2
$\begingroup$

When did derivative mean not only "slope of tangent" but also "instantaneous rate of change"?

Fermat was interested in minima and maxima, and realized these occur when the tangent has zero slope. When was it realized that this slope of tangent was the same thing as an instantaneous rate of change, i.e. velocity?

The post History of the derivative/tangent of a curve discusses the history of the tangent notion, but not the instantaneous rate of change notion.

$\endgroup$
3
  • 1
    $\begingroup$ I'm pretty sure that Newton's definition of a fluxion (derivative) was as the rate of change of the fluent (the changing quantity). For Newton, fluxion was synonymous with velocity. $\endgroup$
    – nwr
    Jun 12, 2023 at 16:57
  • $\begingroup$ Background: Sherry, "On Instantaneous Velocity" (1986) jstor.org/stable/27743785, on the shift from Aristotle in the Middle Ages (in particular by Oresme) that introduced a "language-game" that permitted discussion and analysis of instantaneous velocity, in particular under uniform acceleration. $\endgroup$
    – Michael E2
    Jun 12, 2023 at 17:44
  • $\begingroup$ The relation between tangency and velocities/rates of change was known long before derivatives. Archimedes uses it implicitly in drawing a tangent to the spiral, Oresme and Oxford calculators made it explicit for simple graphs, Galileo likely picked it up from them. His student Torricelli used it systematically to find velocities of projectiles with power equations of motion via tangents to higher parabolas. By Newton's time, the correspondence was well-established, see Boyer, History of Calculus, ch. IV. $\endgroup$
    – Conifold
    Jun 12, 2023 at 19:32

1 Answer 1

3
$\begingroup$

"Fermat was interested in minima and maxima, and realized these occur when the tangent has zero slope. When was it realized that this slope of tangent was the same thing as an instantaneous rate of change, i.e. velocity?"

As far as Fermat is concerned, it should be noted that he did not work with any notion of "slope" at all, as far as I am aware. He did realize that at minima and maxima of an expression (he didn't use functions, either), the change of the expression is only quadratic in the change $E$ in the variable.

While he of course found the tangent line for many curves, he did not think of them as representing the rate of change as we do today. Some recent publications on Fermat can be consulted at https://u.math.biu.ac.il/~katzmik/fermat.html

$\endgroup$
2
  • $\begingroup$ This is news to me! I looked up the references on adequality, thanks. So did Fermat connect minima/maxima/adequality to tangents in any way? Or was finding tangents a completely separate problem of interest? $\endgroup$ Jun 13, 2023 at 17:42
  • 2
    $\begingroup$ The technique of adequality was used by Fermat in a variety of contexts, including finding minima and maxima, finding tangents, finding centers of mass, etc. He was an unimaginable genius and well ahead his time. Lagrange, more than a century later, would consider him the founder of infinitesimal calculus. $\endgroup$ Jun 13, 2023 at 17:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.