## Introductio in analysin infinitorum.

Lausanne: M.-M. Bousquet, 1748.

First edition, entirely untouched in the printer's interim boards, the finest copy we have seen, of Euler’s great textbook on analysis, the only work of Euler listed in *Printing and the Mind of Man*. “In his ‘Introduction to Analysis’ Euler did for modern analysis what Euclid had done for ancient geometry” (PMM). The editors of the first volume of the *Introductio* in *Opera Omnia* (Ser. 1, Vol. 8) emphasize that the work “still today deserves to be not only read, but studied with devotion. No mathematician will put it aside without immense benefit,” and that the *Introductio* “marks the beginning of a new epoch and that this work has become influential for the whole development of the mathematical science by virtue of not only its content, but also its language.” The eminent historian of mathematics, Carl Boyer, in his address to the International Congress of Mathematicians in 1950, called it the greatest modern textbook in mathematics. Boyer cited Euclid’s Geometry as the greatest mathematical textbook of the classical period, perhaps of all time, appearing in over one thousand editions. For the medieval period, he chose the less well-known Al-Khowarizmi, largely devoted to algebra. But for “modern” times, Boyer made the case for Euler’s *Introductio* as the greatest modern textbook—and, appropriately, this time a text in analysis. Gauss said, “The study of Euler’s works will remain the best school for the various fields of mathematics, and nothing can replace it.” In 1979 André Weil remarked that he was trying to convince “the mathematical community that our students of mathematics would profit much more from a study of Euler’s *Introductio in analysis infinitorum*, rather than of the available modern textbooks” (Alexanderson, p. 635). “The *Introductio* contained expositions of algebra, trigonometry, analytical geometry (both plane and solid), infinite series and number theory, as well as a thorough and highly important investigation into the concept of function. Euler gave the first precise definition of ‘functions’, classified functions in the way generally used today, and presented the first clear statement of the idea that mathematical analysis is a science of functions. The work also standardized the still somewhat uncertain field of mathematical notation, putting into general use the symbols sin, cos, e and π; with few exceptions, the notation Euler used is the modern notation” (Norman). “In Euler’s mathematical work, first place belongs to analysis, which at the time was the most pressing need in mathematical science; seventeen volumes of the *Opera Omnia* are in this area. Thus, in principle, Euler was an analyst. He contributed numerous particular discoveries to analysis, systematized its exposition in his classical manuals, and, along with all this, contributed immeasurably to the founding of several large mathematical disciplines: the calculus of variations, the theory of differential equations, the elementary theory of functions of complex variables, and the theory of special functions” (DSB).

“Leonhard Euler (1707-83), the great Swiss mathematician, was born at Basel where he became the pupil of Johann Bernoulli, whose sons persuaded Catherine the Great to invite him to St. Petersburg. In 1747 he was invited by Frederick the Great to Berlin. In 1766 he became blind, but this did not interfere with his colossal output of work. Altogether he produced thirty-two books in Latin, German and French, and approximately nine hundred memoirs (making him the most productive mathematician who ever lived). He covered practically the entire field of pure and applied mathematics, but some of his best and most enduring work was done in analysis, which he established as an independent science.

“In his ‘Introduction to Analysis’ Euler did for modern analysis what Euclid had done for ancient geometry. It contains an exposition of algebra, trigonometry and analytical geometry, both plane and solid, a definition of logarithms as exponents, and important contributions to the theory of equations. He evolved the modern exponential treatment of logarithms, including the fact that each number has an infinity of natural logarithms. In the early chapters there appears for the first time the definition of mathematical function, one of the fundamental concepts of modern mathematics. From Euler’s time mathematics and physics tended to be treated algebraically, and many of his principles are still used in teaching mathematics” (PMM).

“The topics in the *Introductio* include: the definition of a function and the distinction between single-valued and many-valued “functions”, or odd and even functions. There are questions of polynomial factorization related to the Fundamental Theorem of Algebra; substitutions, infinite series, exponentials and logarithms, and expansions into partial fractions; multiple angle formulas; evaluations of ζ(2*k*), for *k* a positive integer (the famous so-called Basel problem); infinite series-infinite product formulas; and partitions of integers. Much of the latter part of volume 1 is devoted to generating functions and continued fractions. Volume 2 is largely concerned with classification of curves and the study of the conics, and higher degree curves and surfaces. D. J. Struik referred to it as the first textbook in analytic geometry.

“What makes this remarkable as a textbook is that the later topics are closely related to Euler’s research, and in some cases their appearance here, and sometimes in earlier papers, opened up whole new fields of research. We cannot say that of most “elementary” modern textbooks. Whether an idea appeared first in the *Introductio *is not very important. To be sure, Leibniz and Johann Bernoulli had come close to our modern definition of function, but it was here that Euler spelled it out clearly …

“This is one of the first mathematics books that “looks modern”. Except for “*xx*” to denote “*x*^{2}” and a few other conventions carried over from earlier times, it introduced modern notation: *f*(*x*) for function,* e* for the base of the natural logarithm (appearing in print here for the first time but used earlier by Euler in a manuscript not published until 1862 in his *Opera posthuma mathematica et physica*), and π (though it had been used by Christian Goldbach two years earlier). Euler proved that *e* is irrational in the *Introductio*, though the fact could be deduced from work in a paper of 1737. The proof in the *Introductio* using continued fractions was not, however, to everyone’s satisfaction. In 1748 Euler still used √−1 and did not introduce “*i*” for this number until 1777. Other problems covered in the *Introductio *had occurred earlier in Euler’s reports to the St. Petersburg Academy and in published papers. The genius exhibited in the *Introductio* was in the way he formulated the problems and the methods he introduced. In the problem of partitions he developed powerful methods using generating functions. Again, using infinite series and products he came up with the relationships that inspired Riemann’s work on the ζ-function. It is often assumed that the formula usually called “Euler’s formula”, relating the exponential and the trigonometric functions, appeared first in the *Introductio*, but the formula was probably observed earlier by Roger Cotes. As was often the case, however, Euler did more with it. He was a master expositor, so when he took a subject he molded it into something that people could understand, thereby prompting subsequent development.

“Beyond the beautiful mathematics contained in the book, however, there is the beauty of the original edition of this set of volumes as physical objects. Most of Euler’s books were published by the press of the Academy in St. Petersburg. Those are rather sober, straightforward products of the printer’s art. When Euler wrote the *Introductio*—probably around 1745—he was working in Berlin. Unfortunately the Berlin Academy published journals but not books, and the St. Petersburg Academy was having financial problems. So Euler had to look for a publisher elsewhere. In 1744 his *Methodus inveniendi lineas curvas* had been published by the firm of Marc-Michel Bousquet, located in Lausanne and Geneva. So he approached them about publishing the *Introductio*. This firm had also published in 1742 a beautifully decorated set of four volumes, the *Opera Omnia* of Jean Bernoulli. Handsome though these earlier volumes were, fortunately for us, the publisher seems to have given this set of two volumes an even more generous budget, so the book is decorated with an elegant engraving for a frontispiece, a title page in two colors, a fold-out, engraved dedication portrait of the physicist-astronomer-geologist Jean Jacques Dortous de Mairan, who was a patron of the publisher, secretary of the Académie Royale des Sciences and member of various other scientific societies.

“The engraved frontispiece is the work of Pierre Soubeyran, based on the design of a member of the de La Monce family of portrait painters. It shows two classically attired women under an arch flanked by columns. Hovering overhead is a putto. One woman is gesticulating while the other is writing mathematical formula in a book and the putto is offering inspiration. There are mathematical instruments on the floor and a book open to pages with “Calcul Differentiel” and “Table des Sinus”. The inscription at the top of the frontispiece gives the title of the *Introductio* in French, *Analyse des infiniments petits*, which, curiously, happens to be the name of L’Hôpital’s famous text of 1696. The title of the first French translation of the *Introductio* in 1796, on the other hand, was *Introduction de l’analyse infinitesimale*” (Alexanderson, pp. 636-7).

PMM 196; Norman 732. Alexanderson, ‘Euler’s *Introductio in analysis infinit*orum,’ *Bulletin of the American Mathematical Society* 44 (2007), pp. 635-8.

Two vols. 4to (247 x 203mm), pp. [vi], iii-xvi, [3] 4-320, folding printed table after p. 224; [iv] [3] 4-398 [2], titles printed in red and black with engraved title vignette, engraved frontispiece by Soubeyran after De la Monce, engraved folding portrait of dedicatee Jean Jacques Dortous de Mairan (1678-1771) by Etienne Frequet (1719-94) after J. Tocquet, and 40 folding engraved plates (a few gatherings a little browned). Absolutely untouched in original interim boards as it came from the printer.

Item #5226

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Price:
$25,000.00
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