Real Bodies. cumference together with twice the difference of the Motions mean longitudes of the second and third satellites, so that of the the sine of the first difference is the same as the sine of Heavenly double the second difference, but with a contrary sign. The inequality produced by the third satellite in the motion of the second, will therefore have the same sign, and will follow the same law as the inequality observed in that motion. It is, therefore, very probable that this inequality is the result of two inequalities depending on the first and third satellite. If in the course of ages, the preceding relation between the mean longitudes of these three satellites should cease to exist, these two inequalities, at present compounded, would separate, and their respective values might be discovered." The inequality relative to the third satellite in its eclipses, compared with the respective positions of the second and third, offers the same relations with the inequality of the second compared with the respective situations of the two first. There exists then in the motion of the third satellite, an inequality which at its maximum amounts to 268". If we suppose a star whose angular motion is equal to the excess of the mean synodical motion of the second satellite, above twice the mean synodical motion of the third, the inequality of the third satellite will in its eclipses be proportional to the motion of this fictitious star. But in consequence of the relation which exists between the mean longitude of the three satellites, the sine of this motion is the same (except its sign), with that of the motion of the first ficti tious star which we formerly considered. Therefore the inequality of the third satellite in its eclipses has the same period, and follows the same laws, with the inequalities of the two first satellites: such are the laws of the principal irregularities of the three first satellites of Jupiter. 329 Satellites Let us now consider the satellites of Saturn, which of Saturn. are seven in number. The satellites of Saturn have not as yet proved so useful to astronomy or geography as those of Jupiter; principally because they cannot be seen unless very powerful telescopes be used. Five of those satellites were discovered in the year 1685, by Cassini and Huygens, who used telescopes consisting of two simple lenses, hut upwards of 100 feet in length; and those were called 1st, 2d, 3d, &c. reckoning from the planet. Two others were discovered by Dr Herschel in the years 1787 and 1788, and these are smaller and nearer to the planet, on which account they ought to have been called the first and second, at the same time that the other five ought to have been called 3d, 4th, 5th, 6th, and 7th; but, imagining that this might create some confusion in the reading of old astronomical books, the five old satellites have been suffered to retain their numerical names, and the two new satellites are now called the 6th and the 7th; so that the 7th is the nearest to the planet, then comes the 6th, then the Ist; and this is followed by the 2d, 3d, 4th, and 5th. The inclinations of the orbits of the 1st, 2d, 3d, and 4th satellites, to the ecliptic, are from 30° to 31°. That of the 5th is from 17° to 18°. Of all the satellites of the solar system, none, except the 5th of Saturn, has been observed to have any spots, from the motion of which the rotation of the satellite round its own axis might be determined. Then the 5th satellite of Saturn, as Dr Herschel has discovered, turns round its own axis; and it is remarkable, that, like our moon, it revolves round its axis exactly in the same time that it revolves round its primary. The following table states the particulars which have been ascertained with respect to the satellites of Saturn. Real Motions of the Heavenly Bodies. Seventh d. h. m. 0 22 40 46 S. Sixth I 8 53 9 First Second Third Fourth Fifth I 21 18 27 107,000 135,000 170,000 0 57 I 14 I 27 2 17 41 22 217,000 152 79 4 12 25 12 15 22 41 13 18 7 48 8 236 704,000 6 18 54 2,050,000 17 4 330 303,000 Herschel. The planet Herschel, with its six satellites, have been Satellites of entirely discovered by Dr Herschel. The planet itself may be seen with almost any telescope; but its satellites cannot be perceived without the most powerful instruments, and the concurrence of all other favourable circumstances. One of these satellites Dr Herschel found to revolve round its primary in 8d. 17h. 1m. 19 sec.; the period of another he found to be 13d. 11h. 5m. 1.5 sec. The apparent distance of the former from the planet is 33"; that of the second 44". Their orbits are nearly perpendicular to the plane of the ecliptic. The other four satellites were discovered a considerable time after, and of course Dr Herschel has had less time to make observations upon them. They are altogether very minute objects; so that the following parti culars must be considered as being not accurate but probable. Admitting the distance of the interior satel lite to be 25′′.5, its periodical revolution will be 5d. 21h. 25m. 66 "If the intermediate satellite be placed at an equal distance between the two old satellites, or at 38".57, its period will be 10d. 23h. 4m. The nearest exterior satellite is about double the distance of the farthest old one; its periodical time will therefore be about 38d. Jh. 49m. The most distant satellite is full four times as far from the planet as the old second satellite; it will therefore take at least 107d. 16h. 14m. to complete one revolution. All these satellites perform their revolu. tions in their orbits contrary to the order of the signs; : that is, their real motion is retrograde." PART Theory of tion. 330 Motion. 332 Moving forces. 333 Composi tion of forces. PART IV. OF THE THEORY OF UNIVERSAL GRAVITATION. HAVING in the last two parts of this treatise given an account of the apparent and real motions of the heavenly bodies, it only remains for us to compare these motions with the laws established by mathematicians, in order to ascertain the forces that animate the solar system, and to acquire notions of the general principle of gravitation on which they depend. To develope this part of the subject properly, three particulars claim our attention. We must in the first place lay down the laws of motion as established by mathematicians; in the second place, we must apply these laws to the heavenly bodies, which will furnish us with the theory of gravitation; and, in the third place, we must apply this theory to the planetary system, and demonstrate that the whole motions of the heavenly bodies are explicable by that theory, and merely cases of it. These particulars shall be the subject of the three following chapters. CHAP. I. Of the Laws of Motion. THE laws of motion, by which all matter is regulated, and to which it is subject notwithstanding the variety of phenomena which it continually exhibits, constitute the first principles of mechanical philosophy. They will claim a separate place hereafter in this work, under the title of DYNAMICS; but some notions of them are requisite in order to understand the theory of gravitation. We shall satisfy ourselves in this place with the following short sketch. A body appears to us to move when it changes its situation with respect to other bodies which we consider as at rest. Thus in a vessel sailing down a river, bodies are said to be in motion when they correspond successively to different parts of the vessel. But this motion is merely relative. The vessel itself is moving along the surface of the river, which turns round the axis of the earth, while the centre of the earth itself is carried round the sun, and the sun with all its attendant planets is moving through space. This renders it necessary to refer the motion of a body to the parts of space, which is considered as boundless, immoveable, and penetrable. A body then is said to be in motion when it corresponds successively to different parts of space. Matter, as far as we know, is equally indifferent to motion or rest. When in motion it moves for ever unless stopt by some cause, and when at rest it remains so, unless put in motion by some cause. The cause which puts matter in motion is called a force. The nature of moving forces is altogether unknown, but we can measure their effects. Whenever a force acts upon matter it puts it in motion, if no other force prevent this effect; the straight line which the body describes, is called the direction of the force. Two forces may act upon matter at the same time. If their direction be the same, they increase the motion; if their direction be opposite they destroy each other; and the motion is nothing if the two forces be Theary of Universal Gravita equal; it is merely the excess of the one force above the other if the motions be unequal. If the directions of the two forces make with each other any angle whatever, the resulting motion will be in a direction between the two. And it has been demonstrated, that if lines be taken to represent the direction and amount of the forces, if these lines be converted into a parallelogram by drawing parallels to them; the diagonal of that parallelogram will represent the direction and quantity of the resulting motion. This is called the composition of forces. For two forces thus acting together, we may substi tute their result, and vice versa. Hence we may decompose a force into two others, parallel to two axes situated in the same plane, and perpendicular to each other. Thus finding that a body A, fig. 117. has moved from A to C, we may imagine either that the body has been impelled by a single force in the direc tion of AC, and proportionate to the length of AC, or that it has been impelled by two forces at once, viz. by one in the direction of AD, and proportionate to the length of AD; and by another force in the direc tion of AB or DC, and proportionate to AB or DC. Therefore, if two sides of any triangle (as AD and DC) represent both the quantities and the directions of two forces acting from a given point, then the third side (as AC) of the triangle will represent both the quantity and the direction of a third force, which acting from the same point, will be equivalent to the other two, and vice versa. Thus also in fig. 118. finding that the body A has moved along the line AF from A to F in a certain time; we may imagine, 1st, that the body has been impelled by a single force in the direction and quantity represented by AF; or 2dly, that it has been impelled by two forces, viz. the one represented by AD, and the other represented by AE; or thirdly, that it has been impelled by three forces, viz. those represented by AD, AB, and AC; or lastly, that it has been impelled by any other number of forces in any directions; provided all these forces be equivalent to the single force which is represented by AF. This supposition of a body having been impelled by two or more forces to perform a certain course; or, on the contrary, the supposition that a body has been inpelled by a single force, when the body is actually known to have been impelled by several forces, which are, however, equivalent to that single force; has been called the composition and resolution of forces. 334 tion. The knowledge of these principles gives mathema-Resolution ticians an easy method of obtaining the result of any of forces. number of forces whatever acting on a body. For every particular force may be resolved into three others, parallel to three axes given in position, and perpendicular to each other. It is obvious, that all the forces parallel to the same axis are equivalent to a single force, equal to the sum of all those which act in one direction, diminished by the sum of those which act tion. Theory of act in the opposite direction. Thus the body will be Universal acted on by three forces perpendicular to each other: Gravita if the direction of these forces be represented by the sides of a parallelopiped, the resulting force will be represented by the diagonal of that parallelopiped. The indifference of a material body to motion or rest, and its perseverance in either state when put into it, is called the vis inertiæ of matter. This property is considered as the first law of motion. Hence, whenever the state of a body changes, we ascribe the change to the action of some cause: hence the motion of a body when not altered by the action of some new force, must be uniform and in a straight line. 335 Velocity. 336 Accelerat In such uniform motions the space passed over is proportional to the time: but the time employed to describe a given space will be longer or shorter according to the greatness of the moving force. This difference in the time of traversing the same space gives us the notion of velocity, which in uniform motions is the ratio between the space and the time employed in traversing it. As space and time are heterogeneous quantities, they cannot indeed be compared together; it is the ratio between the numbers representing each that constitutes velocity. A unity of time, a second for instance, is chosen, and in like manner a unity of space, as a foot. Thus, if one body move over 20 feet in one second, and another only 10, then the velocity of the first is double that of the second; for the ratio between 20 and I is twice as great as the ratio of 10 to I. When the space, time, and velocity, are represented by numbers, we have the space equal to the velocity multiplied by the time, and the time equal to the space divided by the time. The force by which a body is moved is proportional to the velocity, and therefore is measured by the velocity. This has been disputed by some philosophers, but has been sufficiently established. We shall consi der it, therefore, as a matter of fact, referring the reader for a discussion of the subject to the article Dy NAMICS. When a body is put in motion by forces which not ing forces only act at first, but which continue to act uniformly, it will describe a curve line, the nature of which depends upon the forces which occasion the motion. Gravitation is an instance of a force which acts in this manner. Let us consider it a little. It appears to act in the same manner in a body at rest and in motion. A body abandoned to its action acquires a very small velocity the first instant; the second instant it acquires a new velocity equal to what it had the first instant; and thus its velocity increases every instant in proportion to the time. Suppose a right-angled triangle, one of the sides of which represents the time, and the other the velocity. The fluxion of the surface of the triangle being equal to the fluxion of the time multiplied by that of the velocity, will represent the fluxion of the space. Hence the whole triangle will represent the space described in a given time. But the triangle increasing as the square of either of its sides, it is obvious, that in the accelerated motion produced by gravitation, the velocities increase with the times, and the heights from which a body falls from rest increase as the squares of the times or of the velocities. Hence, if we denote by 1 the space through which a body falls VOL. III. Part I. the first second, it will fall 4 in 2", 9 in 3", and so on; Theory of so that every second it will describe spaces increasing Universul as the odd numbers 1, 3, 5, 7, &c. This important Gravitapoint will perhaps be rendered more intelligible by the following diagram. Let AB, fig. 119. represent the time during which a body is descending, and let BC represent the velocity acquired at the end of that time. Complete the triangle ABC, and the parallelogram ABCD. Also suppose the time to be divided into innumerable particles ei, im, mp, po, &c. and draw ef, ik, mn, &c. all parallel to the base BC. Then, since the velocity of the descending body has been gradually increasing from the commencement of the motion, and BC represents the ultimate velocity; therefore the parallel lines ef, ik, mn, &c. will represent the velocities at the ends of the respective times Ae, Ai, Am, &c. Moreover, since the velocity during an indefinitely small particle of time may be considered as uniform; therefore the right line ef will be as the velocity of the body in the indefinitely small particle of time ei; ik will be as the velocity in the particle of time im, and so forth. Now the space passed over in any time with any velocity is as the velocity multiplied by the time; viz. as the rectangle under that time and velocity; hence the space passed over in the time ei with the velocity ef, will be as the rectangle if; the space passed over in the time im with the velocity ik, will be as the rectangle mk; the space passed over in the time mp with the velocity mn, will be as the rectangle pn, and so on. Therefore the space passed over in the sum of all those times, will be as the sum of all those rectangles. But since the particles of time are infinitely small, the sum of all the rectangles will be equal to the triangle ABC. Now since the space passed over by a moving body in the time AB with a uniform velocity BC, is as the rectangle ABCD, (viz. as the time multiplied by the velocity) and this rectangle is equal to twice the triangle ABC (Eucl. p. 31. B. I.) therefore the space passed over in a given time by a body falling from rest, is equal to half the space passed over in the same time with an uniform velocity, equal to that which is acquired by the descending body at the end of its fall. Since the space run over by a falling body in the time represented by AB, fig. 120. with the velocity BC is as the triangle ABC, and the space run over in any other time AĎ, and velocity DE, is represented by the triangle ADE; those spaces must be as the squares of the times AB AD; for the similar triangles ABC, and ADE, are as the squares of their homologous sides, viz. ABC is to ADE as the square of AB is to the square of AD, (Eucl. p. 29. B. VI.). When a body is placed upon an inclined plane, the force of gravity which urges that body downwards, acts with a power so much less, than if the body descended freely and perpendicularly downwards, as the elevation of the plane is less than its length. The space which is described by a body descending freely from rest towards the earth, is to the space which it will describe upon the surface of an inclined plane in the same time as the length of the plane is to its elevation, or as radius is to the sine of the plane's inclination to the horizon. If upon the elevation BC, fig. 121. of the plane BD, as tion. 337 Of the The time of a body's descending along the whole length of an inclined plane, is to the time of its descending freely and perpendicularly along the altitude of the plane, as the length of the plane is to its altitude; or as the whole force of gravity is to that part of it which acts upon the plane. A body by descending from a certain height to the same horizontal line, will acquire the same velocity whether the descent be made perpendicularly or obliquely, over an inclined plane, or over many successive inclined planes, or lastly over a curve surface. From these propositions, which have been sufficiently established by mathematicians, it follows, that in the circle ABC (fig. 122), a body will fall along the diameter from A to B, or along the chords CB, DB, in exactly the same line by the action of gravity. When a body is projected in any line whatever not perpendicular to the earth's surface, it does not continue in that line, but continually deviates from it, describing a curve, of which the primary line of direction is a tangent. The motion of the body relative to this line is uniform. But if vertical lines be drawn from this tangent to the curve, it will be perceived that its velocity is uniformly accelerated in the direction of these verticals. They are proportional to the squares of the corresponding parts of the tangent. This property shows us that the curve in which the body projected moves is a parabola. The oscillations of the pendulum are regulated likependulum. wise by the same law of gravitation. The fundamental proportions respecting pendulums are the follow ing: If a pendulum be moved to any distance from its natural and perpendicular direction, and there be let go, it will descend towards the perpendicular; then it will ascend on the opposite side nearly as far from the perpendicular, as the place whence it began to descend; after which it will again descend towards the perpendicular, and thus it will keep moving backwards and forwards for a considerable time; and it would continue to move in that manner for ever, were it not for the resistance of the air, and the friction at the point of suspension, which always prevent its ascending to the same height as that from which it lastly began to descend. The velocity of a pendulum in its lowest point is as the chord of the arch which it has described in its descent. The very small vibrations of the same pendulum are performed in times nearly equal; but the vibrations through longer and unequal arches are performed in times sensibly different. As the diameter of a circle is to its circumference, so is the time of a heavy body's descent from rest through half the length of a pendulum to the time of one of the smallest vibrations of that pendulum. It is from these propositions, and the experiments made with pendulums, that the space described by a tion. The late Mr John Whitehurst, an ingenious mem- Gravita ber of the Royal Society, seems to have contrived and performed the least exceptionable experiments relatively on this subject. The result of his experiments shews, that the length of the pendulum which vibrates seconds in London, at 113 feet above the level of the sea, in the temperature of 60° of Fahrenheit's thermometer, and when the barometer is at 30 inches, is 39,1196 inches; whence it follows that the space which is passed over by bodies descending perpendicularly, in the first second of time, is 16,087 feet. This length of a second pendulum is certainly not mathematically exact, yet it may be considered as such for all common purposes; for it is not likely to differ from the truth by more thanth part of an inch. By these propositions, also, the variations of gravity in different parts of the earth's surface and on the tops of mountains has been ascertained. Newton also has shown, by means of the pendulum, that gravity does not depend upon the surface nor figure of a body. 338 The motion of bodies round a centre affords another of central well known instance of a constant force. As the mo-forces tion of matter left to itself is uniform and rectilinear, it is obvious that a body moving in the circumference of a curve, must have a continual tendency to fly off at a tangent. This tendency is called a centrifugal force, while every force directed towards a centre is called a central or centripetal force. In circular motions the central force is equal, and directly contrary, to the centrifugal force. It tends constantly, to bring the body towards the centre, and in a very short interval of time, its effect is measured by the versed sine of the small arch described. Let A (fig. 123.) be the centre of a force. Let a body in B be moving in the direction of the straight line BC, in which line it would continue to move if undisturbed; but being attracted by the centripetal force towards A, the body must necessarily depart from this line BC; and being drawn into the curve line BD, must pass between the lines AB and BC. It is evident, therefore, that the body in B being gradually turned off from the straight line BC, it will at first be convex towards that line, and concave towards A. And that the curve will always continue to have this concavity towards A, may thus appear: In the line BC, near to B, take any point, as E, from which the line EFG may be so drawn as to touch the curve line BD in some point, as F. Now, when the body is come to F, if the centripetal power were immediately to be suspended, the body would no longer continue to move in a curve line, but, being left to itself, would forthwith reassume a straight course, and that straight course would be in the line FG; for that line is in the direc tion of the body's motion of the point F. But the centripetal force continuing its energy, the body will be gradually drawn from this line FG so as to keep in the line FD, and make that line, near the point F, to be concave towards the point A; and in this manner the body may be followed in its course throughout the line BD, and every part of that line be shown to be concave towards the point A. Again, the point A (fig. 124.) being the centre of a centripetal force, let a body at B set out in the direction Gravitation. towards A and convex towards BC, it is more and Theory of more turned off from that line: so that in the point H, Universal the line AK will be more obliquely inclined to the Gravita Theory of rection of the straight line BC, perpendicular to the Universal line AB. It will be easily conceived, that there is no other point in the line BC so near to A as the point B; that AB is the shortest of all the lines which can be drawn from A to any part of the line BC; all others, as AD or AE, being longer than AB. Hence it follows, that the body setting out from it, if it moved in the line BC, would recede more and more from the point A. Now, as the operation of a centripetal force is to draw a body towards the centre of that force, if such a force act upon a resting body, it must necessarily put that body so into motion as to cause it move towards the centre of the force: if the body were of itself moving towards that centre, it would accelerate that motion, and cause it to move faster down; but if the body were in such a motion that it would of itself recede from the centre, it is not necessary that the action of a centripetal power should make it immediately approach the centre from which it would otherwise have receded; the centripetal force is not without ef fect if it cause the body to recede more slowly from that centre than otherwise it would have done. Thus, the smallest centripetal power, if it act on the body, will force it out of the line BC, and cause it to pass in a bent line between BC and the point A, as has been already explained. When the body, for instance, has advanced to the line AD, the effect of the centri"petal force discovers itself by having removed the body out of the line BC, and brought it to cross the line AD somewhere between A and D, suppose at F. Now, AD being longer than AB, AF may also be longer than AB. The centripetal power may indeed be so strong, that AF shall be shorter than AB; or it may be so evenly balanced with the progressive motion of the body that AF and AB shall be just equal; in which case the body would describe a circle about the centre A: this centre of the force being also the centre of the circle. If now the body, instead of setting out in the line BC perpendicular to AB, had set out in another line BG more inclined towards the line AB, moving in the curve line BH; then, as the body, if it were to continue its motion in the line BG, would for some time approach the centre A, the centripetal force would cause it to make greater advances towards that centre: But if the body were to set out in the line BI, reclined the other way from the perpendicular BC, and were to be drawn by the centripetal force into the curve line BR; the body, notwithstanding any centripetal force, would for some time recede from the centre; since some part at least of the curve line BK lies between the line BI and the perpendicular BC. Let us next suppose a centripetal power directed toward the point A (fig. 109.), to act on a body in B, which is moving in the direction of the straight line BC, the line BC reclining off from AB. If from A the straight lines AD, AE, AF, are drawn to the line CB, prolonged beyond B to G, it appears that AD is inclined to the line GC more obliquely than AB, AE more obliquely than AD, and AF than AE; or, to speak more correctly, the angle under ADG is less than that under ABG, that under AEG is less than ADG, and AFG less than AEG. Now suppose the body to move in the curve line BHIK, it is likewise evident that the line BHIK being concave curve line BHIK than the same line AHD is inclined to BC at the point D; at the point I the inclination of the line AI to the curve line will be more different from the inclination of the same line AIE to the line BC at the point IE; and in the points K and F the difference of inclination will be still greater; and in both, the inclination at the curve will be less oblique than at the straight line BC. But the straight line AB is less obliquely inclined to BG than AD is inclined towards DG: therefore, although the line AH be less obliquely inclined towards the curve HB than the same line AHD is inclined towards DG, yet it is possible, that the inclination at H may be more oblique than the inclination at B. The inclination at HI may indeed be less oblique than the other, or they may be both the same. This depends upon the degree of strength wherewith the centripetal force exerts itself during the passage of the body from B to H: and in like manner the inclinations at I and K depend entirely on the degree of strength wherewith the centripetal force acts on the body in its passage from H to K: if the centripetal force be weak enough, the lines AH and AI drawn from the centre A to the body at H and at I, shall be more obliquely inclined to the curve than the line AB is inclined towards BG. The centripetal force may be of such a strength as to render all these inclinations equal; or if stronger, the inclination at I and K will be less oblique than at B; and Sir Isaac Newton has particularly shown, that if the centripetal power decreases after a certain manner without the increase of distance, a body may describe such a curve line, that all the lines drawn from the centre to the body shall be equally inclined to that curve line. tion. 339 round a We must further remark, that if the centripetal Revolution power, while the body increases its distance from the of a body centre, retain sufficient strength to make the lines centre exdrawn from the centre to the body to become at plained. length less oblique to the curve: then, if this diminution of the obliquity continue, till at last the line drawn from the centre to the body shall cease to be obliquely inclined to the curve, and become perpendicular thereto; from this instant the body shall no longer recede from the centre, but in its following motion shall again descend, and describe a curve in all respects like that which it has described already, provided the centripetal power, everywhere at the same distance from the body, acts with the same strength. This return of the body may be proved by the following proposition: That if the body in any place, suppose at I, were to be stopped, and thrown directly backward with the velocity wherewith it was moving forward in that point I, then the body, by the action of the centripetal force upon it, would move back again over the path IHB, in which it had before advanced forward, and would arrive again at the point B in the same space of time as was taken up in its passage from B to I; the velocity of the body at its return from the point B being the same as that wherewith it first set out from that point. The truth of this proposition may be illustrated in the following manner. Suppose, in fig. 110. that a P 2 body |