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 too

320 THE MATHEMATICAL PRINCIPLES [BoOK IL

narrow for the magnitude of the pendulous globe, and by its narrowness

obstructed 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 and

lower oscillated in the water, and the greater and higher was fastened to

the thread just above the water, and, by oscillating in the air, assisted the

motion of the pendulum, and continued it longer. The experiments made

by this contrivance proved according to the following table.

Arc descr. in first descent . .16. 8.4.2.1.1. i

Arc descr. in last ascerit . . 12 . 6 . 3 . H . | . | , j\

Diff. of arcs, proport. io ) ^ 2 1 i i i i

motion lost \ ' ' •248*i6

Number of oscillations .. . 3f . 6i . 12yV. 21 1 . 34 . 53 . 62i

In comparing the resistances of the mediums with each other, I also caused iron pendulums to oscillate in quicksilver. The length of the iron wire was about 3 feet, and the diameter of the pendulous globe about \ of an inch. To the wire, just above the quicksilver, there was fixed au other leaden globe of a bigness sufficient to continue the motion of the pendulum for some time. Then a vessel, that would hold about 3 pounds of quick- silver, was filled by turns with quicksilver and common water, that, by making the pendulum oscillate successively in these two different fluids, I might .find the proportion of their resistances ; and the resistance of the quicksilver 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 made use of a pendulous globe something bigger, as of one whose diameter was about 1 or I of an inch, the resistance of the quicksilver proved to be to the resistance of the water as about 12 or 10 to 1. But the former experi- ment is more to be relied on, because in the latter the vessel was too nar- row in proportion to the magnitude of the immersed globe ; for the vessel ought to have been enlarged together with the globe. I intended to have repeated these experiments with larger vessels, and in melted metals, and other liquors both cold and hot ; but I had not leisure to try all : and be- sides, from what is already described, it appears sufficiently that the resist- ance of bodies moving swiftly is nearly proportional to the densities of the fluids in which they move. I do not say accurately ; for more tena- cious fluids, of equal density, will undoubtedly resist more than those that are more liquid ; as cold oil more than warm, warm oil more than rain- water, and water more than spirit of wine. But in liquors, which are sen- sibly fluid enough, as in air, in salt and fresh water, in spirit of wine, of turpentine, and salts, in oil cleared of its faeces by distillation and warmed, in oil of vitriol, and in mercury, and melted metals, and any other such like, that are fluid enough to retain for some time the motion impressed

Sec. VI.] OF natural philosophy. 321

upon tliem by the agitation of the vessel, and which being poured out are easily resolved into drops, I doubt not but the rule already laid down may be accurate enough, especially if the experiments be made with larger pendulous bodies and more swiftly moved.

Lastly, since it is the opinion of some that there is a certain sethereal medium extremely rare and subtile, which freely pervades the pores of all bodies ; and from such a medium, so pervading the pores of bodies, some re- sistance must needs arise; in order to try whether the resistance, which we experience in bodies in motion, bS made upon their outward superficies only, or whether their internal parts meet with any considerable resistance upon their superficies, T thought of the following experiment. I suspended a round deal box by a thread 11 feet long, on a steel hook, by means of a ring of the sam.e metal, so as. to make a pendulum of the aforesaid length. The hook had a sharp hollow edge on its upper part, so that the upper arc of the ring pressing on the edge might move the more freely ; and the thread was fastened to the lower arc of the ring. The pendulum being thus pre- pared, I drew it aside from the perpendicular to the distance of about 6 feet, and that in a plane perpendicular to the edge of the hook, lest the ring, while the pendulum oscillated, should slide to and fro on the edge of the hook : for the point of suspension, in which the ring touches the hook, ought to remain immovable. I therefore accurately noted the place to which the pendulum was brought, and letting it go, I marked three other places, 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 pos- sible. Then I filled the box with lead and other heavy metals that were near at hand. But, first, I weighed the box when empty, and that part of the thread that went round it, and half the remaining part, extended be- tween the hook and the suspended box ; for the thread so extended always acts upon the pendulum, when drawn aside from the perpendicular, with half its weight. To this weight I added the weight of the air contained in the box. And this whole weight was about -^^ of the weight of the box when filled with the metals. Then because the box when full of the metals, by ex- tending the thread with its weight, increased the length of the pendulum, J shortened the thread so as to make the length of the pendulum, when os- cillating,, the same as before. Then drawing aside the pendulum to the place first marked, and letting it go, I reckoned about 77 oscillations before the box returned to the second mark, and as many afterwards before it came to the third mark, and as many after that before it came to the fourth mark. From whence I conclude that the whole resistance of the box, when full, 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, by reason of its vis insUa, which was 78 times greater than the vis insita of the same when empty, ought to have continued its oscillating motion so

21

322 THE MATHEMATICAL PRINCIPLES [BoOK II.

mucli the longer, and therefore to have returned to those marks at the end of 78 oscillations. But it returned to them at the end of 77 oscillations.

Let, therefore, A represent the resistance of the box upon its external superficies, 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 the matter, or the number of particles that are resisted, then 78B will be the resistance made to the internal parts of the box, when full ; and therefore the whole resistance A + B of the empty box will be to the whole resist- ance A -f 78B of the full box as 77 to 78, and, by division, A + B to 77B as 77 to 1 ; and thence A + B to B as 77 X 77 to 1, and, by division again, A to B as 5928 to 1. Therefore the resistance of the empty box in its internal parts will be above 5000 times less than the resistance on its external superficies. This reasoning depends upon the supposition that the greater 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 I had 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 retarded sooner. Tlie cause I found to be, that the hook was not strong enough to bear the weight of the box ; so that, as it oscillated to and fro, the hook was bent sometimes this and sometimes that way. I therefore procured a hook of sufficient strength, so that the point of suspension might remain unmoved, and then all things happened as is above described.

Sec. YIL] of natural philosophy. 323

SECTION VII.

Of the motion of fluids^ and the resistance made to projected bodies,

PROPOSITION XXXII. THEOREM XXVL

Suppose two similar systems of bodies consistino; of an equal number of particles, and let the correspondent particles be similar and propor- tional, each in one system to each in the other, and have a like situa- tion among themselves, and the same given ratio of density to each other ; and let them begin to m^ove amo7ig them^selves in proportional times, and loith like m^otions [that is, those in one system among one another, and those in the other among one another). And if the par- ticles that are in the same system do not touch one another, except in the m^oments of reflexion ; nor attract, nor repel each other, except with accelerative forces that are as the diameters of the correspondent parti- cles inversely, and the squares of the velocities directly ; I say, that the particles of those systems will continue to move among themselves with like m^otions and in proportional times.

Like bodies in like situations are said to be moved among themselves with like motions and in proportional times, when their situations at the end of those times are always found alike in respect of each other ; as sup- pose we compare the particles in one system with the correspondent parti- cles in the other. Hence the times will be proportional, in which similar and proportional parts of similar figures will be described by correspondent particles. Therefore if we suppose two systems of this kind, the corre- spondent particles, by reason of tlie similitude of the motions at their beginning, will continue to be moved with like motions, so long as they move 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 do agitate one another with some certain forces, and those forces are as the diameters of the correspondent particles inversely and the squares of the velocities directly, then, because the particles are in like situations, and their forces are proportional, the whole forces with which correspondent particles are agitated, and which are compounded of each of the agitating forces (by Corol. 2 of the Laws), will have like directions, and hUve the same effect as if they respected centres placed alike among the particles ; and those whole forces will be to each other as the several forces which compose them, that is, as the diameters of the correspondent particles in- versely, and the squares of the velocities directly : and therefore will cause

324 THE MATHEMATICAL PRINCIPLES [BoOK II.

correspondent particles to continue to describe like figures. These things will be so (by Cor. 1 and 8, Prop. lY., Book 1), if those centres are at rest ; but if they are moved, yet, by reason of the similitude of the translations, their situations among the particles of the system will remain similar , so that the changes introduced into the figures described by the particles will still be similar. So that the motions of correspondent and similar par- ticles will continue similar till their first meeting with each other ; and thence will arise similar collisions, and similar reflexions: which will again beget similar motions of the particles among themselves (by what was just now shown), till they mutually fall upon one another again, and so on ad infinitum.

Cor. 1. Hence if any two bodies, which are similar and in like situations to the correspondent particles of the systems, begin to move amongst them in like manner and in proportional times, and their magnitudes and densi- ties be to each other as the magnitudes and densities of the corresponding particles, these bodies will continue to be moved in like manner and in proportional times ; for the case of the greater parts of both systems and of the particles is the very same.

Cor. 2. And if all the similar and similarly situated parts of both sys- tems be at rest among themselves ; and two of them, which are greater than the rest, and mutually correspondent in both systems, begin to move in lines alike posited, with any similar motion whatsoever, they will excite similar motions in the rest of the parts of the systems, and will continue to move among those parts in like manner and in proportional times ; and will therefore describe spaces proportional to their diameters.

PROPOSITION XXXIII. THEOREM XXVII.

The same things being supposed, I say, that the greater parts of the systems are resisted in a ratio com^poiinded of the duplicate ratio of their velocities, and the duplicate ratio of their diameters, and the sim- ple ratio of the density of the parts of the systems. For the resistance arises partly from the centripetal or centrifugal forces with which the particles of the system mutually act on each other, partly from the collisions and reflexions of the particles and the greater parts. The resistances of the first kind are to each other as the whole motive forces from which they arise, that is, as the whole accelerative forces and the quantities of matter in corresponding parts ; that is (by the sup- position), as the squares of the velocities directly, and the distances of the corresponding particles inversely, and the quantities of matter in the cor- respondent parts directly : and therefore since the distances of the parti- cles in one system are to the correspondent distances of the particles of the -jther as the diameter of one particle or part in the former system to the

Sec. YIL] of natural philosophy. 325

•

diameter of the correspondent particle or part in the other, and since the quantities of matter are as the densities of the parts and the cubes of the diameters ; the resistances are to each other as the squares of the velocities and the squares of the diameters and the densities of the parts of the sys- tems. Cl.E.D. The resistances of the latter sort are as the number of correspondent reflexions and the forces of those reflexions conjunctly ; but the number of the reflexions are to each other as the velocities of the cor- responding parts directly and the spaces between their reflexions inversely. And the forces of the reflexions are as the velocities and the magnitudes and the densities of the corresponding parts conjunctly ; that iS; as the ve- locities and the cubes of the diameters and the densities of the parts. And, joining all these ratios, the resistances of the corresponding parts are to each other as the squares of the velocities and the squares of the diameters and the densities of the parts conjunctly. Q..E.D.

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 pro- portional in magnitude and density to the parts of the fluids, and similarly situated among those parts, be any how projected in the direction of lines similarly posited ; and the accelerative forces with which the particles of the fluids mutually act upon each other are as the diameters of the bodies projected inversely and the squares of their velocities directly ; those bodies will excite similar motions in the fluids in proportional times, and will de- scribe similar spaces and proportional to their diameters.

Cor. 2. Therefore in the same fluid a projected body that moves swiftly meets with a resistance that is, in the duplicate ratio of its velocity, nearly. For if the forces with which distant particles act mutually upon one another should be augmented in the duplicate ratio of the velocity, the projected body would be resisted in the same duplicate ratio accurately ; and therefore in a medium, whose parts when at a distance do not act mu- tually with any force on one another, the resistance is in the duplicate ra- tio of the velocity accurately. Let there be, therefore, three mediums A, B, C, consisting of similar and equal parts regularly disposed at equal distances. Let the parts of the mediums A and B recede from each other with forces that are among themselves as T and V ; and let the parts of the medium C be entirely destitute of any such forces. And if four equal bodies D, E, F, G, move in these mediums, the two first D and E in the two first A and B, and the other two F and G in the third C ; and if the velocity of the body D be to the velocity of the body E, and the velocity of the body F to the velocity of the body G, in the subduplicate ratio of the force T to the force V ; the resistance of the body D to the resistance of the body E, and the resistance of the body F to the resistance of the body G, will be in the duplicate ratio of the velocities ; and therefore the resistance of the body D will be to the resistance of the body F as the re-

326 THE MATHEMATICAL PRINCIPLES [BoOK 11.

sistance of the body E to the resistance of the body G. Let the bodies D and F be equally swift, as also the bodies E and G ; and, augmenting the velocities of the bodies D and F in any ratio, and diminishing the forces of the particles of the medium B in the duplicate of the same ratio, the medium B will approach to the form and condition of the medium C at pleasure ; and therefore the resistances of the equal and equally swift bodies E and G in these mediums will perpetually approach to equality, so that their diiference will at last become less than any given. There- fore since the resistances of the bodies D and F are to each other as the resistances of the bodies E and G, those will also in like manner approach to the ratio of equality. Therefore the bodies D and F, when they move with very great swiftness, meet with resistances very nearly equal ; and therefore since the resistance of the body F is in a duplicate ratio of the velocity, the resistance of the body D will be nearly in the same ratio.

Cor. 3. The resistance of a body moving very swift in an elastic fluid is almost the same as if the parts of the fluid were destitute of their cen- trifugal forces, and did not fly from each other ; if so be that the elasti- city of the fluid arise from the centrifugal forces of the particles, and the velocity be so great as not to allow the particles time enough to act.

Cor. 4. Therefore, since the resistances of similar and equally swift bodies, in a medium whose distant parts do not fly from each other, are as the squares of the diameters, the resistances made to bodies moving with very great and equal velocities in an elastic fluid will be as the squares of the diameters, nearly.

Cor. 5. And since similar, equal, and equally swift bodies, moving through mediums of the same density, whose particles do not fly from each mother mutually, will strike against an equal quantity of matter in equal, times, whether the particles of which the medium consists be more and smaller, or fewer and greater, and therefore impress on that matter an equal quantity of motion, and in return (by the 3d Law of Motion) sufier an equal 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 with extreme swiftness, their resistances are nearly equal, whether the fluids consist of gross parts, or of parts ever so subtile. For the resistance of projectiles moving with exceedingly great celerities is not much diminished by the subtilty of the medium.

' Cor. 6. All these things are so in fluids whose elastic force takes its rise from the centrifugal forces of the particles. But if that force arise from some other cause, as from the expansion of the particles after the manner of wool, or the boughs of trees, or any other cause, by which the particles are hindered from moving freely among themselves, the resistance, by reason of the lesser fluidity of the medium, will be greater than in the Corollaries above.

Sec. VII.

OF NATURAL PHILOSOPHY.

327

K

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KD>>

b li F

(X]

o

I c

\l

PROPOSITION XXXIV. THEOREM XXVIII.

If in a rare medium^ consisting of equal particles freely disposed at equal distances from each other, a globe and a cylinder described on equal diameters move with equal velocities in the direction of the axis of the cylinder, the resistance of the globe will be but half so great as that of the cylinder. For since the action of the medi- um upon the body is the same (by Cor. 5 of the Laws) whether the body move in a quiescent medium, or whether the particles of the medium impinge with the same velocity upon the quiescent body, let us consider the body as if it were quiescent, and see with what force it would be im- pelled by the moving medium. Let, therefore, ABKI represent a spherical body described from the centre C with the semi-diameter CA, and let the particles of the medium impinge with a given velocity upon that spherical body in the directions of right lines parallel to AC ; and let FB be one of those right lines. In FB take LB equal to the semi-diameter CB, and draw BD touching the sphere in B. Upon KC and BD let fall the per- pendiculars BE, LD ; and the force with which a particle of the medium, impinging on the globe obliquely in the direction FB, would strike the globe in B, will be to the force with which the same particle, meeting the cylinder ONGQ, described about the globe with the axis ACI, would strike it perpendicularly in b, as LD to LB, or BE to BC. Again ; the efficacy of this force to move the globe, according to the direction of its incidence FB or AC, is to the efficacy of the same to move the globe, according to the direction of its determination, that is, in the direction of the right line BC in which it impels the globe directly, as BE to BC. And, joining these ratios, the efficacy of a particle, falling upon the globe obliquely in the direction of the right line FB, to move the globe in the direction of its incidence, is to the efficacy of the same particle falling in. the same line perpendicularly on the cylinder, to move it in the same direction, as BE- to BC-^. Therefore if in 6E, which is perpendicular to the circular base of the cylinder NAO, and equal to the radius AC, we take 6H equal to

BE- CB

then 6H will be to 6E as the effect of the particle upon the globe to

the effect of the particle upon the cylinder. And therefore the solid which is formed by all the right lines 6H will be to the solid formed by all the right lines UE as the effect of all the particles upon the globe to the effect of all the particles upon the cylinder. But the former of these solids is a

J2S

THE MATHEMATICAL PRINCIPLES

[Book II.

paraboloid whose vertex is 0; its axis CA, and latus rectum CA, and the latter solid is a cylinder circumscribing the paraboloid ; and it is known that a paraboloid is half its circumscribed cylinder. Therefore the whole force of the medium upon the globe is half of the entire force of the same upon the cylinder. And therefore if the particles of the medium are at rest; and the cylinder and globe move with equal velocities, the resistance of the globe will be half the resistance of the cylinder. Q,.E.D.

SCHOLIUM.

By the same method other figures may be compared together as to their resistance ; and those may be found which are most apt to continue their motions in resisting mediums. As if upon the circular base CEBH from the centre O, with the radius OC, and the altitude OD, one would construct a frustum CBGP of a cone, which should meet with less resistance than any other frustum constructed with the same base and altitude, and going forwards towards D in the direction of its axis : bisect the altitude OD in d, and produce OQ, to S, so that QS may be equal to QC. and S will be the vertex of the cone whose frustum is sought.

Whence, by the bye, since the angle CSB is always acute, it follows, that, if the solid ADBE be generated by the convolution of an elliptical or oval figure ADBE about its axis AB, and the generating figure be touched by three right lines FG, GH, HI, in the points F, B, and I, so that GH shall be perpendicular to the axis in the point of contact B, and FG, HI may be inclined to GH in the angles FGB, BHI of 135 degrees : the solid arising from the convolution of the figure ADFGHIE about the same axis AB will be less resisted than the former solid ; if so be that both move forward in the direction of their axis AB, and that the extremity B of each go foremost. Which Proposition I conceive may be of use in the building of ships.

If the figure DNFG be such a curve, that if, from any point thereof, as N, the perpendicular NM be let fall on the axis AB, and from the given point G there be drawn the right line GR parallel to a right line touching^ the figure in N, and cutting the axis produced in R, MN becomes to GR ,as GR=^ to 4BR X GB-, the solid described by the revolution of this figure

Sec. YIL] of natural philosophy. 329

about its axis AB, moving in the before-mentioned rare medium from A towards B, will be less resisted than any other circular solid whatsoever, described of the same length and breadth.

PROPOSITION XXXY. PROBLEM VII.

If a rare medmm consist of very small quiescent particles of equal mag- nitudes, and freely disposed at equal distances from, one another : to find the resistance of a globe moving uniformly forward in this 7nediu?n.

Case 1. Let a cylinder described with the same diameter and altitude be conceived to go forward with the same velocity in the direction of its axis through the same medium ; and let us suppose that the particles of the medium, on which the globe or cylinder falls, fly back with as great a force of reflexion as possible. Then since the resistance of the globe (by the last Proposition) is but half the resistance of the cylinder, and since the globe is to the cylinder as 2 to 3, and since the cylinder by falling perpendicu- larly on the particles, and reflecting them with the utmost force, commu- nicates to them a velocity double to its own ; it follows that the cylinder, in moving forward uniformly half the length of its axis, will communicate a motion to the particles which is to the whole motion of the cylinder as the density of the medium to the density of the cylinder ; and that the globe, in the time it describes one length of its diameter in moving uni- formly forward, will communicate the same motion to the particles ; and in the time that it describes two thirds of its diameter, will communicate a motion to the particles which is to the whole motion of the globe as the density of the medium to the density of the globe. And therefore the globe meets with a resistance, which is to the force by which its whole mo- tion may be either taken away or generated in the time in which it de- scribes two thirds of its diameter moving imiformly forward, as the den- sity of the medium to the density of the globe.

Case 2. Let us suppose that the particles of the medium incident on the globe or cylinder are not reflected ; and then the cylinder falling per- pendicularly on the particles will communicate its own simple velocity to them, and therefore meets a resistance but half so great as in the former case, and the globe also meets with a resistance but half so great.

Case 3. Let us suppose the particles of the medium to fly back from the globe with a force which is neither the greatest, nor yet none at all, but with a certain mean force : then the resistance of the globe will be in the same mean ratio between the resistance in the first case and the resistance in the second. Q,.E.I.

Cor. 1. Hence if the globe and the particles are infinitely hard, and destitute of all elastic force, and therefore of all force of reflexfon ; the resistance of the globe will be to the force by which its whole motion may

330 THE MATHEMATICAL PRINCIPLES [BoOK IT.

be destroyed or generated, in the time that the globe describes four third parts of its diameter, as the density of the medium to the density of the globe.

Cor. 2. The resistance of the globe, cceteris paribus, is in the duplicate ratio of the velocity.

Cor. 3. The resistance of the globe, cceteris paribus, is in the duplicate ratio of the diameter.

Cor. 4. The resistance of the globe is, cceteris paribus, as the density of the medium.

Cor. 5. The resistance of the globe is in a ratio compounded of the du- plicate ratio of the velocity, and the duplicate ratio of the diameter, and the ratio of the density of the medium.

Cor. 6. The motion of the globe and its re- sistance may be thus expounded. Let AB be the time in which the globe may, by its resistance uniformly continued, lose its whole motion. Erect AD, BC perpendicular to AB. Let BC be tiiat whole motion, and through the point C, the asymptotes being AD, AB, describe the hyperbola CF. Produce AB to any point E. Erect the perpendicular EF meeting the hyperbola in F. Complete the parallelogram CBEG, and draw AF meeting BC in H. Then if the globe in any time BE, with its first mo- tion BC uniformly continued, describes in a non-resisting medium the space CBEG expounded by the area of the parallelogram, the same in a resisting medium will describe the space CBEF expounded by the area of the hy- perbola; and its motion at the end of that time will be expounded by EF, the ordinate of the hyperbola, there being lost of its motion the part FG. And its resistance at the end of the same time will be expounded by the length BH, there being lost of its resistance the part CH. All these things appear by Cor. 1 and 3, Prop. V., Book 11.

Cor. 7. Hence if the globe in the time T by the resistance R uniformly continued lose its whole motion M, the same globe in the time ^ in a resisting medium, wherein the resistance R decreases in a duplicate

ratio of the velocity, will lose out of its motion M the part tftt^l the

TM

part ™—— -remaining; and will describe a space which is to the space de-

scribed in the same time t, with the uniform motion M, as the logarithm of

T + ^ the number — pp— multiplied by the number 2,302585092994 is to the

number 7=^, because the hyperbolic area BCFE is to the rectangle BCGE in that proportion.

Sec. YIL]

OF NATURAL PHILOSOPHY.

331

SCHOLIUM.