Etienne Bézout (1739–1783) had a relatively short mathematical career. He was elected to the Académie des Sciences in Paris in 1758. At that time Euler, the dominant figure in the Continental mathematics community, was already 51 years old. Nevertheless, the two men died in the same year. Unlike Euler, Bézout spent a considerable portion of his life engaged in teaching duties. Notwithstanding all of this, Bézout had a significant impact on the development of mathematics in at least two ways.
First of all, he published a six-volume elementary mathematics text, based on the classes he taught to naval and artillery cadets. His Cours de mathématiques a l’usage des Gardes du Pavillon et de la Marine (1764–69) went through many French editions. It was also translated into English by John Farrar of Harvard University. Grabiner tells us that its excellent expository style and its practical orientation “considerably influenced the form and content of American mathematical education in the nineteenth century.”
Bézout was also a researcher and is remembered for the theorem in the theory of equations that bears his name. Bézout’s Theorem says that generically, the graphs of two equations in two variables of degrees m and n intersect in mn points. Thus, two lines typically meet in a single point, two conics in 4 points, and a line generically meets a curve of degree n in n points. Eighteenth century mathematician would have said that the curves meet in at most mn points. A more precise statement is possible: the number of intersections is exactly mn, as long as one also considers complex-valued solutions, solutions at infinity and the multiplicity of solutions. For a modern treatment of Bézout’s Theorem in the complex projective plane, see Bix.
Like the Fundamental Theorem of Algebra, Bézout’s Theorem was widely recognized as true by eighteenth century mathematicians, but a proof remained elusive for many decades. Newton seems to have been the first to give its statement and Euler took at least three cracks at the problem. Bézout and Euler independently resolved the problem in 1764, but Bézout went even further: in his 1779 Théorie Générale des équations algébriques, he showed that the number of solutions of an arbitrary number N of equations in N variables is no greater than the product of their degrees. We should note that although Bézout’s proofs were considered satisfactory in his time, proofs meeting modern standards of rigor had to wait until the use of homogeneous coordinates in the late 1800s.
Eric Feron has produced an excellent translation of Bézout’s 1779 book, the first ever to appear in the English language. General Theory of Algebraic Equations is divided into three parts: a brief introduction to the theory of differences and sums, Book One, in which Bézout considers the problem of determining a “final equation” in one variable, by eliminating all but one of the variables from a system of N of polynomial equations in N variables, and Book Two, in which he considers the solution of systems of simultaneous linear equations, provides justification for some of the claims he made in Book One, and finally considers systems of polynomial equations in which the number of equations exceeds the number of variables. In addition, Feron includes a translation of Bézout’s Preface and a brief Translator’s Foreword.
Unlike some recent translations of mathematical classics, Feron does not supply extensive annotation, preferring to let Bézout speak for himself. In fact, there are no translator’s notes at all within the translated text, and only the briefest of introductory remarks in his Foreword. This is very much in the spirit of Blanton’s Euler translations. Although I generally approve of this minimalist approach of leaving interpretation of the text to the reader, I might have appreciated a somewhat more extensive roadmap to the book than the brief synopsis provided in the Foreword.
The quality of the translation strikes me as first rate. The original French text is available online (for example, go to gallica.bnf.fr and search under “Bezout,” no accent needed) and I compared the original to the translation in quite a number of places. In the text itself, Feron is quite faithful to Bézout’s original, but still provides smooth, readable prose. This is probably a testament both to the clarity of the original text and to the quality of Feron’s translation. His translation of Bézout’s Preface is a little more free, but this entirely appropriate to the more informal style of this non-technical, introductory portion of the book.
Reading the General Theory of Algebraic Equations from cover to cover is a daunting challenge, because it is 337 pages long and is quite technical in places. Unfortunately, it does not lend itself well to random browsing, at least not at first. This is due in large measure to the notation. Bézout was writing in the days before subscripting was in general use. He also did not have the benefit of modern notation for matrices and determinants in his discussion in Book Two. Furthermore, he made some notational choices that seem strange to us; for example, he used the expression (u…n)T to denote a complete polynomial of degree T in n variables.
To get your feet wet, you can start by reading Bézout’s Introduction and Section I of Book One. Over the course of this manageable chunk of 25 pages, we are introduced to much of the notation, we see an overview of the theory of finite differences, and we get a general feeling for the style and content of the book. It culminates with a statement of Bézout’s Theorem, although the key algorithm for the proof is not encountered until Book Two. Once this much of the book has been mastered, it is much easier to jump directly to a later portion of the text.
The General Theory of Algebraic Equations will be of interest to anyone specializing in the history of algebra or analysis. Although the infinitesimal calculus is not the subject matter, the book is still very much a product of its time, a time when the calculus was shedding its geometric roots and being elaborated and justified in algebraic terms by Euler, Lagrange and others. Furthermore, although Bézout’s Introduction does not explicitly concern infinitesimal differences and sums, it is easy to make the transition from finite increments k, l,… to infinitesimal ones dx, dy,…, as Euler did in his differential calculus text. Indeed, Bézout speaks of differentials (différencielles) and differences interchangeably, but never specifically considers the infinitely small. In at least one unfortunate place (page 7), Feron translates différence as “derivative,” which not only commits Bézout to the infinitesimal, but also to the 19th century point of view, rather than the Leibnizian/Eulerian differential, which was surely his metaphysics for the calculus.
It is unfortunate that Feron did not provide an index for this book, but his decision is entirely defensible, because Bézout himself did not give an index.
Such minor quibbles notwithstanding, this is an excellent translation of one of the classics of algebra. It deserves a place in any serious English language collection of original sources for the history of mathematics.
 Grabiner, Judith, “Etienne Bézout,” in Dictionary of Scientific Biography, ed. C C. Gillespie, New York: Scribner, 1972, vol. 2, p. 111-114.
 Bix, Robert, Conics and Cubics, 2nd ed., New York: Springer, 2006, p. 202.
 See Bix, p. 245–49.
 Euler, L., Institutiones Calculi Differentialis, St. Petersburg: Imp. Academy, 1755. English translation: Blanton, J., Foundations of Differential Calculus, New York: Springer, 2000.
Rob Bradley is professor of mathematics at Adelphi University. His main research interest is the history of mathematics. He is currently the president of the Euler Society and past president of the Canadian Society for the History and Philosophy of Mathematics.