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Updated: June 26, 2025
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.
Everything depends on a whim, the state of the weather, the morning meal or the impression which the visitor makes upon them. Still, Krall seems to know, by certain imperceptible signs, that this is not going to be a bad day. Muhamed quivered with excitement, snorts loudly through his nostrils, utters a series of indistinct little whinnyings: excellent symptoms, it appears.
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.
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.
They stood up to each other like two enemies: and the lessons almost assumed the form of a tragic and secret struggle in which the animal's soul rebelled against man's domination. Krall, on the other hand, adores his pupils; and this atmosphere of affection has in a manner of speaking humanized them.
He has not yet gone beyond the rudiments of mathematics; and the early part of his education was particularly difficult. They managed to make him understand the value and meaning of the numbers and of the addition- and multiplication-signs by means of little taps on his sides. Krall speaks to him as a father might speak to the youngest of his sons.
A few months ago, Herr Krall, whom I had promised the year before that I would come and see his wonderful horses, was kind enough to repeat his invitation in a more pressing fashion, adding that his stable would perhaps be broken up after the 15th of September and that, in any case, be would be obliged, by his doctor's orders, to interrupt for an indefinite period a course of training which he found exceedingly fatiguing.
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 found Herr Krall in his goldsmith's shop, a sort of palace of Golconda, streaming and glittering with the most precious pearls and stones on earth. Herr Krall, it is well to remember, in order to dispel any suspicion of pecuniary interest, is a rich manufacturer whose family for three generations, from father to son, have conducted one of the most important jewelry businesses in Germany.
In four months' time, he knew how to extract square and cubic roots; and, soon after, he learnt to spell and read by means of the conventional alphabet devised by Krall. This alphabet, at the first glance, seems rather complicated. For that matter, it is only a makeshift; but how could one find anything better?
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