The term " moment of inertia" was here introduced for the first time. Another of Euler's works, Theoria motus corpus solidorum seu rigidorum (1765), treated the mechanics of solid bodies in the same way by resolving the motion of a solid body into a motion of the center of mass and a rotation about this point, Euler arrived at the general equations of motion. Mechanics and HydrodynamicsĮuler's Mechanica (1736) was the first textbook in which Newtonian particle dynamics was developed using analytical methods. Although Euler failed to prove this assertion, he used successive solutions of the equation to compute approximations to √ a and, reversing the procedure, found solutions of the equation by developing √ a as a continued fraction. Euler made the first attack on the problem, demonstrating the theorem for n = 3 and n = 4.įermat also stated that the Diophantine equation x 2 − ay 2 = 1 always has an infinity of solutions. Fermat's most famous proposition, the general proof of which has defeated the efforts of the ablest mathematicians to the present day, states that the equation x n + y n = z n has no solution in integers for n greater than 2. The first volume takes algebra up to cubic and biquadratic equations, while the second is devoted to the theory of numbers.Įuler proved many of the results that had been stated by Pierre Fermat. Theory of NumbersĪnother of Euler's outstanding textbooks was Vollständige Anleitung zur Algebra (1770). His method was essentially geometrical, and this made the solution of the simpler problems very clear. Although particular problems had been solved by others, it was Euler who developed a general method. Such problems, involving the determination of the form of a curve having a certain maximum or minimum property, were quite different from the ordinary maximum and minimum problems of the differential calculus. This subject grew out of the isoperimetric problems which had created great interest among the mathematicians of the time. His outstanding achievement in this field was the invention of the calculus of variations, described in The Art of Finding Curves Which Possess Some Property of Maximum or Minimum (in Latin 1744). This part of the Introductio really constituted the first treatise on analytical geometry.Įuler wrote two great textbooks on the calculus: Institutiones calculi differentialis (1755) and Institutiones calculi integralis (3 vols., 1768-1770). Turning to the case of three dimensions, Euler gave the first complete classification of surfaces represented by the general equation of the second degree. First Euler considered the general equation of the second degree in two dimensions, showing that it represents the various conic sections the discussion included a treatment of asymptotes, centers of curvature, and curves of higher degree. The second volume of the Introductio contains an analytical study of curves and surfaces. He also resolved the subtle problem of the logarithms of negative and imaginary numbers, and he proved that e is irrational. Warned against the use of divergent series, he himself did not always succeed in avoiding such series. At this time no clear notion of convergence existed it is not surprising, therefore, that although Euler These functions are developed as infinite series. The first volume is devoted to the theory of functions, and in particular the exponential, logarithmic, and trigonometric functions. Moreover, he defined the trigonometric values as ratios and introduced the modern notation.Įuler's first great textbook was Introductio in analysin infinitorum (1748). Angles of a triangle he represented by A, B, C and the corresponding sides by a, b, c, thus simplifying trigonometric formulas. At various times he used the notations f(x), e, π, i, Σ, though he was not in every case the first to do so. Analysis and the CalculusĮuler's textbooks presented all that was known of mathematics in a clear and orderly manner, setting fashions in notation and method which have been influential to the present day. In 1776, having lost his first wife, he married his sister-in-law. Petersburg, Euler became blind but continued to dictate books and papers. Euler was director of mathematics at the Academy of Sciences there until 1766, when he returned to St. In 1741 Frederick the Great called him to Berlin. Petersburg, Russia he became professor of physics in 1730 and professor of mathematics in 1733. In 1727 Catherine I invited him to join the Academy of Sciences in St. He graduated from the University of Basel in 1724. The son of a clergyman, Leonhard Euler, was born in Basel on April 15, 1707. The Swiss mathematician Leonhard Euler (1707-1783) made important original contributions to every branch of mathematics studied in his day.
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