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Proceedings of the Royal Society of Edinburgh. [Sess. 
and 900 respectively, appears to afford some evidence in favour of this 
hypothesis. But the later Greek system of alphabetic numeration was 
unquestionably artificial (whether it was suggested by the ultimate aerology 
of the Herodian system, or was due to other circumstances, does not 
concern us here), and its purposes were definite and obvious; while, on 
the other hand, it will hardly be maintained that the cumbrous and 
unadaptable Roman notation, beginning (as Canon Taylor himself admits) 
with the primitive ideographs I, V, X, was an artificial system ; or that,, 
had it been such, the Romans required to go beyond their own alphabet in 
order to obtain the few signs that were necessary for the completion of 
their system. It is not, of course, denied that a system whose lower values 
are represented by pictographic signs may be arbitrarily extended by the 
adoption of alphabetic or other signs to denote the higher values — -just as 
natives in various parts of the world have extended their own numerical 
systems by the arbitrary adoption of the English word thousand. But the 
acceptance of the pictographic method in explanation of the signs that 
represent the low&r values in any system would at least suggest the 
question : Can the whole system not be similarly accounted for on the 
pictographic method ? 
In regard to the higher values of the Roman system, the alphabetic 
explanation is confronted by a dilemma. For, if it be held that mille was 
the highest denomination of that system, one may inquire why additional 
alphabetic forms were not adopted to denote higher values, as might 
readily have been done if the notation were merely alphabetic and 
arbitrary, and as indeed had been done in the Greek Herodian system, 
where M.(vpioi) = 10,000. If, on the other hand, it be held that the Roman 
system did not stop at mille , one may ask why, on the alphabetic 
hypothesis, the higher values are variously represented by the cumbrous 
duplication of CD (as CD CD CD = 3000; CCDD = 10,000 ; etc.), or by the 
convenient but later devices V, X, M, instead of by a supplementary 
alphabetic character. In fact, the rigid and unadaptable Roman system,* 
* Next to the obscurity of its origin, the most remarkable thing about the Roman 
notation is its continuance and persistence without undergoing curtailment in respect of 
brevity, or evolution in respect of greater adaptability. The Arabian, Assyrian, and 
later Greek systems adopted the device of place-value ( e.g . 4000 ; K | ^ = 10 x 100, 
HIT ^ ]f = 1000 ; pv 7 = 153) ; even the Herodian system had its ingenious device of 
the circumscribing n (f 31 = 50, p 1 = 500, p 1 = 5000) ; while, on the other hand, the Roman 
system which existed contemporaneously and coextensively with the Empire is, if we may 
except the later and restricted employment of such forms as XY = 15,000, as inflexible and 
cumbrous as when first it appeared on the page of history. For mathematical purposes its 
