I< 



SCIENCE. 



[Vol. XIV. No. 344 



astronomefs and geodesists of his day, bat this is no adequate 

 justification for the exaggerated estimate long entertained of the 

 precision of the elements of his spheroid. 



The next step in the approximation was the important one of 

 Clarke in 1866. His new values showed an increase over Bessel's 

 of about half a mile in the equatorial semi-axis and about three- 

 tenths of a mile in the polar semi-axis. Since 1886. General Clarke 

 has kept pace with the accumulating 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 

 1866, which has come to be pretty generally adopted, seems likely 

 to enjoy a justly greater celebrity than that of its immediate pre- 

 decessor. 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 hun- 

 dred 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 approxi- 

 mation be carried .'' The answer to this question is not yet writ- 

 ten, and the indications are not favorable for its speedy announce- 

 ment. The first approximation, as we have seen, requires no 

 knowledge of the interior density and arrangement 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 astron- 

 omy," says Laplace, " rests on the stability of the earth's axis of 

 rotation." 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 not in general coincident with 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 ob- 

 servation, 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 inade- 

 quate knowledge of distribution and extent. 



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

 the deviations of the intensity of gravity from what may be called 

 the spheroidal type. Given a closely spheroidal form of the sea 

 level and it follows from the law of gravitation, as a first approxi- 

 mation, without any knowledge of the distribution of the earth's 

 mass, that the increase of gravity varies as the square of the sine 

 of the latitude in passing from the equator to the poles. This is 

 the remarkable theorem of Stokes, and it enables us to determine 

 the form or ellipticity of the earth, by means of pendulum observa- 

 tions alone. It must be admitted, however, that the values for the 

 ellipticity recently obtained in this way by the highest authorities, 

 Clarke and Helmert, are far from satisfactory, whether we regard 

 them in the light of their discrepancy or in the light of the dif- 

 ferent methods of computing them. In general terms we may say 

 that the difficulty in the way of the use of pendulum observations 

 still hinges on the treatment of local anomalies and on the ques- 

 tion of reduction to sea level. At present, the case is one concern- 

 ing which the doctors agree neither in their diagnosis nor in their 

 remedies. 



Turning attention now from the surface, towards the interior, 

 what can be said of the earth's mass as a whole, of its laws of dis- 

 tribution, and of the pressures that exist at great depth ? Two 

 facts, namely, the mean density and the surface density, are 

 roughly known ; and a third fact, namely, the precession constant, 

 or the ratio of the difference of the two principal moments of 

 inertia to the greater of them, is known with something like pre- 



cision. These facts lie within the domain of observation, and re- 

 quire only the law of gravitation for their verification. Certain in- 

 ferences also from these facts and others have long been and still 

 are held to be hardly less cogent and trustworthy, but before stat- 

 ing them, it will be well to recall briefly the progress of opinion 

 concerning this general subject during the past century and a half. 



The conception of the earth as having been primitively fluid was 

 the prevailing one among mathematicians before Clairaut published 

 his " Theorie de la Figure de la Terre " in 1743. By the aid of 

 this conception Clairaut proved the celebrated theorem which bears 

 his name, and probably no idea in the mechanics of the earth has 

 been more suggestive and fruitful. It was the central idea in the 

 elaborate investigations of Laplace, and received at his hands a de- 

 velopment which his successors have found it about equally difficult to 

 displace or to improve. From the idea of fluidity spring naturally 

 the hydrostatical notions of pressure and level surfaces, or the ar- 

 rangement of fluid masses in strata of uniform density. Hence 

 follows, also, the notion of continuity of increase in density from 

 the surface towards the centre of the earth. All of the principal 

 mechanical properties and effects of the earth's mass, viz., the 

 ellipticity, the surface density, the mean density, the precession 

 constant, and the lunar inequalities, were correlated by Laplace in 

 a single hypothesis, involving only one assumption in addition to 

 that of original fluidity and the law of gravitation. This assump- 

 tion relates to the compressibility of matter, and asserts that the 

 ratio of the increment of pressure to the increment of density is 

 proportional to the density. Many interesting and striking con- 

 clusions follow readily from this hypothesis, but the most interest- 

 ing and important are those relative to density and pressure, espe- 

 cially the latter, whose dominance as a factor in the mechanics of 

 celestial masses seems destined to survive whether the hypothesis 

 stands or falls. The hypothesis requires that while the density in- 

 creases slowly from something less than 3 at the surface to about 

 1 1 at the centre of the earth, the pressure within the mass increases 

 rapidly below the surface, reaching a value surpassing the crushing 

 strength of steel at the depth of a few miles, and amounting at the 

 centre to no less than three million atmospheres. The inferences, 

 then, as distinguished from the facts, are that the mass of the 

 earth is very nearly symmetrically disposed about its centre of 

 gravity, that pressure and density except near the surface are 

 mutually dependent, and that the earth in reaching this stage has 

 passed through the fluid or quasi-fluid state. 



Later writers have suggested other hypotheses for a continuous 

 distribution of the earth's mass, but none of them can be said to. 

 rival the hypothesis of Laplace. Their defects lie either in not 

 postulating a direct connection between density and pressure or in 

 postulating a connection which implies extreme or impossible 

 values for these and other mechanical properties of the mass. 



It is clear from the positiveness of his language in frequent 

 allusions to this conception of the earth, that Laplace was deeply 

 impressed with its essential correctness. " Observations," he says, 

 " prove incontestably that the densities of the strata [cozic/tes] of 

 the terrestrial spheroid increase from the surface to the centre ; " 

 and " the regularity with which the observed variation in length of 

 a seconds pendulum follows the law of the squares of the sines of 

 the latitudes, proves that the strata are arranged symmetrically 

 about the centre of gravity of the earth." The more recent inves- 

 tigations of Stokes, to which allusion has already been made, for- 

 bid our entertaining anything like so confident an opinion of the 

 earth's primitive fluidity or of a symmetrical and continuous ar- 

 rangement of its strata. But, though it must be said that the 

 sufficiency of Laplace's arguments has been seriously impugned, 

 we can hardly think the probability of the correctness of his con- 

 clusions has been proportionately diminished. 



Suppose, however, that we reject the idea of original fluidity. 

 Would not a rotating mass of the size of the earth assume finally 

 the same aspects and properties presented by our planet ? Would 

 not pressure and centrifugal force suffice to bring about a central 

 condensation and a symmetrical arrrangement of strata similar at 

 least to that required by the Laplacian hypothesis .' Categorical 

 answers to these questions cannot be given. But whatever may 

 have been the antecedent condition of the earth's mass, the conclu- 

 sion seems unavoidable that at no great depth the pressure is suffi- 



