Book: Institutiones calculi differentialis

Overview and Purpose
Leonhard Euler's Institutiones calculi differentialis (1755) presents a systematic and pedagogical development of differential calculus as it stood in the mid-18th century. The book organizes rules, techniques, and illustrative problems into a coherent course intended to teach the subject to students and to serve as a reference for working mathematicians and physicists. It emphasizes algebraic manipulation of differentials, general methods for finding derivatives, and the practical use of series expansions and transformations.
The text aims to reduce disparate techniques into a unified analytic approach. Rather than pursuing the later, more formal foundations of limits, the book builds on the operational use of differentials and infinite series, showing how those tools unlock problems in geometry, mechanics, and analysis.

Structure and Contents
The Institutiones is arranged as a sequence of chapters that move from elementary rules to more advanced techniques. Early chapters treat the basic rules for differentiating algebraic and transcendental functions, including the product, quotient, and chain rules, and systematic procedures for implicit and parametric differentiation. Subsequent sections develop higher-order differentials and the manipulation of series to represent functions and compute derivatives.
Later chapters apply these methods to problems of maxima and minima, curvature and tangency of curves, asymptotes, and small oscillations. Numerous worked examples accompany the theoretical development, illustrating how algebraic reduction, substitution, and series expansion yield explicit derivative formulae and approximations.

Key Concepts and Techniques
Euler presents differentials as algebraic entities, using dx and dy to represent infinitesimal increments and building rules for their combination. The book makes extensive use of power series expansions to represent functions locally and to derive Taylor-type approximations; these expansions serve both as a computational device and as justification for manipulation of higher differentials. Euler also treats transcendental functions such as exponential, logarithmic, and trigonometric functions through series and differential identities.
Implicit differentiation and parametric representations receive careful attention, with clear algorithms for obtaining derivatives when functions are given indirectly. The treatment of higher derivatives is practical and explicit, showing patterns and methods for repeated differentiation of complex expressions.

Mathematical Style and Notation
Euler's exposition is notable for its clarity, economy, and striking use of notation that became standard in later analysis. The symbolic approach favors concise formulae and algebraic transformations, and Euler demonstrates an almost engineering sense for selecting representations that simplify computation. Rigorous epsilon-delta arguments are absent; justification rests on algebraic manipulation, convergent series, and illustrative computation rather than on the later axiomatic foundations.
The book popularizes operational habits, such as treating differentials as manipulable quantities and using series expansions freely, that became widespread among analysts and applied mathematicians. The concise notation and systematic presentation helped disseminate these methods across Europe.

Innovations and Highlights
Several contributions in Institutiones stand out: the systematic cataloguing of differentiation rules, the broad use of series to handle transcendental and composite functions, and the explicit handling of higher-order differentials with methods that streamline repeated differentiation. Euler frames many classical problems, finding tangents, curvatures, and extrema, in an analytic way that links algebraic computation with geometric interpretation.
Examples often bridge pure and applied mathematics, showing how derivatives model instantaneous rates and how small-change approximations underpin mechanical and physical reasoning. These practical connections helped establish differential calculus as the principal tool for analysis and modeling.

Legacy and Influence
Institutiones calculi differentialis became a formative textbook for generations of mathematicians, consolidating and extending the Leibnizian tradition into a pedagogical canon. Its methods and notation influenced later textbooks and research, smoothing the path toward more systematic treatments of analysis. While later centuries replaced some heuristic justifications with rigorous foundations, Euler's clear operational calculus remained central to computation and application, and the Institutiones stands as a landmark in the development and dissemination of differential calculus.