But the mass of the sun being 315,000 times that of the earth, it will at the same distance exert an attraction 315,000 times greater. The intensity of the attraction, therefore, which the sun exerts at the earth's mean distance will be

and the intensity of its attraction at the mean distance of the other planets being still inversely as the squares of the distances, will be found by dividing this by a2. So that if G express this attraction, and F the height, in thousandths of an inch, through which a body placed at each distance would fall in one second, we shall have

G=

9 1690a

F= 16095 X 12 XG=193140.

By the numbers given in the column &, it is there to be understood that a mass of matter which, placed upon the surface of the earth, would weigh the number of pounds expressed by the denominators of the fractions severally, would, if submitted only to the sun's attraction at the respective mean distances of the planets, gravitate to the sun with the force of one pound. Thus, a mass which on the earth's surface would weigh 1690 lbs., would weigh only one pound if exposed to the sun's attraction in the absence of the earth. In like manner, a mass which upon the earth's surface would weigh 1,524,652 lbs., or 680 tons, would, if exposed to the sun's attraction at the mean distance of Neptune, weigh only one pound, so extremely is the intensity of solar attraction enfeebled by the enormous increase of distance. (Table V.)

The numbers given in the column F have a more absolute sense, and express in thousandths of an inch the actual spaces through which a body would be drawn in one second of time by the sun's attraction at the mean distances of the planets severally.

773. Calculation of the central force of gravity by the velocity and curvature of a body.-The space through which any central attraction would draw a body in a given time can be easily calculated, if the body in question moves in a circular, or nearly circular, orbit around such a centre, as all the planets and satellites do.

Let E, fig. 104, be the centre of attraction, and E m the distance or radius vector. Let m m'v, the linear velocity. Let m n' and

n m' be drawn at right angles to E m, and therefore parallel to each other. The velocity m m' may be considered as compounded of two forces (M. 169), one in the direction mn' of the tangent,

and the other m n directed towards the centre of attraction E. Now if the body were deprived of its tangential motion mn', it would be attracted towards the centre E, through the space m n, in the unit of time. By means of this space, therefore, the force which the central attraction exerts at m can be brought into direct comparison with the force which terrestrial gravity exerts at the surface of the earth.

It follows, therefore, that if ƒ express the space through which such a body would be drawn in the unit of time, falling freely towards the centre of attraction, we shall have ƒ=m n. But by the elementary principles of geometry,

mnx 2 Em=mm22.

that is, the space through which a body would be drawn towards the centre of attraction, if deprived of its orbital motion, in the unit of time, is found by dividing the square of the linear orbital velocity by twice its distance from the centre of attraction.

The attractive force, or, what is the same, the space through which the revolving body would be drawn towards the centre in the unit of time, can, therefore, be always computed by these formulæ, when its distance from the centre of attraction and its linear or angular velocity are known.

which being substituted for a in the preceding formula, will give

f=412541×

by which the attractive force may always be calculated when the distance and period of the revolving body are known.

774. Law of gravitation shown in the case of the moon. -The attraction exerted by the earth, at its surface, may be compared with the attraction it exerts on the moon by these formulæ.

In the case of the moon v=0·6356 miles, and r=239,000 miles, and by calculations from these data, we find

f=0'0000008459miles=0.0536inch.

The attraction exerted by the earth at the moon's distance would, therefore, cause a body to fall through 536 ten-thousandths of an inch, while at the earth's surface it would fall through 193 inches (M 245).

The intensity of the earth's attraction on the moon is, therefore, less than its attraction on a body at the surface, in the ratio of 1,930,000 to 536, or 3600 to 1, or, what is the same, as the square of 60 to 1.

But it has been shown that the moon's distance from the earth's centre is 60 times the earth's radius. It appears, therefore, that in this case the attraction of the earth decreases as the square of the distance from the attracting centre increases; and that consequently, the same law of gravitation prevails as in the elliptic orbit of a planet.

775. Sun's attraction on planets compared—law of gravitation fulfilled.—In the same manner, exactly, the attractions which the sun exerts at different distances may be computed by the motions and distances of the planets. The distance of a planet gives the circumference of its orbit, and this, compared with its periodic time, will give the arc through which it moves in a day, an hour, or a minute. This, represented by m m', fig. 104, being known, the space m n through which the planet would fall towards the sun in the same time may be calculated, and this being done for any two planets, it will be found that these spaces are in the inverse ratio of the squares of their distances.

Thus, for example, let the earth and Jupiter be compared in this manner. If D express the distance from the sun in miles, P the period in days, A the arc of the orbit in miles described by the planet in an hour, and H the space m n in miles, through which the planet would fall towards the sun in an hour if the tangential force were destroyed, we shall then have

HH

Now, on comparing the numbers in the last column with the squares of those in the first column, we find them in almost exact accordance. Thus,

(91) (4757): 24'402 0·9028.

The difference, small as it is, would disappear, if exact values were taken instead of round numbers.

776. The harmonic law.-A remarkable numerical relation thus denominated, prevails between the periodic times of the planets and their mean distances, or major axes of their orbits. If the squares of the numbers expressing their periods be compared with the cubes of those which express their mean distances, they will be found to be very nearly in the same ratio. They would be exactly so if the masses or weights of the planets were absolutely insignificant compared with that of the sun. But although these masses, as will appear, are comparatively very small, they are sufficiently considerable to affect, in a slight degree, this remarkable and important law.

Omitting for the present, then, this cause of deviation, the harmonic law may be thus expressed. If P, P', P", &c., be a series of numbers which express or are proportional to the periodic times, and r, r', r', &c., to the mean distances of the planets, we

shall have

13

&c.;

that is, the quotients found by dividing the numbers expressing the cubes of the distances by the numbers which express the squares of the periods are equal, subject nevertheless to such deviations from the law as may be due to the cause above mentioned.

777. Fulfilled by the planets.—Method of computing the distance of a planet from the sun when its periodic time is known. To show the near approach to numerical accuracy with which this remarkable law is fulfilled by the motions of the planets composing the solar system, we have exhibited in the following table the relative approximate numerical values of their several distances and periods, and it is evident on comparing the two columns which give the cubes of the distance, and the squares of

the periods, that the quotients found by dividing the column by that of p2 are sensibly equal:—

In general the distance of a planet from the sun can be computed by means of this law, when the distance of the earth and the periodic times of the earth and planet are known.

For this purpose find the number which expresses the periodic time P of the planet, that of the earth being expressed by 1; and let D be the number which expresses the mean distance of the planet from the sun, that of the earth being also expressed by 1. We shall then, according to the harmonic law, have

To find the distance D, therefore, it is only necessary to find the number whose cube is the square of the number expressing the period, or, what is the same, to extract the cube root of the square of the period.

778. Harmonic law deduced from the law of gravitation.— It is not difficult to show that this remarkable law is a necessary consequence of the law of gravitation.

Supposing the orbits of the planets to be circular, which for this purpose they may be taken to be, let the distance, period, and angular velocity of any one planet be expressed by r, P, and a, and those of any other by 'P' and a', and let the forces with which the sun attracts them respectively be expressed by ƒ and f. We shall then, according to what has been proved (773), have