146 MEMOIR OF LEGENDRE. 



volume of more than 100 pages; hut in the quietude of liis licappy retreat ho 

 had turned his thoughts to other subjects. The former professor of mathematics 

 in the military school began anew to occupy himself with the Elements of 

 Gcomdrij. The first edition of his work under this title, a work written with 

 elegant simplicity, and in which all the propositions are disposed in a natm'al 

 and methodical order, appeared in 1794. The author, modelling himself upon 

 Euclid, remands the science to the severity of the Greek school. In this, 

 without perhaps designing it, ho accommodated himself to the spirit of his epoch. 

 Architecture, abandoning the distorted forms of the reign of Louis XV, was 

 returning, more and more, to the elegant simplicity of the Greek style. A few 

 years previous our great painter, David, had inaugurated, l)y his picture of the 

 Homtli, a con:iplete revolution in painting, whichj after his example, reverted 

 likewise to the imitation of the ancients. 



The work attained at once the first rank among classical books. In less 

 than 30 years fourteen editions were published, of which the last has under- 

 gone a large number of impressions : more than 100,000 copies of it have been 

 sold in France alone. Legendre's Elements of Geometrij have l)een reproduced 

 in the princij)al languages of Europe, and have been even translated into Arabic 

 for the schools established in Egypt by the viceroy, ]\Iehemet-Aii. 



The author, prepossessed with the method of Euclid, has perhaps somewhat 

 unduly availed himself of the reditcfio ad cibsurdum, which might often be 

 replaced by more facile demonstrations ; but his work has served to excite a 

 sort of vigorous intellectual gymnastics by which mathematical studies have 

 been invigorated, and its influence has been undoubtedly salutar}'. Among other 

 things, M. Legendre here demonstrates, in a novel manner, the equality of vol- 

 ume of two symmetrical polyhedrons formed of equal plane faces, adjusted 

 under the same angles, but with an inverse arrangement which does not admit 

 of their being superposed. The first editions did not contain the excellent trea- 

 tise on trigonometry which the author has added to subse(pient ones. He has 

 also enriched these with notes, in which he treats analytically certain parts of 

 geometry on a new system, as where he demonstrates that the ratios of the 

 circumference to the diameter and to its square are irrational numbers. 



The ratio of the circumference to the diameter, being an irrational number, is 

 not susceptible of iK'ing exactly expressed by any fraction, however great the 

 whole numbers which form the numerator and denominator. Hence results the 

 impossihility of ever finding the quadrature of the circle, and it was in conse- 

 quence of a proposition of M. Legendre, based on this demonstrated impossi- 

 bility, that the Academy renounced all further attention to a problem, the 

 importance of which is in some sort axiomatic among persons little versed in 

 matliematics. 



But whatever might be the success of his Elements, M. Legendre did not 

 question the feasibility of using other methods with success, and himself con- 

 tributed, in 1802, to the publication of a new edition of Clairaut's Elements of 

 Geometry, to which he added notes derived probably from his memoranda of the 

 military school. Geometry is further indebted to him for a method, directly 

 demonstrated by himself, of inscribing in the circle a regular polygon of 17 

 sddes. Algebra, properly so called, owes to him, among other things, two 

 different methods for the solution of numerical equations, methods which make 

 known with much rapidity all the roots, whether real or imaginary, of those 

 equations. 



So highly was M. Legendre appreciated as a skillful calculator, that rarely was 

 any great series of numerical operations undertaken in France without recourse 

 being had to his services. In 1787 he had been called to take part in the com- 

 mission charged with connecting trigonometrically Dunkirk and Greenwich. 

 For the same reason M. de Prony, placed in the j^ear II (1794) at the head of the 

 cadastre, (registry of the survey of lands,) did not deem it expedient to dispense 

 with his services. The decimal division of the circle, then regarded as a neces- 



