224 THE POPULAR SCIENCE MONTHLY. 



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 thesebe arranged, 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 arrange- 

 ment used. 



It is more difficult to correctly estimate the number of dots ar- 

 ranged 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 : 



Practice, which is naught 



% but patient and correct repe- 



_ q 9 tition, will, however, even here 



# - make perfect. However, it 



$9 # may be regarded as proved by 



the case of Dase, before re- 

 ferred to, that practice, how- 

 ever 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 ir- 

 regular 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 addi- 

 tion. 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 fingrer. 



In order to quickly attain the faculty of counting unconsciously, a 



