419 



Now scrupulus = — j-r-. scrupulus; because — j-r zz igin 



J , . scrupulus 



. . scrupulus zz cams . — — . 



calcts 



But ~ — zz siliqua .*. scrupulus = calcis . siliqua. 



\.^A^^^' '7' • scrupulus . . scrupulus 

 and 5^?we5 . stliqua =z *^?W25 . =^-: — zz semiuncia . ^^- 



z: semiuncia, " . . scrupulus, 

 calcis ^ 



But, by the integral notation, -^ = "f''""-'f^ = 



calcis andras . cat c^* 



2i^«w andras igin imfi 



—-7 — • — 7-:- = — 1 — . ■ . z= semiuncia . duella. 

 andras calcis andras orpiis 



Therefore, 



5^?wz5 . siliqua zz semiuncia . semiuncia . duella . scrupulus* 



We have now to reduce the 



semiuncia . semiuncia zz —^^7 — . — ^^ — =z -^^r — = sicilms, 

 andras andras arbas 



and, semis . siliqua z= 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.\ 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 lb. col. 30. 



