For the rest, much light is thrown on this question by Binet’s ingenious experiments. Their principal result may be summarised as follows.[28] A little girl of four does not know how to read or count; she has simply learnt a few figures and applies them exactly to one, two, or three objects; above this she gives chance names, say six or twelve, indifferently to four objects. If a group of fifteen counters, and another group of eighteen, of the same size, are thrown down on the table, without arranging them in heaps, she is quick to recognise the most numerous group. The two groups are then modified, adding now to the right, now to the left, but so that the ratio fourteen to eighteen is constant. In six attempts the reply is invariably exact. With the ratio seventeen-eighteen, the reply is correct eight times, wrong once. If, however, the groups are found with counters of unequal diameter, everything is altered. Some (green) measure two and one-half centimetres, others (white) measure four centimetres. Eighteen green counters are put on one side, fourteen white counters on the other. The child then makes a constant error, and takes the latter group to be the more numerous, and the group of fourteen may even be reduced to ten without altering her judgment. It is not until nine that the group of eighteen counters appear the more numerous. This fact can only be explained by supposing that the child appreciates by _space_, and not by number, by a perception of continuous and not by discontinuous size—a supposition which agrees with other experiments by the same author to the effect that, in the comparison of lines, children can appreciate differences of length. At this intellectual stage, numeration is accordingly very weak, and restricted to the narrowest limits. As soon as these are exceeded, the distribution between minus and plus rests, not upon any real numeration, but upon a difference of mass, felt in consciousness. In children, _reasoning_ prior to speech is, as with animals, practical, but well adapted to its ends. No child, if carefully watched, will fail to give proof of it. At seventeen months, Preyer’s child, which could not speak a word, finding that it was unable to reach a plaything placed above its reach in a cupboard, looked about to the right and left, found a small travelling trunk, took it, climbed up, and possessed itself of the desired object. If this act be attributed to imitation (although Preyer does not say this), it must be granted that it is imitation of a particular kind,—in no way comparable with a servile copy, with repetition pure and simple,—and that it contains an element of invention.