The Geometry of the Complex Domain

As an example of (a) we may ask how to find a geometrical representation of the complex points of a line, a circle, or a plane. Question (b) leads to mathematical considerations of a very different order. We usually assume that whatever is true in the real domain is true in the complex one also the properties of the complex portion of a curve are inferred from those of its real trace. If we are asked for our grounds for this erroneous belief, we are inclined to reply Continuity' or 'analytic continuation' or what not. But these vague generalities do not by any means exhaust the question. There are more things in Heaven and Earth than are dreamt of in our philosophy of reals. What, for instance, can be said about the totality of points in the plane such that the sum of the squares of the absolute values of their distances from two mutually perpendicular lines is equal to unity? This is a very numerous family of points indeed, depending on no less than three real parameters, so that it is not contained completely in any one curve, nor is any one curve contained completely therein; it is an absolutely different variety from any curve or system of curves in the plane.