tinguish 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 numerals 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 together; 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 attentively 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 window-panes in a large house, was even more remarkable than the accuracy 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 improved 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 conviction 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 number 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 consciously count in these attempts; nor will it answer to attempt analysis from memory, after the objects are again hidden from view; all this would consume too much time. It is, in fact, necessary to do
