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_: "