Euler and Modern Science

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Brand new. We distribute directly for the publisher. This book is an English translation of a collection of papers in Russian from a conference held in Moscow and St. Petersburg ... in 1983, 200 years after Euler's death. Two of the Russian papers in the collection are themselves translations from the German. The present English translation appears in 2007, 300 years after his birth. We speak of the age of Euler. A justification of this term is provided by a list of scientific terms connected with Euler's name and his many contributions to pure mathematics, well-known in the mathematical community and, in part, covered in this volume. What makes this collection remarkable, though, is the extensive treatment of Euler's contributions outside pure mathematics, astronomy, celestial mechanics, ballistics, history of science, instruments and technology, physics, mechanics, hydromechanics, mechanics of elastic systems, variational principles of mechanics, and a mathematical theory of music. Euler was also actively involved in the preparation and sale of scientific books, and while in Berlin, of scientific almanacs.In addition to this survey of his contributions to science, we find here material otherwise very difficult to find detailed accounts of Euler's family life and the careers pursued by his children and grandchildren. Readers otherwise well-informed about Euler and his work will find here much to enhance their appreciation of this extraordinary scientist and human being. Read more Show Less

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We speak of the age of Euler. A justification of this term is provided by a list of scientific terms connected with Euler's name and his many contributions to pure mathematics, well-known in the mathematical community and, in part, covered in this volume. What makes this collection remarkable, though, is the extensive treatment of Euler's contributions outside pure mathematics - astronomy, celestial mechanics, ballistics, music and many other areas. In addition to this survey of his contributions to science, we find also rare, detailed accounts of Euler's family life and the careers pursued by his children and grandchildren. Readers otherwise well-informed about Euler and his work will find here much to enhance their appreciation of this extraordinary scientist and human being.

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Leonhard Euler: his life and work
A. P. Yushkevich

Of all famous scientists of all times and peoples, there is scarcely one who is recalled more often in today's mathematics courses than Leonhard Euler: In the differential calculus the theorem on homogeneous functions bears his name; in the integral calculus the substitution used to rationalize quadratic irrationalities and the Euler integrals of the first and second kind" are due to him; in the theory of ordinary differential equations, two classes of linear differential equations with variable coefficients were introduced by him, as was the method of approximate integration serving as the basis for Cauchy's well-known theorem on the existence of solutions; in the calculus of variations there is the differential equation used to find the functions at which a given functional takes on extreme values, and one of the so-called direct" methods; in the calculus of finite differences the Maclaurin-Euler summation formula; in analytic function theory we have the Cotes-Euler formula relating the exponential function to trigonometric functions, and also the d'Alembert-Euler equations, better known as the Cauchy-Riemann equations; in the theory of infinite series Euler invented one of the methods for summing divergent series and for increasing the rate of convergence of convergent series; in differential geometry we have his formula for the curvature of a normal section of a surface; and in topology the fundamental Euler characteristic of a topological complex. This list is far from complete: for instance one might add to it Euler's function from number theory, Euler's constant, Eulerian numbers, the Euler angles, and so on.We have included here only those instances evoking the great scholar's name in present-day syllabi at the university level or that of higher technical colleges; in fact many of Euler's methods and formulae are usually expounded without acknowledgement, or, worse still, with names of later mathematicians attached. For example, Euler had derived analytically the so-called Fourier formulae of the theory of trigonometric series 30 years before Fourier. To Euler is due the present standard exposition of logarithms and trigonometry in high school. Many of the mathematical symbols suggested by Euler have become firmly established, for instance the imaginary unit i, the natural logarithmic base e, the symbol D indicating a finite difference, and S denoting summation.
All of these formulae, theorems, methods, and symbols partially reflect Euler's enormous contribution just to mathematics; in the present essay I will for the most part not consider mechanics, astronomy, physics, geography, and technology, in all of which his achievements are remarkable. The 18th century is often called the century of enlightenment. As far as the exact sciences are concerned it would be more appropriate to call it the century of Euler, even though there were then also active such outstanding scientists as Clairaut, d'Alembert, Lagrange, Cramer, Lambert, de Moivre, Stirling, Maclaurin, and, in Russia, Lomonosov.
In my talk I shall attempt to describe Euler's life briefly, and to give an overall characterization of his contribution to science, with the emphasis mainly on mathematics.
Euler's life can be divided into four periods: his first 20 years, spent in Basel, the next 14 years working in St. Petersburg, then 25 years in Berlin, and, finally, a second 17-year-long Petersburg period.
Leonhard Euler was born on April 5, 1707, in the city of Basel, into the family of a pastor of modest means, and passed his childhood in the town of Riehen, where his father had obtained a living. His first mathematics lessons were taught him by his father, who had in his student years attended the lectures of Professor Jakob Bernoulli, the nearest mathematical successor to Leibniz. Later a special mathematics tutor was hired, albeit a theologian by profession.
At that time Euler studied the difficult, although by then outdated, Algebra of Ch. Rudolff 1(1522), as revised by M. Stifel2 , another prominent 16th century algebraist. In the autumn of 1720, at the age of thirteen and a half, Euler became a student in the faculty of liberal arts of the University of Basel. The university had at that time no faculty of physics and mathematics, and offered no special preparation for a career in mathematics. Students graduating from the liberal arts faculty might continue their education in any of the faculties of divinity, law, or medicine.
The mathematics lectures attended by Euler covered only the elements of mathematics and astronomy. However they were delivered by Johann I Bernoulli, the world's most powerful mathematician of the period—if we exclude the aging Newton—, pupil of his elder brother Jakob, and the latter's successor to the Chair in Basel. Euler was completely absorbed by mathematics, and appealed to his professor to tutor him privately. Bernoulli refused to give separate lessons, but, having noticed the boy's exceptional gift, helped him in a different way. He systematically recommended specific mathematical literature for him to work through, and allowed him to come to his home on Saturdays to discuss difficult questions. For Euler this was the best possible scientific education; in his case no further help was necessary. He soon became close to the sons of his mentor: Nicolaus II, Daniel, and Johann II, and also to Nicolaus I Bernoulli, the nephew of Johann I. These all entertained a lively interest in mathematics, taking it as their calling. Euler was an excellent student, and also actively participated in university life. A year after graduating he gave a lecture in which he compared the views of Newton and Descartes on natural philosophy, following which he was awarded the Master of Arts degree.
His father thought that it would be best for his son to take up, like himself, the career of pastor, and, bowing to his father's wish, the young Euler enrolled in the Faculty of Divinity. However there he was bored, and his studies were not very successful. His father did not try to oppose his son's inclinations further, and Euler totally immersed himself in mathematics. In 1726-1727 his first two papers were published in the international journal Acta eruditorum. These both dealt with problems in analysis, which was then the main focus of research of the Bernoullis and other researchers. It was then also that he took part in the competition of the Paris Academy to find the optimal placement of a ship's masts. Though not awarded the prize, Euler's entry was well received, and was published in Paris in 1728. (I note here parenthetically that between 1738 and 1772 Euler was awarded the prize of the Paris Academy 12 times for a variety of entries on problems of applied mathematics and technology.)
At the time when the 20-year-old Euler became adjunct of the Petersburg Academy—where he was immediately given the opportunity of working not in physiology but in the mathematical sciences—, intensive scientific work was already being carried out there. Nowhere else in the world were conditions more favorable for Euler's further scientific development than in St. Petersburg. In the first place, he found himself a member of a cholarly collective with mutual interests and aims, providing stimulus to all concerned. Twice a week scientific colleagues gathered at a collegial conference where they announced and discussed their latest work and letters from foreign scholars, as well as current questions of academic life, such as the content of the Commentarii of the Academy, a yearly publication of which the first issue after 1726 appeared in 1728. (The title page of an issue of the Commentarii is reproduced below.) Of especial importance for Euler were his regular discussions with Daniel Bernoulli, with whom he shared an apartment until Bernoulli's departure in the spring of 1733. Apart from Bernoulli, there were among the Academicians the noted mathematician J. Hermann, a former student of Jakob Bernoulli, the geometer F.-Ch. Meyer, the physicist and mathematician G. W. Kraft, Ch. Goldbach, of wide education and extraordinary perspicacity, especially in number theory, the well-known astronomer J. N. Delisle, and others.
Euler's scientific correspondence began immediately to be of importance—especially that with his former teacher Johann Bernoulli, which continued for over 20 years from the autumn of 1727. In view of the almost complete lack of scientific journals and international conferences, which so enhance the scientific life of today's scholars, for those living in different towns or countries in the 17th and 18th centuries scientific correspondence provided the most important means of exchanging information quickly. Many such letters were in effect detailed critiques of current research. When in 1728 Goldbach had to transfer to Moscow for a few years, he and Euler began a correspondence of great substance, comprising 196 letters, which continued until Goldbach's death in 1764. The departure of Daniel Bernoulli likewise led to an epistolary exchange of ideas and problems between him and Euler lasting many years; 90 letters from this correspondence have been preserved. In fact Euler corresponded with nearly all contemporary mathematicians of note: with, in addition to those already mentioned, Stirling, Clairaut, d'Alembert, Lagrange, Cramer, Lambert, and many others. Characteristically, he took great pains over the preservation of his correspondence. As he put it: If anyone should take the trouble to read it [his correspondence], he will find in it many important things whose publication would be more to the public taste than works of the deepest conception". It is worth noting that the Petersburg Academy defrayed for its members the cost of their postage, at that time quite substantial.
An exceptionally agreeable condition of Euler's employment in St. Petersburg consisted in the possibility of regular publication of his papers in the Academy's journal Commentarii (in Latin, the lingua franca of the 18th century scholarly world) and of certain of his books. Euler remains the most productive mathematician of all time, and it is remarkable that his literary output did not decrease with time. I once had occasion to count by decades the number of works he prepared for publication, not distinguishing voluminous books from short articles, and leaving aside a relatively few undated items. Of the overall number of about 850 individual works (including more than 20 large monographs), the distribution of percentages of items prepared for publication in successive decades is as follows:

1726—1734 approx. 5% 1755—1764 approx. 14%
1735—1744 approx. 10 1765—1774 approx. 18%
1745—1754 approx. 19% 1775—1783 approx. 34%

Incidentally here one should take into account the circumstance that during Euler's second period in St. Petersburg, when he was almost blind, he was greatly helped by first-rate scientific secre world would he have been able to publish his work on such a scale as in Russia, where the publications of the Academy were so generously financed by the state.
Of course the above statistics are insufficient for determining the course of Euler's mental development. As is often the case with mathematicians, many of Euler's interests and ideas were formed in his youth, although even in his declining years he invented new approaches and methods and remained fully receptive to the discoveries of his younger contemporaries. Over the span of decades he returned again and again to problems which had at one time piqued his interest, but had for one reason or another been set aside or else solved in a way unsatisfactory to him, and his scientific notebooks, which he kept from 1725 to 1783 (12 notebooks totaling 4000 pages are preserved in the Archive of the Academy of Sciences of the USSR), bear witness to an untiring accumulation of material for further treatment. All his life he was unable to keep pace in written form with the scientific ideas teeming in his mind. Many of the research themes he brought to perfection in the mid-18th century or even later, can be traced back to the early part of his first Petersburg period or even to his years in Basel. For example, it was in Basel that he conceived the project of reformulating in terms of the infinitesimals of the Leibnizian school Newton's mechanics of point-particles, expounded in the latter's Philosophiae naturalis principia mathematica (1687) and in Hermann's Phoronomia (1716), where, as Newton himself asserted, the proofs are couched in synthetic-geometric terms not useful for obtaining uniform solutions of subsequent problems. Euler realized these intentions ten years later in his two-volume work Mechanics, that is, the science of motion, expounded analytically (SPb., 1736), which became the starting point for the whole of the future development of mechanics. The corresponding claim may be made also for his investigations in the theory of music, set out in his An attempt at a new theory of music, clearly expounded on the basis of the truest elements of harmony (SPb., 1739).
As a state institution, the Petersburg Academy of Sciences was called upon to provide answers to important practical questions. The university and gymnasium attached to the Academy were engaged in preparing national cadres of scientists; many Russian scholars of the 18th century, including the great M. V. Lomonosov, were students at these academic institutions, which were closed only at the beginning of the 19th century, having become redundant in connection with the reorganization of the whole system of national education. At that time only the graduate program, as we would now call it, remained linked to the Academy, and that only briefly. (It was revived under Soviet rule.) The Academy was also charged with providing technical expertise of many kinds, but especially important among its responsibilities was a complete exploration of the little known regions of the enormous Russian empire, in particular all of Siberia
including Kamchatka. The famous academic expeditions of the 18th century
were of first importance in this regard.
Euler took an active part in many such enterprises, which incidentally facilitated his learning Russian well, so that, unlike certain other foreign academicians, he was able to communicate with those Russians not familiar with any other language. For the students of the Academy's gymnasium he wrote a Guide to arithmetic, published in German in 1738, and subsequently translated into Russian, which greatly influenced the teaching of that subject as well as authors of later texbooks. For several years Euler, in collaboration with J. N. Delisle and the academician G. Heinsius, astronomer and geographer, devoted a considerable amount of time to cartography, even drawing certain maps himself. The importance of having accurate geographical maps, hitherto ignored, had become clear to the government, in particular in connection with the need to define the borders with neighboring foreign countries accurately. Euler's cartographical work was later reflected in his theoretical investigations: 40 years on, in the 1770s, he was the first to apply the theory of functions of a complex variable to cartography.
As mentioned earlier, already in Basel Euler had worked on problems related to shipbuilding. In St. Petersburg in the mid-1730s he returned to this broad theme, and in 1740 undertook to compose a special treatise on the topic. It is hardly necessary to explain the timeliness of this work for Russia, which had emerged under Peter I as a major sea-power. Euler completed the fundamental two-volume work Naval science, or a treatise on shipbuilding and navigation in Berlin, whence he sent the manuscript back to St. Petersburg where it was published in 1749. This work was of fundamental importance for mechanics as a whole, and not just for fluid mechanics and the kinematics and dynamics of a rigid body. However by the character of its exposition it was not suitable as a textbook, and almost a quarter of a century later Euler wrote for the students of Russian naval colleges the more accessible The complete theory of shipbuilding and navigation, which first appeared in French in 1773, and soon afterwards, in 1776, was republished in Paris, and then in Russian, English and Italian translations. The Russian version of 1778 was prepared with useful supplementary explanations by M. E. Golovin, a former student of Euler's and a nephew of Lomonosov.
Euler was able to successfully combine the fulfilment of these and other such commitments with work on purely theoretical problems. He presented his first scientific communication at the Academy's regular conference of August 5th, 1727 and thereafter became its most frequent speaker. It was then that his articles began to appear, in Latin, in the annual of the Academy, which went under a succession of names: Notes (Commentarii), New Notes, Proceedings, and New Proceedings. From the second volume of the Commentarii of 1727, which appeared in 1729, Euler continued throughout his entire life to contribute articles to the Academy's journals, sometimes as many as ten to a single issue. Over his first Petersburg period he prepared over 80 papers and published around 50 on various questions of pure and applied mathematics. Some of these continued or brought to completion investigations begun in Basel; however in the majority of cases these works struck out in new directions in which Euler's predecessors had taken but the first steps; for instance his solution of several problems of the calculus of variations, results on the integration of ordinary differential equations, the introduction of the gamma-, beta-, and zeta-functions, investigations into infinite series, including asymptotic expansions, results on continued fractions, number theory, topology, and so on. This was the period also of his first original researches in astronomy. And of course at the same time he was working on the new problems that would occupy him over the next decade.
Except for one misfortune—the unexpected loss of sight in his right eye
in 1738—Euler's personal life was also going well at this time. At the beginning of January 1734 he married Catharina Gsell, the daughter of a painter of the Petersburg Academy of Sciences, also from Switzerland. In that same year their son Johann-Albrecht was born, and in 1740 a second son Karl. Euler acquired a house on Vasil'evskii Island, not far from the Academy, in which the family of his younger brother Heinrich, another artist, also took up residence. It might seem that Euler had put down strong roots in the Russian capital. However after the death in 1740 of the empress Anna Ioannovna and the proclamation of the three-month-old Ioann VI as emperor, the political situation in St. Petersburg became unstable on account of the struggle for power by rival court factions. At first Biron, the favorite of the late empress, became regent, but was very soon sent into exile, being replaced by the mother of Ioann VI, Anna Leopol'dovna. However her hold on the reins of power also turned out to be insecure. The discontent of the Petersburg nobility and guard over the undue influence of foreigners at court presaged further complications. This political uncertainty affected the activities of the Academy of Sciences. Somewhat earlier, in the summer of 1740, Euler had received an invitation from the Prussian king Friedrich II to transfer to Berlin, where he planned to organize an academy of sciences of no less repute than the academies of Paris and St. Petersburg. The invitation was renewed in the Winter of 1741, and an anxious Euler answered in the affirmative. He left St. Petersburg, accompanied by his whole family, on June 19th, and by July 25 they were already in Berlin. In the meantime in Russia, at the very beginning of 1742, following on a palace revolution, Elizaveta Petrovna, daughter of Peter I, ascended the throne.
Translation of excerpts of the letter from Euler to Teplov reproduced on these two pages:

My Dear Sir
...........................................

He is fully competent to be in charge of the printing of my book on navigation and shipbuilding, and for this reason I humbly entreat you to allow me to keep my book here with me while Mr. Oechlitz(?) is on his way to you, since I wanted to elucidate to him personally many circumstances surrounding that book, which would be very difficult to do in writing. I am expecting a response from him this week, which I shall at once communicate to you so that he can leave shortly after receipt of the payment for his trip. And I humbly request that following on his arrival the publication of my book not be delayed, for I fear that that my conceptions will with time cease to be new as French mathematicians are trying very hard in their research on this material, and have already published some important discoveries, which I had made many years earlier than they. I am very unhappy that in the negotiations with Mr. Kies I was unlucky, since if the declared final conditions had arrived within two or three months, he would have accepted them immediately; for Mr. Braun will tell you that at that time he was out of favor with our President, but is now back in favor with him. I asked him to convey his opinion to me in writing, and I am attaching his letter to this. In the meantime I have discouraged his hopes so that he not remain in suspense, since it seems to me that he would certainly not refuse. All of my family humbly bows to you, and I beg that you accept my deepest respect. Convey to his highness the Count my most complete veneration.
My dear Sir
Your most obedient servant
L. Euler.

Berlin
9 April, 1748."

Thus ended the first Petersburg period of Euler's life. In 14 years he had
achieved a great deal in furthering science in Russia and the world at large. At the same time he had a clear understanding of the invitation to St. Petersburg as being of decisive significance in his life, and expressed this in one of the letters he sent to the Petersburg Academy in 1749; after all, it was only in St. Petersburg that research in mathematics, which from his youth he saw as his main vocation, could have become his chief occupation and develop so successfully.
Euler arrived in Berlin as a world-renowned 34-year-old scholar, whom his old teacher Johann Bernoulli called with complete justification, in a letter to Euler written in 1745, princeps mathematicorum"—chief mathematician. In Berlin Euler actively assisted\break Friedrich in organizing the new Academy of Sciences. Although it is true that that they did not have to start from scratch, since as early as 1700 there had been established in Berlin, on Leibniz' initiative, a Science Society, the members of this Society were not active scientists, and furthermore Leibniz himself lived not in Berlin, but in Hannover. Under the father of Friedrich II—the uneducated and despotic soldier-king Friedrich Wilhelm I—the Berlin Science Society had eked out a pitiful existence. Friedrich II, acceding to the throne in 1740, was as militaristic a ruler as his father—it is enough to recall that 15 of the first 23 years of his reign were spent in warfare: the 8 years of the war of the Austrian succession (1740—1748), and then the seven years' war (1756—1763). However at the same time he was a typical representative of the enlightened absolutism of the 18th century, providing support—partly from considerations of prestige—for science and the arts, and permitting, within bounds consistent with absolute monarchy, moderate philosophical freedom of thought. In his personal tastes Friedrich II was an adherent of French culture, corresponded amiably with Voltaire (but subsequently quarreled with him), wrote chiefly in French, and made French the official language of the new Academy. And while the first president of the Petersburg Academy had been the German doctor to the imperial family L. Blumentrost (born, it is true, in Moscow), to take up the post of first president of the Berlin Academy of Sciences and Literature, which replaced the old Science Society in 1744, Friedrich II invited the French scholar P. L. Moreau de Maupertuis, who especially impressed the king with his courtly manners. Maupertuis took up his duties in 1746, establishing from the start the best of relations with Euler, who in 1744 had been appointed director of the mathematics department. With Friedrich, however, Euler's relations were of another sort. The whole of his upbringing had been such that the king of mathematicians was in his essence a typical Bürger of Basel, utterly incapable of playing the role of salon philosopher, moreover a pious Bürger and therefore opposed to freethinking, especially the French variant patronized by the king of Prussia. Friedrich II was disdainful of mathematics, unless it proved useful in solving problems of exclusively practical importance. In essence their relations were such that while neither felt any empathy for the other, each tolerated the other out of necessity.
In Berlin Euler had to do a great deal of work of an organizational character, and in addition fulfill a variety of the king's personal commissions. Only his extraordinary capacity for work and ability to rationalize his time allowed him to simultaneously cope with the many duties imposed on him and carry out his scientific research at an ever-increasing rate.
As a member of the directorate of the Berlin Academy, and assuming in effect the function of president during Maupertuis' frequent absences, Euler was occupied with the building and equipping of the observatory, supervising the preparation and publication of maps, the ordering of seeds and plants for the botanical gardens, hiring, firing, and pension arrangements of the Academy's employees, and in addition to all this, the publication of a series of annual calendars, the income from which accounted for most of the Academy's budget. Taking the king's interests into account, Euler translated from English the best text on ballistics of that time, written by B. Robins and published in 1742. Euler provided his German version of The New Elements of Artillery, which appeared in 1745, with explanations and fresh applications that significantly increased the size—and price—of the book; these additions were subsequently incorporated into a new English edition and a French translation. I note incidentally that Euler had already become acquainted with problems of ballistics in St. Petersburg, where in 1727 he and D. Bernoulli were present at a testing of artillery weapons; his brief essay on these trials, first printed in 1862, was for a long time preserved in the Archive of the Academy of Sciences of the USSR.
By commission from the king, Euler was obliged to occupy himself also with hydraulic engineering, in particular to act as consultant in connection with the work of leveling the Finow canal between the rivers Havel and Oder, and with the water supply of the royal residence at Potsdam with its many fountains on various levels. Through an exchange of letters with J. A. Segner, professor of mathematics first at Göttingen and then Halle, Euler became familiar with the details of an hydraulic machine invented by Segner, the simplest form of which is the Segner wheel", familiar to schoolchildren. Over the period 1750—1753 Euler made several significant improvements to this machine, and concurrently laid the foundations of the theory of hydraulic turbines. In conjunction with all of these activities he produced a long series of theoretical articles on the theory of mechanisms and machines, and most importantly on fluid mechanics.
Questions about the mechanics of fluids (or hydrodynamics) had been investigated in ancient times—it is enough to recall Archimedes' law concerning bodies immersed in a liquid. Fresh progress in fluid mechanics was made in the 16th and 17th centuries by Stevin, Galileo, Torricelli, Pascal, Newton, and others, first in hydrostatics, and then in hydrodynamics, ultimately resulting from the need to solve practical problems concerning navigation, and the construction of canals, dams, pumps, water mills, etc. Euler's notebooks show that as early as 1727 he had conceived the idea of writing a substantial work on hydraulics, but
then refrained in order not to compete with Daniel Bernoulli, who was already well launched in that direction and while still in St. Petersburg had completed the first version of a large treatise, the second version of which was printed in Strasbourg in 1738 under the title Hydrodynamics. In the 1730s Euler was also very much taken up with questions from fluid mechanics, on both the theoretical plane and with applications in view, one outcome of which was the treatise Naval
Science noted above. However a more profound treatment of fluid mechanics became possible only later, during Euler's time in Berlin, when towards the end of the 1740s, following on the work of d'Alembert, he began to create the apparatus of the theory of partial differential equations. I shall return below to the decisive first step, taken in the context of the theory of elasticity, by d'Alembert and Euler; for now I merely note the series of classic articles on fluid mechanics written by Euler in the period 1757-1761 published in Berlin and St. Petersburg. In these papers Euler provided the foundations of the modern theory of equilibrium and motion of an ideal fluid; they contain the equation of continuity and the general differential equations of both hydrostatics and hydrodynamics, which with complete justification bear the name of Euler.
While I shall not go into detail concerning the other area of mathematical
physics close to fluid mechanics, namely aerodynamics, where Euler also made important contributions, I shall describe his researches in celestial mechanics. These investigations, including in particular his work on the orbits of Jupiter and Saturn awarded a prize by the Paris Academy, include results of an essentially mathematical nature, for the most part having to do with integration of differential equations, expansions in series (in particular trigonometric series), and with numerical methods in analysis—which incidentally do not figure in Euler's strictly mathematical publications. It is appropriate to dwell on Euler's role in the development of the theory of the moon's motion, which attracted the particular attention of the scientific world because of the discrepancy between the moon's observed orbit and the orbit calculated by Newton using his Universal Law of Gravitation. The disagreement between observation and computation based on theory was in fact so large, that at one time d'Alembert, Clairaut, and Euler all considered that it might be necessary to make a correction to Newton's universal law. However when towards the end of 1748 Clairaut concluded that the discrepancy arose as a result of insufficient accuracy in the basic approximations, Euler advised the Petersburg Academy of Sciences to announce a competition for the best essay on the motion of the moon. At the same time he verified Clairaut's conclusion using a method of his own devising. Clairaut's essay, awarded the prize of the Petersburg Academy on Euler's recommendation, was published in 1752, and a paper by Euler on the same topic appeared in Berlin in 1753, for which he received special compensation from St. Petersburg. In addition to this Euler received a portion of the prize money set aside by the British parliament for the highly accurate lunar tables compiled by the G\"ottingen astronomer J. T. Mayer, who had used Euler's results with his consent. At that time, and for that matter for a long time thereafter, lunar tables were used to determine longitude on the open sea, and Mayer's tables were included in marine almanacs. Incidentally, the very substantial correspondence between these two scholars has been preserved, and is published.
I note also two areas of Euler's activity in Berlin closely related to his involvement with practical problems. First, there were the calculations he made in connection with the organization of state lotteries—which served as a supplementary source of revenue for the Prussian treasury—, and with problems relating to insurance and demography, involving questions in probability theory and mathematical statistics. Much later a textbook on the insurance business and the organization of lotteries, furnished with tables, was prepared in St. Petersburg under Euler's supervision by his student N. I. Fuss, who will be mentioned several more times below; this book was published in 1776. Secondly, there was his interest in optics, extending over many years. Newton's corpuscular theory of light and colors has the consequence that an increase in the strength of an optical apparatus necessarily results in chromatic aberration of the images of objects and imposes, therefore, an impassable limit to the perfectibility of refracting telescopes, as opposed to the reflecting kind. On the basis of his own theory of light, different from Newton's and also from Huygens' wave theory (but not, it must be said, surviving into modern physics), Euler concluded that achromatic lenses of arbitrary refractive power are after all possible in principle, if made from transparent material having appropriate optical characteristics. Because of technical limitations, Euler's own attempts at manufacturing such instruments were largely unsuccessful, but soon afterwards, in 1758, the Englishman George Dollond was successful in making achromatic lenses of high power from an alloy of crown glass and flint glass. This represented a decisive step forward in the technology of the manufacture of telescopes and microscopes. Euler carried out detailed calculations relating to various dioptrical systems, and expounded the results of this research in a three-volume treatise entitled Dioptrics, composed largely in Berlin, but completed in St. Petersburg, where it was published in 1769—1771. Also in St. Petersburg, the talented mechanic and designer I. P. Kulibin took on the task of building a powerful microscope under Euler's supervision, and N. I. Fuss, on the basis of the Dioptrics, composed a detailed manual for master opticians, published in 1784.
Before leaving St. Petersburg Euler had reached a formal agreement with the Russian Academy allowing him to retain the title of foreign member, which then carried with it the payment of a large yearly pension of 200 rubles. Of course the value of the ruble in terms of actual goods has gone through a great deal of fluctuation over the past two and a half centuries. Although it would appear to be impossible to provide a single index of prices over such a period, some indication of the worth of 200 rubles at the time in question may be inferred from the fact that, in terms of the price of bread, during the reign of Catherine II one ruble would buy the same amount as 8 rubles at the beginning of the 20th century. Euler undertook to complete certain scientific projects begun in St. Petersburg (in particular his Naval Science mentioned earlier), and to submit articles to the St. Petersburg Commentarii; he also regularly agreed to fulfill commissions of various kinds, even as ordinary as the acquisition of books and scientific instruments. Notwithstanding the fact of his official position in Berlin, Euler was in essence not so much a foreign member of the Petersburg Academy as an actual member who happened to reside in Berlin—so that he also functioned as an active intermediary between the Petersburg and Berlin Academies. The three volumes of his correspondence with employees of the Petersburg Academy—comprising around 800 letters sent in both directions during his 25-year sojourn in Berlin, i.e., just under three letters a month on average—bear witness to the enduring nature of his connection with Russia. During the seven years' war the volume of this correspondence was substantially reduced without ceasing entirely; the letters were sent via persons residing in German principalities which, unlike Prussia, were non-participants in the war. Euler's correspondence with M. V. Lomonosov is preserved, though not in its entirety; Euler highly esteemed the latter's work in physics, and continued supporting him in the face of transparent hints to the effect that he not do so from influential enemies of Lomonosov in the administration of the Petersburg Academy.
From Berlin Euler edited the mathematical section of the St. Petersburg
Commentarii, wrote reports on the work—sent to him in Berlin—of adjuncts and Russian students of the Academy, compiled lists of topics for international competitions and wrote reports on submitted essays (the competition on the topic of the moon's motion was mentioned above), communicated news of scientific life in Germany and Western Europe generally, and so on. He was asked several times by the administration of the Petersburg Academy to suggest possible candidates for vacant positions in view of the continuing dearth of suitable specialists in Russia. Thus it was at Euler's suggestion that the eminent physicist F. U. T. Aepinus was invited in 1754, the renowned physiologist C. F. Wolff in 1767, and the astronomer and geographer G. M. Lowitz in 1768. The lengthy mathematical apprenticeship which the three Russian adjuncts of the mathematics department of the Petersburg Academy S. K. Kotel'nikov, S. Ya. Rumovskiui, and M. Sofronov served with Euler was of great significance for Russia: under Euler's tutelage these three, together with Euler's eldest son Johann-Albrecht, substantially improved their qualifications in higher mathematics and mechanics, and subsequently became prominent
representatives of science and the enlightenment in Russia.
Euler's scientific work at this time is extraordinary for its quantity and
for the wide variety of problems tackled in both pure and applied mathematics; on average he published 10 papers a year during this period, approximately half appearing in Latin in the Commentarii of the Petersburg Academy and half in French in the corresponding journal of the Berlin Academy. I have already discussed his research in mechanics but to complete the picture of his activity in this area mention should also be made of his fundamental treatise. The theory of motion of rigid bodies, published in 1765, complementing the Mechanics of 1736
devoted to the mechanics of point-particles. The period from the end of the 1740s onwards witnessed a growing interest on Euler's part in mathematical physics—which of course was to undergo a magnificent flowering in the 19th and 20th centuries. This interest was stimulated by rivalry first with d'Alembert and Daniel Bernoulli, and somewhat later with Lagrange. The solution of the new and difficult problems of natural science that arose required herculean efforts in connection with the perfection of analysis in various ways—not only among its several branches, but also in its relations with other areas such as number theory, algebra, and differential geometry. New chapters in the differential calculus appear as part of Euler's output at this time: the differential calculus of functions of several variables, elliptic integrals and an addition theorem for them, special improper integrals, important classes of ordinary differential equations, the theory of integrating factors and many types of partial differential equations of

the second and higher order, the calculus of variations, numerical methods, and so on.
Historical interdependencies such as those I have noted glancingly above,
rarely occur in mathematics as it is pursued in modern universities. I limit myself to a single example. The Fundamental Theorem of Algebra was discovered in the 17th century in the course of the natural evolution of that subject, but its earliest formulations were both incomplete and unclear in view of the lack of a fully worked-out theory of complex numbers. By the 18th century both this theory and the Fundamental Theorem of Algebra had become essential for solving the problem of integration of rational functions, to which the integration in closed form of many other kinds of functions reduces. At that time no mathematician of note doubted that every polynomial with real coefficients could be decomposed as a product of real first degree and quadratic factors: such was the case for instance with Leibniz, Nicolaus I Bernoulli, and Goldbach. Only d'Alembert and Euler, independently of one another, saw the need for a more thorough investigation of complex numbers in the form a+bv-1, and around 1750 produced the first proofs (their proofs were different) of the Fundamental Theorem of Algebra—from our point of view not, it is true, complete, but correct in essence.
The foregoing inventory of Euler's mathematical activities, far from complete, gives some idea of his work only in those areas of mathematics serving the natural sciences. Although in Euler's work the relationship between the so-called pure and applied domains of mathematics was immediate, nevertheless it was in mathematics itself that his deepest interest lay, and from which he derived the greatest satisfaction. More often than not the theorems and methods which he discovered in the course of solving an applied problem later became the starting point of a pertinaceous, systematic, purely theoretical development, and thus turned out to constitute the first links of a new theory, or even a whole discipline. In this respect Euler differed greatly from such contemporaries as for instance Daniel Bernoulli, who, once having found a mathematical solution of some problem or other from the natural sciences, refrained from realizing all the possibilities latent in his solution, even though his mathematical gifts were outstanding. Often Bernoulli confined himself to purely physical arguments, regarding them as sufficient or even preferring them to a precise mathematical treatment of the problem at hand. When Euler was occupied with applied problems, he remained in essence a mathematician; Bernoulli, on the other hand, used mathematics only insofar as it was needed and so remained in essence a physicist. It is characteristic of Euler that at the same time as he was occupied with the mathematical sciences noted above, he was with ever-increasing absorption carrying out investigations in number theory, the subject of three decades of correspondence with Goldbach, and more than a decade with Lagrange. In Berlin, he elaborated, in particular, the theory of Diophantine equations, and of residues of powers. The initial impulse in this direction came to him from Fermat's discoveries in number theory.
Unfortunately, lack of time does not permit me to linger over all of the scientific debates of that period in which Euler participated; I mention just two of them: first, the continuing disagreement with d'Alembert concerning the properties of logarithms of negative numbers, which already at the beginning of the 18th century had been the subject of fruitless polemics exchanged between Leibniz and Johann Bernoulli, at root resulting from the vagueness surrounding the concept of the logarithm itself at the time. Leibniz had suggested that logarithms of negative numbers should be imaginary, in some indefinite sense of the word, while on the other hand Bernoulli tried to prove that the logarithm of a negative quantity must be the same as the logarithm of the absolute value of that quantity. D'Alembert invented more and more reasons supporting Bernoulli's opinion. Euler, however, was the first to give the modern definition of the logarithmic function, and on the basis of his study of complex numbers created the complete theory of this function in the complex domain.
The second debate I wish to mention is the celebrated one concerning the nature of the arbitrary functions that feature in solutions of the equations of mathematical physics, in which Euler, d'Alembert, Daniel Bernoulli, Lagrange—for that matter just about every prominent mathematician of the second half of the 18th century—participated, and which was to have such an enormous influence on the progress not just of mathematical physics, but of analysis as a whole. This debate is often called the argument over the vibrating string", since it began in 1749 with the analysis proposed by d'Alembert of the problem concerning the transverse vibrations of an ideal string under prescribed initial and boundary conditions, yielding the expression of the general solution of the partial differential equation of the problem as a sum of two arbitrary functions. Without going into great detail, I note merely that d'Alembert imposed on the initial conditions, i.e., on the functions giving the initial position of the string and the initial distribution of velocities of the points of the string, severe restrictions as to their differentiability, essential, in his opinion, for an analytic solution of the problem to be possible. Euler, taking into account the special nature of the problem and bringing in geometric considerations, found d'Alembert's conditions unjustifiably restrictive, and to some extent anticipated the idea of solving the problem when the initial functions are non-smooth in some sense or other—an idea which was to be extensively elaborated upon and rigorously formulated in the 20th century in S. L. Sobolev's theory of generalized functions (1935), and then developed in detail by L. Schwartz as the theory of distributions". Of course Euler's treatment of this theme cannot be considered satisfactory from the viewpoint of 20th century science, and perhaps even of his own time. All the same one cannot but admire the boldness and perspicacity of his mathematical thought. Note that a summation of the argument over the vibrating string, relative to the period of the 18th century, was given in an interesting essay by the French scholar Louis Arbogast on the nature of arbitrary functions featuring in solutions of partial differential equations, which earned a prize in a special competition organized by the Petersburg Academy of Sciences and was published in French in St. Petersburg in 1791.
A new method of mathematical physics consisting in the expansion of solutions of certain boundary-value problems in series of eigenfunctions, was proposed by Jean-Baptiste-Joseph Fourier at the beginning of the 19th century in his theory of heat, reviving an idea conceived in a particular case by Daniel Bernoulli, who believed it possible to represent every solution of the vibrating string problem as the sum of an infinite trigonometric series in sines and cosines. However in support of this idea Bernoulli could only offer physical analogies, which neither Euler nor d'Alembert considered proofs, and to which only Fourier managed to give analytic form, leading to an intensive development of the theory of trigonometric series, and in direct connection with this the theory of the integral and ultimately the general theory of functions of a real variable.
A little earlier I spoke of Euler's extraordinary boldness and keenness of vision as evinced by his mathematical ideas. A further instance of this is afforded by his thoroughgoing elaboration, in the middle of the 18th century, of the conception of the sum of an infinite series. The distinction between convergent and divergent series was at that time generally well known, and many mathematicians of the 17th century considered only the former to be of practical significance. However Euler, having obtained by means of certain divergent series several remarkable results—for example in the theory of the zeta-function—considered the exclusion of such series from use inexpedient, and worked out—to the extent that the contemporary analytic apparatus allowed—an original theory resting on a certain generalization of the concept of the sum of a series and a special method for transforming series, allowing under suitable conditions the calculation of such generalized sums, and also an improvement in the rate of convergence of a series. In the 19th century such applications of divergent series were subject to criticism and were rejected by several first-rate mathematicians, including N. H. Abel. It thus became the order of the day to formulate a general theory of convergence of series, hitherto non-existent. However later, when Cauchy and his followers had produced this theory—resulting in the intensive development of the theory of analytic functions—Euler's ideas were provided with the firm foundation not vouchsafed him to furnish, and became part of the general theory of summability of series, which has flourished for the last hundred years.
The vast accumulation of mathematical knowledge by the middle of the 18th century, most of all in analysis, was in need of systematization, the filling of gaps, the improvement in precision of basic concepts, and so on. Euler took upon himself the composition of a series of monographs providing an account of all the work in this area, with the aim of facilitating the efforts of others in furthering its development. This project had ripened in Euler's mind over many years, and even in St. Petersburg he had started work on it, although the actual publication of such a series of monographs began only during his Berlin period and was not completed till after his return to St. Petersburg.
On leaving St. Petersburg, Euler had undertaken to finish a certain essay, which in official documents was called Higher Algebra". In Berlin Euler did in fact do work in algebra, however it is clear from various documents that the essay in question was really an extensive treatise on mathematical analysis. In 1744 Euler published in Switzerland A method for finding curves possessing maximal and minimal properties3 actual, Latin, title was Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti. This work contained the first known method for finding extrema of certain classes of definite integrals (now called functionals") whose value depends on the choice of unknown functions entering into them, by reducing the problem to the solution of a certain partial differential equation. It is remarkable that Euler achieved this reduction by means of a method nowadays termed "direct"; such so-called direct" methods, which allow one to bypass integration of the corresponding differential equations, were systematically developed only in the 20th century. Several particular problems of this type had been posed and solved by Johann and Jakob Bernoulli, but both were very far from a providing general treatment of the question, this being due entirely to Euler.
In appendices to the above-mentioned book Euler provided solutions of
several problems in mechanics, incidentally giving a precise formulation of the principle of least action, Maupertuis' version of which had been very special; it was here that he derived his formula for the critical loading of columns, which in the theory of resistance of materials bears his name. In the mid-1750s the young Lagrange, in introducing the concept and associated symbolism of the variational calculus, proposed a new, formally analytic presentation of the general problem independent of the geometric reasoning used in Euler's treatment, and extending readily to wider classes of functionals. Euler in turn wrote a more widely accessible exposition of Lagrange's method. Thus were laid the classical foundations of the calculus which, following Euler, came to be called variational.
At the time he was working on this book, Euler was also preparing a multi-volume work intended to encompass all areas of mathematical analysis. In the year of publication of {\it A method for finding curves}, he sent, again to Switzerland, the first part of this work, published in 1748 in two volumes under the title {\it Introduction to the analysis of the infinite}. The first volume contained a purely analytical treatment of the elements of analysis considered as a general study of functions, developed without recourse to the differential calculus as far as that was then feasible. The chief means for achieving this were infinite power series (as well as infinite products and sums of simple fractions). The unusually
elegant and at the same time accessible presentation of the theory of the elementary functions (excluding the logarithm function, to which, apart from a few cursory remarks in Volume II of the Introduction, he devotes a separate work), for the first time considered as functions not only of a real variable but also of a complex variable, together with the beauty of the examples, make this one of the most distinguished works of the whole literature of mathematical analysis. Even today, after an interval of over two and a half centuries, one can recommend this book as absorbing reading to any novice lover of mathematics. The second volume of the Introduction is geometrical, for the most part devoted to the study of curves of degrees 2 and 3, and to the theory, expounded in detail for the first time, of surfaces of degree 2, with excursions into the realm of plane transcendental curves. A few years later there appeared his fundamental Differential calculus, whose publication in 1755 in Berlin was financed by the Petersburg Academy. In Berlin Euler also prepared a large portion of the manuscript of his three-volume Integral calculus4 which was completed and
published in St. Petersburg between 1768 and 1770. To describe briefly the
riches contained in both of these works is impossible. It must be explained, however, that by the integral calculus Euler understood not only the computation of integrals in the narrow sense of that phrase—which in fact occupies only the first half of the first volume of the three-volume treatise of 1768—1770—but also the solution of both ordinary and partial differential equations; in this work he also provides a new, more complete exposition of the calculus of variations. Neither of these books contains geometric applications; these are to be found in his numerous articles.
This six-volume trilogy of Euler's played an extraordinarily large part in the development of analysis and of mathematics as a whole. Even by itself it fully justifies the words of Laplace: Lisez Euler, lisez Euler, c'est notre maître à tous5 Euler's monographs have been widely used by authors of practically every mathematics textbook right up to modern times.
The circumstances surrounding Euler's personal life in Berlin with his family had turned out very favorably. In 1753 he acquired a farm at Charlottenburg, of which his mother, come from Basel, became manager. His son Johann-Albrecht was elected member of the Berlin Academy of Sciences, and his youngest son Christofor, born in Berlin, became an officer in the Prussian army. It was at this time also that Euler was awarded many prizes by the Paris Academy of Sciences, and in 1755 he was made a foreign member. The Royal Society of London had elected him much earlier, in 1746. Euler's relations with the Petersburg Academy were arranged to his satisfaction, and in the Berlin Academy he enjoyed very great scientific authority, in particular in connection with the running of that institution as Maupertuis' deputy during the years of the latter's absences. Even his relations with the king, who was for the most part preoccupied with war operations,~were~satisfactory.
However after Maupertuis' death the situation began to change for the worse. When d'Alembert refused the post of president of the Academy, the king took the directorship upon himself, and from 1762 onwards more and more frequently and to a greater and greater extent began to interfere in its affairs, in particular in the appointment of new members, where he clearly showed his personal pro-French sympathies. At the end of the seven years' war, when the king took up permanent residence near Berlin, his relations with Euler steadily worsened. In 1763 Euler felt it prudent to sell the farm at Charlottenburg and to strengthen his contacts with the Petersburg Academy and with representatives of the Russian government. On examining the finances of the Berlin Academy, the management of which had fallen to Euler, the king expressed his dissatisfaction with their state and set up a special commission to audit them; in his opinion the revenue from the sale of the calendars should have been greater. For Euler these actions were oppressive and offensive.
By 1766 the discord between scholar and king had become such that Euler felt he had to resign. This decision was made easier by the ever more frequent invitations he was receiving to return to St. Petersburg. The king was reluctant to give up so useful a consultant and organizer, but with the weight of the Russian government behind Euler, had to release him. On June 9, 1766 Euler, together with all of his family except the youngest son, obliged to continue serving in the army, left for St. Petersburg, where they arrived on June 28. Some time later the king also released Christofor. Euler's position in the Berlin Academy was filled by Lagrange, who remained there till 1787, when he left permanently for Paris.
Thus Euler returned to Russia at the age of 60, with a rich experience of life behind him, a large number of unpublished and unfinished works in hand, and an abundant store of creative energy. He was received with the joy and esteem that his genius merited. Upon his arrival he was almost at once received by Empress Catherine II, and together with his son Johann-Albrecht appointed to the advisory council of the director" of the Petersburg Academy of Sciences, Count V. G. Orlov, at that time deputizing for the official president count K. G. Razumovskii, who had withdrawn from the imperia l court. When a few years later both Euler and his son resigned their positions on Orlov's council over a disagreement with him, this did not at all reflect adversely on their situation as a whole; in particular Johann-Albrecht remained in the post of conference secretary assigned to him in 1769. In fact this time marks the beginning of a period of almost a hundred years during which the Euler family wielded great influence in the running of the Academy as a whole. Upon Johann's death in 1800, the post of conference secretary, i.e., the permanent secretary of the Academy, fell to Euler's grandson-in-law N. I. Fuss, and on Fuss' death, in turn, to his son, the mathematician P. N. Fuss, who filled the post till his death in 1855.
The Russian government was unchanging in the generosity of its financial support for Euler. To house his family, which now counted 16 people, a large house was built on the bank of the Neva, not far from the premises of the Academy of Sciences. When this house burnt down in the spring of 1771, it was rebuilt, and in a somewhat reconstructed form is preserved to this day. However two unhappy events clouded the last 17 years of Euler's life. In the autumn of 1766 he almost completely lost the vision of his hitherto healthy remaining left eye. From this time on he was only able to distinguish bulky objects and read large letters written in chalk on a blackboard. Although, fortunately, this did not affect his creative activity, it did change the way in which he worked. He now prepared his works with the help of a secretary, a position requiring a qualified specialist capable under his supervision of making necessary calculations and editing texts dictated to them. First Euler's son Johann-Albrecht worked in this capacity, then the physicist and academician L. Yu. Kraft, the son of his colleague from the 1730s, then the talented mathematician A. J. Lexell, and finally his students M. E. Golovin and N. I. Fuss, mentioned earlier, the latter having been invited from Basel as a youth on the recommendation of Daniel Bernoulli. These arrangements allowed Euler, with his constant freshness of intellect and his extraordinary memory, still fully intact, to continue working till the end of his days. The only change was a sudden falling off in the volume of his correspondence, in particular with Lagrange, since he was unable to re-read and verify the complicated arguments and computations contained in the latter's letters.
The second tragic event of these years was the death of his wife, with whom he had lived for almost 40 years. However his large family required a wife to manage it, and three years later Euler married Salome-Abigail Gsell, a sister of his first wife. Generally speaking, the family lived in complete contentment. The eldest son, a highly qualified, if not eminent, scholar, occupied, as they used to say, a responsible post at the Academy. The second son Karl became a very successful doctor, and the youngest, Christofor, enlisted in the Russian army, and for many years worked as director of an arms factory at Sestroretsk, near St. Petersburg, ending his career with the rank of general. Euler's sons became Russian citizens (although Euler himself remained a citizen of Basel for the whole of his life); to this day there are direct descendants of Euler living in
Leningrad6 and Moscow. For a long time Euler maintained excellent health and capacity for work. It was only a little before his death that he began to suffer from dizziness, and on September 18, 1783 died suddenly from a stroke. On September 22 at a general meeting of the Academy the oldest of the academicians, J. von Stählin, gave a funeral oration, and on November 3 N. I. Fuss delivered a eulogy, distinguished by its high seriousness. Finally, the great scientist was accorded a special posthumous honor: On January 25, 1785 a bust in his likeness, prepared not long before by the well known French sculptor J.-D. Rachette, was solemnly placed on a pedestal in the large conference hall of the Academy. At the present time7 this bust adorns the premises of the Presidium of the Academy of Sciences of the USSR8 in Moscow.
Euler was buried in the Smolensk Lutheran cemetery, where in 1837 the Petersburg Academy of Sciences erected a huge monument with the inscription, in Latin: To Leonhard Euler—Petersburg Academician". In 1957, in connection with celebrations of Euler's 250th birthday, both tomb and monument were moved to the Leningrad Mausoleum, where they were installed next to the burial site of M. V. Lomonosov, and a memorial marble plaque was affixed to Euler's house.
Over the second Petersburg period of Euler's life, from 1767 to 1783, he
published around 250 works, including several lengthy books which had been partially composed in Berlin and were now completed and edited. However the publication of this output in the Commentarii of the Academy and other periodicals proved unmanageable. At Euler's death about 300 of his papers remained unpublished; of these about 200 eventually appeared in the Commentarii at the initiative of N. I. Fuss, and in 1862 P. N. Fuss had a further 50 or so published in the two-volume Posthumous essays in mathematics and physics of Euler. Many of these works opened fresh perspectives for research in mathematics and mechanics, receiving due attention and development only in the 19th century; some of them, long unnoticed by posterity, contained results rediscovered later. By way of example I note again Euler's purely analytic derivation of the so-called Cauchy-Riemann equations", his calculation of many special integrals using functions of a complex variable (a method which Laplace arrived at about the same time), and also his elementary derivation of the formulae for the so-called Fourier coefficients in the theory of trigonometric series. It is curious that Fourier derived them anew and in a different way, clearly unaware of Euler's work.
I now turn to some of the substantial monographs published by Euler in
this last period of his life, apart from the three-volume Integral calculus and the three-volume Dioptrics, already discussed. Over the two years 1768-1769 there appeared in print a Russian translation of his two-volume Introduction to algebra (the German original was published later, in 1770); Euler had dictated the whole of this work to a young servant of German background. While on the one hand this highly original exposition of algebra to a very large extent defined—of course
in abbreviated form—the content of all successive textbooks in algebra at the level of the gymnasium, on the other hand it contained many of Euler's own discoveries in the theory of Diophantine equations, reaching far beyond gymnasium level. The French edition of this work (translated by a member of the Bernoulli family), which appeared, with valuable additions by Lagrange, in 1774 in Lyon, represented a substantial leap forward in the development of Diophantine analysis. At about the same time, between 1768 and 1772, there appeared in French the three volumes of Letters to a German princess on various questions of physics and philosophy, published simultaneously in the Russian translation of S. Ya. Rumovskii These Letters became the most widely read of all of Euler's works, going through tens of editions in French, English, German, Russian, Dutch, Swedish, Danish, Spanish, and Italian. Written in the early 1760s, the Letters represented a popular exposition of the most fundamental questions of physics, philosophy, logic, ethics, theology, etc. On the
scientific side this popularization answered to the highest standards of the time. It also reflected Euler's deep religious beliefs and his attitude to certain philosophical systems of the 17th and 18th centuries. As is well known, he was an opponent of Leibniz' monadology. In the Letters he expressed a negative attitude also towards subjective idealism and solipsism, and took up an intermediate position in the argument between the adherents of the rival natural philosophies of Newton and Descartes (in many respects, however, closer to that of Descartes). It cannot be doubted that the Letters exerted some influence on Kant in the first period of his philosophical creativity. I would be reluctant to say that Euler presented in his Letters some kind of complete philosophical system. Incidentally, the Letters to a German princess are to be discussed in two lectures of the present conference—by A. T. Grigor'ian and V. S. Kirsanov, and by K. Grau.
To the list of large monographs it is appropriate to add the Theory of the motion of the moon, worked out using a new method, published in 1772. This work was of even greater significance for the development of celestial mechanics than his book of 1753 devoted to that topic. It was prepared for the press under Euler's general supervision by three academicians: his eldest son Johann-Albrecht, and Kraft and Lexell, who were compelled to carry out very laborious calculations.
In conclusion I shall try to characterize Euler's output along general lines—an output which astounds anyone familiar with it by its volume, variety, and originality. Not for nothing did d'Alembert once call him a "man-devil". The Helvetic Society of Natural Scientists began to publish a complete works in 1911, and over the past 70 or so years there have appeared 69 thick volumes divided into three series: "Mathematical works", "Works in mechanics and astronomy", and "Works in physics, and others", and there remain four more volumes to be published9. Incidentally, two volumes from the first series were prepared under the editorship of A. M. Lyapunov. In 1975 the publication of a fourth series was begun with the participation of the Academy of Sciences of the USSR. This fourth series is subdivided into two subsets. The eight volumes of the first of these are devoted to Euler's scientific correspondence; of these three have appeared so far10, a further one is in press, and the preparation of another for publication is almost complete. The second subset will consist of five or six volumes devoted to Euler's unpublished scientific manuscripts and fragments; this project is as yet in its initial stages and will take some years to complete.
While it is indeed unlikely that any mathematician or physicist ever
published as many works as Euler, one is not any the less astonished by their extraordinary breadth. They embrace literally all areas of mathematics and the mathematical sciences, as well as a great many problems of technology, philosophy, and even theology. If one counts individual volumes, whose contents, incidentally, are often quite diverse in character, then almost 43% of the total are devoted to mathematics and the same percentage to mechanics, including astronomy, making a total of 86% of his published works11. However here it is essential to take into account the fact that Euler's works in mechanics and astronomy are replete with solutions of differential equations, expansions of functions in convergent or asymptotic series, etc., and very often contain completely original purely mathematical results not given a separate, self-contained treatment.
Thus notwithstanding the thematic variety of his researches, Euler was first and foremost a mathematician. The academician A. N. Krylov remarked a half-century ago that in essence Euler transformed mechanics from a physical science into a mathematical one. Following in Euler's footsteps, Lagrange announced in 1787 that mechanics had become a new branch of analysis, and then Fourier in 1822 that analysis was as vast as nature herself.
Naturally, in the majority of his mathematical works Euler's approach was that of an analyst. Of all his purely mathematical works, those in analysis account for around 60%, followed by geometry, chiefly differential, at 17%, then algebra, combinatorics and probability theory at 13%, and finally number theory, occupying 10%. These statistics—which are of course only approximate—indicate the organic connection of Euler's work with the investigation of the natural world. He was led to many of his new methods by the search for solutions to problems of the natural sciences amenable to mathematical formulation; however it was precisely in this that the special nature of Euler's mathematical genius consisted that he did not limit himself to solving individual concrete problems, whether applied or purely mathematical, but, returning constantly to a further deepening and generalizing investigation of the question to hand, was often able to create a new and fully independent mathematical theory. In this way not only did he extend enormously the boundaries of the analysis of Newton, Leibniz, and the elder Bernoulli brothers, but also created completely new branches of that subject: the variational calculus, the elements of complex function theory, the most important part of the theory of special functions, a whole system of approaches to the solution of differential equations, and so on. In terms of rigor, Euler's proofs remained within the bounds imposed by 18th century standards; however his rejection of excessive rigor, for which he was often reproached in the first half of the 19th century, turned out to be historically justified. His intuition shielded him from major error, and many of his bold ideas—such as for example the methods he invented for summing divergent series—became capable of being evaluated and developed on a new, more complete and rigorous basis only at the end of the 19th century and the beginning of the 20th. And it would be a great mistake to think that only applied problems served as the source of Euler's discoveries: a great many of his ideas arose in the course of pondering problems of the most purely mathematical character, beginning with his introduction of such important classes of functions as the beta-
gamma- and zeta-functions, and ending with all those problems of number theory with which he occupied himself unceasingly from the early 1730s till the end of his life.
It is perhaps Euler's attitude to number theory that is especially persuasive as evidence of the essentially mathematical style of his thinking. Following on the ingenious insights of Fermat, for many decades number theory suffered neglect. It failed to interest such outstanding mathematicians as D. Bernoulli, A. Clairaut, J. d'Alembert, and indeed the majority of the contemporaries of Euler, who, according to Chebyshev, was the first to turn number theory into a science in its own right. Of course number theory attracted Euler by its beauty together with the difficulty of many of its theorems, so easily formulated and yet requiring for their discovery acute discernment and for their proof means of extraordinary refinement. However here the chief consideration was Euler's awareness of the profound organic interdependence of all areas of mathematics; he understood mathematics as a single whole, of which number theory forms an integral part, and considered progress in that subject a precondition of the advance of mathematics over its entire front.
There are various approaches to defining a typology of mathematicians.
On the one hand there have always been mathematicians of a definitely applied bent, such as Ch. Huygens and Newton, or D. Bernoulli, Clairaut, and d'Alembert, already noted, or J. B. Fourier and S. D. Poisson, M. V. Ostrogradskii and V. A. Steklov, etc. On the other hand the "pure" mathematical tradition has existed for a very long time, some prominent representatives of which are: N. H. Abel and E. Galois—although the latter, it is true, died very young—, B. Bolzano, and somewhat later R. Dedekind and G. Cantor and L. Brouwer, E. I. Zolotarev and I. M. Vinogradov, N. N. Luzin and P. S. Aleksandrov. However Euler belongs to that category of mathematician combining these two tendencies organically, immanently mathematical and applied, of which some representatives were, in ancient times Archimedes, and in the last few hundred years J.-L. Lagrange, C. F. Gauss, A. Cauchy, B. Riemann, P. L. Chebyshev, H. Poincaré, D. Hilbert, etc. One should doubtless delineate also the category of those mathematicians of a philosophical disposition, such as R. Descartes and G. W. Leibniz, as well as some of those already mentioned; I would include also among these N. I. Lobachevskii.

It is appropriate to add a few words about Euler's relations with his
contemporaries. As a rule, if one excludes a few heated discussions, his attitude to other scholars was always benevolent, and he never allowed any sense of his superiority to emerge. He was without envy towards those who preceded him in some discovery or other, and—to use the expression applied by B. Fontenelle to Leibniz—took pleasure in observing how plants springing from seeds that he had supplied, flourished in others' gardens. Always a student with the widest interests, throughout his life he was prepared to learn from others, and in his own works often expounded others' discoveries in more convenient and accessible form. However of course overall he gave to others immeasurably more than he took from them. He influenced the work of many generations of mathematicians; in particular the St. Petersburg mathematics school of the second half of the 19th century and the first quarter of the 20th, founded by P. L. Chebyshev, was very close in spirit to Euler. As the late B. N. Delone put it some years ago as a participant in the Euler Days of 1957, the guiding principle in the work of this school was that of Euler and Chebyshev: Confronted with a difficult problem—arising as often happens from a sister science or from technology—, construct a broad and deep mathematical theory, and once a solution has been obtained, convert it back into the numerical form required by the original practical problem.
I would like to end with the words pronounced 200 years ago by N. I. Fuss
at the meeting of the Petersburg Academy of Sciences dedicated to the just-deceased Euler: Such were the works of Mr. Euler, such is his right to immortality. His name will perish only with the end of science itself". To which I add: That will never be!
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Introduction; Translator's note; From the editors; 1. Opening speech of the symposium 'Modern Developments of Euler's Ideas' October 24, 1983 A. P. Aleksandrov; 2. Leonhard Euler: his life and work A. P. Yushkevich; 3. Leonhard Euler, active and honored member of the Petersburg Academy of Sciences Yu. Kh. Kopelevich; 4. The part played by the Petersburg Academy of Sciences (the Academy of Sciences of the USSR) in the publication of Euler's collected works E. P. Ozhigova; 5. Leonhard Euler and the Berlin Academy of Sciences K. Grau; 6. Was Leonhard Euler driven from Berlin by J. H. Lambert? K.-R. Biermann; 7. Euler's mathematical notebooks E. Knobloch; 8. On Euler's surviving manuscripts and notebooks G. P. Matvievskaya; 9. The manuscript materials of Euler on number theory G. P. Matvievskaya and E. P. Ozhigova; 10. Euler's contribution to algebra I. G. Bashmakova; 11. Diophantine equations in Euler's works T. A. Lavrinenko; 12. The foundations of mechanics and hydrodynamics in Euler's works G. K. Mikhaĭlov and L. I. Sedov; 13. Leonhard Euler and the variational principles of mechanics V. V. Rumyantsev; 14. Leonhard Euler and the mechanics of elastic systems N. V. Banichuk and A. Yu. Ishlinskiĭ; 15. Euler's research in mechanics during the first Petersburg period N. N. Polyakhov; 16. The significance of Euler's research in ballistics A. P. Mandryka; 17. Euler and the development of astronomy in Russia V. K. Abalakin and E. A. Grebenikov; 18. Euler and the evolution of celestial mechanics N. I. Nevskaya and K. V. Kholshevnikov; 19. New evidence concerning Euler's development as an astronomer and historian of science N. I. Nevskaya; 20. Leonhard Euler in correspondence with Clairaut, d'Alembert and Lagrange A. P. Yushkevich and R. Taton; 21. Letters to a German Princess and Euler's physics A. T. Grigor'ian and V. S. Kirsanov; 22. Euler and I. P. Kulibin N. M. Raskin; 23. Euler and the history of a certain musical-mathematical idea E. V. Gertsman; 24. Euler's music-rheoretical manuscripts and the formation of his conception of the theory of music S. S. Tserlyuk-Askadskaya; 25. An unknown portrait of Euler by J. F. A. Darbés G. B. Andreeva and M. P. Vikturina; 26. Eulogy in memory of Leonhard Euler Nikolaĭ Fuss; 27. Leonhard Euler's family and descendants I. R. Gekker and A. A. Euler; Index.

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