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。 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。
12
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。
But the lesson suddenly ends with a joke carried rather too far by the pupil； who catches his good master by the seat of his trousers； into which he plants disrespectful teeth。 He is severely reprimanded； deprived of his carrots and sent back in disgrace to his private apartments。
13
Next comes Bette； who is like a big； sleek Norman horse。 He makes the calm； dignified； peaceful entrance of a blind giant。 His large； dark； brilliant eyes are quite dead； deprived of any reflex power。 He feels about with his hoof for the board on which he is to rap his answers。 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。 He explains to him fondly the easy sums which I suggest his doing： two plus three； eight minus four； four times three； he says：
〃Mind！ It's not plus three or minus three this time； but four multiplied by three！〃
Berto hardly ever makes a mistake。 When he does not understand the question； he waits for it to be written with the finger on his side； and the careful way in which he works it out like some backward and afflicted child is an infinitely pathetic sight。 He is much more zealous and conscientious than his fellow…pupils； and we feel that； in the darkness wherein he dwells； this work is； next to his meals； the only spark of light and interest in his existence。 He will certainly never rival Muhamed； for instance； who is the arithmetical prodigy； the Inaudi； of horses； but he is a valuable and living proof that the theory of unconscious and imperceptible signs； the only one which the German theorists have hitherto seriously considered； is now clearly untenable。
I have not yet spoken of Zarif。 He is not in the best of tempers； and besides； in arithmetic； he is only a less learned and more capricious Muhamed。 He answers most of the questions at random； stubbornly raising his foot and declining to lower it； so as clearly to mark his disapproval； but he solves the last problem correctly when he is promised a panful of carrots and no more lessons for that morning。 The groom enters to lead him away and makes some movement or other at which the horse starts； rears and shies。
〃That's his bad conscience；〃 says Krall； gravely。
And the expression assumes a singular meaning and importance in this hybrid atmosphere； steeped in an indefinable something from another world。
But it is half…past one； the sacred German dinner…hour。 The horses are taken back to their racks and the men separate； wishing one another the inevitable Mahlzeit。
As he walks with me along the quays of the black and muddy Wupper； Krall says：
〃It is a pity that you did not see Zarif in one of his better moods。 He is sometimes more startling than Muhamed and has given me two or three surprises that seem incredible。 One morning； for instance； I came to the stable and was preparing to give him his lesson in arithmetic。 He was no sooner in front of the spring…board than he began to stamp with his foot。 I left him alone and was astounded to hear a whole sentence； an absolutely human sentence； come letter by letter from his hoof： 'Albert has beaten Hanschen；' was what he said to me that day。 Another time； I wrote down from his dictation； 'Hanschen has bitten Kama。' Like a child seeing its father after an absence； he felt the need to inform me of the little doings of the stable； he provided me with the artless chronicle of a humble and uneventful life。〃
Krall； for that matter； living in the midst of his miracle； seems to think this quite natural and almost inevitable。 I； who have been immersed in it for only a few hours； accept it almost as calmly as he does。 I believe without hesitation what he tells me； and； in the presence of this phenomenon which； for the first time in man's existence； gives us a sentence that has not sprung from a human brain； I ask myself whither we are tending； where we stand and what lies ahead of us。。 。 。
14
After dinner； the experiments begin again； for my host is untiring。 First of all； pointing to me； he asks Muhamed if he remembers what his uncle's name is。 The horse raps out an H。 Krall is astonished and utters fatherly reprimands：
〃Come； take care！ You know it's not an H。〃
The horse raps out an E。 Krall becomes a little impatient： he threatens； he implores； he promises in turn； carrots and the