Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/66

The particular case of this theorem in which

$n=2$, $x_{1}=ix$, $x_{2}=0$, $p_{1}=p_{2}=1$,

shews that $e^{ix}+1$ cannot be zero if a is an algebraic number, and thus that, since $e^{i\pi }+1=0$, it follows that the number $\pi$ is transcendental.

From the general theorem there follow also the following important results:

(1) Let $n=2$, $p_{1}=1$, $p_{2}=-a$, $x_{1}=x$, $x_{2}=0$; then the equation $e^{x}-a$ cannot hold if $x$ and $a$ are both algebraic numbers and $x$ is different from zero. Hence the exponential $e^{x}$ is transcendent if $x$ is an algebraic number different from zero. In particular $e$ is transcendent. Further, the natural logarithm of an algebraic number different from zero is a transcendental number. The transcendence of $i\pi$ and therefore of $\pi$ is a particular case of this theorem.

(2) Let $n=3$, $p_{1}=-i$, $p_{2}=i$, $p_{3}=-2a$, $x_{1}=ix$, $x_{2}=-ix$, $x_{3}=0$; it then follows that the equation $\sin {x}=a$ cannot be satisfied if $a$ and $x$ are both algebraic numbers different from zero. Hence, if $\sin {x}$ is algebraic, $x$ cannot be algebraic, unless $x=0$, and if $a$ is algebraic, $\sin ^{-1}{x}$ cannot be algebraic, unless a = 0.

It is easily seen that a similar theorem holds for the cosine and the other trigonometrical functions.

The fact that $\pi$ is a transcendental number, combined with what has been established above as regards the possibility of Euclidean constructions or determinations with given data, affords the final answer to the question whether the quadrature or the rectification of the circle can be carried out in the Euclidean manner.

The quadrature and the rectification of a circle whose diameter is given are impossible, as problems to be solved by the processes of Euclidean Geometry, in which straight lines and circles are alone employed in the constructions.

It appears, however, that the transcendence of $\pi$ establishes the fact that the quadrature or the rectification of a circle whose diameter is given are impossible by a construction in which the use only of algebraic curves is allowed.

The special case (2) of Lindemann's theorem throws light on the interesting problems of the rectification of arcs of circles and of the quadrature of sectors of circles. If we take the radius of a circle to be unity then $\textstyle 2\sin {{\frac {1}{2}}x}$ is the length of the chord of an arc of which the length is $x$. It has been shewn that $\textstyle 2\sin {{\frac {1}{2}}x}$ and $x$ cannot both be algebraic, unless $x=0$. We have therefore the following result: