i 



contained in the Thii'd Book of the Mecanique Celeste. 431 



ing from the centre to the surface ; Huygens proposed the 

 perpendicularity of gravity to the outer surface. These are 

 properties which undoubtedly both belong to every fluid body 

 i7i equilibrio; and it deserves to be remarked, that either of 

 them is sufficient for the equilibrium of a homogeneous fluid 

 if we suppose that it has the figure of an elliptical spheroid. But 

 it was soon found that there are some figilres which would be in 

 eqiiilibriobythe one principle, and not so according to the other. 

 Bouguer therefore imagined that the two principles must both 

 concur in every case of true equilibrium. Clairaut afterwards 

 showed that, even with this restriction, we are not sure that 

 an equilibrium will always take place. Abandoning all pre- 

 carious suppositions, that able geometer, in the first part of his 

 work on the Figure of the Earth, has deduced a more satisfac- 

 tory theory from the laws of hydrostatics, which, without any 

 change in its principles at least, has ever since been univer- 

 sally adopted. It is however extremely remarkable that, in 

 the second part of his work, Clairaut has not been able to in- 

 vestigate the figure of the earth from his theory of equilibrium, 

 but is obliged to adopt the supposition of an elliptical sphe- 

 roid. It may be said, indeed, that in his time mathematical 

 knowledge was not sufficiently advanced to overcome the dif- 

 ficulties of so intricate a problem. But it is still true at the 

 present day, that, what Maclaurin rigorously proved by means 

 of the synthetic geometry of the ancients, has never been de- 

 duced, except by approximation and in a very particular case, 

 from the modern theory of the equilibrium ot fluids. We may 

 add, that the deduction not only has not been made, but that 

 it never can be accomplished ; because there is wanting a 

 principle necessary for determining the figure of equilibrium. 

 Clairaut's theory of fluids is in the same predicament with the 

 principles advanced by Newton and Huygens ; though just 

 and accurate, it is incomplete, at least in the question of the 

 figure of the planets. 



The theory in the Mecanique Celeste applies only to sphe- 

 roids little different from spheres. It is founded on the equi- 

 librium of fluids usually received. The results obtained, at 

 least in a first approximation, are exact ; for they agree with 

 what has been proved with undoubted evidence by Maclaurin 

 and Clairaut. But as the theory of fluids on which the inves- 

 tigation proceeds is in general insufficient to determine the 

 true figures of equilibrium, it follows that there must be some 

 peculiarity in the case considered, something hidden probably 

 under the analytical operations employed, which corrects tiie 

 general defect, and leads to a true result in particular circum- 

 stances. In every physical investigation, it is of great impor- 

 tance 



