419 



_, , calcis , , calcis . . 



Mow scrupulus ~ — r-r. scrupulus; because — z—r zzimn 

 x calcis calcis 



. 7 . scrupulus 

 . . scrupulus = calcis . y-: — . 



calcis 



_ scrupulus ... . , 7 . .,. 



But =j—. — — sihqua . . scrupulus = calcis . siliqua. 



calcis 



, . scrupulus . . scrupulus 



axia semis . siliqua — semis . V - : — =: semiuncia . :-: — 



calcis calcis 



igin 7 



= semiuncia. " . . scrupulus. 

 calcis 



_, , , , . iWra andras . is^'w 



But, by the integral notation, — j-r = — : ^y-r = 



^ ° calcis andras .calcis 



igin andras igin igin . . , „ 



-^g — . — r - r - — —£-: — . . =z semiuncia . duella. 

 andras calcis andras ormis 



Therefore, 



se?m.s . siliqua ~ semiuncia . semiuncia . duella . scrupulus. 



We have now to reduce the 



semiuncia . semiuncia — 



igin igin igin 



andras ' andras ~ arbas ~~ 



and, semis . siliqua — sicilius . duella . scrupulus. 



We have thus the required quantity in a very compli- 

 cated form. To effect the reduction of this is the work of 

 another long operation, which it is unnecessary to repeat here, 

 as the above will sufficiently serve for an example of the la- 

 borious, though ingenious, plan which was pursued. 



Gerbert, in his Treatise on Geometry,* makes use of this 

 notation, and adds, quod abacistce facillimum est.f The MS. 

 Burney, 213, in the British Museum contains an express 

 treatise on it. Added to which, the Alabaldine contractions 

 are constantly occurring in mathematical manuscripts, and 

 till very lately no one had the slightest idea of their true 

 nature. 



Pezri Thesaurus, t. i. p. ii. col. 13. f Ib - col< 30, 



