Calcolo differenziale e principii di calcolo integrale is Giuseppe Peano’s systematic treatise on the infinitesimal calculus, pairing a treatment of differential calculus with a foundational account of the principles of integral calculus. It belongs to his early period as a lecturer in infinitesimal calculus at the University of Turin, and it reads as a course text: a structured exposition written for students, moving from the basic objects of the calculus toward the definition and justification of its central operations. Its two broad concerns are announced in its title — the differential calculus itself, and the principles (rather than the full technical apparatus) of the integral calculus.
The work is characteristic of the tradition of rigorous analysis that descended from Cauchy. Its interest lies less in computational drill than in the careful construction of the general theory of functions: the treatment of functions, limits, continuity, and the derivative is pursued with an eye to precise definitions and justified propositions, and the integral is approached through its foundations. This is the quality that gives the book its historical place; it is regarded, together with Peano’s later Lezioni di analisi infinitesimale, as one of the important works on the development of the general theory of functions after Cauchy. The book thus stands as a contribution to the effort to place the calculus on a secure footing, treating the subject as a branch of analysis to be reasoned about rather than a collection of manipulative techniques.
Because it is a treatise on principles, the integral-calculus portion is deliberately introductory in scope: it establishes what the integral is and why its basic properties hold, rather than developing the extensive catalogue of integration methods and applications found in more elementary manuals. The differential-calculus portion carries the main weight of the exposition, building the theory of the derivative and its uses on the underlying notions of function, limit, and continuity.
The book is a mathematical text, not a work of logic; its notation and exposition belong to analysis, and its arguments are those of the calculus. Its importance for Peano’s later reputation lies in the rigour and generality of its mathematical standpoint, which anticipates the formal care that would mark his subsequent work on the foundations of mathematics.
The derivative is the organizing notion of the differential-calculus portion: it is introduced as a limit and then used to define the differential, to formulate the mean-value theorems and Taylor expansion, and to develop the elementary theory of maxima and minima. The exposition is constructed so that the hypotheses on limits and continuity are explicit; it is concerned with when a derivative exists and with the conditions under which the usual theorems about sums, products, composites, and inverse functions are legitimate. Its rigour is therefore structural rather than ornamental: the sequence of definitions and propositions is meant to exhibit the calculus as a deductive development from the notion of function.
The integral-calculus portion is correspondingly foundational. It sets up the definite integral and its basic properties, and it treats the fundamental theorem as the bridge between integration and differentiation. Because the scope is the “principles” of the integral calculus, the aim is not a complete catalogue of integration techniques but the construction of the integral as a well-defined object within the same analytic framework as the derivative.
The book’s deeper significance lies in this conception of the subject: it presents the calculus as part of the general theory of functions, with theorems stated for broad classes of functions and proofs depending on exact hypotheses. It registers the movement from a calculus organized around manipulative rules and geometric intuition toward one organized around limits, definitions, and justified propositions. That mathematical standpoint—rather than any extensive technical novelty—is what gives the work its place in the development of rigorous analysis.
