Mr. Halliwell on the Boetian and Arabic Numerals. 449 



lently improbable as to induce one to reject it on the grounds 

 of no immediate evidence*. 



Boetius immediately precedes the passage de ratione abaci 

 by definitions of numbers, digiti, arliculi, and composiii, — a 

 division which, I may observe, could only have been subser- 

 vient to the use of the Abacus, and the very same which 

 Alexander de Villa Dei and others, expressly using the Arabic 

 calculus, have employed in their works -j-. This is a strong 

 argument; and in MS. Trin. Coll. Cantab. R. xv. 16, we find 

 the Boetian digital forms identical with the middle- Arabic 

 with the exception of a very slight deviation in ornus, and 

 having arbas written on one side. 



I venture to exhibit the following conclusions, which I hope 

 will not be considered too premature. 



1. That the Boetian contractions are wholly independent 

 of the middle-Arabic in their introduction into Europe. 



2. That the Boetian contractions formed a distinct system 

 of decimal numeration, borrowed from the East, and intro- 

 duced through the Latins into Eastern Europe. 



3. That the Boetian notation was anterior to the introduc- 

 tion of the middle- Arabic numerals through Spain. 



Professor Davies, in a key to the new edition of Hutton's 

 Course of Mathematics, now on the eve of publication, has 

 given some entirely new views relative to the period of change 

 between the abacal and concentrated modes of operation, and 

 it is on that account that I defer entering into that part of 

 the subject, because his arguments are so forcible, so con- 

 clusive, and agree so well with an examination of early 

 manuscripts, that an abridgement would sensibly deteriorate 

 their value; moreover, as the work itself will ere long be so 

 accessible to every one, there can be no necessity. 



I hope shortly to be able to put in a form fit for publica- 

 tion some researches on the Leopoldine Numerical Contrac- 

 tions, which form a system of numeration that has hitherto 

 entirely escaped the researches of every writer on the history 

 of arithmetic. Thus within the space of a few months will a 

 completely new face be laid on the history of the Hindoo 

 arithmetic in Europe during the middle ages. 



* I entered into a full exjjlanation of the results of this conjecture in a 

 paper read before the Royal Society of Literature, but which is not yet 

 printed. 



f Rara Mathematica, p. 2, 74; Professor Peacock's History of Arith- 

 metic, in Encyc. Metrop. 



Phil. Mag. S. 3. Vol. 15. No. 98. Dec. 1839. 2 G 



