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 your imagination, I will place before you material and visible figures about which my reasoning will be approximately true. From these I must ask you to try to rise upward to the imagination of their archetypes, the immaterial realities.” What shall we reply to our overbearing mathematician who in this abrupt and audacious manner introduces the non-existent and imaginary creatures of his brain as being “realities”? Shall we deride him, and the arithmetician likewise? Shall we bid the latter exchange his calculations in abstract numbers for manifestly useful sums about sacks of wheat and casks of beer? Shall we bid the mathematician descend from his high geometrical theories to the practical measurements of agriculture? Pouring scorn on his avowal that the objects of his reasoning are “invisible, ideal, and imaginary,” shall we decline to study a science that is confessedly—so we can word it—visionary and illusive? If we do, he will not be without a reply, somewhat after this fashion: “My practical friends, it will be the worse for you if you despise these invisible, ideal and imaginary objects. I say nothing about the mental training and development to be derived from the study of these things; for to this argument you do not appear to me to be at present accessible: but I will take your own line—the practical. Do you then want to measure your fields with ease and to make accurate maps and charts; to construct houses that shall stand longer, ships that shall sail faster, cannon that shall shoot further, engines that shall pull harder, than any known before; do you want to utilize electricity for lighting, gas for motion, water for pressure; in a word do you wish to make yourselves lords over the material world and to have all the forces of Nature at your beck and call? If you do, you must not despise the