LAPLACE. 137 



the latter was characterized by a defect of a still more serious nature ; it 

 supposed the density of the earth, during the original state of fluidity, 

 to be homogeneous.* 



When in attempting the solution of great problems we have recourse 

 to such simplifications; when in order to elude difficulties of calculation 

 we depart so widely from natural and physical conditions, the results 

 relate to an ideal world, thej are in reality nothing more than llights of 

 the imagination. In order to apply mathematical analysis usefully to 

 the determination of the figure of the earth, it was necessary to aban- 

 don all idea of homogeneity, all constrained resemblance between the 

 forms of the superposed and unequally dense strata ; it was necessary 

 also to examine the case of a central solid nucleus. This generality in- 

 creased tenfold the difficulties of the problem ; neither Clairaut nor 

 D'Alembert was, however, arrested by them. Thanks to the efforts of 

 these two eminent geometers, thanks to some essential developments 

 due to their immediate successors, and especially to the illustrious Le- 

 gendre, the theoretical determination of the figure of the earth has 

 attained all desirable perfection. There now reigns the most satisfac- 

 tory accordance between the results of calculation and those of direct 

 measurement. The earth, then, was originally fluid ; analysis has 

 enabled us to ascend to the earliest ages of our planet.t 



In the time of Alexander, comets were supposed by the majority of 

 the Greek philosophers to be merely meteors generated iu our atmos- 

 phere. Daring the Middle Ages, persons, without giving themselves much 

 concern about the nature of those bodies, supposed them to prognosti-. 



in the year 1674, but the account of his observations with the pendulum during hi.« residence 

 there was not published until 1679, nor is there to be found any allusion to them during the 

 intermediate interval, either iu the volumes of the Academy of Sciences or any other pub- 

 lication. We have no means of ascertaining hovv Newton was first induced to suppose that 

 the tigure of the earth is spheroidal, but we know, upon his own authority, that as early as 

 the year 16G7 or 1668, he was led to consider the effects of the centrifugal force iu diminish- 

 ing the weight of bodies at the equator. With respect to Huyghens, he appears to have 

 formed a conjecture respecting the spheroidal figure of the earth independently of Newton; 

 but his method for computing the ellipticity is founded upon that given in the Principia. — 

 Translator. 



* Newton assumed that a homogeneous fluid mass of a spheroidal form would be in 

 equilibrium if it were endued with an adequate rotatory motion and its constituent par- 

 ticles attracted each other in the inverse proportion of the square of the distance. Maclaurin 

 first demonstrated the truth of this theorem by a rigorous application of the ancient geome- 

 try.— Translator. 



t The results of Clairaut's researches on the figure of the earth are mainly embodied in a 

 remarkable theorem discovered by that geometer, and which may be enunciated thus : The 

 sum of the fractions expressing the ellipticity and the increase of gravity at the pole is 

 equal to two and a half times the fraction expressing the centrifugal force at the equator, 

 the unit of force being represented by the force of gravity at the equator. This theorem is 

 independent of any hypothesis with respect to the law of the densities of the successive 

 strata of the earth. Now the increase of gravity at ilie pole may be ascertained by means 

 of observations with the pendulum iu different latitudes. Hence it is plain that Clairaut's 

 theorem furnishes a practical method for determining the value of the earth's ellipticity. — 

 Translator. 



