Mr. Halli well’s ’Ne^BesearcJiesonthe Boetian Contractions, 137 
(arcus singularis), and in the upper part of the same arc is 
placed chalcus quasi fundamentum multiplicationis.” But 
in the actual multiplication recourse is had to the common 
Roman notation, and the result of the multiplication of arbas 
and chalcus in the singular arc is xxiiij. Then the system of 
articuli comes into operation, and the articuliis of this number 
(24?) is andras, which, by the principle of local position and of 
no othcr^ is placed in the decenal arc. Now I would ask M. 
Libri, in reply to every one of his arguments, how can we 
possibly suppose a rule of this nature with its full explanation 
to exist, without allowing its author to have possessed the 
knowledge of the value of local position ? The decenal arc is 
made use of in a simple but masterly manner, and the articulate 
system is invented to avoid the principal difficulty. The digit 
arbas, it is almost unnecessary to observe, is placed in the 
singular arc, and thus we have the complete number repre- 
sented. 
In higher numbers the centenal, millenal, and other arcs 
come into use. The following rule is a fair specimen of the 
methods employed : — 
Cum autem per decenum multiplicabis singularem, dices 
hanc regulam deceni ; — Decenus quemcunque arcum multi- 
plicat, in secundo ab eo pone digitum in ulterior! articulum,” 
fob 2, the reason of which is obvious. Thus, in the MS., 
the operation for finding the square of twelve is as follows: — 
proceeding in a most complicated manner, but merely using 
the simple formula 
m , [np) = m np, or, 12 x 12 = 12 X 2 x 6 = 24 x 6. 
in which latter case the above rule is applicable. This rule 
is afterwards generalized. 
‘‘ His patefactis, oculus mentis aperiatur ad subtilitatem 
divisionis;” but as the same system is carried out, precisely 
similar to the methods of Johannes de Sacro-Bosco, I do not 
consider it at all necessary to repeat them. 
Gerbert uses the Boetian fractional notation*, and I con- 
sider this fact a grand argument for his acquaintance with the 
Boetian contractions, if, indeed, the passage in the geometry 
* Pezii Thesaurus, tom. i. pars ii. col. 13. “ Quod abacistae facillimum 
est.” col.30. 
6 
1 
4 
4 
2 
4 
