In this book, Netz argues that the history of mathematics should consist not only of a catalog of which mathematicians worked on what problems when, but also of an analysis of how they conceived of the problems they were solving. He observes that many modern textbooks, sourcebooks, and even more scholarly works on the history of mathematics obscure the line of development when they almost automatically translate earlier works into modern notation. The example he develops throughout is a problem studied by Archimedes: how do you cut a sphere, with a plane going through one of the latitude lines, so that the volume of the bigger part has a certain given ratio to the volume of the smaller part? For Archimedes, this is essentially a geometric problem, to be solved by manipulating geometric objects and their ratios — lines, rectangles, similar triangles, and so on. But for Omar Khayyam, the 11th-12th century mathematician perhaps better known for The Rubaiyat, the sphere-cutting problem is an algebraic problem about squares and cubes.1 Modern treatments of Archimedes almost invariably say “here is where Archimedes solves cubic equations.”2 This is mathematically correct, in that the problem can in fact be expressed in terms of a cubic equation in one variable, to be solved for x. But N’s point is that this is historically false: Khayyam was thinking (more or less) of cubic equations, but Archimedes was not and could not. The book shows how the geometric style of posing and solving problems gives way, in the Arab world, to an algebraic style much more familiar to modern mathematicians.
At this point, regular readers of BMCR may be wondering what this review is doing here. It is a mathematical book, to be sure. But this book, like N’s earlier The Shaping of Deduction in Greek Mathematics (reviewed here, 2000-02-17), has much to tell us about how the Greeks saw the world. Although Greek mathematics is an ancestor of modern mathematics, it is also radically different, highly foreign. N is correct that we lose much of the “Greekness” of the ancient works when we think only in modern terms.
Where The Shaping of Deduction was primarily about Greek mathematicians’ use of language, however, the present book is much more about the actual mathematics. N argues that the choice of techniques for solving a given problem is neither arbitrary nor neutral. Naturally, mathematicians can only use techniques they know about; neither Archimedes nor Khayyam can solve this problem with integral calculus as we might today.3 But within the tool-kit of available techniques, classical Greek mathematicians, in general, make a point of choosing a different method from the last person to work on the problem, while late antique and Arab mathematicians, on the whole, look for the general, over-arching principles behind the problem (p. 187-190).
N suggests that among the Hellenistic Greeks (Archimedes, Dionysodorus, Diocles, as well as their predecessors and contemporaries; 3rd-2nd centuries BC mathematics is a competitive sport. The goal is to demonstrate that you have a better solution than other people: more general, more complete, or more elegant. As N puts it, “the space of [mathematical] communication is an arena for confrontation, rather than for solidarity. The relation envisaged between works is that of polemic. A Greek mathematical text is a challenge” (p. 62). While this was not necessarily true at the very beginning, by the Hellenistic period it does seem to be a significant part of the style of mathematical discourse. To a modern mathematician, used to building on theorems proven by others, even collaborating on joint papers, this seems an unusual way of doing mathematics. One implication of this combative style, according to N, is that mathematics comes to consist of a series of problems to be solved, not a theory, language, or system as we conceive of it.
On the other hand, present-day mathematicians do not behave precisely like the next groups N studies either. These are the mathematicians of late antiquity in the West, and then those of the Arab world. The main figures here are Eutocius (6th c. AD who wrote a commentary on Archimedes which is still one of our main sources for his work, and a series of Arab mathematicians in the ninth through the twelfth centuries. For N, both of these groups have the same aim; their mathematics is fundamentally deuteronomic. By this he means that these writers crucially depend on previous writings. Eutocius, for example, is a commentator on Archimedes in much the same way as Servius is a commentator on Virgil. For a deuteronomic mathematician, the goal is not to find a better solution to a known problem, but instead to standardize and systematize the existing solutions and also their presentations. N points out that this is one of the ways we get “second-order terminology” for mathematical objects and problems (p. 122), terms for describing and classifying, such as characterizing problems as linear, planar, or solid based on the kinds of objects involved in their solutions. Mathematicians like Archimedes want to create an “individual aura” around their solutions (p. 125, et passim), while “Eutocius aims at contextualization, which is the removal of aura” (p. 125). That is, while the earlier mathematicians are competing to find the best or cleverest solution, Eutocius and his contemporaries and successors are concerned with explaining the various solutions and ensuring all the cases are covered.
N suggests that the Arab mathematicians’ project is deuteronomic as well, though not in quite the same way as that of Eutocius. They are attempting to fill in the gaps in Archimedes’ presentation, to complete his work, but not to compete with him (p. 131). That is, when a Greek mathematician, Diocles for example, reads a solution to the sphere-cutting problem that is not fully general, his initial impulse is to say “This is wrong and I can do better.” An Arab mathematician, according to N, might instead say “This is missing something which should be filled in, and I can help.” In N’s terms, this sort of research is just as dependent on the text of Archimedes as is Eutocius’ sort; the difference is more of attitude. It also helps, of course, that Khayyam and the other Arab mathematicians N discusses were considerably more creative as mathematicians than Eutocius was.
The Greek solutions to the sphere-cutting problem are all fundamentally geometric. Archimedes begins by reducing the problem to one involving ratios of rectangles in the plane and rectangular boxes in three dimensions. He then uses similar triangles and conic sections to construct a solution. The largest mathematical difference in the Arab solutions is that they are algebraic: Al-Khwarizmi’s book On the Art of Al-Jabr wa l-Mukabala, published in the ninth century, gave the Arab world another way to look at calculations. The two nouns in the title refer to the operations involved in posing and solving problems like “a given number of quantities plus another given number of roots equals another number,” or, in more modern terms, ax 2 + bx = c. The Greeks never talked about solving an equation for a numerical answer; for them, problems are always posed in terms of lines, squares, rectangles, and ratios. Al-Khwarizmi starts from lines and rectangles, but goes on to describe a method of calculating the measure of the basic line x — and then leaves the idea of lines and measures behind.4 Moreover, he and his successors are interested in determining all the possible problem-patterns of a given class (for example, involving sums of “quantities” (squares), “roots” (the things the quantities are the squares of), and numbers), and figuring out how to solve them.
By the time we get to Khayyam, nearly 400 years after Al-Khwarizmi, the basics of the Art of Al-Jabr have become routine. Khayyam’s Algebra claims to be a complete synthesis of the field, with particular attention to the most difficult problems (p. 146). N analyzes the structure of Khayyam’s text, which is simultaneously a scientific autobiography, a survey of what algebra is, and a series of theorems with their proofs and commentary. Khayyam divides, classifies, lists, and categorizes his theorems and problems, working through each list in a careful and systematic way. He makes references to many earlier mathematicians, both Arab and Greek. As N points out, the method of making exhaustive lists of kinds of problems, then treating them in order, also operates at the level of the individual proofs (p. 153), which are often divided into cases: for example, one length or quantity may be larger than, smaller than, or equal to another one. As each of these three cases may require a different diagram (with the point or line representing the first number being to the right, to the left, or on top of the one representing the other), it is convenient to treat them separately, as mathematicians do to this day.
Archimedes and Khayyam, then, approach the sphere-cutting problem in essentially different ways. “Archimedes’ problem arises, as it were, in ‘real-life geometry,’ and its shape is determined by the demands of this ‘real-life geometry.’ Khayyam’s problem arises from its position in a list of problems — the list deriving not from an external, geometrical investigation, but from its own independent listing principle” (p. 183). More fundamentally, Khayyam steps back from the particular task of cutting a particular sphere and looks at an entire class of similar problems. Algebra itself, N argues, is a way to “step back” from geometry and look more abstractly at problems; it is intrinsically deuteronomic, in N’s sense. In short, “Hellenistic Greek mathematics, whose practice may be summed up by the aura, foregrounded the local characteristics of configurations, giving rise to the problem; medieval mathematics, whose practice may be summed up by deuteronomy, foregrounded the global characteristics of relations, giving rise to the equation” (p. 190; italics original). The cultural differences between Greek and later mathematicians and their preferred approach to mathematics produce significant differences in the kinds of mathematics they end up doing. Simply stating, as some modern books do, that Archimedes and Khayyam are both solving cubic equations, while true at some level, obscures this cultural difference. Mathematical progress involves not only the ability to solve more problems, but also the ability (and even the desire) to pose different problems, solve them in different ways, and compare the solutions.
N’s style throughout is engaging, almost conversational, with many rhetorical questions and other addresses to the reader. The text is very well organized: each of the three chapters begins with an overview, and section titles and running heads keep the structure clear for the reader. N has translated all the Greek and Arabic texts; not a word of Arabic, and very little Greek, appears in the book (and the few Greek words are transliterated). As a result this book is a more mathematical companion piece to The Shaping of Deduction, which was largely concerned with the language of the Greek texts. Here N is more concerned with the mathematics itself than the words in which it is presented, so translation is a reasonable choice and makes the book accessible to a larger group of potential readers. He does, of course, give references to standard editions of the various texts.
The history of mathematics is generally studied in mathematical terms, looking at questions like who solved a given problem first, whether someone’s announced or rumored proof really existed and was sound, whether various people worked together or knew each other’s work. As N points out, often the mathematics in question is expressed in modern terms, however different they might be from those used by the scholars under study.5 N’s program, applying the tools of philology to historical mathematical texts, is highly original and beautifully creative; it shows off features of Greek mathematics that many modern mathematicians have ignored. The present book is useful reading not only for historians of mathematics, but for anyone interested in how the Greeks understood the world.
1. I give all Arabic words, including proper names, in N’s transliteration.
2. For example, Ivor Thomas, in the Loeb Greek Mathematical Works (Cambridge: 1941, rpt. 1963), vol. 2, p. 126-163, gives the text under the title “Solution of a Cubic Equation,” and gives an interpretation in modern notation in a footnote. Similarly, a standard introductory textbook in history of mathematics talks of Archimedes “reducing his cubic equation” to a quadratic (Carl B. Boyer, A History of Mathematics (New York: Wiley, 1968), p. 147).
3. Although Archimedes uses several arguments that look very much like limit processes, infinite sums, or integrals, he never relies on these for a proof. They are part of the analysis, not the synthesis; part of figuring out the answer, but not part of demonstrating its correctness. Examples and details in Thomas.
4. As is well known, Al-Khwarizmi’s name has become the standard word for a method of working a problem or calculating something: an algorithm. And the method of Al-Jabr, reducing a problem to canonical form by moving a term from one side of the equation to the other, gives its name to algebra.
5. This is not invariably true, particularly for more recent, less radically foreign mathematics. Studies of the development of differential calculus, for example, generally mention the parallel development of notation for referring to derivatives and differentials. See for example Boyer p. 441, 533.