That the notion of imaginary points is of supreme importance in Geometry, will be seen by any one who reflects that the circular points are imaginary, and that the reduction of metrical to projective Geometry, which is one of Cayley's greatest achievements, depends on these points. But to discuss adequately their philosophical import is difficult to me, since I am unacquainted with any satisfactory philosophy of imaginaries in pure Algebra. I will therefore adopt the most favourable hypothesis, and assume that no objection can be successfully urged against this use. Even on this hypothesis, I think, no case can be made out for imaginary points in Geometry. In the first place, we must exclude, from the imaginary points considered, those whose coordinates are only imaginary with certain special systems of coordinates. For example, if one of a point's coordinates be the tangent from it to a sphere, this coordinate will be imaginary for any point inside the sphere, and yet the point is perfectly real. A point, then, is only to be called imaginary, when, whatever real system of coordinates we adopt, one or more of the quantities expressing these coordinates remains imaginary. For this purpose, it is mathematically sufficient to suppose our coordinates Cartesian--a point whose Cartesian coordinates are imaginary, is a true imaginary point in the above sense. To discuss the meaning of such a point, it is necessary to consider briefly the fundamental nature of the correspondence between a point and its coordinates. Assuming that elementary Geometry has proved--what I think it does satisfactorily prove--that spatial relations are susceptible of quantitative measurement, then a given point will have, with a suitable system of coordinates, in a space of _n_ dimensions, _n_ quantitative relations to the fixed spatial figure forming the axes of coordinates, and these _n_ quantitative relations will, under certain reservations, be unique--_i.e._, no other point will have the same quantities assigned to it. (With many possible coordinate systems, this latter condition is not realized: but for that very reason they are inconvenient, and employed only in special problems.) Thus given a coordinate system, and given any set of quantities, these quantities, _if they determine a point at all_, determine it uniquely. But, by a natural extension of the method, the above reservation is dropped, and it is assumed that to _every_ set of quantities some point must correspond. For this assumption there seems to me no vestige of evidence. As well might a postman assume that, because every house in a street is uniquely determined by its number, therefore there must be a house for every imaginable number. We must know, in fact, that a given set of quantities can be the coordinates of some point in space, before it is legitimate to give any spatial significance to these quantities: and this knowledge, obviously, cannot be derived from operations with coordinates alone,