THE LATEST ARITHMETICAL PRODIGY. 67 



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 compre- 

 hend 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 mathe- 

 maticians of every kind and every age, he has learned that most 

 of them have a visual image of the figures during their calcula- 

 tions ; 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 visuat 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- 



