Fagnani discovered that the two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and proved that any arc of a lemniscate, like that of a circle, may be multiplified any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. What he had accomplished with respect to the arcs of the lemniscates, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. Landen showed that the arcs of the hyperbola may be reduced, by a proper transformation, to those of an ellipse. Lagrange furnished us 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. Legendre distributed the elliptic functions into distinct classes, and reduced them to a regular theory, developing many of their properties which were before unknown, and introducing many important additions and improvements in the theory. Mr. Abel of Christiana happity conceived the idea of expressing the amplitude of an elliptic function in terms of the function itself, which led to the discovery of many new and useful properties. Mr. Jacobi proved, by a different method, that an elliptic function may be transformed in innumerable ways into another similar function, to which it bears constantly the same proportion. But his demonstrations require long and complicated calculations; and the train of deductions he pursues does not lead naturally to the truths which are proved, nor does it present in a connected view all the conclusions which the theory embraces. The author of the present paper gives a comprehensive view of the theory in its full extent, and deduces all the connected truths from the same principle. He finds that the sines or cosines of the amplitudes, used in the transformations, are analogous to the sines or cosines of two circular arcs, one of which is a multiple of the other; so that the former quantities are changed into the latter when the modulus is supposed to vanish in the algebraic expression. Hence he is enabled to transfer to the elliptic transcendents the same methods of investigation that succeed in the circle: a procedure which renders the demonstrations considerably shorter, and which removes most of the difficulties, in consequence of the close analogy that subsists between the two cases.
On the theory of the elliptic transcendents
