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simpler will be the operations performed under its use, but the 

 more complex or extended will be the expressions of the re- 

 sults, as is strikingly shown by the examples of the binary 

 scale, in which the decimal number 1810 is expressed by 

 11 1000 100 10, consisting of eleven digits : on the contrary, the 

 higher the radix, the heavier will become the operations, but 

 the results will be correspondingly simple ; as, for instance, in 

 the scale to radix 30 (which of course would require thirty 

 characters), the same decimal number 1810 would be ex- 

 pressed by 20x (where x stands for the digit ten), of three 

 digits only. But another consideration must be attended 

 to in selecting a radix : it must be a composite number, 

 involving some of the simple prime factors 2, 3, 5, etc., so as 

 to accommodate the greatest possible number of finite frac- 

 tions ; and therefore undoubtedly the number 30=2. 3. 5 

 would constitute a very perfect radix in this respect, but 

 would prove quite unmanageable in the actual performance 

 of the common arithmetical operations. Some number 

 greater than 2, and much less than 30, must therefore be 

 sought, which may combine a mean of the three advanta- 

 ges required ; and one of the first that strikes us as adapted 

 to our purpose is the number 6=2.3, which fulfils very 

 well our third condition, while the operations, to be per- 

 formed with only the six characters 1, 2, 3, 4, 5, 0, would 

 also be sufficiently easy ; but in the expression of the deci- 

 mal number 1810, namely, 12214 in this scale, we see an 

 inconvenient extension of the result of the operations, 

 which would rapidly accumulate with larger numbers, and 

 are thereby advised to pass the senary, and close at once 

 upon the denary scale, our cherished digitary number 

 10=2.5. Although 5 may not be quite so acceptable a 

 factor as 3, it certainly is preferable to 6=2. 3, which enters 

 into the number 12=2. 2. 3; a number that has frequently 

 been proposed as a substitute for 10, but which, as a radix, 

 thus unnecessarily loads us with useless repetition of the 

 factor 2, and at the same time renders the operatical pro- 

 Trans. viii.\ 10 



