In the autumn of 1720, being not yet fourteen, Euler entered the University of Basel in the department of arts to get a general education before specializing. In 1723 he received his master’s degree in philosophy.

Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.

In the autumn of 1720, being not yet fourteen, Euler entered the University of Basel in the department of arts to get a general education before specializing. In 1723 he received his master’s degree in philosophy.

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the center it shows his polyhedral formula, in English written as "v − e + f = 2".

Leonhard Euler was a Swiss mathematician and physicist, one of the founders of pure mathematics. He not only made decisive and formative contributions to the subjects of geometry, calculus, mechanics, and number theory but also developed methods for solving problems in observational astronomy and demonstrated useful applications of mathematics in technology and public affairs.

Background

Leonhard Euler was born on April 15, 1707, in Basel, Switzerland. His forebears settled in Basel at the end of the sixteenth century. His great-great-grandfather, Hans Georg Euler, had moved from Lindau, on the Bodensee (Lake Constance). They were, for the most part, artisans; but the mathematician’s father, Paul Euler, graduated from the theological department of the University of Basel. He became a Protestant minister, and in 1706 he married Margarete Brucker, the daughter of another minister. Euler had two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich. In 1708 the family moved to the village of Riehen, near Basel, where Euler spent his childhood.

Education

Euler’s father gave his son his elementary education, including mathematics. In the brief autobiography dictated to his eldest son in 1767, Euler recollected that for several years he diligently and thoroughly studied Christoff Rudolf’s Algebra, a difficult work which only a very gifted boy could have used. Euler later spent several years with his maternal grandmother in Basel, studying at a rather poor local Gymnasium; mathematics was not taught at all, so Euler studied privately with Johann Burckhardt, an amateur mathematician.

In the autumn of 1720, being not yet fourteen, Euler entered the University of Basel in the department of arts to get a general education before specializing. The university was small; it comprised only a few more than a hundred students and nineteen professors. But among the latter was Johann Bernoulli, who had followed his brother Jakob, late in 1705, in the chair of mathematics. During the academic year, Bernoulli delivered daily public lectures on elementary mathematics; besides that, for additional pay he conducted studies in higher mathematics and physics for those who were interested. Euler laboriously studied all the required subjects, but this did not satisfy him.

In the summer of 1722, Euler delivered a speech in praise of temperance, “De temperantia,” and received his prima laurea, a degree corresponding to the bachelor of arts. The same year he acted as opponent (respondens) at the defense of two theses - one on logic, the other on the history of law. In 1723 Euler received his master’s degree in philosophy. This was officially announced at a session on June 8, 1724; Euler made a speech comparing the philosophical ideas of Descartes and Newton.

Some time earlier, in the autumn of 1723, he had joined the department of theology, fulfilling his father’s wish. His studies in theology, Greek, and Hebrew were not very successful, however; Euler devoted most of his time to mathematics. He finally gave up the idea of becoming a minister but remained a wholehearted believer throughout his life. He also retained the knowledge of the humanities that he acquired in the university; he had an outstanding memory and knew by heart the entirety of Vergil’s Aeneid. At seventy he could recall precisely the lines printed at the top and bottom of each page of the edition he had read when he was young.

At the age of eighteen, Euler began his independent investigations. His first work, a small note on the construction of isochronous curves in a resistant medium, appeared in Acta eruditorum (1726); this was followed by an article in the same periodical on algebraic reciprocal trajectories (1727). The problem of reciprocal trajectories was studied by Johann Bernoulli, by his son Nikolaus II, and by other mathematicians of the time. Simultaneously Euler participated in a competition announced by the Paris Academie des Sciences which proposed for 1727 the problem of the most efficient arrangement of masts on a ship. The prize went to Pierre Bouguer, but Euler’s work received the accessit. Later, from 1738 to 1772, Euler was to receive twelve prizes from the Academy.

Euler received the invitation to serve as adjunct of physiology in St. Petersburg in the autumn of 1726. and he began to study this discipline, with an effort toward applying the methods of mathematics and mechanics. He also attempted to find a job at the University of Basel. A vacancy occurred in Basel after the death of a professor of physics, and Euler presented as a qualification a small composition on acoustics, Dissertatio physica de sono (1727). Vacancies were then filled in the university by drawing lots among the several chosen candidates. In spite of a recommendation from Johann Bernoulli, Euler was not chosen as a candidate, probably because he was too young - he was not yet twenty. But, as O. Spiess has pointed out, this was in Euler’s favor; a much broader field of action lay ahead of him.

On April 5, 1727, Euler left Basel for St. Petersburg, arriving there on May 24. From this time his life and scientific work were closely connected with the St. Petersburg Academy and with Russia. He never returned to Switzerland, although he maintained his Swiss citizenship.

In spite of having been invited to St. Petersburg to study physiology, Euler was at once given the chance to work in his real field and was appointed an adjunct member of the Academy in the mathematics section. He became professor of physics in 1731 and succeeded Daniel Bernoulli, who returned to Basel in 1733 as a professor of mathematics. The young Academy was beset with numerous difficulties, but on the whole the atmosphere was exceptionally beneficial for the flowering of Euler’s genius. Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer, and specialist in theoretical mechanics Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F. Maier, working in trigonometry; and the astronomer and geographer J.-N. Delisle.

In St. Petersburg, Euler began his scientific activity at once. No later than August 1727 he started making reports on his investigations at sessions of the Academy; he began publishing them in the second volume of the academic proceedings, Commentarii Academiae scientiarum imperialis Petropolitanae (1727). The generous publication program of the Academy was especially important for Euler, who was unusually prolific.

In addition to conducting purely scientific work, the St. Petersburg Academy from the very beginning was also obliged to educate and train Russian scientists, and with this aim a university and a Gymnasium were organized. The former existed for nearly fifty years and the latter until 1805. The Academy was also charged to carry out for the government a study of Russian territory and to find solutions for various technological problems. Euler was active in these projects. From 1733 on, he successfully worked with Delisle on maps in the department of geography. From the middle of the 1730s he studied problems of shipbuilding and navigation, which were especially important to the rise of Russia as a great sea power. He joined various technological committees and engaged in testing scales, fire pumps, saws, and so forth. He wrote articles for the popular periodical of the Academy and reviewed works submitted to it (including those on the quadrature of the circle), compiled the Einleitung zur Rechen-Kunst for Gymnasiums, and also served on the examination board.

Euler’s main efforts, however, were in the mathe-matical sciences. During his fourteen years in St. Petersburg he made brilliant discoveries in such areas as analysis, the theory of numbers, and mechanics. By 1741 he had prepared between-eighty and ninety works for publication.

As is usual with scientists, Euler formulated many of his principal ideas and creative concepts when he was young. Neither the dates of preparation of his works nor those of their actual publication adequately indicate Euler’s intellectual progress, since a number of the plans formulated in the early years in St. Petersburg were not realized until much later. For example, the first drafts of the theory of motion of solid bodies, finished in the 1760s, were made during this time. Likewise Euler began studying hydromechanics while still in Basel, but the most important memoirs on the subject did not appear until the middle of the 1750s; he imagined a systematic exposition of differential calculus on the basis of calculus of finite differences in the 1730s but did not realize the intention until two decades later; and his first articles on optics appeared fifteen years after he began studying the subject in St. Petersburg.

In November 1740 Anna Leopoldovna, mother of the infant Emperor Ivan VI, became regent, and the atmosphere in the Russian capital grew troubled. According to Euler’s autobiography, “things looked rather dubious.” At that time Frederick the Great, who had succeeded to the Prussian throne in June 1740, decided to reorganize the Berlin Society of Sciences, which had been founded by Leibniz but allowed to degenerate during Frederick’s father’s reign. Euler was invited to work in Berlin. He accepted, and after fourteen years in Russia he sailed with his family on June 19, 1741, from St. Petersburg. He arrived in Berlin on July 25.

Euler’s energy in middle age was inexhaustible. He was working simultaneously in two academies - Berlin and St. Petersburg. He was very active in transforming the old Society of Sciences into a large academy - officially founded in 1744 as the Académie Royale des Sciences et des Belles Lettres de Berlin. Euler was appointed director of the mathematical class of the Academy and member of the board and of the committee directing the library and the publication of scientific works. He also substituted for the president, Maupertuis, when the latter was absent. When Maupertuis died in 1759, Euler continued to run the Academy, although without the title of president. Euler’s friendship with Maupertuis enabled him to exercise great influence on all the activities of the Academy, particularly on the selection of members.

Euler’s administrative duties were numerous: he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal, which was built in 1744 to join the Havel and the Oder. At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.

In 1749 and again in 1763 he advised on the organization of state lotteries and was a consultant to the government on problems of insurance, annuities, and widows’ pensions. Some of Euler’s studies on demography grew out of these problems. An inquiry from the king about the best work on artillery moved Euler to translate into German Benjamin Robins’ New Principles of Gunnery. Euler added his own supplements on ballistics, which were five times longer than the original text (1745). These supplements occupy an important place in the history of ballistics; Euler himself had written a short work on the subject as early as 1727 or 1728 in connection with the testing of guns.

From his very first years in Berlin, Euler kept in regular working contact with the St. Petersburg Academy. This contact was interrupted only during military actions between Prussia and Russia in the course of the Seven Years’ War - although even then not completely. Before his departure from the Russian capital, Euler was appointed an honorary member of the Academy and given an annual pension; on his part he pledged to carry out various assignments of the Academy and to correspond with it. During the twenty-five years in Berlin, Euler maintained membership in the St. Petersburg Academy a tous les titres, to quote N. Fuss. On its commission he finished the books on differential calculus and navigation begun before his departure for Berlin; edited the mathematical section of the Academy journal; kept the Academy apprised, through his letters, of scientific and technological thought in Western Europe; bought books and scientific apparatus for the Academy; recommended subjects for scientific competitions and candidates to vacancies; and served as a mediator in conflicts between academicians.

Euler’s participation in the training of Russian scientific personnel was of great importance, and he was frequently sent for review the works of Russian students and even members of the Academy. For example, in 1747 he praised most highly two articles of M. V. Lomonosov on physics and chemistry; and S. K. Kotelnikov, S. Y. Rumovski, and M. Sofronov studied in Berlin under his supervision for several years. Finally, Euler regularly sent memoirs to St. Petersburg. About half his articles were published there in Latin, and the other half appeared in French in Berlin.

During this period, Euler greatly increased the variety of his investigations. Competing with d’Alembert and Daniel Bernoulli, he laid the foundations of mathematical physics; and he was a rival of both A. Clairaut and d’Alembert in advancing the theory of lunar and planetary motion. At the same time, Euler elaborated the theory of motion of solids, created the mathematical apparatus of hydrodynamics, successfully developed the differential geometry of surfaces, and intensively studied optics, electricity, and magnetism. He also pondered such problems of technology as the construction of achromatic refractors, the perfection of J. A. Segner’s hydraulic turbine, and the theory of toothed gearings.

In the 1740s and 1750s Euler took part in several philosophical and scientific arguments. In 1745 and after, there were passionate discussions about the monadology of Leibniz and of Christian Wolff. German intellectuals were divided according to their opinions on monadology. As Euler later wrote, every conversation ended in a discussion of monads.

In 1751 a sensational new argument began when S. König published some critical remarks on Maupertuis’s principle of least action (1744) and cited a letter of Leibniz in which the principle was, in König’s opinion, formulated more precisely. Submitting to Maupertuis, the Berlin Academy rose to defend him and demanded that the original of Leibniz’ letter be presented. When it became clear that the original could not be found, Euler published, with the approval of the Academy, “Exposé concernant l’examen de la lettre de M. de Leibnitz” (1752), where, among other things, he declared the letter a fake.

The conflict grew critical when later in the same year Voltaire published his Diatribe du docteur Akakia, médecin du pape, defending König and making laughingstocks of both Maupertuis and Euler. Frederick rushed to the defense of Maupertuis, quarreling with his friend Voltaire and ordering the burning of the offensive pamphlet. His actions, however, did not prevent its dissemination throughout Europe. The argument touched not only on the pride of the principal participants but also on their general views: Maupertuis and, to a lesser degree, Euler interpreted the principle of least action theologically and teleologically; König was a follower of Wolff and Voltaire - the greatest ideologist of free thought.

Three other disputes in which Euler took part were much more important for the development of mathematical sciences: his argument with d’Alembert on the problem of logarithms of negative numbers, the argument with d’Alembert and Daniel Bernoulli on the solution of the equation of a vibrating siring, and Euler’s polemics with Dollond on optical problems.

As mentioned earlier, after Maupertuis died in 1759, Euler managed the Berlin Academy, but under the direct supervision of the king. But relations between Frederick and Euler had long since spoiled. They differed sharply, not only in their views but in their tastes, treatment of men, and personal conduct. Euler as manager of the Academy, the king did not intend to give him the post of president. In 1763 it became known that Frederick wanted to appoint d’Alembert, and Euler thus began to think of leaving Berlin. He wrote to G. F. Muller, secretary of the St. Petersburg Academy, which had tried earlier to bring him back to Russia. Catherine the Great then ordered the academicians to send Euler another offer.

D’Alembert’s refusal to move permanently to Berlin postponed for a time the final decision on the matter. But during 1765 and 1766 grave conflicts over financial matters arose between Euler and Frederick, who interfered actively with Euler’s management of the Academy after the Seven Years’ War. The king thought Euler inexperienced in such matters and relied too much on the treasurer of the Academy. For half a year Euler pleaded for royal permission to leave, but the king, well-aware that the Academy would thus lose its best worker and principal force, declined to grant his request. Finally he had to consent and vented his annoyance in crude jokes about Euler. On June 9, 1766, Euler left Berlin, spent ten days in Warsaw at the invitation of Stanislas II, and arrived in St. Petersburg on July 28. Euler’s three sons returned to Russia also. Johann Albrecht became academician in the chair of physics in 1766 and per-manent secretary of the Academy in 1769. Christoph, who had become an officer in Prussia, successfully resumed his military career, reaching the rank of major-general in artillery. Both his daughters also accompanied him.

Euler settled in a house on the embankment of the Neva, not far from the Academy. Soon after his return he suffered a brief illness, which left him almost completely blind in the left eye; he could not now read and could make out only outlines of large objects. He could write only in large letters with chalk and slate. An operation in 1771 temporarily restored his sight, but Euler seems not to have taken adequate care of himself and in a few days he was completely blind. Shortly before the operation, he had lost his house and almost all of his personal property in a fire, barely managing to rescue himself and his manuscripts.

Euler’s blindness did not lessen his scientific activity. Only in the last years of his life did he cease attending academic meetings, and his literary output even increased - almost half of his works were produced after 1765. His memory remained flawless, he carried on with his unrealized ideas, and he devised new plans. He naturally could not execute this immense work alone and was helped by active collaborators: his sons Johann Albrecht and Christoph; the academicians W. L. Krafft and A. J. Lexell; and two new young disciples, adjuncts N. Fuss, who was invited in 1772 from Switzerland, and M. E. Golovin, a nephew of Lomonosov. Sometimes Euler simply dictated his works; thus, he dictated to a young valet, a tailor by profession, the two-volume Vollständige Anleitung zur Algebra (1770), first published in Russian translation.

But the scientists assisting Euler were not mere secretaries; he discussed the general scheme of the works with them, and they developed his ideas, calculated tables, and sometimes compiled examples. The enormous, 775-page Theoria motuum lunae (1772) was thus completed with the help of Johann Albrecht, Krafft, and Lexell - all of whom are credited on the title page. Krafft also helped Euler with the three-volume Dioptrica (1769-1771). Fuss, by his own account, during a seven-year period prepared 250 memoirs, and Golovin prepared seventy. Articles written by Euler in his later years were generally concise and particular. For example, the fifty-six works prepared during 1776 contain about the same number of pages (1,000) as the nineteen works prepared in 1751.

Besides the works mentioned, during the second St. Petersburg period Euler published three volumes of Institutiones calculi integralis (1768-1770), the principal parts of which he had finished in Berlin, and an abridged edition of Scientia navalis - Theorie complette de la construction et de la manoeuvre des vaisseaux (1773). The last, a manual for naval cadets, was soon translated into English, Italian, and Russian, and Euler received for it large sums from the Russian and French governments.

Euler continued his participation in other functions of the St. Petersburg Academy. Together with Johann Albrecht he was a member of the commission charged in 1766 with the management of the Academy. Both resigned their posts on the commission in 1774 because of a difference of opinion between them and the director of the Academy, Count V. G. Orlov, who actually managed it.

On September 18, 1783 Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o’clock in the afternoon he suffered a brain hemorrhage and uttered only “I am dying,” before he lost consciousness. He died about eleven o’clock in the evening.

Much of what is known of Euler's religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Defense of the Divine Revelation against the Objections of the Freethinkers. These works show that Euler was a devout Christian who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of scripture.

Views

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and the philosophy of Christian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic."

Quotations:
"Madam, I have come from a country where people are hanged if they talk."

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."

"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be."

"It would be a considerable invention indeed, that of a machine able to mimic speech, with its sounds and articulations. I think it is not impossible."

"Nothing takes place in the world whose meaning is not that of some maximum or minimum."

"Logic is the foundation of the certainty of all the knowledge we acquire."

Membership

Euler was a member of the Saint Petersburg Academy of Sciences (1730), the Royal Prussian Academy of Sciences (1741), the Royal Society, the French Academy of Sciences (1750), and the American Academy of Arts and Sciences (1782).

Personality

Euler was a simple man, well disposed and not given to envy. One can also say of him what Fontenelle said of Leibniz: “He was glad to observe the flowering in other people’s gardens of plants whose seeds he provided.”

Physical Characteristics:
Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye, but Euler rather blamed the painstaking work on cartography he performed for the St. Petersburg Academy for his condition.

He later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, he was rendered almost totally blind. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional memory.

Quotes from others about the person

François Arago: "Euler calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air."

William Dunham: "Somebody said 'Talent is doing what others find difficult. Genius is doing easily what others find impossible.' By that definition, Euler was a genius. He could do the seemingly impossible, and he did it throughout his long and illustrious life."

Frederick the Great: "Euler calculated the force of the wheels necessary to raise the water in a reservoir. My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!"

Ferdinand Georg Frobenius: "Euler lacked only one thing to make him a perfect genius: He failed to be incomprehensible."

Carl Friedrich Gauss: "The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it."

Pierre-Simon Laplace: "Read Euler: he is our master in everything."

Richard Mankiewicz: "Euler was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children."

Eli Maor: "If we compared the Bernoullis to the Bach family, then Leonhard Euler is unquestionably the Mozart of mathematics."

Thomas Reid: "It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics."

William Whewell: "The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler."

Interests

Philosophers & Thinkers

René Descartes, Isaac Newton

Connections

At the end of 1733 Euler married Katharina Gsell, a daughter of Georg Gsell, a Swiss who taught painting at the Gymnasium attached to the St. Petersburg Academy. Johann Albrecht, Euler’s first son, was born in 1734, and Karl was born in 1740.

Euler lived in Berlin for twenty-five years. In 1744 he moved into a house, still preserved, on the Behrenstrasse. The family increased with the birth of a third son, Christoph, and two daughters; eight other children died in infancy. In 1753 Euler bought an estate in Charlottenburg, which was then just outside the city. The estate was managed by his mother, who lived with Euler after 1750. He sold the property in 1763.

In November 1773 Euler’s wife died, and three years later he married her half sister, Salome Abigail Gsell.