For the purpose of present illustration of the workings of genius it seems well to choose a kind of ability which is quite indisputable, and which also admits of some degree of quantitative measurement. I would choose the higher mathematical processes, were data available; and I may say in passing how grateful I should be to receive from mathematicians any account of the mental processes of which they are conscious during the attainment of their highest results. Meantime there is a lower class of mathematical gift which by its very specialisation and isolation seems likely to throw light on our present inquiry. During the course of the present century,--and alas! the scientific observation of unusual specimens of humanity hardly runs back further, or so far,--the public of great cities has been from time to time surprised and diverted by some so-called "calculating boy," or "arithmetical prodigy," generally of tender years, and capable of performing "in his head," and almost instantaneously, problems for which ordinary workers would require pencil and paper and a much longer time. In some few cases, indeed, the ordinary student would have no means whatever of solving the problem which the calculating boy unriddled with ease and exactness. The especial advantage of the study of arithmetical prodigies is that in their case the subjective impression coincides closely with the objective result. The subliminal computator feels that the sum is right, and it _is_ right. Forms of real or supposed genius which are more interesting are apt to be less undeniable. An American and a French psychologist[28] have collected such hints and explanations as these prodigies have given of their methods of working; methods which one might naturally hope to find useful in ordinary education. The result, however, has been very meagre, and the records left to us, imperfect as they are, are enough to show that the main and primary achievement has in fact been subliminal, while conscious or supraliminal effort has sometimes been wholly absent, sometimes has supervened only after the gift has been so long exercised that the accesses between different strata have become easy by frequent traversing. The prodigy grown to manhood, who now recognises the arithmetical artifices which he used unconsciously as a boy, resembles the hypnotic subject trained by suggestion to remember in waking hours the events of the trance.