152 MEMOIR OF LEGENDRE. 



■terms, cacli of wliicli is only integvable by arcs of conic sections. This important 

 discovery led the illustrious geometer to compare, in a manner more general than 

 had been done before, not only the arcs of the same ellipsis or the same hyper- 

 bola, but in general all the transcendents of which the differential approximates 

 to those of these two curves, in presenting, like them, a rational algebraic func- 

 tion of the variable divided by the square root of an algebraic polynome of the 

 fourth degree.* One of the results of this comparison was, that the integration 

 by arcs of the hyperbola may always be reduced to integration by arcs of the 

 ellipsis. 



From this time Euler foresaw that by means of a suitable notation the cal- 

 culation of arcs of the ellipsis and other analogous transcendents might become 

 of almost as general use as that of arcs of the circle and of logarithms ; but, with 

 the exception of the English geometer Landen, who demonstrated, in a memoir 

 of 1775, that every arc of' the h>/perboIa is immediatchj rectified J)ij means of two 

 arcs of the ellipsis,] no one but M. Legendre recognized the importance of real- 

 izing the prevision of Euler ; and it may be said that our learned colleague alone 

 occupied himself with this subject from the year 1786, when he published his 

 first researches on integrations by arcs of the ellipsis, until the year 1825, when 

 his Treatise of ElVq)'ic Functions appeared. 



Arcs of the ellipsis, being after arcs of the circle and logarithms one of the 

 most simple transcendents, might become in some sort a new instrument of cal- 

 culation, if we were once familiarized with their properties and possessed ready 

 means of calculating them with precision. ]\I. Legendre applied himself to this 

 important subject in two memoirs inserted in the volume of the Academy of 

 Sciences for 1786. In both of them the author demonstrates, by means pecu- 

 liar to himself, that the rectification of the hyberbola depends on that of the 

 ellipsis and presents no special transcendent, and in the second he shows that in 

 an infinite series of ellipses formed after the same law we can reduce the rectifica- 

 tion of one of these ellipses to that of two others taken at choice in the same 

 series. This, he says with characteristic modesty, is one stej) more in a difficult 

 path. 



In the first memoir IM. Legendre gives convergent series adapted for the easy 

 calculation of the length of an arc of an ellipsis, whether in the case in which 

 the elipsis but slightly eccentric approximates to a circle, or in that when, 

 greatly elongated, it recedes but little from its greater axis; and in the second 

 he adds: 



If the zeal of calculators could furnish us with tables of arcs of the ellipsis! for different 

 degrees of amplitude and eccentricity, and each arc were accompanied by the coefScieut of 

 its partial difterence, we should have the means of integri\ting by these tables a very largo 

 number of differentials, and especially all those which MM. d'Alembert and Euler have 

 referred to the arcs of conic sections. 



j\L Legendre had then attained the age of 34 years; he knew not that it 

 Avould be permitted him to labor till that of 80 years, and that unassisted he 

 would himself accomplish the task of which he here traces the programme. 



In the course of these two memoirs, and particularly towards the end of the 

 second, he indulges himself in a just tribute of praise to the learned geometers 

 (Euler, Landen, and Fagnani) who, before himself, had demonstrated, in a differ- 



* R being a radi^'al of (he form in question and P a rational algebraic function, all these 



/P 1 

 — 77—. — Legendre, Meinoirc sur Its Transcend- 



nntes ellipliqiies, p. 4. 



t Landen published his researches in the Philosophical Transactions, and still later in a 

 special work entitled, Mutheniatical Memoirs Kespecliug a Vurutij of Subjects, by John Lan- 

 den, F. R. S. : London, 178U, 



i See the volume of the Academy of Sciences for 1786, pp. 618 and 644. 



