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XXVI. Ne"(S) ^Researches on the true nature of the Boetian Con-- 
tractions^ especially ^ith reference to the Ex plantation ginen hy 
M. Chasles. By J. O. Halliwell, Esq., F.R.S., F.S.A., 
F.R.A.S., ^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 1 am 
fulfilling the wish of the patriarch of English Literaturef. 
The manuscript referred to, sometimes called the Mentz 
Manuscript, 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 Ran^a Mathematica, 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 articidi, 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 
clumsy, 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 Haliam’s Introduction to the Literature of Europe during the Middle 
Ages, vol. i. p. 151. 
