[338] The most noteworthy of these was for a long time Cantor (_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even believed that Pythagoras had known them. Cantor says (_Die roemischen Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein Codex bereits in Jahre 821 im Kloster Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu Mantua in die Haende Gerbert's gelangte, von welcher mannigfache Handschriften noch heute vorhanden sind." But against this opinion of the antiquity of MSS. containing these numerals is the important statement of P. Tannery, perhaps the most critical of modern historians of mathematics, that none exists earlier than the eleventh century. See also J. L. Heiberg in _Philologus, Zeitschrift f. d. klass. Altertum_, Vol. XLIII, p. 508. Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his argument for authenticity appearing in the _Revue Archeologique_ for 1856-1857, and in his treatise _Les signes numeraux_ etc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numeration decimale ecrite qui fait usage de neuf chiffres prenant les valeurs de position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre systeme de numeration," _Comptes rendus_, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la geometrie de Boece, relatif a un nouveau systeme de numeration," in his work _Apercu historique sur l'origine et le developpement des methodes en geometrie_, 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 roemischen 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 Einfuehrung der jetzigen Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, _Etudes_, 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 privee 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, [Forged numerals