Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/56

he set up the relations

$\cos {x}={\frac {e^{ix}+e^{-ix}}{2}}$, $\sin {x}={\frac {e^{ix}-e^{-ix}}{2}}$,

which can also be written

$e^{ix}=\cos {x}+i\sin {x}$, $e^{-ix}=\cos {x}-i\sin {x}$.

The relation $e^{i\pi }=-1$, which Euler obtained by putting $x=\pi$, is the fundamental relation between the two numbers $\pi$ and $e$ which was indispensable later on in making out the true nature of the number $\pi$.

In his very numerous memoirs, and especially in his great work, Introductio in analysin infinitorum (1748), Euler displayed the most wonderful skill in obtaining a rich harvest of results of great interest, largely dependent on his theory of the exponential function. Hardly any other work in the history of Mathematical Science gives to the reader so strong an impression of the genius of the author as the Introductio. Many of the results given in that work are obtained by bold generalizations, in default of proofs which would now be regarded as completely rigorous; but this it has in common with a large part of all Mathematical discoveries, which are often due to a species of divining intuition, the rigorous demonstrations and the necessary restrictions coming later. In particular there may be mentioned the expressions for the sine and cosine functions as infinite products, and a great number of series and products deduced from these expressions; also a number of expressions relating the number $e$ with continued fractions which were afterwards used in connection with the investigation of the nature of that number.

Great as the progress thus made was, regarded as preparatory to a solution of our problem, nothing definite as to the true nature of the number $\pi$ was as yet established, although Mathematicians were convinced that $e$ and $\pi$ are not roots of algebraic equations. Euler himself gave expression to the conviction that this is the case. Somewhat later, Legendre gave even more distinct expression to this view in his Éléments de Géométrie (1794), where he writes: " It is probable that the number $\pi$ is not even contained among the algebraical irrationalities, i.e. that it cannot be a root of an algebraical equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly."