Biography of M. Le Comte Lagrange. 251 



of limits and of vanishing, and demonstrates the lawfulness 

 of the abreviations permitted in these two calculs, which 

 Jae also frees from all difficulties, and from all paradoxes 

 that had sprung up in an imperfect and suspected meta- 

 physique. 



5. The demonstration of a curious theorem on primal 

 numbers; a demonstration that no one had been able to 

 find, and the more difficult, as we know how to express 

 algebraically propositions of this kind. 



6. The integration of partial differences of the first order, 

 by a fruitful principle, sufficient for the greater part of 

 cases where this integration is possible. 



7. A purely analytical solution of the problem of the 

 rotation of the body of any figure, wherein he at last sur- 

 mounts difficulties that had long stopped him, and by 

 which geometers seemed to expect, with curiosity, some 

 ulterior developments, that they hoped to find in the second 

 volume of his new Mechanique Analytique. 



In addition tO these, he wrote many memoirs on the 

 obscure and difficult theory of probabilities, wherein we 

 admire the integral that forms its base, the number and 

 importance of the problems it resolves ; the application 

 that the author 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 observa- 

 tions ; and wherein is found this remarkable property, 

 and so favourable to the circles of Borda, that each of the 

 even numbers states as probable, by the odd number im- 

 mediately above, that the error will be comprised within 

 certain limits. M. le Comte Laplace had on his part 

 laboured on the same theory. M. Lagrange resumed it, on 

 his part, by means which extend to equations of all orders. 

 Of these, they give finite integrals, and facilitate, in all 

 cases, the determination of arbitrary functions. 



He made then a similar attempt for the problem of 

 eclipses ; he found that the methods, somewhat prolix, of 

 Dusejou, had neither the simplicity nor the facility that 

 ought to have been expected from the actual state of 

 analysis. He exhibits, in this work, all bis resources and 

 all his address. 



(To be continued.) 



