child and never able to exercise even ordinary intelligence in other directions or to learn much 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 ignorant.
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 remember 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 hundred 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 intelligence 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 aptitude in individuals otherwise defective, but these will suffice 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 astonishing.
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 congenital. 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 stigmata of degeneration in such cases. Thirdly, much of the faculty is due to the increased power of visualization—to great development of certain parts of the sight centers. Most of us, in mental arithmetic, compute by means of visual images. We, who have
