186 THE MATHEMATICAL THEORIES OF THE EARTH. 



ill the ])o]ar semi axis. Since 186(5. General (Jlarke has kept jiace with 

 the accuimilatiug data and given us so many different elements for our 

 spheroid that it is necessary to affix a date to any of his values we may 

 use. The later values, however, differ but slightly from the earlier 

 ones, so that the spheroid of 1860, which has come to be pretty gener- 

 ally adopted, seems likely to enjoy a justly greater celebrity than that 

 of its immediate predecessor. The probable error of the axes of this 

 spheroid is not much greater than the hundred thousandth part,* and 

 it is not likely that new data will change their lengths by more than a 

 few hundred feet. 



In the present state of science, therefore, it may be said that the first 

 order of approximation to the form and dimensions of the Earth has 

 been successfully attained. The question which follows naturally and 

 immediately is, how much further can the approximation be carried ! 

 The answer to this question is not yet written, and the indications are 

 not favorable for its speedy announcement. The first approximation, 

 as we have seen, requires no knowledge of the interior density and ar- 

 rangement of the earth's mass; it proceeds on the simple assumption 

 that the sea surface is closely spheroidal. The second approximation, 

 if it be more than a mere interpolation formula, requires a knowledge 

 of both the density and arrangement of the constituents of the earth's 

 mass, and especially of that part called the crust. "All astronomy," 

 says Laplace, "rests on the stability of the earth's axis of rotation." t 

 In a similar sense we may say all geodesy rests on the direction of the 

 plumb line. The simple hypothesis of a spheroidal form assumes 

 that the plumb line is everywhere coincident with the normal to the 

 spheroid, or that the surface of the spheroid coincides with the level 

 of the sea. But this is not quite correct. The plumb line is uoc in 

 general coincident xrith the normal, and the actual sea level or geoid 

 must be imagined to be an irregular surface lying partly above and 

 partly below the ideal spheroidal surface. The deviations, it is true, 

 are relatively small, but they are in general much greater than the 

 unavoidable errors of observation and they are the exact numerical 

 expression of our ignorance in this branch of geodesy. It is well 

 known, of course, that deflections of the plumb line can sometimes be 

 accounted for by visible masses, but on the whole it must be admitted 

 that we possess only the vaguest notions of their cause and a most in- 

 adequate knowledge of their distribution and extent. 



What is true of plumb-line deflections is about equally true of the de- 

 viations of the intensity of gravity from what may be called the sphe- 

 roidial type. Given a closely spheroidal form of the sea level and it 

 follows from the law of gravitation, as a first approximation, without 



* Clarke, Col. A. R., Geodesy, Oxford, 1880, p. 319. 



f'Toiite I'Astronoiuie repose sur I'lnvariabilit^ de I'axe de rotation de la Terre d la 

 surface dn sph<iroide terrestre et snr l'uuiformit6 de cette rotation." Mecaniqiic Ce- 

 leste (Paris, 1882), Tome v. p,22. 



