Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/58

It has frequently been stated that the first rigorous proof of Lambert's results is due to Legendre (1752—1833), who proved these theorems in his Éléments de Géométrie (1794), by the same method, and added a proof that $\pi ^{2}$ is an irrational number. The essential rigour of Lambert's proof has however been pointed out by Pringsheim (Münch. Akad. Ber., Kl. 28, 1898), who has supplemented the investigation in respect of the convergence.

A proof of the irrationality of $\pi$ and $\pi ^{2}$ due to Hermite (Crelle's Journal, vol. 76, 1873) is of interest, both in relation to the proof of Lambert, and as containing the germ of the later proof of the transcendency of $e$ and $\pi$.

A simple proof of the irrationality of $e$ was given by Fourier (Stainville, Mélanges d' analyse, 1815), by means of the series

$1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\ldots$

which represents the number. This proof can be extended to shew that $e^{2}$ is also irrational. On the same lines it was proved by Liouville (1809—1882) (Liouville's Journal, vol. 5, 1840) that neither $e^{2}$ nor $e^{2}$ can be a root of a quadratic equation with rational coefficients. This last theorem is of importance as forming the first step in the proof that $e$ and $\pi$ cannot be roots of any algebraic equation with rational coefficients. The probability had been already recognized by Legendre that there exist numbers which have this property.

Existence of transcendental numbers

The confirmation of this surmised existence of such numbers was obtained by Liouville in 1840, who by an investigation of the properties of the convergents of a continued fraction which represents a root of an algebraical equation, and also by another method, proved that numbers can be defined which cannot be the root of any algebraical equation with rational coefficients.

The simpler of Liouville's methods of proving the existence of such numbers will be here given.

Let $x$ be a real root of the algebraic equation

$ax^{n}+bx^{n-1}+cx^{n-2}+\ldots =0$,

with coefficients which are all positive or negative integers. We shall assume that this equation has all its roots unequal; if it had equal roots we might suppose it to be cleared of them in the usual manner.