In this simple manner he tried to get the horse to understand that the three belongs to the two, and that both together make five. The image of the five pins as it was known from previous experience, was to be associated with the combined groups of two and three, and conversely, it was to be reproduced when these groups were presented. Later the cloth and pins were omitted and the question was asked: "How much is two and three?". The horse tapped five times. It had learned how to add. Still this could be regarded only as a mechanical process, if the horse were able to add only those numbers which had been presented together one or more times in the manner just described. And so long as we remained within the first decade, we could get twenty-five binary combinations whose sum does not exceed 10 (counting inverted orders we would have forty-five binary permutations),--all of which might have been practised separately. But as a matter of fact, Mr. von Osten did not take this course, for as he himself says, he allowed Hans to discover a great deal for himself. "Hans had to develop the multiplication table for himself."--With larger numbers and more addends, the number of combinations becomes so great that there can be no doubt they were not practised separately. Since, after all this preliminary instruction, Hans really began to give solutions of new problems, the master believed that this was proof that he had succeeded in inculcating the inner meaning of the number concepts, and not merely an external association of memory images with certain movement responses. But he always remained within the sphere of the ideas thus developed, and adhered closely to the customary vocabulary and its usage. Every new concept, each additional word was explained anew.