[64]. Similarly, coinage and double-entry book-keeping play analogous parts in the money-thinking of the Classical and the Western Cultures respectively. See Vol. II, pp. 610 et seq.
[65]. The same may be said in the matter of Roman Law (see Vol. II, pp. 96 et seq.) and of coinage (see Vol. II, pp. 616 et seq.).
[66]. That is, “it is impossible to part a cube into two cubes, a biquadrate into two biquadrates, and generally any power above the square into two powers having the same exponent.” Fermat claimed to possess a proof of the proposition, but this has not been preserved, and no general proof has hitherto been obtained.—Tr.
[67]. Thus Bishop Berkeley’s Discourse addressed to an infidel mathematician (1735) shrewdly asked whether the mathematician were in a position to criticize the divine for proceeding on the basis of faith.—Tr.
[68]. From the savage conjuror with his naming-magic to the modern scientist who subjects things by attaching technical labels to them, the form has in no wise changed. See Vol. II, pp. 116 et seq., 322 et seq.
[69]. See Vol. II, pp. 137 et seq.
[70]. A beginning is now being made with the application of non-Euclidean geometries to astronomy. The hypothesis of curved space, closed but without limits, filled by the system of fixed stars on a radius of about 470,000,000 earth-distances, would lead to the hypothesis of a counter-image of the sun which to us appears as a star of medium brilliancy. (See translator’s footnote, p. 332.)
[71]. That only one parallel to a given straight line is possible through a given point—a proposition that is incapable of proof.
[72]. It is impossible to say, with certainty, how much of the Indian mathematics that we possess is old, i.e., before Buddha.
[73]. The technical difference (in German usage) between Grenz and Grenzwert is in most cases ignored in this translation as it is only the underlying conception of “number” common to both that concerns us. Grenz is the “limit” strictly speaking, i.e., the number a to which the terms a1₁, a2₂, a₃ ... of a particular series approximate more and more closely, till nearer to a than any assignable number whatever. The Grenzwert of a function, on the other hand, is the “limit” of the value which the function takes for a given value a of the variable x. These methods of reasoning and their derivatives enable solutions to be obtained for series such as (1⁄m¹,) (1⁄m²,) (1⁄m³,) ... (1⁄mx) or functions such as