was that of Mondeux and Colburn, and of all who have given clear explanations of themselves. With this, nothing is easier than to account for the faculty of mental calculation—that is, of calculating without reading or writing anything. Whenever any one has a clear and sure visual memory, he does not need to have the figures before his eyes to read them and write them out in order to be able to combine them; he can turn away his eyes from the slate, because they are written as if with chalk on the tablet which his memory presents to him. This explanation appears so satisfactory that Bidder, one of the greatest mental calculators of the century, wrote in his autobiography that he could not comprehend the possibility of mental calculation without this faculty of representing the figures to himself as if he was looking at them.
This interpretation has been confirmed by the researches of Mr. Galton. Inquiring of a large number of calculators and mathematicians of every kind and every age, he has learned that most of them have a visual image of the figures during their calculations; the natural series of figures is presented in a straight line, or follows the bendings of a curved line. With some persons the figures appear placed as if in relation to the rounds of a ladder; with others they are inclosed in squares or circles. Mr. Galton calls these images number-forms. The visual image must be very clear for it to be possible to recognize so many details. M. Taine, who has studied the phenomenon of the image with much care, has discovered a resemblance between mental calculators and checker-players who do not have to look at their boards. He explains their faculty by the clearness of their visual images. "It is evident," he says, "that every move, the figure of the whole checker-board, with the order of the different pieces, is presented to them as in an inner mirror; else they would not be able to foresee the probable consequences of the move that has been made upon them and of the one they are about to order." The direct testimony of players confirms this interpretation. "With my eyes turned to the wall," says one of them, "I see at once the whole board and all the pieces as they really stand. . . . I see the pieces exactly as the turner has made them—that is, I see the checker-board in front of my adversary, and not some other checker-board."
In the light of so many facts we are naturally led to believe that all mental calculators work by the considerable development of their visual memory. But the study of M. Inaudi shows that we can not draw a general conclusion from them, and that there are other means than mental vision that seem to have the same efficaciousness and power. M. Inaudi declares that no figure is presented to him under a visual form. When he endeavors to retain a series of twenty-four figures that have just been pro-