## Institutionun calculi integralis. Volumen primum [secundum, tertium].

St. Petersburg: Academiae Imperialis Scientiarum, 1768-69-70.

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 editor and printer for the manuscript; he recognised that for any press printing this book would be difficult. By December 1763, the manuscript of the text had been completed … Although originally projected to require two volumes, the text contained so many emendations since 1759 that three volumes were now necessary. The title page lists Euler as vice director of the Imperial Academy as well as a member of the Paris Academy and the Royal Society of London. In December 1767 Lagrange expressed delight that the Imperial Academy had agreed to have Euler’s *Institutionum calculi integralis* printed and called it a ‘true service’ to scholars in the sciences.

“Between 1768 and 1770, the academic press published the three volumes in quarto of the *Institutionum calculi integralis* … they contained most elementary solutions of differential equations and all basic cases of integrability. The *Institutionum calculi integralis* was a novel work offering a comprehensive catalogue of solutions to partial and ordinary differential equations by an evolving concept of elementary functions and by power series … In his *Introductio* Euler had already given the general theory of algebraic and elementary transcendental functions; he represented functions by a single analytical expression composed of variables and constants. Before him there had been few methods of solution and few applications. He refined, simplified, and extended each of these and added hundreds of innovations in the theories of those equations that are particularly helpful in their application to mechanics” (Calinger, pp. 457-459).

“Euler defined integration as anti-differentiation, in other words, he regarded integration as the inverse operation of differentiation by which one returned from the differential to the function generating the differential. This definition is rather problematic. First, Euler did not provide any proof of the existence of the anti-differential of a given function. Second, Eulerian functions consisted only of elementary functions [algebraic, exponential and logarithmic, trigonometric] and their composition … Euler was aware that many simple functions could not be integrated by means of elementary functions. He tackled the question in the introduction to *Institutionum calculi integralis*, where he compared integration with inverse arithmetical operations. He stated that analytical operations are always opposed in pairs. In the same way as addition is opposed to subtraction, multiplication to division, or the raising to a power to the extraction of a root, differentiation and integration are also opposed. In certain cases, the inverse operations of subtraction, division and extraction of roots led to new ‘quantities’, namely negative, rational and irrational numbers. Similarly, integration was not always successful in finding the function generating the differential. In such cases, integration led to new transcendental ‘quantities’” (Ferrero, p. 80).

Leibniz and Newton had introduced integration as the problem of the finding areas, by adding infinitely many infinitesimal rectangles. “According to Euler, the concept of the integral as the sum of an infinite number of infinitesimals was similar to the concept of lines as aggregates of infinitely-many points. Both could be admitted (or, better, tolerated), as long as there was reference to the true principles of the calculus and geometry. In other words, it was possible to take into account non-null infinitesimals, but it constituted only an imprecise and approximate version of the notion of integral, which nevertheless had useful applications” (*ibid*., pp. 80-81). The proper resolution of this problem was only found in the next century.

“Volume I of the *Institutionum calculi integralis* consists of an introduction describing the general nature of integral calculus, followed by three sections that comprise what might be termed 10 chapters. Each chapter is the elaboration of several problems with solutions and several corollaries with proofs. The first section integrates elementary functions, especially the transcendental logarithmic, exponential, and trigonometric, and examines expansion by infinite products. The second section covers the integration of second-order differential equations via multipliers, approximations, and infinite series. [In the third section use is made of a new variable, *p = dy/dx*, in solving some more difficult first order differential equations.]

“Chapter 6 in the second section of volume I dealt with elliptic integrals, and here Euler was extending the work of Count Giulio Carlo de’ Toschi di Fagnano. Like Fagnano, he emphasized the relations between integrals and algebraic results. Of the elliptic integrals, the arc lengths of the ellipse, hyperbola, and lemniscates gave the typical cases … Perhaps he alone saw here the germ of an entirely new branch of analysis” (Calinger, pp. 457-460).

“Euler termed the integral of a differential equation containing an adequate number of arbitrary constants a complete integral. He gave the name particular integral to a relation between the variables that satisfies the equations and that does not contain any new constant quantity. Euler observed that the complete integral yields infinitely-many particular integrals, however there were particular integrals that are not contained in the complete integral [now called singular integrals] … Euler provided a criterion to distinguish the singular solution from a particular integral without knowledge of the complete integral” (Ferrero, p. 89).

“The second volume was divided into two parts. The first treated ordinary differential equations of second order, and the second part differential equations of third and higher orders, their construction by the quadrature of curves, and elliptic integrals. After receiving the two volumes, the first from the printer and the second from Formey [Samuel Formey, secretary of the Berlin Academy], Lagrange lauded Euler’s research as erudite, ingenious, and – as always – fruitful” (Calinger, p. 460).

“In the third volume, Euler dealt with the integration of functions of two variables. In 1765 he had stated that the integration of a function of two variables is a new and little developed field of the integral calculus, which differs very much from the common integral calculus, where functions of only one variable occur” (Ferraro, p. 96). Euler noted that when one integrates a function of two variables with respect to one of the variables, the result involves an arbitrary function, rather than the arbitrary constant that occurs when integrating functions of one variable. “According to Euler, it was not necessary that [the arbitrary function] was one of the functions accepted in analysis, which were given by means of a single analytical expression. Instead, it could be thought of as the ordinate of a curve which could be traced by a free stroke of the hand or composed of more than one continuous curve. Therefore Euler termed these quantities as discontinuous functions. In opposition, functions composed by only one analytical formula were termed continuous functions. In this way the notion of discontinuity, which had previously been considered [in the *Introductio*] only in reference to curves, was extended to functions” (*ibid*.).

“A seven-chapter Appendix to volume III refines what Euler called a new branch of integral calculus, the calculus of variations, following the analytic methods of Lagrange, whom he called the creator of this field. For it Euler offered a higher degree of perfection of its methods and computations” (Calinger, p. 460). Euler had treated the calculus of variations in his *Methodus inveniendi lineas curvas maximi minive proprietate gaudentes* (1744), prior to the publication of his calculus trilogy. Euler’s derivation in the *Methodus inveniendi* of the fundamental equation of the calculus of variations, or Euler equation as we know it today, was based on polygonal approximations and was not completely adequate, as he himself realized. After receiving a letter from Lagrange in 1755 showing him a better method for deriving the equation, Euler asked his colleague to present the method to the Berlin academy and to further develop the field. Euler presented Lagrange’s approach in the Appendix.

“The *Integral Calculus*, the *Differential Calculus* and the first part of the *Introductio* are texts in pure analysis, so much so that Euler does not even deal with applications to geometry, let alone physics. This is, perhaps, especially surprising in the *Integral Calculus* since the original motivation for the solution of differential equations came from physical questions, questions that in fact led Euler to some of these methods of solution in the 1730s and 1740s … in the *Integral Calculus* there is no mention of the vibrating string problem or various other vibration problems that had led Euler to ‘invent’ the trigonometric functions in the 1730s, nor is there any calculation of areas nor any material on lengths of curves, or volumes, or surface areas of solids. And then, although Euler presented an extraordinary number of methods to find anti-derivatives, the central technique of modern texts for determining areas, the fundamental theorem of calculus, did not appear. That is not to say that Euler did not know how to calculate areas using anti-derivatives. Euler in fact did so in various papers. But since geometrical ideas are not present in the calculus texts, there is no definition of the area under a curve as a function, and therefore no call for the derivative of such a function. And until an independent definition of area could be provided, as Cauchy did in the 1820s, this fundamental relationship between derivatives and integrals, discovered by Newton and Leibniz, could not be ‘fundamental.’

“It appears that, with the exception of the second part of the *Introductio*, which was filled with graphs, Euler had an abiding belief that ‘pure mathematics’ had no need of diagrams. One could understand everything that was needed by pure manipulation of symbols according to the rules that he and others developed … Euler seemed further to believe, like Euclid two thousand years earlier, that it was unnecessary to help students learn analysis by showing them the motivations for the various techniques … That is not to say that Euler was not a good pedagogue. He was very patient with his readers, frequently explaining every step in an argument while also demonstrating the same result in several ways. And he certainly used motivations from the sciences in many of his other papers …

“The techniques that Euler developed to determine derivatives and integrals continued to appear in other texts, but his use of infinitesimals as a basis for the calculus was gradually replaced by the idea of a limit, beginning with the ideas of Jean d’Alembert as expressed in his articles in the French *Encyclopédie*” (Katz, pp. 231-232).

A second edition, in four volumes, was issued by the same publisher in 1792-94, and a third, also in four volumes, in 1824-45.

Eneström 342, 366, & 385; Norman 734; Poggendorff I, 690; Roller-Goodman I, 374; Sotheran 1252. Calinger, *Leonhard Euler*, 2016. Ferraro, ‘Euler’s treatises on infinitesimal analysis,’ pp. 39-101 in: *Euler reconsidered* (Baker, ed.), 2007. Katz, ‘Euler’s analysis textbooks,’ pp. 213-234 in: *Leonhard Euler*: *Life, Work and Legacy* (Bradley & Sandifer, eds.). 2007. Vellando, ‘… and so Euler discovered Differential Equations,’ *Foundations of Science* 24 (2009), pp. 343-374.

Three vols., 4to (250 x 192 mm), pp. [iv], 542; [iv], 526, [8]; [viii], 639, with one folding engraved plate (light marginal damp stains to a few leaves, small red ink stamp on verso of each title page – as in the Norman copy). A few, mostly marginal, corrections in an early hand, two sheets of early manuscript calculations tipped in at pp. 326 and 338 of vol. I. Contemporary boards with manuscript paper spine labels (rubbed, capitals worn). A very good unrestored copy in original state.

Item #5364

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Price:
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