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Home > Category Index for Science Quotations > Category Index I > Category: Integral Calculus

Integral Calculus Quotes (7 quotes)

I see with much pleasure that you are working on a large work on the integral Calculus [ ... ] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics.
Letter to S. F. Lacroix, 1792. Quoted in S. F. Lacroix, Traité du calcul differentiel et du calcul integral (1797), Vol. 1, xxiv, trans. Ivor Grattan-Guinness.
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I’m very good at integral and differential calculus,
I know the scientific names of beings animalculous:
In short, in matters vegetable, animal, and mineral,
I am the very model of a modern Major-General.
In The Pirates of Penzance (1879), Act 1.
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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 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”.
In 'A Plea for the Mathematician', Nature, 1, 238 in Collected Mathematical Papers, Vol. 2, 655-56.
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Please forget everything you have learned in school; for you haven’t learned it. … My daughters have been studying (chemistry) for several semesters already, think they have learned differential and integral calculus in school, and even today don’t know why x · y = y · x is true.
From 'Vorwort für den Lernenden', Grundlagen der Analysis (1930), vi. A later edition of the German text, published in the United States (1951, 1965), included Prefaces (only) translated into English by F. Steinhardt. In the latter text, the quote appears in English on pages 7-8, and in the original German on pages 15-16: “Bitte vergiß alles was Du auf der Schule gelernt hast; denn Du hast es nicht gelernt. … meine Töchter … schon mehrere Semester studieren (Chemie), schon auf der Schule Differential- und Integral-rechnung gelernt zu haben glauben und heute noch nicht, wissen, warum x · y = y · x ist.”
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Science is the Differential Calculus of the mind. Art the Integral Calculus; they may be beautiful when apart, but are greatest only when combined.
Quoted in The Complete Poems of Hugh MacDiarmid 1920-76 (1978), Vol. 2, 1360.
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Surely the claim of mathematics to take a place among the liberal arts must now be admitted as fully made good. Whether we look at the advances made in modern geometry, in modern integral calculus, or in modern algebra, in each of these three a free handling of the material employed is now possible, and an almost unlimited scope is left to the regulated play of fancy. It seems to me that the whole of aesthetic (so far as at present revealed) may be regarded as a scheme having four centres, which may be treated as the four apices of a tetrahedron, namely Epic, Music, Plastic, and Mathematic. There will be found a common plane to every three of these, outside of which lies the fourth; and through every two may be drawn a common axis opposite to the axis passing through the other two. So far is certain and demonstrable. I think it also possible that there is a centre of gravity to each set of three, and that the line joining each such centre with the outside apex will intersect in a common point the centre of gravity of the whole body of aesthetic; but what that centre is or must be I have not had time to think out.
In 'Proof of the Hitherto Undemonstrated Fundamental Theorem of Invariants', Collected Mathematical Papers (1909), Vol. 3, 123.
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The great artifice of regarding small deviations from the truth as being the truth itself is at the same time the foundation of wit, where the whole thing would often collapse if we were to regard these deviations in a spirit of philosophical rigor.
Aphorism from Georg Christoph Lichtenberg and R.J. Hollingdale (trans.) 'Notebook A: 1765-1770', The Waste Books (1990), 4. Also seen translated as, “The great trick of regarding small departures from the truth as the truth itself—on which is founded the entire integral calculus—is also the basis of our witty speculations, where the whole thing would often collapse if we considered the departures with philosophical rigour,” for example, as quoted in FractalVision: Put Fractals to Work For You (1992), 5, citing Aphorisms: 1764-1799.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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