enticity 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

[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