[445] See Bubnov, loc. cit., p. x.
[446] Olleris, loc. cit., p. 361, l. 15, for Bernelinus; and Bubnov, loc. cit., p. 381, l. 4, for Richer.
[447] Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100. [Propagation, p. 246.] "Et prima quidem trium spaciorum superductio unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also Alfred Nagl, "Der arithmetische Tractat des Radulph von Laon" (Abhandlungen zur Geschichte der Mathematik, Vol. V, pp. 85-133), p. 97.
[448] Weissenborn, loc. cit., p. 239. When Olleris (Œuvres de Gerbert, Paris, 1867, p. cci) says, "C'est à lui et non point aux Arabes, que l'Europe doit son système et ses signes de numération," he exaggerates, since the evidence is all against his knowing the place value. Friedlein emphasizes this in the Zeitschrift für Mathematik und Physik, Vol. XII (1867), Literaturzeitung, p. 70: "Für das System unserer Numeration ist die Null das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er selbst schrieb alle Zahlen mit den römischen Zahlzeichen und man kann ihm also nicht verdanken, was er selbst nicht kannte."
[449] E.g., Chasles, Büdinger, Gerhardt, and Richer. So Martin (Recherches nouvelles etc.) believes that Gerbert received them from Boethius or his followers. See Woepcke, Propagation, p. 41.
[450] Büdinger, loc. cit., p. 10. Nevertheless, in Gerbert's time one Al-Manṣūr, governing Spain under the name of Hishām (976-1002), called from the Orient Al-Beġānī to teach his son, so that scholars were recognized. [Picavet, p. 36.]
[451] Weissenborn, loc. cit., p. 235.
[452] Ibid., p. 234.
[453] These letters, of the period 983-997, were edited by Havet, loc. cit., and, less completely, by Olleris, loc. cit. Those touching mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.
[454] He published it in the Monumenta Germaniae historica, "Scriptores," Vol. III, and at least three other editions have since appeared, viz. those by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877.