one of the cases there is, as it were, a chain, and the particular link rises as if an unseen hand had lifted it.  The Forms are sometimes variously coloured, occasionally very brilliantly (see Plate IV.).  In all of these the definition and illumination vary much in different parts.  Usually the Forms fade away into indistinctness after 100; sometimes they come to a dead stop.  The higher numbers very rarely fill so large a space in the Forms as the lower ones, and the diminution of space occupied by them is so increasingly rapid that I thought it not impossible they might diminish according to some geometrical law, such as that which governs sensitivity.  I took many careful measurements and averaged them, but the result did not justify the supposition.

It is beyond dispute that these forms originate at an early age; they are subsequently often developed in boyhood and youth so as to include the higher numbers, and, among mathematical students, the negative values.

Nearly all of my correspondents speak with confidence of their Forms having been in existence as far back as they recollect.  One states that he knows he possessed it at the age of four; another, that he learnt his multiplication table by the aid of the elaborate mental diagram he still uses.  Not one in ten is able to suggest any clue as to their origin.  They cannot be due to anything written or printed, because they do not simulate what is found in ordinary writings or books.

About one-third of the figures are curved to the left, two-thirds to the right; they run more often upward than downward.  They do not commonly lie in a single plane.  Sometimes a Form has twists as well as bends, sometimes it is turned upside down, sometimes it plunges into an abyss of immeasurable depth, or it rises and disappears in the sky.  My correspondents are often in difficulties when trying to draw them in perspective.  One sent me a stereoscopic picture photographed from a wire that had been bent into the proper shape.  In one case the Form proceeds at first straightforward, then it makes a backward sweep high above head, and finally recurves into the pocket, of all places!  It is often sloped upwards at a slight inclination from a little below the level of the eye, just as objects on a table would appear to a child whose chin was barely above it.

It may seem strange that children should have such bold conceptions as of curves sweeping loftily upward or downward to immeasurable depths, but I think it may be accounted for by their much larger personal experience of the vertical dimension of space than adults.  They are lifted, tossed and swung, but adults pass their lives very much on a level, and only judge of heights by inference from the picture on their retina.  Whenever a man first ventures up in a balloon, or is let, like a gatherer of sea-birds’ eggs, over the face of a precipice, he is conscious of having acquired a much extended experience of the third dimension of space.