*CURIOUS SYSTEMS OF NOTATION.*

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tribes, does not furnish a good system of notation, because five is an odd number. The first ten numbers would be 1, 2, 3, 4, 10, 11, 12, 13, 14, 20.

The six scale is theoretically superior in some respects to the decimal, because its radix can be divided by three, but it is objectionable for the same reason as the decimal. Its radix admits of only one bisection. Its notation would be somewhat simpler than that now in use, but more places of figures would be required. Its first ten numbers would be 1, 2, 3, 4, 5, 10, 11, 12, 13, 14.

Of the seven, nine, and eleven scales, it needs only to be said that they present no merits, since the numbers upon which they are founded are odd numbers.

None of the scales to which we have briefly referred have been advocated as practicable systems, but the duodenary or twelve scale has many striking advantages, and is used to some extent for certain classes of calculation. Its radix is divisible by two, three, four, and six. It can be bisected twice. The system has not only been the favorite with many who have theorized upon the subject, but it has been used to a great extent by different nations in the practical affairs of life. The Scandinavian nations have a preference for this scale. Traces of its use appear in our words *dozen, gross,* and *great gross*. It also appears in quite a number of the primary divisions in our weights and measures. Its use is quite common among mathematicians in long arithmetical computations. The additional mental labor required to compute in this scale is not very great, while the manual labor is somewhat less than in using the decimal system. The scale has always been a favorite one with those who object to the decimal notation.

The sexagenary system, founded upon the number sixty, deserves a passing mention on account of its historical interest. It was used for a long time by the Greeks in astronomical and other calculations. Our subdivisions of time and the circle are made with reference to it; but for practical operations it is very laborious and complicated.

The octonary system, founded upon the number eight, most completely presents the qualities which are desired in a system of notation. Eight is without doubt theoretically the best number of all to be used. It is a cube, and admits of indefinite subdivisions by halving. The system appears to have all the merits of the sixteen scale, while it avoids the disadvantages of a large radix. It is much easier to use than the decimal scale. It requires only seven digits. The figures 8 and 9 do not appear, and its tables of addition and multiplication are much simpler than those now in use.^{[1]} The danger of error in computations is

- ↑ The following multiplication-table, given by the author of an octonary scale, will show how simple must be the mental work of calculating in that system:
1 2 3 4 5 6 7 2 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6 11 . . . . . . . . . . . . . . . . . . . . . . . . 4 10 14 20 . . . . . . . . . . . . . . . . . . 5 12 17 24 31 . . . . . . . . . . . . 6 14 22 30 36 44 . . . . . . 7 16 25 34 43 52 61