Newton's fluxion notation, $\dot{x}$, names a rate. Leibniz's $dy/dx$ names a ratio of differentials, and that is the difference that matters here: written as a ratio, the chain rule $$\frac{dz}{dx} \;=\; \frac{dz}{dy}\cdot\frac{dy}{dx}$$ reads as cancellation and composes without bookkeeping. A notation that composes can be applied mechanically to an arbitrary composite, which is precisely what an automatic-differentiation engine does: it walks a composition graph and multiplies local derivatives. Reverse-mode is that walk taken from the output backwards, and reverse-mode on a scalar loss is backpropagation.
Nothing in that chain is new mathematics, and this volume claims none. What it claims is that the corpus already stands on it: G = U ∘ F ∘ K ∘ C is a composite map, its order-dependence is a statement about composition, and the derivative of that composite is the Leibniz rule. Book XVII ch 1 locates the standard statement at Deisenroth p. 146 and automatic differentiation at p. 164.
| ch | title | state |
|---|---|---|
| 1 | A Ratio, Not a Rate — why Leibniz's notation composes and Newton's does not, and what that cost each tradition | blocked — no Leibniz |
| 2 | The Rule Itself — the chain rule for composite and vector-valued maps, Jacobians, and the corpus's operator chain as an instance | writable |
| 3 | Mechanising It — forward and reverse mode, why reverse mode wins on a scalar loss, and what backpropagation actually is | writable |
| 4 | Euler in the Middle — what the eighteenth century did with the notation before anyone could run it | planned |
Corrected the same day. An earlier pass reported no calculus textbook of any kind. Two arrived within the hour and the ledger was re-run: Loomis & Sternberg, Advanced Calculus (revised edition, 592 pp) and Knill, Introduction to Calculus (Harvard Math 1a, 311 pp). The earlier statement was true of its snapshot and false within sixty minutes — the standing hazard of quoting a ledger instead of re-running it.
What remains not held is Leibniz himself — not the 1684 Nova Methodus, not a collected works, not a secondary study. A narrower gap, and it binds one chapter.
Also held: Newton's Principia (594 pp), Bradley's Leonhard Euler (543 pp), and Deisenroth, where the rule is stated on p. 146 and mechanised on p. 164.
Chapters 2, 3 and 4 can now be written — the rule, its mechanisation and the eighteenth century all have founding texts with addresses. Chapter 1 still cannot. Its argument is about what Leibniz's notation does that Newton's does not, and that needs Leibniz, not a modern textbook restating him. [OPEN]