In this address, Professor Cayley devoted most of his time to non-Euclidean systems. Non-Euclidean _spaces_, he declared, seemed to him mistaken _à priori_[59]; but non-Euclidean _Geometries_, here as in his mathematical works, were accepted as flowing from a change in the definition of distance. This view has been already discussed, and need not, therefore, be further criticised here. What I wish to speak about, is the question with which Cayley himself opened his address, namely, the geometrical use and meaning of imaginary quantities. From the manner in which he spoke of this question, it becomes imperative to treat it somewhat at length. For he said (pp. 8-9): "... The notion which is the really fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole notion of modern analysis and Geometry, [is] that of imaginary magnitude in analysis, and of imaginary space (or space as the _locus in quo_ of imaginary points and figures) in Geometry: I use in each case the word imaginary as including real.... Say even the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it." =42.= 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.