Page:Popular Science Monthly Volume 13.djvu/437

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of groups as he can express. But all his groups are of the first order, and consist of units. He must make great advance in intelligence before he can take the next logical step in counting, by grouping the groups of five and then indicating these groups of the second order by a symbolic act.

In nearly all instances the method of grouping connects itself with the number of fingers on one or both hands, or the number of fingers and toes. Classification by pairs is also common. This is the simplest method, and was probably the first that was used. It arose, without doubt, from the common use of the hands in separating and combining articles in pairs. But the bases found most commonly in use are five, ten, and twenty. So universal is the selection of these numbers, that systems founded upon them have been termed the natural systems. There can be no doubt that the use of them arose from the number of fingers and toes. But, as has been said, these systems are natural only in the sense that ignorance is natural. They originated among the most ignorant races, without alphabet or figures. They were selected in crude attempts by unlettered savages to count game, or the days as they passed. The fingers formed the most convenient counting-board, and were therefore used.

The number of the fingers upon one hand was probably used in counting before the device of using the number upon both hands was thought of. In many of the Oriental languages the name for five means also hand. Vestiges of a scale of five are found in the decimal systems of many countries.

But the quinary system usually passes into the decimal for numbers above twenty, and frequently at some higher point into a third system in which twenty is a basis. Some of the Celtic dialects present a strange mixture of the three. The French language shows the vicinary scale in parts of its notation, and the use of this scale is much more common than is usually supposed. The Greenlanders give to twenty a name which means "a man." Our word "score" is probably a vestige of this scale. Its use was at one time very common for numbers between sixty and one hundred, where a similar counting now obtains in French. There can be little doubt that our Teutonic ancestors formerly used the vicinary scale for a portion of their counting. There are other instances where the vicinary has preceded the decimal system; but there is no example where the twenty scale has been carried to groups of the second order. Usually, like the five scale, it has been superseded by the denary system, which is now universally used.

With devices for numeration, there have been developed different systems of notation. By these, the attempt has been made to express numbers by written signs or symbols. As a general rule, practical methods of numeration have preceded the use of written symbols. The different systems of notation which have been developed and used, exhibit different degrees of excellence. The Greek and the Roman