PHYSICAL.

The whole science of Astronomy may be reduced to two general problems. The first is to express the position of all the heavenly bodies in terms of the time reckoned from a given instant, either in the past or the future duration of the world. The same may be otherwise stated by saying, that the thing required is, to express the position of any one of the heavenly bodies in a function of the time, the time being considered as the only variable quantity, though combined with other known quantities, which enter into the function as the co-efficients of the different terms. This is the most general view of that which is usually called Descriptive, or sometimes Geometrical Astronomy. The solution of this problem enables us to determine for any time the places of the heavenly bodies, relatively to one another, and relatively to any point on the earth's surface. It contains under it an endless variety of subordinate problems, embracing a long series of successive generalizations, from the first observations, to the determination of the orbits of the heavenly bodies, and the final reduction of all that concerns their motions into the form of astronomical tables. The second problem is, to compare the laws of motion in the heavens, as discovered from the preceding investigations, with the laws of motion as already known on the surface of the earth, in order to find out whether or not they are the same; and, if not, in what their difference consists. The solution of this problem constitutes what is called Physical Astronomy; it is the same with inquiring into the causes of the celestial motions: for by causes we mean the general facts concerning the motion of bodies which are observed to take place on the surface of the earth.

Though the first of these two problems goes necessarily before the second, for the solution of which it affords the data, yet, after this solution is obtained, it affords great assistance to many of the researches involved in the first, and exemplifies, in a most remarkable manner, the use of theory in the investigation of facts, and the re-action, as it were, of the second problem on the first.

Taking for granted the solution of the first problem, as given under the article Astronomy in the Encyclopedia, we are now to consider the second, and to explain the manner in which it has been resolved by Newton and the philosophers who have come after him.

The history of the first of these two problems is long and interesting, beginning from the remotest period to which the records or the traditions of mankind have ventured to ascend, and coming down to the present time; and, in the ages to come, it is never likely to know any limit but the moveable instant which separates the past from the future,—as long, at least, as science and civilization are inhabitants of the earth.

The history of the second comes within small compass; because between the first rude effort, and the last refined investigation, there is hardly any intermediate step but one.

VOL. I. PART II. II. That the planets describe ellipses, each of which has one of its foci in the same point, viz. the centre of the sun.

III. That the squares of the times of the revolutions of the planets are as the cubes of their mean distances from the sun.

Section I.

Of the Forces which retain the Planets in their Orbits.

1. If a body gravitating to a fixed centre, have a motion communicated to it in a direction not passing through that centre, it will move in a curve, and the straight line drawn from the body to the centre will describe areas proportional to the times.

Let S (fig. 1.) be the centre to which the body A gravitates, at the same time that a motion is communicated to it in the direction AB. And first, let the gravitating or centripetal force be supposed to act not continually but at intervals, producing instantaneously, at the beginning of each interval, the same velocity that it would have produced by acting continually during the whole of that time; let AC be the space which the body would describe by the action of this force alone; also let AB be the space which it would describe in the same time by the projectile force acting on it alone. It will therefore describe the line AD, the diagonal of the parallelogram contained by AB and AC, and at the end of the first interval will be in D. If, then, no new impulse of gravity were to act on it, it would in the second interval of time go on in the direction AD, and describe DF equal to AD. But, if, at the beginning of the second interval, an impulse of the centripetal force be instantaneously impressed, sufficient to carry the body in that time from D to E, in the line DS, it will describe the line DG, the diagonal of the parallelogram contained by DE and DF. The same is true of the third interval, in which the body will go from G to L, and of every subsequent interval. Join SB, SF, SK, &c. The areas of the triangles ABS, ADS are equal, the triangles being on the same base AS, and between the same parallels AS and BD. For the same reason, the triangles DGS, DFS are equal, and DFS is equal to ADS, because they have equal bases and the same altitude. For the same reason, the triangle SGL is equal to SDG, or to ADS; and the same is true of all the other triangles that are described in the equal intervals of time by the line drawn from the body to the centre S. This holds, however short the intervals may be, and however great their number; and therefore, it is true, when the intervals are infinitely small, and their number infinitely great, that is, when the action of the centripetal force is continual.

But when the intervals of time become infinitely small, the rectilineal figure ADGL passes into a curve. For when these intervals diminish, the lines AB, DF, &c. the lengths of the parallelograms, diminish in the same proportion, but the lines AC, DE, &c. the breadths, diminish in a greater proportion, viz. in that of the squares of these intervals. Hence, the angles which AD, DG, GL, the diagonals, make with the sides AB, DF, GK, continually diminish; and therefore, the angles ADG, DGL, or the angle which each diagonal makes with that which is contiguous, increases without limit, so that, as the diagonals diminish in length, the angles they make with one another become greater than any finite rectilineal angles, and therefore the figure becomes a curve line.

That the lines AC, &c. or the supposed effect of the centrifugal force diminish as the squares of the times, is evident from the laws of the descent of heavy bodies, as explained under the head of Dynamics.

2. Hence, conversely, if a body move in a curve, so that the line drawn from it to a fixed point, describe areas proportional to the times, the body gravitates to that point, or tends continually to descend to it.

For, since it does not move in a straight line, it must be continually acted on by a deflecting force; and the direction of the deflecting force must always pass through the same point, otherwise the areas described about that point would not be proportional to the time.

3. Corollary. The velocities of a body in different points of the curve, which it describes about a centre of force, are inversely as the perpendiculars drawn from the centre to the tangents of the curve at these points. Let ACA', fig. 2., be the curve which a body describes about the centre S. Let AA' and A'a' be two arches of the curve, described in the same indefinitely small portion of time. Join Sa, Sa', then the areas of the triangles ASa, A'Sa' are equal by this proposition. At A and A', draw the tangents AB' A'B', and from S let fall on them the perpendiculars SB and SB'. Because the areas of the triangles ASa, A'Sa', are equal, AA × SB = A'a' × SB'; or AA : A'a' :: SB' : SB; but AA is to A'a' as the velocity of the body describing the curve at A to its velocity at A'; therefore, these velocities are inversely as the perpendiculars SB, SB'.

The straight line AB, (fig. 1.), according to which the projectile motion was impressed on the body, is a tangent to the curve at the point A.

4. On comparing the first and second of these propositions, with the first of Kepler's laws, as just enumerated, it is evident that the primary planets all gravitate to the sun, and that the secondary planets gravitate every one to its primary. The next thing, therefore, is to discover the law observed by this force, or the function of the distance to which it is proportional; and also, whether, in that function, other variable quantities are not involved beside the distance. The general fact that the orbits, or curves described by the planets round the sun are ellipses, may assist in this investigation, and in expressing the velocity of a planet, in terms of the radius vector, or its distance from the sun.

5. Let ADBE (fig. 3.) be the orbit of a planet, S the focus in which the sun is placed, AB the transverse, and DE the conjugate axis, C the centre, and F the superior focus. Let the planet be anywhere at P; draw a tangent to the orbit in P, on which from the foci let fall the perpendiculars SG, FH. Draw also DK touching the orbit in D, and let SK be perpendicular to it. Let the velocity of the planet, when at the mean distance, or at D, be = c, and when at Physical Astronomy.

P = v. Join SP, FP. Then, by the corollary to the last proposition, the velocity at D is to the velocity at P as SG to SK, that is, \( c : c' : SG : DC \), or \( v = \frac{DC}{SG} \).

But because the triangles SGP, FHP, are equiangular, having right angles at G and H, and also the angles SPG, FPH equal, from the nature of the ellipse, SP : PF :: SG : FH, and, therefore, also SP : PF : SG² : SG × GH. But SG × FH = CD², therefore SP : PF :: SG² : CD², and \( \frac{CD^2}{SG^2} = \frac{PF}{SP} \).

Now \( v = c \cdot \frac{CD}{SG} \), or \( v' = c' \cdot \frac{CD}{SG} \), and therefore \( v'^2 = c'^2 \cdot \frac{PF}{SP} \).

Hence, as the distance of a planet from the sun, at any point in its orbit to its distance from the superior focus, so the square of its velocity at its mean distance from the sun to the square of its velocity at the point just mentioned.

6. If SL be taken in the greater axis equal to SP, and FN = PF, so that SN = the transverse axis AB,

\( v^2 = c^2 \cdot \frac{NL}{LS} \). Then as SN is a given line, \( v \) is expressed in terms where SP, the distance from the sun, is the only variable quantity.

If, from the velocity of the revolving body thus expressed, in terms of the distance, a transition can be made to that of a body descending in a straight line, the law of the centripetal force will be easily investigated. This will be facilitated by the following proposition:

An equal approach to the centre of force, produces an equal increase of the square of the velocity, whether the body revolve in a curve about the centre, or descend to it in a straight line. In like manner, equal recesses from the centre of force, produce equal diminutions of the square of the velocities, in whatever lines the bodies move.

Let ABC (fig. 4.) be a curve which a body describes about a centre, S, to which it gravitates, while another body descends in a straight line AS, to that centre. Let BC be any arch of the curve ABC, and let BD, CH, be arches of circles described from the centre S, intersecting the line AS in D and H; the square of the velocity of the body, which describes the arch BC, will be as much increased as the square of the velocity of that which falls through DH.

From the centre S, describe the arch bd, indefinitely near to BD, and draw Ef perpendicular to the arch Bd. Also let the centripetal force at B or D, be called G. Now, the part of this force which is in the direction Bd, and which is employed in accelerating the body moving in that line, is \( G \times \frac{Bf}{BE} \) and the increment of the space being Bb, therefore, \( 2G \times \frac{Bf \times Bb}{BE} \) is the momentary increment of the square of the velocity of the body at B. But \( Bf \times Bb = BE^2 \), because \( BEb \) is a right-angled triangle, and Ef the perpendicular on the hypotenuse. Therefore \( 2G \times \frac{Bf \times Bb}{BE} = 2G \times \frac{BE^2}{BE} = 2G \times BE = 2G \times Dd \). But \( 2G \times Dd \) is the momentary increment of the square of the velocity of the body at D, or the increment of that square, while the body falls from D to d. These momentary increments therefore are equal; and as the same may be shown for the next and every subsequent instant, the whole increase of the square of the velocities of the bodies in moving over BO and DH are equal.

If the bodies moved in the opposite directions, the one from C to B, and the other from H to O, it would be proved, in the same manner, that the squares of their velocities would be equally diminished.

7. Hence it is evident, that, if the velocities of the revolving and of the falling body, are equal in any one instance when they are equally distant from the centre, their velocities will always be equal when they are equally distant from that point. For equal quantities receiving equal increments continue equal.

8. Suppose now, that a planet revolves in the elliptical orbit APB (fig. 3.), it will have at A, the higher apsis, a velocity \( = c \times \sqrt{\frac{AF}{AS}} \), or (if AN in the axis produced, be taken equal to AF) \( = c \times \sqrt{\frac{AN}{AS}} \).

Let a body at A begin to descend towards S with the same velocity, then if SL = SP, the velocity of the planet at P, will be the same with that of the falling body at L. But the velocity of the planet at P is \( c \times \sqrt{\frac{PF}{PS}} = c \times \sqrt{\frac{NL}{SL}} \), therefore, a body descending from A, and falling directly to the sun, under the action of the same centripetal force which urges the planet, would at any point L in its fall have its velocity \( = c \times \sqrt{\frac{LN}{LS}} \). Hence at the point N, its velocity would be equal to 0, or the body must begin to fall from N, in order that its velocity may be everywhere equal to that which the planet has in its orbit, when at the same distance from the sun.

The law, therefore, according to which the planets gravitate is such, that any body under the influence of the same force, and falling direct to the sun, will have its velocity at any point equal to a certain velocity, multiplied into the square-root of the distance it has fallen through, divided by the square-root of the distance between it and the sun's centre.

This is a fact with respect to the law of gravity in the solar system, of which, though there be no direct example, yet is it no less certain than the ellipticity of the planetary orbits, of which it is a necessary consequence.

From the law thus found to regulate the velocity of bodies falling in straight lines to the sun, the law of the force by which that velocity is produced, may be derived by help of reasoning which is quite elementary.

Let C (fig. 5.) be the centre to which the falling body gravitates, A the point from which it begins to fall, Physical and let its velocity at any point B, be to its velocity in the point G, which bisects AC as \( \frac{AB}{BC} = 1 \); it is required to find the law of the force with which the body gravitates to C.

Let DEF be a curve, such, that if AD be an ordinate or a perpendicular to AC, meeting the curve in D, and BE any other ordinate, AD is to BE, as the force at A to the force at B; then will twice the area ABED be equal to the square of the velocity which the body has acquired in B. If, therefore, the velocity at B be v, that at the middle point b being c, \( v = c \sqrt{\frac{AC}{BC}} \), by hypothesis, and, therefore, \( 2ABED = c^2 \cdot \frac{AB}{BC} \); and since \( AB = AC - BC \), \( 2ABED = c^2 \cdot \frac{AC - BC}{BC} = c^2 \left( \frac{AC}{BC} - 1 \right) \).

For the same reason \( 2BbED = c^2 \left( \frac{AC}{bc} - 1 \right) \) and, therefore, the difference of those areas, or \( 2BbE \), that is, \( 2EB \times Bb = c^2 \left( \frac{AC}{bc} - \frac{AC}{BC} \right) = c^2 \cdot \frac{AC \cdot Bb}{BC^2} \). Therefore, dividing by Bb, \( 2EB = c^2 \cdot \frac{AC}{BC^2} \), or \( EB = c^2 \cdot \frac{AG}{BC} \); now \( c^2 \) and AG are given, therefore, EB is inversely as BC, that is, the centripetal force at B is inversely as the square of BC, the distance from the centre of force. In the planetary system, therefore, the force with which any planet gravitates to the sun, varies in the inverse ratio of the square of the distance of the planet from the sun's centre.

10. The line CG is the same with the mean distance of the planet, in an orbit of which AC is the length of the transverse axis, and if the gravitation at that distance \( F \), and the mean distance itself \( a \), \( F = c^2 \cdot \frac{a^2}{c^2} = \frac{c^2}{a} \), or \( aF = c^2 \).

Let it next be required, the elliptic orbit of a planet being given, to find the time in which the planet will revolve round the sun.

If \( a \) be the mean distance, or the semitransverse axis, \( b \) the semiconjugate, then \( ab = \) the area of the orbit. But as \( c \) is the velocity at the mean distance, or the elliptic arch which the planet moves over in a second when it is at D, the vertex of the conjugate axis, therefore \( \frac{1}{2} bc \) is the area described in that second by the radius vector; and since this area is the same for every second of the planet's revolution, therefore the area of the orbit divided by \( \frac{1}{2} bc \) will give the number of seconds in which the revolution is completed, which is therefore \( \frac{ab}{\frac{1}{2} bc} = \frac{2ab}{c} \), or since \( c^2 = aF \), the time of a revolution \( = \frac{2\pi a}{\sqrt{aF}} = 2\pi \sqrt{\frac{a}{F}} \).

11. Hence it is easy to compare the times of the revolutions of any two planets of which the mean distances are known. Let \( t \) and \( t' \) be the times of revolution for two different planets, of which the mean distances are \( a \) and \( a' \), and the gravitation at those distances \( F \) and \( F' \), and, by what has just been shown \( t : t' :: a^{\frac{3}{2}} : a'^{\frac{3}{2}} \), or \( t^2 : t'^2 :: \frac{a}{F} : \frac{a'}{F'} \). But \( F : F' :: \frac{a^2}{a'^2} : \frac{a^2}{a'^2} \), or \( t^2 : t'^2 :: a^3 : a'^3 \), that is, the squares of the times of revolution of any two planets, are as the cubes of their mean distances from the sun. Thus the third law of Kepler is explained by the conclusions deduced from the other two.

12. The share which this third law has in establishing the principle of universal gravitation, does not seem to have been always clearly apprehended. From the elliptical orbit of a planet, it is fairly inferred that, over all the circumference of that orbit, gravitation is inversely as the square of the distance from the centre of the sun. That force is shown to be \( \frac{c^2a}{x^2} \), and the same is true of every individual planet; but whether \( c^2a \) was a constant quantity, or one which retained the same value through the whole planetary system, could not be known without comparing the periods of different planets, with their distances from the sun. It was indeed highly probable, that \( c^2a \) was a given quantity, or the same for every part of our system; but it could not be considered as a thing demonstrated till the evidence of the third law was introduced.

13. These laws hold of the secondary planets relatively to their primary, just as with the primary planets relatively to the sun. Each system of secondary planets, however, has a different numerator to the fraction which expresses gravity; that is, the quantity \( c^2a \) is the same for all the satellites of Jupiter, but it is a different quantity from that which belongs to the satellites of Saturn, and different from that which belongs to the primary planets. The quantity \( c^2a \) seems therefore to depend on the central body of each system of planets, and the precise nature of this connection requires to be farther examined into.

14. Let the centripetal force tending to a given centre \( s \) be inversely as the squares of the distances, and let the intensity of that force at any given distance from the centre be also given; then, if a body be projected from a given point, with a given velocity, and in a given direction, it is required to determine the comic section which it will describe.

Let the semitransverse, or the mean distance to be found \( = a \), the semiconjugate \( = b \), the velocity at the distance \( a = c \), and at the given distance \( d \) let the centripetal force \( = f \); and first let the direction of the initial motion be at right angles to the radius vector, so that the point of projection is either the higher or the lower apsis. Let the velocity of the projection \( = v \), and the radius vector at the point of projection \( = r \).

Because the areas described in equal times are equal, \( bc = rv \); and if \( F \) denote the centripetal force at the distance \( a \), \( c^2 = aF \), and \( F = \frac{c^2}{a} \). But \( F = \frac{d^2f}{a^3} \), therefore \( \frac{c^2}{a} = \frac{d^2f}{a^3} \), and \( c = d \sqrt{\frac{f}{a}} \). Hence,

by substituting for \( c \), \( bd \sqrt{\frac{f}{a}} = rv \), and \( b^2d^2f = ar^2v^2 \). But \( b^2 = AS \times SB = r(2a - r) \), wherefore \( r(2a - r)d^2f = ar^2v^2 \), and \( a = \frac{rd^2f}{2d^2f - rv^2} \).

Thus \( a \), the semitransverse axis, and therefore the transverse axis itself is found, and thence with the focus \( S \) and the apsis \( A \), the conic section may be described.

15. The conic section will be a circle, when \( a = r \), that is, when \( 2d^2f - v^2r = d^2f \), or when \( d^2f = v^2r \), and \( v^2 = \frac{d^2f}{r} \).

16. But if \( 2d^2f = rv^2 \), or \( v^2 = \frac{d^2f}{r} \), the denominator vanishes, and \( r \) becomes infinite, so that the trajectory is a parabola, of which the focus is \( S \), the vertex \( A \), and the parameter \( 4r \). The square of the velocity which determines the trajectory to be a parabola, is, therefore, double of the square of the velocity which determines it to be a circle.

17. When \( 2d^2f > v^2r \), the value of \( a \) is affirmative, and the conic section is an ellipse, and this ellipsis has its higher apsis at \( A \), if \( v^2 < \frac{d^2f}{r} \), but when \( v^2 > \frac{d^2f}{r} \), and less than \( \frac{2d^2f}{r} \), \( A \) is the lower apsis.

18. When \( v^2 \) goes beyond this last limit, or when \( v^2 > 2d^2f \), the value of \( a \) is negative, and the trajectory become a hyperbola.

19. Next, let the body be projected from \( B \) (fig. 6.) with the velocity \( v \), in the direction \( BD \), oblique to \( BS \). Find the distance from which a body must fall to acquire at \( B \) the velocity \( v \), and let \( OB \) taken in \( SB \) produced, be equal to this distance; then is \( SO \) equal to the transverse axis. Let \( BE \) be drawn, making with \( BD \) the same angle that \( SB \) makes with \( BG \), and let \( BE = BO \), then is \( E \) the higher focus. Produce \( SE \) to \( N \), so that \( SN = SO \), and bisect \( EN \) in \( A \), then is \( A \) the higher apsis, and if \( SP \) be made equal to \( EA \), \( P \) is the lower apsis, and \( AP \) the transverse axis, and therefore the foci and the transverse axis being given, the elliptic orbit may be described.

20. From what has been shown at Art 9. it is evident, that the primary planets gravitate to the sun with forces that are inversely as the squares of the distances, and that the secondary gravitate toward the primary, according to the same law. This inference, however, does not apply exactly to the moon, which, being a single satellite, does not by comparison with any other, afford a proof that, in bodies revolving round the earth, the squares of the periodic times are as the cubes of the distances. The centripetal force at the moon, however, from our knowledge of her periodic time, may be compared with the force of gravity at the earth's surface, and will determine whether that force decrease as we recede from the earth in the inverse ratio of the squares of the distances.

Let \( a \) be the distance of the moon from the centre of the earth, \( r \) the radius of the earth, \( g \) the velocity acquired by a heavy body at the earth's surface, by falling during one second; let \( t \) be the period of the moon's revolution in seconds, and \( c \) the velocity of her motion. Then by sect. 14, \( ac^2 = r^2g \), and, therefore, \( c = r \sqrt{\frac{g}{a}} \).

Now, the circumference of the circle described by the moon is \( 2\pi a \), and this, divided by \( c \), gives the periodic time of the moon in seconds, or \( \frac{2\pi a}{c} \times \sqrt{\frac{a}{g}} = t \), so that \( t^2 = \frac{4\pi^2a^3}{rg} \), and \( a^3 = \frac{rgt^2}{4\pi^2} \). Hence \( \frac{a^3}{r^3} = \frac{gt^2}{4\pi^2r} \), and \( \frac{a}{r} = \left( \frac{gt^2}{4\pi^2r} \right)^{\frac{1}{3}} \). Hence as \( g \), \( r \), and \( t \), are known, we may find \( \frac{a}{r} \) or the ratio of the moon's distance to the radius of the earth, which, if it come out the same that it is known to be, from observations of the moon's parallax, will prove, that the force which retains the moon in her orbit, is the same that causes bodies to fall at the surface of the earth, but diminished in the same ratio that the square of the moon's distance is greater than the square of the radius of the earth.

Now \( g = 32.166 \) feet, \( r = 3481279.4 \) fathoms, or \( 20887676.4 \) feet, and \( t = 2360591.5 \) seconds. Hence \( \frac{a}{r} = 60.218 \).

Now the mean equatorial parallax of the moon is found by observation, \( = 57'11'' \), from which the mean distance in semidiameters of the equator is found \( = 60.121 \).

But it is in mean semidiameters of the earth, that the moon's distance is given in the former computation; therefore, to reduce the last measure to the same scale, it must be increased by a 600th part, as the mean radius of the globe is about that much less than the radius of the equator; the distance 60.121, then becomes 60.221, which agrees with the former number to the small fraction .008 of the earth's radius.

Thus, from the theory of gravity, combined with the time of the moon's sidereal revolution, her distance from the earth is found to be less than a twenty-thousandth part of the whole.

21. It is, therefore, a general proposition, derived from the most rigorous induction, that the primary planets gravitate to the sun, and the secondary planets to the primary, with forces which are inversely as the squares of the distances. But since, in all communication of motion, the reaction is equal to the action, when a planet gravitates to the sun, analogy forces us to conclude, that the sun gravitates to the planet, in such Physical attraction in the direction EC is \( \frac{2\pi m}{x^3} \times c = \frac{2\pi m}{x^3} \), the fluent of which taken so as to vanish when DC = 0, or when \( x = a \), is \( 2\pi m - \frac{2\pi ma}{x} = 2\pi m \left(1 - \frac{a}{x}\right) \) = the attraction of the circle DKH.

Therefore, when \( x = AE \), the whole attraction of the plate, or the whole force which it exerts on the particle E, is \( 2\pi m \left(1 - \frac{EC}{ED}\right) \).

23. Next, let ADB (fig. 8.) be a circle of which the centre is C, and E a particle of matter anywhere in the diameter AB produced. Draw ED to any point D in the circumference; draw also DC, and let DF be at right angles to AB. Then, when the whole figure revolves about EB, the semicircle ADB will generate a sphere, and DF a circle perpendicular to the plane ABD, and having its centre in F. If all the particles of the sphere attract the particle E with forces inversely as the squares of their distances from it, then, by the last proposition, the attraction of the circular plate, of which the centre is F, will be \( 2\pi m \left(1 - \frac{EF}{ED}\right) \).

Let CE = \( a \), AC = \( r \), ED = \( x \), EF = \( y \), and the attraction above will be \( 2\pi m \left(1 - \frac{y}{x}\right) \); and if \( x \) and \( y \) be variable, the quantity \( m \) in this formula, or the thickness of the circular plate will be \( y \), and therefore the attraction of the plate \( = 2\pi y \left(1 - \frac{y}{x}\right) \).

In order to integrate this quantity, \( y \) must be expressed in terms of \( x \), or \( x \) in terms of \( y \).

Now, because AE = \( a - r \), and AF = \( y - a + r \), FB = \( 2r - y + a - r = a + r - y \), and AF × FB = \( (y - a + r)(a + r - y) = r^2 - a^2 + 2ay - y^2 \). Hence \( r^2 - a^2 + 2ay = x^2 \), or \( y = \frac{x^2 - a^2 + 2ay}{2a} \), and therefore \( \dot{y} = \frac{\dot{x}}{a} \). By substituting these values of \( y \) and \( \dot{y} \) in the expression for the attraction of the circular plate, that attraction

\[ = \frac{2\pi m}{a} \left(1 - \frac{a^2 - r^2 + x^2}{2ax}\right) \]

\[ = \frac{2\pi m}{a} \left(\frac{a^2 - a^2 + r^2 - x^2}{a^2}\right) + C. \]

But the attraction of this circular plate may be considered as the fluxion of the attraction of the spherical segment, generated by the revolution of the arch AD, and therefore the fluent of the above fluxionary quantity will give the attraction of that segment. Now, Here C must be so determined, that the fluent may be equal to 0, when the arch AD = 0; or when \( x = y = a - r \). Therefore \( C = \frac{1}{a^2} (ax^2 - ax^2 + \frac{3}{2} r^2) \); and the attraction is found \( = \frac{4\pi r^5}{3a^2} \). But \( \frac{4\pi r^3}{3} \) is the solid content of the sphere; therefore the attraction of the sphere, on any particle E, is as the quantity of matter in the sphere, divided by the square of the distance of its centre from E. Hence also the sphere attracts any particle without it, as if all its matter were united in its centre. The sphere, it is also obvious, would attract another sphere, just in the ratio of its quantity of matter, divided by the distance of the centres of the spheres.

24. Thus, supposing that the particles of matter attract one another, with forces which are inversely as the squares of the distances, it is certain that the spherical bodies composed of these particles would do so likewise, or would attract one another with forces directly as their quantities of matter, and inversely as the squares of the distances of their centres. Since, therefore, it has been found that round or spherical bodies, such as the sun and the planets, do attract other bodies with forces that are inversely as the squares of the distances, it is reasonable to suppose, that these bodies are composed of particles gravitating towards one another, or attracting one another with forces inversely as the squares of the distances. Gravitation, therefore, is not to be considered as a force residing in the centres of the planets, but as a force belonging to all the particles of matter, and as universally diffused throughout the universe.

And, as it has been shown, that, between spherical bodies constituted of such particles, the force of attraction is as the quantity of matter in the attracting body, divided by the square of the distance between its centre and that of the attracted body; if \( m \) be the mass or quantity of matter in the former body, and \( x \) the distance of the centres, \( \frac{m}{x^2} \) is the value of \( f \), the accelerating force with which it attracts the other body.

25. Hence the masses of any two planets which have bodies revolving round them, may be compared with one another. Let \( a \) and \( a' \) be the mean distances at which satellites revolve about any two planets, \( m \) and \( m' \) the quantities of matter in those planets, \( t \) and \( t' \) their periods of revolution; it has been shown that

\[ t = \frac{2\pi a^{\frac{3}{2}}}{a'^{\frac{3}{2}}} = \frac{2\pi a^{\frac{3}{2}}}{m^{\frac{3}{2}}} \text{, and consequently } t : t' : \frac{a^{\frac{3}{2}}}{m^{\frac{3}{2}}} : \frac{a'^{\frac{3}{2}}}{m'^{\frac{3}{2}}} \]

The masses, therefore, of any two planets are as the cubes of the mean distances at which their satellites revolve, divided by the squares of the periodic times of those satellites.

26. In this way, the masses of the four planets which have satellites may be compared with one another, and with the mass of the sun.

When this calculation is undertaken with the most correct data, it is found that the mass of the sun being

| Planet | Mass | |------------|------| | Earth | 1 | | Jupiter | 1/1067.1 | | Saturn | 1/3359.4 | | Uranus | 1/19504 |

Or if we make the mass of the Earth 1, that of the Sun = 329630; of Jupiter, 308.9; of Saturn, 98.122; and of Uranus, 16.94. From this also may be derived the densities of the sun, and of the four planets just mentioned. Seen from a distance equal to the mean radius of the earth's orbit, the diameter of the sun subtends an angle of 1914"; that of the earth would subtend 17".4; of Jupiter, 186".8; of Saturn, 177".7; and of Uranus, 74". The real diameters, therefore, are in the proportion of these numbers, and the bulk in the proportion of their cubes. By dividing the quantities of matter by the bulks, we have the densities; and if that of the earth be 4.713, which is its mean density, that of water being = 1.

| Planet | Density | |------------|---------| | Sun | 1.1775 | | Earth | 4.713 | | Jupiter | 1.1673 | | Saturn | 0.4055 | | Uranus | 1.0348 |

The mean density of the earth, in respect of water, is here taken from the experiments made at Schehallien. Phil. Trans. 1811, p. 376.

27. It has been already observed, that, because action is always accompanied by an equal reaction, when the sun attracts a planet, the planet also attracts the sun, and that the velocities impressed on the bodies by their mutual attraction are in the inverse ratio of their masses.

In consequence of this mutual action, the sun and the planet must both move, and must describe orbits about their common centre of gravity, the only point which the mutual action of those bodies has no tendency to put in motion.

In the solar system, therefore, the centre of gravity of the whole is the focus about which all the orbits are described. Thus if C be that centre (fig. 9.), S the sun, and P a planet, while P describes the elliptic arch PP' about C, S describes the arch SS' similar to PP', and having to it the ratio that SC has to CP, or the ratio which the mass of the planet has to the mass of the sun.

The true orbits, therefore, are all described about the same immovable point; but the orbit of any of the planets may be referred to the sun as a centre, by supposing a body placed in that centre equal to the sum of the masses of the sun and of the planet. This is true, because the bodies appear to approach one another, or to recede from one another, with a force that is equal to the sum of the forces with which they tend towards their centre of gravity. Thus if S denote the mass of the sun, and E that of the earth, the distances from the centre being CP and CS, the orbit which each of the two bodies will appear to describe round the other, is that which would be described about an immoveable centre C, with a centripetal force \( \frac{S + P}{SP^2} \).

Thus we have arrived at the knowledge of the principle of universal gravitation, a power which pervades all nature, extending to an unlimited distance, and determining the condition of every body in the universe, at any instant from its state in the former instant, and from the relations in which it stands to all other bodies. Whether this force can be explained upon any principle more general than itself, is yet undecided, though, from the bad success which has hitherto attended all attempts towards that object, it seems probable that such explanation is not within the reach of the human understanding. This much, however, we know with certainty, that the law of Gravity, as just announced, may be considered as a very accurate expression of all the phenomena of the planetary motions.

Section II.

Of the Forces which disturb the Elliptic Motion of the Planets.

1. Of the force by which the Sun disturbs the motion of the Moon round the Earth.

The motion of the moon in an elliptic orbit round the earth is disturbed by the action of the sun; the gravity of the moon to the earth is increased at the quadratures, and diminished at the syzygies, and the areas described by the radius vector, except near these four points, are never exactly proportional to the times.

Let ADBC (fig. 10.) be the orbit, nearly circular, in which the moon M revolves, in the direction CADB round the earth E. Let S be the sun, and let SE, the radius of the earth's orbit, be taken to represent the force with which the earth gravitates to the sun. Then \( \frac{1}{SE^2} : \frac{1}{SM^2} : : SE : SM = \) the force by which the sun draws the moon in the direction MS. Take MG \( = \frac{SE^3}{SM^2} \), and let the parallelogram KF be described, having MG for its diagonal, and having its sides parallel to EM and ES. The force MG may be resolved into the two MF, and MK, of which MF, directed towards E, the centre of the earth, increases the gravity of the moon to the earth, and does not hinder the areas described by the radius vector from being proportional to the times.

The other force MK draws the moon in the direction of the line joining the centres of the sun and earth. It is, however, only the excess of this force above the force represented by SE, or that which Physical draws the earth to the sun, which disturbs the relative position of the moon and earth. This is evident; for if KM were just equal to ES, no disturbance of the moon relatively to the sun could arise from it. If, then, ES be taken from MK, the difference HK is the whole force in the direction parallel to SE, by which the sun disturbs the relative position of the moon and earth. Now, if in MK, MN be taken equal to HK, and if NO be drawn perpendicular to the radius vector EM produced, the force MN may be resolved into two, MO and ON; the first, lessening the gravity of the moon to the earth, and the second, being parallel to the tangent of the moon's orbit in M, accelerates the moon's motion from C to A, retards it from A to D, and so alternately in the other two quadrants.

Thus the whole solar force directed to the centre of the earth is composed of the two parts MF and MO, which are sometimes opposed to one another, but which never affect the uniform description of the areas about E. Near the quadratures, the force MN vanishes, and the force MF, which increases the gravity of the moon to the earth, coincides with CE or DE. As the moon approaches the conjunction at A, the force MO prevails over MF, and lessens the gravity of the moon to the sun. In the opposite point of the orbit, when the moon is in opposition at B, the force with which the sun draws the moon is less than that with which the sun draws the earth, so that the effect of the solar force is to separate the moon and earth, or to increase their distance; that is, it is the same as if, conceiving the earth not to be acted on, the sun's force drew the moon in the direction from E to B. This force is negative, therefore, in respect of the force at A, and the effect in both cases is to draw the moon from the sun, in a direction perpendicular to the line of the quadratures.

The analytical values of these forces must be found, if a more exact estimate is to be made of their effects. Let SE, considered as constant, \( = a \); EM, the radius vector of the moon's orbit, \( = r \); the angle CEM \( = \phi \); the mass of the sun \( = m \). The force SE then, which retains the earth in its orbit, is \( \frac{ma}{a^2} \), and the sun's force in the direction SM, if ML be drawn perpendicular to ES, is \( \frac{m}{SM^2} = \frac{m}{SL^2 + LM^2} = \frac{m}{(a - r \sin \phi)^2 + r^2 \cos^2 \phi} = \frac{m}{a^2 - 2ar \sin \phi + r^2} \). The part of this force, which is in the direction ES or MK, is therefore \( \frac{ma}{(a^2 - 2ar \sin \phi + r^2)} \). By raising the denominator to the power \( -\frac{3}{2} \), rejecting the terms which involve the higher powers of \( r \), and multiplying \( ma \) by those that are left, the force MK comes out \( = \frac{m}{a^2} \left( 1 + \frac{3r}{a} \sin \phi \right) \) nearly. Taking away from this ES or MH \( = \frac{m}{a^2} \), there remains the force MN \( = \frac{m}{a^2} \times 3r \sin \phi \). Hence the force $MO = \frac{m}{a^3} \cdot 3r \sin \varphi^2$; and the force $NO$ at right angles to the radius vector $= \frac{m}{a^3} \cdot 3 \sin \varphi \times \cos \varphi = \frac{m}{a^3} \cdot \frac{3r}{2} \sin 2\varphi$. Also the force $MF = \frac{mr}{a^3}$, rejecting such terms as involve the square and higher powers of $r$. Therefore $MF - MO$, or the whole solar force increasing or diminishing at any point, the moon's tendency to the earth is $\frac{mr}{a^3} (1 - 3 \sin \varphi^2)$.

30. At the quadratures where $\varphi$ vanishes, this force is $\frac{mr}{a^3}$, and is affirmative, increasing the moon's gravity to the earth. At a certain point, between the quadratures and the syzygies, when $3 \sin \varphi^2 = 1$, or $\sin \varphi = \frac{1}{\sqrt{3}}$, that is, when $\varphi = 35^\circ 15' 5''$, the same force becomes equal to 0, and at this point in each quadrant, the moon's gravity to the earth is neither increased nor diminished. From these points to the conjunction and opposition, as $\sin \varphi$ increases, the quantity $1 - 3 \sin \varphi^2$ is negative, and the moon's gravity to the earth suffers a diminution. At the opposition and conjunction $\sin \varphi = 1$, and therefore the disturbing force is $-\frac{2mr}{a^3}$, and by this quantity the moon's gravitation is diminished.

The mean quantity of the force which is thus continually directed to, or from the centre of the earth, may also be easily computed. Since for any point in the moon's orbit, where the radius vector makes the angle $\varphi$ with the line of the quadratures, this force $= \frac{mr}{a^3} (1 - 3 \sin^2 \varphi)$; multiplying by $\dot{\varphi}$, we have $\frac{mr}{a^3} (\dot{\varphi} - 3 \dot{\varphi} \sin^2 \varphi)$, the fluent of which $= \frac{mr}{a^3} (-\frac{1}{2} \dot{\varphi} + \frac{3}{2} \sin \varphi \cos \varphi)$, and this, when $\varphi$ is an entire circumference or four right angles, is $\frac{mr}{a^3} \times -\frac{\pi}{2}$. This is the sum of the forces for an entire revolution, and when divided by $\pi$, gives the mean force $-\frac{mr}{2a^3}$, which being negative, shows that the solar force, on the whole, diminishes the gravitation of the moon to the earth.

Thus it appears, that at the quadratures the gravity of the moon to the earth is increased by a quantity equal to the mass of the sun, multiplied into the radius of the moon's orbit, and divided by the cube of the sun's distance from the earth: at the syzygies it is diminished by twice this quantity; and the effect on the whole is a diminution by one half of the same quantity.

If $\frac{mr}{a^3}$ be reduced to its numerical value, supposing the moon's gravitation to the earth to be 1, it is found $= \frac{1}{174}$ nearly. Hence the mean disturbing force of the sun is nearly $= \frac{1}{583}$ of the moon's gravity to the earth.

31. From the disturbing force of the sun arise two kinds of inequalities which affect the lunar motions. The one kind affects the form and position of the orbit of that planet, the other immediately affects the motion of the planet in the orbit. When any of these inequalities is expressed numerically, the measure it so obtained is, in the language of astronomy, called an Equation.

32. The line in which the plane of the moon's orbit cuts the ecliptic is called the line of the nodes, and this line is subject to change its position continually, in such a manner as to go back annually $19^\circ 18' 11''$. The way in which this effect is produced may be thus conceived. That part of the solar force which is parallel to the line joining the centres of the sun and earth, is not in the plane of the moon's orbit except when the sun itself is in that plane, or when the line of the nodes, being produced, passes through the sun. In all other cases, it is oblique to the plane of the orbit, and may be resolved into two forces, one of which is at right angles to that plane, and is directed towards the ecliptic. This force of course draws the moon continually towards the ecliptic, or produces a continual deflection of the moon from the plane of her own orbit towards that of the earth. Hence the moon meets the plane of the ecliptic sooner than it would have done if that force had not acted. At every half revolution, therefore, the point in which the earth meets the ecliptic advances in a direction contrary to that of the moon's motion, or contrary to the order of the signs. This retrograde motion is such that, in its mean quantity, it amounts to $19^\circ 18' 11''$ in a year. The manner of deducing it from the theory of gravity is explained by Newton, Princip. Lib. iii. Prop. 31. This motion is subject to many inequalities, depending on the changes in the quantity and direction of the solar force.

If the earth and the sun were at rest, the effect of the deflecting force just described would be to produce a retrograde motion of the line of the nodes till that line was brought to pass through the sun, and of consequence the plane of the moon's orbit to do the same, after which they would both remain in their position, there being no longer any force tending to produce a change in either. The motion of the earth carries the line of the nodes out of this position, and produces, by that means, its continual retrogradation.

33. The same force produces a small variation in the inclination of the moon's orbit, giving it an alternate increase and decrease within very narrow limits.

34. The line of the moon's apsides, that is, the longer axis of her orbit, has also a slow angular motion round the centre of the earth, which is progressive, or in the same direction with the motions of the moon itself. To conceive the cause of this phenomenon, we may begin with supposing the moon at the lower apsis, or perigee; and, it is plain, if that planet were urged by no other force than its gravitation to the earth, that after the radius vector had moved over 180°, the moon would be at the higher apsis, where its motion would be at right angles to the said radius. But as the mean disturbing force in the direction of the radius vector, tends, on the whole, to diminish the gravitation of the moon to the earth, the portion of her path, described in any instant, will be less bent or deflected from the tangent, than if this disturbing force did not exist. The actual path of the moon, therefore, will be less incurved than the elliptic orbit, that would be described under the influence of gravity alone, and will not be brought to intersect the radius vector at right angles, till this last have moved over an arch of more than 180°.

Hence, the solar force, by lessening the moon's gravity to the earth, produces a progressive motion in the apsides of the lunar orbit. If the disturbing force had increased the moon's gravity to the earth, the motion of the apsides would have been in antecedentia.

The precise quantity of the motion of the apsides is not however easily determined. Newton left this part of the lunar theory almost untouched, and the only investigation he has entered into having any reference to it, assigned a measure only the half of that which is known from observation to belong to it. Several years afterwards, when Clairaut attempted a more accurate investigation of the lunar inequalities than was to be obtained by the method which Newton had followed, he at first encountered the same difficulty, and found that his calculus gave the motion of the apogee only half of the real quantity. He began, therefore, to suspect that gravity does not follow so simple a law as the inverse of the squares of the distances; but one which is more complex, and such as cannot be expressed but by a formula of two terms. The second of these terms he supposed to be inversely as the fourth power of the distance, and proceeded to inquire what must be the co-efficient of that term, in order to make this new supposition represent the true motion of the apsides. In order to this, he found it necessary to carry his approximation farther than he had yet done, and to include terms which he had before neglected. When these terms were included, he found, that the coefficient he was seeking for came out equal to 0; the necessary inference from which was, that there was no such term; and that the Newtonian law of gravity, when the approximation was carried far enough, was quite sufficient to explain the motion of the apsides. This doubt concerning the law of gravity terminated, therefore, in the confirmation of it.

35. When the doubts excited by Clairaut's first attempt were made known, and before his final solution of the difficulty was fully understood, there were several mathematicians who, still following the method of Newton, endeavoured to deduce the true motion of the moon's apsides from the theory of gravity. Among those who were most successful in this attempt, were Dom. Walmesley, and afterward Dr Matthew Stewart, Professor of Mathematics in the University of Edinburgh. In his Mathematical and Physical Tracts, he has demonstrated this remarkable theorem.

Let \( r \) be the radius of the moon's orbit, supposing it to be a circle, and the moon to be acted on only by \( F \), her gravity to the earth. If the mean disturbing force by which the sun diminishes the moon's gravity be \( f \); then will the greatest distance to which the moon will recede from the earth be \( r \times \frac{F-3f}{F-5f} \); and the cube of this distance will be to the cube of \( r \), in the duplicate ratio of the angle described by the moon, from one apsis to the same apsis again, to four right angles.

Hence the angle described by the radius vector from one apsis to the same apsis, is \( 360^\circ \times \left( \frac{F-3f}{F-5f} \right)^{\frac{3}{2}} \).

This proposition, which is demonstrated by Dr Stewart in the 4th of his Tracts, in a manner somewhat prolix, on account of his rigorous adherence to the manner of the ancient geometry, but in a way perfectly clear and elementary, is employed by him to deduce the mean disturbing force, from the motion of the apsides as ascertained by observation. But when the mean disturbing force is known from other phenomena, the same proposition may be employed to deduce the motion of the apsides from that force. Accordingly, if the disturbing force be taken \( = \frac{1}{357.7} \), the motion of the apsides will come out \( = 3^\circ 1' 20'' \) for a sidereal revolution of the moon, very near the quantity actually observed.

36. Having determined the sun's mean disturbing force from the motion of the apsides, Dr Stewart proceeded to determine, from the former of these, the sun's distance from the earth. The result of a very nice investigation gave the sun's parallax \( 6''.9 \), a quantity that is no doubt too small, and makes, of course, the sun's distance too great. It is indeed but an inconsiderable part of the sun's disturbing force into which the parallax enters as an element, and therefore any deduction founded on it must be liable to this inaccuracy, that a small error in the data will produce a great one in the result.

37. After the inequalities which are conceived as belonging to the moon's orbit, come those which directly affect the place of the moon in that orbit. The most considerable of these, after what is called the equation of the centre, arising from the elliptic figure of the lunar orbit, and independent of all disturbance, is the equation or inequality called the excentric, which was discovered by the Greek astronomers. This depends on the position of the transverse axis of the moon's orbit in respect of the line of the syzygies. When that axis is in the line just mention- because the quantity by which the solar force diminishes the gravitation of the moon in the syzygies is *ceteris paribus* proportional to her distance from the earth, it is greatest when the moon is in the apogee, and least when in the perigee. In this situation of the orbit, therefore, the greatest diminution is made from the quantity of the moon's gravitation, which is already the least, and the least from that which is already the greatest; the gravitation at the perigee, and therefore the difference, is augmented, and the orbit appears to have its eccentricity increased. When the line of the apsides is in the quadratures, the contrary happens: the gravitation at the apogee is most augmented, and at the perigee least; the difference is therefore diminished, and the eccentricity of the lunar orbit seems also to be diminished. This is conformable to observation; and when the evocation is accurately deduced from the theory of gravitation, it appears

\[ (1^\circ 25' 5'' \frac{3}{4}) \sin \left( 2(\zeta - \odot) - a \right) \]

where \( \zeta \) is the mean longitude of the moon, \( \odot \) that of the sun, and \( a \) the mean anomaly of the moon counted from the perigee.

38. The moon's variation is an inequality which was discovered by Tycho, and found to depend on the angular distance of that planet from the sun. It is derived from that part of the sun's disturbing force which is at right angles to the radius vector, and which accelerates the motion of the moon from the quadratures to the syzygies, and retarding it from the syzygies to the quadratures. The effect of this force is found, from the theory of gravity, to be represented by three terms, which, if \( \Delta \) be the angular distance of the moon from the sun, are,

\[ + (33' 47'') \sin 2 \Delta \] \[ + (0' 2'') \sin 3 \Delta \] \[ + (0' 14'') \sin 4 \Delta \]

39. The lunar inequality, called the annual equation, arises from the variation of the sun's disturbing force according to the place which the earth occupies in its orbit. It is shown above that the sun's disturbing force is *ceteris paribus* as the cube of his distance from the earth, so that, when the earth is in its perihelion, this force is the greatest, and at the aphelion the least, its effect varying at the same rate with the equation of the sun's centre, or having everywhere the same ratio to that equation. Hence this equation is nearly \((11' 12'') \times \sin\) sun's mean anomaly, with a contrary sign to the equation of the sun's centre.

40. These inequalities are all phenomena which were observed before the explanation of them was known. To them may be added a fourth inequality, known by the name of the moon's acceleration. It appeared to Astronomers as a continual increase in the velocity of the moon, or in the rate of her mean motion, amounting to about \(10''\) in a century; and its effect, like that of all other constant accelerations, accumulating as the squares of the times. It did not seem to be periodical, like the other lunar inequalities, but to be a constant increase of the velocity, and a corresponding diminution of the periodical time of the moon, which must in the end change entirely the relation of that body to the earth.

It is but within these few years that La Place discovered it to be a periodic inequality, though requiring, in order to accomplish the series of its changes, a length of time which science has not yet ventured to calculate. For many centuries to come it may be expressed by this formula, taking \( n \) to denote the number of centuries reckoned from the year 1700, viz.

\[ 10'' 1816127 \times n^2 + 0.018538441 \times n^3. \]

The first term includes all that was known from observation previously to the discovery of La Place. This, however, must be considered not as the true form of the equation, which must include the sines or cosines of certain angles, but merely a provisional formula, to serve till the true one can be rigorously assigned.

This inequality has, in its cause, a great affinity to the annual equation.

Whatever changes the form of the earth's orbit has an effect on the disturbing force of that body on the moon, which is in the inverse ratio of the cube of the distance between the sun and earth. But it is found that though the mean distance remains invariable, the eccentricity of the earth's orbit changes, on account of the action of the other planets, and in fact has been diminishing, from a more remote antiquity than that to which the history of astronomy extends. From this cause La Place has deduced the supposed acceleration of the moon's mean motion.

41. All these inequalities have been pointed out by observations, and have been explained in the most satisfactory manner by the principle of universal gravitation. But when all these were reduced into equations and arranged in tables, yet the places of the moon calculated from them were never quite exact; and there seemed a cause of error or a mass of small inequalities unknown in their magnitude and form, to which this inaccuracy was to be ascribed, and which operated, as it may be said, like a mist which concealed the true place of the moon from the calculator, and prevented his results from agreeing completely with those of the observer. The most likely way to discover these inequalities, if they arose from gravity, was to push the approximation to the moon's place still farther, and to try if the terms hitherto neglected in the approximation would not, when taken into account, afford a complete analysis of the circle of confusion, which might be said to surround the moon on all occasions.

The problem on which mathematicians now entered, and which Clairaut, already mentioned, Euler, and D'Alcambert, all three resolved nearly about the same time, has been called the Problem of Three Bodies. The thing proposed is, Three bodies which attract one another with forces directly as their quantities of matter, and inversely as the squares of their distances, being given, and any motions whatever being impressed on them, to find the orbits which they will describe round their common centre of gravity. It is, however, only in certain cases, that this general problem admits of solution, and one of these is, when one of the bodies is at a vast distance from the other two. This is exactly the case with the moon and earth in respect of the sun, the orbit of the earth being nearly the same. as if there only existed the sun and earth; and the orbit of the moon relatively to the earth being nearly the same as if there were only the moon and earth. This solution of the problem, however, in this direct way, leads to far more exact conclusions than can be obtained from the more simple but more indirect method which Newton followed. The general view which leads to the most exact estimate of the merit of the two solutions, is, that the motions of the moon, when analytically and fully expressed, necessarily form a number of different series, each of which converges with more or less rapidity. The prosecution of the direct method, allows the terms of these series to be computed to an indefinite extent, or till the quantities omitted are too small to affect observation. The method of Newton can go no farther than to compute the first, or at most a few of the leading terms of each of the series. Its accuracy is therefore limited; that of the other knows no limits. Though this be a true estimate of the value of the methods, yet that of the original inventor possesses infinite merit, as having first led the way to this arduous investigation, and as still serving to carry the imagination better along with it than the other, and to keep the mechanical principles more directly in view.

The complete solution of the problem of the three bodies, has accordingly discovered a great number of new equations, each individually small, which would sometimes nearly destroy one another, and, at other times, having many of them the same sign, would accumulate to a considerable amount. This was the triumph of the theory, and the strongest evidence of its truth. The effect of these irregularities varied so much, and depended on so many elements, that it may be doubted whether the most accurate and most constant observation would ever have enabled astronomers to discover their precise quantities, and to separate them from one another.

The tables of the moon in the state to which they are now (1816) brought, contain twenty-eight equations for the longitude, twelve for the latitude, and thirteen for the horizontal parallax of the moon. Of the first of these, twenty-three have been deduced from theory alone; of the second nine, and of the third eleven. This applies to the tables of De Bürgh; those since published by Burckhardt contain more equations, and are still more accurate.

2. Of the Disturbance in the Motion of the Primary Planets, produced by their action on one another.

42. It is evidently necessary, in this inquiry, to know the quantities of matter in the different planets, or which comes to the same, the intensity of the attraction of each at a given distance from its centre. With respect to those planets which have satellites, the Earth, Jupiter, Saturn, and Uranus, their masses or quantities of matter have been already determined. The masses of Venus and Mars have been estimated by M. La Place, from the effects which they appear to produce on the earth's motion. The mass of Mercury has been estimated on the supposition that the densities of that planet and of the earth are inversely as their mean distances from the sun. This law holds with respect to the Earth, Jupiter, and Saturn, and analogy renders it probable that the same law includes the other planets. Thus, the mass of the Sun being 1, that of Mercury is $\frac{1}{2025810}$, of Venus $\frac{1}{383137}$, and of Mars $\frac{1}{1846082}$, the masses of the other planets being as already stated.

43. The effects of the action of the planets on one another is more difficult to be investigated than the effects of the sun's action on the moon, because the disturbing forces are not only more numerous, but because the distance of the disturbing from the disturbed body, is not so great that the quantities divided by higher powers of that distance, can be so safely rejected. The general principle, however, according to which the solar action on the moon was resolved into forces either in the direction of the radius vector, or at right angles to it, is applicable to both questions.

Thus supposing $P$ and $P'$ (fig. 11.) to be two planets revolving in orbits, nearly circular, about the sun at $S$, in order to find how the motion of $P'$ is affected by the action of $P$, let $PP'$, $PS$, and $P'S$ be drawn, and let the line $A$ denote the force with which $P$ attracts a particle of matter at the distance $PS$, then the force with which it attracts a particle at the distance $PP'$, will be $A \times \frac{PS^2}{PP'^2}$. Let $P'R = A \times \frac{PS^2}{PP'^2}$, and if $P'R$ be resolved into two forces, $P'M$ and $P'N$, the one in the direction of the radius vector $P'S$, and the other parallel to $PS$; take $NO = A$, then the remaining forces $OP'$ and $P'M$ are those which disturb the motion of $P'$, as was proved in the case of the moon. The former of these, $OP'$, may be resolved into $OQ$ and $P'Q$, of which $P'Q$ diminishes the gravity of the planet to the sun, and $OQ$ accelerates its motion in a direction perpendicular to the radius vector. Therefore, as the force $P'M$ always increases the planet's gravity to the sun, $P'M - P'Q$ is the whole force increasing or diminishing the gravity of $P'$ to $S$; and the force directly employed in increasing or diminishing the angular motion of $P$ about $S$, is $OQ$ or $P'T$. The analytical values of these quantities may be found, as in the theory of the moon, though not with equal simplicity, because $SP$ cannot always be supposed great in respect of $SP'$.

44. In consequence of these actions, the orbit of every planet may be considered as an ellipse, which is undergoing slowly certain changes in its form, magnitude, and position, or in what are called its elements. By the elements of the orbit of any heavenly body, are meant the quantities that determine the position and magnitude of that orbit, viz. the position of the line of the nodes, the inclination of the plane of the orbit to the plane of the ecliptic, the position of the line of the apsides, the eccentricity, and the mean distance. These are all quantities independent of one another, and from them may be deduced all other circumstances with respect to the elliptic orbit. Of these five elements, which would be invariable, if the planet only gravitated to the sun, all, except the mean distance, are subject to slow, but perpetual changes. 45. The line of the nodes in every one of the planets has a retrograde angular motion, which goes on continually, and of which the amount, when calculated as due to each planet, agrees very well with observation. The plane of the orbit also varies its inclination by certain small periodical changes which alternately increase and diminish it, as in the case of the moon. The line of the apsides, from the same cause as in the planet just mentioned, has a continued motion forward, or according to the order of the signs. Thus, in Mercury, the node goes back about $9^\circ$ annually. The aphelion goes forward about $6^\circ 2''$, and the inclination of the orbit in the course of a century, increases about $2^\circ 1''$, which, in the course of succeeding ages, will be compensated by an equal diminution, so as to preserve it always nearly of the same quantity. In the same planet the equation of the centre, which depends on the eccentricity, increases about $20^\circ 1\frac{1}{2}$ in a century, indicating a small increase of eccentricity. These variations in the orbit of Mercury, arise from the action of Venus, the Earth, Mars, Jupiter, and Saturn; the effects of the first of these planets, on account of its vicinity, being by much the most considerable. The mean distance, however, of Mercury from the Sun, does not, any more than that of the other planets, undergo any change whatever.

46. Similar conclusions apply also to the orbit of Venus. The orbit of the earth also is subject to similar changes, the line of the apsides moving forward annually at the rate of $11^\circ 8''$, in respect to the fixed stars. The earth's eccentricity is also diminishing, and the secular variation of the greatest equation of the centre is $-17''.66$.

The motion of the earth is subject to another inequality on account of the action of the moon; for, to speak strictly, it is not the centre of the earth, but the centre of gravity of the moon and earth, which describes equal areas in equal times about the centre of the sun. It is evident, that, on this account, the earth will be sometimes advanced before, and sometimes will fall behind, the point which describes the circumference of the ellipse, in conformity with the general law of the planetary motions. From the same cause also, as the moon does not move in the plane of the ecliptic, the earth will be forced out of that plane, in order to preserve a position diametrically opposite to the moon. These irregularities, however, are inconsiderable. By observers on the earth's surface, they are transferred to the sun, but in an opposite direction. The sun, therefore, has a motion in longitude, by which he alternately advances before the point which describes the elliptic orbit in the heavens, and falls behind it, and also a motion in latitude, by which he alternately ascends above, and descends below the plane of the ecliptic. As the mass of the moon, however, is not more than $\frac{1}{68.5}$ part of that of the earth, the distance of the centre of gravity of the moon and earth, from the centre of the latter, must be less than a semidiameter, and therefore the inequality thus produced in the sun's longitude, must be less than his horizontal parallax. The alteration in latitude can hardly amount to a second. This inequality in the sun's motion is called the menstrual parallax, and was first mentioned by Smeaton, Phil. Trans. 1768.

47. In the orbit of Mars, the node moves backward $23^\circ 1'$ annually, and the line of the apsides moves forward $16^\circ 3'$, both in respect of the fixed stars. The eccentricity of the orbit is diminishing, and the secular variation of the greatest equation of the centre is $-37''$.

In the case of this planet, however, the elliptic orbit is not only changed by these quantities, but the place of the planet in that orbit is sensibly affected by the action of Venus, Jupiter, and the Earth. The effect of the action of Venus is expressed by this formula $5''.7 \sin (\text{long. } \varphi - 3 \text{ long. } \delta)$; of the earth, $7''.2 \sin (\text{long. } \odot - \text{long. } \varphi)$. Several inequalities are produced by Jupiter.

48. The inequalities of the small planets Vesta, Juno, Ceres, and Pallas, have not yet been computed; the disturbances which they must suffer from Mars and Jupiter, are no doubt considerable, and, on account of their vicinity, though their masses are small, they may somewhat disturb the motions of one another. Their action on the other bodies in the system is probably insensible.

As two of these planets have nearly the same periodic time, they must preserve nearly the same distance, and the same aspect with regard to one another. This offers a new case in the computation of disturbing forces, and may produce equations of longer periods than are yet known in our system.

The motion of the apsides and the change of eccentricity in the orbits of Jupiter and Saturn are chiefly produced by their action on one another, but a part also depends on the action of the other planets. The node of Jupiter moves backward annually $19\frac{1}{2}'$, and his aphelion forward $6''.58$. The secular change in the inclination of the orbit is $27''$, and in the first and last of these inequalities the action of Venus has the principal share. The equation of the centre increases $56\frac{1}{2}''$ in a century, of which nearly the whole arises from the action of Saturn. In Saturn again the node goes back at the rate of $21''$ annually, and the aphelion forward at the rate of $16''$, the secular change of the inclination is $-23''$, and the secular diminution of the equation of the centre $1' 50''$.

There is, beside these variations in the orbits, an inequality in the motion of each of these planets, which it has been found very difficult to explain, and has only lately been fully accounted for, according to the theory of gravity, by the profound investigations of La Place. These inequalities are both of a long period, viz. $918.76$ years, which is the time that they take to run through all their changes. If $n$ express a number of years reckoned from the beginning of 1750, $S$ the mean longitude of Saturn, and $I$ that of Jupiter, reckoned from the same time, then the equation which must be applied to the mean longitude of Jupiter, or the amount of this inequality, is

$$+ (20' 49''.5 - n \times 0''.042733) \times \sin (5 S - 2 I + 5' 34''.8 - n \times 58''.88)$$

and that which must be applied to $S$, is

$$-(48' 44''. - n \times 0''.1) \times \sin (5 S - 2 I + 5' 34''.8 - n \times 58''.88).$$ These two equations are to one another nearly in the ratio of 3 to 7. The reason of the long period above mentioned is, that the argument $5S - 21 = n \times 58^\circ.38$, requires all that time to increase from 0° to 360°.

Uranus, on account of his great distance, suffers hardly any disturbance in his motion, but from Saturn and Jupiter. The node moves backward at the rate of $34^\circ.4$ annually, and the aphelion forward at that of $2^\circ.55$. The eccentricity is increasing, and the secular variation of the greatest equation of the centre is $11^\circ.03$.

There is also an inequality in the longitude of this planet depending on the action of Saturn. If $S$ be the longitude of this last planet, $U$ the longitude of Uranus, and $A$ the longitude of the aphelion of Saturn, the inequality in question amounts to $2^\circ.30' \times \sin(S - 2U + A)$.

Of all these inequalities, and of many other smaller ones which theory has discovered, it must be observed that they are periodical, each returning after a certain time to run through the same series of changes which it had formerly exhibited.

Another general remark is, that one element in every planetary orbit, viz. the mean distance, is exempted from all change; and since on the mean distance depends the time of revolution, that time remains also unchanged. From the invariability of the mean distance, and the periodical revolution of all the inequalities, it follows that the actual condition of the planetary system can never deviate far from the mean, about which, we may, therefore, conceive it to be continually making small oscillations, which, in the course of ages, compensate one another; and, therefore, produce nothing like disorder or permanent change. It is in this manner that the stability of the planetary system is provided for by the wisdom of its Author.

Comets, in describing their elliptic orbits round the sun, have been found to be disturbed by the action of the larger planets, Jupiter and Saturn; but the great eccentricity of their orbits makes it impossible, in the present state of mathematical science, to assign the quantity of that disturbance for an indefinite number of revolutions, though it may be done for a limited portion of time, by considering the orbit as an ellipse, the elements of which are continually changing. This is the method of La Grange, and is followed in the *Mecanique Celeste*, Part II, chap. 9.

Dr Halley, when he predicted the return of the comet of 1682, took into consideration the action of Jupiter, and concluded that it would increase the periodic time of the comet a little more than a year; he therefore fixed the time of the re-appearance to the end of the year 1758, or the beginning of 1759. He professed, however, to have made this calculation hastily, or, as he expresses it, *lexi calamo*. (Synopsis of the Astronomy of Comets.)

The effects both of Jupiter and Saturn on the return of the same comet were afterwards calculated more accurately by Clairaut, who found that it would be retarded 511 days by the action of the former planet, and 100 by the action of the latter; in consequence of which, the return of the comet to its perihelion would be on the 15th of April 1759. He admitted, at the same time, that he might be out a month in his calculation. The comet actually reached its perihelion on the 13th of March, just 33 days earlier than was predicted; affording, in this way, a very striking verification of the theory of gravity, and the calculation of disturbing forces. The same comet may be expected again about the year 1835.

In some instances, the effect which the planets produce on the motion of comets are far more considerable than in this example. A comet which was observed in 1770 had a motion which could not be reconciled to a parabolic orbit, but which could be represented by an elliptic orbit of no great eccentricity, in which it revolved in the space of five years and eight months. This comet, however, which had never been seen in any former revolution, has never been seen in any subsequent one. On tracing the path of this comet, Mr Burekhardt found that, between the year 1767 and 1770 it had come very near to Jupiter, and had done so again in 1779. He therefore conjectured, that the action of Jupiter may have so altered the original orbit as to render the comet for a time visible from the earth; and that the same cause may have so changed it, after one revolution, as to restore the comet to the same region in which it had formerly moved. This is the greatest instance of disturbance which has yet been discovered among the bodies of our system, and furnishes a very happy, as well as an unexpected, confirmation of the theory of gravity.

Though the comets are so much disturbed by the action of the planets, yet it does not appear that their re-action produces any sensible effect. The comet of 1770 came so near to the earth as to have its periodic time increased by two days .246 according to La Place's computation, and if it had been equal in mass to the earth it would have augmented the length of the year by not less than two hours and forty-eight minutes. It is certain that no such augmentation took place, and therefore that the disturbing force by which the comet diminished the gravity of the earth is insensible, and the mass of the comet, therefore, less than $\frac{3}{10}$th of the mass of the earth. The same comet also passed through the middle of the satellites of Jupiter. Hence it is reasonable to conclude, that no material or even sensible alteration has ever been produced in our system by the action of a comet.

3. Of the disturbances which the satellites of Jupiter suffer from their action on one another.

51. The same resolution of the forces by which one satellite acts upon another, into two, one directed to the centre of the primary, and the other at right angles to it, serves to explain the irregularities which had been observed in their motions, and to reduce under known laws, several other inequalities, of which the existence only is indicated by observation.

An instance of this we have in the very remarkable relation which takes place between the mean motions of the first three satellites; the mean motion of the first satellite, together with twice the mean motion of the third, being equal to three times the mean motion of the second. La Place has shown that, if the primitive mean motions of these satellites were nearly in this proportion, the mutual action of these bodies on one another must, in time, have brought about an accurate conformity to it.

The first satellite moves nearly in the plane of Jupiter's equator, and has no eccentricity, except what is communicated from the third and fourth, the irregularities of one of these small planets producing similar irregularities in those that are contiguous to it. The first satellite has beside an inequality, chiefly produced by the action of the second, and circumscribed by a period of 437.659 days.

52. The orbit of the second satellite moves on a fixed plane, to which it is inclined at an angle of 27° 13', and on which its nodes have a retrograde motion, so that they complete a revolution in 29.914 years. The motion of the nodes of this satellite, is one of the principal data used for determining the masses of the satellites themselves, which are so necessary to be known for computing their disturbances. This satellite has no eccentricity but that which it derives from the action of the third and fourth. The third satellite moves on a fixed plane, to which it is inclined at an angle of 12° 20', and its nodes make a tropical revolution backwards in 141.739 years. The equator of Jupiter is inclined to the plane of his orbit at an angle of 3° 5' 27". The fixed planes on which the planes of the orbits move are determined by theory, and could not have been discovered by observation alone.

The orbit of the third satellite is eccentric, but appears to have two distinct equations of the centre; one which really arises from its own eccentricity, and another which theory shows to be an emanation, from the equation of the centre of the fourth satellite. The first equation is referable to an apsis, which has an annual motion of 2° 36' 39" forward in respect of the fixed stars; the second equation is referable to the apsides of the fourth satellite.

These two equations may be considered as forming one equation of the centre, referable to an apsis that has an irregular motion. The two equations coincided in 1682, and the sum of their maxima was 13° 16'. In 1777, the equations were opposed, and their difference was 5° 6'.

The last two inequalities were perceived by Mr Wargentin by observation alone, but their exact amount, and the law which they observe in their changes, he could not discover. The orbit of the fourth satellite moves on a fixed plane, to which it is inclined at an angle of 14° 58', and its nodes complete a sidereal revolution backward in 551 years. The fixed plane on which the orbit moves is inclined at an angle of 24° 33' to the equator of Jupiter; the orbit is sensibly elliptical, and its greater axis has an annual motion of 43° 35'. The motion of this axis is one of the principal data from which the quantities of matter of the different satellites have been determined.

If the mass of Jupiter be supposed unity, the mass of the 1st Satellite = .0000173281. Of the 2d = .0000232355. Of the 3d = .0000854972. Of the 4th = .0000426591.

If the mass of the earth be supposed unity, that of the third satellite will be found = .027337; and, as the mass of the moon is \( \frac{1}{68.5} = .014599 \), the quantity of matter in the third satellite is about twice as great as that in the moon. The fourth satellite is therefore nearly equal to the moon; the second about one half, and the first somewhat more than one third.

53. The general result of this investigation concerning the inequalities in the motion of the planets, both primary and secondary, is, that in every one of these orbits, two things remain secure against all disturbance, the mean distance and the mean motion; or, which is the same, the transverse axis of the orbit, and the time of the planet's revolution. Another result is, that all the inequalities in the planetary motions are periodical, and observe such laws, that each of them, after a certain time, runs through the same series of changes. This last conclusion follows from the fact, that every inequality is expressed by terms of the form \( A \sin nt \) or \( A \cos nt \), where \( A \) is a constant co-efficient, and \( n \) a certain multiplier of \( t \) the time, so that \( nt \) is an arch of a circle, which increases proportionally to the time. Now, in this expression, though \( nt \) is capable of indefinite increase, yet, since \( nt \) never can exceed the radius or 1, the maximum of the inequality is \( A \). Accordingly, the value of the term \( A \sin nt \) first increases from 0 to \( A \), and then decreases from \( A \) to 0; after which it becomes negative, extends to — \( A \), and passes from thence to 0 again. If, when the inequality was affirmative, it was an addition to the mean motion, when negative, it will become a diminution of it; and the sum of all these increments and decrements, after \( nt \) has passed over an entire circumference or \( 360^\circ \), is equal to 0; so that, at the end of that period, the planet is in the same position as if it had moved on regularly all the while, at the rate of the mean motion. As this happens to every one of the inequalities, the deviation of the system from its mean state can never go beyond certain limits, each inequality in a certain course of time destroying its own effect.

It would be far otherwise, if into the value of any inequalities, a term entered of the form \( A \times \tan nt \), \( \frac{A}{\sin nt} \). The inequalities so expressed would continually increase with the time, so as to go beyond any assignable limit, and of consequence to destroy entirely the order of any system to which they belonged.

La Grange and La Place, who discovered and demonstrated, that no such terms as these last can enter into the expression of the disturbances which the planets produce by their action on one another, made known one of the most important truths in physical science. They proved that the planetary system is stable, that it does not involve any principle of destruction in itself, but is calculated to endure for ever, or till the action of an external power shall put a period to its existence. After the knowledge of the principle of gravitation, this may be fairly considered as the greatest discovery to which men have been led by the study of the heavens. The accurate compensation, just remarked, depends on three conditions, belonging to the primitive or original constitution of our system, but not necessarily determined, inasmuch as we know, by any physical principle. The first of these conditions is, that the eccentricities of the orbits are all inconsiderable, or contained within very narrow limits, not exceeding in any instance \( \frac{1}{10} \) or \( \frac{1}{5} \) part of the mean distance. The second condition is, that the planets all move in the same direction, or from west to east. This is true both of the primary and secondary planets, with the exception only of the satellites of Uranus, which may be accounted retrograde, but their planes being nearly at right angles to the orbit of their primary, the direction of their motion, whether retrograde or otherwise, can have little effect. Lastly, the planes of the orbits of the planets are not much inclined to one another. This is true of all the larger planets, though it does not hold of some of the new and smaller ones; of which, however, the action on the whole system must be wholly insensible.

Unless these three conditions were united in the constitution of the solar system, terms of the kind just mentioned, admitting of indefinite increase, might enter into the expression of the inequalities, which would indicate a gradual and unlimited departure from the original order and constitution of the universe.

Now, the three conditions just enumerated, do not necessarily arise out of the nature of motion, or of gravitation, or from the action of any physical cause with which we are acquainted. Neither can they be considered as arising from chance, for the probability is almost infinite to one, that, without a cause particularly directed to that object, such a conformity could not have arisen in the motions of thirty-one different bodies, scattered over the whole extent of the solar system. The only explanation, therefore, that remains is, that all this is the work of intelligence and design, directing the original constitution of our system, and impressing such motions on the parts as were calculated to give stability to the whole.

For some farther particulars, connected with Physical Astronomy, see Earth, Figure of, in this Supplement.

PRACTICAL. See the article Observatory, in this Supplement.