Review of Unknown Quantity

Well, anything's possible, I suppose.  I must say, though, I am not a fan of conspiracy theories, and these speculations by friends and colleagues seems a bit far-fetched to me.  I don't know Victor Katz, but I know people who do, and they assure me he is an honest man and a conscientious researcher.  Probably he is left-wing--most academics are--but I prefer to believe that professionals bring to their side activities, like book reviewing, the same objectivity that makes math and science possible.  (And which I try to bring to my books.  I defy you to find any conservative political sentiment in Unknown Quantity or Prime Obsession.)

So far as I am concerned--and I speak from many years of reviewing books myself--Katz is entitled to his opinion.  And he did catch me out in some plain errors of fact (Cardano, Khayyam).  These errors are corrected in the paperback.  The rest of his objections, when there is anything to them, really amount to saying: "I would not have made these selections, or presented the material like this."  Which is fine:  Prof. Katz is at liberty to write his own book. 

Looking through Unknown Quantity, I do not think I should have given the Indians and Chinese more space than I did, I do not think my stress on literal symbolism is misplaced, nor my coverage of Descartes inadequate,... and so on.  These are fair matters of opinion, and I respect the reviewer's opinions.  I shall do my best to correct typos and errors in future editions, but Katz's implied suggestion that I should totally rewrite the book to suit his inclinations is just impertinent.  As I have already said, he is free to write his own book.

I think fundamentally Katz did not understand what kind of book I was writing--a pop-math book for non-mathematicians.  His closing declaration that he will not recommend the book to his students leaves me unruffled.  Unknown Quantity was not intended as a study text for college students, though I hope it will find acceptance as a supplementary text.  In regard to the Babylonians, for instance, I thought I should tell the reader that they did primitive algebra; then give the reader an actual example, translated and illustrated; then say how this all came to light, with a 20th-century anecdote for flavor.  Next!  This is not the kind of book to present and contrast current theories about the motives of the Babylonians, which are in any case unsettled.  I have no space to do that, only to present the essentials, with such emphases and omissions as bring the book to its commissioned length, and as I, the author, feel appropriate to the kind of book I am writing.

     ---J.D., September 2006]

A Faulty Survey of Algebra's Roots

By Victor Katz

"Mathematics is, let's face it, a dry subject, with little in the way of glamour or romance." Thus John Derbyshire begins chapter 11 of Unknown Quantity, his new history of algebra. Of course, Derbyshire [a systems analyst, columnist, and writer (1)] does not really believe this, or he would not even attempt to tell the story of algebra in a manner accessible to the "curious nonmathematician." In fact, the book demonstrates quite the opposite, that algebra in particular is a very exciting subject, developed by interesting characters and full of exciting ideas--many of which lie at the base of modern science.

Derbyshire's clear prose takes us from the Alexandria of Diophantus to the Baghdad of al-Khwārizmī, from the Isfahan of Omar Khayyam to the Pisa of Leonardo, from the Milan of Girolamo Cardano to the Paris of François Viète, and then on to such mathematicians as Niels Henrik Abel, Augustus De Morgan, Èvariste Galois, Emmy Noether, Saunders Mac Lane, and Alexander Grothendieck. Punctuating the historical chapters, and helping readers better understand the mathematical ideas, are brief sections called "Math Primers." In these, Derbyshire presents some basic algebraic ideas, including introductions to numbers and polynomials, cubic and quartic equations, roots of unity, vector spaces and algebras, field theory, and projective geometry.

In general, the book succeeds in its aim of enlightening the non-expert on what algebra is today, giving good summaries of recent work in such fields as algebraic topology, algebraic geometry, and even category theory. Unfortunately, Derbyshire's history of algebra through the 17th century has many shortcomings.

First, a history of algebra should begin by telling us what algebra is. But instead of giving us a definition with some meaning, Derbyshire offers as his working definition of "algebra" in the modern American high school and early college curriculum that it is "the part of advanced mathematics that is not calculus." As with his phrase in chapter 11, it appears that Derbyshire does not really believe what he wrote. Taken literally, this definition would include such topics as Euclidean geometry, trigonometry, probability, and statistics--all of which are generally taught in high school and early college mathematics and none of which are, to my mind, either calculus or algebra. But in fact, through the first seven chapters (which more or less deal with the algebra currently taught in high school) Derbyshire mainly considers the solution of equations, and that topic is, I would argue, the working definition of algebra held by most high school students.

If algebra is about solving equations, then certainly both the Egyptians and Mesopotamians should get some credit. Derbyshire briefly mentions both civilizations, but he unfortunately has not read any modern research on the algebraic material in Mesopotamian civilization. He quotes Otto Neugebauer's 1945 translation of one problem from a clay tablet (YBC 6967) without noting that recent researchers have concluded that there was a distinct geometric flavor to the instructions on the tablet, that in fact the Mesopotamians were performing algebra by manipulating squares and rectangles (2). In other words, for them a "square" was really a geometric square and a number and its reciprocal could always be thought of as two sides of a rectangle whose area was 1. Thus, the earliest algebra had a geometric flavor. Classical Greek mathematics also contained some "geometric algebra," of which proposition 28 from book six of Euclid's Elements (referred to somewhat uncomfortably by Derbyshire) is an example. But proposition 85 of Euclid's Data is easier to understand: If two straight lines contain a given area in a given angle, and if the sum of them be given, then shall each of them be given (i.e., determined) (3). Here Euclid is asking simply to find two straight lines, say x and y, when their sum, x y, and the area of the rectangle determined by them, xy, are known (think of the "given angle" as a right angle).

Derbyshire, however, is reluctant to call the Greek and Mesopotamian material algebra, because he believes that the key to algebra is the use of symbolism. Thus the author credits Diophantus with being the "father of algebra" on the grounds that he used some symbolism, while being "disappointed" with al-Khwārizmī's work in 9th-century Baghdad because that is a "sliding back" from Diophantus. Yet al-Khwārizmī was able to solve complicated algebraic problems without symbolism, and his Islamic successors (including Abū Kāmil, al-Karajī, and al-Samaw'al) were able to deal with even more sophisticated problems even though everything was written out in words. For example, if the following problem of Abū -Kāmil involving three unknowns is not algebra, then I am not sure what is: "One says that ten is divided into three parts, and if the small one is multiplied by itself and added to the middle one multiplied by itself, it equals the large one multiplied by itself, and when the small is multiplied by the large, it equals the middle multiplied by itself " (4). Abū Kāmil was able to solve this problem using techniques of manipulation that have become standard, even though these techniques did not involve symbolism. Certainly it is easier to solve this problem using symbols, but I would venture to guess that even so today's high school students would have trouble with it. It should also be noted that both al-Khwārizmī and Abū Kāmil gave geometric demonstrations of their methods for solving quadratic equations, the former in a naïve form similar to the Mesopotamian methods and the latter quoting from Euclid's Elements.

Derbyshire also fails to mention most of the more advanced algebraic techniques developed in Islam, including polynomial algebra and the law of exponents. And in his limited treatment of Omar Khayyam's solution of cubic equations, he claims that Omar could only solve four of the fourteen types of cubic equations by geometrical means. If Derbyshire had consulted Omar's algebra text (5), he would have seen that Omar solved all fourteen by using the intersection of carefully selected conic sections.

Derbyshire is completely silent on the issue of whether there was algebra in India. So although the Indians knew how to solve quadratic equations by 700 CE, and even used some limited symbolism, there is no mention of these accomplishments in the book. In fact, Brahmagupta, in the 7th century, could solve x² - 10x = -9 using essentially the same algorithm we use today and, besides, made use of the negative numbers that Derbyshire claims were a "fruit of the European Renaissance." We also find Indian mathematicians working out methods for solving (in integers) what has become known as the Pell equation, Dx² + b= y2, where D and b are given integers. For example, in the 12th century, Bhaāskara showed that the smallest solution to 67x² + 1 = y² was x = 5967, y = 48,842.

Derbyshire is on somewhat firmer ground once he moves to Europe and the developments of the 16th and 17th centuries. But even here, there are some serious errors. For example, Cardano, who wrote out the first complete study of the algebraic solution of cubic equations in his famous Ars Magna of 1545, did not "realize that there must always be three solutions" to a cubic. In fact, we read in the first chapter of that work that "if x³ + 6x = 20, there is no other solution than 2" (6). For us today, this equation has two complex solutions, but Cardano, having only a very vague notion of complex numbers, certainly did not assert that a cubic equation could have one real and two complex solutions.

François Viète's attitude toward complex numbers was hardly "retrograde." In fact, he realized that to solve the irreducible case of a cubic (where Cardano's procedure requires complex numbers), he could avoid complex numbers and their uncertain status by using trigonometry. And despite Derbyshire's claim that Viète only dealt with cubics trigonometrically, in the very book (De equationem emendatione) mentioned by the author, Viète presents formulas for the solution of cubic equations closely related to Cardano's procedures (7). Viète could do this, and this needs to be emphasized, because he used letters to represent coefficients in equations, whereas Cardano could only give an algorithm and an example with specific numerical coefficients. As Derbyshire notes, Viète did understand the basics of the relationship between the coefficients and the roots of equations. It is then very curious that while Derbyshire mentions that Albert Girard, in his 1629 New Discoveries in Algebra, generalized Viète's result--because he understood complex solutions--he fails to note that as part of this result, Girard states the fundamental theorem of algebra, the theorem that every polynomial equation of degree n has precisely n solutions, assuming one counts multiples (8). Derbyshire claims that Descartes was the first to state this result, in 1637, and gave it very tentatively. Girard was considerably more assertive about the result, even though (of course) neither he nor Descartes could prove their claim.

But the most serious failing of Unknown Quantity in dealing with the 17th century is the almost total absence of any consideration of the development of analytic geometry by Descartes and Fermat. The realization that geometric curves could be represented by algebraic equations was a crucial step in the major developments in mathematics and in physics toward the end of that century and into the next one. Recall that Galileo showed, using Greek geometrical methods, that a projectile traveled on a parabolic curve, whereas Kepler found, through long calculations, that the planets traveled in ellipses with the sun at one focus. Neither Kepler nor Galileo used algebra at all. But once algebra in its symbolic form was brought into the service of geometry, it was easy to represent parabolas and ellipses; they could be studied and their properties developed. In fact, algebra changed dramatically at this time. Instead of just being able to give the solution to an equation as a number, one could now ask for and find the solution to a problem as a curve. And there were many problems, especially those derived from Newton's physical principles, that required finding curves. Curiously, Newton himself used little algebra in his Principia, but his 18thcentury successors used it extensively.

There is much good reading in Unknown Quantity, and the intended audience can probably learn a lot. That is why it is especially disappointing that Derbyshire failed to check his facts on so many occasions and, evidently, did not have his manuscript read by an expert in the history of the subject. I had hoped, on receiving the book, that I could recommend it to my students. Unfortunately, until the major errors are corrected in a new edition, that is not possible.

References and Notes

1. J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (Joseph Henry, Washington, DC, 2003). Reviewed by B. Conrey, Science 302, 60 (2003).
2. J. Høyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Springer-Verlag, New York, 2002). See especially pp. 55-58.
3. For information on Euclid's Data, see G. L. McDowell, M. A. Sokolik, The Data of Euclid (Union Square, Baltimore, MD, 1993).
4. This translation from Mordecai Finzi's Hebrew version of Abū Kāmil's algebra appears in M. Levey, The Algebra of Abū Kāmil (Univ. Wisconsin Press, Madison, WI, 1966), pp. 186-192.
5. D. S. Kasir, The Algebra of Omar Khayyam (Columbia Teachers College, New York, 1931).
6. G. Cardano, The Great Art: Or, the Rules of Algebra, T. R. Witmer, Transl. (MIT Press, Cambridge, MA, 1968), p. 11.
7. F. Viète, The Analytic Art, T. R. Witmer, Transl. (Kent State Univ. Press, Kent, OH, 1983), pp. 286-289. The trigonometric solution can be found on pp. 174-175.
8. E. Black, Transl., The Early Theory of Equations: Their Nature and Constitution: Translations of Three Treatises by Viéte, Girard, and de Beaune (Golden Hind, Annapolis, MD, 1986). The fundamental theorem of algebra is the first sentence of Girard's theorem II (p. 139), and the result relating symmetric functions of roots to the coefficients is given in the remainder of the theorem.
9. O. Neugebauer, A. Sachs, Mathematical Cuneiform Texts (American Oriental Society, New Haven, CT, 1945), pl. 17.
 

Victor Katz is at the Department of Mathematics, University of the District of Columbia, Washington, DC 20008, USA.  E-mail:  vkatz@udc.edu.