Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/67

If the chord of a circle bears to the diameter a ratio which is algebraic, then the corresponding arc is not rectifiable by any construction in which algebraic curves alone are employed; neither can the quadrature of the corresponding sector of the circle be carried out by such a construction.

The method employed by Hermite and Lindemann was of a complicated character, involving the use of complex integration. The method was very considerably simplified by Weierstrass^[1], who gave a complete proof of Lindemann's general theorem.

Proofs of the transcendence of $e$ and $\pi$, progressively simple in character, were given by Stieltjes^[2], Hilbert, Hurwitz and Gordan^[3], Mertens^[4], and Vahlen^[5].

All these proofs consist of a demonstration that an equation which is linear in a number of exponential functions, such that the coefficients are whole numbers, and the exponents algebraic numbers, is impossible. By choosing a multiplier of the equation of such a character that its employment reduces the given equation to the equation of the sum of a non-vanishing integer and a number proved to lie numerically between 0 and 1 to zero, the impossibility is established.

Simplified presentations of the proofs will be found in Weber's Algebra, in Enriques' Questions of Elementary Geometry (German Edition, 1907), in Hobson's Plane Trigonometry (second edition, 1911), and in Art. IX. of the "Monographs on Modern Mathematics," edited by J. W. A. Young.

Proof of the transcendence of π

The proof of the transcendence of $\pi$ which will here be given is founded upon that of Gordan.

(1) Let us assume that, if possible, $\pi$ is a root of an algebraical equation with integral coefficients; then $i\pi$ is also a root of such an equation.

Assume that $i\pi$ is a root of the equation

$C(x-\alpha _{1})(x-\alpha _{2})\ldots (x-\alpha _{s})=0$,

where all the coefficients

$\textstyle C,C\sum a,C\sum \alpha _{r}\alpha _{s},\ldots ,C\alpha _{1}\alpha _{2}\ldots \alpha _{s}$

1. Ber. Akad. Berlin, 1885.
2. Comptes Rendus, Paris Acad. 1890
3. These proofs are to be found in the Math. Annalen, vol. 43 (1896), by Hilbert, Hurwitz and Gordan.
4. Wiener Ber. Kl. cv. IIa (1896).
5. Math. Annalen, vol. 53 (1900).