Newton's law has been shown to hold also for smaller bodies at less distances apart. Within the range of possible experimental accuracy the law has been verified for bodies whose distances apart are as small as three or four centimeters. Whether the same law holds for bodies as small as molecules, and at distances as small as molecu lar distances, or not, cannot be said. Stated mathematically, Newton's law is that the gravi tation force between two particles of masses m, and separated by a distance r, is given by the formula G m m F — where G is a constant expressing the fact that F is proportional to and is called the 'gravi tative constant.' The actual mechanical force between two bodies of known masses at a known distance apart has been measured by different observers, first by Cavendish in 1798. Thus G may be determined. It has been shown by Mac kenzie and Poynting to be the same for crystalline as for isotropic bodies; and, so far as is known, it is independent of the intervening medium and a true constant of nature. In the C. G. S. system its value, as determined by Boys, is 0: 000000066576, or 6.6576 X The same law may he applied to a body falling toward the earth; viz.
F m2 In this case, if m is the mass of the falling body, F = mg, where g is the acceleration of such a body, and nearly equals 980 on the C. G. S. sys tem. It was shown by Newton, and later by Bessel, that the value of g is independent of the kind of matter which is falling, and it has been proved to be independent of the mass. g varies
of course from point to point on the earth's sur face, owing both to its rotation and to its spher oidal shape. Formulas have been calculated for g as a function of the geographical latitude 4'; one of the best is g = 97.989 (1 + 0.0052 sin' c13). In the formula m, is the mass of the earth, and r is the radius of the earth if it is assumed that the gravitation action of the earth is the same as if it were all concentrated at the centre. (A homogeneous sphere, or one made up of homo geneous spherical shells, would have this action.) Therefore g G or g r2 If this G is the same as in the previous formula —an assumption for or against which there is no evidence—the three quantities on the right hand side of the equation are known; and so m„ the mass of the earth, may be determined. The average density of the earth, then, is this mass divided by the volume of the earth, which is ap proximately That is, calling A this density, 3q 47rrG Assuming the above value for G; viz. G = 6.6576 X , this gives A = 5.5270.
For full details as to the law of gravitation, its history and its verification, reference should be made to The Laws of Gravitation, in "Scien tific Memoir Series," edited by Mackenzie, vol. ix. (New York, 1900).