he right, even though seen from his own side, it would
appear to be the left. Hans would also walk toward the persons or things
that he was asked to point out, and he would bring from a row of colored
cloths, the piece of the particular color demanded. Taking into account
his limited means of expression, his master had translated a large
number of concepts into numbers; e. g.:--the letters of the alphabet,
the tones of the scale, and the names of the playing cards were
indicated by taps. In the case of playing cards one tap meant "ace," two
taps "king," three "queen," etc.

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