There was certainly something peculiar in my calculating faculty. It began to show itself at between five and six, and lasted about three years.... I soon got to do the most difficult sums, always in my head, for I knew nothing of figures beyond numeration. I did these sums much quicker than any one could upon paper, and I never remember committing the smallest error. When I went to school, at which time the passion wore off, I was a perfect dunce at ciphering, and have continued so ever since.

Still more remarkable, perhaps, was Professor Safford's loss of power. Professor Safford's whole bent was mathematical; his boyish gift of calculation raised him into notice; and he is now a Professor of Astronomy. He had therefore every motive and every opportunity to retain the gift, if thought and practice could have retained it. But whereas at ten years old he worked correctly in his head, in one minute, a multiplication sum whose answer consisted of 36 figures, he is now, I believe, neither more nor less capable of such calculation than his neighbours.

Similar was the fate of a personage who never rises above initials, and of whose general capacity we know nothing.

"Mr. Van R., of Utica," says Dr. Scripture on the authority of Gall, "at the age of six years distinguished himself by a singular faculty for calculating in his head. At eight he entirely lost this faculty, and after that time he could calculate neither better nor faster than any other person.

He did not retain the slightest idea of the manner in which he performed his calculations in childhood".

Turning now to the stupid or uneducated prodigies, Dase alone seems to have retained his power through life. Colburn and Mondeux, and apparently Prolongeau and Mangiamele, lost their gift after childhood.

A few hints as to processes have been gleaned from this group; - the most interesting point being that Colbum was for some years unable, but afterwards to some extent able, to explain his own processes. "His friends tried to elicit a disclosure of the methods by which he performed his calculations, but for nearly three years he was unable to satisfy their inquiries. He positively declared that he did not know how the answers came into his mind."1 Later on he did give an account of his artifices, which, however, showed no great ingenuity.

But on the whole the ignorant prodigies seldom appear to have been conscious of any continuous logical process, while in some cases the separation of the supraliminal and subliminal trains of thought must have been very complete. "Buxton would talk freely whilst doing his questions, that being no molestation or hindrance to him."2 Fixity and clearness of inward visualisation seems to have been the leading necessity in all these achievements; and it apparently mattered little whether the mental blackboard (so to say) on which the steps of the calculation were recorded were or were not visible to the mind's eye of the supraliminal self.

I have been speaking only of visualisation; but it would be interesting if we could discover how much actual mathematical insight or inventiveness can be subliminally exercised. Here, however, our materials are very imperfect. From Gauss and Ampére we have, so far as I know, no record. At the other end of the scale, we know that Dase (perhaps the most successful of all these prodigies) was singularly devoid of mathematical grasp. "On one occasion Petersen tried in vain for six weeks to get the first elements of mathematics into his head." "He could not be made to have the least idea of a proposition in Euclid. Of any language but his own he could never master a word." Yet Dase received a grant from the Academy of Sciences at Hamburg, on the recommendation of Gauss, for mathematical work; and actually in twelve years made tables of factors and prime numbers for the seventh and nearly the whole of the eighth million, - a task which probably few men could have accomplished, without mechanical aid, in an ordinary lifetime.

He may thus be ranked as the only man who has ever done valuable service to Mathematics without being able to cross the Ass's Bridge.

On the other hand, in the case of Mangiamele, there may have been real ingenuity subliminally at work. Our account of this prodigy is authentic, but tantalising from its brevity.

1 Scripture, op. cit., p. 50.

2 Scripture, op. cit., p. 54.

In the year 1837 Vito Mangiamele, who gave his age as 10 years and 4 months, presented himself before Arago in Paris. He was the son of a shepherd of Sicily, who was not able to give his son any instruction. By chance it was discovered that by methods peculiar to himself he resolved problems that seemed at the first view to require extended mathematical knowledge. In the presence of the Academy Arago proposed the following questions: "What is the cubic root of 3,796,416?" In the space of about half a minute the child responded 156, which is correct. "What satisfies the condition that its cube plus five times its square is equal to 42 times itself increased by 40? " Everybody understands that this is a demand for the root of the equation x3 + 5 x2 - 42 x - 40 = o. In less than a minute Vito responded that 5 satisfied the condition; which is correct. The third question related to the solution of the equation x5 - 4 x - 16779 = 0. This time the child remained four to five minutes without answering: finally he demanded with some hesitation if 3 would not be the solution desired. The secretary having informed him that he was wrong, Vito, a few moments afterwards, gave the number 7 as the true solution.

Having finally been requested to extract the 10th root of 282,475,249 Vito found in a short time that the root is 7.

At a later date a committee, composed of Arago, Cauchy, and others, complains that "the masters of Mangiamele have always kept secret the methods of calculation which he made use of."1

There is another point on which something might have been learnt from the study of so marked a group of automatists - utilisers of subliminal capacity - as these "prodigies" form. Their bodily characteristics might have been examined with a view to tracing such physical concomitants as may go with this facility of communication between psychical strata. We have, however, few data available for this purpose. Colburn inherited supernumerary digits, and Mondeux is reported to have been hysterical. On the other hand the "prodigies" of whose lives after childhood anything is known seem to have been free from nervous taint. No one, for instance, could well be more remote from hysteria than the elder Bidder; - or than his son, the late Mr. Bidder, Q.C., or than Mr. Blyth of Edinburgh, perhaps the best living English representative of what we may call the calculating diathesis.