[Illustration:  PLATE I. Examples of Number-Forms.]

“All numbers are to me as images of figures in general; I see them in ordinary Arabic type (except in some special cases), and they have definite positions in space (as shown in the Fig.).  Beyond 100 I am conscious of coming down a dotted line to the position of 1 again, and of going over the same cycle exactly as before, e.g. with 120 in the place of 20, and so on up to 140 or 150.  With higher numbers the imagery is less definite; thus, for 1140, I can only say that there are no new positions, I do not see the entire number in the place of 40; but if I think of it as 11 hundred and 40, I see 40 in its place, 11 in its place, and 100 in its place; the picture is not single though the ideas combine.  I seem to stand near 1.  I have to turn somewhat to see from 30-40, and more and more to see from 40-100; 100 lies high up to my right and behind me.  I see no shading nor colour in the figures.”

Figs. 2 to 6 are from returns collected for me by the Rev. A.D.  Hill, science-master of Winchester College, who sent me replies from 135 boys of an average age of 14-15.  He says, speaking of their replies to my numerous questions on visualising generally, that they “represent fairly those who could answer anything; the boys certainly seemed interested in the subject; the others, who had no such faculty either attempting and failing, or not finding any response in their minds, took no interest in the inquiry.”  A very remarkable case of hereditary colour association was sent to me by Mr. Hill, to which I shall refer later.  The only five good cases of Number-Forms among the 135 boys are those shown in the Figs.  I need only describe Fig. 2.  The boy says:—­“Numbers, except the first twenty, appear in waves; the two crossing-lines, 60-70, 140-150, never appear at the same time.  The first twelve are the image of a clock, and 13-20 a continuation of them.”

Figs. 7, 8, are sent me by Mr. Henry F. Osborn of Princeton in the United States, who has given cordial assistance in obtaining information as regards visualising generally.  These two are the only Forms included in sixty returns that he sent, 34 of which were from Princeton College, and the remaining 26 from Vassar (female) College.  Figs. 9-19 and Fig. 28 are from returns communicated by Mr. W.H.  Poole, science-master of Charterhouse College, which are very valuable to me as regards visualising power generally.  He read my questions before a meeting of about 60 boys, who all consented to reply, and he had several subsequent volunteers.  All the answers were short, straightforward, and often amusing.  Subsequently the inquiry extended, and I have 168 returns from him in all, containing 12 good Number-Forms, shown in Figs. 9-19, and in Fig. 28.  The first Fig. is that of Mr. Poole himself; he says, “The line only represents position; it does not exist in my mind.  After 100, I return to my old starting-place, e.g. 140 occupies the same position as 40.”