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XXVI. Ne^ Researches on the true nature of the Boetian Con- 

 tractions^ especially with reference to the Explanatioyi given by 

 M. Chasles. By J. O. Halliwell, Esq., RR.S., F.S.A., 

 F.R.A.S., 4-c.* 



I HAVE the pleasure of placing before the readers of the 

 Philosophical Magazine a complete explanation of the first 

 tract in No. 343 of the Arundel MSS. in the British Museum ; 

 and that I have been able to accomplish this desideratum af- 

 fords me the greater gratification, because in so doing I am 

 fulfilling the wish of the patriarch of English Literaturef. 



The manuscript referred to, sometimes called the Mentz 

 Man?iscript, is a small quarto of the twelfth century, on vellum ; 

 and the first tract, entitled de Arte Numerandi, consists of four 

 leaves only, unfortunately being imperfect at the end. A frag- 

 ment from the recto of the first folio is lithographed in the 

 appendix to the Rara Mathematical which serves to show the 

 style of the manuscript and the forms and names of the con- 

 tractions. 



The treatise itself commences with an explanation of the 

 increasing value of igin, andras, &c., in the different abacal 

 compartments; in point of fact, a definition of abacal nume- 

 ration dependent upon the principle of local decimal value. 

 It is important to notice that, after this explanation, the com- 

 piler gives the usual definitions of digiti and ariiculi, clearly 

 showing by that his comprehension of their future value. It 

 is remarkable that everything stated is subservient to multi- 

 plication and division, no notice whatever being taken of addi- 

 tion, subtraction, duplation, or mediation ; — a plain proof, if 

 any were needed, that when the boundaries were abolished, 

 and when an attempt at a generalization of the local system 

 was made, artificial methods were adopted to come to the same 

 conclusions. Now I would ask M. Libri, or any one who 

 agrees with him, how he can possibly account for such a 

 clums}', primitive, yet most ingenious, method of avoiding 

 abacal difficulties, if we suppose that the writers of the thir- 

 teenth and following centuries derived their arithmetical know- 

 ledge direct from the Arabs ? 



And now for the modus operandi: and in order to render it 

 intelligible to every reader, let us take the first example in 

 multiplication, on account of its great simplicity : — 



" Sint ergo iiij. pedes equi, unusquisque habens vj. clavos." 



Arbas is to be placed in the lower part of the singular arc 



* Communicated by the Author. 



t Hallam's Introduction to the Literature of Europe during the Middle 

 Ages, vol. i. p. 151. 



