450 APPENDIX 



" How much does the second root from 1 6 give ? " Muhamed 

 answers 82 and then 6. We are not surprised to learn that the 

 above explanation is repeated in an altered form, and then ^16=4. 

 is written up. It is pointed out to the horse that the second root 

 of 1 6 is 4. "Then 4 2 is how much?" Muhamed: 16. 

 4 2 = is then written up. " You see you have there the number 

 which is placed under the sign of the root, now deal with it." 

 Apparently there is no result, but ^25 = is written up in the 

 same way. " I point to the open space behind the sign of equality : 

 Here must be the number which, multiplied by itself, gives 25. 

 So ?" Muhamed answers 5. 



" You have done that well. Once again." Muhamed : 5. 

 After this, with one or two errors, Muhamed gives >y/49 as 7, but 

 fails to extract the root of 36, so that the conclusion is that he has 

 not yet conceived the matter rightly. On the evening of the 

 same day, however, the answer 4 followed at once when he was 

 asked for the root of 16, and the understanding of Muhamed 

 adapted itself to this kind of sum in a short time, so that he 

 could soon do the sum ^81 + ^49=16, or +/# 1^/25 = 4. 



Thus, if we are to believe that Muhamed's answers are given 

 by genuine process of thought, we must suppose that in a single 

 lesson, by dint of sheer explanation, he advances from the power 

 to count 4 to that of adding 3 numbers and understanding multi- 

 plication ; while again, in a single lesson, he was able to grasp the 

 meaning of a root and the general notion of an inverted operation. 

 Apart altogether from the miraculous results achieved by the horse, 

 the method of instruction would suggest a totally different conclu- 

 sion. It convinces us that, in these lessons at any rate, whatever 

 might be the case afterwards, the horse must in reality have been 

 responding to voluntary or involuntary signs indicating which foot 

 he is to use, 1 when he is to begin to paw, and when he is to stop. 



Such are the broad facts, which might be indefinitely multiplied, 

 and how are they to be explained ? The obvious hypothesis is 

 that the horse responds to a sign, voluntary or involuntary, on the 

 part of his master or of the groom, or, if someone else is conduct- 

 ing the experiment, of the experimenter himself. As to this it 

 must be remarked first of all that the theory of conscious fraud is 

 not suggested as against Herr Krall, even by his bitterest opponents. 

 All observers appear to have come away convinced of Herr Krall's 

 honesty, and the most that can be said by his critics is that he is a 

 fanatic, who may have the failings of fanaticism but who undoubt- 



1 The horses give units first, reckoning them with their right foot. 

 They then give the tens with their left foot, then the hundreds with the 

 right, and so on. Confusions as to the foot account for some of the errors 

 in their computations, e.g., we easily get 45 instead of 54 by this method. 



