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 shouldbe equal, and have contrary directions. Therefore since the impressionsare as the contiguous superficies, and as their translations from one another^the translations will be inversely as the superficies, that is, inversely as thesquares of the distances of the superficies from the centre. But the differences of the angular motions about the axis are as those translations appliedto 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 SABCDEQSEC. IX.j OF NATURAL PHILOSOPHY. 373there be erected the perpendiculars Aa, B6. Cc, Dd, Ee, c.; reciprocallyproportional to the cubes of SA5 SB, SO, SD, SE, etc., the sums of thedifferences, that is, the whole angular motions will be as the correspondingsums of the lines A#, B&, Cc, DC/, Ee, <fcc., that is (if to constitute an uniformly fluid medium the number of the orbs be increased and their thickness diminished in infinitum), as the hyperbolic areas AaQ, B&Q,, CcQ,DtfQ,, EeQ,, etc., analogous to the sums ; and the periodic times being reciprocally proportional to the angular motions, will be also reciprocallyproportional to those areas. Therefore the periodic time of any orb DIOis reciprocally as the area Dt/Q,, that is (by the known methods of quadratures), directly as the square of the distance SD. Which was first to bedemonstrated.CASE 2. From the centre of the sphere let there be drawn a great number of indefinite right lines, making given angles with the axis, exceedingone another by equal differences; and, by these lines revolving about theaxis, conceive the orbs to be cut into innumerable annuli; then will everyannulus have four annuli contiguous to it, that is, one on its inside, one onits outside, and two on each hand. Now each of these annuli cannot beimpelled equally and with contrary directions by the attrition of the interior and exterior annuli, unless the motion be communicated according tothe law which we demonstrated in Case 1. This appears from that demonstration. And therefore any series of annuli, taken in any right lineextending itself in infinitum from the globe, will move according to thelaw of Case 1, except we should imagine it hindered by the attrition of theannuli on each side of it. But now in a motion, according to this law, nosuch is, and therefore cannot be, any obstacle to the motions perseveringaccording to that law. If annuli at equal distances from the centrerevolve either more swiftly or more slowly near the poles than near theecliptic, they will be accelerated if slow, and retarded if swift, by theirmutual attrition; and so the periodic times will continually approach toequality, according to the law of Case 1. Therefore this attrition will notat all hinder the motion from going on according to the law of Case 1, andtherefore that law will take place ; that is, the periodic times of the severalannuli 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 intoinnumerable particles constituting a substance absolutely and uniformlyfluid ; and because these sections do not at all respect the law of circularmotion, but only serve to produce a fluid substance, the law of circular motion will continue the same as before. All the very small annuli will eitheinot at all change their asperity and force of mutual attrition upon accountof these sections, or else they will change the same equally. Therefore theproportion of the causes remaining the same, the proportion of the effects3r4 THE MATHEMATICAL PRINCIPLES [BOOK II.will remain the same also ; that is, the proportion of the motions and tinperiodic times. Q.E.D. But now as the circular motion, and the centrifugal force thence arising, is greater at the ecliptic than at the poles, theremust be some cause operating to retain the several particles in their ciicles;otherwise the matter that is at the ecliptic will always recede from thecentre, 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 circulation.COR. 1. Hence the angular motions of the parts of the fluid about theaxis of the globe are reciprocally as the squares of the distances from thecentre of the globe, and the absolute velocities are reciprocally as the samesquares applied to the distances from the axis.COR. 2. If a globe revolve with a uniform motion about an axis of agiven position in a similar and infinite quiescent fluid with an uniformmotion, it will communicate a whirling motion to the fluid like that of avortex, and that motion will by degrees be propagated onward in infinitnm ;and this motion will be increased continually in every part of the fluid, tillthe periodical times of the several parts become as the squares of the distances from the centre of the globe.COR. 3. Because the inward parts of the vortex are by reason of theirgreater velocity continually pressing upon and driving forward the externalparts, and by that action are perpetually communicating motion to them,and at the same time those exterior parts communicate the same quantityof motion to those that lie still beyond them, and by this action preservethe quantity of their motion continually unchanged, it is plain that themotion is perpetually transferred from the centre to the circumference ofthe vortex, till it is quite swallowed up and lost in the boundless extent ofthat circumference. The matter between any two spherical superficiesconcentrical to the vortex will never be accelerated ; because that matterwill be always transferring the motion it receives from the matter nearerthe centre to that matter which lies nearer the circumference.COR. 4. Therefore, in order to continue a vortex in the same state ofmotion, some active principle is required from which the globe may receivecontinually the same quantity of motion which it is always communicatingto the matter of the vortex. Without such a principle it will undoubtedlycome to pass that the globe and the inward parts of the vortex, being always propagating their motion to the outward parts, and not receiving anynew motion, will gradually move slower and slower, and at last be carriedround no longer.COR. 5. If another globe should be swimming in the same vortex at acertain distance from its centre, and in the mean time by some force revolveconstantly about an axis of a given inclination, the motion of Jiis globewill drive the fluid round after the manner of a vortex and at first thisSEC. IX.] OF NATURAL PHILOSOPHY. 375new and small vortex will revolve with its globe about the centre of theother; and in the mean time its motion will creep on farther and farther,and by degrees be propagated in iiifinitum, after the manner of the firstvortex. And for the same reason that the globe of the new vortex watcarried about before by the motion of the other vortex, the globe of thisother will be carried about by the motion of this new vortex, sc that thetwo globes will revolve about some intermediate point, and by reason ofthat circular motion mutually fly from each other, unless some force restrains them. Afterward, if the constantly impressed forces, by which theglobes persevere in their motions, should cease, and every thing be left toact according to the laws of mechanics, the motion of the globes will languish by degrees (for the reason assigned in Cor. 3 arid 4), and the vorticesat last will quite stand still.COR. 6. If several globes in given places should constantly revolve withdetermined velocities about axes given in position, there would arise fromthem as many vortices going on in infinitum. For upon the same accountthat any one globe propagates its motion in itifinitum, each globe apartwill propagate its own motion in infiidtwtn also ; so that every part of theinfinite fluid will be agitated with a motion resulting from the actions ofall the globes. Therefore the vortices will not be confined by any certainlimits, but by degrees run mutually into each other ; and by the mutualactions of the vortices on each other, the globes will be perpetually movedfrom their places, as was shewn in the last Corollary ; neither can theypossibly keep any certain position among themselves, unless some force restrains them. But if those forces, which are constantly impressed uponthe globes to continue these motions, should cease, the matter (for the reason assigned in Cor. 3 and 4) will gradually stop, and cease to move invortices.COR. 7. If a similar fluid be inclosed in a spherical vessel, and, by theuniform rotation of a globe in its centre, is driven round in a vortex ; andthe globe and vessel revolve the same way about the same axis, and theirperiodical times be as the squares of the semi-diameters ; the parts of thefluid will not go on in their motions without acceleration or retardation,till their periodical times are as the squares of their distances fromthe centre of the vortex. No constitution of a vortex can be permanentbut this.COR. 8. If the vessel, the inclosed fluid, and the globe, retain this motion, and revolve besides with a common angular motion about any givenaxis, because the mutual attrition of the parts of the fluid is not changedby this motion, the motions of the parts among each other will not bechanged ;for the translations of the parts among themselves depend uponthis attrition. Any part will persevere in that motion in which its attri376 THE MATHEMATICAL PRINCIPLES [BOOK II.tion on one side retards it just as much as its attrition on the other sideaccelerates it.COR. 9. Therefore if the vessel be quiescent, and the motion of theglobe be given, the motion of the fluid will be given. For conceive a planeto pass through the axis of the globe, and to revolve with a contrary motion ; and suppose the sum of the time of this revolution and of the revolution of the globe to be to the time of the revolution of the globe as thesquare of the semi-diameter of the the square of the semi-diameterof the globe ; and the periodic times of the parts of the fluid in respect ofthis plane will be as the squares of their distances from the centre of theglobe.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 willbe given. For if from the whole system we take away the angular motionof the vessel, all the motions will remain the same among themselves asbefore, 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 revolveswith an uniform motion, that motion will be propagated by degrees throughthe 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 continuallyaccelerated till their periodic times become equal to the periodic times ofthe globe. If the vessel be either withheld by some force, or revolve withany constant and uniform motion, the medium will come by little andlittle to the state of motion defined in Cor. 8, 9, 10, nor will it ever persevere in any other state. But if then the forces, by which the globe andvessel revolve with certain motions, should cease, and the whole system beleft to act according to the mechanical laws, the vessel and globe, by meansof the intervening fluid, will act upon each other, and will continue topropagate their motions through the fluid to each other, till their periodictimes become equal among themselves, and the whole system revolves together like one solid body.SCHOLIUM.In all these reasonings I suppose the fluid to consist of matter of uniformdensity and fluidity ;I mean, that the fluid is such, that a globe placedany where therein may propagate with the same motion of its own, at distances from itself continually equal, similar and equal motions in the fluidin the same interval of time. The matter by its circular motion endeavoursto recede from the axis of the vortex, and therefore presses all the matterthat lies beyond. This pressure makes the attrition greater, and theSeparation of the parts more difficult; and by consequence diminishesthe fluidity of the matter. Again ;if the parts of the fluid are in any oneplace denser or larger than in the others, the fluidity will be less in that[lace, because there are fewer superficies where the parts can be separatedfclC IX.] Or NATURAL PHILOSOPHY. 3?<from each other. In these cases I suppose the defect of the fluidity to besupplied by the smoothness or softness of the parts, or some other condition ; otherwise the matter where it is less fluid will cohere more, and bemore sluggish, and therefore will receive the motion more slowly, and propagate it farther than agrees with the ratio above assigned. If the vesselbe riot spherical, the particles will move in lines not circular, but answering to the figure of the vessel ; and the periodic times will be nearly as thesquares of the mean distances from the centre. In the parts between thecentre and the circumference the motions will be slower where the spacesare wide, and swifter where narrow ; but yet the particles will not tend to thecircumference at all the more for their greater swiftness ;for they thendescribe arcs of less curvity, and the conatus of receding from the centre isas much diminished by the diminution of this curvature as it is augmented by the increase of the velocity. As they go out of narrow into widespaces, they recede a little farther from the centre, but in doing so are retarded ; and when they come out of wide into narrow spaces, they are againaccelerated ; and so each particle is retarded and accelerated by turns forever. These things will come to pass in a rigid vessel ; for the state ofvortices in an infinite fluid is known by Cor. 6 of this Proposition.I have endeavoured in this Proposition to investigate the properties ofvortices, that I might find whether the celestial phenomena can be explained by them; for the phenomenon is this, that the periodic times of theplanets revolving about Jupiter are in the sesquiplicate ratio of their distances from Jupiter s centre ; and the same rule obtains also among theplanets that revolve about the sun. And these rules obtain also with thegreatest accuracy, as far as has been yet discovered by astronomical obsertion.Therefore if those planets are carried round in vortices revolvingabout Jupiter and the sun, the vortices must revolve according to thatlaw. But here we found the periodic times of the parts of the vortex tobe in the duplicate ratio of the distances from the centre of motion ; andthis ratio cannot be diminished and reduced to the sesquiplicate, unlesseither the matter of the vortex be more fluid the farther it is from the centre, or the resistance arising from the want of lubricity in the parts of thefluid should, as the velocity with which the parts of the fluid are separatedgoes on increasing, be augmented with it in a greater ratio than that inwhich the velocity increases. But neither of these suppositions seem reasonable. The more gross and less fluid parts will tend to the circumference, unless they are heavy towards the centre. And though, for the sakeof demonstration, I proposed, at the beginning of this Section, an Hypothesis that the resistance is proportional to the velocity, nevertheless, it is intruth probable that the resistance is in a less ratio than that of the velocity ; which granted, the periodic times of the parts of the vortex will bein 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 slower378 THE MATHEMATICAL PRINCIPLES [BOOK ITto a certain limit, then again swifter near the circumference, certainlyneither the sesquiplicate, nor any other certain and determinate ratio, canobtain in them. Let philosophers then see how that phenomenon of thesesquiplicate ratio can be accounted for by vortices.PROPOSITION LIII. THEOREM XLI.Bodies carried about in a vortex, and returning- in the same orb, are ofthe same density with the vortex, and are moved according to thesame law with the parts of the vortex, as to velocity and direction ojmotion.For if any small part of the vortex, whose particles or physical pointspreserve a given situation among each other, be supposed to be congealed,this particle will move according to the same law as before, since no changeis made either in its density, vis insita, or figure. And again ;if a congealedor solid part of the vortex be of the same density with the rest of the vortex,and be resolved into a fluid, this will move according to the same law asbefore, except in so far as its particles, now become fluid, may be movedamong themselves. Neglect, therefore, the motion of the particles amongthemselves as not at all concerning the progressive motion of the whole, andthe motion of the whole will be the same as before. But this motion will bethe same with the motion of other parts of the vortex at equal distancesfrom the centre; because the solid, now resolved into a fluid, is becomeperfectly like to the other parts of the vortex. Therefore a solid, if it beof the same density with the matter of the vortex, will move with the samemotion as the parts thereof, being relatively at rest in the matter that surrounds it. If it be more dense, it will endeavour more than before to recede from the centre ; and therefore overcoming that force of the vortex,by which, being, as it were, kept, in equilibrio, it was retained in its orbit,it will recede from the centre, and in its revolution describe a spiral, returning no longer into the same orbit. And, by the same argument, if itbe more rare, it will approach to the centre. Therefore it can never continually go round in the same orbit, unless it be of the same density withthe fluid. But we have shewn in that case that it would revolve according to the same law with those parts of the fluid that are at the same orequal distances from the centre of the vortex.COR. 1. Therefore a solid revolving in a vortex, and continually goinground in the same orbit, is relatively quiescent in the fluid that carries it.COR. 2. And if the vortex be of an uniform density, the same body mayrevolve at any distance from the centre of the vortex.SCHOLIUM.Hence it is manifest that the planets are not carried round in corporealvortices ; for, according to the Copernican hypothesis, the planets goingSEC. IX.] OF NATURAL PHILOSOPHY. 379round the sun revolve in ellipses, having the sun in their common focus ;and by radii drawn to the sun describeareas proportional to the times. Butnow the parts of a vortex can never revolve with such a motion. Let AD,BE, CF, represent three orbits described about the sun S, of which let theutmost circle CF be concentric to thesun ; and let the aphelia of the two innermost be A, B j and their periheliaD, E. Therefore a body revolving inthe orb CF, describing, by a radiusdrawn to the sun, areas proportional tothe times, will move with an uniform motion. And, according to the lawsof astronomy, the body revolving in the orb BE will move slower in itsaphelion B, and swifter in its perihelion E ; whereas, according to thelaws of mechanics, the matter of the vortex ought to move more swiftly inthe narrow space between A and C than in the wide space between D andF ; that is, more swiftly in the aphelion than in the perihelion. Now thesetwo conclusions contradict each other. So at the beginning of the sign ofVirgo, where the aphelion of Mars is at present, the distance between the*orbits of Mars and Venus is to the distance between the same orbits, at thebeginning of the sign of Pisces, as about 3 to 2 ; and therefore the matterof the vortex between those orbits ought to be swifter at the beginning ofPisces than at the beginning of Virgo in the ratio of 3 to 2 ; for the narrower the space is through which the same quantity of matter passes in thesame time of one revolution, the greater will be the velocity with which itpasses through it. Therefore if the earth being relatively at rest in thiscelestial matter should be carried round by it, and revolve together with itabout the sun, the velocity of the earth at the beginning of Pisceswould be to its velocity at the beginning of Virgo in a sesquialteral ratio.Therefore the sun s apparent diurnal motion at the beginning of Virgoought to be above 70 minutes, and at the beginning of Pisces less than 48minutes; whereas, on the contrary, that apparent motion of the sun isreally greater at the beginning of Pisces than at the beginning of Virgo;as experience testifies; and therefore the earth is swifter at the beginningof Virgo than at the beginning of Pisces ; so that the hypothesis of vortices is utterly irreconcileable with astronomical phenomena, and ratherserves to perplex than explain the heavenly motions. How these motions are performed in free spaces without vortices, may be understoodby the first Book j and I shall now more fully treat of it in the followingBook.BOOK IIIBOOK III.IN the preceding Books I have laid down the principles of philosophy ,principles not philosophical, but mathematical : such, to wit, as we maybuild our reasonings upon in philosophical inquiries. These principles arethe laws and conditions of certain motions, and powers or forces, whichchiefly have respect to philosophy : but, lest they should have appeared ofthemselves dry and barren, I have illustrated them here and there withsome philosophical scholiums, giving an account of such things as are ofmore general nature, and which philosophy seems chiefly to be founded on ;such as the density and the resistance of bodies, spaces void of all bodies,and the motion of light and sounds. It remains that, from the same principles, I now demonstrate the frame of the System of the World. Uponthis subject I had, indeed, composed the third Book in a popular method,that it might be read by many ; but afterward, considering that such ashad not sufficiently entered into the principles could not easily discern thestrength of the consequences, nor lay aside the prejudices to which they hadbeen many years accustomed, therefore, to prevent the disputes which mightbe raised upon such accounts, I chose to reduce the substance of this Bookinto the form of Propositions (in the mathematical way), which should beread by those only who had first made themselves masters of the principlesestablished in the preceding Books : not that I would advise any one to theprevious study of every Proposition of those Books ; for they abound withsuch as might cost too much time, even to readers of good mathematicallearning. It is enough if one carefully reads the Definitions, the Laws ofMotion, and the first three Sections of the first Book. He may then passon to this Book, and consult such of the remaining Propositions of thefirst two Books, as the references in this, and his occasions, shall require.384 THE MATHEMATICAL PRINCIPLES [BOOK III.RULES OF REASONING IN PHILOSOPHY,RULE I.We are Io admit no more causes of natural things than such as are bothtrue and sufficient to explain their appearances.To this purpose the philosophers say that Nature does nothing in vain,and more is in vain when less will serve ;for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.RULE II.Therefore to the same natural effects we must, as far as possible, assignthe same causes.As to respiration in a man and in a beast; the descent of stones in Europeand in America ; the light of our culinary fire and of the sun ; the reflection of light in the earth, and in the planets.RULE III.The qualities of bodies, which admit neither intension nor remission ojdegrees, and which are found to belong to all bodies within the reachof our experiments, are to be esteemed the universal qualities of allbodies whatsoever.For since the qualities of bodies are only known to us by experiments, weare to hold for universal all such as universally agree with experiments ;nnd such as are not liable to diminution can never be quite taken away.We are certainly not to relinquish the evidence of experiments for the sakeof dreams and vain fictions of our own devising ;nor are we to recede fromthe analogy of Nature, which uses to be simple, and always consonant toitself. We no other way know the extension of bodies than by our senses,nor do these reach it in all bodies; but because we perceive extension inall that are sensible, therefore we ascribe it universally to all others also.That abundance of bodies are hard, we learn by experience ; and becausethe hardness of the whole arises from the hardness of the parts, we thereforejustly infer the hardness of the undivided particles not only of the bodieswe feel but of all others. That all bodies are impenetrable, we gather notfrom reason, but from sensation. The bodies which we handle we find impenetrable, and thence conclude impenetrability to be an universal property