# Was Newton's method of finding derivatives of his fluents based on applying the chain rule?

Can Newton’s process for finding the derivative of y as a function of x resulting in $\frac{dy}{dx}$ be thought of as an application of the chain rule?

My thinking is: If y is a (continuous?) function of x then as a particle moves along the curve traced out, the particle has a velocity $\dot y (\frac{dy}{dt})$ in the y direction and a velocity $\dot x (\frac{dx}{dt})$ in the x direction. Hence the gradient function, $\frac{dy}{dx}$ is $\frac{\dot y}{\dot x} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dt}.\frac{dt}{dx} = \frac{dy}{dx}$ ?

Separately, was the terminology “fluent” used because Newton was interested in using (differential) calculus to model the motion of objects?

• Yes, Newton's terminology is derived from his ardent work on the foundations of mechanics – DanielC Feb 22 '18 at 22:38
• Agreed with Conifold's answer... especially for the "buts". See Newton, Two Treatises, page 1-2: "I consider mathematical quantities as described by a continued motion. [...] Fluxions are very nearly as the arguments of the fluents generated in equal but very small particles of time." 1/2 – Mauro ALLEGRANZA Feb 23 '18 at 10:09
• Thus, Newtons' symbolism: $x=\dot x o$ is: $dx= \dot x dt$ and thus when he computes $\dfrac {\dot y} {\dot x}$ he is computing $\dfrac {dy/dt}{dx/dt}$ but without chain rule. – Mauro ALLEGRANZA Feb 23 '18 at 10:13
• That's not really the chain rule. – Carl Witthoft Feb 23 '18 at 13:34

The answer is more of a yes, but with many buts. Newton did not have the modern concept of function, it was introduced by Dirichlet in the 19th century, or even its predecessor as assignment of values according to a "law", as Euler defined it in the 18th. Fluent was closer to what was later called variable quantity, something like a temporally changing ("evanescent") magnitude, and it was conceived on a kinematic model based on Aristotelian "fluid" continuum. Newton, of course, did not use the modern Dedekind-Cantor notion of the continuum composed of points either, his geometric line was not assembled from "real numbers". Here is from On the Quadrature of Curves:

"I don't here consider Mathematical Quantities as composed of Parts extremely small, but as generated by a continual motion. Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the Rotation of their Legs, Time by a continual flux, and so in the rest."

Fluxion was the "derivative" of a fluent only speaking anachronistically. Newton, of course, could not use predicates and nested quantifiers of modern analysis to define limits and then derivatives. He originally used infinitesimals in the definition, but grew dissatisfied with their haziness and unreliability, and replaced them in Principia with "first and last ratios". These were kinematic substitutes for limits, but the best he could describe with was just an analogy, to velocities, see What was the notion of limit that Newton used? Still, this allowed to connect them to geometry, which was on a much firmer footing than infinitesimals. Newton even uses the word "limit", but fluxions of evanescent quantities are no limits of functions. Here is from Book I of Principia:

"For by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be."

Since Newton did not have functions he could not have had compositions and the chain rule either. But he did have fluents traversing curves and those curves did have Cartesian equations like $$y=f(x)$$, and their tangents had slopes. Of course, Newton did not "apply" the "chain rule", that would have been circular even aside from the difference in concepts, but he did use a procedure somewhat like the one described in the OP to find slopes of curves using fluxions. The procedure was based on their analogy to velocities and geometry. Here is Friedman's description:

"We start with fluents or "flowing quantities" $$x, y$$, conceived as continuous functions of time. We can then form the fluxions or time-derivatives $$x, y$$, because continuously changing quantities obviously have well-defined instantaneous velocities or rates of change. If we are then given a curve or figure $$y = f(x)$$ generated by independent motions in rectangular coordinates of the fluents $$x, y$$ the derivative (slope of the tangent line) will be $$dy/dx = \dot{y}/\dot{x}$$ by the "parallelogram of velocities". Finally, we can recover the integral from the derivative via the Fundamental Theorem (which is also understood temporally). So all the basic notions of the calculus are explained without ever appealing to differentials or infinitely small quantities.

[...] Moreover, although the kinematic interpretation of the calculus certainly does not meet modern standards of rigor, it is also not afflicted with the obvious problems about consistency and coherence facing an interpretation based on differentials, infinitesimals, and infinitely small quantities. Indeed, when the kinematic interpretation was explicitly criticized by mathematicians like D'Alembert and l'Huilier in the late eighteenth century, it was not on grounds of coherence and consistency but because it was thought to import a "foreign" or "physical" element into pure mathematics."

• It's not true that Dirichlet gave the modern definition of function, nor is it true that Euler defined a function as an assignment of values according to a law. Instead, for both of these men (and basically for all mathematicians/physicist up to 1900) a function was a variable quantity that depended on another quantity. (Check their definitions.) So although they used the word function, I don't think that they had a fundamentally different notion than Newton. cont. – Michael Bächtold Feb 25 '18 at 16:24
• Hence a claim like "Since Newton did not have functions he could not have composition and the chain rule" would also apply to Leibniz, Euler, Lagrange and Dirichlet – Michael Bächtold Feb 25 '18 at 16:26
• Concerning the chain rule in times of Leibniz and Euler, I find the comment of the user NotNotLogical under this question relevant. – Michael Bächtold Feb 25 '18 at 17:49
• @MichaelBächtold Dirichlet wrote "If now a unique finite $y$ corresponding to each $x$, and moreover in such a way that when $x$ ranges continuously over the interval from $a$ to $b$, $y = f ( x )$ also varies continuously, then $y$ is called a continuous function of $x$ for this interval... it is not necessary that it be regarded as a dependence expressed using mathematical operations". There is a controversy as to whether this is exactly "modern", but it at least closer to it compared to Newton or Euler. – Conifold Feb 26 '18 at 20:31
• It's not modern in the sense that Dirichlet is neither calling the "correspondence" nor $f$ the function . What he calls a function is the variable $y$. I think there is no controversy about that. My personal interpretation is that the phrase "$y$ is a function of $x$" is synonymous with "$y$ is determined by $x$" or "$y$ depends on $x$". I don't know how Newton expressed this idea, but I suspect that he had a way of saying it. – Michael Bächtold Feb 27 '18 at 7:34

Both Newton and Liebniz used the notion of infinitesimals which Newton called fluxions. Given the philosopher, Bishop Berkleys acid criticisms of 'the ghosts of departed quantities' mathematicians began searching for a more rigorous approach culminating in Cauchy's definition of a limit. However, Liebniz already had this idea. Today, the pendulum has turned full circle with approaches much closer to Newton and Liebniz's notion of an infinitesimal now finding a rigorous foundation with synthetic differential geometry.

According to the Encyclopedia Brittanica, both Newton and Liebniz wete aware of the chain rule. However, what was essential was their understanding of infinitesimals as moments of change.