152 MEMOIR OF LEGENDRE. 



terms, eacli of wliicli is only intcgrahle l>y 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 h^'per- 

 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 nota^tion the cal- 

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

 of almost as general use as that of ai'cs of the circle and of logarithms ; but, with 

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

 of 1775, that ercr>/ arc of the Jii/pcrbola is immcdiaMij rcdijicd liij means of two 

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

 izing the prevision of Euler ; and it may bo 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 

 Ins Treatise of Ellip'ic Functions appeared. 



Arcs of tlie 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 fomiliarized with their properties and possessed ready 

 means of calculating them with precision. M. 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 rectilication 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, ho says with characteristic modesty, is one step more in a difiicult 

 path. 



In the first memoir M. 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 coefSeieut of 

 its partial difference, we should have the means of integrating by tbese tables a very large 

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

 referred to the arcs of conic sections. 



M. Legendre had then attained the age of 34 years ; he knew not that it 

 would 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 indvdges himself in a just tribute of praise to the learned geometers 

 (Euler, Landen, and Fagnani) who, bef(n-e himself, had demonstrated, in a differ- 



* R being a radical of the form in question and P a rational algebraic function, all these 



/P ^/x 

 — TT-- — Legendre, Memoirc sur Us Transccnd- 



antcs elliptiques, p. 4. 



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

 special work entitled, Mathematical Memoirs Respecting a Variety of Subjects, by John Lan- 

 den, F. K. S. : London, 1780. 



t See the volume of the Academy of Sciences for 178G, pp. 618 and 644. 



