Serafina CUOMO, Ancient Mathematics. London: Routledge, 2001. Pp. xii+290. ISBN 0-415-16495-8.

Review by Roderick Gow

University College
Dublin

University degree courses in mathematics nowadays often include a module on the history of mathematics. There are various reasons why this seems to be sensible pedagogical practice. One is that prospective teachers of mathematics who have an appreciation of the way in which mathematical ideas were discovered should be able to teach the subject more effectively. Another is that such history modules make individuals aware of the contributions of people of differing cultures to the growth and development of mathematical ideas. A striking example of such a contribution is the Hindu-Arabic system of numerals, including the symbol for zero, which slowly displaced the system of Roman numerals during the fifteenth and sixteenth centuries.

The history of ancient Greek geometry is a popular choice for inclusion in a history of mathematics course. Since the 19th century, several such histories have been published, including Greek geometry from Thales to Euclid (1889) by George J. Allman, professor of mathematics at Queen's College Galway for more than four decades. In his book, Allman commented:

It is pleasing to see that the number of students of the history of mathematics is ever increasing; and that the centres in which the subject is cultivated are becoming more numerous; it is particularly gratifying to observe that the subject has at last attracted attention in England.

At the time Allman wrote, much of the scholarly work on the history of mathematics originated in Germany and France. The Danish historian of mathematics Johan Heiberg had just competed an edition of the works of Archimedes, and was currently working on a complete edition of the works of Euclid. In 1906, he discovered a palimpsest containing a lost work of Archimedes on The Method. (The auction sale of this palimpsest for two million dollars in October 1998 in New York caused something of a stir in the usually sedate world of the history of mathematics, partly because the ownership of the work was in question.) The English scholar Thomas L. Heath, who maintained a lifelong interest in the history of Greek mathematics while earning his living in the Treasury in London, was about to issue a work on Diophantus of Alexandria. Among several other books on eminent Greek mathematicians, Heath?s most enduring work is his edition in English of the thirteen books of Euclid?s Elements (1908), comprising three volumes. He discusses, among numerous other things, how the Elements were transmitted to us through the ages, our indebtedness to commentators such as Proclus, translations into various languages, such as that of Gherard of Cremona from Arabic into Latin in the twelfth century, and the history of printed editions. The first one hundred and fifty pages of Heath's work make fascinating reading, and could well form the basis of a topic in a history of mathematics module.

This brings us to an evaluation of the book under review. The history of mathematics books described above was devoted to the highest achievements of Greek geometry and algebra, as exemplified in the works of Euclid, Archimedes and Diophantus. They gave, as Cuomo puts it, an account of the advanced, high-brow practices. On the whole, they offered no indication of how ordinary people used mathematics in their lives, or how surveyors measured the area of land for administrative and taxation purposes. There may have been good reasons for this, as it does not make particularly interesting reading in comparison with, say, the discovery of incommensurable magnitudes or the classification of Platonic solids. Practising mathematicians probably have an elitist view of what is significant in mathematics, and concentrate on study of the most advanced and imaginative aspects of the subject. Cuomo?s intention is different, as she informs us in the introduction that she will give an account of:

. . . ?lower? and more basic levels of mathematics, such as counting and measuring. . . . such a choice will pay off in at least two senses: we will achieve a better-balanced picture, with a full spectrum of activities rather than an isolated upper end; and, since counting and measuring affect a greater section of the population than squaring the parabola or trisecting the angle, looking at them will give us more insight into everyday, everyperson, ?popular? views of mathematics.

I confess that I prefer to read Heath?s introductory account of Euclid?s Elements compared with, say, how arithmetic was used in the Athenian judicial system, as the mathematics involved in the latter is elementary and routine. Serafina Cuomo has published a few research papers, including one on Julius Sextus Frontinus. He wrote on land surveying and on aqueducts, and his work is included in the Corpus Agrimensorum Romanorum. He was in charge of the Roman water supply around the year 100 and sought to account for losses in the system. In the book under review, Cuomo explains how he used arithmetic to estimate the quantity of water entering the system and the quantity arriving in Rome. His accounting identified large-scale fraud in the water supply. This makes reasonably interesting reading, but the mathematical content is negligible. As is well known, there is nothing much to report about the Roman contribution to theoretical mathematics, as the Romans were only interested in practical mathematics suitable for calculation and measurement.

Cuomo is at pains to remind the reader of the paucity of reliable contemporary sources regarding the history of Greek mathematics. On p.39, she writes:

The second section [of Chapter 2] will tackle a historiographical issue: how later ancient sources depict early Greek mathematics, and what can be done with them. It will be, I am afraid, an exercise in scepticism.

On p. 50 (in Chapter 2), she writes:

In an attempt to introduce the novice to the raw business of squeezing reliable evidence out of unlikely informers, I will focus on seven ancient sources (Archimedes, Philodemus, Plutarch, Diogenes Laertius, Proclus, Simplicius and Eutocius), rather than discussing modern reconstructions of early Greek mathematics, especially since they inevitably use those same ancient sources anyway.

She proceeds to give a somewhat polemical discussion of why we should believe what Archimedes writes because we trust him as a mathematician, we trust him as an historian. Yet, as she says:

Nobody I know (not even me) doubts Archimedes? testimony on Democritus and Eudoxus.

She points out possible weaknesses in Proclus?s account of Greek geometry, suggesting that he might have concocted names and lines of transmission of knowledge, to justify his picture of the gradual movement towards perfection of mathematics.  Some of the names he mentions occur nowhere else. Eudemus, a pupil of Aristotle, wrote a history of geometry that is now lost. Proclus claimed Eudemus as one of his sources, but Cuomo sees it as unlikely that Eudemus?s text would have survived until the time of Proclus (mid-fifth century). She questions whether the Eudemus text ever existed, and asks whether it might have been a forgery, presumably to show us not to trust too naively.

As I see it, if we decide that nothing much in the historical sources on ancient Greek mathematics is reliable, and that what has survived is questionable because it sought to promote certain points of view, we will simply be left with dull descriptions of how people surveyed land or counted tax revenue. I certainly found her sustained scepticism a little tedious at times. Thomas Heath presumably used the usual sources, and must be suspect under Cuomo?s harsher criteria of belief, but I his account of the Elements nonetheless. (With regard to reliable sources, evidence of mathematics of the Hellenistic period is much more detailed. Entire treatises have survived, letters are preserved on papyrus, and a papyrus concerned with geometric problems has been found. The earliest extant papyrus fragment seemingly related to Euclid dates from the second century BCE.)

Cuomo?s book is divided into eight chapters, and the subject matter concerns the mathematics of four periods: early Greek, Hellenistic, Graeco-Roman, and late ancient. An odd-numbered chapter presents material evidence of the mathematics of a given period, a succeeding even numbered chapter asks questions about sources. As Cuomo is the author of a book entitled Pappus of Alexandria and the Mathematics of Late Antiquity (CUP, 2000), which has been favourably reviewed in the scientific literature, we would expect her to give some attention to the later ancient period, but she certainly does not do it to excess. Her introduction states clearly what her working method will be, so we cannot really complain too much about the contents, even if they are not to our taste. Indeed, she writes in the introduction:

Moreover, there is a risk that much of what I discuss in the book will not be recognized as mathematics by some of the readers.

There are a few stylistic points that I found annoying. She writes of ?big guys?, as opposed to ?little people?, of a ?dodgy field?, as well as of a ?whiz kid?. This type of language often serves to date a work after a few years. She is also apparently a little too defensive of her subject, which fairly obviously will not appeal to the great mass of people. She writes for instance in the introduction:

In sum, to put it in brutal terms, ancient mathematics can be perceived as mostly incomprehensible and largely irrelevant: a double challenge which this volume will strive to address.

There is little point in even alluding to this possibility, as any academic discipline can be dismissed similarly by those unacquainted with its intricacies and intellectual challenges.

In conclusion, this is not the book that I would choose to recommend to students. If we accept the point of view taken by the author, the book is reasonably well written and remarkably free from typographical errors. I just find the subject matter too dull in parts, but admit that the author is free to develop and illustrate her choice of subjects without following meekly in the footsteps of the previously accepted authorities. There is a good bibliography of twenty pages, which is useful to anybody who wishes to pursue the subject further.