416 



"It would be impossible, with the few materials yet 

 brought to light, to conjecture with any great probability 

 how far these Boetian contractions may have influenced the 

 introduction, or cooperated with the Arabic system to the 

 formation of our present numerical notation. It appears to 

 me highly probable that the two systems became united, 

 because the middle age forms of the figure Jive coincide 

 with the Boetian mark for the same numeral, and those of 

 two others are very similar. The idea of local position, 

 again, may have had an independent European origin ; the 

 inconveniences of the abacus on paper would have suggested 

 it by destroying the distinguishing boundaries, and invent- 

 ing an arbitrary hieroglyphic for the representation of an 

 empty square." — Rara Mathsmatica, p. 111. 



Now, from the Mentz MS. in the Arundel Collection, it is 

 very evident that their mode of operation with the abacus 

 had received one great improvement on the Greek and 

 Roman system, the abolishing the necessity of motion, and 

 only using the table cum pulvere, as a means for distinguish- 

 ing position. Thus, in the following addition sum, using the 

 names instead of the cabalistic characters, we have an ex- 

 ample of their method of proceeding: — 



Andras 



Igin 



Calcis 



Andras 



Tenis 



Arbas 



Ormis 



Celentis 



Celentis 



Calcis 





Igin 



Thus making a total of — celentis, calcis, zero, igin. Now, it 

 is evident, that in order to do away with the necessity of 

 this table, supposing the contractive marks again substituted 

 for the latter words, we have only to imagine an arbitrary 

 character for the deficiency under the ormis, and the modus 

 operandi is synonymous with our own present form. 



