Leonhard Euler (born April 15, 1707,Basel, Switzerland—died September 18, 1783,St. Petersburg, Russia) was aSwiss mathematician andphysicist, one of the founders of puremathematics. He not only made decisive and formative contributions to the subjects ofgeometry,calculus,mechanics, andnumber theory but also developed methods for solving problems in observationalastronomy and demonstrated useful applications of mathematics in technology and public affairs.

From St. Petersburg to Berlin

Euler’s mathematical ability earned him the esteem ofJohann Bernoulli, one of the first mathematicians in Europe at that time, and of his sons Daniel and Nicolas. In 1727 he moved to St. Petersburg, where he became an associate of the St. Petersburg Academy of Sciences and in 1733 succeededDaniel Bernoulli to the chair of mathematics. By means of his numerous books and memoirs that he submitted to the academy, Euler carriedintegralcalculus to a higher degree of perfection, developed the theory of trigonometric and logarithmic functions, reducedanalytical operations to a greater simplicity, and threw new light on nearly all parts of pure mathematics. Overtaxing himself, Euler in 1735 lost the sight of one eye. Then, invited byFrederick the Great in 1741, he became a member of the Berlin Academy, where for 25 years he produced a steady stream of publications, many of which he contributed to the St. Petersburg Academy, which granted him a pension.

Foundational works and new concepts

In 1748, in hisIntroductio in analysin infinitorum, he developed the concept offunction in mathematicalanalysis, through which variables are related to each other and in which he advanced the use of infinitesimals andinfinite quantities. He did for modernanalytic geometry andtrigonometry what theElements ofEuclid had done for ancient geometry, and the resulting tendency to render mathematics andphysics in arithmetical terms has continued ever since. He is known for familiar results in elementary geometry—for example, the Euler line through the orthocentre (the intersection of the altitudes in a triangle), the circumcentre (the centre of the circumscribed circle of a triangle), and the barycentre (the “centre of gravity,” or centroid) of a triangle. He was responsible for treating trigonometric functions—i.e., the relationship of an angle to two sides of a triangle—as numerical ratios rather than as lengths of geometric lines and for relating them, through the so-called Euler identity (e^iθ = cos θ +i sin θ), with complex numbers (e.g., 3 + 2Square root of√−1). He discovered the imaginarylogarithms of negative numbers and showed that eachcomplex number has an infinite number of logarithms.

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Numbers and Mathematics

Euler’s textbooks in calculus,Institutiones calculi differentialis in 1755 andInstitutiones calculi integralis in 1768–70, have served asprototypes to the present because they contain formulas of differentiation and numerous methods of indefiniteintegration, many of which he invented himself, for determining thework done by aforce and for solving geometric problems, and he made advances in the theory of linear differential equations, which are useful in solving problems in physics. Thus, he enriched mathematics with substantial new concepts and techniques. He introduced many current notations, such as Σ for the sum; the symbole for the base ofnatural logarithms;a,b andc for the sides of a triangle and A, B, and C for the opposite angles; the letterf and parentheses for a function; andi forSquare root of√−1. He also popularized the use of the symbolπ (devised by British mathematician William Jones) for the ratio of circumference todiameter in a circle.

Later years

AfterFrederick the Great became less cordial toward him, Euler in 1766 accepted the invitation ofCatherine II to return toRussia. Soon after his arrival at St. Petersburg, acataract formed in his remaining good eye, and he spent the last years of his life in totalblindness. Despite this tragedy, his productivity continued undiminished, sustained by an uncommon memory and a remarkable facility in mental computations. His interests were broad, and hisLettres à une princesse d’Allemagne in 1768–72 were an admirably clear exposition of the basic principles of mechanics,optics, acoustics, and physical astronomy. Not a classroom teacher, Euler nevertheless had a morepervasivepedagogical influence than any modern mathematician. He had fewdisciples, but he helped to establish mathematical education in Russia.

Euler devoted considerable attention to developing a more perfect theory of lunarmotion, which was particularly troublesome, since it involved the so-calledthree-body problem—the interactions ofSun,Moon, andEarth. (The problem is still unsolved.) His partial solution, published in 1753, assisted the British Admiralty in calculating lunar tables, of importance then in attempting to determine longitude at sea. One of the feats of his blind years was to perform all the elaborate calculations in his head for his second theory of lunar motion in 1772. Throughout his life Euler was much absorbed by problems dealing with the theory ofnumbers, which treats of the properties and relationships of integers, or whole numbers (0, ±1, ±2, etc.); in this, his greatest discovery, in 1783, was the law ofquadratic reciprocity, which has become an essential part of modernnumber theory.

Legacy

In his effort to replacesynthetic methods byanalytic ones, Euler was succeeded byJoseph-Louis Lagrange. But, where Euler had delighted in special concrete cases, Lagrange sought for abstract generality, and, while Euler incautiously manipulated divergent series, Lagrange attempted to establish infinite processes upon a sound basis. Thus it is that Euler and Lagrange together are regarded as the greatest mathematicians of the 18th century, but Euler has never been excelled either in productivity or in the skillful and imaginative use of algorithmic devices (i.e., computational procedures) for solving problems.

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Since 1993 the U.S.-based Institute of Combinatorics and its Applications has awarded a medal named for Euler that recognizes lifetime achievement incombinatorics.