## Institutiones Calculi Differentialis cum eius usu in Analysi Finitorum ac Doctrina Serierum.

St. Petersburg: Academiae Imperialis Scientiarum, 1755.

First edition, rare, of the second part of Euler’s monumental trilogy on analysis, “the first textbook on the differential calculus which has any claim to be regarded as complete” (Rouse Ball, p. 368). “In 1755 Euler published another masterwork in mathematics, his two-part *Institutiones calculi differentialis*, the second part of his trilogy on calculus. The book was probably begun around 1727, but it was mostly finished by 1748 and completed two years later, when Euler was forty-three. For the previous decade, he had worked on it steadily. *Institutiones calculi differentialis* is the first textbook to organize systematically the hundreds of important discoveries made since the time of Leibniz and Newton. Today it is mainly remembered for the definition of the concept of a function, which stressed not the role of formulae but the more general idea of a formal correspondence between two sets of numbers. This new definition, which looked forward to the modern concept of a mapping between two sets, was probably motivated by the controversy with d’Alembert and Daniel Bernoulli over the vibrating string. The book began with the first didactic presentation of the calculus of finite differences when the differences become ‘infinitely small’. This was a sound idea, but Euler did not possess a formal theory of limits, so … [he] had to resort to the idea that the differential, an infinitely small quantity, is ‘a true zero’ and to formalize differential calculus as a ‘calculus of zeroes’ … His vague definition of infinitesimals as quantities smaller than any fixed number looked back to the ideas of Johann I Bernoulli, and it would remain the accepted formulation of calculus for several decades … The second part of the *Institutiones calculi differentialis* contains an impressive array of important results, many of them found by Euler himself. Chapters 5 and 6 elaborate his summation formula for the Basel Problem [finding an exact formula for the sum of the reciprocal squares of the integers] and what would later be called the Euler-Maclaurin Formula. The results on the Bernoulli numbers were many, starting from their generating formula and going on to their application to the summation of power series and connection with the Riemann zeta function. Euler found several properties of these numbers, which had applications in many fields of mathematics, and their computation provides a challenging problem even today. With them Euler obtained exact sums of power series of even reciprocals. Among the equations which Euler studied in chapter 6 are the partial sums of the harmonic series, the Euler constant γ, the value of π, and approximate formulae for large factorials. The book was extremely influential and is now regarded as one of the most important scientific texts of the eighteenth century” (Calinger, pp. 395-6). “[Euler] thoroughly elaborated formulas of differentiation under substitution of variables; revealed his theorem on homogeneous functions, stated as early as 1736 …, deduced the necessary condition for an exact differential; applied Taylor’s series to finding extrema [of functions of a single variable]; and investigated extrema of [functions of two variables]” (DSB). Copies in good condition in contemporary bindings are rare on the market. The Norman copy, in modern binding with library stamps, realized $3450 in 1998.

Euler “He started to write this book already in Saint Petersburg and finished it around 1750 in Berlin, where it was published under the auspices of the Saint Petersburg Academy of Sciences. The existence of an early Latin manuscript ‘Calculi differentialis’, conserved in the Archives of the Russian Academy of Sciences in Saint Petersburg, shows that Euler worked over a very long period to present his modern view of the differential calculus. An account of his scientific manuscripts dated this one to the 1730s, while A.P. Yushkevich considered that it was written even earlier, around 1727. We consider that its comparison with the book of 1755 reveals the evolution of the calculus during these 20 years (to a great extent due to Euler himself) and the modification of his orientation: while the manuscript reveals his approach to the infinitesimals as a pupil of Johann Bernoulli, in the book of 1755 he founded the calculus on his own ‘calculus of zeros’ …

“The exposition, which is very succinct, comprises two Parts, each with its own sequence of numbered chapters and articles. Despite the diversity of the topics and the impressive size, it is a complete, well-organized treatise. Many of the results are Euler’s own. The first Part is devoted to the differential calculus and its foundations, and the second Part contains applications of the differential calculus related to analysis and algebra. At the end of the first Part and in the last chapters of the second Part he states his intention to write a third Part, devoted to the geometrical applications of the differential calculus; but he never realizes it …

“In the extended introduction Euler explains the purpose of calculus, including, in particular, his famous ‘expanded’ conception of a mathematical function: ‘if some quantities depend on others in such a way as to undergo variation when the latter are varied, then the former are called functions of the latter’. This formulation has an extensive character; it embraces all the ways by which one quantity can be determined by means of others, and anticipates the definitions of later mathematicians such as N. I. Lobachevsky and J. P. G. Dirichlet. However, in his book Euler’s conception is not utilized in practice: functions are mainly considered as analytical expressions, including infinite series. His introduction also includes a very concise and schematic historical essay, a criticism of the foundation of the calculus on the infinitesimals, and a very brief survey of the book’s contents” (Demidov, p. 192).

The work is divided into two Parts. In the first Part, consisting of nine chapters, Euler begins with his general definition of a function:

‘If some quantities so depend on other quantities that if the latter are changed the former undergo change, then the former quantities are called functions of the latter. This denomination is of the broadest nature and comprises every method by means of which one quantity could be determined by others. If, therefore, *x* denotes a variable quantity, then all quantities which depend upon *x *in any way or are determined by it are called functions of it.’

In this first part, Euler “attempts to place differential calculus on a more rigorous basis… He rejects geometry as the proper foundation, emphasizing instead algebraic functions and arithmetic. He begins with the calculus of finite differences, treating differential calculus as a limiting case, arising when differences approach 0. In an effort to address earlier criticisms of the calculus, largely aimed at the concept of infinitesimals, he sets all infinitesimals precisely at 0, then elaborates a calculus of ‘zeroes of different order,’ *dx ^{2}*, for instance, being a zero of higher order than

*dx*. He also develops methods of differentiation through substitution of variables, and proves Nikolaus Bernoulli’s theorem on the irrelevance of the order of partial differentiation” (Parkinson,

*Breakthroughs,*p. 175). Other important results derived by Euler in the first Part include his famous theorem on homogeneous functions, and the condition for a differential to be exact.

The second Part, consisting of eighteen chapters, is devoted to the applications of differential calculus to the theory of infinite series. The first four chapters treat the general theory of finite and infinite series, and methods of expressing given functions as series, including but not limited to Taylor’s series (Chapter 4 contains the first example, in print, of a Fourier series). In chapters 5 and 6 Euler presents his way of finding sums of series, both finite and infinite, via his discovery of the Euler-Maclaurin summation formula. Chapter 9 deal with Newton’s method, and improvements thereof, for solving nonlinear equations. Chapters 10 and 11 apply differential calculus to the determination of maxima and minima of functions (of one and several variables), Chapter 12 with criteria for algebraic equations to have only real roots, and Chapter 15 with the indeterminate forms 0/0, ∞/∞ and ∞ - ∞.

“The entire exposition has an abstract arithmetic-algebraic character, and not without pride Euler wrote in its introduction: ‘all the exposition is bounded in the frame of pure analysis, so for the exposition of all the rules we did not use even one figure’ … such an exposition, thanks to the emancipation of the analysis from geometrical and mechanical ideas, liberates it from erroneous conclusions imposed by them… [It was also] a model for the future books on analysis and prepared the way for its arithmetization. At the same time [the present work] became a mine of concepts for several generations of mathematicians in the 18^{th} and 19^{th} centuries, for example, asymptotic developments, divergent series, and the zeta-function” (Demidov, pp. 196-7).

Calinger, *Leonhard Euler*, 2016. Demidov, ‘Leonhard Euler, Treatise on the Differential Calculus (1755),’ Chapter 14 in *Landmark Writings in Western Mathematics* (Grattan-Guinness, ed.), 2005. Enestrom 212. Honeyman 1069. Norman 733. Parkinson, *Breakthroughs,* p. 175. Poggendorff I, 690. Rouse Ball, *A Short Account of the History of Mathematics*, Macmillan, 1888. Sotheran 7686.

One vol. bound in two, 4to (250 x 194 mm) pp. xxiv, 278; [2], 281-880 (light uniform browning). Contemporary calf, spines richly gilt with two red lettering-pieces (rubbed with slight loss at extremities).

Item #5321

**
Price:
$13,500.00
**