1887.] *^^^ [Taylor. 



time ; and the world llms put in possession of this grand invention eight 

 centuries earlier than it was by the Arabic importation.* 



In our survey of the principal scales, fnun which alone a selection 

 could be made for popular uses, we liave found that there are certain in- 

 cidental, but opposing advantages, incompatible with each other ; and 

 that no scale, therefore, could possibly furnish a maximum of every con- 

 dition that might be thought desirable. Thus the binary scale affords so 

 admirable a simplicity, beauty and facility, that it would have to be re- 

 garded the perfect system, if its redundant employment of figures (the 

 necessary consequence of its simplicity), did not render it unsuited to the 

 small and constant calculations required in the daily course of trade. On 

 the other hand the manifold divisions permitted by the sexagenary scale 

 give it convenient qualities, impossible to the lower scales ; but here we 

 find a complication so onerous that it would appall the most inveterate of 

 calculating monomaniacs. 



The conditions, however, that are really most essential to an arithmeti- 

 cal radix, are so few and precise, and their requirements so imperative, 

 that there is little difficulty in deciding upon "the best possible scale of 

 numeration." The first consideration would naturally have regard to the 

 size of the radix, in order to assign certain limits within which our scale 

 is to be found. To realize a maximum convenience, it must be neither too 

 large, nor too small. We have seen that while the notation of places (and 

 the consequent labor of transcription) diminishes very slowly with the 

 ascending scales — the tax upon the mental faculties increases in a far 

 more rapid ratio. The labor of mere calculation, which may be estimated 

 at zero for the binary scale, advances materially, and in a compound ratio 

 with every figure added to the radix. Were we then required to choose 

 between any two scales — separated by a considerable interval, that is, be- 

 tween a very small one and a very large one (no other insuperable objec- 

 tion being supposed), we should adopt, unhesitatingly, the smaller one. 

 The advantage imagined by some, of the great expressiveness of a rapid 

 increase of value, is wholly illusory. It needs comparatively very few 

 figures, in any case, to carry us not only beyond all true conceptions of 



*Thc Greeks, like the Hebrews, Arabs, and all other nations excepting the Hindoos, 

 employed an alpliabetic numeral ; and it is a somewhat curions circumstance that our 

 modern character for the cipher was derived not from India or Arabia, but from Alex- 

 andria. Tlie Hindoos indicated thecij)her place by a simple dot (.), and the Arabians, in 

 ■borrowing their system, did the same; until the sexagenary system, introduced by Ptolemy 

 so many centuries before, supplied tliem with a new character. This philosopher, find- 

 ing a frequent occasion to mark the absence of a particular denomination (as " no miu- 

 utes," or " no seconds "), in order to avoid mistake employed the first vacant letter of the 

 alphabet for that purpose. As the Greek numeral for 60 is the letter c, all those which 

 followed would be useless for the sexagenary scale ; Jieuce the next letter, o (omicron), 

 naturally became the empty counter. This notation became established by long habit 

 among the astronomers of Alexandria, Constantinople, and Arabia ; and tinally crept 

 into the IHndoo system of numerals. Thus to the accidental position of tlie Greek letter 

 omicron, which happened to represent seventy, we are indebted for the present form of 

 our modern cipher as a circle, instead of a decimal period. 



