"one" and "two," I think I see before me dimly "one" or "two" dots or small strokes, and I perceive that two and one of these dots or strokes make up three dots or strokes. When I speak of "twenty" and "thirty," I do not see any images of these existences; and when I say that "twenty" and "thirty" make "fifty," I do not realize the process of addition at all visibly; I merely repeat the statement on the authority of previous observations and reasonings mostly made by others and not by myself. But so far as I approximate to the realization of an abstract number, I do it by a kind of negative imagination. And in any case we can hardly deny that all arithmetical propositions, since they employ terms that denote mere imaginary ideas, must be regarded as based on the imagination.
It is the same with Geometry. The whole of what we call "Euclid" is based upon a most aerial effort of the Imagination. We have to imagine lines without thickness, straightness that does not deviate the billionth part of an inch from perfect evenness, perfectly symmetrical circles, and—climax of audacity!—points that have "no parts and no magnitude!" Obviously these things have no existence except in the dreams of Imagination; yet Euclid's severe reasoning applies to none but these things. If you step from your ideal triangle in Dreamland into your material triangle in chalk-land, you step from absolute truth into statements that are not absolutely true. The angles at the base of your chalk isosceles triangle are not exactly equal, if you measure them with sufficient accuracy. In a word the whole of Geometry is an appeal to the Imagination in which the geometer says to us, "I know that my propositions are not exactly true except with respect to invisible, ideal, and imaginary figures, planes, and solids. These ideas, therefore, you must endeavour to imagine. In order to relieve the strain on
