250 THE HORSE OF MR. VON OSTEN 



unfortunate, however, that he did not utilize these same 

 signs for purposes of counter-testing also, as, for in- 

 stance, 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 be- 

 longing 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 em- 

 phatic 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 



