Note 42 — Introductory material

This right it is now my purpose to demonstrate. But for fear non-mathematicians should miss the point of Cayley's remark (which has sometimes been erroneously supposed to refer to non-Euclidean spaces), I may as well explain, at the outset, that this question is radically distinct from, and only indirectly connected with, the validity or import of Metageometry. An imaginary quantity is one which involves √–1 : its most general form is a + √–1 b where a and b are real; Cayley uses the word imaginary so as to include real, in order to cover the special case where b = 0. It will be convenient, in what follows, to exclude this wider meaning, and assume that b is not zero. An imaginary point is one whose coordinates involve √–1, i.e. whose coordinates are imaginary quantities. An imaginary curve is one whose points are imaginary—or, in some special uses, one whose equation contains imaginary coefficients. The mathematical subtleties to which this notion leads need not be here discussed; the reader who is interested in them will find an excellent elementary account of their geometrical uses in Klein's Nicht-Euklid, II. pp. 38–46. But for our present purpose, we may confine ourselves to imaginary points. If these are found to have a merely technical import, and to be destitute of any philosophical meaning, then the same will hold of any collection of imaginary points, i.e. of any imaginary curve or surface.