[329] J. Keane, The Evolution of Geography, London, 1899, p. 38.
[330] The Arabs who lived in and about Mecca.
[331] S. Guyard, in Encyc. Brit., 9th ed., Vol. XVI, p. 597.
[332] Oppert, loc. cit., p. 29.
[333] "At non credendum est id in 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-géometrie de Boèce," 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 über 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 römischen 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 Hände 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.