Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/65

The criterion thus obtained is sufficient, whenever it can be applied, to determine whether a proposed Euclidean problem is a possible one or not.

In the case of the rectification of the circle, we may assume that the data of the problem consist simply of the two points $(0,0)$ and $(1,0)$, and that the point to be determined has the coordinates $(\pi ,0)$. This will, in accordance with the criterion obtained, be a possible problem only if $\pi$ is a root of an algebraic equation with rational coefficients, of that special class which has roots expressible by means of rational numbers and numbers obtainable by successive operations of taking the square roots. The investigations of Abel have shewn that this is only a special class of algebraic equations.

As we shall see, it is now known that $\pi$, being transcendental, is not a root of any algebraic equation at all, and therefore in accordance with the criterion is not determinable by Euclidean construction. The problems of duplication of the cube, and of the trisection of an angle, although they lead to algebraic equations, are not soluble by Euclidean constructions, because the equations to which they lead are not in general of the class referred to in the above criterion.

The transcendence of π

In 1873 Ch. Hermite^[1] succeeded in proving that the number $e$ is transcendental, that is that no equation of the form

$ae^{m}+be^{n}+ce^{r}+\ldots =0$

can subsist, where $m,n,r,\ldots a,b,c,\ldots$ are whole numbers. In 1882, the more general theorem was stated by Lindemann that such an equation cannot hold, when $m,n,r,\ldots a,b,c,\ldots$ are algebraic numbers, not necessarily real; and the particular case that $e^{ix}+1=0$ cannot be satisfied by an algebraic number $x$, and therefore that $\pi$ is not algebraic, was completely proved by Lindemann^[2].

Lindemann's general theorem may be stated in the following precise form:

If $x_{1},x_{2},\ldots x_{n}$ are any real or complex algebraical numbers, all distinct, and $p_{1},p_{2},\ldots p_{n}$ are $n$ algebraical numbers at least one of which is different from zero, then the sum

$p_{1}e^{x_{1}}+p_{2}e^{x_{2}}+\ldots +p_{n}e^{x_{n}}$

is certainly different from zero.

1. "Sur la fonction exponentielle," Comptes Rendus, vol. 77, 1873.
2. Ber. Akad. Berlin, 1882.