418 
before the French Institute, relative to the bearing of certain 
points in my Essay on the Boetian contractions, has not es- 
tablished a single important fact, save that the knowledge of 
local value was apparent in the integral abacal operations of 
these contractions. On the question of the period of the 
introduction of the Boetian zero, confessedly the most curious 
and difficult point to be established, none of the continental 
writers, M. Chasles, M. Libri, or M. Vincent, have ven- 
tured on more than random conjectures. 
The Boetian fractional notation, or the Alabaldine nota- 
tion,* was first explained in the above-mentioned Essay, 
previously to which no rational conjectures respecting them 
had been made. I amnow enabled to prove that this notation 
was not only recognized, but commonly employed throughout 
the middle ages. 
A passage at the end of the second book of “ Boetii Geo- 
metria,” de minutiis,} proves that the system was contempo- 
rary with that writer. Bede, in his Treatise on Arithmetic, 
has given a whole chapter to it. Next comes the Arundel 
MS. of the twelfth century, from which I am enabled to 
give a most exceedingly curious specimen of their modus 
operandi :— 
Question.§ It is required to multiply semis (zs) into 
siliqua (Ga): What is the result ? 
Sotution. Semis = as .semiuncia, but as = igin; there- 
fore, semis = igin.semiuncia = semiuncia; because igin 
is the Boetian unity. 
* So called from its presumed inventor. 
+ MS. Lansd. 842, B. &c. ¢ No. 343, in the British Museum. 
§ It is almost unnecessary to observe that this is much simplified and abridged 
from the MS, 
