1887.] 



317 



[Taylor. 



symbols of local value (designating only the orders of units), we may ex- 

 hil)it in a tabular form, a series of scales, with the successive increments 

 of value for each place, according to the radix, or ratio of geometrical 

 progression selected. In the following table the letter "U" in the top 

 line denotes a "unit;" that is, any figure which may occupy a single 

 place : 



Table of Arithmetical Scales. 



The most striking feature displayed by such a comparison of the differ- 

 ent scales is the rapid increase of value in the higher ratios, as compared 

 with the lower. While the ternary scale, for example, requires four fig- 

 ures to express so small a number as 27, the tricenary scale expresses one 

 thousand times as many, by the use of no more places. The very first 

 inquiry would, therefore, naturally be (in the absence of any other con- 

 sideration), which would be found more convenient — a very small radix, or 

 a very large one. 



The first and lowest scale of the series — the Unary — presents, with some 

 disadvantages, many very remarkable advantages. In the first place it 

 requires but a single figure, 1 (together with the cipher for determining 

 its place), to express with facility and precision all the values within the 

 reach of figures.* According to the law of the Hindoo notation, by 



* It was in reference to this curious property of the scale, that a medal .struck in honor 

 of Leibnitz, and to commemorate his invention of the binary system, bore on its re- 

 verse, the striking inscription : "Omnibus ex nihilo ducendis suflicit Unum." Unity 

 being very commonly regarded as the symbol of the Deity {Peacock's ArithincUc, Eucy- 

 clopcdia Metropolitaua, Vol. ii, page 392). 



