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extreme old age, he commenced and finished, with all the vigour and 

 activity of youth, a third volume, expressly devoted to the discussion 

 and classification of these ultra -elliptic functions, and to point out 

 their analogy with, and relation to other classes of transcendents 

 which he had himself already considered, or to which they would 

 naturally lead. 



M. LeGendre was the author of a justly celebrated treatise or essay 

 on the Theory of Numbers, which first reduced the numerous and 

 disconnected discoveries of Fermat, Euler and Lagrange to syste- 

 matic order. He was the proper author, amongst many other dis- 

 coveries, of the law of reciprocity between any two prime numbers, 

 one of the most fertile and important in this theory, though its com- 

 plete establishment was reserved for Gauss, whose work on this sub- 

 ject has gained him so just a reputation. Notwithstanding, however, 

 the labours of these great men, this most important department of 

 analysis still continues to be too much insulated, both in its form 

 and its treatment, from the other branches of algebra, though much 

 has been done to reunite them by the very valuable and original re- 

 searches of that distinguished analyst M. Libri, of Florence, who has 

 been recently naturalized in France, and who has succeeded M. Le- 

 Gendre in his place in the Institute. 



The work of M. LeGendre, on Geometry, has enjoyed a singular 

 reputation, and has been most extensively used, particularly on the 

 ■continent of Europe, in the business of education. It may be 

 doubted, however, whether this work has altogether merited the high 

 character which it has obtained : it has rather increased than cleared 

 away the difficulties of the theory of parallels, which have so long 

 embarrassed the admirers of ancient geometry and of the Elements 

 of Euclid ; and it has not succeeded, at least in any essential degree, 

 in adding to the simplicity of the demonstrations, or to the clear and 

 logical connexion and succession of the propositions of that unrivalled 

 •and unique elementary work, which has alone maintained its place 

 amongst all civilized nations for more than two thousand years. It 

 is proper, however, to observe that the notes appended to this work 

 •are full of valuable and original remarks, and are justly celebrated 

 for the elegance of the demonstrations which they furnish of many 

 important propositions. 



M. LeGendre was the author of many other works and memoirs, 

 containing many valuable series of investigations, and very important 

 •discoveries. He first attacked the great problem of the determination 

 •of the orbits of comets by general methods, which display all the re- 

 sources of his analysis ; though astronomers have not found it expe- 

 dient to make use of his methods in the actual calculation of their 

 elements, which is the only proper test of their practical value, though 

 it may not be decisive of their theoretical perfection. He was the 

 ■author of the method of the least squares of the errors, for the pur- 

 pose of determining the most probable mean amongst the results of 

 a great number of observations, of which such extensive use is now 

 made in practical astronomy : a celebrated and most useful theorem 



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