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 et in ultimo figuraO, sipos nomine. Quae licet numeruni 

 nullum significet : tum ad alia quasdam utilis est." 



MS. Lansd. 842, is a contractive mark for a sipos, outside 

 the drawing of the abacus. 



MS. Hatton. 1 1 2. 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 explicit of the fifteen books, the number 1 5 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 Alahaldine Notation 

 considered as an Argument in Favour of the early Introduce 

 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. 



