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From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra

Jacqueline Stedall
European Mathematical Society
Publication Date: 
Number of Pages: 
Heritage of European Mathematics
[Reviewed by
Fernando Q. Gouvêa
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Many accounts of the history of algebra focus most of their attention on the problem of solving polynomial equations in one variable. Starting with the quadratic problems of Ancient Mesopotamia, they go on to talk about Al-Khwarizmi’s book on al-jabr w’al-muqabala and perhaps talk about other important texts in Arabic. Then it’s Renaissance Italy and the shenanigans of Tartaglia and Cardano. At that point, methods (not “formulas”!) had been found for solving equations of degrees two through four. Maybe there follows a nod to Viète and Descartes for their creation of a proper algebraic notation. And then it’s on to the unsolvability of the quintic equation, with Ruffini, Abel, and Galois taking center stage and changing the very meaning of “algebra.”

The gap in Stedall’s title is precisely the period between the solution of the cubic and quartic equations (the time of Cardano, in the middle of the 16th century) and the first work to make a serious contribution to understanding why the quintic was resisting solution (Lagrange’s Reflections, late in the 18th century). Stedall shows us what was going on in between, paying attention in particular to the themes that led to Lagrange’s ideas.

The book has two parts which are organized differently. The first, “from Cardano to Newton,” is organized chronologically. This is the period when algebraic notation acquired its modern form; most importantly, for the first time one could write equations involving both unknowns and parameters. It is during this time that the subject takes on features we all take for granted. The general quadratic becomes \(ax^2+bx+c=0\) instead of being expressed in words such as “squares equal to numbers and things.” It is now possible to write \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\]

It is also in this period that the relationship between the roots and the coefficients of an equation becomes clear. Harriot, Descartes, and Newton all knew that if the roots of \[ x^n-s_1x^{n-1}+s_2x^{n-2}-\dots \pm s_n=0\] are \(r_1,r_2,\dots,r_n\), then \(s_1=r_1+r_2+\dots+r_n\), \(s_2=r_1r_2+r_1r_3+\dots+r_{n-1}r_n\), etc. Newton added that some other symmetric functions of the roots could be expressed in terms of these.

The existence of \(n\) roots of some kind was taken for granted. Descartes and Newton both considered the nature of the roots: how many are positive and how many negative? Can one tell a priori, from the coefficients, how many roots were imaginary? Descartes’ “rule of signs” gave a precise answer to the first question as long as all the roots were known to be real. But Descartes provided no proof of his rule. Newton’s rule for counting imaginary roots was only approximate. As a result, both problems were left for future mathematicians to consider.

Part II deals with the period from Newton to Lagrange, which is basically the 18th century. Here the organization is topical, which allows Stedall to usefully highlight what she feels were the central themes of 18th century algebra. First we read about work on the nature of the roots, then on their form (were the roots always sums of radicals? Of what form?). Functions of the roots (e.g., the symmetric functions above) begin to be considered, and elimination theory is developed by Euler, Cramer, and Bézout. The last two connect directly to Lagrange’s Reflections, which is examined in some detail.

There is a short chapter on numerical solutions, dealing both with approximate solutions and with the problem of finding upper and lower bounds for solutions. Newton’s method is discussed here (and Raphson’s improvement); one of the things we learn is that the original formulation of the method did not involve the derivative and (in retrospect, obviously) did not come with the usual picture of “descending along the tangent.”

Inevitably, there are a few typos and (fewer) mistakes. On page 109, the condition for choosing the cube roots that appear in Cardano’s solution of the cubic is stated incorrectly, and it is unclear whether the mistake is Stedall’s or Euler’s. But in general the mathematics is clearly explained. In keeping with our monoglot times, Stedall has provided translations of all texts, with the originals usually given as footnotes.

The writing is very clear, and occasionally beautiful. I particularly liked this description of Waring’s Meditationes Algebraicae:

The book has all the qualities of a fertile but wandering mind: ideas arise, intermingle, and coalesce apparently at random, appearing brilliant for a time but then subsiding into obscurity or lengthy algebraic calculations. The usual structures of good mathematical writing are entirely missing: there is no sense here of building from basic principles or easy examples to more general theorems. Instead, problems, lemmas, corollaries and examples tumble over one another without apparent order or reason so that the reader is left without any sense of either starting point or direction.

I think I’m going to use that quote when I next give my students a writing assignment!

Stedall certainly fills a gap in the history of the theory of polynomial equations in one variable, but I question whether she has succeeded completely when it comes to filling the gap in the history of algebra, because she gives too little attention to work in several variables. This is, after all, the period when Euler first realized that systems of \(n\) equations in \(n\) unknowns do not always have unique solutions, when Cramer discovered his “rule” for solving systems of linear equations (and hence the determinant) and Bézout realized the connection between determinants and elimination theory. Stedall’s focus on equations in one variable means that the chapter on elimination theory is uncharacteristically weak, missing such things as the discovery of determinants (which were to become central in the following century) and Bézout’s theorems on linear combinations of polynomials.

Despite not quite fulfilling the promise of its title, this is a very good book, one that provides us with real understanding of the theory of equations in this period.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and editor of MAA Reviews. He is very interested in the history of algebra and number theory in the 19th century.

I From Cardano to Newton: 1545 to 1707

1 From Cardano to Viète
2 From Viète to Descartes
3 From Descartes to Newton

II From Newton to Lagrange: 1707 to 1771

4 Discerning the nature of the roots
5 Roots as sums of radicals
6 Functions of the roots

7 Elimination theory
8 The degree of a resolvent
9 Numerical Solutions
10 The insights of Lagrange
11 The outsiders

III After Lagrange

​12 Dissemination and new directions