CURIOUS SYSTEMS OF NOTATION. 421 



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 



