2 34 POPULAR SCIENCE MONTHLY. 



child and never able to exercise even ordinary intelligence in other 

 directions or to learn mucli of anything. He had supernumerary 

 digits on both hands and feet, and was a degenerate. 



Vito Mangiamele, son of a Sicilian shepherd, born in 1827, was 

 exhibited as a calculating boy, but was otherwise dull and ig- 

 norant. 



Dase, a German, born in 1824, extremely stupid and dull in 

 other directions, never able to master a word of any language 

 but his own, was a mathematical genius. As an instance of his 

 power, he multiplied correctly, in fifty- four seconds, 79,532,853 by 

 93,758,479. 



Grandmange, a Frenchman, born without legs or arms in 

 1836, was another example of a mathematical prodigy. 



Mondeux, a Frenchman, son of a woodcutter, born in 1826, 

 possessed an extraordinary arithmetical faculty, although he 

 could neither read, nor write, nor cipher. He could not remem- 

 ber a name or address. He solved this problem in a few seconds : 

 How many quarts of water in a fountain from which a group 

 of people draw as follows: The first person takes one hundred 

 quarts and one thirteenth of the remainder ; the second, two hun- 

 dred quarts and one thirteenth of the remainder ; the third, three 

 hundred quarts and one thirteenth of the remainder; and so on 

 until the fountain was emptied ? 



Dr. Heim cites the instance of a woman of very limited intelli- 

 gence and deficient in language, who could give the number of 

 minutes a person had lived as soon as the age was told her. 



There are other examples in literature of mathematical apti- 

 tude in individuals otherwise defective, but these will sufiice to 

 illustrate the character of the cases under consideration. When 

 we remember the deficiency of idiots in general as regards even 

 the simplest kinds of calculations such as counting, addition, 

 subtraction, multiplication, and division the contrast of powers 

 in the exceptional instances mentioned becomes even more as- 

 tonishing. 



We may deduce from a study of such cases several facts which 

 are noteworthy. First, the mathematical aptitude in idiocy is 

 never of a high order. The faculty consists entirely of excessive 

 powers in mental arithmetic in simple calculation, which is a 

 better term to apply to it. Secondly, it is instinctive and con- 

 genital. It is observed only in the congenital variety of idiots, 

 imbeciles, and degenerates ; and on careful examination we shall 

 find anatomical and physiological as well as psychological stig- 

 mata of degeneration in such cases. Thirdly, much of the faculty 

 is due to the increased power of visualization to great develop- 

 ment of certain parts of the sight centers. Most of us, in mental 

 arithmetic, compute by means of visual images. We, who have 



