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A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that "E is dense in E" does not mean the same thing as "E is dense in itself."
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
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Ancient Mathematics. Serafina Cuomo. 2001. xii, 290 pp., illus. maps, hardback $ 125.00, paperback $ 37.95, e-book (eISBN 0-203-99573-2). ISBN 0-415-16494-X. New York: Routledge: www.routledge-ny.com).
If nothing else, the reader of Ancient Mathematics will discover that people living and working in the Greek and Latin speaking Mediterranean world of ancient times could no more avoid mathematics in their lives than we can today. Contractors and surveyors, merchants, lawyers, and farmers, not to ignore teachers, students, and divines are just a few of the practioners of mathematics within the time-frame of this book, the thousand years that ended about 500 A.D. Not only does Serafina Cuomo write of “the usual suspects” (Euclid and Archimedes, Pappus, Diophantus, and Ptolemy), but she also introduces people not so often encountered in standard histories of mathematics like Vitruvius, Nipsus, and Balbus, Cicero and Saint Augustine. She sets definite goals for the book and chooses appropriate means to accomplish them. The primary goal was to demonstrate that ancient mathematics is comprehensible and relevant within its historical context. She accomplished this by a careful selection of examples and direct quotations from documents and treatises about counting, measuring, and applying mathematics that are not in the mainstream of the history of mathematics. Additionally, she fit some “high end” mathematics into her discussion by showing how it was part and parcel of contemporary life. The mathematics she reviews encompasses early Greek (the peninsula), Hellenistic (Greater Greece), Graeco-Roman (Roman Empire), and late Ancient Mathematics (first centuries A.D.). Holding her material together is a unique format: the evidence found in each time division and questions prompted by the evidence, are considered in that order.
The questions are clearly stated or discernible. For the first division she seeks answers to (1) “What were the political functions of early Greek mathematics?” (2) “How did later ancient sources depict early Greek mathematics?” and (3) “What can be done with them?” For Hellenistic mathematics, she inquiries (1) “What can really be said about the authorship of Euclid and the Elements ?” (2) “Who were the members of ‘communities of mathematicians’?” and (3) “How did they relate to ‘the wider context of Hellenistic culture’?” The questions developed from Graeco-Roman evidence are (1) “How did ‘Cicero, Plutarch, the myth of Archimedes and the Greek/Roman divide’ (the mathematics that divided the two cultures) impact the history of mathematics?” (2) “What boundaries did [the mathematicians] draw?” and (3) “How did they see themselves?” The final section, late Ancient Mathematics, investigates (1) “What did ‘Christian authors’ offer mathematics?” and (2) “What is the relationship between the mathematics of this period and what preceded?”.
Dr. Serafina Cuomo of Imperial College, London: www3.imperial.ac.uk/people/s.cuomo, organized a wealth of information about the four corners of her topic, judiciously allowing for meaningful overlap of topics and personalities. In addition to the personalities noted above, she enhances our appreciation of Plato and Aristotle, Iamblicus, Hero of Alexandria, and Proclus. While a number of passages read as though they were part of or entire lectures, she keeps one’s interest flowing. Perhaps she may want to let the evidence speak for itself, rather than to inform the reader of her skepticism (p. 39) before presenting the evidence. A later edition would do well to include in the glossary numerous technical terms, particularly those from the Latin or Greek. Because of the descriptive passages from ancient sources, many of which are her own translations, the book would be a worthwhile addition to a school library.
Barnabas Hughes,O.F.M, Professor Emeritus, California State University, Northridge