1887.] *^1^ [Taylor. 



two groups of five, than as a single group of ten" (Treatise on Arit7i- 

 metic. Book i, chap, i, p. 5-6). 



The gradual and successive development of these scales, is so well set 

 forth in Mr. Peacock's valuable treatise, that perhaps no apology is neces- 

 sary for a somewhat lengthened extract from it, even at the cost of some 

 repetition. 



"The decimal scale of numeration is not the only one which may be 

 properly characterized as a natural scale. In numbering with the fingers 

 we might, very naturally, pause at the completion of the fingers on one 

 hand ; and registering this result by a counter, or by any other means, 

 . we might proceed over the fingers of the same hand again, or with the 

 fingers of the second hand, and register the result by another counter, 

 or replace the former by a new counter which should become the rep- 

 resentative of ten. * * * Again, the scale of numeration by 

 twenties has its foundation iu nature, equally with the quinary and 

 denary scales. In a rude state of society, before the discovery of other 

 methods of numeration, men might avail themselves, for this purpose, 

 not merely of the fingers on the hands, but likewise of the toes of the 

 naked feet ; such a practice would naturally lead to the formation of a 

 vicenary scale of numeration, to which the denary, or the denary with 

 the quinary, or the quinary alone, might be subordinate. * * * 

 Of other systems of numeration, the binary might be considered as 

 natural, from the use of the two hands iu separating objects into pairs, 

 and from the prevalence of binary combinations in the members of the 

 human body ; but the scale of its superior units increases too slowly to 

 embrace within moderate limits the numbers which are required for the 

 ordinary wants of life, even in the infancy of society. * * * 



As the necessity of numeration is one of the earliest and most urgent of 

 those wants which are not essential to the support and protection of life, 

 we might naturally expect that the discovery of expedients for that pur- 

 pose should precede the epoch of civilization, and the full development 

 and fixing of language. That such has been the case, we shall find very 

 fully and clearly established, by an examination of the numerical words 

 of diflferent languages ; for, without any exception which can be well 

 authenticated, they have been formed upon regular principles, having 

 reference to some one of those three systems which we have character- 

 ized as natural ; the quinary scale, whenever any traces of it appear, 

 being generall}' subordinate to the denary, and, in some cases, both the 

 quinary and denary scales being subordinate to the vicenary. In some 

 cases, also, we shall find, from an examination of primitive numerical 

 words conveying traces of obsolete methods of numeration, that the 

 quinary, and even the vicenary scales have been superseded altogether 

 by the denary" (Encyclopedia Meti'opoUtana, art. "Arithmetic," Vol. i, 

 p. 371). 



Decimal arithmetic thus appears to be coeval and coextensive with the 

 human race. It is, indeed, perhaps, the most universal of human insti- 



PROC. AMER, PHILOS. SOC. XXIV. 12G. 2n. PRINTED NOV. 21, 1887. 



