ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 247 



consideration ; other sources of information will be cited in their proper 

 places. 



(3) A good account (especially from the German standpoint) is given 

 of the early development of the theory of functions by Brill and Nother 

 (loc. cit.). I shall here consider very briefly only such parts of the theory 

 of elliptic functions that have a direct bearing upon this report, omitting 

 as far as possible what has already been given by Ellis. 



(4) The contributions towards the advancement of the elliptic 

 functions by Tschirnhaus (1683-1700), the Bernoullis (1690-1730), 

 Fagnano ('Produzioni Matematiche,' Pesaro, 1750) are discussed by 

 Enneper (' EUiptische Functionen '). 



Two works which must have exercised great influence upon subsequent 

 writers are Maclaurin, 'A Treatise on Fluxions,' Edinb. 1742, and 

 d'Alembert, ' Recherches sur le calcul integral ' (' Histoire de I'Acad. de 

 Berlin,' 1746, pp. 182-224). 



(5) Euler extended and systematised the work that Fagnano had 

 begun. It was known that the expressions for sin (a+/?), sin (a — ^), etc., 

 gave a means of adding or subtracting the arcs of circles, and that between 

 the limits of two integrals that express lengths of arc of a lemniscate an 

 algebraical relation exists, so that the arc of a lemniscate, although a 

 transcendent of higher order, may be doubled or halved just as the arc of 

 a circle by means of geometric construction. 



It was natural to inquire if the ellipse, hyperbola, etc., did not have 

 similar properties ; investigating such questions, Euler made the remark- 

 able discovery of the addition-theorem of elliptic integrals {cf, ' Nov. 

 Comm.' Petrop. vi. pp. 58-84, 1761 ; vii. p. 3 ; vii. p. 83). 



Euler showed that if 



•where <f>{i) is a rational integral function of the fourth degree in i, there 

 exists between the upper limits x, y, and a of the integrals an algebraic 

 relation which is the addition-theorem of the arcs of an ellipse and is the 

 algebraic solution of the differential equation ^ 



^^ + ^'^ =0 

 n/</'(^) V^W 



Euler stated that the above results were obtained, not by any regular 

 method, but pothis tentando, vel divinando, and suggested that mathe- 

 maticians seek a direct proof. The numerous discoveries of Euler are 

 systematised in his work, ' Institutiones calculi integralis.' 



The fourth volume (p. 446) contains an extension of the addition- 

 theorem to integrals of the second and third kinds, as they were sub- 

 sequently classified and named by Legendre, 



In each case geometrical application of the formulae are made for the 

 comparison of elliptic arcs. 



(6) The addition-theorem for elliptic integrals gave a similar meaning in 

 higher analysis to the elliptic functions as the cyclometric and logarithmic 

 functions had had for a long time. See Enneper (' EUipt. Funct.,' p. 541 

 et seq.) regarding the position occupied by Euler in the development of 

 the elliptic functions, and for a statement regarding Legendre's work in 



' Euler, Nov. Comm, vol. x. pp. 3-50. 



