Euler, Lagrange, and Laplace

proprietate gaudentes, 1744, which, displaying an amount of mathematical genius seldom rivalled, contained his researches on the calculus of variations (a subject afterwards improved by Lagrange), to the invention of which Euler was led by the study of isoperimetrical curves, the brachistochrone in a resisting medium, and the theory of geodesics (subjects which had previously engaged the attention

of the elder Bernoullis and others); the Theoria motuum planetarum et cometarum, 1744, Theoria motus lunæ, 1753, Theoria motuum lunæ, 1772, are his chief works on astronomy; Ses lettres à une princesse d'Allemagne sur quelques sujets de Physique et de Philosophie, 1770, was a work which enjoyed great popularity.

We proceed to mention the principal innovations and inventions of Euler. He treated trigonometry as a branch of

analysis, introduced (simultaneously with Thomas Simpson in

England) the now current abbreviations for trigonometric functions, and simplified formulæ by the simple expedient of designating the angles of a triangle by $A$, $B$, $C$, and the

opposite sides by $a$, $b$, $c$, respectively. He pointed out the relation between trigonometric and exponential functions. In a paper of 1737 we first meet the symbol $\pi$ to denote $3.14159\dots$.21

Euler laid down the rules for the transformation of co-ordinates

in space, gave a methodic analytic treatment of plane curves and of surfaces of the second order. He was the first to

discuss the equation of the second degree in three variables, and to classify the surfaces represented by it. By criteria analogous to those used in the classification of conics he obtained five species. He devised a method of solving biquadratic equations by assuming $x=\sqrt{p}+\sqrt{q}+\sqrt{r}$, with the

hope that it would lead him to a general solution of algebraic equations. The method of elimination by solving a series of

linear equations (invented independently by Bézout) and the method of elimination by symmetric functions, are due to him.20

Far reaching are Euler's researches on logarithms. Leibniz

and John Bernoulli once argued the question whether a

negative number has a logarithm. Bernoulli claimed that since $(-a{)}^2=(+a{)}^2$, we have $\log (-a{)}^2=\log (+a{)}^2$ and $2\log (-a)=2\log (+a)$, and finally $\log (-a)=\log (+a)$. Euler proved that $a$ has really an infinite number of logarithms, all of which are imaginary when $a$ is negative, and all except one when $a$ is positive. He then explained how $\log (-a{)}^2$ might equal $\log (+a{)}^2$, and yet $\log (-a)$ not equal $\log (+a)$.

The subject of infinite series received new life from him.