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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 bringing these natural measures into commensurable relations. By sexagesimal divisions (60 being the first number divisible both by 10 and 12), the Babylonian 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 tendency 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 13-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 facilitated 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 facilities which duodecimal division gives. It needs only to introduce two additional 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 instead 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 numbering 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 predominant 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 ( shilling), the 6d. piece ( 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 gravitate into a system of doubling and again doubling and redoubling; and when until recently there existed the Ad. 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