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General Theory of Algebraic Equations

Princeton University Press
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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.”[1]

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.[2]

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.[3]

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 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 (un)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.[4] 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.


[1] Grabiner, Judith, “Etienne Bézout,” in Dictionary of Scientific Biography, ed. C C. Gillespie, New York: Scribner, 1972, vol. 2, p. 111-114.

[2] Bix, Robert, Conics and Cubics, 2nd ed., New York: Springer, 2006, p. 202.

[3] See Bix, p. 245–49.

[4] 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.

Date Received: 
Wednesday, June 28, 2006
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Etienne Bézout
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Translator's Foreword xi
Dedication from the 1779 edition xiii
Preface to the 1779 edition xv

Introduction: Theory of differences and sums of quantities 1

Definitions and preliminary notions 1
About the way to determine the differences of quantities 3
A general and fundamental remark 7
Reductions that may apply to the general rule to differentiate quantities when several differentiations must be made. 8
Remarks about the differences of decreasing quantities 9
About certain quantities that must be differentiated through a simpler process than that resulting from the general rule 10
About sums of quantities 10
About sums of quantities whose factors grow arithmetically 11
Remarks 11
About sums of rational quantities with no variable divider 12

Book One

Section I

About complete polynomials and complete equations 15
About the number of terms in complete polynomials 16
Problem I: Compute the value of N(u . . . n)T 16
About the number of terms of a complete polynomial that can be divided by certain monomials composed of one or more of the unknowns present in this polynomial 17
Problem II 17
Problem III 19
Remark 20
Initial considerations about computing the degree of the final equation resulting from an arbitrary number of complete equations with the same number of unknowns 21
Determination of the degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns 22
Remarks 24

Section II

About incomplete polynomials and first-order incomplete equations 26
About incomplete polynomials and incomplete equations in which each unknown does not exceed a given degree for each unknown. And where the unknowns, combined two-by-two, three-by-three, four-by-four etc., all reach the total dimension of the polynomial or the equation 28
Problem IV 28
Problem V 29
Problem VI 32
Problem VII: We ask for the degree of the final equation resulting from an arbitrary number n of equations of the form (u a . . . n)t = 0 in the same number of unknowns 32
Remark 34
About the sum of some quantities necessary to determine the number of terms of various types of incomplete polynomials 35
Problem VIII 35
Problem IX 36
Problem X 36

Problem XI 37
About incomplete polynomials, and incomplete equations, in which two of the unknowns (the same in each polynomial or equation) share the following characteristics:
(1) The degree of each of these unknowns does not exceed a given number (different or the same for each unknown);
(2) These two unknowns, taken together, do not exceed a given dimension;
(3) The other unknowns do not exceed a given degree (different or the same for each), but, when combined groups of two or three among themselves as well as with the first two, they reach all possible dimensions until that of the polynomial or the equation 38
Problem XII 39
Problem XIII 40
Problem XIV 41
Problem XV 42

Problem XVI 42
About incomplete polynomials and equations, in which three of the unknowns satisfy the following characteristics:
(1) The degree of each unknown does not exceed a given value, different or the same for each;
(2) The combination of two unknowns does not exceed a given dimension, different or the same for each combination of two of these three unknowns;
(3) The combination of the three unknowns does not exceed a given dimension.
We further assume that the degrees of the n - 3 other unknowns do not exceed given values; we also assume that the combination of two, three, four, etc. of these variables among themselves or with the first three reaches all possible dimensions, up to the dimension of the polynomial 45
Problem XVII 46
Problem XVIII 47
Summary and table of the different values of the number of terms sought in the preceding polynomial and in related quantities 56
Problem XIX 61
Problem XX 62
Problem XXI 63

Problem XXII 63
About the largest number of terms that can be cancelled in a given polynomial by using a given number of equations, without introducing new terms 65
Determination of the symptoms indicating which value of the degree of the final equation must be chosen or rejected, among the different available expressions 69
Expansion of the various values of the degree of the final equation, resulting from the general expression found in (104), and expansion of the set of conditions that justify these values 70
Application of the preceding theory to equations in three unknowns 71
General considerations about the degree of the final equation, when considering the other incomplete equations similar to those considered up until now 85
Problem XXIII 86
General method to determine the degree of the final equation for all cases of equations of the form (u a . . . n)t = 0 94
General considerations about the number of terms of other polynomials that are similar to those we have examined 101
Conclusion about first-order incomplete equations 112

Section III

About incomplete polynomials and second-, third-, fourth-, etc. order incomplete equations 115
About the number of terms in incomplete polynomials of arbitrary order 118
Problem XXIV 118
About the form of the polynomial multiplier and of the polynomials whose number of terms impact the degree of the final equation resulting from a given number of incomplete equations with arbitrary order 119
Useful notions for the reduction of differentials that enter in the expression of the number of terms of a polynomial with arbitrary order 121
Problem XXV 122
Table of all possible values of the degree of the final equations for all possible cases of incomplete, second-order equations in two unknowns 127
Conclusion about incomplete equations of arbitrary order 134

Book Two

In which we give a process for reaching the final equation resulting from an arbitrary number of equations in the same number of unknowns, and in which we present many general properties of algebraic quantities and equations 137

General observations 137
A new elimination method for first-order equations with an arbitrary number of unknowns 138
General rule to compute the values of the unknowns, altogether or separately, in first-order equations, whether these equations are symbolic or numerical 139
A method to find functions of an arbitrary number of unknowns which are identically zero 145
About the form of the polynomial multiplier, or the polynomial multipliers, leading to the final equation 151
About the requirement not to use all coefficients of the polynomial multipliers toward elimination 153
About the number of coefficients in each polynomial multiplier which are useful for the purpose of elimination 155
About the terms that may or must be excluded in each polynomial multiplier 156
About the best use that can be made of the coefficients of the terms that may be cancelled in each polynomial multiplier 158
Other applications of the methods presented in this book for the General Theory of Equations 160

Useful considerations to considerably shorten the computation of the coefficients useful for elimination. 163
Applications of previous considerations to different examples; interpretation and usage of various factors that are encountered in the computation of the coefficients in the final equation 174
General remarks about the symptoms indicating the possibility of lowering the degree of the final equation, and about the way to determine these symptoms 191
About means to considerably reduce the number of coefficients used for elimination. Resulting simplifications in the polynomial multipliers 196
More applications, etc. 205
About the care to be exercised when using simpler polynomial multipliers than their general form (231 and following), when dealing with incomplete equations 209
More applications, etc. 213
About equations where the number of unknowns is lower by one unit than the number of these equations. A fast process to find the final equation resulting from an arbitrary number of equations with the same number of unknowns 221
About polynomial multipliers that are appropriate for elimination using this second method 223
Details of the method 225
First general example 226

Second general example 228
Third general example 234
Fourth general example 237
Observation 241
Considerations about the factor in the final equation obtained by using the second method 251
About the means to recognize which coefficients in the proposed equations can appear in the factor of the apparent final equation 253
Determining the factor of the final equation: How to interpret its meaning 269
About the factor that arises when going from the general final equation to final equations of lower degrees 270
Determination of the factor mentioned above 274
About equations where the number of unknowns is less than the number of equations by two units 276

Form of the simplest polynomial multipliers used to reach the two condition equations resulting from n equations in n - 2 unknowns 278
About a much broader use of the arbitrary coefficients and their usefulness to reach the condition equations with lowest literal dimension 301
About systems of n equations in p unknowns, where p < n 307
When not all proposed equations are necessary to obtain the condition equation with lowest literal dimension 314
About the way to find, given a set of equations, whether some of them necessarily follow from the others 316
About equations that only partially follow from the others 318
Re exions on the successive elimination method 319
About equations whose form is arbitrary, regular or irregular. Determination of the degree of the final equation in all cases 320
Remark 327

Follow-up on the same subject 328
About equations whose number is smaller than the number of unknowns they contain. New observations about the factors of the final equation 333

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Thursday, January 22, 2009