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LXIV. On the Theory of the Figure of the Planets contained 

 in the Third Book of the Mecanique Celeste. By J. Ivory, 

 Esq. M.A. F.R.S. 



"VTO part of the philosophy of Newton has advanced more 

 ■^^ slowly than that which treats of the figure of the pla- 

 nets. This speculation, depending more immediately upon 

 the principle of a mutual attraction between all the particles 

 of matter, was entirely new, and required every thing to be 

 created. Much is done in the Principia, the greatest monu- 

 ment of human genius that has yet appeared ; and much has 

 been added in many researches by all the great mathematicians 

 who have laboured to fill up the outlines traced with such in- 

 imitable skill in that illustrious work. It is not the present 

 intention to enter upon any detail of the steps by which this 

 branch of philosophy has advanced to its present state ; it 

 will be sufficient to remark that all our knowledge on this sub- 

 ject naturally falls under two different heads. The first of 

 these comprehends the discoveries of Maclaurin, with which 

 I arrange all that is contained in the second part of Clairaut's 

 excellent treatise on the Figure of the Earth. In the view 

 of the subject taken under this head, it is not proposed to in- 

 vestigate a priori the figures of equilibrium ; it is merely de- 

 monstrated that the elliptical spheroid of revolution will fulfill 

 those conditions. Newton was under the necessity of making 

 some gratuitous assumptions in order to evade the great diffi- 

 culties that occurred in treating of the figure of the earth. 

 He supposed that it was composed of a homogeneous fluid, 

 and that it had the form of an elliptical spheroid of small ob- 

 lateness at the poles ; and upon these suppositions he investi- 

 gated the degree of the oblateness, or the proportion between 

 the equatorial and the polar diameters. The researches both 

 of Maclaurin and Clairaut were undertaken for the purpose 

 of inquiring into the legitimacy of the assumptions made by 

 Newton. They both succeeded in demonstrating the truth of 

 the suppositions on which the investigation in the Principia is 

 founded. Maclaurin's demonstration, in which the method of 

 the ancient geometiy is followed, is very elegant and accurate, 

 and is at this day the only case of the problem that has been 

 solved in a manner strictly rigorous and without having re- 

 course to approximations. Clairaut used the modern analysis; 

 and admitting the supposition of elliptical spheroids of small 

 oblateness which allowed the square of the excentricity to be 

 neglected, he extended his researches to the case when the 

 earth is composed of fluid strata varying in density from the 



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