no adequate explanation of their mathematical agility. This hypothesis is further weakened by the recently developed fact that Inaudi, the ruling French mathematical wonder, is not a visionnaire at all, but a distinct auditaire who hears all his numbers.
Referring now to the accompanying forms, Figs. 1, 2, and 4 demand no further explanation. In Fig. 3 we have an interesting double form, the one to the left showing how the numbers from 1 to 15 appear when thought of by themselves or in connection with one another. But when any number below 15 is thought of in connection with any number above 15, it is seen as shown in the form to the right. Above 15 the numbers are unalterably fixed. The possessor of this form writes me as follows:
I do not believe I can think of a number apart from this outline. I refer all numbers to it, however large. One million is located where 1,000 is, and so of 1,000,000,000; 550 would be at 55 on the circle; 1,235 is at 35. You will notice that of the last two numbers I mention, the first is located at the point indicated by the first two figures, viz., 55; but the last number, 1,235, is located at 35, the last two figures. I can not explain this, but simply state it as a fact. I think possibly in large uneven numbers, I really, though almost unconsciously, separate the number into parts, in 1,235 the 1,200 either being ignored and my mind directed to 35, or else I in some manner connect the two locations but direct my attention more to one than the other. I stated above that I did not believe I could think of a number apart from this outline, and that is true when I think of some one number by itself and in adding and subtracting small numbers. If any one should ask me how many hours intervened from 3 to 11 o'clock, I would say 8, because I see that many on my number form, which immediately appears before my mind's eye, but I could not subtract 37 from 89 in that way. I would immediately locate the two numbers but I could not determine how many numbers intervened, and I find that in adding, subtracting, and multiplying odd numbers, and numbers beyond 15 say, I do it abstractly without referring to my form; but as I said, in thinking of any one number by itself, it is always connected with some point along that outline. This number form, by the way, stands upright and is about two feet in height—that is, the number 100 is two feet above 18 and about six inches to the right.
Among the seventy-five young men and women interrogated in the first experiment, was a rather diffident young woman who communicated to a classmate that while, she had no number form, there were certain associations that she always made with the nine digitis. Learning this, I questioned her, and she consented to write out the associations, which I reproduce here exactly as given:
1 = a child about two years old.
2 = a boy, ten or twelve years old, brown hair and eyes, frank, active, noisy, always ready to help.
3 = a girl, short hair, black, curly; sharp features, not pretty; slight; awful temper; shrill voice; bangs and slams around generally.
