On Transformations ( - 1899)

"Given that annihilation of nature in its entirety is impossible, and that death and dissolution are not appropriate to the whole mass of this entire globe or star, from time to time, according to an established order, it is renewed, altered, changed, and transformed in all its parts." (Giordano Bruno, "The Ash Wednesday Supper", 1584)

"Concerning the theory of equations, I have tried to find out under what circumstances equations are solvable by radicals, which gave me the opportunity of investigating thoroughly, and describing, all transformations possible on an equation, even if it is the case that it is not solvable by radicals." (Évariste Galois, [letter to Auguste Chevalier] 1832)

"[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"[…] the quantities of heat which must be imparted to, or withdrawn from a changeable body are not the same, when these changes occur in a non-reversible manner, as they are when the same changes occur reversibly. In the second place, with each non-reversible change is associated an uncompensated transformation […] I propose to call the magnitude S the entropy of the body […] I have intentionally formed the word entropy so as to be as similar as possible to the word energy […]" (Rudolf Clausius, "The Mechanical Theory of Heat", 1867)

"The second fundamental theorem [the second law of thermodynamics], in the form which I have given to it, asserts that all transformations occurring in nature may take place in a certain direction, which I have assumed as positive, by themselves, that is, without compensation […] the entire condition of the universe must always continue to change in that first direction, and the universe must consequently approach incessantly a limiting condition. […] For every body two magnitudes have thereby presented themselves - the transformation value of its thermal content [the amount of inputted energy that is converted to 'work'], and its disgregation [separation or disintegration]; the sum of which constitutes its entropy." (Rudolf Clausius, "The Mechanical Theory of Heat", 1867)

"Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number." (George H Howison, "The Departments of Mathematics, and their Mutual Relations", Journal of Speculative Philosophy Vol. 5, No. 2, 1871)

"Instead of the points of a line, plane, space, or any manifold under investigation, we may use instead any figure contained within the manifold: a group of points, curve, surface, etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these figures should depend, the number of dimensions of the line, plane, space, etc. is likewise arbitrary and depends only on the choice of space element. But so long as we base our geometrical investigation on the same group of transformations, the geometrical content remains unchanged. That is, every theorem resulting from one choice of space element will also be a theorem under any other choice; only the arrangement and correlation of the theorems will be changed. The essential thing is thus the group of transformations; the number of dimensions to be assigned to a manifold is only of secondary importance." (Felix Klein, "A comparative review of recent researches in geometry", Bulletin of the American Mathematoical Society 2(10), 1893)

"In our century the conceptions substitution and substitution group, transformation and transformation group, operation and operation group, invariant, differential invariant and differential parameter, appear more and more clearly as the most important conceptions of mathematics." (Sophus Lie, Leipziger Berichte No. 47, 1896)