Mostly, of course, the subconscious upsurgings come to them when they are awake, sometimes in momentary gleams of insight, sometimes continuing through comparatively long periods, when they write, compose, or develop valuable discoveries without conscious effort. In fact, there even is one type of genius—although by no means the most useful—in which, within a certain limited field, the subconscious is perpetually in evidence, and perpetually responsive to the demands of the upper consciousness. I refer to the so-called “lightning calculators,” those prodigies whose mathematical feats, performed without the aid of pencil and paper, have been a source of unending surprise to the world, and have at times been so remarkable as to be well-nigh incredible. Thus, Zerah Colburn, an American lightning calculator, when only six years old, unable to read, and ignorant of the name and value of any numeral set down on paper, is known to have stated correctly the number of seconds in a period as long as two thousand years, and to have returned the correct answer (9,139,200) to the question, “Supposing I have a corn-field, in which are 7 acres, having 17 rows to each acre, 64 hills to each row, 8 ears on a hill, and 150 kernels on the ear, how many kernels in the corn-field?” A little later, having been taken by his father to England, it is recorded that, in the presence of a number of witnesses: “He undertook and succeeded in raising the number 8 to the sixteenth power, 281,474,976,780,656. He was then tried as to other numbers, consisting of one figure, all of which he raised as high as the tenth power, with so much facility that the person appointed to take down the results was obliged to enjoin him not to be too rapid. With respect to numbers of two figures, he would raise some of them to the sixth, seventh, and eighth power, but not always with equal facility; for the larger the products became the more difficult he found it to proceed. He was asked the square root of 106,929, and before the number could be written down he immediately answered 327. He was then requested to name the cube root of 268,336,125, and with equal facility and promptness he replied 645.” Henri Mondeux, Vito Mangiamele, Jacques Inaudi, Zacharias Dase, Jedediah Buxton, Truman Safford, André Ampère, Karl Gauss, George Bidder and his son of the same name, were other world famous calculators. From some of them direct evidence as to the subconscious character of their calculations has been forthcoming. One of the most remarkable in this group, the elder Bidder, in a paper contributed to a scientific journal, declared, “Whenever I feel called upon to make use of the stores of my mind, they seem to rise with the rapidity of lightning.” In a later issue of the same journal it is asserted regarding him: