Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/57

CHAPTER IV

The Third Period

The irrationality of π and e

The third and final period in the history of the problem is concerned with the investigation of the real nature of the number $\pi$. Owing to the close connection of this number with the number $e$, the base of natural logarithms, the investigation of the nature of the two numbers was to a large extent carried out at the same time.

The first investigation, of fundamental importance, was that of J. H. Lambert (1728—1777), who in his "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques" (Hist. de l'Acad. de Berlin, 1761, printed in 1768), proved that $e$ and $\pi$ are irrational numbers. His investigations are given also in his treatise Vorläufige Kenntnisse für die, so die Quadratur und Rektification des Zirkels suchen, published in 1766.

He obtained the two continued fractions

${\frac {e^{x}-1}{e^{x}+1}}={\frac {1}{2/x+}}{\frac {1}{6/x+}}{\frac {1}{10/x+}}{\frac {1}{14/x+\ldots }}$,

$\tan {x}={\frac {1}{1/x-}}{\frac {1}{3/x-}}{\frac {1}{5/x-}}{\frac {1}{7/x-\ldots }}$,

which are closely related with continued fractions obtained by Euler, but the convergence of which Euler had not established. As the result of an investigation of the properties of these continued fractions, Lambert established the following theorems:

(1) If $x$ is a rational number, different from zero, $e^{x}$ cannot be a rational number.

(2) If $x$ is a rational number, different from zero, $\tan {x}$ cannot be a rational number.

If $\textstyle x={\frac {1}{4}}\pi$, we have $\tan {x}=1$, and therefore $\textstyle {\frac {1}{4}}\pi$, or $\pi$, cannot be a rational number.