168 Proceedings of the Royal Society of Edinburgh. [Sess. 
“ Entstehung der romischen Zalilzeichen,” was published in the Sitzb. der 
k. Preussischen Akademie (1887, pp. 1011-28). His account is complicated 
by incidental discussion regarding the relative priority of the different 
forms of the symbols ; but his essential argument admits of brief and 
simple statement. He mentions that the words decussis, decussare, 
decussatio, decussatim occur in the works of Cicero, Vitruvius, Columella, 
and others; and he affirms that striking evidence for the existence of a 
principle of decussation is furnished by the following forms (“ Einen 
schlagenden Beweis dafiir, dass dies Princip des decussare wirklich bestand, 
bieten die folgenden Formen, welche sich bis in die romische Kaiserzeit 
hinein finden”): R=20, HI = 30, HR = 40. When the principle of 
decussation has been thus asserted, the next step in the logical, though 
not in the numerical, process is to apply it to the interpretation of the sign 
for 1000 ; because, as already stated, that sign is the crux of all hypotheses. 
Now M, as used by the Romans, was merely an abbreviation for mille 
or millia, and had no connection with the sign for 1000 (although, after 
the invention of printing, the M came to represent the numerical sign 
direct; because the sign CD, written hurriedly as (~Q, was, by the early 
printers, represented by the approximately similar form of the Gothic m, 
viz. and thus later by M). Zangemeister holds that the original sign 
was CO or CXO, and that the various other forms, oo and the later CD, 
were derived therefrom. Without being able to point to intermediate 
forms in its previous evolution, he considers that this sign OO arose from 
an earlier combination of decussated strokes, as X( = lxl0 x 10x10) = 
1000. The half of this sign CXh ot M, gives v? which gradually became 
assimilated to D ( = 500) ; in any case, he thinks, the sign D cannot be de- 
rived from CD ( = 1000), as the latter is relatively later. He asserts that 
the uncommon sign X may, with great probability, be taken as denoting 
100 ; and (on the analogy of = 1000) he interprets this sign X as 
being ( = 1 X 10 x 10), and supposes that the X gradually vanished, 
while the upright dash obligingly bowed to the receding X, until it finally 
became recognised as the C for which he was searching. The reason why 
the form X did not continue in use, but gave place to the C, was possibly 
the fact that an X ( = 10) could, by the addition of a single upright stroke, 
be falsified or altered to X ( = 100) (“Warum das X sich nicht dauernd im 
Gebrauch erhielt, sondern dem andern Zeichen weichen musste, dafiir lasst 
sich der Grund denken, dass ein X = 10 leicht durch Zufiigung eines 
senkrechten Striches zu X = 100 gefalscht werden konnte ”). From that 
doubtful and ultimately discarded form, however, he derives the sign for 
50, by the simple process of halving it, as V ; or again, as an alternative 
