APPENDIX. 109 



Lansdown collection (842) in the British Museum, is a 

 very beautiful MS. of the whole works of Boetius : what 

 renders it more interesting in the present inquiry, is the 

 contraction for the sipos without the drawing of an 

 abacus, which curiously illustrates the difficulty of the 

 transition from numerical operations, by means of that in- 

 strument, to local position without distinguishing boun- 

 daries. 



5. The Metz MS. in the Arundel collection, referred 

 to by Mr. Hallam, is probably an abridgment of one or 

 more extensive treatises on the subject : the author says, 

 quicquid ab abacistis excerpere potui compendiose collegi, 

 componens inde mihi certas regulas, que volentibus ad 

 hanc disciplinam attingere non inutiles. He quotes 

 Boetius. 



6. The following verses occur in a MS.* of the four- 

 teenth century on arithmetic : 



Unus adest igin ; andras duo ; tres reor armin ; 

 Quatuor est arbas ; et per quinque fore quinas ; 

 Sex calcis ; septem zenis ; octo zenienias ; 

 Novem celentis ; per deno sume priorem. 



And a list of the contractions is given on the preceding 

 page of the same volume. 



7. The fractional notation appears to be as curious as 

 the integral, but the contractions are not quite so arbi- 

 trary, and a regular system is evidently followed up 

 throughout. It is for the most part merely an adapta- 



* Bib. Trin. Coll. Cant, inter MSS. Gal. O. 2. 45. f. 33b. Chasles has 

 given verses to the same import, but he does not mention the source whence 

 he has obtained them. There is every reason to think that the sipos was a 

 later improvement, and the Metz MS. contains no allusion to it. Mr. Barn- 

 well, of the British Museum, informs me, that a short tract on celentis was 

 once pointed out to him in a MS. volume belonging to that establishment, 

 but no reference to it being in any of the catalogues, I have not been 

 fortunate enough to find it. 



