222 THE POPULAR SCIENCE MONTHLY. 



tin<nrish three marbles from four, must first add each marble to the 

 other in this way many learn to count before knowing the numerals. 

 From this it follows that, in order to count, a knowledge of the nu- 

 merals is not a necessity ; even untrained deaf-mutes, who can neither 

 read nor write, are capable of counting, without figures, merely by the 

 aid of their fingers. 



From the action of a child who has learned the meaning of the 

 numerals, it furthermore follows that it is only by practice, that is by 

 oft-repeated counting of actual objects, that surety is gained in the art 

 of counting small numbers unconsciously. An idiot, or whoever does 

 not practice, can not count three without adding one by one, and will 

 never rise above the lowest plane of mental development. 



Now, however, as is well known, no one can tell in a moment how 

 many objects are lying before him, provided the number of these 

 objects is somewhat large approximates, say, fifty. Some persons can 

 count more rapidly than others ; a broker's apprentice will make 

 groups of three, of five, of ten coins, and then add the groups to- 

 gether ; the experienced money-broker is able to determine in a few 

 seconds what the amount is, and this, perhaps, without even touching 

 the coins. But he too, as well as every one else, must count atten- 

 tively as soon as the number of pieces exceeds a certain limit. But 

 what is this limit ? 



Dase, the well-known calculator, who died in 1861, stated that he 

 could distinguish some thirty objects of a similar nature in a single 

 moment as easily as other people can recognize three or four, and his 

 claim was often verified by tests. The rapidity with which he would 

 name the number of sheep in a herd, of books in a book-case, of win- 

 dow-panes in a large house, was even more remarkable than the accu- 

 racy with which he solved mentally the most difficult problems. Not 

 before or after his time has such perfection been attained ; but as 

 every one possesses this faculty to a small extent, and as it can be im- 

 proved by practice, it is not impossible that in future other experts in 

 this line may appear. The only trouble is that so few know how easy 

 it is to practice. 



In the first place, one can by a few trials readily gain the convic- 

 tion that, without practice, not every one can distinguish six and seven 

 objects as easily as three and four. 



In order to learn that it is a comparatively easy matter to estimate 

 up to six and seven, and then up to nine, as correctly as from three to 

 five, one need only make a few trials in guessing at an unknown num- 

 ber of matches or pins that are concealed beneath a sheet of paper, 

 and are then exposed to view but for a second. 



Great care must be exercised, however, that one does not con- 

 sciously count in these attempts ; nor will it answer to attempt analy- 

 sis from memory, after the objects are again hidden from view ; all 

 this would consume too much time. It is, in fact, necessary to do 



