350 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
ellipse. Every part of analysis is indebted to Lagrange, who enriched this 
particular branch with a general method for changing an elliptic function into 
another having a different modulus, a process which greatly facilitates the 
numerical calculation of this class of integrals. An elliptic function lies 
between an arc of the circle on one hand, and a logarithm on the other, 
approaching indefinitely to the first when the modulus is diminished to zero, 
and to the second when the modulus is augmented to unit, its other limit. 
By repeatedly applying the transformation of Lagrange, we may compute 
either a scale of decreasing moduli reducing the integral to a circular arc, or 
a scale of increasing moduli bringing it continually nearer to a logarithm. 
The approximation is very elegant and simple, and attains the end proposed 
with great rapidity. 
The discoveries that have been mentioned occurred in the general cultivation 
of analysis ; but Legendre has bestowed much of his attention and study upon 
this particular branch of the integral calculus. He distributed the elliptic 
functions in distinct classes, and reduced them to a regular theory. In 
a Memoire sur les Transcendantes Elliptiques, published in 1793, and in 
his Exercices de Calcul Integral, which appeared in 1817, he has developed 
many of their properties entirely new ; investigated the easiest methods of 
approximating to their values ; computed numerical tables to facilitate their 
application ; and exemplified their use in some interesting problems of geo- 
metry and mechanics. In a publication so late as 1825, the author, returning 
to the same subject, has rendered his theory still more perfect, and made many 
additions to it which further researches had suggested. In particular we find 
a new method of making an elliptic function approach as near as we please to 
a circular arc, or to a logarithm, by a scale of reduction very different from 
that of which Lagrange is the author, the only one before known. This step 
in advance would unavoidably have conducted to a more extensive theory of 
this kind of integrals, which, nearly about the same time, was being discovered 
by the researches of other geometers. 
M. Abel of Christiana, and M. Jacobi of Konigsberg, entirely changed the 
aspect of this branch of analysis by the extent and importance of their disco- 
veries. The first of these geometers, whom, to the great loss of science, a pre- 
mature death cut off in the beginning of a career of the highest expectations. 
