Mr. Halliwell ow the Boetian and Arabic Numerals. 't4<7 



well as of siliceous sandstones and slates, which I refer to an 

 upper group of the Devonian rocks, and of these I have pre- 

 pared sections and details for the Dublin Geological Society. 

 The lower figure in the accompanying plate will, it is 

 hoped, sufficiently explain my views of the order of succes- 

 sion. 

 Dublin, October 23, 1839. 



LX VI I. On the Connexion between the Boetian a?id the Middle- 

 Arabic numerical Forms. By J. O. Halliwell, Fsq.t 

 F.R.S., F.S.A., c^c* 



IN the Comptes Rendus des Seances de V Academic des Sci- 

 ences for the 7th of October, M. Libri and M. Chasles 

 have commented at some length on certain portions of an 

 essay on the Boetian numerical contractions which I ventured 

 to place before the notice of the public some months ago. I 

 hasten to explain some opinions there given, which 1 regret 

 have been found to be expressed in too ambiguous terms to 

 give sufficient satisfaction to the inquiries of those zealous and 

 learned mathematicians. 



With respect to the Bodleian manuscripts, I believe I have 

 erred in asserting that in all three treatises local position is 

 clearly -pointed out, for in the tract of Berhelinus this is only 

 to be inferred ; but whatever doubt may arise in that case, I am 

 fully persuaded that in the first Hattonian manuscript (No. 7), 

 and in the treatise immediately preceding that of Berhelinus 

 in the second Hattonian MS. (No, 112), not only is local 

 position clearly pointed out, but there is quite as complete a 

 system of Arithmetic as we find in the treatises o'i Johannes 

 de Sacro-Bosco, or Alexander de Villa Dei. For, after all, 

 the arithmetic of those authors is nothing more than the sim- 

 plest adaptation of abacal principles, and every one knows 

 that in contemporary MSS. of those treatises, nnd even in 

 transcripts of as recent a period as the close of the 15th cen- 

 tury, all the marginal operative examples are given in the 

 abacal form, and only one step removed from the Arundel 

 manuscript, where a blank space is used instead of the sipos. 

 Again, even in the Arundel manuscript, which does not possess 

 one higher principle than the passage in Boetius, alt^iough 

 local position is not clearly pointed out, yet, as I have else- 

 where stated, the writer was evidently acquainted with the 

 decuple value which a digit receives by its dtuation on the left 

 of another^. 



* Communicated by the Author. 



t Appendix to a Life of Sir Samuel Murland, p. 26. 



