On Calculus (1850-1899)

"The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore, the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind." (James C Maxwell, 1850)

"[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values." (Auguste Comte, "Philosophy of Mathematics", 1851)

"The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the master keys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments." (Thomas Hill, "The Imagination in Mathematics", North American Review Vol. 85, 1857)

"There are no limits in mathematics, and those that assert there are, are infinite ruffians, ignorant, lying blackguards. There is no differential calculus, no Taylor's theorem, no calculus of variations, [...] in mathematics. There is no quackery whatever in mathematics." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma." (Henry J S Smith, Nature, Volume 8, 1873)

"The invention of a new symbol is a step in the advancement of civilisation. Why were the Greeks, in spite of their penetrating intelligence and their passionate pursuit of Science, unable to carry Mathematics farther than they did? and why, having formed the conception of the Method of Exhaustions, did they stop short of that of the Differential Calculus? It was because they had not the requisite symbols as means of expression. They had no Algebra. Nor was the place of this supplied by any other symbolical language sufficiently general and flexible; so that they were without the logical instruments necessary to construct the great instrument of the Calculus." (George H Lewes "Problems of Life and Mind", 1873)

"The rules of Arithmetic operate in Algebra; the logical operations supposed to be peculiar to Ideation operate in Sensation, There is but one Calculus, but one Logic; though for convenience we divide the one into Arithmetic the calculus of values, and Algebra the calculus of relations; the other into the Logic of Feeling and the Logic of Signs." (George H Lewes "Problems of Life and Mind", 1873)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

"If one looks at the different problems of the integral calculus which arise naturally when he wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, "Sur les Equations aux Dérivées Partielles de la Physique Mathématique", American Journal of Mathematics Vol. 12, 1890)

"Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define." (Giuseppe Peano, "Geometric Calculus", 1895)