Between Leibniz's notation (1684) and the first machine that could run it lies the eighteenth century, and in the middle of it is Euler. According to Victor Katz's reading of Euler's textbooks, Euler never stated the chain rule. He did not need to. His notation, and one rule for handling it, applied the chain rule for him whenever it came up. That rule turns out to be the arithmetic a forward-mode engine runs on.
Euler's three analysis textbooks are the Introductio in analysin infinitorum (1748), the Institutiones calculi differentialis (1755) and the Institutiones calculi integralis (1768–70). Katz calls them the first “modern” texts, readable today without translating fluxions into current terms, because “Euler introduced much of our current notation for the calculus” (p. 213) CITED. The Differential Calculus is also pure analysis, with no geometry, no physics and no diagrams. Katz's summary of Euler's view (p. 231) is that one could understand everything needed “by pure manipulation of symbols according to the rules that he and others developed.”
That sentence describes a program, a century and a half before anyone could run one. A calculus that is carried out entirely by rules applied to symbols is, in principle, mechanical.
Euler treated the calculus as “nothing but a special case of the method of differences … wherein the differences are infinitely small” (Katz p. 224). To differentiate y = xn he expands (x + dx)n by the binomial theorem and subtracts xn. He then declares that “the second term and all succeeding terms vanish in the presence of the first term”, which leaves d(xn) = nxn−1dx. For logarithms he expands ln(1 + dx/x) and dispenses with the higher-order terms, which leaves d(ln x) = dx/x (p. 225).
Make that rule into arithmetic: numbers of the form a + b·dx, multiplied as usual, except that dx·dx = 0. In that arithmetic the script computes (1.3 + dx)5 = 1.35 + 14.2805·dx, and 14.2805 = 5 × 1.34 COMPUTED. The derivative falls out of multiplication. Nothing was differentiated.
Katz is precise about what Euler did not do (p. 224): “Euler did not give an explicit statement of the modern chain rule, but did deal with special cases as the need arose. Thus if p is a function of x whose differential is dp, then d(pn) = npn−1dp.”
That special case is the chain rule, and the notation delivers it without being asked. Writing dp where the power rule had dx is the whole step. In the dx² = 0 arithmetic the step is not even a decision. Let p = x² + 1, computed on x + dx. It arrives already carrying dp = 2x·dx, and raising it to the fifth power returns 5p4·dp, which at x = 1.3 is 680.694862. A central difference agrees COMPUTED. The inner derivative is carried along by the arithmetic; nobody multiplied it in.
Euler's rule works because the higher terms really are negligible for the functions he used, not because they are zero. The eighteenth century had no account of why they could be dropped. Katz notes that the texts rest on “infinitely large and infinitely small quantities” and lack the modern theory of the real numbers (pp. 231–232). That account came later, and Chapter 2's Loomis and Sternberg is where this volume holds it: the differential as a linear map, with the dropped terms controlled by the definition of differentiability. Euler had the algebra without the foundation. A machine needs only the algebra, which is why the algebra could be mechanised long before most users of it could state the theorem underneath.
Laplace's advice survives in Katz's chapter (p. 232): “Read Euler; read Euler. He is the master of us all.” Katz adds that it is hard to say who actually did. The Differential Calculus had a single eighteenth-century translation, into German in 1790.
The lineage the volume index set out is now written at three of its four points. Chapter 2 is the rule as a theorem, Chapter 3 is the rule as a machine, and this chapter is the rule as an algebra that was already mechanical in form. Chapter 1, on why Leibniz's notation composes and Newton's does not, waits for Leibniz's 1684 paper and Newton's Method of Fluxions to be held. OPEN Euler's own Institutiones calculi differentialis would also let this chapter quote Euler directly rather than through Katz. WANTED