Let us turn now to some of his specific accomplishments. He had, apparently, completely mastered the cardinal numbers from 1 to 100 and the ordinals to 10, at least. Upon request he would count objects of all sorts, the persons present, even to distinctions of sex. Then hats, umbrellas, and eyeglasses. Even the mechanical activity of tapping seemed to reveal a measure of intelligence. Small numbers were given with a slow tapping of the right foot. With larger numbers he would increase his speed, and would often tap very rapidly right from the start, so that one might have gained the impression that knowing that he had a large number to tap, he desired to hasten the monotonous activity. After the final tap, he would return his right foot--which he used in his counting--to its original position, or he would make the final count with a very energetic tap of the left foot,--to underscore it, as it were. "Zero" was expressed by a shake of the head. But Hans could not only count, he could also solve problems in arithmetic. The four fundamental processes were entirely familiar to him. Common fractions he changed to decimals, and _vice versa_; he could solve problems in mensuration--and all with such ease that it was difficult to follow him if one had become somewhat rusty in these branches. The following problems are illustrations of the kind he solved.[E] "How much is 2/5 plus 1/2?" Answer: 9/10. (In the case of all fractions Hans would first tap the numerator, then the denominator; in this case, therefore, first 9, then 10). Or again: "I have a number in mind. I subtract 9, and have 3 as a remainder. What is the number I had in mind?"--12. "What are the factors of 28?"--Thereupon Hans tapped consecutively 2, 4, 7, 14, 28. "In the number 365287149 I place a decimal point after the 8. How many are there now in the hundreds place?"--5. "How many in the ten thousandths place?"--9. It will be noticed, therefore, that he was able to operate with numbers far exceeding 100, indeed he could manipulate those of six places. We were told that this, however, was no longer arithmetical computation in the true sense of the term; Hans merely knew after the analogy of 10 and 100 that the thousands take the fourth place, the ten-thousands the fifth, etc. If an error entered into Hans' answer, he could nearly always correct it immediately upon being asked: "By how many units did you go wrong?" [Footnote E: All examples mentioned are cited from extant works of various observers.]