EULER, LEONHARD (170 7–I 783) , Swiss mathematician, was born at Basel on Apr. 15, 1707. His father, a good mathema tician, was a Lutheran pastor. In 1723 he graduated at Basel, where he studied geometry under Jean Bernoulli, at that time one of the first mathematicians in Europe, and became a close friend of his sons, Daniel and Nicolas. He then took up theology and oriental languages, and medicine. In 1727 on the invitation of Catherine I., Euler joined his friends in St. Petersburg, where he became professor of physics in 1730 and three years later of mathematics, in succession to Daniel Bernouilli. The severity of the climate and close application to study affected his hearth and in 1735 he lost the sight of one eye. In 1741 Euler went to Berlin at the command of Frederick the Great, and during the next 25 years contributed many memoirs to the Prussian Academy. During this period he continued to contribute memoirs to the academy of St. Petersburg, and in 1766 he obtained, though with difficulty, permission to return to Russia. Soon afterwards a cataract formed in his left eye, which left him almost blind ; with the help of his sons and of Krafft and Lexell, however, he con tinued his labours. In the next seven years he sent in 70 memoirs to the Academy, and left in his papers some 200 more. He died of apoplexy on Sept. 18, 1783.
Importance of His Work.—Euler's greatest work was done in pure mathematics, and he must be regarded as one of the founders of the modern science. In his Introductio in analysin in finitorum (1748), of which a full analysis is given in Cantor's Geschichte der Mathematik (vol. iii.), he provided an introduction to pure analytical mathematics. In the first part he gave the bulk of the matter to be found in treatises on algebra, the theory of equations and trigonometry ; the second was devoted to analytical geometry. "In the algebra he paid particular attention to the expansion of various functions in series, and to the summation of given series; and for the first time we find the rule laid down that an infinite series cannot be safely employed unless it is convergent" (W. W. R. Ball, Short History of Mathematics). Euler treated trigo nometry as a branch of analysis. He introduced (at the same time as Thomas Simpson) the abbreviations now used for the trigo nometric functions and made use of the symbols a and ir. In the second he made many investigations which were new in his time; he discussed the general equation of the second degree in three dimensions, and classified the surfaces represented by it; he showed that the conic sections were represented by the general equation of the second degree in two dimensions.
His next important works were the Institutiones calculi di$eren tialis (1755) and the Institutiones calculi integralis (1768-70), which may be said to be the first complete and accurate treatises on the calculus of that time. Beta and Gamma Functions and other original investigations were contained in these works. His Methodus inveniendi lineas curvas maxim.i minimive pro prietate gaudentes (1744) is an earlier attempt to elaborate the calculus of variations afterwards perfected by Lagrange. His Anleitung zur Algebra (1770; translated and extended by Lagrange 1795) is in two parts, the first treating of determinate and the second of indeterminate algebra.
But though Euler's most important work was done in pure mathematics he was a man of wide culture, interested in many branches of applied mathematics and science. He made im portant contributions to astronomy, hydrodynamics and optics. His T heoria motuum lunae (17 7 2) was based on earlier work of his on the subject. It was completed under terrible difficulties. His house bad been burned down and some of his papers de stroyed ; he was nearly blind, and had to carry all the elaborate computations involved in his head. The complicated work on lunar motion in this treatise formed the basis of the lunar tables subsequently constructed by Mayer. His researches in optics were collected by him in the three vols. of his Dioptrica (I 771).
A general view of the principal facts of mechanics, optics, acous tics and physical astronomy is provided in his Lettres a une prin cesse d'Allemagne stir quelques sujets de physique et de philoso phie (1768-72), written for the use of the princess of Anhalt Dessau.
A catalogue of Euler's works was drawn up by Fuss, the secretary of the St. Petersburg Academy, which issued some of his papers under the title Opera Postuma in 1862. A complete edition of his works was begun in 1926. See M. Cantor, Geschichte der Mathematik (1906).