D’Alembert.[62] They no longer understood the deep and complicated forms of the Classical thought, and their own were confused and new, far as yet from urban clarity and tidiness. Their Culture was in the _Gothic_ condition, as all Cultures have been in their youth—as even the Classical was in the early Doric period which is known to us now only by its Dipylon pottery. Only in Baghdad and in the 9th and 10th Centuries were the young ideas of the age of Diophantus carried through to completion by ripe masters of the calibre of Plato and Gauss. ----- Footnote 48: As also those of law and of money. See Vol. II, pp. 68 et seq., pp. 616 et seq. Footnote 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 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._ Footnote 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._ Footnote 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._ Footnote 52: See Vol. II, pp. 11 et seq. Footnote 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. Footnote 54: