Other work done by Euler on infinite series included the introduction of his famous Euler's constant, in 1735, which he showed to be the limit of
1/1 + 1/2 + 1/3 + ... + 1/n - logen
as n tends to infinity. He calculated the constant to 16 decimal places. Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result
/2 - x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ...
in a letter to Goldbach. Like most of Euler's work there was a fair time delay before the results were published; this result was not published until 1755.
Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series. In particular he wrote [60]:-
Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.
He then goes on to describe what is now called the Euler- Maclaurin summation formula. Two years later Stirling replied telling Euler that Maclaurin:-
... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.
Euler replied:-
... I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.
Some of Euler's number theory results have been mentioned above. Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3. Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form a + b-3 for integers a and b. Although there were problems with his approach this eventually led to Kummer's major work on Fermats Last Theorem and to the introduction of the concept of a ring.
One could claim that mathematical analysis began with Euler. In 1748 in Introductio in analysin infinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he stated that mathematical analysis was the study of functions. This work bases the calculus on the theory elementary functions rather than on geometric curves, as had been done previously. Also in this work Euler gave the formula
eix= cos x + i sin x.
In Introductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positive values although he had discovered the formula
ln(-1) = i
in 1727. He published his full theory of logarithms of complex numbers in 1751.
Analytic functions of a complex variable were investigated by Euler in a number of different contexts, including the study of orthogonal trajectories and cartography. He discovered the Cauchy- Riemann equations in 1777, although d'Alembert had discovered them in 1752 while investigating hydrodynamics.
In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculus of finite differences. The work makes a thorough investigation of how differentiation behaves under substitutions.
In Institutiones calculi integralis (1768-70) Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions. He also studied beta and gamma functions, which he had introduced first in 1729. Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function by Binet and Gauss respectively. As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work.
The calculus of variations is another area in which Euler made fundamental discoveries. His work Methodus inveniendi lineas curvas ... published in 1740 began the proper study of the calculus of variations.
It is noted that Carathéodory considered this as:-
... one of the most beautiful mathematical works ever written.
Problems in mathematical physics had led Euler to a wide study of differential equations. He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions.
Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces. Many unpublished results by Euler in this area were rediscovered by Gauss. Other geometric investigations led him to fundamental ideas in topology such as the Euler characteristic of a polyhedron.
In 1736 Euler published Mechanica which provided a major advance in mechanics. As Yushkevich writes:-
The distinguishing feature of Euler's investigations in mechanics as compared to those of his predecessors is the systematic and successful application of analysis. Previously the methods of mechanics had been mostly synthetic and geometrical; they demanded too individual an approach to separate problems. Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.
In Mechanica Euler considered the motion of a point mass both in a vacuum and in a resisting medium. He analysed the motion of a point mass under a central force and also considered the motion of a point mass on a surface. In this latter topic he had to solve various problems of differential geometry and geodesics.
Mechanica was followed by another important work in rational mechanics, this time Euler's two volume work on naval science. It is described as:-
Outstanding in both theoretical and applied mechanics, it addresses Euler's intense occupation with the problem of ship propulsion. It applies variational principles to determine the optimal ship design and first establish the principles of hydrostatics ... Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.
Of course hydrostatics had been studied since Archimedes, but Euler gave a definitive version.
In 1765 Euler published another major work on mechanics Theoria motus corporum solidorum in which he decomposed the motion of a solid into a rectilinear motion and a rotational motion. He considered the Euler angles and studied rotational problems which were motivated by the problem of the precession of the equinoxes.
Euler's work on fluid mechanics is also quite remarkable. He published a number of major pieces of work through the 1750s setting up the main formulas for the topic, the continuity equation, the Laplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid. In 1752 he wrote:-
However sublime are the researches on fluids which we owe to Messrs Bernoulli, Clairaut and d'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ...
Euler contributed to knowledge in many other areas, and in all of them he employed his mathematical knowledge and skill. He did important work in astronomy including:-
... determination of the orbits of comets and planets by a few observations, methods of calculation of the parallax of the sun, the theory of refraction, consideration of the physical nature of comets, .... His most outstanding works, for which he won many prizes from the Paris Académie des Sciences, are concerned with celestial mechanics, which especially attracted scientists at that time.
In fact Euler's lunar theory was used by Tobias Mayer in constructing his tables of the moon. In 1765 Tobias Mayer's widow received 3000 from Britain for the contribution the tables made to the problem of the determination of the longitude, while Euler received 300 from the British government for his theoretical contribution to the work.
Euler also published on the theory of music, in particular he published Tentamen novae theoriae musicae in 1739 in which he tried to make music:-
... part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing.
However, according to the work was:-
... for musicians too advanced in its mathematics and for mathematicians too musical.
Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy's geography section in 1735. He had the specific task of helping Delisle prepare a map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration and it appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudly remarked that this work put the Russians well ahead of the Germans in the art of cartography.
J J O'Connor and E F Robertson