Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/62

Euclidean construction, for example by bisecting the segment of the line on which two of the given points are incident, and then determining a perpendicular to that segment. We may therefore assume that a given set of points, the data of a Euclidean problem, are specified by means of a set of numbers, the coordinates of these points.

The determination of a required point $P$ is, in a Euclidean problem, made by means of a finite number of applications of the three processes, (1) of determining a new point as the intersection of straight lines given each by a pair of points already determined, (2) of determining a new point as an intersection of a straight line given by two points and a circle given by its centre and one point on the circumference, all four points having been already determined, and (3) of determining a new point as an intersection of two circles which are determined by four points already determined.

In the analytical interpretation we have an original set of numbers $a_{1},a_{2},\ldots a_{2r}$ given, the coordinates of the $r$ given points; ($r\geq 2$). At each successive stage of the geometrical process we determine two new numbers, the coordinates of a fresh point.

When a certain stage of the process has been completed, the data for the next step consist of numbers $(a_{1},a_{2},\ldots a_{2n})$ containing the original data and those numbers which have been already ascertained by the successive stages of the process already carried out.

If (1) is employed for the next step of the geometrical process, the new point determined by that step corresponds to numbers determined by two equations

$Ax+By+C=0$, $A'x+B'y+C'=0$,

where $A$, $B$, $C$, $A'$, $B'$, $C'$ are rational functions of eight of the numbers $(a_{1},a_{2},\ldots a_{2n})$. Therefore $x$, $y$ the coordinates of the new point determined by this step are rational functions of $a_{1},a_{2},\ldots a_{2n}$.

In order to get the data for the next step afterwards, we have only to add to $a_{1},a_{2},\ldots a_{2n}$ these two rational functions of eight of them.

If case (2) is employed, the next point is determined by two equations of the form

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

$y=mx+n$,

where $m$, $n$ are rational functions of four of the numbers $a_{1},a_{2},\ldots a_{2n}$. On elimination of $y$, we have a quadratic equation for $x$; and thus $x$ is determined as a quadratic irrational function of $(a_{1},a_{2},\ldots a_{2n})$, of