A Text-Book of Astronomy

by George C. Comstock

f = k mB/r²,

where r is the distance of its center from the plumb bob and B is its mass which we may suppose, for illustration, to be a ton. In consequence of this attraction the plumb line will be pulled a little to one side, as shown by the dotted line, and if we represent by l the length of the plumb line and by d the distance between the original and the disturbed positions of the plumb bob we may write the proportion

F : f :: l : d;

and introducing the values of F and f given above, and solving for E the proportion thus transformed, we find

E = B · l/d · (R/r)².

In this equation the mass of the ball, B, the length of the plumb line, l, the distance between the center of the ball and the center of the plumb bob, r, and the radius of the earth, R, can all be measured directly, and d, the amount by which the plumb bob is pulled to one side by the ball, is readily found by shifting the ball over to the other side, at 2, and measuring with a microscope how far the plumb bob moves. This distance will, of course, be equal to 2 d.

By methods involving these principles, but applied in a manner more complicated as well as more precise, the mass of the earth is found to be, in tons, 6,642 × 10¹⁸—i. e., 6,642 followed by 18 ciphers, or in kilogrammes 60,258 × 10²⁰. The earth's atmosphere makes up about a millionth part of this mass.

If the length of the plumb line were 100 feet, the weight of the ball a ton, and the distance between the two positions of the ball, 1 and 2, six feet, how many inches, d, would the plumb bob be pulled out of place?

Find from the mass of the earth and the data of [§ 40] the mass of the sun in tons. Find also the mass of Mars. The computation can be very greatly abridged by the use of logarithms.

46. Precession.—That the earth is isolated in space and has no support upon which to rest, is sufficiently shown by the fact that the stars are visible upon every side of it, and no support can be seen stretching out toward them. We must then consider the earth to be a globe traveling freely about the sun in a circuit which it completes once every year, and rotating once in every twenty-four hours about an axis which remains at all seasons directed very nearly toward the star Polaris. The student should be able to show from his own observations of the sun that, with reference to the stars, the direction of the sun from the earth changes about a degree a day. Does this prove that the earth revolves about the sun?