Taylor.] <J^^ [Oct. 21, 



two additional integers, it would offer however a considerable increase 

 of complexity and mental labor ; while the economy of places in nota- 

 tion could scarcely be regarded as appreciable — 25 of the duodenary 

 being required for 27 of the denary. As compared with the octonary, 

 it would require 5 places, where the latter would require 6 ; so that 

 while its digits are more by fifty per cent, the excess of the other in 

 places is only twenty per cent. But there are far more important con- 

 siderations than these, whicli unfortunately opppse themselves to the 

 adoption of this system, as the best substitute for the denary, notwith- 

 standing its admitted features of superiority. 



The most fatal objection to the radix 12, is that it permits only a single 

 bisection beyond that given by the radix 10. The quality of continued 

 divisibility, we regard as paramount to all others ; not merely for the 

 convenience of art and trade, universal as their requirements are, but 

 even for many scientific purposes ; and however valuable the property 

 of a varied subdivision (as that furnished by the duodenary scale), ex- 

 perience has fully demonstrated, what is clearly seen by theory, that no 

 aliquot parts can ever be as widely useful as the binal fractions. An- 

 other objection to the 12 scale, somewhat allied to this, is that the num- 

 ber is not a power of any integer — a point, as we shall discover, of no 

 slight importance. In this respect, it may be remarked, the number nine 

 (awkward and inconvenient as it undoubtedly would be as the basis of 

 an arithmetic) would have several advantages over the number ten, 

 and even over the number twelve. A third objection to the scale under 

 consideration, which, though not so striking, is yet no less real : the radix 

 is too large. On the simple score of size, there must be somewhere in 

 the indefinite range of scales, a point where we should expect to find 

 the most convenient medium between the inexpediences of opposing diffi- 

 culties ; and although this most advisable limit of magnitude may not 

 admit of very precise determination, the question is one of too great con- 

 sequence in the comparisons we are making, not to deserve a special 

 attention. 



The Seni-denary scale presents many excellent points, the number 16 

 being both a square, and a fourth power, and admitting of indefinite 

 division by two. Its only disadvantage is the incommodious number of 

 digits it would require ; while its notation would yet economize only a 

 single place of figures in every six places required by the denary scale. 



The Vicenary scale furnishes no single point of merit which could rec- 

 ommend it to our acceptance, unless its divisibility by four should be 

 regarded as giving it a superiority to the denary. "With an exceedingly 

 troublesome and unwieldy range of digits, it would reduce the extent of 

 our common notation only from 13 to 10 places. Man was, however, un- 

 fortunately born with 20 extremities, or branches to his limbs, and hence 

 traces of what may be designated a rudimentary vicenary scale, are to be 

 met with among many nations, both ancient and modern. In ancient 

 Phoenicia and Palmyra, the system of numbering by twenties, as far as 



