=43.= The fact that the fiction _is_ convenient, however, may be thought to indicate that it is more than a fiction. But this presumption, I think, can be easily explained away. For all the fruitful uses of imaginaries, in Geometry, are those which begin and end with real quantities, and use imaginaries only for the intermediate steps. Now in all such cases, we have a real spatial interpretation at the beginning and end of our argument, where alone the spatial interpretation is important: in the intermediate links, we are dealing in a purely algebraical manner with purely algebraical quantities, and may perform any operations which are algebraically permissible. If the quantities with which we end are capable of spatial interpretation, then, and only then, our result may be regarded as geometrical. To use geometrical language, in any other case, is only a convenient help to the imagination. To speak, for example, of projective properties which refer to the circular points, is a mere _memoria technica_ for purely algebraical properties; the circular points are not to be found in space, but only in the auxiliary quantities by which geometrical equations are transformed. That no contradictions arise from the geometrical interpretation of imaginaries, is not wonderful: for they are interpreted solely by the rules of Algebra, which we may admit as valid in their application to imaginaries. The perception of space being wholly absent, Algebra rules supreme, and no inconsistency can arise. Wherever, for a moment, we allow our ordinary spatial notions to intrude, the grossest absurdities do arise--every one can see that a circle, being a closed curve, cannot get to infinity. The metaphysician, who should invent anything so preposterous as the circular points, would be hooted from the field. But the mathematician may steal the horse with impunity. Finally, then, only a knowledge of space, not a knowledge of Algebra, can assure us that any given set of quantities will have a spatial correlate, and in the absence of such a correlate, operations with these quantities have no geometrical import. This is the case with imaginaries in Cayley's sense, and their use in Geometry, great as are its technical advantages, and rigid as is its technical validity, is wholly destitute of philosophical importance.