Miscellaneous.

Progress, Volume VI, Issue 5, 1 March 1911, Page 575

Miscellaneous.

The World's Figures. . A Comparison of the Sextal and Decimal Scales of Notation. A Paper read before the Wellington Philosophical Society, on July 3, 1907, by C. W. Adams. (Abstract.) It is admitted by all who have studied the subject, that a duo-decimal notation would be much superior to the decimal notation at present in use, as 12 can be divided by 2,3, and 6, while 10 can be divided only by 2 and 5. But the duodecimal notation would require the addition of two new characters for 10 and 11, as 12 in the duo-decimal scale would, of course, be represented by '' 10,'' and would be called "ten," as in any scale of notation, the base is always represented by unity followed by 0. It has been ingeniously suggested that the two new characters (for 10 and 11) would not require the invention of any new forms, but could bo represented by two of our present figures turned upside down, in the same way that we at present represent 9 by a 6 turned upside down. The two figures selected would be 2 and 3 : A 2 reversed becomes 5, which is the first letter of " Sen," and 3 becomes 8, which is the first letter of "81even." This gets over all difficulty as regards symbols, but there is an insuperable difficulty in regard to nomenclature, which most writers on the subject seem to ignore. Firstly, we should have to invent new names for 10 and 11, as "ten" would be appropriated for "12"; and "eleven" and "twelve " would mean 12 + 1 and 12 +2, and so on up to "nineteen," which would mean 12 9; but what should we call 12 + 5 and 12 +8? and what should we call, 24 + 5 and 24 +B, and so on ? In fact we should have to invent new terms for two numbers in every twelve, in any consecutive numeration. Besides which the difficulties of the first four rules of arithmetic would be greatly increased. So the duo-decimal system of notation may be dismissed from our consideration. But in considering the claims of 6 as a base of notation, there are no fresh symbols to invent, nor any fresh nomenclature. We should merely have to discard the figures 6,7, 8 and 9, as all the figures required in the sextal rotation would be I, 2, 3. 4,5, and 0. These six figures would have the same names as at present, and 6 would be represented by unity followed by 0, and would be called "ten." So also 2 x 6 = 20, 3 x 6 = 30, 4 x 6 = 40, 5X6 = 50, and 6X6, or 6=(10)' 2 100. The counting from 1 to 100 in the sextal notation would embrace only 36 numbers. and would be as follows:—1, 2,3, 4,5, 101 11, 12, 13, .14, 15, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44; • 45, 50, 51, 52, 53, 54, 55, 100. Now just consider the wonderful simplicity of all calculations. The troubles of the multiplication table would entirely disappear. The little children could then master it in less than a quarter of the time now spent on it; as taking the present multiplication table of

144 squares, the new tables would comprise only 36 squares cut out of the easiest corner of the old table. Multiplication Table in the Sextal Notation.

In studying this table, it must be kept in mind that 10, 20, 30, 40, 50, and 100 in the sextal notation, are respectively equal to 6, 12, 18, 24, 30, and 36 in the decimal notation. In some games of cards, the smallest numbers are thrown out, retaining only those of the higher denominations. But in the sextal notation we use (excluding unity) only 2,3, 4,5, four small figures, while in the decimal notation we use double the number, including the figures 6,7, 8, 9. The sextal notation is not quite so concise as the decimal, as a number expressed by 4 figures in the decimal scale would, ordinarily require 5 figures in the sextal notation; but this slight want of conciseness would be more than counterbalanced by the simplicity and ease with which the calculations would be made. If the French had introduced this great reform in the 18th century, instead of their absurd metric system, they would have covered themselves with glory. It is not too late to make the change .even now. Both systems could be taught for a time in our schools, and when it was seen how much easier and simpler the new system was, the old one could be dropped, and the new one adopted.

1 2 3 4 5 10 2 4 10 12 14 20 3 10 13 20 23 30 4 12 20 24 32 40 5 14 23 32 41 50 10 20 30 40 50 100