the same book demands the possibility of describing a
circle for every "centre" and "distance." Having employed the circle for
the construction and demonstration of several propositions in Books I.
and II. Euclid devotes his third book entirely to theorems and problems
relating to the circle, and certain lines and angles, which he defines
in introducing the propositions. The fourth book deals with the circle
in its relations to inscribed and circumscribed triangles,
quadrilaterals and regular polygons. Reference should be made to the
article GEOMETRY: _Euclidean_, for a detailed summary of the Euclidean
treatment, and the elementary properties of the circle.

_Analytical Geometry of the Circle._

Cartesian co-ordinates.

In the article GEOMETRY: _Analytical_, it is shown that the general
equation to a circle in rectangular Cartesian co-ordinates is
x^2+y^2+2gx+2fy+c=0, i.e. in the general equation of the second degree
the co-efficients of x^2 and y^2 are equal, and of xy zero. The
co-ordinates of its centre are -g/c, -f/c; and its radius is
(g^2+f^2-c)^1/2. The equations to the chord, tangent and normal are
readily derived by the ordinary methods.

Consider the two circles:--

x^2+y^2+2gx+2fy+c=0, x^2+y^2+2g'x+2f'y+c'=0.

Obviously these equations show that the curves intersect in four
points, two of which lie on the intersection of the line, 2(g - g')x +
2(f - f')y + c - c' = 0, the radical axis, with the circles, and the
other two where the lines x squared + y squared = (x + iy) (x - iy) = 0 (where i =
sqrt -1) intersect the circles. The first pair of intersections may be
either real or imaginary; we proceed to discuss the second pair.

The equation x squared + y squared = 0 denotes a pair of perpendicular imaginary
lines; it follows, therefore, that circles always intersect in two
imaginary points at infinity along these lines, and since the terms
x squared + y squared occur in the equation of every circle, it is seen that all
circles pass through two fixed points at infinity. The introduction of
these lines and points constitutes a striking achievement in geometry,
and from their association with circles they have been named the
"circular lines" and "circular points." Other names for the circular
lines are "circulars" or "isotropic lines." Since the equation to a
circle of zero radius is x squared + y squared = 0, i.e. identical with the circular
lines, it follows tha