the aid of vision and hearing. They have come to it through the
sense of touch alone. Everything depends upon whether or not the
predisposition for it is present. And it was quite rational that Mr. von
Osten should have chosen counting and arithmetical calculation as the
processes by which to make his attack upon the animal mind, for as a
matter of fact, nowhere else is it so easy to bridge the gap between
perception and conception and nowhere else can the sign of success or
failure be perceived so readily as in the handling of numbers. It is
unfortunate, however, that he did not utilize these same signs for
purposes of counter-testing also, as, for instance, by inquiring for the
cube root of 729. But he was prevented from doing this by his close
adherence to his pedagogical principle and by his unquestioning faith in
the soundness of the entire procedure.

In teaching multiplication the counting machine was used. Two of the ten
balls on one of the rods were pushed far to the left, thus: 00. "How
many are there?" Two taps. "Very well. That is once two." Another group
of two was pushed to the left, at a short interval from the first group,
thus: 00 00. "How many times two balls are there?" was asked, with a
decided movement of the hand toward the two groups. Two taps. "How many,
therefore, are two times two?" Four taps.

The horse was supposed to learn the meaning of the word "times" by means
of the spatial separation of the groups; he was to be taught to notice
and to count the groups, and also the number of units in a single group.
Three times two then meant three groups with two units in each group.
The horse was supposedly aided by the following factors: the relative
nearness of the units belonging to one group, as over against the space
interval between the groups themselves; also that the groups were
pointed out as wholes in connection with the emphatic enunciation of the
words 'once, twice,' etc.; and finally the touching and raising of the
horse's foot by means of the hand until all the desired associations of
the ideas with one another and with the corresponding tapping movements
were quite perfect.

Subtraction was taught in the following manner. Five pins were set up;
the horse tapped five times. Mr. von Osten then removed two of them and
said emphatically: "I take away,--minus. How many are still standing?"
The horse tapped three times. Here, too, there was at first some
assistance by means of the hand to g