THE ELLIPTIC FUNCTIONS DEFINED INDEPEND- 

 ENTLY OF THE CALCULUS 



BY FEANK H. LOUD. 



Among the minor works of the eminent mathematician, C. G. 

 J. Jacobi, is one containing a geometrical construction for the 

 addition-theorem of elliptic integrals, — a construction, as the 

 author remarks, which is not without advantage over that given 

 by liagrange by the use of the spherical triangle. The sugges- 

 tion of Jacobi has been followed by several writers of text- 

 books upon the elliptic functions: it appears, e. g., in brief on 

 pp. 28 and 29 of Cayley's "Elementary Treatise," and is much 

 more fully handled by Dur^ge, " Theorie der Elliptischen 

 Functionen" pp. 168-190. The object which I propose at 

 present is to apply the Jacobian construction to a still more 

 primary use; viz., to produce from it a definition of the elliptic 

 functions which shall be independent of differentiation and 

 integration, and being based on the methods of elementary 

 mathematics alone, or those anterior to the calculus, shall enable 

 the student to form a preliminary conception of the elliptic 

 functions analogous to that which he acquires of the sine, 

 cosine, etc., in elementary trigonometry. 



The memoir of Jacobi here cited appeared in the third 

 volume of Crelle's Journal; and is to be found in his collected 

 works, (Berlin, 1881,) on pp. 277-293. It bears the date of April 

 1st, 1828, and is entitled, "Upon the Application of the Elliptic 

 Transcendents to a known Problem of Elementary Geometry; 

 viz., to find the Eelation between the Kadii and the Distance 

 between the Centers of two Circles, one of which is inscribed in 

 an Irregular Polygon, and the other is circumscribed about the 



