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The disadvantages also of this scale are many. It requires a multi- 

 plication-table for all numbers from one to sixteen. The mental labor 

 of calculating would, therefore, be increased. The number of symbols 

 required would not be quite so large, but the advantage from this source 

 would be slight. It may be noted, in passing, that in the Hindoo nota- 

 tion, the smaller the radix the greater is the number of symbols required 

 to express any number, but the easier the mental work of calculating. 

 The binary scale, which has the smallest possible radix, is an extreme 

 example under this rule. For instance, the lady who would be called 

 23 under the tonal system would have to confess to no less than 100011 

 summers were she living among people who counted with a binary 

 scale ! On the other hand, the larger the radix the less the number of 

 symbols required, but the greater the difficulty of computation. Thus 

 the tonal system expresses numbers more compendiously than the deci- 

 mal, but the difficulty of its many tables would make the use of it a 

 continual and severe strain upon the mind. 



Its author proposed also a tonal unit of linear, superficial, and cubical 

 measurement, a tonal watch, a tonal compass, tonal wet and dry meas- 

 ures, a tonal currency for the world, a tonal division of time, tonal 

 thermometers and barometers, and tonal postage-stamps. There is not 

 opportunity in this paper to describe these schemes. 



But other numbers might be used as radices, though most of them 

 will be found to be ill adapted to the purpose. The number three 

 would furnish a system which would possess no merits whatever. Its 

 scale would present only two digits, and the first ten numbers would 

 be 1, 2, 10, 11, 12, 20, 21, 22, 100, 101. But three is an odd number, 

 and the first bisection would result in an endless fraction. The same 

 is true of all systems in which odd numbers form the radix. 



The number four, however, would furnish a practicable scale. It is 

 a square, and can be bisected indefinitely without producing an odd 

 number except at unity. The notation would be simple, and the tables 

 of combinations easy to learn. Theoretical^, the scale would be an 

 excellent one, but calculations in it would require much manual labor, 

 and consequently be more tedious than similar computations in our 

 system. There would be three digits and the first ten numbers would 

 be 1, 2, 3, 10, 11, 12, 13, 20, 21, 22. 



The five scale, which is in use to a very limited extent among savage 



