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Updated: May 16, 2025
We are limited in the application of Mathematical Laws, even within the sphere of Pure Mathematics; general equations of the fifth degree having until recently resisted all attempts to solve them; and fields yet remain into which we cannot advance.
It is to Fourier, also, that we owe certain theorems, by means of which we may ascend from the differential equations to the integrals, and push the solutions in the majority of cases to the final numerical applications. The first memoir of Fourier on the theory of heat dates from the year 1807.
Equations, conic sections, differential calculus, constitute a skull and cross-bones to which I allow as wide a berth as possible." The weary dissatisfied expression of her large, luminous eyes, belied the sneer in her voice and the curl of her thin lip, and it cost her an effort to answer his next question.
And Du Bois-Reymond: "We can imagine the knowledge of nature arrived at a point where the universal process of the world might be represented by a single mathematical formula, by one immense system of simultaneous differential equations, from which could be deduced, for each moment, the position, direction, and velocity of every atom of the world."
Thus in all physical phenomena we should only have to regard the quantities of energy brought into play, and all the equations which link the phenomena to one another would have no meaning but when they apply to exchanges of energy. For energy alone can be common to all phenomena.
M. Duhem, however, arrives at the idea that the establishment of the equations of thermodynamics presupposes, among other hypotheses, one which is entirely arbitrary, namely: that when the state of the system is given, external actions capable of maintaining it in that state are determined without ambiguity, by equations termed conditions of equilibrium of the system.
In that superstitious age, when emperors and princes daily investigated the future, by alchemy, by astrology, and by books of fate, filled with formula; as gravely and precisely set forth as algebraical equations; when men of every class, from monarch to peasant, implicitly believed in supernatural portents and prophecies, it was not singular that a somewhat striking appearance, observed in the sky some weeks previously to the battle of Mookerheyde, should have inspired many persons with a shuddering sense of impending evil.
These equations, "actio the practical, passio = the theoretical function," are soon limited and modified, however. The natural appetites and affections are forms of volition, it is true, but not free products of the mind, for they take their origin in its connection with the body.
He took the sheet of paper, looked at one side, turned it over and looked at the other, and then raised his eyes to Dexter's, which avoided his gaze directly. "What is this?" he cried. "The equations, sir," said Dexter humbly. "Tut tut tut!" ejaculated Mr Limpney. "Was there ever such a boy? plus where it ought to be minus, and why, what's this!" "This, sir?" said Dexter. "Half-crowns."
Sheet after sheet of paper was filled with equations and calculations, and the table was covered with reference books. After two hours of intense study and hot discussion Crane's face took on a look of dawning comprehension, which changed to amazement and then to joy. For the first time in Seaton's long acquaintance with him, his habitual calm was broken.
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