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XX. On the Theory of the Elliptic Transcendents. By James Ivory, A.M. 
F.R.S. Instit. Reg. Sc. Paris , et Soc. Reg. Sc. Gottin., Corresp. 
Read June 9, 1831. 
THE branch of the integral calculus which treats of elliptic transcendents 
originated in the researches of Fagnani, an Italian geometer of eminence. He 
discovered that 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 he proved that any arc of the lemniscata, like that of a circle, may 
be multiplied any number of times, or may be subdivided into any number of 
equal parts, by finite algebraic equations. These are particular results ; and 
it was the discoveries of Euler that enabled geometers to advance to the inves- 
tigation of the general properties of the elliptic functions. An integral in 
finite terms deduced by that geometer from an equation between the diffe- 
rentials of two similar transcendent quantities not separately integrable, led 
immediately to an algebraic equation between the amplitudes of three elliptic 
functions, of which one is the sum, or the difference, of the other two. This 
sort of integrals, therefore, could now be added or subtracted in a manner 
analogous to circular arcs, or logarithms ; the amplitude of the sum, or of the 
difference, being expressed algebraically by means of the amplitudes of the 
quantities added or subtracted. What Fagnani had accomplished with respect 
to the arcs of the lemniscata, which are expressed by a particular elliptic inte- 
gral, Euler extended to all transcendents of the same class. To multiply a 
function of this kind, or to subdivide it into equal parts, was reduced to 
solving an algebraic equation. In general, all the properties of the elliptic 
transcendents, iri which the modulus remains unchanged, are deducible from 
the discoveries of Euler. Landen enlarged our knowledge of this kind of 
functions, and made a useful addition to analysis, by showing that the arcs of 
the hyperbola may be reduced, by a proper transformation, to those of the 
