iraculous bridge which； in this instance； spans the gulf between the animal and man is much more the expression of thought than thought itself。 We may go further and grant that certain elementary calculations； such as little additions； little subtractions of one or two figures； are； after all； conceivable； and I； for my part； am inclined to believe that the horse really executes them。 But where we get out of our depth； where we enter into the realm of pure enchantment is when it becomes a matter of mathematical operations on a large scale； notably of the finding of roots。 We know； for instance； that the extraction of the fourth root of a number of six figures calls for eighteen multiplications； ten subtractions and three divisions and that the horse does thirty…one sums in five or six seconds； that is to say； during the brief； careless glance which he gives at the black…board on which the problem is inscribed； as though the answer came to him intuitively and instantaneously。
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。
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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 whileand； 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。〃
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Truth to say； the theory of intelligence in the animal would be so extraordinary as to be almost untenable。 If we are determined； at whatever cost； to pin our faith to it； we are bound to call in the aid of other ideas； to appeal； for instance； to the extremely mysterious and essentially uncomprehended and incomprehensible nature of numbers。 It is almost certain that the science of mathematics lies outside the intelligence。 It forms a mechanical and abstract whole； more spiritual than material and more material than spiritual； visible only through its shadow and yet constituting the most immovable of the realities that govern the universe。 From first to last it declares itself a very strange force and； as it were； the sovereign of another element than that which nourishes our brain。 Secret； indifferent； imperious and implacable； it subjugates and oppresses us from a great height or a great depth； in any case； from very far； without telling us why。 One might say that figures place those who handle them in a special condition。 They draw the cabalistic circle around their victim。 Henceforth； he is no longer his own master； he renounces his liberty； he is literally 〃possessed〃 by the powers which he invokes。 He is dragged he knows not whither； into a formless； boundless immensity； subject to laws that have nothing human about them； in which each of those lively and tyrannical little signs which move and dance in their thousands under the pen represents nameless； but eternal； invincible and inevitable verities。 We think that we are directing them and they enslave us。 We become weary and breathless following them into their uninhabitable spaces。 When we touch them； we let loose a force which we are no longer able to control。 They do with us what they will and always end by hurling us； blinded and benumbed； into blank infinity or upon a wall of ice against which every effort of our mind and will is shattered。
It is possible； therefore； in the last resort； to explain the Elberfeld mystery by the no less obscure mystery that surrounds numbers。 This really only means moving to another spot in the gloom； but it is often just by that moving to another spot that we end by discovering the little gleam of light which shows us a thoroughfare。 In any case； and to return to more precise ideas； more than one instance has been cited to prove that the gift of handling great groups of figures is almost independent of the intelligence proper。 One of the most curious is that of an Italian shepherd boy； Vito Mangiamele； who was brought before the Paris Academy of Science in 1837 and who； at the age of ten； though devoid of the most rudimentary education； was able in half a minute to extract the cubic root of a number of seven figures。 Another； more striking still； also mentioned by Dr。 Clarapede in his paper on the learned horses； is that of a man blind from birth； an inmate of the lunatic…asylum； at Armentieres。 This blind man； whose name is Fleury； a degenerate and nearly an idiot； can calculate in one minute and fifteen seconds the number of seconds in thirty…nine years； three months and twelve days； not forgetting the leap…years。 They explain to him what a square root is； without telling him the conventional method of finding it； and soon he extracts almost as rapidly as Inaudi himself； without a blunder； the square roots of numbers of four figures； giving the remainder。 On the other hand； we know that a mathematical genius like Henri Pomcare confessed himself incapable of adding up a column of figures without a mistake。
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》From the maybe enchanted atmosphere that surrounds numbers we shall pass more easily to the even more magic mists of the final theory； the only one remaining to us for the moment： the mediumistic or subliminal theory。 This； we must remember； is not the telepathic theory proper which decisive experiments have made us reject。 Let us have the courage to venture upon it。 When one can no longer interpret a phenomenon by the known； we must needs try to do so by the unknown。 We； therefore； now enter a new province of a great unexplored kingdom； in which we shall find ourselves without a guide。
Mediumistic phenomena； manifestations of the secondary or the subliminal consciousness； between man and man； are； as we have more than once had occasion to assure ourselves； capricious； undisciplined； evasive and uncertain； but more frequent than one thought and； to one who examines； them seriously and honestly； often undeniable。 Have similar manifestations been discovered between man and the animals？ The study of these manifestations； which is very difficult even in the case of man； becomes still more so when we question witnesses doomed to silence。 There are； however； some animals which are looked upon as 〃psychic；〃 which； in other words； seem indisputably to be sensitive to certain subliminal influences。 One usually classes the cat； the dog and the horse in this somewhat ill…defined category。 To these superstitious animals one might perhaps add certain birds； more or less birds of omen； and even a few insects； notably the bees。 Other animals； such as； for instance； the elephant and the monkey； appear to be proof against mystery。 Be this as it may； M。 Ernest Bozzano； in an excellent article on Les Perceptions psychiques des animaux；＇1＇ collected in 1905 sixty…nine cases of telepathy； presentiments and hallucinations of sight or hearing in which the principal actors are cats； dogs and horses。 There are； even among them； ghosts or phantoms of dogs which； after their death； return to haunt the homes in which they were happy。 Most of these cases are taken from the Proceedings of the S。 P。 R。； that is to say； they have nearly all been very strictly investigated。 It is impossible； short of filling these pages with often striking and touching but rather cumbersome anecdotes； to enumerate them here； however briefly。 It will be sufficient to note that sometimes the dog begins to howl at the exact moment when his master loses his life； for instance； on a battlefield； hundreds of miles from the place where the dog is。 More commonly； the cat； the dog and the horse plainly manifest that they perceive； often before men do； telepathic apparitions； phantasms of the living or the dead。 Horses in particular seem very sensitive to places that pass as haunted or uncanny。 On the whole； the result of these observations is that we can hardly dispute that these animals communicate as much as we do and perhaps in the same fashion with the mystery that lies around us。 There are moments at which； like man； they see the invisible and perceive events； influences and emotions t