BERNOULLI, Jakob. Analysin magni problematis isoperimetrici in actis erud. Lips. m. Maj. 1697 pag. 214 propositi ... sub praesidio Jacobi Bernoulli... publice defendendam suscipit Joh. Jacobus Episcopius.
Basel: Typis Joh. Conradi à Mechel, 1701.

Very rare first separate edition of Bernoulli’s solution of the isoperimetric problem, which represents the beginnings of the calculus of variations, later to be developed extensively by Euler and Lagrange.

In 1696 Johann Bernoulli had posed as a challenge to the mathematicians of Europe the problem of finding the curve of quickest descent, the so-called brachistochrone. Solutions were given by L’Hospital, Leibniz, Newton, and by Johann’s brother Jakob in the May 1697 issue of Acta Eruditorum, At the end of this memoir Jakob raised the isoperimetric problem as a challenge to his readers, but especially to Johann (he even offered Johann a prize of 50 silver ecus for a satisfactory solution within three months). The isoperimetric problem asks for the curve of a given length joining two fixed points in the plane for which a certain integral is a maximum (a special case would be to maximize the area contained between the curve and the straight line joining the two points). “Johann underestimated the complexity of this problem by failing to perceive its variational character; and he furnished an incomplete solution... and thereby brought upon himself the merciless criticism of his brother. This was the beginning of alienation and open discord between the brothers – and also the birth of the calculus of variations. A comparison of Jakob’s solution [in the offered work] with Johann’s analysis of the problem... clearly shows Johann’s to be inferior. Nevertheless, Jakob was not able to enjoy his triumph, since – for reasons that remain mysterious – the sealed envelope containing Johann’s solution was not opened by the [Paris] Academy until 17 April 1706, the year following Jakob’s death” (DSB, under Johann Bernoulli). After solving the isoperimetric problem, “Jakob was able to furnish the proof [also contained in the offered work], which Johann and Leibniz had been seeking in vain, that the inexpansible and homogeneous catenary is the curve of deepest center of gravity between the points of suspension” (ibid., under Jakob Bernoulli).

This separate edition was published in March 1701, two months before its appearance in the May 1701 issue of Acta Eruditorum. It was dedicated to L’Hospital, Leibniz, Newton and Fatio de Duillier.

OCLC lists 6 copies worldwide, none in America.

4to: 179 x 150 mm. Pp. 18,(2). Old limp vellum, end papers renewed, fine and clean. Small tape repair to page 18.