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“Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic 31 geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence.”

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James Joseph Sylvester

James Joseph Sylvester. "A Plea for the Mathematician, Nature," Vol. 1, p. 238; Collected Mathematical Papers, Vol. 2 (1908), pp. 655, 656.

“It has been said that to appreciate what virtue and morals mean, men must live virtuous and moral lives. It is equally true, that a knowledge of the objects of science is not to be attained by any scheme of definitions, however carefully contrived. He who would know what geometry is, must venture boldly into its depths and learn to think and feel as a geometer.”

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James Joseph Sylvester

J. J. Sylvester. "A Probationary Lecture on Geometry", Collected Mathematical Papers, Vol. 2 (1908), p. 9 https://babel.hathitrust.org/cgi/pt?id=miun.aas8085.0002.001;view=1up;seq=25

“It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock.”

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James Joseph Sylvester

James Joseph Sylvester, Collected Mathematical Papers, Vol. 2 (1908), p. 214.
Bigeometric Calculus: A System with a Scale-Free Derivative by Michael Grossman, p. 31.

“As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.”

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James Joseph Sylvester

"A plea for the mathematician", Nature, Vol. 1, p. 261.

“As a public teacher of mere striplings, I am often amazed by the facility and absence of resistance with which the principles of the infinitesimal calculus are accepted and assimilated by the present race of learners. When I was young, a boy of sixteen or seventeen who knew his infinitesimal calculus would have been almost pointed at in the streets as a prodigy, like Dante, as a man who had seen hell. Now-a-days, our Woolwich cadets at the same age, talk with glee of tangents and asymptotes and points of contrary flexure and discuss questions of double maxima and minima, or ballistic pendulums, or motion in a resisting medium, under the familiar and ignoble name of sums.”

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James Joseph Sylvester

J. J. Sylvester. "Additional Notes to Prof. Sylvester's Exeter British Association Address", Collected Mathematical Papers, Vol. 2 (1908), pp. 717–718 https://babel.hathitrust.org/cgi/pt?id=miun.aas8085.0002.001;view=1up;seq=732