418 



before the French Institute, relative to the bearing of certain 

 points in my Essay on the Boetian contractions, has not es- 

 tablished a single important fact, save that the knowledge of 

 local value was apparent in the integral abacal operations of 

 these contractions. On the question of the period of the 

 introduction of the Boetian zero, confessedly the most curious 

 and difficult point to be established, none of the continental 

 writers, M . Chasles, M. Libri, or M. Vincent, have ven- 

 tured on more than random conjectures. 



The Boetian fractional notation, or the Alabaldine nota- 

 tion,* was first explained in the above-mentioned Essay, 

 previously to which no rational conjectures respecting them 

 had been made. I am now enabled to prove that this notation 

 was not only recognized, but commonly employed thronghout 

 the middle ages. 



A passage at the end of the second book of ^' Boetii Geo- 

 metria," de minutiis,^ proves that the system was contempo- 

 rary with that writer. Bede, in his Treatise on Arithmetic, 

 has given a whole chapter to it. Next comes the Arundel 

 MS.J of the twelfth century, from which I am enabled to 

 give a most exceedingly curious specimen of their modus 

 operandi : — 



QuESTi0N.§ It is required to multiply semis (-^ into 

 siliqua (ttt) • What is the result ? 



Solution. Semis =z as . semiuncia, but as zz igin; there- 

 fore, semis zr i^in . semiuncia = semiuncia ; because igin 

 is the Boetian unity. 



• So called from its presumed inventor. 



t MS. Lansd. 842, B. &c. X No. 343, in the British Museum. 



§ It is almost unnecessary to observe that this is much simplified and abridged 

 from the MS, 



