Most mathematicians who know anything about the history of their subject know the standard story about Greek mathematics. David Fowler describes that story like this:

The early Pythagoreans based their mathematics on commensurable magnitudes (or on rational numbers, or on common fractions m/n), but their discovery of the phenomenon of incommensurability (or the irrationality of the square root of 2) showed that this was inadequate. This provoked problems in the foundation of mathematics that were not resolved before the discovery of the proportion theory that we find in Book V of Euclid's *Elements* (p. 356).

As he points out, we have all heard, and some of us have told, that story.

It may be somewhat surprising, then, to discover that many, if not most, historians of ancient mathematics disbelieve some or all of this story. In *The Mathematics of Plato's Academy*, David Fowler gives a convincing account of the reasons for rejecting the standard story, and offers a very interesting alternative reconstruction of the history of early Greek mathematics.

Fowler's case against the usual story is complex. It involves a mathematical reconstruction, an assessment of the available evidence, a careful examination of papyrological evidence for everyday arithmetic in ancient Greece, and lots more. His book is an education in and of itself, leading us through various approaches to studying the history of ancient mathematics, and arguing forcefully for his version of that history.

The focus here is on *early* Greek mathematics, and particularly the mathematics attached to such names as Theaetetus, Eudoxus, and Archytas, all of which are associated to Plato's Academy, and all of which are thought to have produced some of (or much of, depending on whom you ask) the material now to be found in Euclid's *Elements*. Fowler focuses mostly on their work on the notions of ratio and proportion, which of course is directly connected to the issue of incommensurability. On the Pythagoreans, Fowler for the most part accepts the (largely negative) conclusions of Walter Burkert in his famous *Lore and Science in Ancient Pythagoreanism*. On later Greek mathematics, Fowler basically only comments on what is relevant to the reconstruction of the early material. The *Elements* (especially books II, X, and XIII) are used as a source for material that is presumed to be early; a tantalizingly short appendix gives some hints of Fowler's views on how the Elements came to be compiled.

One of the important themes in Fowler's book is that one must carefully assess the available sources on early Greek mathematics. He demonstrates that most of the elements of the "standard story" either come from very late Greek sources (500 years or more after the time of Plato) or are reconstructions by modern historians who are really reflecting nineteenth and twentieth century concerns about foundational issues.

He calls our attention, then, to two kinds of evidence: very early material, especially from Plato and Aristotle, and what he calls "homogeneous slabs of early material" to be found embedded in, for example, the *Elements*. Examples of such "homogeneous slabs" are Books II, X, and XIII: the first contains the so-called "geometric algebra", the second (sometimes called "the cross of the mathematicians" because it is so hard to understand) contains a complicated classification of various kinds of incommensurable lines, and the third classifies lines that arise from the construction of the regular polygons and polyhedra. Another important source is Plato's mathematical curriculum as explained in the *Republic*. Fowler argues that a good reconstruction must offer a rationale for these blocks as coherent wholes, and he finds that the standard story doesn't offer such an interpretation. For example, the standard story would suggest that the crucial distinction would be between commensurable and incommensurable lines, but that isn't the way Book X organizes the world at all.

Fowler's counterproposal is very interesting. I would summarize it in four assertions:

**Greek geometry was not arithmetized.** In other words, the way we automatically connect the notion of "length" or "area" to numbers is something completely foreign to Greek mathematics. This is perhaps what makes it so hard for us to think mathematics in the Greek way. The idea that a length is a number is so deeply ingrained in our thought that it takes a conscious effort to conceive of an approach to geometry that does not make such an assumption. It is such an arithmetized interpretation that led historians to describe Book II of the elements as "geometric algebra". Fowler argues that Greek geometry was completely non-arithmetized. The strongest evidence comes from his analysis of the very difficult Book X, where he shows, I think successfully, that the way Euclid (or Theaetetus?) structures the argument precludes an arithmetical approach.

Most historians would agree with this description of Greek geometry, I think. Many, however, see this characteristic as *something to be explained*. Those committed to the "standard story" would probably argue that the only way one can arrive at a non-arithmetized geometry is by *de-arithmetizing* a "more normal", arithmetized geometry (specifically, Babylonian geometry). Fowler turns this on its head, and argues that in fact the non-arithmetized nature of Greek geometry should be read as evidence that Greek geometry does *not* derive from Babylonian sources. Essentially, Fowler argues that this approach to geometry is just as "natural" as the arithmetized one that we now use, and that it therefore it's not necessary to explain why the Greeks used it.

**Greek arithmetic was entirely done with "parts".** By "parts", Fowler means what are usually called "unit fractions", or sometimes "Egyptian fractions". An extensive review of the papyrological data shows that, contrary to what some older historians asserted, there is no evidence that the Greeks had any sort of notion of "common fraction". Instead, their everyday arithmetic was done in the Egyptian style, in terms of "parts", using "division tables" which contain lines such as

of the 2 the 9' is 6' 18'

which means that the ninth part of 2 is 1/6+1/18. There are two important consequences. First, there is no evidence for a concept of "common fraction" (which the standard story assumes). Second, this system makes arithmetic (especially multiplication and division) much less transparent than we think it is.

**The early Greek mathematicians worked with several different concepts of "ratio", the most important of which was the "anthyphairetic".** Fowler's most crucial contention is that the mathematicians surrounding Plato actually had a notion (in fact, several notions) of ratio (and not merely a notion of proportion). This is a crucial distinction. A modern mathematician moves almost effortlessly from a definition of proportion ("A is to B as C is to D") to the definition of ratio as a kind of equivalence class. Fowler argues that this move simply never happened in mathematics until the nineteenth century, and hence we have to investigate what the Greeks meant by a ratio. Euclid's definition is vague. Fowler argues that there were three competing definitions: one coming from music theory, one coming from astronomy (this is the one that underlies Book V of the *Elements*), and one based on "anthyphairesis", or reciprocal subtraction. It's the last one that Fowler thinks is most interesting. He argues that the Greek mathematicians would understand, say, the ratio 22:6 by saying

You can subtract 6 from 22 **three times** and have 4 remaining.

You can subtract 4 from 6 **once** and have 2 remaining.

You can subtract 2 from 4 **twice**, exactly.

And so the ratio 22:6 would be defined as the sequence "three times, once, twice exactly", which Fowler denotes by [3,1,2]. This allows one to define and compare ratios without requiring a concept of rational (much less real) numbers.

**Behind Books II and X of the ***Elements* stands an anthyphairetic investigation of quadratic ratios. If one tries to compute the ratio between the diagonal of a square and its side in the above manner, one gets "once, twice, twice, twice, ....". It's not hard to give a proof, in a Greek geometric style, that the sequence goes on forever (and the incommensurability follows!). Fowler suggests that an investigation of the ratios between sides of squares is behind the theorems in Book II, and that the distinction between numbers that have a nice anthyphairesis and those that do not is behind Book X. He also argues that when Plato complains about the fact that people have neglected solid geometry, he is referring to the analogous problem for sides of *cubes*, which is, as Fowler points out, essentially unsolved even today.

This reconstruction is fascinating, and the chapters in which Fowler gives an account of all this in "Greek style" are quite exciting. It all "fits" (As Fowler himself says, it fits almost too well!), and gives a coherent account of what is behind books II and X of the Elements. The resulting picture is quite different from the traditional one. In particular, incommensurability is seen not as a foundational crisis but rather as an interesting discovery that led to significant mathematics.

From a modern point of view, "anthyphairesis" is basically the theory of "continued fractions". Fowler includes a fascinating chapter on the later history of this theory. This includes a survey of the very important role continued fractions played in early modern mathematics, and a conjecture about the reason why they seem to have lost the importance they once had. This chapter also includes a (difficult) section on the recently-discovered algorithms for doing arithmetic with continued fractions.

Should we believe Fowler's reconstruction of early Greek mathematics? I certainly don't know, but I think it makes for a fascinating story! One thing is for sure, though: after reading this book, there is no returning to a naive view of Greek mathematics. And that, I think, is a good thing.

Why should you read this book? The best reason is simply that it's an exciting, fascinating book, full of neat ideas, interesting mathematics, and valuable information. It contains some serious mathematics that is interesting in and of itself. In fact, Fowler suggests that this material is a good way to get beginners started:

This route... is one of the quickest into the mathematical way of thinking. The *Parmenides* algorithm applied to \(\sqrt{n}:\sqrt{m}\) — or any of its equivalent manifestations — leads directly into subtle, surprising, and useful mathematics whose exploration will initiate and train a gifted novice (p. 398).

I think he way well be right, and those of us who teach undergraduates should take notice.

A further reason to read this book is that it'll change the way you think about the history of mathematics. Fowler's approach takes both history and mathematics seriously in the way it makes an effort to think in a different way, to shed the biases and concerns of our time and try to reconstruct the questions and issues of the ancient Greeks. The book may even get you talking to the Classicist in the other building…

This is the second edition of Fowler's book. I don't have access to the first edition, so I can't say for sure, but there seems to be substantial new material here, both throughout the book and also concentrated in a final chapter. This chapter contains an assortment of new material added for this edition, including a long section specifically on incommensurability which Fowler recommends should be read before anything else. The additions are quite valuable, and are full of interesting hints about further directions of study. I suspect that even if you do have the older version, you'll want to get (or, given the price, have your library get) a copy of the second edition.

For some reason, the publisher has chosen to print the book on glossy paper. This kind of paper is often used for plates and photographs, but it feels strange as a choice for the whole text. It is also quite heavy, giving the book significantly more heft than its size suggests. Other (relatively minor) production issues are the placement of the plates (given the choice of paper, they could be anywhere, so why not place them near the pages that refer to them?) and the frontspiece (a reproduction of Raphael's fresco of "The School of Athens"), which is too small to be useful, particularly given the details in the accompanying note. The quality of the content would have justified more care with the physical package.

Should we regret losing the old story about incommensurability? I don't think so, because the various "new" stories are much more interesting. We can no longer argue that the foundational issues that disturbed the Greeks were the same ones that led Dedekind to finally define what real numbers are -- but that was never really an honest argument. Instead, Fowler gives us a different story, one that includes some very interesting mathematics that leads quickly to problems that are still unsolved and a very different account of the complexities surrounding the notion of real numbers. On top of that, we get a much deeper way of thinking about the whole corpus of Greek mathematics.

I'll be using ideas from *The Mathematics of Plato's Academy* over and over in my classes; it's that kind of book: generous and inspiring, full of ideas and information, a book to which one returns over and over. And I'll be on the lookout for more of Fowler's writing! This one is a must-read if you have even the slightest interest in Greek mathematics, and a recommended read even if you don't.

Fernando Q. Gouvêa ([email protected]), an alumnus of the MAA-sponsored *Institute for the History of Mathematics and its use in Teaching*, is associate professor of Mathematics at Colby College in Waterville, ME, and editor of MAA Online.