will immediately, and without stopping to count, realize that there are ten hearts on a ten-spot of that suit, but he will not be able to give as correctly the number of hearts or of dots if thesebearranged, for in-stance, in the form of a cross.
Hence it follows that it is not the symmetry of arrangement that facilitates the estimating, but acquaintance with the manner of arrangement used.
It is more difficult to correctly estimate the number of dots arranged in the form of a cross than to determine them if arranged as on cards and in similar ways.
It is more easy to estimate the dots if arranged as on dominoes; the dots must not be too small, and must be made a deep black on a white ground, or the reverse.
The estimation of the number of dots is most difficult if they are grouped in an irregular manner, as, for instance, in the following figures:
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Practice, which is naught but patient and correct repetition, will, however, even here make perfect. However, it may be regarded as proved by the case of Dase, before referred to, that practice, however long continued, can not aid beyond a certain limit. It seems that, for the rapid estimation or the unconscious counting of dots placed in unknown symmetrical arrangement, and for objects grouped into irregular forms, twenty is the limit.
Probably already, when the number of the objects exceeds twenty—undoubtedly, when it exceeds thirty—accuracy in estimating can no longer be attained, even after the greatest amount of practice, in which Dase for one certainly was not wanting.
However, this is not to say that more than thirty dots can not, under any circumstances, be simultaneously determined; but in order that this may be done they must be presented in some well-known manner of arrangement, which must, as it were, have been fairly learned by heart. Thus, very skillful card and domino players are able at a glance to take in as many as forty points, in nines, tens, fives, sixes, etc. This they do so rapidly as not to be conscious of any addition. Bat in such cases it is no longer the seeing of single dots, but seeing the pictures they form, which makes the feat possible. As no one on seeing the number 8 will count from one to eight, so no card-player will stop to count on seeing an eight of hearts, for instance. A child, however, not yet familiar with the appearance of cards, will count each heart separately, perhaps even touching each one in turn with his finger.
In order to quickly attain the faculty of counting unconsciously, a
