Leonhard Euler “. . .was the key figure in eighteenth-century mathematics and the dominant theoretical physicist of the century, the man who should be ranked with Archimedes, Newton, and Gauss ...”(1) Euler was the most prolific mathematician in history, with the Swiss edition of his complete works consisting of 74 volumes.(2) It has been computed that his publications during his working life averaged 800 pages a year. Euler, who worked mainly in St. Petersburg and Berlin, was a prolific and original contributor to all branches of mathematics.(3) 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. Euler created much of the calculus that is taught in colleges today, including many specific sequences and series and much of standard notation, such as f(x) for a function. More than a dozen well-known results bear his name, and e, the base of the natural logarithms, is called Euler’s number.

​Early Years

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At 20, in 1727, Euler moved to St. Petersburg, Russia, where he became an associate of the St. Petersburg Academy of Sciences, and in 1733, at age 26, succeeded Daniel Bernoulli to the chair of mathematics. “By means of his numerous books and memoirs that he submitted to the academy, Euler carried integral calculus to a higher degree of perfection, developed the theory of trigonometric and logarithmic functions, reduced analytical operations to a greater simplicity, and threw new light on nearly all parts of pure mathematics.”(4) Overtaxing himself, Euler in 1735, at age 28, lost the sight of one eye. Then, invited by Frederick 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. “He did for modern analytic geometry and trigonometry what the Elements of Euclid had done for ancient geometry, and the resulting tendency to render mathematics and physics in arithmetical terms has continued ever since.”(5)

Euler’s work of 1748 titled, Introductio in analysis infinitorum (Introduction to 

Analysis of the Infinite), is the key text that presents modern analytic geometry and trigonometry. According to Carl Boyer, an historian of mathematics, Euler’s Introductio is the most influential textbook of modern times.(6) Here, in effect, Euler accomplished for analysis what Euclid and Al-Khwarizmi had done for synthetic geometry and elementary algebra respectively. It contains the earliest systematic graphical study of functions of one and two independent variables, including the recognition of the quadrics as constituting a single family of surfaces. The Introductio was first also in the algorithmic treatment of logarithms as exponents and in the analytic treatment of the trigonometric functions as numerical ratios.

Finally, Introductio examined the relation between the exponential and trigonometric functions for complex numbers, leading to the formula: eiØ = cos Ø + i sin Ø

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From this, setting Ø = π  radians = 180˚, it's possible to derive the famous equation:

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eiπ + 1 = 0

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relating the two enigmatic constants e and π, and the imaginary number i. Here e = 2.718 .... is the base of natural logarithms and i is the symbol that Euler introduced for the square root of minus one, still standard today. This formula connects the five most important constants of mathematics (and also the three most important mathematical operations--addition, multiplication, and exponentiation).(7) These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 0 and 1; algebra, by i; geometry, by π; and analysis, by e. Now that complex analysis is better understood, this relationship does not come as much of a surprise, but in Euler's day it was amazing. Trigonometric functions came from the geometry of circles and the measurements of triangles; the exponential function came from the mathematics of compound interest and the calculating tool of logarithms. Why should these things be so intimately connected?

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Beginnings of Topology

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Euler can be considered the founding father of topology, a separate branch of mathematics, which the eighteenth century knew as geometria situs. Euler's classic paper on the branch of topology we now call 'graph theory' was written in 1736 to solve a particular puzzle - that of the seven bridges of Konigsberg (now Kaliningrad ) spanning the River Pregel where tributaries meet at an island as shown in the figure below. The problem was to devise a route around the city which would cross each of the seven bridges once only. It turns out that such a journey cannot be completed across the seven bridges of Konigsberg.

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The seven bridges of Konigsberg

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Another of Euler's topological discoveries is his polyhedra formula illustrated in the stamp at the beginning of this biography.

Think of a triangle. Now think of a square, followed by a pentagon, a hexagon, and so on. These shapes are called polygons. "Poly" is the Greek word for "many" and "gon" is the Greek word for "angle." Now move up a dimension. Think of a cube, a pyramid, or perhaps an octahedron. These are all polyhedra ("hedra" is the Greek word for "base"). A polyhedron is an

object made up of a number of flat polygonal faces. The sides of the faces are called edges and the corners of the polyhedron are called vertices. The Platonic solids are examples of polyhedra.  From left to right we have the tetrahedon with four faces, the cube with six faces, the octahedron with eight faces, the dodecahedron with twelve faces, and the icosahedron with

twenty faces. 

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Now imagine counting the number V of vertices, the number E of edges and the number F of faces of a polyhedron. It turns out that, as long as your polyhedron is convex (has no sticky out bits) and has no holes running through it, the number of vertices minus the number of edges plus the number of faces, V - E + F, is always equal to 2. Your polyhedron could be a cube or an octahedron. 

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In 1750, Euler wrote in a letter to Christian Goldbach (1690-1764), "I cannot yet give an entirely satisfactory proof of the following proposition: In every solid enclosed by plane faces the aggregate of the number of faces and the number of solid angles exceeds by two the number of edges." In other words, given any polyhedron with V vertices (solid angles), E edges, and F faces, then V + F = E + 2, or, more familiarly, V - E + F = 2.

In the same letter Euler wrote "it astonishes me that these general properties of stereometry have not, as far as I know, been noticed by anyone else."(8) A completely correct proof was given in 1794 by Adrien-Marie Legendre (1752-1833). It was not until the late nineteenth and early twentieth centuries, however, that these facts and certain others were systematically studied and finally turned into the subject of topology.(9)    

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Mathematics–Calculus, Symbols

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Euler’s textbooks in calculus, Institutiones calculi differentialis (Methods of the Differential Calculus, 1755) and Institiones calculi integralis (Methods of the Integral Calculus, 1768–70), have served as prototypes to the present because they contain formulas of differentiation and numerous methods of indefinite integration, many of which he invented himself, for determining the work done by a force and for solving geometric problems.(10) His textbooks in calculus also freed the new science from its geometric origins and established it as algebraic calculus, which was subsequently called “analysis.” The term “analysis” thus became the name of one of the main branches of mathematics.(11) 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; ∫ n for the sum of divisors of n; the symbol e for the base of natural logarithms; a, b, and c for the sides of a triangle and A, B, and C for the opposite angles; the letter “f” and parentheses for a function, (f); and i for √-1. Euler chose the Greek letter π (pi) for the ratio of the circumference to the diameter of a circle. This symbol had been used before, but Euler’s use of it in textbooks made it widely known. His textbooks on algebra, calculus, complex analysis, and other topics standardised mathematical notation and terminology.(12) He even brought together Newton's and Leibniz's notations in differential calculus. Overall, he influenced modern notation more than any other mathematician in history.(13)

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Later Years

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After Frederick the Great became less cordial toward him, Euler in 1766, at age 59, accepted the invitation of Catherine II to return to Russia. Soon after his arrival at St. Petersburg, a cataract formed in his remaining good eye, and he spent the last years of his life in total blindness. Despite this tragedy, his productivity continued undiminished, sustained by an uncommon memory and a remarkable facility in mental computations. Not a classroom teacher, Euler nevertheless had a more pervasive pedagogical influence than any modern mathematician.(14) He had few disciples, but he helped to establish mathematical education in Russia.

Astronomy, Number Theory

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Euler devoted considerable attention to developing a more perfect theory of lunar motion, which was particularly troublesome, since it involved the so-called three-body problem—the interactions of sun, moon, and earth. (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. Throughout his life, Euler was much absorbed by problems dealing with the theory of numbers, which focuses on the properties and relationships of integers, or whole numbers (0, ±1, ±2, etc.). In this, his greatest discovery, in 1783, was the law of quadratic reciprocity, which has become an essential part of modern number theory. Gauss later proved this law.

Euler was succeeded by Joseph-Louis LaGrange. But where Euler favored special concrete cases, Lagrange sought for abstract generality.

Footnotes:

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(1) Morris Kline, Mathematical Thought From Ancient To Modern Times (New York, 1972), p. 401.

(2) Lloyd Motz and Jefferson Hane Weaver, The Story of Mathematics (New York, 1993), p. 97. 

(3) Judy Pearsall and Bill Trumble (editors), The Oxford Encyclopedic English Dictionary (New York, 1996), p. 483.

(4) Encyclopaedia Britannica, Micropaedia, Volume 4, 1993, 15th Edition, p. 596.

(5) Ibid., p. 596.

(6) www-history.mcs.st-andrews.ac.uk/Extras/Boyer_Foremost_Text.html.

(7) Ibid., p. 596.

(8) Victor J. Katz, A History of Mathematics - An Introduction, 3rd Edition (Boston, 2009), p. 702. 

(9) Ibid., p. 702.

(10) The field of mathematics can be divided in various ways. For this work the divisions are: (a) Algebra; (b) Geometry; (c) Calculus/Analysis; (d) Foundations; and (e) Applied Mathematics.

(11) Eli Maor, e: The Story of a Number (Princeton, 1994), p. 160.

(12) Ian Stewart, Significant Figures - The Lives and Work of Great Mathematicians (New York, 2017), p. 84.

(13) Carl B. Boyer, A History of Mathematics, 2nd Edition (New York, 1991), p. 442.

(14) Ibid., p. 596.

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