Principia Mathematica (Motte Translation, 1848) — part 28 of 45

For, letting those things stand which were constructed in Prop. XLYII, if any physical line, as EF, describing the space PS in each vibration, be acted on in the extremities P and S of every going and return that it makes by an elastic force that is equal to its weight, it will perform its several vibrations in the time in which the same might oscillate in a cy- cloid Avhose whole perimeter is equal to the length PS ; and that because equal forces will impel equal corpuscles through equal spaces in the same or equal times. Therefore since the times of the oscillations are in the subduplicate ratio of the lengths of the pendulums, and the length of the pendulum is equal to half the arc of the whole cycloid, the time of one vi- bration would be to the time of the oscillation of a pendulum whose length is A in the subduplicate ratio of the length -^-PS or PO to the length A. But the elastic force with which the physical lineola EG is urged, when it is found in its extreme places P, S, was (in the demonstration of Prop. XL VII) to its whole elastic force as HL — KN to V, that is (since the point K now falls upon P), as HK to Y: and all that force, or which is the same thing, the incumbent weight by which the lineola EG is com^ pressed, is to the weight of the lineola as the altitude A of the incumbent weight to EG the length of the lineola ; and therefore, ex cequo, the force

Sec. YIIL]

OF NATURAL PHILOSOPHY.

367

with which the lineola EG is urged in the places P and S is to the weight of that lineola as HK X A to Y X EG ; or as PO X A to VV; because HK was to EG as PO to Y. Therefore since the times in which equal bodies are impelled through equal spaces are reciprocally in the subduplicate ratio of the forces, the time of one vibration, produced by the action of that elastic force, will be to the time of a vi- bration, produced by the impulse of the weight in a subdu- plicate ratio of YV to PO X A, and therefore to the time of the oscillation of a pendulum whose length is A in the subduplicate ratio of YY to PO X A, and the subdupli- cate ratio of PO to A conjunctly ; that is, in the entire ra- tio of Y to A. But in the time of one vibration composed of the going and re- turning of the pendulum, the pulse will be propagated right onward through a space equal to its breadth BC. There- fore the time in which a pulse runs over the space BC is to the time of one oscillation composed of the going and returning of the pendulum as Y to A, that is, as BC to the circumference of a circle whose radius is A. But the time in which the pulse will run over the space BC is to the time in which it will run over a length equal to that circumference in the same ratio ; and therefore in the time of such an oscillation the pulse will run over a length equal to that circumference. Q.E.D.

Cor. 1. The velocity of the pulses is equal to that which heavy bodies acquire by falling with an equally accele- rated motion, and in their fall describing half the alti- tude A. For the pulse will, in the time of this fall, sup-

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posing it to move with the velocity acquired by that fall, run over a space that will be equal to the whole altitude A ; and therefore in the time of one oscillation composed of one going and return, will go over a space equal to the circumference of a circle described with the radius A ; for the time of the fall is to the time of oscillation as the radius of a circle to its circumference.

Cor. 2. Therefore since that altitude A is as the elastic force of the fluid directly, and the density of the same inversely, the velocity of the pulses will be in a ratio compounded of the subduplicate ratio of the den- sity inversely, and the subduplicate ratio of the elastic force directly.

36S THE MATHEMATICAL PBINCIPLES fBoOK 11.

PROPOSITION L. PROBLEM XII.

To find the distances of the pulses. Let the number of the Yibrations of the body, by whose tremor the pulses are produced, be found to any given time. By that number divide the space which a pulse can go over in the same time, and the part found will be the breadth of one pulse. Q..E.I.

SCHOLIUM.

The last Propositions respect the motions of light and sounds ; for since light is propagated in right lines, it is certain that it cannot consist in ac- tion alone (by Prop. XLI and XLIl). As to sounds, since they arise from tremulous bodies, they can be nothing else but pulses of the air propagated through it (by Prop. XLIII) ; and this is confirmed by the tremors which sounds, if they be loud and deep, excite in the bodies near them, as we ex- perience in the sound of drums ; for quick and short tremors are less easily excited. But it is well known that any sounds, falling upon strings in unison with the sonorous bodies, excite tremors in those strings. This is also confirmed from the velocity of sounds ; for since the specific gravities of rain-water and quicksilver are to one another as about 1 to 13f, and when the mercury in the barometer is at the height of 30 inches of our measure, the specific gravities of the air and of rain-water are to one another as about 1 to 870, therefore the specific gravity of air and quick- silver are to each other as 1 to 11890. Therefore when the height of the quicksilver is at 30 inches, a height of uniform air, whose weight would be sufficient to compress our air to the density we find it to be of, must be equal to 356700 inches, or 29725 feet of our measure ; and this is that very height of the medium, which I have called A in the construction of the foregoing Proposition. A circle whose radius is 29725 feet is 186768 feet in circumference. And since a pendulum 39} inches in length com- pletes one oscillation, composed of its going and return, in two seconds of time, as is commonly known, it follows that a pendulum 29725 feet, or 356700 inches in length will perform a like oscillation in 190| seconds. Therefore in that time a sound will go right onwards 186768 feet, and therefore in one second 979 feet.

But in this computation we have made no allowance for the crassitude of the solid particles of the air, by which the sound is propagated instan- taneously. Because the weight of air is to the weight of water as 1 to 870, and because salts are almost twice as dense as water ; if the particles of air are supposed to be of near the same density as those of water or salt, and the rarity of the air arises from the intervals of the particles ; the diameter of one particle of air will be to the interval between the centres

Sec. VIIL] of natural philosophy. 369

of the particles as 1 to about 9 or 10, and to the interval between the par- ticles themselves as 1 to 8 or 9. Therefore to 979 feet, which, according to the above calculation, a sound will advance forward in one second of time, we may add ^l^, or about 109 feet, to compensate for the crassitude of the particles of the air : and then a sound will go forward about 1088 feet in one second of time.

Moreover, the vapours floating in the air being of another spring, and a different tone, will hardly, if at all, partake of the motion of the true air in which the sounds are propagated. Now if these vapours remain unmov- ed, that motion will be propagated the swifter through the true air alone, and that in the subduplicate ratio of the defect of the matter. So if the atmosphere consist of ten parts of true air and one part of vapours, the motion of sounds will be swifter in the subduplicate ratio of 11 to 10, or very nearly in the entire ratio of 21 to 20, than if it were propagated through eleven parts of true air : and therefore the motion of sounds above discovered must be increased in that ratio. By this means the sound will pass through 1142 feet in one second of time.

These things will be found true in spring and autumn, when the air is rarefied by the gentle warmth of those seasons, and by that means its elas- tic force becomes somewhat more intense. But in winter, when the air is condensed by the cold, and its elastic force is somewhat remitted, the mo- tion of sounds will be slower in a subduplicate ratio of the density ; and, on the other hand, swifter in the summer.

Now by experiments it actually appears that sounds do really advance in one second of time about 1142 feet of English measure, or 1070 feet of French measure.

The velocity of sounds being known, the intervals of the pulses are known also. For M. Sauveur, by some experiments that he made, found that an open pipe about five Paris feet in length gives a sound of the same tone with a viol-string that vibrates a hundred times in one second. Therefore there are near 100 pulses in a space of 1070 Paris feet, which a sound runs over in a second of time ; and therefore one pulse fills up a space of about 1 0y\ Paris feet, that is, about twice the length of the pipe. From whence it is probable that the breadths of the pulses, in all sounds made in open pipes, are equal to twice the length of the pipes.

Moreover, from the Corollary of Prop. XL VII appears the reason why the sounds immediately cease with the motion of the sonorous body, and why they are heard no longer when we are at a great distance from the sonorous bodies than when we are very near them. And besides, from the foregoing principles, it plainly appears how it comes to pass that sounds are so mightily increased in speaking-trumpets ; for all reciprocal motion uses to be increased by the generating cause at each return. And in tubes hin- dering the dilatation of the sounds, the motion decays more slowly, and

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recurs more forcibly ; and therefore is the more increased by the new mo- tion impressed at each return. And these are the principal phasnomena of sounds.

SECTION IX.

Of tlie circular motion of fluids,

HYPOTHESIS.

The resistance arising from the want of lubricity in the parts of a fluid, is, cseteris paribus, proportional to the velocity loith ivhich the parts of the fluid are separated from each other.

PROPOSITION LI. THEOREM XXXIX.

If a solid cylinder infinitely long, in aji uniforwj and infinite fluid, revolve with an uniformj inotion about an axis given in position, and the fluid be forced round by only this i?npulse of the cylinder, and every part of the fluid persevere uniformly in its m,otion ; I say, that the periodic iim£S of the parts of the fluid are as their distances from the axis of the cylinder.

Let AFL be a cylinder turning uni- formly about the axis S, and let the concentric circles BGM, CHN, DIO, EKP, (fee, divide the fluid into innu- merable concentric cjliudric solid orbs of the same thickness. Then, because the fluid is homogeneous, the impres- sions which the contiguous orbs make upon each other mutually will be (by the Hypothesis) as their translations from each other, and as the contiguous superficies upon which the impressions If the impression made upon any orb be greater or less on its concave than on its convex side, the stronger impression will prevail, and will either accelerate or retard the motion of the orb, according as it agrees with, or is contrary to, the motion of the same. Therefore, that every orb may persevere uniformly in its motion, the impressions made on both sides must be equal and their directions contrary. Therefore since the impres- sions are as the contiguous superficies, and as their translations from one another, the translations will be inversely as the superficies, that is, inversely as the distances of the superficies from the axis. But the differences of

Sec. IX.] OF natural philosophy. 371

the angular motions about the axis are as those translations applied to the distances, or as the translations directly and the distances inversely ; that is, joining these ratios together, as the squares of the distances inversely. Therefore if there be erected the lines Aa, Bb, Cc, Dd, Ee, (fee, perpendic- ular to the several parts of the infinite right line SABCDEQ, and recip- rocally proportional to the squares of SA, SB, SC, SD, SE, ifec, and through the extremities of those perpendiculars there be supposed to pass an hyperbolic curve, the sums of the differences, that is, the whole angular motions, will be as the correspondent sums of the lines Aa, Bb, Cc, Dd, Ee, that is (if to constitute a medium uniformly fluid the number of the orbs be increased and their breadth diminished i?i wfinitiun), as the hyperbolic areas AaQ,, BbQi, CcQ., D(f Q, EeQ, (fee, analogous to the sums ; and the times, reciprocally proportional to the angular motions, will be also recip- rocally proportional to those areas. Therefore the periodic time of any particle as D. is reciprocally as the area DrfQ,, that is (as appears from the known methods of quadratures of curves), directly as the dis- tance SD. a.E.D.

%JoR. 1. Hence the angular motions of the particles of the fluid are re- ciprocally as their distances from the axis of the cylinder, and the absolute velocities are equal.

Cor. 2. If a fluid be contained in a cylindric vessel of an infinite length, and contain another cylinder within, and both the cylinders revolve about one common axis, and the times of their revolutions be as their semi- diameters, and every part of the fluid perseveres in its motion, the peri- odic times of the several parts will be as the distances from the axis of the cylinders.

Cor. 3. If there be added or taken away any common quantity of angu- lar motion from the cylinder and fluid moving in this manner ; yet because this new motion will not alter the mutual attrition of the parts of the fluid, the motion of the parts among themselves will not be changed ; for the translations of the parts from one another depend upon the attrition. Any part will persevere in that motion, which, by the attrition made on both sides with contrary directions, is no more accelerated than it is re- tarded.

Cor. L Therefore if there be taken away from this whole system of the cylinders and the fluid all the angular motion of the outward cylinder, we shall have the motion of the fluid in a quiescent cylinder.

Cor. 5. Therefore if the fluid and outward cylinder are at rest, and the inward cylinder revolve uniformly, there will be communicated a circular motion to the fluid, which will be propagated by degrees through the whole fluid ; and will go on continually increasing, till such time as the several parts of the fluid acquire the motion determined in Cor. 4.

Cor. 6. And because the fluid endeavours to propagate its motion gtill

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farther, its impulse will carry the outmost cylinder also about with it, un- less the cylinder be violently detained ; and accelerate its motion till the periodic times of both cylinders become equal among themselves. But if the outward cylinder be violently detained, it will make an effort to retard the motion of the fluid ; and unless the inward cylinder preserve that mo- tion by means of some external force impressed thereon, it will make it cease by degrees.

All these things will be found true by making the experiment in deep standing water.

PROPOSITION LII. THEOREM XL.

If a solid sphere, in an uniform and infinite fluid, revolves about an axis given in position loith an uniform motion, and the fluid he forced round by only this impulse of the sphere ; and every part if the fluid perse- veres uniformly in its motion ; I say, that the periodic times of the parts of the fluid are as tJie squares of their distances from the centre of the sphere.

Case 1. Let AFL be a sphere turn- ing uniformly about the axis S, and let the concentric circles BGM, CHN, DIO, EKP, &c.; divide the fluid into innu- merable concentric orbs of the same thickness. Suppose those orbs to be solid ; and, because the fluid is homo- geneous, the impressions which the con- tiguous orbs make one upon another will be (by the supposition) as their translations from one another, and the contiguous superficies upon which the impressions are made. If the impression upon any orb be greater or less upon its concave than upon its convex side, the more forcible impression will prevail, and will either accelerate or retard the velocity of the orb, ac- cording as it is directed with a conspiring or contrary motion to that of the orb. Therefore that every orb may persevere uniformly in its motion, it is necessary that the impressions made upon both sides of the orb should be equal, and have contrary directions. Therefore since the impressions are as the contiguous superficies, and as their translations from one another^ the translations will be inversely as the superficies, that is, inversely as the squares of the distances of the superficies from the centre. But the differ- ences of the angular motions about the axis are as those translations applied to the distances, or as the translations directly and the distances inversely ] that is, by compounding those ratios, as the cubes of the distances inversely. Therefore if upon the several parts of the infinite right line SABCDEQ.

Sec. IX.] OF natural philosophy. 373

there be erected the perpendiculars Aa, B6, Cc, Dc?, Ee, &c., reciprocally proportional to the cubes of SA^ SB, SO, SD, SE, &c., the sums of the differences, that is, the whole angular motions will be as the corresponding sums of the lines Aa, Bb, Cc, Del, Ee, (fee, that is (if to constitute an uni- formly fluid medium the number of the orbs be increased and their thick- ness diminished in infinitum), as the hyperbolic areas AaQ,, BbQ, CcQ, DdGi, EeQ, &c., analogous to the sums ; and the periodic times being re- ciprocally proportional to the angular motions, will be also reciprocally proportional to those areas. Therefore the periodic time of any orb DIO is reciprocally as the area BdQ,, that is (by the known methods of quadra- tures), directly as the square of the distance SD. Which was first to be demonstrated.

Case 2. From the centre of the sphere let there be drawn a great num- ber of indefinite right lines, making given angles with the axis, exceeding one another by equal differences ; and, by these lines revolving about the axis, conceive the orbs to be cut into innumerable annuli ; then will every annulus have four annuli contiguous to it, that is, one on its inside, one on its outside, and two on each hand. Now each of these annuli cannot be impelled equally and with contrary directions by the attrition of the inte- rior and exterior annuli, unless the motion be communicated according to the law which we demonstrated in Case 1. This appears from that dem- onstration. And therefore any series of annuli, taken in any right line extending itself in infinitum from the globe, will move according to the law of Case 1, except we should imagine it hindered by the attrition of the annuli on each side of it. But now in a motion, according to this law, no such is, and therefore cannot be, any obstacle to the motions persevering according to that law. If annuli at equal distances from the centre revolve either more swiftly or more slowly near the poles than near the ecliptic, they will be accelerated if slow, and retarded if swift, by their mutual attrition ; and so the periodic times will continually approach to equality, according to the law of Case 1. Therefore this attrition will not at all hinder the motion from going on according to the law of Case 1, and therefore that law will take place ; that is, the periodic times of the several annuli will be as the squares of their distances from the centre of the globe. Which was to be demonstrated in the second place.

Case 3. Let now every annulus be divided by transverse sections into innumerable particles constituting a substance absolutely and uniformly fluid ; and because these sections do not at all respect the law of circular motion, but only serve to produce a fluid substance, the law of circular mo- tion will continue the same as before. All the very small annuli will either not at all change their asperity and force. of mutual attrition upon account of these sections, or else they v^^ill change the same equally. Therefore the proportion of the causes remaining the same, the proportion of the effects

3r4 THE MATHEMATICAL PRINCIPLES [BoOK II.

will remain tlie same also ; that is, the proportion of the motions and the periodic times. Q,.E.D. But now as the circular motion, and the centri- fugal force thence arising, is greater at the ecliptic than at the poles, there must be some cause operating to retain the several particles in their circles ; otherwise the matter that is at the ecliptic will always recede from the centre, and come round about to the poles by the outside of the vortex, and from thence return by the axis to the ecliptic with a perpetual circu- lation.

Cor. 1. Hence the angular motions of the parts of the fluid about the axis of the globe are reciprocally as the squares of the distances from the centre of the globe, and the absolute velocities are reciprocally as the same squares applied to the distances from the axis.

Cor. 2. If a globe revolve with a uniform motion about an axis of a given position in a similar and infinite quiescent fluid with an uniform motion, it will communicate a whirling motion to the fluid like that of a vortex, and that motion will by degrees be propagated onward in infinitum ; and this motion will be increased continually in every part of the fluid, till the periodical times of the several parts become as the squares of the dis- tances from the centre of the globe.

Cor. 3. Because the inward parts of the vortex are by reason of their greater velocity continually pressing upon and driving forward the external parts, and by that action are perpetually communicating motion to them, and at the same time those exterior parts communicate the same quantity of motion to those that lie still beyond them, and by this action preserve the quantity of their motion continually unchanged, it is plain that the motion is perpetually transferred from the centre to the circumference of the vortex, till it is quite swallowed up and lost in the boundless extent of that circumference. The matter between any two spherical superficies concentrical to the vortex will never be accelerated ; because that matter will be always transferring the motion it receives from the matter nearer the centre to that matter which lies nearer the circumference.

Cor. 4. Therefore, in order to continue a vortex in the same state of motion, some active principle is required from which the globe may receive continually the same quantity of motion which it is always communicating to the matter of the vortex. Without such a principle it will undoubtedly come to pass that the globe and the inward parts of the vortex, being al- ways propagating their motion to the outward parts, and not receiving any new motion, will gradually move slower and slower, and at last be carried round no longer.

Cor. 5. If another globe should be swimming in the same vortex at a certain distance from its centre, and in the mean time by some force revolve constantly about an axis of a given inclination, the motion of this globe will drive the fluid round after the manner of a vortex ; and at first this

Sec. IX.] OF natural philosophy. 375

new and small vortex will revolve with its globe about the centre of the other ; and in the mean time its motion will creep on farther and farther, and by degrees be propagated in infinitum^, after the manner of the first vortex. And for the same reason that the globe of the new vortex was carried about before by the motion of the other vortex, the globe of this other will be carried about by the motion of this new vortex, so that the two globes will revolve about some intermediate point, and by reason of that circular motion mutually fly from each other, unless some force re- strains them. Afterward, if the constantly impressed forces, by which the globes persevere in their motions, should cease, and every thing be left to act according to the laws of mechanics, the motion of the globes will lan- guish by degrees (for the reason assigned in Cor. 3 and 4), and the vortices at last will quite stand still.

Cor. 6. If several globes in given places should constantly revolve with determined velocities about axes given in position, there would arise from them as many vortices going on in injinitum. For upon the same account that any one globe propagates its motion in injinitum, each globe apart will propagate its own motion in injinitwm also ; so that every part of the infinite fluid will be agitated with a motion resulting from the actions of all the globes. Therefore the vortices will not be confined by any certain limits, but by degrees run mutually into eacli other ; and by the mutual actions of the vortices on each other, the globes will be perpetually moved from their places, as was shewn in the last Corollary ; neither can they possibly keep any certain position among themselves, unless some force re- strains them. But if those forces, which are constantly impressed upon the globes to continue these motions, should cease, the matter (for the rea- son assigned in Cor. 3 and 4) will gradually stop, and cease to move in vortices.

Cor. 7. If a similar fluid be inclosed in a spherical vessel, and, by the uniform rotation of a globe in its centre, is driven round in a vortex ; and the globe and vessel revolve the same way about the same axis, and their periodical times be as the squares of the semi-diameters ; the parts of the fluid will not go on in their motions without acceleration or retardation, till their periodical times are as the squares of their distances from the centre of the vortex. No constitution of a vortex can be permanent but this.

Cor. 8. If the vessel, the inclosed fluid, and the globe, retain this mo- tion, and revolve besides with a common angular motion about any given axis, because the mutual attrition of the parts of the fluid is not changed by this motion, the motions of the parts among each other will not be changed ; for the translations of the parts among themselves depend upon this attrition. Any part will persevere in that motion in which its attri-

376 THE MATHEMATICAL PRINCIPLES [BoOK H.

tion on one side retards it just as mucli as its attrition on the other side accelerates it.

Cor. 9. Therefore if the vessel be quiescent, and the motion of the globe be given, the motion of the fluid will be given. For conceive a plane to pass through the axis of the globe, and to revolve with a contrary mo- tion ; and suppose the sum of the time of this revolution and of the revolu- tion of the globe to be to the time of the revolution of the globe as the square of the semi-diameter of the vessel to the square of the semi-diameter of the globe ; and the periodic times of the parts of the fluid in respect of this plane will be as the squares of their distances from the centre of the globe.

Cor. 10. Therefore if the vessel move about the same axis with the globe, or with a given velocity about a different one, the motion of the fluid will be given. For if from the whole system we take away the angular motion of the vessel, all the motions will remain the same among themselves as before^ by Cor, 8, and those motions will be given by Cor. 9.

Cor. 11. If the vessel and the fluid are quiescent, and the globe revolves with an uniform motion, that motion will be propagated by degrees through the whole fluid to the vessel, and the vessel will be carried round by it, unless violently detained ; and the fluid and the vessel will be continually accelerated till their periodic times become equal to the periodic times of the globe. If the vessel be either withheld by some force, or revolve with any constant and uniform motion, the medium will come by little and little to the state of motion defined in Cor. 8, 9, 10, nor will it ever perse- vere in any other state. But if then the forces, by which the globe and vessel revolve with certain motions, should cease, and the whole system be left to act according to the mechanical laws, the vessel and globe, by means of the intervening fluid, will act upon each other, and will continue to propagate their motions through the fluid to each other, till their periodic times become equal among themselves, and the whole system revolves to- gether like one solid body.

SCHOLIUM.

In alj these reasonings I suppose the fluid to consist of matter of uniform density and fluidity ; I mean, that the fluid is such, that a globe placed any where therein may propagate with the same motion of its own, at dis- tances from itself continually equal, similar and equal motions in the fluid in the same interval of time. The matter by its circular motion endeavours to recede from the axis of the vortex, and therefore presses all the matter that lies beyond. This pressure makes the attrition greater, and the Reparation of the parts more difficult; and by consequence diminishes the fluidity of the matter. Again ; if the parts of the fluid are in any one place denser or larger than in the others, the fluidity wiU be less in that place, because there are fewer superficies where the parts can be separated

$EC. IX.] OF NATURAL PHILOSOPHY. 377

from each other. In tjiese cases I suppose the defect of the fluidity to be supplied by the smoothness or softness of the parts, or some other condi- tion ; otherwise the matter where it is less fluid will cohere more, and be more sluggish, and therefore will receive the motion more slowly, and pro- pagate it farther than agrees with the ratio above assigned. If the vessel be not spherical, the particles will move in lines not circular, but answer- ing to the figure of the vessel ; and the periodic times will be nearly as the squares of the mean distances from the centre. In the parts between the centre and the circumference the motions will be slower where the spaces are wide, and swifter where narrow ; but yet the particles will not tend to the circumference at all the more for their greater swiftness ; for they then describe arcs of less curvity, and the conatus of receding from the centre is as much diminished by the diminution of this curvature as it is augment- ed by the increase of the velocity. As they go out of narrow into wide spaces, they recede a little farther from the centre, but in doing so are re- tarded ; and when they come out of wide into narrow spaces, they are again accelerated ; and so each particle is retarded and accelerated by turns for ever. These things will come to pass in a rigid vessel ; for the state of vortices in an infinite fluid is known by Cor. 6 of this Proposition.

I have endeavoured in this Proposition to investigate the properties of vortices, that I might find whether the celestial phsenomena can be explain- ed by them ; for the phasnomenon is this, that the periodic times of the planets revolving about Jupiter are in the sesquiplicate ratio of their dis- tances from Jupiter's centre ; and the same rule obtains also among the planets that revolve about the sun. And these rules obtain also with the greatest accuracy, as far as has been yet discovered by astronomical obser- tion. Therefore if those planets are carried round in vortices revolving about Jupiter and the sun, the vortices must revolve according to that law. But here we found the periodic times of the parts of the vortex to be in the duplicate ratio of the distances from the centre of motion ; and this ratio cannot be diminished and reduced to the sesquiplicate, unless either the matter of the vortex be more fluid the farther it is from the cen- tre, or the resistance arising from the want of lubricity in the parts of the fluid should, as the velocity with which the parts of the fluid are separated goes on increasing, be augmented with it in a greater ratio than that in which the velocity increases. But neither of these suppositions seem rea- sonable. The more gross and less fluid parts will tend to the circumfer- ence, unless they are heavy towards the centre. And though, for the sake of demonstration, I proposed, at the beginning of this Section, an Hypoth- esis that the resistance is proportional to the velocity, nevertheless, it is in truth probable that the resistance is in a less ratio than that of the velo- city ; which granted, the periodic times of the parts of the vortex will be in a greater than the duplicate ratio of the distances from its centre. If, as some think, the vortices move more swiftly near the centre, then slower

378 THE MATHEMATICAL PRINCIPLES [BoOK II.

to a certain limit, then again swifter near the circumference, certainly neither the sesquiplicate, nor any other certain and determinate ratio, can obtain in them. Let philosophers then see how that phasnomenon of the sesquiplicate ratio can be accounted for by vortices.

PROPOSITION LIIL THEOREM XLI.

Bodies carried about in a vortex, and returning in the same orb, are of the same density with the vortex, and are vioved according to the same law with the parts of the vortex, as to velocity and direction of 'motion.