**EULER, LEONHARD** (1707–1783), Swiss mathematician,
was born at Basel on the 15th of April 1707, his father Paul
Euler, who had considerable attainments as a mathematician,
being Calvinistic pastor of the neighbouring village of Riechen.
After receiving preliminary instructions in mathematics from
his father, he was sent to the university of Basel, where geometry
soon became his favourite study. His mathematical genius
gained for him a high place in the esteem of Jean Bernoulli, who
was at that time one of the first mathematicians in Europe,
as well as of his sons Daniel and Nicolas Bernoulli. Having
taken his degree as master of arts in 1723, Euler applied himself,
at his father’s desire, to the study of theology and the Oriental
languages with the view of entering the church, but, with his
father’s consent, he soon returned to geometry as his principal
pursuit. At the same time, by the advice of the younger
Bernoullis, who had removed to St Petersburg in 1725, he applied
himself to the study of physiology, to which he made a happy
application of his mathematical knowledge; and he also attended
the medical lectures at Basel. While he was engaged in
physiological researches, he composed a dissertation on the nature
and propagation of sound, and an answer to a prize question
concerning the masting of ships, to which the French Academy
of Sciences adjudged the second rank in the year 1727.

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 Daniel Bernoulli in the chair of mathematics. At the commencement of his new career he enriched the academical collection 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 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 St 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 *Anleitung*
*zur Algebra* (1770), 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 d’Allemagne*
*sur quelques sujets de physique et de philosophie* (3 vols., 1768–1772).
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 W. L. Krafft
and A. J. Lexell, and pursued his researches until he had
constructed the new tables, which appeared in his *Theoria motuum*
*lunae* (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 co-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 an operation 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 *Acta*
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. His last subject of investigation was the motion of
balloons, and the last subject on which he conversed was the
newly discovered planet Herschel (Uranus). He died of apoplexy
on the 18th of September 1783, whilst he was amusing himself
at tea with one of his grandchildren.

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 the marquis de
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* (St Petersburg, 1736, in 2 vols. 4to); *Einleitung in*
*die Arithmetik* (ibid., 1738, in 2 vols. 8vo), in German and Russian;
*Tentamen novae theoriae musicae* (ibid. 1739, in 4to); *Methodus*
*inveniendi lineas curvas, maximi minimive proprietate gaudentes*
(Lausanne, 1744, in 4to); *Theoria motuum planetarum et cometarum*
(Berlin, 1744, in 4to); *Beantwortung*, &c., or Answers to Different
Questions respecting Comets (ibid., 1744, in 8vo); *Neue Grundsätze*,
&c., or New Principles of Artillery, translated from the English of
Benjamin Robins, with notes and illustrations (ibid., 1745, in 8vo);
*Opuscula varii argumenti* (ibid., 1746–1751, in 3 vols. 4to); *Novae*
*et correctae tabulae ad loca lunae computanda* (ibid., 1746, in 4to);
*Tabulae astronomicae solis et lunae* (ibid., 4to); *Gedanken*, &c., or
Thoughts on the Elements of Bodies (ibid. 4to); *Rettung der*
*göttlichen Offenbarung*, &c., Defence of Divine Revelation against
Free-thinkers (ibid., 1747, in 4to); *Introductio in analysin infinitorum*
(Lausanne, 1748, in 2 vols. 4to); *Scientia navalis, seu tractatus de*
*construendis ac dirigendis navibus* (St Petersburg, 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 differentialis, cum ejus*
*usu in analysi Infinitorum ac doctrina serierum* (ibid., 1755, in 4to);
*Constructio lentium objectivarum*, &c. (St Petersburg, 1762, in 4to);
*Theoria motus corporum solidorum seu rigidorum* (Rostock, 1765,
in 4to); *Institutiones calculi integralis* (St Petersburg, 1768–1770, in
3 vols. 4to); *Lettres à une Princesse d’Allemagne sur quelques sujets de*
*physique et de philosophie* (St Petersburg, 1768–1772, in 3 vols. 8vo);
*Anleitung zur Algebra*, or Introduction to Algebra (ibid., 1770, in
8vo); *Dioptrica* (ibid., 1767–1771, in 3 vols. 4to); *Theoria motuum*
*lunae nova methodo pertractata* (ibid., 1772, in 4to); *Novae tabulae*
*lunares* (ibid., in 8vo); *Théorie complète de la construction et de la*
*manœuvre des vaisseaux* (ibid., 1773, in 8vo); *Éclaircissements sur* *établissements en faveur tant des veuves que des morts*, without a date; *Opuscula analytica* (St Petersburg, 1783–1785, in 2 vols. 4to).

See Rudio, *Leonhard Euler* (Basel, 1884); M. Cantor, *Geschichte der Mathematik*.