

Lagrange's Memoirs. 77 



importance of the problems it resolves ; the application that the au- 

 thor makes of it to the question, recurring every day in astronomy ; 

 of the degree of confidence that can be allowed to the mean result 

 of a great number of observations ; and wherein is found this re- 

 markable property, and so favorable to the circles of Borda, that 

 each of the even numbers states as probable, by the odd number 

 immediately above, that the error will be comprised within certain % 

 limits. M. le Comte Laplace had on his part labored on the same 

 theory. M. Lagrange resumed it, on his part, by means which ex- 

 tend to equations of all orders. Of these, they give finite integrals, 

 and facilitate, in all cases, the determination of arbitrary functions. 



Maclaurin had treated, after the method of the ancients, the at- 

 traction of elliptic spheroids. Lagrange thought this work compara- 

 ble with all that Archimedes had left of ingenuity and excellence ; 

 he showed then that analysis can treat this difficult subject with the 

 same success ; he succeeded in it, but was stopped at the same point 

 as the English geometer. M. Legend re and M. Laplace have since 



been farther. But Ivory has just shewn us, that an extremely sim- 

 ple view can render useless many calculations, and reach even theo- 

 rems to which the most tedious calculations lead only with difficulty. 

 Formerly, geometers, in every question, tried at first to gain these 

 insights, that could simplify them or reduce them to questions al- 

 ready solved, thus shortening the calculations or rendering them 

 even entirely useless. Since the discovery of the infinitesimal cal- 

 culus, the facility, the generality of the method, which often dis- 

 penses with the calculator's having genius, has caused, that in the 

 most difficult cases, the object chiefly in view was to perfect the 

 universal instrument. But now, that the resources of this kind have 

 been entirely exhausted, by the labors of Euler, of Lagrange, and 

 of their worthy rivals, it will be perhaps time to return to the an- 

 cient method, and to imitate D. Bernouilli, whom Condorcet has 

 praised with having distinguished himself upon calculations. La- 

 grange made more constantly another use of his sublime talents ; 

 he drew all from analysis. Yea, it is still more true, to say that he 

 has united each method to the highest degree. The proof of it is 

 in the calculus of variations, to which cannot be compared, either 

 for greatness or for generality, any of the most happy ideas of other 

 geometers. But if it is a question of these ingenious glimpses, of 

 which all before was limited to simplify a single question, it is thus 

 that from the first steps he had reduced the phenomena of sound to 



