i 9 4 POPULAR SCIENCE MONTHLY. 



nite multiples one of another, and where they were approximate multiples 

 the numbers of these were irregular would not conform to any system. 

 But there early began, as among the Chaldeans, arrangements for bring- 

 ing these natural measures into commensurable relations. By sexagesimal 

 divisions (60 being the first number divisible both by 10 and 12), the Baby- 

 lonian cubit was brought into relation with the Babylonian foot. The 

 stages of change from nation to nation and from age to age can not, of 

 course, be traced, but it suffices to recognize the fact that the tendeucy has 

 been toward systems of easily-divisible quantities the avoirdupois pound 

 of 16 ounces, for instance, which is divisible into halves, into quarters, into 

 eighths. But, above all, men have gravitated toward a 12-division, because 

 12 is more divisible into aliquot parts than any other number halves, 

 quarters, thirds, sixths and their reason for having' in so many cases 

 adopted the duodecimal division is that this divisibility has greatly facili- 

 tated their transactions. When counting by twelves instead of by tens, 

 they have been in far fewer cases troubled by fragmentary numbers. 

 There has been an economy of time and mental effort. These practical 

 advantages are of greater importance than the advantages of theoretical 

 completeness. Thus, even were there no means of combining the benefits 

 achieved by a method like that of decimals with the benefits achieved by 

 duodecimal division, it would still be a question whether the benefits of the 

 one with its evils were or were not to be preferred to the benefits of the 

 other with its evils a question to be carefully considered before making 

 any change. 



But now the important fact, at present ignored, and to which I draw 

 your attention, is that it is perfectly possible to have all the facilities which 

 a method of notation like that of decimals gives, along with all the facili- 

 ties which duodecimal division gives. It needs only to introduce two addi- 

 tional digits for 10 and 11 to unite the advantages of both systems. The 

 methods of calculation which now go along with the decimal system of 

 numeration would be equally available were 12 made the basic number in- 

 stead of 10. In consequence of the association of ideas established in them 

 in early days and perpetually repeated throughout life, nearly all people 

 suppose that there is something natural in a method of calculation by tens 

 and compoundings of tens. But I need hardly say that this current notion 

 is utterly baseless. The existing system has resulted from the fact that we 

 have five fingers on each hand. If we had had six on each there would 

 never have been any trouble. No man would ever have dreamt of num- 

 bering by tens, and the advantages of duodecimal division with a mode of 

 calculation like that of decimals would have come as a matter of course. 



Even while writing I am still more struck with the way in which pre- 

 dominant needs have affected our usages. Take our coinage as an example. 

 Beginning at the bottom we have the farthing ( penny), the halfpenny 

 and penny (or one-twelfth of a shilling); next we have the threepenny 

 piece (i shilling), the Qd. piece (i shilling), and the shilling; and then 

 above them we have the eighth of a pound (2s. 6d.), the quarter of a pound 

 (5s.), and half pound (10s.). That is to say, daily usage has made us gravi- 

 tate into a system of doubling and again doubling and redoubling; and 

 when until recently there existed the 4d. piece we had the convenience of 

 a third as well as a half and a quarter" a convenience which would have 

 been retained but for the likeness of the 3d. and 4d. coins. And observe 



