The gross statistical result from the schoolboys is as follows:  —­Total returns, 337:  viz.  Winchester 135, Princeton 34, Charterhouse 168; the number of these that contained well-defined Number-Forms are 5, 1, and 12 respectively, or total 18—­that is, one in twenty.  It may justly be said that the masters should not be counted, because it was owing to the accident of their seeing the Number-Forms themselves that they became interested in the inquiry; if this objection be allowed, the proportion would become 16 in 337, or one in twenty-one.  Again, some boys who had no visualising faculty at all could make no sense out of the questions, and wholly refrained from answering; this would again diminish the proportion.  The shyness in some would help in a statistical return to neutralise the tendency to exaggeration in others, but I do not think there is much room for correction on either head.  Neither do I think it requisite to make much allowance for inaccurate answers, as the tone of the replies is simple and straightforward.  Those from Princeton, where the students are older and had been specially warned, are remarkable for indications of self-restraint.  The result of personal inquiries among adults, quite independent of and prior to these, gave me the proportion of 1 in 30 as a provisional result for adults.  This is as well confirmed by the present returns of 1 in 21 among boys and youths as I could have expected.

I have not a sufficient number of returns from girls for useful comparison with the above, though I am much indebted to Miss Lewis for 33 reports, to Miss Cooper of Edgbaston for 10 reports from the female teachers at her school, and to a few other schoolmistresses, such as Miss Stones of Carmarthen, whose returns I have utilised in other ways.  The tendency to see Number-Forms is certainly higher in girls than in boys.

Fig. 20 is the Form of Mr. George Bidder, Q.C.; it is of much interest to myself, because it was, as I have already mentioned, through the receipt of it and an accompanying explanation that my attention was first drawn to the subject.  Mr. G. Bidder is son of the late well-known engineer, the famous “calculating boy” of the bygone generation, whose marvellous feats in mental arithmetic were a standing wonder.  The faculty is hereditary.  Mr. G. Bidder himself has multiplied mentally fifteen figures by another fifteen figures, but with less facility than his father.  It has been again transmitted, though in an again reduced degree, to the third generation.  He says:  —­

“One of the most curious peculiarities in my own case is the arrangement of the arithmetical numerals.  I have sketched this to the best of my ability.  Every number (at least within the first thousand, and afterwards thousands take the place of units) is always thought of by me in its own definite place in the series, where it has, if I may say so, a home and an individuality.  I should, however, qualify this by saying that when I am multiplying together two large numbers, my mind is engrossed in the operation, and the idea of locality in the series for the moment sinks out of prominence.”