"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. 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."[29] 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 Ampère 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.