"But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not." (Johann I Bernoulli, cca. 1691–1692
"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)
"It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum." (Joseph-Louis Lagrange, "Leçons sur le calcul des fonctions", 1806)
"It was long accepted as a fact that a metrical character could be described by means of a quadratic differential form, but the fact was not clearly understood. Riemann many years ago pointed out that the metrical groundform might, with equal right, essentially, be a homogeneous function of the fourth order in the differentials, or even a function built up in some other way. and that it need not even depend rationally on the differentials. But we dare not stop even at that point." (Hermann Weyl, "Space, Time, Matter", 1922)
"The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a linear transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem "linear" by reverting to infinitely small quantities." (Hermann Weyl, "Space - Time - Matter", 1922)
"Mathematicians call this combination [space and time] a quadratic form of the differentials of four variables, but we may call it more briefly, with Minkowski, ‘the Universe’." (Émile Borel, "Space and Time", 1926)
"The great advantage of infinitesimals in general and differentials in particular is that they make calculations easier. They provide shortcuts. They free the mind for more imaginative thought, just as algebra did for geometry in an earlier era. […] The only thing wrong with infinitesimals is that they don’t exist, at least not within the system of real numbers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not." (Johann I Bernoulli [attributed])
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