[720]. It is not only a decided preference for synthesis and a complete denial of general methods which characterizes the ancient mathematics as against our newer science [modern mathematics]: besides this external formal difference there is another real, more deeply seated, contrast, which arises from the different attitudes which the two assumed relative to the use of the concept of variability. For while the ancients, on account of considerations which had been transmitted to them from the philosophic school of the Eleatics, never employed the concept of motion, the spatial expression for variability, in their rigorous system, and made incidental use of it only in the treatment of phonoromically generated curves, modern geometry dates from the instant that Descartes left the purely algebraic treatment of equations and proceeded to investigate the variations which an algebraic expression undergoes when one of its variables assumes a continuous succession of values.—Hankel, Hermann.
Untersuchungen über die unendlich oft oszillierenden und unstetigen Functionen; Ostwald’s Klassiker der exacten Wissenschaften, No. 153, pp. 44-45.
[721]. Without doubt one of the most characteristic features of mathematics in the last century is the systematic and universal use of the complex variable. Most of its great theories received invaluable aid from it, and many owe their very existence to it.—Pierpont, J.
History of Mathematics in the Nineteenth Century; Congress of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. 474.
[722]. The notion, which is really the fundamental one (and I cannot too strongly emphasise the assertion), underlying and pervading the whole of modern analysis and geometry, is that of imaginary magnitude in analysis and of imaginary space in geometry.—Cayley, Arthur.
Presidential Address; Collected Works, Vol. 11, p. 434.
[723]. The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast.—Russell, Bertrand.
The Study of Mathematics; Philosophical Essays (London, 1910), p. 77.
[724]. Induction and analogy are the special characteristics of modern mathematics, in which theorems have given place to theories and no truth is regarded otherwise than as a link in an infinite chain. “Omne exit in infinitum” is their favorite motto and accepted axiom.—Sylvester, J. J.