Still, if we admit the theory of intelligence, we must also admit that the horse knows what he is doing, since it is not until after learning what a squared number or a square root means that he appears to understand or that, at any rate, he gradually works out correctly the ever more complicated calculations required of him. It is not possible to give here the details of this instruction, which was astonishingly rapid. The reader will find them on pages 117 et seq. of Krall's book, Denkende Tiere. Krall begins by explaining to Mohammed that 2 squared is equal to 2 X 2 = 4; that 2 cubed is equal to 2 X 2 X 2 = 6; that 2 is the square root of 4; and so on. In short, the explanations and demonstrations are absolutely similar to those which one would give to an extremely intelligent child, with this difference, that the horse is much more attentive than the child and that, thanks to his extraordinary memory, he never forgets what he appears to have understood. Let us add, to complete the magical and incredible character of the phenomenon that, according to Krall's own statement, the horse was not taught beyond the point of extracting the square root of the number 144 and that he spontaneously invented the manner of extracting all the others. 27 Must we once more repeat, in connection with these startling performances, that those who speak of audible or visible signals, of telegraphy and wireless telegraphy, of expedients, trickery or deceit, are speaking of what they do not know and of what they have not seen? There is but one reply to be made to any one who honestly refuses to believe: "Go to Elberfeld---the problem is sufficiently important, sufficiently big with consequences to make the journey worth while--and, behind closed doors, alone with the horse, in the absolute solitude and silence of the stable, set Mohammed to extract half-a dozen roots which, like that which I have mentioned, require thirty-one operations. You must yourself be ignorant of the solutions, so as to do away with any transmission of unconscious thought. If he then gives you, one after the other, five or six correct solutions, as he did to me and many others, you will not go away with the conviction that the animal is able by its intelligence to extract those roots, because that conviction would upset too thoroughly the greater part of the certainties on which your life is based; but you will, at any rate, be persuaded that you have been for a few minutes in the presence of one of the greatest and strangest riddles that can disturb the mind of man; and it is always a good and salutary thing to come into contact with emotions of this order." 28