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... numbers have neither substance, nor meaning, nor qualities. They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession.

"Mathematics and the Laws of Nature" in The Armchair Science Reader, New York: Simon and Schuster, 1959.

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# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation

## A Reader's Guide to Servois' 'Essay'

This chapter is designed to be a friendly guide to understanding Servois’ journey through the differential calculus. We propose two different ways to read Servois’ “Essay”:

1. to gain a general understanding of Servois’ notation, algebraic foundations (which are the hallmark of the “Essay”), and the derivation of the rules of calculus, or
2. to examine the mathematical details in depth.

For those interested in a basic understanding of Servois’ “Essay,” we recommend reading Sections 1-12, in which Servois gives his notation and derives the algebraic properties of operators. Then, using this reader’s guide to make the transition, skip ahead to Section 18, where Servois develops the rules of the differential calculus. Readers who want to examine the full mathematical detail of Servois’ derivations will want to read Sections 13-17 and 19 in their entirety. This guide is designed to aid both audiences. Enjoy the journey!

Figure 3. Title page of Servois’ “Essay” (public domain).

Servois begins the “Essay” by giving vital definitions and notation. He denotes a function of $$z$$ by $$fz$$ and it follows that the composition of two functions of $$z$$ would be denoted by $$fgz$$. For simplicity, we use the modern notation $$f(z)$$ and $$f(g(z))$$ for function and function composition, respectively, throughout the reader’s guide. Except for this lack of parentheses, Servois’ basic definitions look familiar to modern readers. For example, he uses $$f^2$$, $$f^3$$, $$\ldots$$ for repeated composition of a function $$f$$ and $$f^{-1}$$ for the inverse function.
Following the notation of Arbogast [1800], Servois defines the varied state $E(z)=\varphi(x + \alpha, y + \beta, \ldots),$ and, based on it, $E^{-1}(z) = \varphi(x - \alpha, y - \beta, \ldots) \quad \rm{and}$ $E^{n}(z) = \varphi(x + n\alpha, y + n\beta, \ldots),$ where $$z = \varphi(x,y, \ldots)$$ is a function of many variables and $$\alpha, \beta, \ldots$$ are “invariable” or constant increments. Using the varied state, Servois defines the difference of a function, $\Delta (z) = E(z) - z = \varphi(x + \alpha, y + \beta, \ldots) - \varphi(x, y, \ldots).$