KEPLER'S THIRD LAW.

274. After comparing in various ways the times of the planets and their distances, Kepler discovered his third law of planetary motion:

The squares of the times are as the cubes of the mean distances from the sun.

This most remarkable law, applying as it does to all the planets in their circuits about the sun; to the satellites, as they revolve about their primaries; even to the members of the far-off stellar systems in the remote regions of the universe, proves that all these objects have a similar origin and are subject to the same government. Nature works with uniformity in all her vast domain.

This law is practically useful in determining the mean distance of a newly-discovered planet. The rate of motion of the stranger would be first observed; from this its time of revolution computed, and its distance obtained. Thus, if a planet were found whose period is 5 years,

12: 52:: 13 : x3 .. x = =/25 =

2.924+.

The distance of the planet from the sun would be 2.924 times the mean radius of the earth's orbit.

Distances obtained by Kepler's third law are deemed more reliable than those derived from other sources.

275. Actual distances not yet found.-As yet we have found only the relative distances of the planets, when compared with the distance of the earth from the sun, taken as a unit of measure. One of these distances positively known, would help as to all the rest. When Mars is nearest the earth, his distance from the sun is about 12, and, therefore, from the earth about one-half the distance from the earth to the sun. When so near, his parallax may be found.

276. Observations of Mars.-From 1700 to 1761, astronomers observed Mars with the greatest care, and obtained the best results which could be given by instruments which were reliable only to two seconds of arc. In 1719, Maraldi found the parallax of Mars to be 27". The distance of the planet from the sun was at that time 1.37; from the earth, .37. But parallax is the angle which the radius of the earth subtends to an observer at the distant object (178): it is therefore inversely in proportion to the distance of the object. Hence,

Sun's dis. Mars' dis. :: Mars' par. : Sun's par.;

:

10.37 27" 9".99, nearly 10".

TRANSITS OF VENUS.

277. In 1725, Dr. Halley explained a method of finding solar parallax by observations of the transits of Venus, taken from remote points on the earth. The next transits of Venus occurred in 1761 and 1769. The problem to be solved was deemed so important that the governments of France, England, and Russia sent expeditions to various parts of the earth to secure observations. It was while engaged in this business that the celebrated navigator, Cook, lost his life at Hawaii. Le Gentil went to India

Fig. 90.

to observe the transit of 1761, but, by detentions on the voyage, he arrived too late. He waited the eight years for the next transit, and was then disappointed by the passage of a cloud over the sun at the critical time.

V

278. Halley's method. Suppose that two persons, each provided with a suitable telescope and an astronomical clock, are at distant places on the earth, A and B, looking for an expected transit. When Venus comes to the position V, the observer at A sees her apparently touch the sun: he notes the time of contact. A little later, the observer at B marks the time at which Venus seems to touch the sun in the position V'. Between the two observations, the earth has moved over the arc EE', which measures the angle ASA, and Venus has moved over the arc VV', opposite a somewhat larger angle ASB. The value of each angle is found from the known rates at The difference between these

which the planets move. two angles, the small angle ASB, or the amount of angular motion which Venus gained in order to make the contact visible at B, is the parallactic angle sought, opposite the base line AB.

The observation is repeated by noting the time

Fig. 91.

SUN

of external and internal contact on each side of the sun's disc.

279. Second method.-To an observer at A, Venus seems to cross the sun's disc on the line ab; to one at B, on the line ef. The two lines AD and BC, from the

Fig. 92.

E

images on the sun to the observers, cross at V, making the angles at V equal: the three bodies, E, V, and the sun, are in the same plane, and the lines AB and CD, perpendicular to that plane, are parallel. Hence, the triangles AVB and CVD are equiangular; but equiangular triangles have their like sides proportional; hence,

AB: CD: AV : VD.

AD=1; VD=.723 (273); therefore AV-1-.723-.277. Hence,

AB: CD :: .277 : .723.

Put for AB the distance in miles between the two places of observation, reduce, and we have the value of CD, the distance between the two chords ab and ef, on the sun's surface, in miles.

During this observation, the sun has an apparent westward motion, at a certain rate; Venus has an eastward motion, at a different rate; the sum of the two rates gives the apparent rate of the planet over the disc of the sun-so many seconds of arc in one second of time. Having noted carefully the time occupied in crossing the sun's disc, the length of the line of passage is known in seconds of arc. Construct the rightangled triangle COb, in which Cb, half the line ab, and Ob, the radius of the sun's disc, are known in seconds; by construction, or better by trigonometry, the triangle is solved, and CO is found. In the same way, from the triangle DOf, DO is found. DO taken from CO leaves CD, the

a

Fig. 93.

D

distance between the chords, in seconds.

We know now how many seconds a certain number of miles will subtend at the distance of the sun.

The sun's parallax is the angle which the radius of the earth subtends at the distance of the sun (191); hence,

CD in miles: CD in seconds ::

Earth's Rad. in miles: Sun's hor. par.

280. Results.

From the transit of Venus of 1769, the solar parallax was computed to be 8".578. This gives the mean distance of the sun equal to 23984 times the radius of the earth, or a little more than 95 millions of miles, (95,298,000). Multiply the numbers which express the planetary distances, as already found (273), by this quantity; the products will be the distances of the planets from the sun in miles, approximately. As we are studying only the plan of this grand mechanism, the solar system, it is necessary for us to remember