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lly certain that this is possible between human subliminals。 Is it I who see the solution and transmit it to the horse； who only repeats it to me？ But； suppose that it is a problem which I myself am incapable of solving？ Whence does the solution come； then？ I do not know if the experiment has been attempted； under the same conditions； with a human medium。 For that matter； if it succeeded； it would be very much the same as the no less subliminal phenomenon of the arithmetical prodigies； or lightning calculators； with which； in this rather superhuman atmosphere； we are almost forced to compare the riddle of the mathematical horses。 Of all the interpretations； it is the one which； for the moment； appears to me the least eccentric and the most natural。
We have seen that the gift of handling colossal figures is almost foreign to the intelligence proper； one can； even declare that； in certain cases； it is evidently and completely independent of such intelligence。 In these cases； the gift is manifested prior to any education and from the earliest years of childhood。 If we refer to the list of arithmetical prodigies given by Dr。 Scripure；＇1＇ we see that the faculty made its appearance in Ampere at the age of three； in Colburn at six； in Gauss at three； in Mangiamele at ten； in Safford at six； in Whateley at three； and so on。 Generally； it lasts for only a few years； becoming rapidly enfeebled with age and usually vanishing suddenly at the moment when its possessor begins to go to school。
＇1＇ American Journal of Psychology； 1 April 1891。
When you ask those children and even most of the lightning calculators who have come to man's estate how they go to work to solve the huge and complicated problems set them； they reply that they know nothing about it。 Bidder； for instance； declares that it is impossible for him to say how he can instinctively tell the logarithm of a number consisting of seven or eight figures。 It is the same with Safford； who； at the age of ten； used to do in his head； without ever making a mistake； multiplication…sums the result of which ran into thirty…six figures。 The solution presents itself authoritatively and spontaneously； it is a vision； an impression； an inspiration； an intuition coming one knows not whence； suddenly and indubitably。 As a role； they do not even try to calculate。 Contrary to the general belief； they have no peculiar methods； or； if method there be； it is more a practical way of subdividing the intuition。 One would think that the solution springs suddenly from the very enunciation of the problem； in the same way as a veridical hallucination。 It appears to rise； infallible and ready…done； from a sort of eternal and cosmic reservoir wherein the answers to every question lie dormant。 It must； therefore； be admitted that we have here a phenomenon that occurs above or below the brain； by the side of the consciousness and the mind； outside all the intellectual methods and habits； and it is precisely for phenomena of this kind that Myers invented the word 〃subliminal。〃＇1＇
＇1＇ I have no need to recall the derivation of the term subliminal： beneath （sub） the threshold （limen） of consciousness。 Let us add； as M。 de Vesme very rightly remarks； that the subliminal is not exactly what classical psychology calls the subconsciousness； which latter records only notions that are normally perceived and possesses only normal faculties； that is to say； faculties recognized to…day by orthodox science。
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Does not all this bring us a little nearer to our calculating horses？ From the moment that it is demonstrated that the solution of a mathematical problem no longer depends exclusively on the brain； but on another faculty； another spiritual power whose presence under various forms has been ascertained beyond a doubt in certain animals； it ceases to be wholly rash or extravagant to suggest that perhaps； in the horse； the same phenomenon is reproduced and developed in the same unknown； wherein moreover the mysteries of numbers and those of subconsciousness mingle in a like darkness。 I am well aware that an explanation laden to such an extent with mysteries explains but very little more than silence does； nevertheless； it is at least a silence traversed by restless murmurs； and sedulous whispers that are better than the gloomy and hopeless ignorance to which we would have perforce to resign ourselves if we did not； in spite of all； to perform the great duty of man； which is to discover a spark in the darkness。
It goes without saying that objections are raised from every side。 Among men； arithmetical prodigies are looked upon as monsters； as a sort of extremely rare teratological phenomenon。 We can count； at most； half…a…dozen in a century； whereas； among horses； the faculty would appear to be almost general； or at least quite common。 In fact； out of six or seven stallions whom Krall tried to initiate into the secrets of mathematics； he found only two that appeared to him too poorly gifted for him to waste time on their education。 These were； I believe； two thoroughbreds that were presented to him by the Grand…duke of Mecklenburg and sent back by Krall to their sumptuous stables。 In the four or five others； taken at random as circumstances supplied them； he met with aptitudes unequal； it is true； but easily developed and giving the impression that they exist normally； latent and inactive； at the bottom of every equine soul。 From the mathematical point of view； is the horse's subliminal consciousness then superior to man's？ Why not？ His whole subliminal being is probably superior to one； of greater range； younger； fresher； more alive and less heavy； since it is not incessantly attacked； coerced and humiliated by the intelligence which gnaws at it； stifles it； cloaks it and relegates it to a dark corner which neither light nor air can penetrate。 His subliminal consciousness is always present； always alert； ours is never there； is asleep at the bottom of a deserted well and needs exceptional operations； results and events before it can be drawn from its slumber and its unremembered deeps。 All this seems very extraordinary； but； in any case； we are here in the midst of the extraordinary； and this outlet is perhaps the least hazardous。 It is not a question； we must remember； of a cerebral operation； an intellectual performance； but of a gift of divination closely allied to other gifts of the same nature and the same origin which are not the peculiar attribute of man。 No observation； no experiment enables us； up to the present； to establish a difference between the subliminal of human beings and that of animals。 On the contrary； the as yet restricted number of actual cases reveals constant and striking analogies between the two。 In most of those arithmetical operations； be it noted； the subliminal of the horse behaves exactly like that of the medium in a rate of trance。 The horse readily reverses the figures of the solution； he replies； 〃37；〃 for instance； instead of 〃73；〃 which is a mediumistic phenomenon so well…known and so frequent that it has been styled 〃mirror…writing。〃 He makes mistakes fairly often in the most elementary additions； and subtractions and much less frequently in the extraction of the