In 1727, on the invitation of Catherine I., Euler took up his residence in St Petersburg, and was made an associate of the Academy of Sciences. In 1730 he became professor of physics, and in 1733 he succeeded his friend Daniel Bernoulli in the chair of mathematics. At the commencement of his new career he enriched the academical collec tion with many memoirs, which excited a noble emulation between him and the Bernoullis, though this did not in any way affect their friendship. It was at this time that he carried the integral calculus to a higher degree of perfection, invented the calculation of sines, reduced analytical operations to a greater simplicity, and threw new light on nearly all parts of abstract or pure mathematics. In 1735 a problem proposed by the academy, for the solution of which several eminent mathematicians had demanded the space of some months, was solved by Euler in three days, but the effort threw him into a fever which endangered his life and deprived him of the use of his right eye. The Academy of Sciences at Paris in 1738 adjudged the prize to his memoir on the nature and properties of fire, and in 1740 his treatise on the tides shared the prize with those of Colin Maclaurin and Daniel Bernoulli,—a higher honour than if he had carried it away from inferior rivals.

In 1741 Euler accepted the invitation of Frederick the Great to Berlin, where he was made a member of the Academy of Sciences and professor of mathematics. He enriched the last volume of the Mélanges or Miscellanies of Berlin with five memoirs, and these were followed, with an astonishing rapidity, by a great number of important researches, which are scattered throughout the annual memoirs of the Prussian Academy. At the same time he continued his philosophical contributions to the Academy of St Petersburg, which granted him a pension in 1742. The respect in which he was held by the Russians was strikingly shown in 1760, when a farm he occupied near Charlottenburg happened to be pillaged by the invading Russian army On its being ascertained that the farm belonged to Euler, the general immediately ordered compensation to be paid, and the empress Elizabeth sent an additional sum of four thousand crowns. In 1766 Euler with difficulty obtained permission from the king of Prussia to return to Petersburg, to which he had been originally invited by Catherine II. Soon after his return to St Petersburg a cataract formed in his left eye, which ultimately deprived him almost entirely of sight. It was in these circumstances that he dictated to his servant, a tailor’s apprentice, who was absolutely devoid of mathematical knowledge, his Elements of Algebra, a work which, though purely elementary, displays the mathematical genius of its author, and is still reckoned one of the best works of its class. Another task to which he set himself immediately after his return to St Petersburg was the preparation of his Lettres à une Princesse eΓ Allemagne sur quelques sujets de Physique (3 vols., 1768-72). They were written at the request of the princess of Anhalt-Dessau, and contain an admirably clear exposition of the principal facts of mechanics, optics, acoustics, and physical astronomy. Theory, however, is frequently unsoundly applied in it, and it is to be observed generally that Euler’s strength lay rather in pure than in applied mathematics. In 1755 Euler had been elected a foreign member of the Academy of Sciences at Paris, and some time afterwards the academical prize was adjudged to three of his memoirs Concerning the Inequalities in the Motions of the Planets. The two prize-questions proposed by the same academy for 1770 and 1772 were designed to obtain a more perfect theory of the moon’s motion. Euler, assisted by his eldest son Johann Albert, was a competitor for these prizes, and obtained both. In the second memoir he reserved for further consideration several inequalities of the moon’s motion, which he could not determine in his first theory on account of the complicated calculations in which the method he then employed had engaged him. He afterwards reviewed his whole theory with the assistance of his son and Krafft and Lexell, and pursued his researches until he had constructed the new tables, which appeared, together with the great work, in 1772. Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by mathematical principles, he reduced them to the three ordinates which determine the place of the moon; and he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit. The inherent difficulties of this task were immensely enhanced by the fact that Euler was virtually blind, and had to carry all the elaborate computations it involved in his memory. A further difficulty arose from the burning of his house and the destruction of the greater part of his property in 1771. His manuscripts were fortunately preserved. His own life was only saved by the courage of a native of Basel, Peter Grimmon, who* carried him out of the burning house.

Some time after this the celebrated Wenzell, by couching the cataract, restored Euler’s sight; but a too harsh use of the recovered faculty, along with some carelessness on the part of the surgeons, brought about a relapse. With the assistance of his sons, and of Krafft and Lexell, however, he continued his labours, neither the loss of his sight nor the infirmities of an advanced age being sufficient to check his activity. Having engaged to furnish the Academy of St Petersburg with as many memoirs as would be sufficient to complete its acts for twenty years after his death, he in seven years transmitted to the academy above seventy memoirs, and left above two hundred more, which were revised and completed by another hand.

Euler’s knowledge was more general than might have been expected in one who had pursued with such unremitting ardour mathematics and astronomy as his favourite studies. He had made very considerable progress in medical, botanical, and chemical science, and he was an excellent classical scholar, and extensively read in general literature. He was much indebted to an uncommon memory, which seemed to retain every idea that was conveyed to it, either from reading or meditation. He could repeat the Aeneid of Virgil from the beginning to the end without hesitation, and indicate the first and last line of every page of the edition which he used. Euler’s constitution was uncommonly vigorous, and his general health was always good. He was enabled to continue his labours to the very close of his life, so that it was said of him that he ceased to calculate and to breathe at nearly the same moment. His last subject of investigation was the motion of balloons, and the last subject on which he conversed wife the neλvly discovered planet Herschel. On the 18th September 1783, whilst he was amusing himself at tea with one of his grandchildren, he was struck with apoplexy, which terminated his illustrious career at the age of seventy-six. Euler’s genius was great, and his industry still greater. His works, if printed in their completeness, would occupy from 60 to 80 quarto volumes. He was simple and upright in his character, and had a strong religious faith. He was twice married, his second wife being a half-sister of his first, and he had a numerous family, several of whom attained to distinction. His éloge was written for the French Academy by Condorcet, and an account of his life, with a list of his works, was written by Von Fuss, the secretary to the Imperial Academy of St Petersburg.

The works which Euler published separately are— Dissertatio physica de Sono, Basel, 1727, in 4to; Mechanica, sive Motus scientia analytice exposita, Petersb., 1736, in 2 vols. 4to; Einleitung in die Arithmetik, ibid. 1738, in 2 vols. 8vo, in German and Russian; Tentamen Novae Thcoriae Mιcsicce, ibid. 1739, in 4to; Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes, Lausanne, 1744, in 4to; Theoriamotuum Planetarum et Cometarum, Berlin, 1744 in 4to; Beantwortung, etc., or Answers to different Questions respecting Comets, ibid. 1744, in 8vo; Neue Grundsätze, etc., or New Principles of Artillery, translated from the English of Benjamin Robins, with notes and illustrations, ibid. 1745, in 8vo; Opuscula varii argumenti, ibid. 1746-51, in 3 vols. 4to; Novae et correctoe Tabulae ad loca Lunae computanda, ibid. 1746, in 4to; Tabulae Astronomic« Solis et Dunce, ibid. 4to; Gedanken, etc., or Thoughts on the Elements of Bodies, ibid. 4to; Rettung der Göttlichen Offenbarung, etc., Defence of Divine Revelation against Freethinkers, ibid, 1747, in 4to; Introductio in Analysin Infinitorum, Lausanne, 1748, in 2 vols. 4to; Scientia Navalis, seu Tractatus de construendis ac dirigendis navibus, Petersb. 1749, in 2 vols. 4to; Theoria motus Lunae, Berlin, 1753, in 4to; Dissertatio de principio minimae actionis, una cum examine Objectionum cl. prof. Koenigii, ibid. 1753, in 8vo; Institutiones Calculi Differentiatis, cum ejus usu in analyst Infinitorum acdoetrina Serierum, ibid. 1755, in 4to; Constructio Lentium Objectivarum, etc., Petersb. 1762, in 4to; Theoria motus corporum solidorum seu rigidorum, Rostock, 1765, in 4to; Institutiones Calculi Integralis, Petersb. 1768-1770, in 3 vols. 4to; Lettres à une Princess d' Allemagne sur quelques sujets de Physique et Philosophie, Petersb. 1768-1772, in 3 vols. 8vo; Anleitung zur Algebra, or Introduction to Algebra, ibid. 1770, in 8vo; Bioptrica, ibid. 1767-1771, in 3 vols. 4to; Theoria motuum Lunae nova methodo pertractata, ibid. 1772, in 4to; Novae Tabulae Lunares, ibid, in 8vo; Théorie complete de la construction et de la manoeuvre des Vaisseaux, ibid. 1773, in 8vo; Éclaircissements sur établissements en faveur tant des Veuves que des Morts, without a date; Opuscula Analytica, Petersb. 1783-1785, in 2 vols. 4to.