A Course in Mathematical Analysis
Rigorous foundations of modern analysis
Description
At the turn of the twentieth century, a French mathematician named Édouard Goursat set out to write down, from the ground up, what a serious student needed to know about mathematical analysis. He was not inventing the subject. Calculus was two centuries old; the theory of functions, the study of differential equations, the machinery of integration were already in use across physics and engineering. What was missing was a single place where all of it held together — where every claim was earned by a proof, and nothing was waved through because it looked obviously true. The result, published in several volumes and known simply as the Cours d'analyse, became one of the enduring reference works of the field.
The ambition was quieter than it sounds. Goursat did not promise a shortcut or a clever reframing. He promised completeness and honesty: to take the reader from the definition of a limit through the deep results on functions of a complex variable, and to justify each step along the way. That is a strange thing to find exciting, until we remember what mathematics had just been through. The nineteenth century had discovered, repeatedly and embarrassingly, that its most trusted intuitions could be wrong. Curves that looked smooth turned out to have no tangent anywhere. Sums that looked convergent added up to nonsense. Analysis needed to be rebuilt on foundations that could not betray it, and Goursat's treatise is one of the buildings that went up on the new ground.
What we find in the work is less a collection of tricks than a temperament — a way of refusing to be satisfied until the argument is airtight. That temperament is the real subject here, more than any single theorem it happens to prove.
The question we’re asking : What does it mean to put a whole branch of mathematics on foundations that cannot betray it — and why did that job fall to a textbook?What we’ll see : How a classic treatise rebuilt analysis proof by proof, from the definition of a limit to the equations that describe the physical world, and what its insistence on rigor left behind.
Table of contents
01Chapter 1 — The moment analysis needed a spine
To understand why Goursat's book matters, we have to understand what had gone wrong. For most of the eighteenth century, calculus worked beautifully and rested on almost nothing. Newton and Leibniz had given the world a method of extraordinary power, and the mathematicians who followed — Euler above all — used it to produce results that were correct far more often than they had any right to be. The reasoning ran on physical intuition and formal manipulation. Infinitesimals were quantities that were somehow both zero and not zero. Series were added, rearranged, and differentiated term by term with cheerful disregard for whether any of it was legitimate.
The trouble was that the intuition eventually lied. Through the nineteenth century, mathematicians kept finding counterexamples that no amount of physical feeling could have predicted. Functions that were continuous everywhere yet differentiable nowhere. Series whose sum depended on the order in which you added the terms. Objects that violated every expectation about how a well-behaved quantity should act. Each of these was a small scandal, and together they made a case that could not be ignored: the subject did not actually know what its own words meant.
02Chapter 2 — From derivatives to the theory of functions
Once the foundations are laid, the treatise climbs. The differential and integral calculus of a single variable comes first, but treated with a care that a first exposure rarely receives: the mean value theorem, Taylor's formula, the conditions under which a function can be expanded as a series, the precise circumstances in which an integral exists. Where an introductory course would state these and move on, Goursat pauses over the hypotheses. He wants the reader to see exactly what each theorem requires, because the counterexamples of the previous century had taught the discipline that a theorem is only as good as the conditions attached to it.
From there the work extends to functions of several variables — partial derivatives, multiple integrals, the changes of variable that let one geometry be traded for another. This is the machinery that physics and engineering actually run on, and Goursat presents it not as a bag of computational rules but as a coherent theory. The technical apparatus of surfaces, line integrals, and the great integral theorems that connect a region to its boundary all appear in their proper logical place, each following from what came before.
03Chapter 3 — Where the equations of the world live
A large part of the work is given to differential equations, and this is where the abstract machinery meets the physical world most directly. A differential equation is a statement about how a quantity changes in relation to its own rate of change — the falling body, the cooling object, the oscillating string, the flow of heat through a bar. These equations are the native language in which nineteenth-century physics wrote its laws, and mastering them was the practical reason many students opened a treatise like this one in the first place.
Goursat's treatment reflects the same discipline as the rest of the book. Before showing how to solve an equation, he asks the prior questions that a less careful text would skip: does a solution exist at all, and if it does, is it the only one? These are not idle worries. A method that produces an answer is worthless if there was no solution to find, or if there were several and the method silently chose one. The existence and uniqueness results, resting on the analytic foundations already established, are what license all the solving that follows. Only once that ground is secure does the treatise develop the techniques — the classification of equation types, the methods of integration, the treatment of linear systems.
04Chapter 4 — The proof was the point
Step back from the specific results and a larger claim comes into focus, one that Goursat never states outright because the entire structure of the treatise states it for him. It is that a mathematical discipline earns its authority not from the truths it arrives at but from its willingness to prove them. The nineteenth century's crisis had not been a shortage of results — Euler and his successors had produced them by the thousand. It had been a shortage of certainty about which results could be trusted and why. What the great rebuilders supplied, and what Goursat systematized, was not more mathematics but a stricter standard for what would count as mathematics at all.
This is why a textbook, of all things, became a landmark. A treatise is where a discipline decides what it considers settled and how it will pass that on. By refusing to state a theorem without its proof, by insisting that every definition be exact and every hypothesis be visible, Goursat was doing more than teaching analysis; he was enforcing a norm about how the subject conducts itself. Generations of students absorbed, along with the content, the expectation that a claim in mathematics is not to be believed on authority or intuition but only on demonstration. That expectation is the real inheritance.
05Conclusion
Édouard Goursat set out to write down what a serious student needed to know about analysis, and in doing so he produced one of the works that fixed the standard for the whole field. The Cours d'analyse takes the reader from the exact definition of a limit through the theory of functions of a complex variable and the equations that describe the physical world, and it never once asks to be believed on faith. Every stage rests visibly on the one before it, so that the finished structure is not a collection of results but a single connected argument about how analysis holds together.