Chapter X: Introduction (6)

9. THE resistance, which arises from the friction of the body against the parts of the fluid, must be very inconsiderable; and the resistance, which follows from the tenacity of the parts of fluids, is not usually very great, and does not depend much upon the velocity of the body in the fluid; for as the parts of the fluid adhere together with a certain degree of force, the resistance, which the body receives from thence, cannot much depend upon the velocity, with which the body moves; but like the power of gravity, its effect must be proportional to the time of its acting. This the reader may find farther explained by Sir ~ISAAC NEWTON~ himself in the postscript to a discourse published by me in THE PHILOSOPHICAL TRANSACTIONS, N^o 371. The principal resistance, which most fluids give to bodies, arises from the power of inactivity in the parts of the fluids, and this depends upon the velocity, with which the body moves, on a double account. In the first place, the quantity of the fluid moved out of place by the moving body in any determinate space of time is proportional to the velocity, wherewith the body moves; and in the next place, the velocity with which each particle of the fluid is moved, will also be proportional to the velocity of the body: therefore since the resistance, which any body makes against being put into motion, is proportional both to the quantity of matter moved and the velocity it is moved with; the resistance, which a fluid gives on this account, will be doubly increased with the increase of the velocity in the moving body; that is, the resistance will be in a two-fold or duplicate proportion of the velocity, wherewith the body moves through the fluid.

10. FARTHER it is most manifest, that this latter kind of resistance increasing with the increase of velocity, even in a greater degree than the velocity it self increases, the swifter the body moves, the less proportion the other species of resistance will bear to this: nay that this part of the resistance may be so much augmented by a due increase of velocity, till the former resistances shall bear a less proportion to this, than any that might be assigned. And indeed experience shews, that no other resistance, than what arises from the power of inactivity in the parts of the fluid, is of moment, when the body moves with any considerable swiftness.

11. THERE is besides these yet another species of resistance, found only in such fluids, as, like our air, are elastic. Elasticity belongs to no fluid known to us beside the air. By this property any quantity of air may be contracted into a less space by a forcible pressure, and as soon as the compressing power is removed, it will spring out again to its former dimensions. The air we breath is held to its present density by the weight of the air above us. And as this incumbent weight, by the motion of the winds, or other causes, is frequently varied (which appears by the barometer;) so when this weight is greatest, we breath a more dense air than at other times. To what degree the air would expand it self by its spring, if all pressure were removed, is not known, nor yet into how narrow a compass it is capable of being compressed. Mr. BOYLE found it by experiment capable both of expansion and compression to such a degree, that he could cause a quantity of air to expand it self over a space some hundred thousand times greater, than the space to which he could confine the same quantity[111]. But I shall treat more fully of this spring in the air hereafter[112]. I am now only to consider what resistance to the motion of bodies arises from it.

12. BUT before our author shews in what manner this cause of resistance operates, he proposes a method, by which fluids may be rendered elastic, demonstrating that if their particles be provided with a power of repelling each other, which shall exert it self with degrees of strength reciprocally proportional to the distances between the centers of the particles; that then such fluids will observe the same rule in being compressed, as our air does, which is this, that the space, into which it yields upon compression, is reciprocally proportional to the compressing weight[113]. The term reciprocally proportional has been explained above[114]. And if the centrifugal force of the particles acted by other laws, such fluids would yield in a different manner to compression[115].

13. WHETHER the particles of the air be endued with such a power, by which they can act upon each other out of contact, our author does not determine, but leaves that to future examination, and to be discussed by philosophers. Only he takes occasion from hence to consider the resistance in elastic fluids, under this notion; making remarks, as he passes along, upon the differences, which will arise, if their elasticity be derived from any other fountain[116]. And this, I think, must be confessed to be done by him with great judgment; for this is far the most reasonable account, which has been given of this surprizing power, as must without doubt be freely acknowledged by any one, who in the least considers the insufficiency of all the other conjectures, which have been framed; and also how little reason there is to deny to bodies other powers, by which they may act upon each other at a distance, as well as that of gravity; which we shall hereafter shew to be a property universally belonging to all the bodies of the universe, and to all their parts[117]. Nay we actually find in the loadstone a very apparent repelling, as well as an attractive power. But of this more in the conclusion of this discourse.

14. BY these steps our author leads the way to explain the resistance, which the air and such like fluids will give to bodies by their elasticity; which resistance he explains thus. If the elastic power of the fluid were to be varied so, as to be always in the duplicate proportion of the velocity of the resisted body, it is shewn that then the resistance derived from the elasticity, would increase in the duplicate proportion of the velocity; in so much that the whole resistance would be in that proportion, excepting only that small part, which arises from the friction between the body and the parts of the fluid. From whence it follows, that because the elastic power of the same fluid does in truth continue the same, if the velocity of the moving body be diminished, the resistance from the elasticity, and therefore the whole resistance, will decrease in a less proportion, than the duplicate of the velocity; and if the velocity be increased, the resistance from the elasticity will increase in a less proportion, than the duplicate of the velocity, that is in a less proportion, than the resistance made by the power of inactivity of the parts of the fluid. And from this foundation is raised the proof of a property of this resistance, given by the elasticity in common with the others from the tenacity and friction of the parts of the fluid; that the velocity may be increased, till this resistance from the fluid’s elasticity shall bear no considerable proportion to that, which is produced by the power of inactivity thereof[118]. From whence our author draws this conclusion; that the resistance of a body, which moves very swiftly in an elastic fluid, is near the same, as if the fluid were not elastic; provided the elasticity arises from the centrifugal power of the parts of the medium, as before explained, especially if the velocity be so great, that this centrifugal power shall want time to exert it self[119]. But it is to be observed, that in the proof of all this our author proceeds upon the supposition of this centrifugal power in the parts of the fluid; but if the elasticity be caused by the expansion of the parts in the manner of wool compressed, and such like bodies, by which the parts of the fluid will be in some measure entangled together, and their motion be obstructed, the fluid will be in a manner tenacious, and give a resistance upon that account over and above what depends upon its elasticity only[120]; and the resistance derived from that cause is to be judged of in the manner before set down.

15. IT is now time to pass to the second part of this theory; which is to assign the measure of resistance, according to the proportion between the density of the body and the density of the fluid. What is here to be understood by the word density has been explained above[121]. For this purpose as our author before considered two distinct cases of bodies moving in mediums; one when they opposed themselves to the fluid by their power of inactivity only, and another when by ascending or descending their weight was combined with that other power: so likewise, the fluids themselves are to be regarded under a double capacity; either as having their parts at rest, and disposed freely without restraint, or as being compressed together by their own weight, or any other cause.

16. IN the first case, if the parts of the fluid be wholly disingaged from one another, so that each particle is at liberty to move all ways without any impediment, it is shewn, that if a globe move in such a fluid, and the globe and particles of the fluid are endued with perfect elasticity; so that as the globe impinges upon the particles of it, they shall bound off and separate themselves from the globe, with the same velocity, with which the globe strikes upon them; then the resistance, which the globe moving with any known velocity suffers, is to be thus determined. From the velocity of the globe, the time, wherein it would move over two third parts of its own diameter with that velocity, will be known. And such proportion as the density of the fluid bears to the density of the globe, the same the resistance given to the globe will bear to the force, which acting, like the power of gravity, on the globe without intermission during the space of time now mentioned, would generate in the globe the same degree of motion, as that wherewith it moves in the fluid[122]. But if neither the globe nor the particles of the fluid be elastic, so that the particles, when the globe strikes against them, do not rebound from it, then the resistance will be but half so much[123]. Again, if the particles of the fluid and the globe are imperfectly elastic, so that the particles will spring from the globe with part only of that velocity wherewith the globe impinges upon them; then the resistance will be a mean between the two preceding cases, approaching nearer to the first or second, according as the elasticity is more or less[124].

17. THE elasticity, which is here ascribed to the particles of the fluid, is not that power of repelling one another, when out of contact, by which, as has before been mentioned, the whole fluid may be rendred elastic; but such an elasticity only, as many solid bodies have of recovering their figure, whenever any forcible change is made in it, by the impulse of another body or otherwise. Which elasticity has been explained above at large[125].

18. THIS is the case of discontinued fluids, where the body, by pressing against their particles, drives them before itself, while the space behind the body is left empty. But in fluids which are compressed, so that the parts of them removed out of place by the body resisted immediately retire behind the body, and fill that space, which in the other case is left vacant, the resistance is still less; for a globe in such a fluid which shall be free from all elasticity, will be resisted but half as much as the least resistance in the former case[126]. But by elasticity I now mean that power, which renders the whole fluid so; of which if the compressed fluid be possessed, in the manner of the air, then the resistance will be greater than by the foregoing rule; for the fluid being capable in some degree of condensation, it will resemble so far the case of uncompressed fluids[127]. But, as has been before related, this difference is most considerable in slow motions.

19. IN the next place our author is particular in determining the degrees of resistance accompanying bodies of different figures; which is the last of the three heads, we divided the whole discourse of resistance into. And in this disquisition he finds a very surprizing and unthought of difference, between free and compressed fluids. He proves, that in the former kind, a globe suffers but half the resistance, which the cylinder, that circumscribes the globe, will do, if it move in the direction of its axis[128]. But in the latter he proves, that the globe and cylinder are resisted alike[129]. And in general, that let the shape of bodies be ever so different, yet if the greatest sections of the bodies perpendicular to the axis of their motion be equal, the bodies will be resisted equally[130].

20. PURSUANT to the difference found between the resistance of the globe and cylinder in rare and uncompressed fluids, our author gives us the result of some other inquiries of the same nature. Thus of all the frustums of a cone, that can be described upon the same base and with the same altitude, he shews how to find that, which of all others will be the least resisted, when moving in the direction of its axis[131]. And from hence he draws an easy method of altering the figure of any spheroidical solid, so that its capacity may be enlarged, and yet the resistance of it diminished[132]: a note which he thinks may not be useless to ship-wrights. He concludes with determining the solid, which will be resisted the least that is possible, in these discontinued fluids[133].

21. THAT I may here be understood by readers unacquainted with mathematical terms, I shall explain what I mean by a frustum of a cone, and a spheroidical solid. A cone has been defined above. A frustum is what remains, when part of the cone next the vertex is cut away by a section parallel to the base of the cone, as in fig. 86. A spheroid is produced from an ellipsis, as a sphere or globe is made from a circle. If a circle turn round on its diameter, it describes by its motion a sphere; so if an ellipsis (which figure has been defined above, and will be more fully explained hereafter[134]) be turned round either upon the longest or shortest line, that can be drawn through the middle of it, there will be described a kind of oblong or flat sphere, as in fig. 87. Both these figures are called spheroids, and any solid resembling these I here call spheroidical.

22. IF it should be asked, how the method of altering spheroidical bodies, here mentioned, can contribute to the facilitating a ship’s motion, when I just above affirmed, that the figure of bodies, which move in a compressed fluid not elastic, has no relation to the augmentation or diminution of the resistance; the reply is, that what was there spoken relates to bodies deep immerged into such fluids, but not of those, which swim upon the surface of them; for in this latter case the fluid, by the appulse of the anterior parts of the body, is raised above the level of the surface, and behind the body is sunk somewhat below; so that by this inequality in the superficies of the fluid, that part of it, which at the head of the body is higher than the fluid behind, will resist in some measure after the manner of discontinued fluids[135], analogous to what was before observed to happen in the air through its elasticity, though the body be surrounded on every side by it[136]. And as far as the power of these causes extends, the figure of the moving body affects its resistance; for it is evident, that the figure, which presses least directly against the parts of the fluid, and so raises least the surface of a fluid not elastic, and least compresses one that is elastic, will be least resisted.

23. THE way of collecting the difference of the resistance in rare fluids, which arises from the diversity of figure, is by considering the different effect of the particles of the fluid upon the body moving against them, according to the different obliquity of the several parts of the body upon which they respectively strike; as it is known, that any body impinging against a plane obliquely, strikes with a less force, than if it fell upon it perpendicularly; and the greater the obliquity is, the weaker is the force. And it is the same thing, if the body be at rest, and the plane move against it[137].

24. THAT there is no connexion between the figure of a body and its resistance in compressed fluids, is proved thus. Suppose A B C D (in fig. 88.) to be a canal, having such a fluid, water for instance, running through it with an equable velocity; and let any body E, by being placed in the axis of the canal, hinder the passage of the water. It is evident, that the figure of the fore part of this body will have little influence in obstructing the water’s motion, but the whole impediment will arise from the space taken up by the body, by which it diminishes the bore of the canal, and straightens the passage of the water[138]. But proportional to the obstruction of the water’s motion, will be the force of the water upon the body E[139]. Now suppose both orifices of the canal to be closed, and the water in it to remain at rest; the body E to move, so that the parts of the water may pass by it with the same degree of velocity, as they did before; it is beyond contradiction, that the pressure of the water upon the body, that is, the resistance it gives to its motion, will remain the same; and therefore will have little connexion with the figure of the body[140].

25. BY a method of reasoning drawn from the same fountain is determined the measure of resistance these compressed fluids give to bodies, in reference to the proportion between the density of the body and that of the fluid. This shall be explained particularly in my comment on Sir ~IS. NEWTON~’s mathematical principles of natural philosophy; but is not a proper subject to be insisted on farther in this place.

26. WE have now gone through all the parts of this theory. There remains nothing more, but in few words to mention the experiments, which our author has made, both with bodies falling perpendicularly through water, and the air[141], and with pendulums[142]: all which agree with the theory. In the case of falling bodies, the times of their fall determined by the theory come out the same, as by observation, to a surprizing exactness; in the pendulums, the rod, by which the ball of the pendulum hangs, suffers resistance as well as the ball, and the motion of the ball being reciprocal, it communicates such a motion to the fluid, as increases the resistance, but the deviation from the theory is no more, than what may reasonably follow from these causes.

27. BY this theory of the resistance of fluids, and these experiments, our author decides the question so long agitated among natural philosophers, whether all space is absolutely full of matter. The Aristotelians and Cartesians both assert this plenitude; the Atomists have maintained the contrary. Our author has chose to determine this question by his theory of resistance, as shall be explained in the following chapter.

~BOOK II.~
CONCERNING THE
SYSTEM of the WORLD.

CHAP. I.

That the Planets move in a space empty of all sensible matter.

I HAVE now gone through the first part of my design, and have explained, as far as the nature of my undertaking would permit, what Sir ~ISAAC NEWTON~ has delivered in general concerning the motion of bodies. It follows now to speak of the discoveries, he has made in the system of the world; and to shew from him what cause keeps the heavenly bodies in their courses. But it will be necessary for the use of such, as are not skilled in astronomy, to premise a brief description of the planetary system.

2. THIS system is disposed in the following manner. In the middle is placed the sun. About him six globes continually roll. These are the primary planets; that which is nearest to the sun is called Mercury, the next Venus, next to this is our earth, the next beyond is Mars, after him Jupiter, and the outermost of all Saturn. Besides these there are discovered in this system ten other bodies, which move about some of these primary planets in the same manner, as they move round the sun. These are called secondary planets. The most conspicuous of them is the moon, which moves round our earth; four bodies move in like manner round Jupiter; and five round Saturn. Those which move about Jupiter and Saturn, are usually called satellites; and cannot any of them be seen without a telescope. It is not impossible, but there may be more secondary planets, beside these; though our instruments have not yet discovered any other. This disposition of the planetary or solar system is represented in fig. 89.

3. THE same planet is not always equally distant from the sun. But the middle distance of Mercury is between ⅕ and ⅖ of the distance of the earth from the sun; Venus is distant from the sun almost ¾ of the distance of the earth; the middle distance of Mars is something more than half as much again, as the distance of the earth; Jupiter’s middle distance exceeds five times the distance of the earth, by between ⅕ and 1/6 part of this distance; Saturn’s middle distance is scarce more than 9½ times the distance between the earth and sun; but the middle distance between the earth and sun is about 217⅛ times the sun’s semidiameter.

4. ALL these planets move one way, from west to east; and of the primary planets the most remote is longest in finishing its course round the sun. The period of Saturn falls short only sixteen days of 29 years and a half. The period of Jupiter is twelve years wanting about 50 days. The period of Mars falls short of two years by about 43 days. The revolution of the earth constitutes the year. Venus performs her period in about 224½ days, and Mercury in about 88 days.

5. THE course of each planet lies throughout in one plane or flat surface, in which the sun is placed; but they do not all move in the same plane, though the different planes, in which they move, cross each other in very small angles. They all cross each other in lines, which pass through the sun; because the sun lies in the plane of each orbit. This inclination of the several orbits to each other is represented in fig. 90. The line, in which the plane of any orbit crosses the plane of the earth’s motion, is called the line of the nodes of that orbit.

6. EACH planet moves round the sun in the line, which we have mentioned above[143] under the name of ellipsis; which I shall here shew more particularly how to describe. I have there said how it is produced in the cone. I shall now shew how to form it upon a plane. Fix upon any plane two pins, as at A and B in fig. 91. To these tye a string A C B of any length. Then apply a third pin D so to the string, as to hold it strained; and in that manner carrying this pin about, the point of it will describe an ellipsis. If through the points A, B the straight line E A B F be drawn, to be terminated at the ellipsis in the points E and F, this is the longest line of any, that can be drawn within the figure, and is called the greater axis of the ellipsis. The line G H, drawn perpendicular to this axis E F, so as to pass through the middle of it, is called the lesser axis. The two points A and B are called focus’s. Now each planet moves round the sun in a line of this kind, so that the sun is found in one focus. Suppose A to be the place of the sun. Then E is the point, wherein the planet will be nearest of all to the sun, and at F it will be most remote. The point E is called the perihelion of the planet, and F the aphelion. In G and H the planet is said to be in its middle or mean distance; because the distance A G or A H is truly the middle between A E the least, and A F the greatest distance. In fig. 92. is represented how the greater axis of each orbit is situated in respect of the rest. The proportion between the greatest and least distances of the planet from the sun is very different in the different planets.

In Saturn the proportion of the greatest distance to the least is something less, than the proportion of 9 to 8, but much nearer to this, than to the proportion of 10 to 9. In Jupiter this proportion is a little greater, than that of 11 to 10. In Mars it exceeds the proportion of 6 to 5. In the earth it is about the proportion of 30 to 29. In Venus it is near to that of 70 to 69. And in Mercury it comes not a great deal short of the proportion of 3 to 2.

7. EACH of these planets so moves through its ellipsis, that the line drawn from the sun to the planet, by accompanying the planet in its motion, will describe about the sun equal spaces in equal times, after the manner spoke of in the chapter of centripetal forces[144]. There is also a certain relation between the greater axis’s of these ellipsis’s, and the times, in which the planets perform their revolutions through them. Which relation may be expressed thus. Let the period of one planet be denoted by the letter A, the greater axis of its orbit by D; let the period of another planet be denoted by B, and the greater axis of this planet’s orbit by E. Then if C be taken to bear the same proportion to B, as B bears to A; likewise if F be taken to bear the same proportion to E, as E bears to D; and G taken to bear the same proportion likewise to F, as E bears to D; then A shall bear the same proportion to C, as D bears to G.

8. THE secondary planets move round their respective primary, much in the same manner as the primary do round the sun. But the motions of these shall be more fully explained hereafter[145]. And there is, besides the planets, another sort of bodies, which in all probability move round the sun; I mean the comets. The farther description of which bodies I also leave to the place, where they are to be particularly treated on[146].

9. FAR without this system the fixed stars are placed. These are all so remote from us, that we seem almost incapable of contriving any means to estimate their distance. Their number is exceeding great. Besides two or three thousand, which we see with the naked eye, telescopes open to our view vast numbers; and the farther improved these instruments are, we still discover more and more. Without doubt these are luminous globes, like our sun, and ranged through the wide extent of space; each of which, it is to be supposed, perform the same office, as our sun, affording light and heat to certain planets moving about them. But these conjectures are not to be pursued in this place.

10. I SHALL therefore now proceed to the particular design of this chapter, and shew, that there is no sensible matter lodged in the space where the planets move.

11. THAT they suffer no sensible resistance from any such matter, is evident from the agreement between the observations of astronomers in different ages, with regard to the time, in which the planets have been found to perform their periods. But it was the opinion of DES CARTES[147], that the planets might be kept in their courses by the means of a fluid matter, which continually circulating round should carry the planets along with it. There is one appearance that may seem to favour this opinion; which is, that the sun turns round its own axis the same way, as the planets move. The earth also turns round its axis the same way, as the moon moves round the earth. And the planet Jupiter turns upon its axis the same way, as his satellites revolve round him. It might therefore be supposed, that if the whole planetary region were filled with a fluid matter, the sun, by turning round on its own axis, might communicate motion first to that part of the fluid, which was contiguous, and by degrees propagate the like motion to the parts more remote. After the same manner the earth might communicate motion to this fluid, to a distance sufficient to carry round the moon, and Jupiter communicate the like to the distance of its satellites. Sir ~ISAAC NEWTON~ has particularly examined what might be the result of such a motion as this[148]; and he finds, that the velocities, with which the parts of this fluid will move in different distances from the center of the motion, will not agree with the motion observed in different planets: for instance, that the time of one intire circulation of the fluid, wherein Jupiter should swim, would bear a greater proportion to the time of one intire circulation of the fluid, where the earth is; than the period of Jupiter bears to the period of the earth. But he also proves[149], that the planet cannot circulate in such a fluid, so as to keep long in the same course, unless the planet and the contiguous fluid are of the same density, and the planet be carried along with the same degree of motion, as the fluid. There is also another remark made upon this motion by our author; which is, that some vivifying force will be continually necessary at the center of the motion[150]. The sun in particular, by communicating motion to the ambient fluid, will lose from it self as much motion, as it imparts to the fluid; unless some acting principle reside in the sun to renew its motion continually. If the fluid be infinite, this gradual loss of motion would continue till the whole should stop[151]; and if the fluid were limited, this loss of motion would continue, till there would remain no swifter a revolution in the sun, than in the utmost part of the fluid; so that the whole would turn together about the axis of the sun, like one solid globe[152].

12. IT is farther to be observed, that as the planets do not move in perfect circles round the sun; there is a greater distance between their orbits in some places, than in others. For instance, the distance between the orbit of Mars and Venus is near half as great again in one part of their orbits, as in the opposite place. Now here the fluid, in which the earth should swim, must move with a less rapid motion, where there is this greater interval between the contiguous orbits; but on the contrary, where the space is straitest, the earth moves more slowly, than where it is widest[153].

13. FARTHER, if this our globe of earth swam in a fluid of equal density with the earth it self, that is, in a fluid more dense than water; all bodies put in motion here upon the earth’s surface must suffer a great resistance from it; where as, by Sir ~ISAAC NEWTON~’s experiments mentioned in the preceding chapter, bodies, that fell perpendicularly down through the air, felt about 1/860 part only of the resistance, which bodies suffered that fell in like manner through water.

14. Sir ~ISAAC NEWTON~ applies these experiments yet farther, and examines by them the general question concerning the absolute plenitude of space. According to the Aristotelians, all space was full without any the least vacuities whatever. DESCARTES embraced the same opinion, and therefore supposed a subtile fluid matter, which should pervade all bodies, and adequately fill up their pores. The Atomical philosophers, who suppose all bodies both fluid and solid to be composed of very minute but solid atoms, assert that no fluid, how subtile soever the particles or atoms whereof it is composed should be, can ever cause an absolute plenitude; because it is impossible that any body can pass through the fluid without putting the particles of it into such a motion, as to separate them, at least in part, from one another, and so perpetually to cause small vacuities; by which these Atomists endeavour to prove, that a vacuum, or some space empty of all matter, is absolutely necessary to be in nature. Sir ~ISAAC NEWTON~ objects against the filling of space with such a subtile fluid, that all bodies in motion must be unmeasurably resisted by a fluid so dense, as absolutely to fill up all the space, through which it is spread. And lest it should be thought, that this objection might be evaded by ascribing to this fluid such very minute and smooth parts, as might remove all adhesion or friction between them, whereby all resistance would be lost, which this fluid might otherwise give to bodies moving in it; Sir ~ISAAC NEWTON~ proves, in the manner above related, that fluids resist from the power of inactivity of their particles; and that water and the air resist almost entirely on this account: so that in this subtile fluid, however minute and lubricated the particles, which compose it, might be; yet if the whole fluid was as dense as water, it would resist very near as much as water does; and whereas such a fluid, whose parts are absolutely close together without any intervening spaces, must be a great deal more dense than water, it must resist more than water in proportion to its greater density; unless we will suppose the matter, of which this fluid is composed, not to be endued with the same degree of inactivity as other matter. But if you deprive any substance of the property so universally belonging to all other matter, without impropriety of speech it can scarce be called by this name.

15. Sir ~ISAAC NEWTON~ made also an experiment to try in particular, whether the internal parts of bodies suffered any resistance. And the result did indeed appear to favour some small degree of resistance; but so very little, as to leave it doubtful, whether the effect did not arise from some other latent cause[154].

CHAP. II.

Concerning the cause, which keeps in motion the primary planets.

SINCE the planets move in a void space and are free from resistance; they, like all other bodies, when once in motion, would move on in a straight line without end, if left to themselves. And it is now to be explained what kind of action upon them carries them round the sun. Here I shall treat of the primary planets only, and discourse of the secondary apart in the next chapter. It has been just now declared, that these primary planets move so about the sun, that a line extended from the sun to the planet, will, by accompanying the planet in its motion, pass over equal spaces in equal portions of time[155]. And this one property in the motion of the planets proves, that they are continually acted on by a power directed perpetually to the sun as a center. This therefore is one property of the cause, which keeps the planets in their courses, that it is a centripetal power, whose center is the sun.

2. AGAIN, in the chapter upon centripetal forces[156] it was observ’d, that if the strength of the centripetal power was suitably accommodated every where to the motion of any body round a center, the body might be carried in any bent line whatever, whose concavity should be every where turned towards the center of the force. It was farther remarked, that the strength of the centripetal force, in each place, was to be collected from the nature of the line, wherein the body moved[157]. Now since each planet moves in an ellipsis, and the sun is placed in one focus; Sir ~ISAAC NEWTON~ deduces from hence, that the strength of this power is reciprocally in the duplicate proportion of the distance from the sun. This is deduced from the properties, which the geometers have discovered in the ellipsis. The process of the reasoning is not proper to be enlarged upon here; but I shall endeavour to explain what is meant by the reciprocal duplicate proportion. Each of the terms reciprocal proportion, and duplicate proportion, has been already defined[158]. Their sense when thus united is as follows. Suppose the planet moved in the orbit A B C (in fig. 93.) about the sun in S. Then, when it is said, that the centripetal power, which acts on the planet in A, bears to the power acting on it in B a proportion, which is the reciprocal of the duplicate proportion of the distance S A to the distance S B; it is meant that the power in A bears to the power in B the duplicate of the proportion of the distance S B to the distance S A. The reciprocal duplicate proportion may be explained also by numbers as follows. Suppose several distances to bear to each other proportions expressed by the numbers 1, 2, 3, 4, 5; that is, let the second distance be double the first, the third be three times, the fourth four times, and the fifth five times as great as the first. Multiply each of these numbers by it self, and 1 multiplied by 1 produces still 1, 2 multiplied by 2 produces 4, 3 by 3 makes 9, 4 by 4 makes 16, and 5 by 5 gives 25. This being done, the fractions ¼, 1/9, 1/16, 1/25, will respectively express the proportion, which the centripetal power in each of the following distances bears to the power at the first distance: for in the second distance, which is double the first, the centripetal power will be one fourth part only of the power at the first distance; at the third distance the power will be one ninth part only of the first power; at the fourth distance, the power will be but one sixteenth part of the first; and at the fifth distance, one twenty fifth part of the first power.

3. THUS is found the proportion, in which this centripetal power decreases, as the distance from the sun increases, within the compass of one planet’s motion. How it comes to pass, that the planet can be carried about the sun by this centripetal power in a continual round, sometimes rising from the sun, then descending again as low, and from thence be carried up again as far remote as before, alternately rising and falling without end; appears from what has been written above concerning centripetal forces: for the orbits of the planets resemble in shape the curve line proposed in § 17 of the chapter on these forces[159].

4. BUT farther, in order to know whether this centripetal force extends in the same proportion throughout, and consequently whether all the planets are influenced by the very same power, our author proceeds thus. He inquires what relation there ought to be between the periods of the different planets, provided they were acted upon by the same power decreasing throughout in the forementioned proportion; and he finds, that the period of each in this case would have that very relation to the greater axis of its orbit, as I have declared above[160] to be found in the planets by the observations of astronomers. And this puts it beyond question, that the different planets are pressed towards the sun, in the same proportion to their distances, as one planet is in its several distances. And thence in the last place it is justly concluded, that there is such a power acting towards the sun in the foresaid proportion at all distances from it.

5. THIS power, when referred to the planets, our author calls centripetal, when to the sun attractive; he gives it likewise the name of gravity, because he finds it to be of the same nature with that power of gravity, which is observed in our earth, as will appear hereafter[161]. By all these names he designs only to signify a power endued with the properties before mentioned; but by no means would he have it understood, as if these names referred any way to the cause of it. In particular in one place where he uses the name of attraction, he cautions us expressly against implying any thing but a power directing a body to a center without any reference to the cause of it, whether residing in that center, or arising from any external impulse[162].

6. BUT now, in these demonstrations some very minute inequalities in the motion of the planets are neglected; which is done with a great deal of judgment; for whatever be their cause, the effects are very inconsiderable, they being so exceeding small, that some astronomers have thought fit wholly to pass them by[163]. However the excellency of this philosophy, when in the hands of so great a geometer as our author, is such, that it is able to trace the least variations of things up to their causes. The only inequalities, which have been observed common to all the planets, are the motion of the aphelion and the nodes. The transverse axis of each orbit does not always remain fixed, but moves about the sun with a very slow progressive motion: nor do the planets keep constantly the same plane, but change them, and the lines in which those planes intersect each other by insensible degrees. The first of these inequalities, which is the motion of the aphelion, may be accounted for, by supposing the gravitation of the planets towards the sun to differ a little from the forementioned reciprocal duplicate proportion of the distances; but the second, which is the motion of the nodes, cannot be accounted for by any power directed towards the sun; for no such can give the planet any lateral impulse to divert it from the plane of its motion into any new plane, but of necessity must be derived from some other center. Where that power is lodged, remains to be discovered. Now it is proved, as shall be explained in the following chapter, that the three primary planets Saturn, Jupiter, and the earth, which have satellites revolving about them, are endued with a power of causing bodies, in particular those satellites, to gravitate towards them with a force, which is reciprocally in the duplicate proportion of their distances; and the planets are in all respects, in which they come under our examination, so similar and alike, that there is no reason to question, but they have all the same property. Though it be sufficient for the present purpose to have it proved of Jupiter and Saturn only; for these planets contain much greater quantities of matter than the rest, and proportionally exceed the others in power[164]. But the influence of these two planets being allowed, it is evident how the planets come to shift continually their planes: for each of the planets moving in a different plane, the action of Jupiter and Saturn upon the rest will be oblique to the planes of their motion; and therefore will gradually draw them into new ones. The same action of these two planets upon the rest will cause likewise a progressive motion of the aphelion; so that there will be no necessity of having recourse to the other cause for this motion, which was before hinted at[165]; viz, the gravitation of the planets towards the sun differing from the exact reciprocal duplicate proportion of the distances. And in the last place, the action of Jupiter and Saturn upon each other will produce in their motions the same inequalities, as their joint action produces in the rest. All this is effected in the same manner, as the sun produces the same kind of inequalities and many others in the motion of the moon and the other secondary planets; and therefore will be best apprehended by what shall be said in the next chapter. Those other irregularities in the motion of the secondary planets have place likewise here; but are too minute to be observable: because they are produced and rectified alternately, for the most part in the time of a single revolution; whereas the motion of the aphelion and nodes, which continually increase, become sensible in a long series of years. Yet some of these other inequalities are discernible in Jupiter and Saturn, in Saturn chiefly; for when Jupiter, who moves faster than Saturn, approaches near to a conjunction with him, his action upon Saturn will a little retard the motion of that planet, and by the reciprocal action of Saturn he will himself be accelerated. After conjunction, Jupiter will again accelerate Saturn, and be likewise retarded in the same degree, as before the first was retarded and the latter accelerated. Whatever inequalities besides are produced in the motion of Saturn by the action of Jupiter upon that planet, will be sufficiently rectified, by placing the focus of Saturn’s ellipsis, which should otherwise be in the sun, in the common center of gravity of the sun and Jupiter. And all the inequalities in the motion of Jupiter, caused by Saturn’s action upon him, are much less considerable than the irregularities of Saturn’s motion[166].

7. THIS one principle therefore of the planets having a power, as well as the sun, to cause bodies to gravitate towards them, which is proved by the motion of the secondary planets to obtain in fact, explains all the irregularities relating to the planets ever observed by astronomers.

8. Sir ~ISAAC NEWTON~ after this proceeds to make an improvement in astronomy by applying this theory to the farther correction of their motions. For as we have here observed the planets to possess a principle of gravitation, as well as the sun; so it will be explained at large hereafter, that the third law of motion, which makes action and reaction equal, is to be applied in this case[167]; and that the sun does not only attract each planet, but is it self also attracted by them; the force, wherewith the planet is acted on, bearing to the force, wherewith the sun it self is acted on at the same time, the proportion, which the quantity of matter in the sun bears to the quantity of matter in the planet. From the action between the sun and planet being thus mutual Sir ISAAC NEWTON proves that the sun and planet will describe about their common center of gravity similar ellipsis’s; and then that the transverse axis of the ellipsis described thus about the moveable sun, will bear to the transverse axis of the ellipsis, which would be described about the sun at rest in the same time, the same proportion as the quantity of solid matter in the sun and planet together bears to the first of two mean proportionals between this quantity and the quantity of matter in the sun only[168].

9. ABOVE, where I shewed how to find a cube, that should bear any proportion to another cube[169], the lines F T and T S are two mean proportionals between E F and F G; and counting from E F, F T is called the first, and F S the second of those means. In numbers these mean proportionals are thus found.

Suppose A and B two numbers, and it be required to find C the first, and D the second of the two mean proportionals between them. First multiply A by it self, and the product multiply by B; then C will be the number which in arithmetic is called the cubic root of this last product; that is, the number C being multiplied by it self, and the product again multiplied by the same number C, will produce the product above mentioned. In like manner D is the cubic root of the product of B multiplied by it self, and the produce of that multiplication multiplied again by A.

10. IT will be asked, perhaps, how this correction can be admitted, when the cause of the motions of the planets was before found by supposing the sun the center of the power, which acted upon them: for according to the present correction this power appears rather to be directed to their common center of gravity. But whereas the sun was at first concluded to be the center, to which the power acting on the planets was directed, because the spaces described round the sun in equal times were found to be equal; so Sir ~ISAAC NEWTON~ proves, that if the sun and planet move round their common center of gravity, yet to an eye placed in the planet, the spaces, which will appear to be described about the sun, will have the same relation to the times of their description, as the real spaces would have, if the sun were at rest[170]. I farther asserted, that, supposing the planets to move round the sun at rest, and to be attracted by a power, which every where should act with degrees of strength reciprocally in the duplicate proportion of the distances; then the periods of the planets must observe the same relation to their distances, as astronomers find them to do. But here it must not be supposed, that the observations of astronomers absolutely agree without any the least difference; and the present correction will not cause a deviation from any one astronomer’s observations, so much as they differ from one another. For in Jupiter, where this correction is greatest, it hardly amounts to the 3000^{th} part of the whole axis.

11. UPON this head I think it not improper to mention a reflection made by our excellent author upon these small inequalities in the planets motions; which contains under it a very strong philosophical argument against the eternity of the world. It is this, that these inequalities must continually increase by slow degrees, till they render at length the present frame of nature unfit for the purposes, it now serves[171]. And a more convincing proof cannot be desired against the present constitution’s having existed from eternity than this, that a certain period of years will bring it to an end. I am aware this thought of our author has been represented even as impious, and as no less than casting a reflection upon the wisdom of the author of nature, for framing a perishable work. But I think so bold an assertion ought to have been made with singular caution. For if this remark upon the increasing irregularities of the heavenly motions be true in fact, as it really is, the imputation must return upon the asserter, that this does detract from the divine wisdom. Certainly we cannot pretend to know all the omniscient Creator’s purposes in making this world, and therefore cannot undertake to determine how long he designed it should last. And it is sufficient, if it endure the time intended by the author. The body of every animal shews the unlimited wisdom of its author no less, nay in many respects more, than the larger frame of nature; and yet we see, they are all designed to last but a small space of time.

12. THERE need nothing more be said of the primary planets; the motions of the secondary shall be next considered.

CHAP. III.

Of the motion of the MOON and the other SECONDARY PLANETS.