I could multiply these examples indefinitely by drawing on the voluminous Elberfeld minutes, which, I may say in passing, have the convincing force of photographic records. All this, it must be agreed, is unexpected and disconcerting, had never been foreseen or suspected and may be regarded as one of the strangest prodigies, one of the most stupefying revelations that have taken place since man has dwelt in this world of riddles, Nevertheless, by reflecting, by comparing, by investigating, by regarding certain forgotten or neglected landmarks and starting-points, by taking into consideration the thousand imperceptible gradations between the greatest and the least, the highest and the lowest, it is still possible to explain, admit and understand. We can, if it comes to that, imagine that, in his secret self, in his tragic silence, our dog also makes similar remarks and reflections. Once again, the miraculous 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.