=208.= Mathematicians assume the right to choose, within the limits of logical contradiction, what path they please in reaching their results.--ADAMS, HENRY. _A Letter to American Teachers of History (Washington, 1910), Introduction, p. v._ =209.= Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself, will some day find it employed against himself.--HERBART, J. F. _Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 105._ =210.= Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics. --PEIRCE, BENJAMIN. _Linear Associative Algebra, American Journal of Mathematics, Vol. 4 (1881), p. 97._ =211.= Mathematics is a science continually expanding; and its growth, unlike some political and industrial events, is attended by universal acclamation.--WHITE, H. S. _Congress of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. 455._ =212.= Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skilful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures.--HERBART, J. F. _Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 101._