D'ALEMBERT. 419 



lie gave this admirable investigation; and in 1755 he 

 followed it up with another first attempted by him, 

 namely, the variation which might occur to the former 

 results, if the earth, instead of being a sphere oblate at 

 the poles, were an elliptic spheroid, whose axes were 

 different. He added an investigation of the Precession 

 on the supposition of the form being any other curve 

 approaching the circle. This is an investigation of as 

 great difficulty perhaps as ever engaged the attention 

 of analysts. It remains to add that Euler, in 1750, 

 entered on the same inquiries concerning Precession 

 and Nutation ; and with his wonted candour, he de- 

 clared that he had read D'Alembert's memoir before 

 he began the investigation. 



The only other works of D'Alembert which it is ne- 

 cessary to mention, are his three papers on the integ- 

 ral calculus. Of these one, in the Berlin Memoirs, is 

 replete with improvements extremely important in the 

 methods of integration, and contains a method of treat- 

 ing linear equations of any order that serve as a foun- 

 dation for the approximate solutions, which are abso- 

 lutely indispensable to physical astronomy in the pre- 

 sent imperfect state of the calculus. The other two 

 are in the French Academy's Memoirs for 1757 and 

 1769, the latter giving the demonstrations of the theo- 

 rems on integration contained in the former. It is in 

 the twenty-first of these that he claims having sug- 

 gested, as we have already seen, to Clairaut his equa- 

 tion of conditions in the case of three variables. The 

 ' Opuscules' contain likewise, especially the 4th, 5th, 

 and- 7th volumes, some most important papers on the 

 calculus. Nor must we omit to record that there is 

 every reason to give him credit for having discovered 

 Taylor's Theorem. It is certain that he first gave this 

 celebrated formula complete, having, in the article 

 ' Series' of the 'Encyclopedic,' first given the remain- 

 ing terms left out by Taylor, and also a demonstration 

 of the whole, better than the original inventor's. Con- 



