Giordano Bruno (born 1548, burned for heresy 1600). His whole life might be expressed as a crusade on behalf of God and the Copernican universe against a degenerated orthodoxy and an Aristotelian world- idea long coagulated in death.—_Tr._ Footnote 55: F. Strunz, _Gesch. d. Naturwiss. im Mittelalter_ (1910), p. 90. Footnote 56: In the “Psammites,” or “Arenarius,” Archimedes framed a numerical notation which was to be capable of expressing the number of grains of sand _in a sphere of the size of our universe_.—_Tr._ Footnote 57: This, for which the ground had been prepared by Eudoxus, was employed for calculating the volume of pyramids and cones: “the means whereby the Greeks were able _to evade_ the forbidden notion of infinity” (Heiberg, _Naturwiss. u. Math. i. Klass. Alter._ [1912], p. 27). Footnote 58: Dr. Anster’s translation.—_Tr._ Footnote 59: See Vol. II, Chapter III. Footnote 60: Oresme was, equally, prelate, church reformer, scholar, scientist and economist—the very type of the philosopher-leader.—_Tr._ Footnote 61: Oresme in his _Latitudines Formarum_ used ordinate and abscissa, not indeed to specify numerically, but certainly to describe, change, i.e., fundamentally, to express functions.—_Tr._ Footnote 62: Alexandria ceased to be a world-city in the second century A.D. and became a collection of houses left over from the Classical civilization which harboured a primitive population of quite different spiritual constitution. See Vol. II, pp. 122 et seq. ----- VIII The decisive act of Descartes, whose geometry appeared in 1637, consisted not in the introduction of a new method or idea in the domain of traditional geometry (as we are so frequently told), but in the definitive conception of _a new number-idea_, which conception was expressed in the emancipation of geometry from servitude to optically- realizable constructions and to measured and measurable lines generally. With that, the analysis of the infinite became a fact. The rigid, so- called Cartesian, system of co-ordinates—a semi-Euclidean method of ideally representing measurable magnitudes—had long been known (witness Oresme) and regarded as of high importance, and when we get to the bottom of Descartes’ thought we find that what he did was not to round off the system but to overcome it. Its last historic representative was Descartes’ contemporary Fermat.[63]