Neglecting the compression, as it is called, i. e., the 27 miles by which the equatorial diameter exceeds the polar, the size of the earth may easily be found by measuring the distance 1 - 2 along the surface and by combining with this the angle 1 O 2 obtained through measuring the meridian altitudes of any star as seen from 1 and 2. Draw on paper an angle equal to the measured difference of altitude and find how far you must go from its vertex in order to have the distance between the sides, measured along an arc of a circle, equal to the measured distance between 1 and 2. This distance from the vertex will be the earth's radius.
Exercise 19.—Measure the diameter of the earth by the method given above. In order that this may be done satisfactorily, the two stations at which observations are made must be separated by a considerable distance—i. e., 200 miles. They need not be on the same meridian, but if they are on different meridians in place of the actual distance between them, there must be used the projection of that distance upon the meridian—i. e., the north and south part of the distance.
By co-operation between schools in the Northern and Southern States, using a good map to obtain the required distances, the diameter of the earth may be measured with the plumb-line apparatus described in [Chapter II] and determined within a small percentage of its true value.
45. The mass of the earth.—We have seen in [Chapter IV] the possibility of determining the masses of the planets as fractional parts of the sun's mass, but nothing was there shown, or could be shown, about measuring these masses after the common fashion in kilogrammes or tons. To do this we must first get the mass of the earth in tons or kilogrammes, and while the principles involved in this determination are simple enough, their actual application is delicate and difficult.
Fig. 26.—Illustrating the principles involved in weighing the earth.
In [Fig. 26] we suppose a long plumb line to be suspended above the surface of the earth and to be attracted toward the center of the earth, C, by a force whose intensity is ([Chapter IV])
F = k mE/R2,
where E denotes the mass of the earth, which is to be determined by experiment, and R is the radius of the earth, 3,963 miles. If there is no disturbing influence present, the plumb line will point directly downward, but if a massive ball of lead or other heavy substance is placed at one side, 1, it will attract the plumb line with a force equal to