EULER, Leonhard.

Vollständige Anleitung zur Algebra. Erster [- Zweiter] Theil.

St. Petersburg: Imperial Academy of Sciences, 1770.

First edition in German, its language of composition, and first obtainable edition, of Euler’s ‘Complete instruction in algebra’, which takes the reader step-by-step from the basics, including negative and imaginary numbers, through to the solution of quadratic, cubic and biquadratic equations, and concluding with a long section on Diophantine problems (i.e., the solution of algebraic equations in whole numbers). “In 1770 Euler published his Vollständige Anleitung zur Algebra . . . The first volume treats of determinate algebra. This contains one of the earliest attempts to place the fundamental processes on a scientific basis: the same subject had attracted D’Alembert’s attention. This work also includes the proof of the binomial theorem for an unrestricted real index which is still known by Euler’s name . . . The second volume of the ‘Algebra’ treats of indeterminate or Diophantine algebra. This contains the solutions of some of the problems proposed by Fermat, and which had hitherto remained unsolved” (Rouse Ball, A Short Account of the History of Mathematics, p. 397). “The entire second section of volume 2 was devoted to Diophantine analysis; it gave his proof of Fermat’s last theorem for the case n = 3 [i.e., that the sum of two cubes of whole numbers cannot be a cube of a whole number]” (Calinger, Leonhard Euler, pp. 477-8). Euler’s proof, “based on the method of infinite descent and using complex numbers of the form a + b√-3, is thoroughly described in his Vollständige Anleitung zur Algebra” (DSB). It was “published in many editions in English, Dutch, Italian, French, and Russian, [and] greatly influenced nineteenth- and twentieth-century texts on the subject” (ibid.). “There is only one other book in the field of mathematics which in the entire history of culture has had a comparable success in sales: the Elements of Euclid, after the Bible the most frequently printed book of all” (Fellmann, Leonhard Euler, p. 121). Probably written in 1765/66 during his Berlin period, the ‘Algebra’ was dictated to an assistant who was not educated in mathematics, but first appeared in 1768/69 after Euler’s second move to St. Petersburg, in a Russian translation by his students Peter Inokhodtsev and Ivan Yudin; only in 1770 was the work published in the original German. Euler’s ‘Algebra’ is designed for self-study and includes many examples, some worked out in the text and other’s left for the reader to solve. Euler took many of his examples from Stifel’s edition of Rudolff’s Coss (1553), probably because this was the book from which he had learned algebra. This German translation is rare on the market: ABPC/RBH lists only two other copies since the Norman sale. The Russian edition is of extreme rarity – we know of no copy having appeared on the market, and OCLC lists just a single copy (in Russia).

“Most likely still in his Berlin period, Euler set out to write his probably most popular mathematical book, his two-volume opus Vollständige Anleitung zur Algebra, which he dictated to his assistant, a mathematically completely unencumbered former tailor, who – allegedly a mediocre fellow – was then to have understood it all. It is a legend, initiated by an editorial ‘pre-report’, that Euler wrote resp. dictated to his assistant the Algebra immediately after his loss of sight in Petersburg for the purpose of selfcontrol, because firstly, Euler absolutely did not need such ‘self-control’, and secondly, he became almost completely blind, as is well known, only after the cataract operation of 1771, when the book had long been out in several editions. Thirdly, in the text of the first volume, there are a few passages which can, if not must, be interpreted as indicating a date 1765/66 of composition.

“This work – especially fascinating in view of Euler’s masterly didactic skill – truly became a bestseller. It appeared 1768/69 first in Russian translation, then 1770 in the original German version, 1774 in the French translation of Johann III Bernoulli, and finally in the English and Dutch language in many editions. The Algebra, as the book is called for short, introduces an absolute beginner step by step from the natural numbers via the arithmetic and algebraic principles and practices through the elementary theory of equations right to the most subtle details of indeterminate analysis (Diophantine equations); it still is – in the judgment of today’s foremost mathematicians – the best introduction into the realm of algebra for a ‘mathematical infant’. With good reason, the ‘great Euler edition’ of 1911 was inaugurated with this volume. The great mathematician Lagrange, Euler’s successor at the Berlin Academy, did not consider it below his dignity to provide the work in the first French edition with valuable additions. They fill there 300 pages, and in the first volume of Euler’s Opera omnia they were reprinted in the original French language.

“In the German-speaking areas, the Algebra experienced the widest circulation through its inclusion in Reclam’s Universal Library, where it figures as the only mathematical book and was printed from 1883 till 1942 in 108,000 copies” (Fellmann, pp. 121-2).

The work consists of two parts. The first part has three sections, and the second part has two sections. Except for the last section, the contents are elementary; namely the four basic rules of numbers and polynomials, exponential and logarithmic functions, theory of the proportion, arithmetic series, geometric series, decimal fractions, and algebraic equations of degree at most four, etc. The second section of the second part is entitled ‘On indeterminate analysis,’ i.e., Diophantine problems, and consists of 15 chapters. Among these, the first three chapters are about linear equations; quadratic equations are treated in chapters 4-7, and cubic and quartic equations in chapters 8-10. Chapter 11 returns briefly to quadratic equations, developing some identities, and chapter 12 solves some quadratic equations in whole numbers. Chapter 13 treats Fermat’s last theorem for the cases of fourth powers, and the two remaining chapters deal with the theorem for cubes.

“In his selection of problems in the Algebra, Euler shows himself familiar with the typical recreational and practical problems of Renaissance and sixteenth-century algebra books. An extensive historical database with algebraic problems, immediately reveals Euler’s use of the Stifel’s edition of Rudolff’s Coss for his repository of problems. This work, published in 1525 in Strassburg, was the first German book entirely devoted to algebra. Stifel used many problems from Rudolff in his Arithmetica Integra of 1544 and found the work too important not to publish his own annotated edition. The table below shows an example of the many textual evidences of Euler’s use of the book.

“The first volume of Euler’s Algebra on determinate equations contains 59 numbered problems. Two thirds of these can be directly matched with the problems from Rudolff. Some are literal reproductions, as the example given. Others were given new values or were slightly reformulated. The second part on indeterminate equations also has 59 problems and although the correlation here is manifest less, many problems still originate from Rudolff.

“While the first sections include some illustrative examples, all the problems appear in the second part on equations, exactly as in Rudolff’s book. The third chapter dealing with linear equations in one unknown has 21 problems. They clearly show how Euler successively selected suitable examples from Rudolff’s book. The problems are put in practically the same order as Rudolff’s. They include well-known problems from recreational mathematics: the legacy problems, two cups and a cover, alligation, division and overtaking problems. The fourth chapter deals with linear problems in more than one unknown, including the mule and ass problem, doubling each other’s money and men who buy a horse. The fifth chapter is on the pure quadratic equation with five problems all taken from Rudolff. The sixth has ten problems on the mixed quadratic equation, of which nine are taken from Rudolff. Chapter eight, on the extraction of roots of binomials, has five problems, none from Rudolff. Finally, the chapter of the pure cubic has five problems, two from Rudolff and on the complete cubic there are six problems, of which four are from Stifel’s addition. Cardano’s solution to the cubic equation was published in 1545, between the two editions of the Coss. While Euler also treats logarithms and complex numbers, no problems on this subject are included.

“Having determined the source for Euler’s problems, the question remains why he went back almost 250 years. The motive could be sentimental. In the Russian Euler archives at St-Petersburg a manuscript is preserved containing a short autobiography dictated by Euler to his son Johann Albrecht on the first of December, 1767. He states that his father Paulus taught him the basics of mathematics using the Stifel edition of Christoff Rudolff’s Coss. The young Euler practiced mathematics for several years using this book, studying over four hundred algebra problems. When he decided to write an elementary textbook, Euler conceived his Algebra as a self-study book, much as he used Rudolff’s Coss, the educational value of which Euler amply recognized” (Heeffer).

Honeyman 1075; Eneström 387 & 388; Norman 735. Heeffer, ‘The origin of the problems in Euler’s Algebra,’ Bulletin of the Belgian Mathematical Society–Simon Stevin 13 (2005), pp. 949–952.

2 vols. 8vo (184 x 115 mm). Bound in one, fine contempoary polished calf with raised bands and richly gilt spine. Pp. [16] [1-3] 4-356 (2 errata); [2] [1-3] 4-532 [2] (complete). Fine and clean, a very nice copy.

Item #5122

Price: $15,000.00