312 



♦ KNOWLEDGE ♦ 



[August 2, 1886. 



degree of proficiency will the exercise of tbis power no 

 longer prove fatiguing. 



In oi'der to practise this kind of counting, dots and small 

 circles were drawn on white paper squares. Some of these 

 dots were arranged symmetrically, others were irregularly 

 placed. These were glanced at for a moment, and proved 

 of considerable aid in acquiring the art. A good deal 

 depends on the arrangement of the dots. A card-player 

 will immediately, and without stopping to count, realise 

 that there are ten hearts on a ten-spot of that suit, but he 

 will not be able to give as coirectly the number of hearts or 

 of dots if these be arranged, for instance, in the form of a 

 cross. 



Hence it follows that it is not the symmetry of arrange- 

 ment that f icilitates 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 easj' 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 : — 



e s 



• •• 



Practice, which is nought but patient and correct repeti- 

 tion, will, however, even here make perfect. However, it 



may be regarded as proved by the case of Dase, before 

 referied to, that practice, however long continued, cannot 

 aid beyond a certain limit. It seems that, for the rapid 

 estimation or the unconscious counting of dots jilaced 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 gi'eatest 

 amount of practice, in which Dase for one certainly was not 

 wanting. 



However, this is not to say that more than thirty dots 

 cannot, under any circumstances, be simultaneously deter- 

 mined ; but in order that this may be done thej' must be 

 presented in some well-known manner of arrangement, 

 which must, as it were, have been fairly learned by heart. 

 Thus, very skilful card and domino players are able at a 

 glance to take in as many as forty points, in nines, tens, 

 fives, sixes, &c. This they do so rapidly as not to be 

 conscious of any addition. But 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 touch- 

 ing each one in turn with his finger. 



In oi'der to quickly attain the faculty of counting uncon- 

 sciously, a book may be used to advantage. If one takes a 

 book, opens the same — the eyes to be kept closed in the 

 meantime — and then casts a rapid glance at a part of the 

 page and tries to estimate how many lines are visible, this 

 way of doing, if often repeated and always tried on difi'erent 

 pages, will soon conduce to great accuracy in estimating. 

 A small child is not able to estimate even three lines 

 correctly, though looking at them for fully a second. 



As the mind develops, it acquires a more simple and rapid 

 process of counting. Something that at first had to be 

 undertaken slowly and with care, perhaps in separate stages, 

 may later on be accomplished much more quickly and with- 

 out requiring any special effort or calling for any great 

 amount of attention, in fact, almost " mechanically." 



One is fully conscious of every perfectly new impression 

 received b}' the brain ; hence the fascination of a novel idea. 

 The more the charm of novelty fades with the recurrence 

 of the same sensation, the less will consciousness be called 

 into play. 



An impression that seemed most startling when first 

 received may, if too often repeated, grow to be trivial. The 

 simple work of counting finally comes to be an unconscious 

 action of the nerve-fibres and cells of the brain. 



On newly-built roads the trains are run but slowly; the 

 longer such roads have been used the more rapidly are trains 

 run on them, and stops at way-stations are no longer 

 needed ; it is even thus with the trains of thought in the 

 human brain. 



And on this rests the practical importance of rapid 

 counting. Whoever can, unconsciously but correctly, 

 count up to twenty, or even up to twelve, has a great 

 advantage over others who cannot, without error, distinguish 

 six from seven in this manner. For such a one can turn 

 his consciousness to other matters, and greatly increase 

 his knowledge, where another would make but slow 

 progress. 



Those movements in man, which take place through some 

 impression received from without, and not aided by any con- 

 scious act of the brain (as, for instance, the contiacting of 

 the pujiil when a bright light strikes the eye), are termed 

 reflex actions. 



