Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his argument for authenticity appearing in the Revue Archéologique for 1856-1857, and in his treatise Les signes numéraux etc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numération décimale écrite qui fait usage de neuf chiffres prenant les valeurs de position," Comptes rendus, Vol. VI, pp. 678-680; "Sur l'origine de notre système de numération," Comptes rendus, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la géométrie de Boèce, relatif à un nouveau système de numération," in his work Aperçu historique sur l'origine et le devéloppement des méthodes en géométrie, of which the first edition appeared in 1837.
[339] J. L. Heiberg places the book in the eleventh century on philological grounds, Philologus, loc. cit.; Woepcke, in Propagation, p. 44; Blume, Lachmann, and Rudorff, Die Schriften der römischen Feldmesser, Berlin, 1848; Boeckh, De abaco graecorum, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn, Abhandlungen, Vol. II, p. 185, his Gerbert, pp. 1, 247, and his Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, Études, p. 17, Entstehung und Ausbreitung, p. 14; Nagl, Gerbert, p. 57; Bubnov, loc. cit. See also the discussion by Chasles, Halliwell, and Libri, in the Comptes rendus, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI, XVII of the same journal.
[340] J. Marquardt, La vie privée des Romains, Vol. II (French trans.), p. 505, Paris, 1893.
[341] In a Plimpton manuscript of the arithmetic of Boethius of the thirteenth century, for example, the Roman numerals are all replaced by the Arabic, and the same is true in the first printed edition of the book. (See Smith's Rara Arithmetica, pp. 434, 25-27.) D. E. Smith also copied from a manuscript of the arithmetic in the Laurentian library at Florence, of 1370, the following forms,
[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'arithmétique 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.]