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 deci- 

 mal, 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 bisec- 

 tion. 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 ad- 

 vocated 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 com- 

 pletely 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 sys- 

 tem 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 sim- 

 pler than those now in use. 1 The danger of error in computations is 



1 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 : 



