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.
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.
The object of pure Physic[s] is the unfolding of the laws of the intelligible world; the object of pure Mathematic[s] that of unfolding the laws of human intelligence.
Number, place, and combination . . . the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred.
James Joseph Sylvester, Collected Mathematical Papers, Vol. 1 (1904), p. 91.
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.
"A plea for the mathematician", Nature, Vol. 1, p. 261.
* 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.
James Joseph Sylvester. "A Plea for the Mathematician, Nature," Vol. 1, p. 238; Collected Mathematical Papers, Vol. 2 (1908), pp. 655, 656.
To know him was to know one of the historic figures of all time, one of the immortals; and when he was really moved to speak, his eloquence equalled his genius.
G. B. Halsted, in F. Cajori's Teaching and History of Mathematics in the United States (Washington, 1890), p. 265.
Professor Sylvester's first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modern algebra. The attempt to lecture on this subject led him into new investigations in quantics.
Florian Cajori, in Teaching and History of Mathematics in the United States (Washington, 1890), p. 264.