Surfing Ancient Mathematics

May 18, 2009

Statue of Xu Guangqi in Shanghai.

Book Review
Philip J. Davis

The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Source Book. Individual sections by Annette Im-hausen, Eleanor Robson, Joseph Dauben, Kim Plofker, and J. Lennart Berggren. Victor Katz, editor, Princeton University Press, Princeton, New Jersey, 2007, 685, $75.

When I was a young and cocky graduate student, I had occasion to be a guest at a lunch at which the eminent archaeologist and biblical scholar William Foxwell Albright of Johns Hopkins was also a guest. During a gap in the conversation, I turned to Albright and asked, What is the point of digging up the remote past? Without answering, he gave me a searing look that said simply, Wise guy kid. This calls to mind the story of a student who asked Euclid what he would gain from learning mathematics. Euclid disdainfully said to an assistant, Give the boy a couple of coins for he wants to profit from what he learns.

What are the uses of history, and in particular the history of mathematics? Skeptics ask a standard question of scholars: What's in it for me (or for society)? What have I and society gained by keeping you scholars in bread and butter while you pursue your odd activities? This question has been asked for many crafts and disciplines, and there are answers galore. History locates us in the flow of time and ideas. Mathematical history shows its material as a human product subject to the strengths and limitations of human understanding and invention, and not as a doctrine that seeps down gradually from a platonic heaven. It displays the development of the manipulative or symbolic sense, as well as that of formal beauty. It teaches us what it means to be human by pointing to common cultural traits and universalities. By making us aware that individual genius exists, it makes us modest. It displays what is considered the best, the most useful of the past.

As a rank amateur in the field of ancient mathematics, I face the problem in writing this review not of dealing directly with the ancient texts in the collection, as explicated by the five scholars mentioned above, but of finding a way to work around the texts. I do this by offering a number of disconnected thoughts that occurred to me while and after "surfing" the book.

Each of the articles in The Mathematics of Egypt . . . is substantial, averaging more than a hundred pages, and each might easily be expanded into an individual book. Each is rich in both the presentation and the interpretation of texts. Selected portions of the ancient texts could be productively built into courses on the history of mathematics or science. The book is also rich in material on the cultural background of the respective periods, conveying how the methods were taught to students and how the texts were recovered, along with stories, opinions, and arguments of modern scholars who have translated and interpreted the texts. This portion of the material is both illuminating and within reach of the non-specialist.

In the introduction, the editor complains that "standard" English-language histories of mathematics give short shrift to the material covered by the collection under review. Correct. I found in Morris Kline's Mathematical Thought from Ancient to Modern Times 38 pages (in a total of 1211) devoted to such material, and in the more recent The Mathematical Sciences: The Rainbow of Mathematics, by Ivor Grattan-Guinness, about 59 of 817 pages.

The amount of mathematics presented in the material under review is amazing. One might say, roughly, but with several exceptions, that all of mathematics now normally taught, up to the calculus, can be found---along with much more. The major exception---and it is significant---is the axiomatic method of demonstration as developed by the Greeks (and, of course, later picked up by Muslim and Chinese mathematicians). And yet, the large collection of texts under review does not contain the full story of the spread of mathematics. It omits, for example, Japanese, Polynesian, North and South American mathematics. And this is not to mention cultures like that of the young, largely uneducated street sellers in Rio de Janeiro, who, spurning the arithmetic they learned in school, have devised their own special methods for dealing with money. Indeed, there have been very few non-numerate cultures or constituencies.

The introductory words of ancient mathematical texts are fascinating in their diversity. An ancient Egyptian papyrus, c. 1000 BC, throws out a challenge:

"Hey, Mapu, vigilant scribe who is at the head of the soldiers, distinguished when you stand at the great gates, bowing beautifully under the balcony. A dispatch has come from the crown prince to the area of Kato please the heart of the Horus of God. An obelisk has been newly made, graven with the name of his majesty, may he live, prosper and be healthy, of 110 cubits in the length of its shaft, its pedestal of 10 cubits, . . ."

The Preface to the Classic of Master Sun (c. 470) reads:

"Master Sun says: mathematics [governs] the length and breadth of the heavens and the earth; [affects] the lives of all creatures; [forms] the alpha and omega of the five constant virtues (i.e., benevolence, righteousness, propriety, knowledge, and sincerity); [acts as] the parents for yin and yang . . . . and determines the principles of the six arts (i.e., propriety, music, archery, charioteership, calligraphy, and mathematics)."

From medieval Islam, the comments of Banu Musa (c. 830) on the Conics of Apollonius begin:

"In the Name of God, the merciful, the forgiving. I have no success except through God."

This sort of acknowledgement apparently was common: pro forma, yet an indication of the supporting role of religion. It puts me in mind of the "secular acknowledgments" placed today as footnotes on the opening page of research papers: This work was performed with the partial support of the Busy Bee Foundation, Grant No. BZ 34279301.

Ancient mathematics as described here is frequently rooted in applications. The positioning of mathematical texts in everyday problems shows mathematics as an integral part of the respective cultures; an example is Su Mi's discussion of the exchange rates for prices of various grains.

Mathematics was developed, perfected, and pursued by scribes and priests, among others, and was not generally public knowledge. Have we improved in this respect? It seems to me that things are not much different today. Mathematics is installed in our daily life by an elite. The general public---even the educated public---doesn't know mathematics, doesn't want to know it, and is hardly aware of where it resides in applications. Mathematics is part of our "hidden" technological infrastructure.

The pure and the abstract are also present in these texts in significant measure. As soon as an arithmetic table is detached from the price of onions or the volume of a basket, mathematics has gone abstract. The abstract often hides behind a facade of the applied. Thus, a problem from ancient Egypt asks for the length and width of a rectangular sail, given its area and the ratio of its sides. A Mesopotamian problem asks for the area of a crescent moon given the lengths of its two arcs. From China: Find the volume of the frustum of a pyramid, given the area of its lower and upper sections and its altitude.

What would be gained if our generally "eurocentric" mathematics histories devoted greater percentages of their totals to material like that presented here? Is there any reason (other than political/social) for them to do so? The "pros" might say that historians who downplay such material present an incomplete picture of our subject. The "antis" would counter that all histories are incomplete and simply reflect the personal predilections (a.k.a. the agendas) of the authors.

Having socialized for years with the members of the (late) Brown History of Mathematics Department, a department with a great international reputation, I have a question that to me is more interesting than the standard one: What's in it for those of you who work on the history of mathematics? What first hooked you? What have been your emotional or other rewards? I know that there is the challenge of the material, of making sense of texts in terms of the cultures in which they were embedded.

But what lies still deeper? I know that what drove Professor Albright was his desire to validate Biblical history via archaeology. Presumably, I might find answers in the case of my personal acquaintances by reading autobiographic material, but such material is scanty. I must go the subjective route by inference. What I have sensed to be operative are the desires (to which I don't attach specific names):

This is no easy task. In her chapter, cuneiformist Eleanor Robson wrote:

"It will never be possible to comprehend this [Mesopotamian] mathematics as it was meant to be read, for we cannot entirely escape our own twenty-first century lives and brains and training, however hard we try. . . ."


As a person who has enjoyed the history of mathematics since my high school days, I dedicate this review to the memory of the unique Department of the History of Mathematics at Brown University, which succumbed to the academic disease of provostitis.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at

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