This chapter was planned as an argument that Leibniz's notation composes and Newton's does not. Both texts are now held, and the argument does not survive them. Newton's method composes too, and his own worked example shows it. What separates the two notations is something smaller and more useful: what each one asks you to name. That difference is the one that later separated forward-mode from reverse-mode differentiation.
The first printed calculus is six pages in the Acta Eruditorum, reproduced in facsimile by William Dunham (p. 22). Its title promises “a new method for maxima and minima, and tangents, which is impeded neither by fractional nor irrational quantities, and a remarkable kind of calculus for them.” It begins by making dx ordinary: “Jam recta aliqua pro arbitrio assumta vocetur dx” — let some straight line, taken at will, be called dx. Then dv is defined as the line that is to dx as v is to the subtangent. In 1684 the differentials are finite lines in a fixed proportion, and the rules of the calculus follow on the same page CITED.
Two sentences after the product rule comes the one this volume is about:
That is the chain rule's license, stated as a rule of notation. A letter may stand for a formula, and every letter has a differential that obeys the same rules. So d can be pushed through any nesting of formulas, and no variable is privileged, not even x. The same passage adds a warning that Chapter 2's corner and every integration course confirm: there is not always a way back from a differential equation, “except with a certain caution, of which elsewhere.”
Newton's Method of Fluxions was written around 1671 and printed in John Colson's English translation of 1736. It sets up a different picture. Quantities are fluents, generated “as if … by continual Increase, after the manner of a Space, which a Body or Thing in Motion describes”. Their velocities of increase are fluxions, written with the same letters “pointed thus” — v̇, ẋ, ẏ, ż (§60, p. 20). One of the quantities, flowing equably, is chosen to play the part of Time (§59) CITED.
Problem I (p. 21) then gives a rule: from an equation relating the fluents, find the relation of their fluxions. The answer comes out as a proportion, for example ẋ : ẏ :: 3y² − ax : 3x² − 2ax + ay. The justification (§§13–18, pp. 24–25) writes each moment as a velocity times an indefinitely small o. It then drops the terms still multiplied by o, because they “will be nothing in respect of the rest. Therefore I reject them.” That is Euler's move of Chapter 4, eighty years earlier.
The volume index claimed that Leibniz's notation composes and Newton's does not. Newton's Example 5 (§12, p. 24) refutes the second half. To handle a curve defined through an intermediate quantity, he introduces an auxiliary fluent z with its own relation. He applies the rule to both relations and then substitutes, “this Value instead of it”. That is composition, done by adding a variable and eliminating it CITED.
The script runs one composite, y = (x² + 1)³ at x = 1.2, both ways:
| procedure | steps | dy/dx |
|---|---|---|
| Leibniz: let z stand for x² + 1; dy = 3z²dz, dz = 2x dx | two substitutions | 42.865920 |
| Newton: fluents y − z³ = 0, z − x² − 1 = 0; Prob. I on each; eliminate | two fluxion equations | 42.865920 |
They agree, and a central difference agrees with both COMPUTED. The work is the same size. The index thesis is withdrawn as stated, and the index now says so. Both notations compose. The difference is what each one makes you name.
Newton names a flow. Every quantity carries its velocity with respect to one equable motion that stands in for time, and the relations among velocities follow. Leibniz names a change, dy, without naming what drives it. Every letter, x included, has its differential “in the same way”, and no variable is the clock.
Written as arithmetic, these are the two directions of Chapter 3. Carry every quantity as a pair (value, fluxion), set ẋ = 1 so that x flows equably as Newton's time, and apply the rules. The result is forward-mode automatic differentiation, and on the composite above it returns 42.865920 COMPUTED. Newton's scheme answers “if this one quantity flows, how fast does everything else move?”, which is exactly forward mode's question. Leibniz's notation, with no clock and every letter's differential available, lets you write dz/dy for any pair. That is what a backward sweep needs, because it asks how the output responds to each intermediate in turn. MODEL — this is a structural correspondence between the notations and the two modes. It is not a claim about what either man intended, and no priority is claimed.
The chapter keeps its title with a changed meaning. “A ratio, not a rate” does not say that Newton could not compose. It says that Newton's derivative is a ratio of two rates against a common clock, while Leibniz's is a ratio of two changes with no clock at all. The first is the natural notation for pushing a derivative forward. The second is the natural one for pulling it back.
Dunham reports Laplace's verdict on Leibniz's symbols, “a very happy notation” (p. 22). He also reports Leibniz's own candour about what dx was: whatever its nature, it could be used as “a tool that has advantages for the purpose of the calculation” (p. 24). Chapter 4 showed Euler running on that tool, and Chapter 2 showed Loomis and Sternberg replacing it with linear maps. Chapter 3 showed a machine that needs neither the infinitesimal nor its replacement, only the rules. The founding paper of backpropagation, Rumelhart, Hinton and Williams (1986), is now held too. Its method is the backward sweep, run through the layers of a network from the output to the weights.