Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/63

the form $A\pm {\sqrt {B}}$, where $A$ and $B$ are rational functions; it is clear that $y$ will be determined in a similar way.

If, in the new step, (3) is employed, the equations for determining $(x,y)$ consist of two equations of the form

$(x-a_{p})^{2}+(y-a_{q})^{2}=(a_{r}-a_{s})^{2}+(a_{t}-a_{u})^{2}$;

on subtracting these equations, we obtain a linear equation, and thus it is clear that this case is essentially similar to that in which (2) is employed, so far as the form of $x$, $y$ is concerned.

Since the determination of a required point $P$ is to be made by a finite number of such steps, we see that the coordinates of $P$ are determined by means of a finite succession of operations on

$(a_{1},a_{2},\ldots a_{2r})$,

the coordinates of the points; each of these operations consists either of a rational operation, or of one involving the process of taking a square root of a rational function as well as a rational operation.

We have now established the following result:

In order that a point $P$ can be determined by the Euclidean mode it is necessary and sufficient that its coordinates can be expressed as such functions of the coordinates $(a_{1},a_{2},\ldots a_{2r})$ of the given points of the problem, as involve the successive performance, a finite number of times, of operations which are either rational or involve taking a square root of a rational function of the elements already determined.

That the condition stated in this theorem is necessary has been proved above; that it is sufficient is seen from the fact that a single rational operation, and the single operation of taking a square root of a number already known, are both operations which correspond to possible Euclidean determinations.

The condition stated in the result just obtained may be put in another form more immediately available for application. The expression for a coordinate x of the point P may, by the ordinary processes for the simplification of surd expressions, by getting rid of surds from the denominators of fractions, be reduced to the form

$x=a+b{\sqrt {c_{1}\pm {\sqrt {c_{2}\pm {\sqrt {c_{3}+\ldots }}}}}}+b'{\sqrt {c_{1}'\pm {\sqrt {c_{2}'\pm {\sqrt {c_{3}'+\ldots }}}}}}+\ldots$,

where all the numbers

$a,b,c_{1},c_{2},\ldots ,b',c_{1}',c_{2}',\ldots$

are rational functions of the given numbers $(a_{1},a_{2},\ldots a_{2r})$ and the number of successive square roots is in every term finite. Let $m$ be