n Autographis contigisse, aut vetustioribus
Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.]

[334] In _Observationes ad Pomponium Melam de situ orbis_. The question was
next taken up in a large way by Weidler, loc. cit., _De characteribus_
etc., 1727, and in _Spicilegium_ etc., 1755.

[335] The best edition of these works is that of G. Friedlein, _Anicii
Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de
institutione musica libri quinque. Accedit geometria quae fertur
Boetii_.... Leipzig.... MDCCCLXVII.

[336] See also P. Tannery, "Notes sur la pseudo-geometrie de Boece," in
_Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in
two books in which are mentioned the numerals. There is a manuscript of
this pseudo-geometry of the ninth century, but the earliest one of the
other work is of the eleventh century (Tannery), unless the Vatican codex
is of the tenth century as Friedlein (p. 372) asserts.

[337] Friedlein feels that it is partly spurious, but he says: "Eorum
librorum, quos Boetius de geometria scripsisse dicitur, investigare veram
inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface,
p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public
Instruction_, 1907, in an article of which a synopsis is given in the
_Jahrbuch ueber die Fortschritte der Mathematik_ for 1907, asserts that the
geometry was written in the eleventh century.

[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 auth