Institutiones calculi integralis. [Volumen primvm. In quo methodus integrandi a primis principiis usque ad integrationem aequationum differentialium primi gradus pertractatur; Volumen secundum. In quo methodus inveniendi functiones unius variabilis ex data relatione differentilium secundi altiorisue gradus pertractatur; Volumen tertium. In quo methodus inveniendi functiones duarum et plurium variabilium, ex data relatione differentialium cuiusuis gradus pertractatur. Una cum appendice de calculo variationum et supplemento, evolutionem casuum prorsus singularium circa integrationem aequationum differentialum continente.]

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THE THIRD PART OF EULER S GREAT TRILOGY ON THE CALCULUS. First edition, very rare when complete with the folding plate, of the third part of Euler s monumental trilogy on analysis, following his Introductio in analysis infinitorum (1748) and Institutiones calculi differentialis (1755). "They were not didactic works in the sense that they did not aim to introduce and explain a consolidated scientific discipline to students. Instead, Euler s main goal in writing them was the constitution of analysis as a new and autonomous branch of mathematics which is independent of geometry and arithmetic" (Ferraro, p. 39). "The methods of indefinite integration in the Institutiones calculi integralis (I, 1768) are described by Euler in quite modern fashion and in a detail that practically exhausts all the cases in which the result of integration is expressible in elementary functions. He invented many of the methods himself; the expression Euler substitution (for rationalization of certain irrational differentials) serves as a reminder of the fact. Euler calculated many difficult definite integrals, thus laying the foundations of the theory of special functions … The Institutiones calculi integralis exhibits Euler s numerous discoveries in the theory of both ordinary and partial differential equations, which were especially useful in mechanics" (DSB). "Euler s contributions to ordinary differential equations, as summarized in this text, are so outstanding that it can be regarded as the main milestone on the subject from which all the related texts can be referred to" (Vellando, p. 344). "The first volume is noteworthy for including Euler s general addition theorem, the discovery of which was his major contribution to the theory of elliptic integrals. The third volume introduced double integrals and established the rule of substitution" (Norman). "In Institutiones calculi integralis(1768-70) Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions. He also studied beta and gamma functions, which he had introduced first in 1729 … As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work" (Mactutor). "The first volume consists of an introduction and three sections. In the introduction, Euler discusses the nature of the integral calculus. In the first section, Euler deals with the integration of elementary functions; in particular he examines approximate integration and integration by series and infinite products. In the second section, Euler investigates differential equations and elliptic integrals. The third section is devoted to differential equations involving higher-order differentials and transcendental functions. The second volume is divided into two sections. The first section is devoted to the integration of second-order ordinary differential equations. The second section deals with differential equations of third and higher order. The third volume consists of two parts along with an appendix and a supplementum. The first two parts deal with partial differential equations. The appendix is devoted to the calculus variations. In the supplementum, Euler discusses some particular differential equations" (Ferraro, pp. 76-78). ABPC/RBH lists only one complete copy since the Norman sale; the Norman copy realised $10,350 in 1998. "By 1766 the Saint Petersburg Imperial Academy agreed to print two multi-volume writings by Euler, and in the next year a third, all of them composed in Berlin. Two were his monumental Institutionum calculi integralis (Foundations of integral calculus) and his Dioptricae (Dioptrics, or general theory of lenses) … On 7 August 1766, at the first conference that he attended upon his return to the Petersburg Imperial Academy, Euler submitted the entire text for his Institutionum calculi integralis on ordinary and partial differential equations. Since 1759 he had searched fruitlessly in German lands for a copy edito. N° de ref. del artículo 5364

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