Cambridge Library Co ll e C t i o n Books of enduring scholarly value Classics From the Renaissance to the nineteenth century, Latin and Greek were compulsory subjects in almost all European universities, and most early modern scholars published their research and conducted international correspondence in Latin. Latin had continued in use in Western Europe long after the fall of the Roman empire as the lingua franca of the educated classes and of law, diplomacy, religion and university teaching. The flight of Greek scholars to the West after the fall of Constantinople in 1453 gave impetus to the study of ancient Greek literature and the Greek New Testament. Eventually, just as nineteenth-century reforms of university curricula were beginning to erode this ascendancy, developments in textual criticism and linguistic analysis, and new ways of studying ancient societies, especially archaeology, led to renewed enthusiasm for the Classics. This collection offers works of criticism, interpretation and synthesis by the outstanding scholars of the nineteenth century. A Short History of Greek Mathematics James Gow’s Short History of Greek Mathematics (1884) provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. Particular attention is given to Pythagorus, Euclid, Archimedes, and Ptolemy, but a host of lesser-known thinkers receive deserved attention as well. Cambridge University Press has long been a pioneer in the reissuing of out-of-print titles from its own backlist, producing digital reprints of books that are still sought after by scholars and students but could not be reprinted economically using traditional technology. The Cambridge Library Collection extends this activity to a wider range of books which are still of importance to researchers and professionals, either for the source material they contain, or as landmarks in the history of their academic discipline. Drawing from the world-renowned collections in the Cambridge University Library, and guided by the advice of experts in each subject area, Cambridge University Press is using state-of-the-art scanning machines in its own Printing House to capture the content of each book selected for inclusion. 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A Short History of Greek Mathematics James Gow Cambridge Universit y PresS Cambridge, New york, Melbourne, Madrid, Cape Town, Singapore, São Paolo, Delhi, Dubai, Tokyo Published in the United States of America by Cambridge University Press, New york www.cambridge.org Information on this title: www.cambridge.org/9781108009034 © in this compilation Cambridge University Press 2010 This edition first published 1884 This digitally printed version 2010 ISBN 978-1-108-00903-4 Paperback This book reproduces the text of the original edition. The content and language reflect the beliefs, practices and terminology of their time, and have not been updated. Cambridge University Press wishes to make clear that the book, unless originally published by Cambridge, is not being republished by, in association or collaboration with, or with the endorsement or approval of, the original publisher or its successors in title. A SHORT HISTORY OF GKEEK MATHEMATICS. C. J. CLAY AND SON, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. : DEIGHTON, BELL, AND CO. ILcipjtg: F. A. BEOCKHATJS. A SHOET HISTOEY OF GREEK MATHEMATICS BY JAMES GOW, M.A. FELLOW OP TBJNITY COLLEGE, CAMBRIDGE, Tlap' EiKXeidjj' ns &pi-d/j.evos yew/ieTpelv, us rb TpQrov $eiipr;/xa tfia6ev, rjpeTo rbv EfeXe/Sijc • rl Si /tot irKtov larai ravra ixavBAvovn; nal b EUKXC£5?/S rbv ira?Sa tcaX^aas' 56s, ^fjt atfrtp Tpi&fiokov, iireiSri 5e? air!}, e£ uv fj.avda.pei, ttepSalvuv, SXOBABUS, Flor. iv. p. 205. EDITED FOR THE SYNDICS OF THE UNIVERSITY PRESS. CAMBRIDGE: AT THE UNIVERSITY PRESS. 1884 PRINTED BY C. J. CLAY, M.A. AND SON, AT THE UNIVEBSITY PRESS. PREFACE. THE history of Greek mathematics is, for the most part, only the history of such mathematics as are learnt daily in all our public schools. And very singular it is that, though England is the only European country which still retains Euclid as its teacher of elementary geometry, and though Cambridge, at least, has, for more than a century, required from all candidates for any degree as much Greek and mathe- matics together as should make this book intelligible and interesting, yet no Englishman has been at the pains of writing, or even of translating, such a treatise. If it was not wanted, as it ought to have been, by our classical professors and our mathematicians, it would have served at any rate to quicken, with some human interest, the melancholy labours of our schoolboys. The work, as usual, has been left to Germany and even to France, and it has been done there with more than usual excellence. It demanded a combination of learning, scholarship and common sense which we used, absurdly enough, to regard as peculiarly English. If anyone still cherishes this patriotic delusion, I would advise him to look at the works of Nessel- mann, Bretschneider, Hankel, Hultsch, Heiberg and Cantor, or, again, of Montucla, Delambre and Chasles, which are so frequently cited in the following pages. To match them we can show only an ill-arranged treatise of Dean Peacock, many brilliant but scattered articles of Prof. De Morgan, and three essays by Dr Allman. I have treated all these writers with VI PREFACE. freedom and have myself added matter which is not to be found in any of them, but they strike me still with humiliation such as a classical scholar feels when he edits a text which Bentley has edited before him. My own book represents part of a collection of notes which I have for many years been making with a view to a general history of the great city of Alexandria. The fact that the history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several pre- ceding centuries, made it necessary that I should extend my inquiries over the whole field of Greek mathematics. In this way, the materials for an account of the Alexandrian Mathematical School grew to exceed the reasonable limits of a chapter, and I have thought it desirable to publish them as a separate essay. I shall treat the Literary School with the same fulness. As a history of Alexandria ought to be interesting to most people, I took especial pains that my treatment of the Mathematical School, which was the oldest, the most con- spicuous and the longest-lived of them all, should not be excessively technical. I have tried to put my account of it generally in such a form as should be useful and attractive to readers of various tastes. As a matter of fact, mathematicians will here find some account of every extant Greek mathe- matical book and a great number of pretty proofs translated from the ipsissima verba of the ancients. Greek scholars will find nomenclature and all manner of arithmetical symbols more fully treated than in any other work. A student of history, who cares little for Greek or mathematics in par- ticular, but who likes to watch how things grow, will be able to extract from these pages a notion of the whole history of mathematical science down to Newton's time, and will find PREFACE. VII some very curious questions raised which it is his especial duty to answer. It was impossible to satisfy the requirements of all readers, but each will perhaps be willing to concede something to the claims of the others, and wherever a subject is introduced but inadequately treated, I have at least given references to sources of fuller information, if any such exist to my knowledge. As the whole book is an endeavour to compromise between conflicting claims, I have allowed myself, with the same intent, some inconsistency in two details. In the first place, I have not drawn a strict line between pure and mixed mathematics, but have given an account of the Phaenomena and Optics of Euclid and the mechanical books of Aristotle and Archimedes, while I have omitted any summary of Ptolemy's astronomical theories. The former are books which are little known, which are short, which came in my way and which are almost purely deductive. A com- plete history of Greek astronomy is tolerably common, is long, is founded for the most part on non-mathematical writers and would consist largely of a history of astronomical observations. In the second place, I have tried to write proper names (following indeed Smith's Dictionary of Greek and Roman Biography) in a way which should generally indicate the Greek form and pronunciation without offending the ordinary eye.