5 o8 THE POPULAR SCIENCE MONTHLY. 



no adequate explanation of their mathematical agility. This hy- 

 pothesis is further weakened by the recently developed fact that 

 Inaudi, the ruling French mathematical wonder, is not a vision- 

 naire 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 interest- 

 ing double form, the one to the left showing how the numbers 

 from 1 to 15 appear when thought of by themselves or in con- 

 nection 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 un- 

 alterably 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; 1,285 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 test 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 immedi- 

 ately locate the two numbers but I could not determine how many numbers in- 

 tervened, 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 w T ay, 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 gen- 

 erally. 



