Page:A History of Mathematics (1893).djvu/269

 opposite sides by $\scriptstyle{a}$, $\scriptstyle{b}$, $\scriptstyle{c}$, respectively. He pointed out the relation between trigonometric and exponential functions. In a paper of 1737 we first meet the symbol $$\scriptstyle{\pi}$$ to denote 3.14159&hellip;.[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 bi-quadratic equations by assuming $\scriptstyle{x=\sqrt{p}+\sqrt{q}+\sqrt{r}}$,|undefined 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 Bezout) 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 $\scriptstyle{(-a)^2=(+a)^2}$, we have $$\scriptstyle{\log(-a)^2=\log(+a)^2}$$ and $\scriptstyle{2\log(-a)=2\log(+a)}$, and finally $\scriptstyle{\log(-a)=\log(+a)}$. Euler proved that $$\scriptstyle{a}$$ has really an infinite number of logarithms, all of which are imaginary when $$\scriptstyle{a}$$ is negative, and all except one when $$\scriptstyle{a}$$ is positive. He then explained how $$\scriptstyle{\log(-a)^2}$$ might equal $\scriptstyle{\log(+a)^2}$, and yet $$\scriptstyle{\log(-a)}$$ not equal $\scriptstyle{\log(+a)}$.

The subject of infinite series received new life from him. To his researches on series we owe the creation of the theory of definite integrals by the development of the so-called Eulerian integrals. He warns his readers occasionally against the use of divergent series, but is nevertheless very careless himself. The rigid treatment to which infinite series are subjected now was then undreamed of. No clear notions existed as to what constitutes a convergent series. Neither Leibniz nor Jacob