[342] Halliwell, in his _Rara Mathematica, _p. 107, states that the disputed passage is not in a manuscript belonging to Mr. Ames, nor in one at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It was the evident corruption of the texts in such editions of Boethius as those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish his work _Sur l'introduction de l'arithmetique indienne en Occident_. [343] They are found in none of the very ancient manuscripts, as, for example, in the ninth-century (?) codex in the Laurentian library which one of the authors has examined. It should be said, however, that the disputed passage was written after the arithmetic, for it contains a reference to that work. See the Friedlein ed., p. 397. [344] Smith, _Rara Arithmetica_, p. 66. [345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507. [346] "Nosse autem huius artis dispicientem, quid sint digiti, quid articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p. 395.] [347] _De ratione abaci._ In this he describes "quandam formulam, quam ob honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus appellabatur abacus." This, as pictured in the text, is the common Gerbert abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an ordinary multiplication table (sometimes called Pythagorean abacus) is given in the illustration. [348] "Habebant enim diverse formatos apices vel caracteres." See the reference to Gerbert on p. 117. [349] C. Henry, "Sur l'origine de quelques notations mathematiques," _Revue Archeologique_, 1879, derives these from the initial letters used as abbreviations for the names of the numerals, a theory that finds few supporters. [350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nuernberg, 1534, f. A4. In England it appeared in the earliest English arithmetical manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most vndirstonde that in this craft ben vsid teen figurys, as here bene writen for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde. Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor." See Smith, _An Early English Algorism_, loc. cit. [351] Friedlein ed., p. 397. [352] Carlsruhe codex of Gerlando. [353] Munich codex of Gerlando. [354] Carlsruhe codex of Bernelinus. [355] Munich codex of Bernelinus. [356] Turchill, c. 1200. [357] Anon. MS., thirteenth century, Alexandrian Library, Rome. [358] Twelfth-century Boethius, Friedlein, p. 396. [359] Vatican codex, tenth century, Boethius.