if the sound of the vowel kept pure in tone, it was certain he did not articulate the figures. The experiment caused M. Inaudi great embarrassment. He was still able to calculate in his head, but it took him four or five times as long as under the usual conditions, and he succeeded in doing it only by cheating a little—that is, he made some articulations of figures in a low voice, the production of which was at once detected on listening attentively to the sound of the sung vowel.
These experiments showed that articulation constitutes an integral part of M. Inaudi's mental calculations, as well as that every experimental artifice that interferes with articulation makes the calculation longer or modifies its accuracy. In other words, M. Inaudi uses auditive and motor images of articulation concurrently. We have no experimental means of determining which is the predominant factor. M. Inaudi thinks that the sound guides him, and that the motion of articulation intervenes only to re-enforce the auditive image. We might be liable to suppose, in view of the part that is played by the memory in mental calculation, that it is the only faculty developed in arithmetical prodigies; and some authors have fallen into this error. But it will be well to guard against such a supposition, which is contrary to the most certain and best established psychological facts. If we take any elementary act of the mind and analyze it, we shall find that it involves the concurrence of a large number of co-ordinated operations; with much stronger reason must such a concurrence be supposed necessary for acts as complex as mental calculations. We have found in our studies of M. Inaudi that a considerable number of his faculties have attained an extreme development, and they are precisely the ones that concur in operations of mental calculations. Perception, attention, and judgment, to the extent and in the shape in which they are needed in his work, have acquired the same perfection as his memory for figures.
It remains to inquire how these aptitudes for calculation have been formed. When we examine the history of these arithmetical prodigies, we are struck by the three facts of their precocity; the impulsive, in a certain sense all-possessing, character of their passion for calculation; and the generally illiterate, often miserable, medium in which they have grown up. Their stories have many traits in common. They are most frequently children of poor and ignorant parents. They are seized with the passion for calculating in their earliest years—at from five to ten years of age on the average—the age when most children are living in the illusions of plays and stories; they begin to combine numbers in their heads, apparently without any exterior provocation, and without the influence of parents or schoolmasters. As they grow up