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. [Illustration: 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/R^{2}, 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 f = k mB/r^{2}, 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_.