Cambridge University Press, ISBN: 9781108833844, 500pp, £34.99
The ancient Greeks played a fundamental role in the history of mathematics and their ideas were reused and developed in subsequent periods all the way down to the scientific revolution and beyond. N. is the Patrick Suppes Professor of Greek Mathematics and Astronomy at Stanford University and is widely regarded as the world expert on one of the key figures in this work, Archimedes. We might expect, as described, a "brilliant, pioneering, breathtakingly ambitious book" which is, as Sir Geoffrey Lloyd sums it up very succinctly on the back cover, "...a veritable joy to read".
Any book claiming to be a "new history" is clearly attempting to move us on from the extant "old history" which, in this case was Heath's History of Greek Mathematics, published in 1921. Across the hundred years since its publication, Heath's efforts have been a reliable guide for many generations of scholars. N. urges us to keep Heath by our side but presses us to develop our understanding of how and why the achievements of the ancient Greek mathematicians actually happened. N. is far more concerned with context than content and he planned this book in seven, approximately chronological, chapters that link the mathematics to the overall cultural developments of the time and beyond ancient Greece to the modern world. Each chapter has an outline plan to provide guidance.
The intent in Chapter 1: To the Threshold of Greek Mathematics is to garner together the threads that led to the establishment of mathematics within Greek culture. The beginnings of mathematics in Mesopotamia are traced through the development of symbols being used to describe number (rather than just objects) and the start of numeracy. N. espouses the view that literacy emerged "piggy-backing on numeracy". Others may disagree. Where the Greeks differed greatly from earlier mathematicians was, in a sense, their playfulness with number. In Babylon, mathematics was largely a tool for accounting in its various guises. In Greece, mathematics was often carried out for the simple pleasure of being mathematics. However, the main revelation from this chapter is that Pythagoras and Thales did no mathematics whatsoever. What we call "Pythagoras' Theorem" was certainly not developed by him. Pythagoras was retrospectively reinvested with a new identity; he was identified as author of what was, in fact, an unwritten oral tradition. The same is true of everything ascribed to Thales.
In Chapter 2: The Generation of Archytas the earliest Greek mathematics is surveyed through its three major figures: Archytas, Theaetetus and Eudoxus. There were, of course, many more mathematicians than these three and they are fitted into the narrative as required. The conclusion of this chapter dwells on the discovery of the mathematics of conic sections (primarily by Menaechmus). Surprisingly, the best known mathematicians among the earlier ancient Greeks neither had a name for zero nor did they use a placeholder like the Babylonians.
N. is moving to familiar territory in Chapter 3: The Generation of Archimedes. Those in between Archytas and Archimedes (the likes of Euclid, Timocharis and Aristarchus) are covered first; Euclid in particular. N. leads the reader through the proofs and arguments of Euclid extremely well. One feels that the pace quickens with the entry of Archimedes. He had real problems to solve and the genius to solve them. In truth, Archimedes was more physicist than mathematician but the the position of the dividing line is a moot point. There are suggestions within the work of Archimedes (and, also, Antiphon) that he took the first steps towards developing calculus in his use of the "method of exhaustion". Simply because of his greatness, the contemporaries of Archimedes appear diminutive and N. does not spare many words for the likes of Apollonius, Eratosthenes and others. You may need to reach for your Heath to flesh out your knowledge.
There is something of a pause after this enthralling chapter with Chapter 4: Mathematics in the World. Here, N. looks at mathematics in day-to-day practice and this melts together the historical narratives of Greece and Rome, with the latter becoming politically dominant. Different themes (number, education, geography, engineering and machinery, military tactics) are covered and contextualised.
Then follows Chapter 5: Mathematics of the Stars. This begins with "What Everyone Knew" and the ethnoastronomy of simple societies around the world and not just in Mesopotamia and the Mediterranean. Easily the most important consequence of this early astronomy was the development of the calendar delineated and regulated by observable events. The relatively simple motion of the stars across the sky is, of course, a backdrop to the apparent motion of the moon, the sun and - for those that observe more closely - the wanderings of the planets. What we now often consider to be Greek astronomy arose from a fusion of that known and taught in Mesopotamia seen through Greek eyes. The achievements of Greek astronomy are astonishing. They made very good estimates for the size of our planet, the distance to the moon and the distance to the sun. This all culminated in the work of Ptolemy and his creation of the Almagest. This view of the universe carried on until Copernicus and Kepler.
N. uses the next, relatively short, chapter to look back at what has gone before and the final chapter (Chapter 7: Into Modern Science) to examine how the legacy of Greek science was then carried forward into the Renaissance and beyond, up to the modern day.
There are many diagrams that accompany the ideas. All, without exception, are accurately and informatively constructed. There are several pages of full-colour plates, including a breathtaking view of the Pleiades rising above the Himalayas. The Antikythera mechanism is described fully in the text and a modern reconstruction is also shown in the plates. The bibliography is excellent and the suggestions for further reading are extremely well-chosen. The index is comprehensive.
In short, this book should be in the "essential" category in any collection. This is the story of some of the finest mathematics ever produced. At £34.99 (in the CUP bookshop) it is extremely good value.
John Timney