Mr. Halli well's Nein) Researches onthe Boetian Contractions. 137 



(arcus singularis), and in the upper part of the same arc is 

 placed chalcns " quasi fundanientum 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 

 articiiU comes into operation, and the articidus of this number 

 (S^) is andras, which, by the principle of local position and of 

 no other, 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 

 banc regulam deceni; — Decenus quemcunque arcum multi- 

 plicat, in secundo ab eo pone digitum in ulterior! articulum," 

 fol. 2, r°, the reason of which is obvious. Thus, in the MS., 

 the operation for finding the square of twelve is as follows: — 





2 











6 



1 



2 



4 





1 



4 

 2 



4 

 4 



proceeding in a most complicated manner, but merely using 

 the simple formula 



m . (np) = mnp, 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 I'epeat 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, torn. i. pars ii. col. 13. "Quod abacistse facillimum 

 est." Ibid. col. 30. 



