Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/55

The work of Euler

Developments of the most far-reaching importance in connection with our subject were made by Leonhard Euler, one of the greatest Analysts of all time, who was born at Basel in 1707 and died at St Petersburg in 1783. With his vast influence on the development of Mathematical Analysis in general it is impossible here to deal, but some account must be given of those of his discoveries which come into relation with our problem.

The very form of modern Trigonometry is due to Euler. He introduced the practice of denoting each of the sides and angles of a triangle by a single letter, and he introduced the short designation of the trigonometrical ratios by $\sin \alpha$, $\cos \alpha$, $\tan \alpha$, &c. Before Euler's time there was great prolixity in the statement of propositions, owing to the custom of denoting these expressions by words, or by letters specially introduced in the statement. The habit of denoting the ratio of the circumference to the diameter of a circle by the letter $\pi$, and the base of the natural system of logarithms by $e$, is due to the influence of the works of Euler, although the notation $\pi$ appears as early as 1706, when it was used by William Jones in the Synopsis palmariorum Matheseos. In Euler's earlier work he frequently used $p$ instead of $\pi$, but by about 1740 the letter $\pi$ was used not only by Euler but by other Mathematicians with whom he was in correspondence.

A most important improvement which had a great effect not only upon the form of Trigonometry but also on Analysis in general was the introduction by Euler of the definition of the trigonometrical ratios in order to replace the old sine, cosine, tangent, &c., which were the lengths of straight lines connected with the circular arc. Thus these trigonometrical ratios became functions of an angular magnitude, and therefore numbers, instead of lengths of lines related by equations with the radius of the circle. This very important improvement was not generally introduced into our text books until the latter half of the nineteenth century.

This mode of regarding the trigonometrical ratios as analytic functions led Euler to one of his greatest discoveries, the connection of these functions with the exponential function. On the basis of the definition of $e$ by means of the series

$1+z+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\ldots$,