GRAVITATION. 157 GRAVITJS. three great Keplcrian laws of motion obtaining in the solar system by reference to an attraetive force residing in tlic sviii. These laws are: (1) That the planets revolve round the sun in ellipses, having the sun for a common focus. (2) Tliat every jjlanet moves in such a waj* that the line drawn from it to the sun sweeps over equal areas in equal times. (3) That the squares of the times occupied by the several planets in their revolutions in their elliptic orbits are projxir- tional to the cubes of their mean distances from their connnon focus, the sun. From the law of equal areas Newton inferred that eveiy planet is retained in its orbit by a force of attraction directed toward- the centre of the sun; from the orbits being elliptical, he inferred that in each case this force varies in intensity according to the inverse square of the body's distance from the sun ; while from the third law he inferred the honiogeneitj' of the central force throughout the solar system. It w-as then, after being familiar with the no- tion of terrestrial gravity and its action, through the researches of Galileo, Huygens. and Hooke, and with the notion of a central force acting in- versely as the square of the distance through his explanations of the laws of Kepler, that he put to himself the question: Is not the force with vvhicli the moon is pulled to the earth the same ■with gravity? A question answered affirmatively on the supposition of gravity, like the sun's at- traction, being a force diminishing with increase of distance and according to the same law. The result was to bring the whole solar system within the range of the law of gravitation. The phe- nomena of double stars justify the extension and the statement of the law as we have given it to universal terms. It may be obsen'ed, in con- elusion, that the Keplerian laws, which may be said to have been the basis of Newton's re- searches, are, owing to perturbations (q.v. ) cau.?ed by the mutual action of the planets, etc., only approximately correct, and that these per- turbations afford, when examined, a further proof of the truth and universality of the law of gravi- tation. 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. Vhether 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 m,, separated by a distance r, is given by the formula P _ G ?«] Wt; ^ r- 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 obsen-ers, first by Cavendish in 1798. Thus G may be determined. It has been showii by IMac- 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: 00(IOOOl)(;('i.")7ri, or G.tJoTC) X in" The same law may be applied to a l)ody falling toward the earth ; viz. F = G '"' '"■-' In this case, if m is the m.iss of the falling body, F = my, where </ is the acceleration of such a body, and nearly equals 980 on the C. G. >S. sys». tcm". It was shown by Newton, and later by Bcssel, 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 <7 as a function of the geograpliical latitude *; one of the best is g = 97.989 { 1 + 0.0052 sin= *) . In the formula nu_ 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 mi jr = G ' , - m, = -i — G If t'ais 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 hi,, 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 firr^ That is, calling A this density, ^ - •iTTcG Assuming the above value for G ; viz. G = 6.6576 X 10"% 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 Lans of flravltation, in "Scien- tific Memoir Series," edited bv Mackenzie, vol. i.x. (New York, 1900). GRA'VIUS, gra've-oos (originally Grat;, or Gkeffe), .JoiiANN Georg (1632-1703'). A distin- guished German philologist, born at Naumburg on the Saale, January 29, l(i.'i2. He studied law at the University of Leipzig, but while on a visit to Holland was led to abandon jurisprudence for philology. He spent two years at Deventer study- ing under Gronovius. In 1050 he was called to the University of Dusburg, nnd two years later was summoned to Deventer at the instance of his former teacher. In 16G1 he became professor of history at Utrecht. There he built up a great reputation, and received flattering offers from the universities of Amsterdam. Heidelberg. Leyden, and Padua, all of which he declined. Pupils crowded to his lectures, not only from all Hol- land, but from all Eurojie. In Germany par- ticularly almost all the great noblemen sent their sons to be educated by him. He died at Utrecht, Januarj- 11, 1703. During his life as a teacher
