But Muhamed gives unmistakable signs of impatience to show that he has had enough of spelling. Thereupon, as a diversion and a reward, his kind master suggests the extraction of a few square and cubic roots. Muhamed appears delighted: these are his favourite problems: for he takes less interest than formerly in the most difficult multiplications and divisions. He doubtless thinks them beneath him.

Krall therefore writes on the blackboard various numbers of which I did not take note. Moreover, as nobody now contests the fact that the horse works them with ease, it would hardly be interesting to reproduce here several rather grim problems of which numerous variants will be found in the accounts and reports of experiments signed by Drs. Mackenzie and Hartkopff, by Overbeck, Clarapede and many others. What strikes one particularly is the facility, the quickness, I was almost saying the joyous carelessness with which the strange mathematician gives the answers. The last figure is hardly chalked upon the board before the right hoof is striking off the units, followed immediately by the left hoof marking the tens. There is not a sign of attention or reflection; one is not even aware of the exact moment at which the horse looks at the problem: and the answer seems to spring automatically from an invisible intelligence. Mistakes are rare or frequent according as it happens to be a good or bad day with the horse; but, when he is told of them, he nearly always corrects them. Not unseldom, the number is reversed: 47, for instance, becomes 74; but he puts it right without demur when asked.

I am manifestly dumbfounded; but perhaps these problems are prepared beforehand? If they were, it would be very extraordinary, but yet less surprising than their actual solution. Krall does not read this suspicion in my eyes, because they do not show it; nevertheless, to remove the least shade of it, he asks me to write a number of my own on the black-board for the horse to find the root.

I must here confess the humiliating ignorance that is the disgrace of my life. I have not the faintest idea of the mysteries concealed within these recondite and complicated operations. I did my humanities like everybody else; but, after crossing the useful and familiar frontiers of multiplication and division I found it impossible to advance any farther into the desolate regions, bristling with figures, where the square and cubic roots hold sway, together with all sorts of other monstrous powers, without shapes or faces, which inspired me with invincible terror. All the persecutions of my excellent instructors wore themselves out against a dead wall of stolidity.

Successively disheartened, they left me to my dismal ignorance, prophesying a most dreary future for me, haunted with bitter regrets. I must say that, until now, I had scarcely experienced the effects of these gloomy predictions; but the hour has come for me to expiate the sins of my youth. Nevertheless, I put a good face upon it: and, taking at random the first figures that suggest themselves to my mind, I boldly write on the black-board an enormous and most daring number. Muhamed remains motionless.

Krall speaks to him sharply, telling him to hurry up. Muhamed lifts his right hoof, but does not let it fall. Krall loses patience, lavishes prayers, promises and threats; the hoof remains poised, as though to bear witness to good intentions that cannot be carried out. Then my host turns round, looks at the problem and asks me:

"Does it give an exact root?"

Exact? What does he mean? Are there roots which. . .? But I dare not go on: my shameful ignorance suddenly flashes before my eyes.

Krall smiles indulgently and, without making any attempt to supplement an education which is too much in arrears to allow of the slightest hope, laboriously works out the problem and declares that the horse was right in refusing to give an impossible solution.

Muhamed receives our thanks in the form of a lordly portion of carrots; and a pupil is introduced whose attainments do not tower so high above mine: Hanschen, the little pony, quick and lively as a big rat. Like me, he has never gone beyond elementary arithmetic: and so we shall understand each other better and meet on equal terms.

Krall asks me for two numbers to multiply. I give him 63 X 7. He does the sum and writes the product on the board, followed by the sign of division: 441 / 7. Instantly Hanschen, with a celerity difficult to follow, gives three blows, or rather three violent scrapes with his right hoof and six with his left, which makes 63, for we must not forget that in German they say not sixty-three, but three-and-sixty. We congratulate him; and, to evince his satisfaction, he nimbly reverses the number by marking 36 and then puts it right again by scraping 63. He is evidently enjoying himself and juggling with the figures. And additions, subtractions, multiplications and divisions follow one after the other, with figures supplied by myself, so as to remove any idea of collusion. Hanschen seldom blunders; and, when he does, we receive a very clear impression that his mistake is voluntary: he is like a mischievous schoolboy playing a practical joke upon his master. The solutions fall thick as hail upon the little spring-board; the correct answer is released by the question as though you were pressing the button of an electric push. The pony's flippancy is as surprising as his skill. But in this unruly flippancy, in this hastiness which seems inattentive there is nevertheless a fixed and permanent idea. Hanschen paws the ground, kicks, prances, tosses his head, looks as if he cannot keep still, but never leaves his spring-board. Is he interested in the problems, does he enjoy them? It is impossible to say; but he certainly has the appearance of one accomplishing a duty or a piece of work which we do not discuss, which is important, necessary and inevitable.