417 
The following are new instances of the appearance of the 
zero without its use :— 
MS. Hatton. 7, we find the following passage :—“ inscri- 
bitur ct in ultimo figura 0, sipos nomine. Que licet numerum 
nullum significet: tum ad alia quadam utilis est.” 
MS. Lansd. 842, is a contractive mark for a sipos, outside 
the drawing of the abacus. 
MS. Hatton. 112. The sipos is given with its contrac- 
tion, but is only used to fill up the space in the abacus. 
Now, at the last page of a very beautiful MS. of the 
translation of Euclid, by Athelard, of the fourteenth century, 
and in the eaplicit of the fifteen books, the number 15 is 
written in these singular contractions, and without a division. 
This MS. is in the Arundel Collection of MSS., and was ac- 
cidentally discovered by me when looking into it for an- 
other purpose. 
The new face thus put upon the question of their gradual 
identity with the present system, and the satisfactory evi- 
dence that the latter portion of my former conjecture is 
correct, is sufficient almost to make me bold enough to 
venture on the truth of the previous one. It must be recol- 
lected, however, that on the last point one document only 
has yet been discovered. 
II. 
The middle-age Knowledge of the Alabaldine Notation 
considered as an Argument in Favour of the early Introduc- 
tion of the Boetian Zero into Western Europe. 
I beg leave to make the following additional observations 
in corroboration of what was stated by me on the same subject 
in a paper read before the Academy on the 13th of January. 
The recent dispute* between M. Chasles and M. Libri, 
_ ® An account of the whole discussion may be found in the Comptes Rendus 
\\Hebdomadaires of the Academy of Sciences, for the 7th and 14th of October last, 
pp. 447-454, and pp- 463-472. 
