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VIII. A few Observations on the Authenticity of the PasS' 

 age in the Treatise of Boetius de Geometria on 'Numerical 

 Contractions. By J. O. Halliwell, Esq.^ F.R.S.f F.S.A., 

 F.R.A.S. 4V.* 



T VENTURE to add the following remarks to what I have 

 previously written f, in consequence of a postscript by 

 M. Libri, who wishes for more substantial evidence on the 

 point of authenticity than has hitherto been produced. Such 

 crude arguments as I am able to furnish must not by any 

 means be considered as the result of a strict or lengthened 

 inquiry, but are rather intended to show that the question is 

 well worthy of much greater attention than has yet been paid 

 to it. 



M. Libri was the first who conjectured that this passage 

 might be an interpolation, and with some justice; for it may 

 be reasonably asked, why does not Boetius allude to the new 

 system in his treatise on arithmetic. Again, from the abacal 

 system employed in that treatise, I should be inclined to think 

 that the articulate and composite divisions were certainly not 

 introduced until after that period. 



In the library of Trinity College, Cambridge, there is a 

 very beautiful quarto MS. of the elevefith cetitury on vellum, 

 containing the treatise of Boetius de Geometria plentifully il- 

 lustrated with neat diagrams. This manuscript, one of the 

 most ancient in this country, does not contain the disputed 

 passage. M. Libri has challenged me to produce such an 

 evidence; accipe si vis — the manuscript may be found under 

 the press mark R. xv. 14, and is briefly mentioned at p. 99 

 of Bernard's Catalogue, No. 491. 



It would, perhaps, be scarcely fair to make its extreme sin- 

 gularity an argument against its authenticity, but we may 

 be permitted to argue on the probability or improbability of 

 such a passage being written at so early a period. Is it likely, 

 that at a time when the arithmetic of the West was a mere 

 geometrical adaptation of quantity, when Boetius himself re- 

 cognised the Roman abacal system, and when it required no 

 inconsiderable depth of foresight to appreciate the advantage 

 of an arbitrary system of digital characters, that Boetius 

 would have inserted so extended an innovation in a treatise 

 written expressly on a science that has no immediate relation 

 with that into which the improvement was introduced? At 

 any rate it is a fair subject for discussion, whether we could 

 reasonably suppose any writer fully acquainted with the merits 



* Communicated by the Author. 

 t Seep. 447 of the present vohime. 

 E2 



