[44]. With, moreover, a “biological horizon.” See Vol. II, p. 34.
[45]. See Vol. II, pp. 327 et seq.
[46]. Also “thinking in money.” See Vol. II, pp. 603 et seq.
[47]. Dynasties I-VIII, or, effectively, I-VI. The Pyramid period coincides with Dynasties IV-VI. Cheops, Chephren and Mycerinus belong to the IV dynasty, under which also great water-control works were carried out between Abydos and the Fayum.—Tr.
[48]. As also those of law and of money. See Vol. II, pp. 68 et seq., pp. 616 et seq.
[49]. Poincaré in his Science et Méthode (Ch. III), searchingly analyses the “becoming” of one of his own mathematical discoveries. Each decisive stage in it bears “les mêmes caractères de brièveté, de soudaineté[soudaineté] et de certitude absolue” and in most cases this “certitude” was such that he merely registered the discovery and put off its working-out to any convenient season.—Tr.
[50]. One may be permitted to add that according to legend, both Hippasus who took to himself public credit for the discovery of a sphere of twelve pentagons, viz., the regular dodecahedron (regarded by the Pythagoreans as the quintessence—or æther—of a world of real tetrahedrons, octahedrons, icosahedrons and cubes), and Archytas the eighth successor of the Founder are reputed to have been drowned at sea. The pentagon from which this dodecahedron is derived, itself involves incommensurable numbers. The “pentagram” was the recognition badge of Pythagoreans and the ἄλογον (incommensurable) their special secret. It would be noted, too, that Pythagoreanism was popular till its initiates were found to be dealing in these alarming and subversive doctrines, and then they were suppressed and lynched—a persecution which suggests more than one deep analogy with certain heresy-suppressions of Western history. The English student may be referred to G. J. Allman, Greek Geometry from Thales to Euclid (Cambridge, 1889), and to his articles “Pythagoras,” “Philolaus” and “Archytas” in the Ency. Brit., XI Edition.—Tr.
[51]. Horace’s words (Odes I xi): “Tu ne quæsieris, scire nefas, quem mihi quem tibi finem di dederint, Leuconoë, nec Babylonios temptaris numeros ... carpe diem, quam minimum credula postero.”—Tr.
[52]. See Vol. II, pp. 11 et seq.
[53]. In the only writing of his that survives, indeed, Aristarchus maintains the geocentric view; it may be presumed therefore that it was only temporarily that he let himself be captivated by a hypothesis of the Chaldaean learning.