1887.] 'JU _ [Taylor. 



Newton in the invention of the " DlfTerential Calculus," proposed this 

 system and zealously urged its adoption ; although he thought that for 

 more common purposes it would be found too prolix. " De Lagny 

 took the trouble of constructing logarithms on the principles of this arith- 

 metic, as being more natural than those usually employed. * - * 

 He even proposed to substitute binary arithmetic for logarithms, afflrni- 

 ing that it was more simple and expeditious, and conducted to the object 

 in view in a less indirect manner" {Anderson' s ■ Article on Arithmetic, 

 in Brewster's Edinhurgli Encyclopedia, Vol. ii, pp. 376 and 409). The 

 same writer adds that " Dangicourt has applied the binary notation with 

 greater success to progressions, and proved that the laws of a series 

 may be detected by it more easily than by any other scale." This results, 

 it maybe as well to state, from the fact that " circulating periods " of 

 figures return far more frequently in this scale than in any other. 



The Ternary scale, although it is also a very simple scale, has nothing 

 else to recommend it ; being incapable of integral bisection, and having 

 very nearly the redundancy of the binary scale, without one of its advan- 

 tages. It may be regarded as one of the most objectionable of all the 

 scales ; and indeed none of the odd numbers could, for a moment, be 

 accepted as a suitable radix of notation. 



The Quaternary scale, as derived from the second power of the binary 

 scale, has many of its excellences. While it employs less than half the 

 number of digits, of the common or denary scale, to task the memory 

 and attention, it requires only about five places of figures, for three of the 

 latter. It combines, therefore, great simplicity of structure, with a mode- 

 rate range of notation, and would form a very convenient and practicable 

 system of numeration ; while it would furnish an admirable scale of 

 division for weights and measures of all kinds. It is said by Balbi, that 

 a very low and ignorant tribeof Indians in South America — the Guranos — 

 had names for only four digits, and that after counting these a second time 

 (to eight) they were unable to proceed any further. The correctness of 

 this account appears, however, to be exceedingly doubtful. It is remark- 

 able, too, that Aristotle mentions a tribe of Thrace as being unable to 

 count beyond four — a statement equally incredible. 



Tlie Quinary scale, whose notation would require ten places for seven 

 of the denary, has nothing to recommend it ; and yet from the accident 

 of man being afllicted with five fingers, it has generally formed the basis 

 of the scale in common use, and traces of it are to be found in perhaps a 

 majority of the nations of the earth. The numerals of Malay and Java 

 were anciently , for the most part, quinary, in subordination to the vice- 

 nary grouping. A trace of this system is also seen among the ancient 

 Greeks, in their word -s/x-a^soOui (to count by fives) ; as it is among the 

 Romans in their notation of numbers above 5, 15, etc. The Persian term 

 for "five" is pendj(t ; and pentcha signifies the expanded hand. Among 

 the South Sea Islanders, the inhabitants of New Caledonia and the Hebri- 

 des, as Avell as the barbarous tribes of Northeastern Asia, the quinary 



