OF GREEK MATHEMATICS BY SIR THOMAS HEATH K.C.B., K.C.V.O., F.R.S. SC. D. CAMB.; HON. D.Sc. OXFORD HONORARY FELLOW (FORMERLY FELLOW) OF TRINITY COLLEGE, CAMBRIDGE An independent world, Created out of pure intelligence.' WORDSWORTH. VOLUME I FROM THALES TO EUCLID 1843 OXFORD AT THE CLARENDON PRESS QA22 H43 OXFORD UNIVERSITY PRESS London Edinburgh Glasgow Copenhagen New York Toronto Melbourne Cape Town Bombay Calcutta Madras Shanghai HUMPHREY MILFORD Publisher to the University 1829 658355 PREFACE THE idea may seem quixotic,. but it is nevertheless the author's confident hope that this book will give a fresh interest to the story of Greek mathematics in the eyes both of mathematicians and of classical scholars. For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring. The interest of the subject for the classical scholar is no doubt of a different kind. Greek mathematics reveals an important aspect of the Greek genius of which the student of Greek culture is apt to lose sight. Most people, when they think of the Greek genius, naturally call to mind its masterpieces in literature and art with their notes of beauty, truth, freedom and humanism. But the Greek, with his insatiable desire to know the true meaning of everything in the universe and to be able to give a rational explanation of it, was just as irresistibly driven to natural science, mathematics, and exact reasoning in general or logic. This austere side of the Greek genius found perhaps its most complete expression in Aristotle. Aristotle would, however, by no means admit that mathematics was divorced from aesthetic; he could conceive, he said, of nothing more beautiful than the objects of mathematics. Plato delighted in geometry and in the wonders of numbers; ἀγεωμέτρητος μηδεὶς εἰσίτω, said the inscription over the door of the Academy. Euclid was a no less typical Greek. Indeed, seeing that so much of Greek is mathematics, 83826 མ it is arguable that, if one would understand the Greek genius fully, it would be a good plan to begin with their geometry. The story of Greek mathematics has been written before. Dr. James Gow did a great service by the publication in 1884 of his Short History of Greek Mathematics, a scholarly and useful work which has held its own and has been quoted with respect and appreciation by authorities on the history of mathematics in all parts of the world. At the date when he wrote, however, Dr. Gow had necessarily to rely upon the works of the pioneers Bretschneider, Hankel, Allman, and Moritz Cantor (first edition). Since then the subject has been very greatly advanced; new texts have been published, important new documents have been discovered, and researches by scholars and mathematicians in different countries have thrown light on many obscure points. It is, therefore, high time for the complete story to be rewritten. It is true that in recent years a number of attractive histories of mathematics have been published in England and America, but these have only dealt with Greek mathematics as part of the larger subject, and in consequence the writers have been precluded, by considerations of space alone, from presenting the work of the Greeks in sufficient detail. The same remark applies to the German histories of mathematics, even to the great work of Moritz Cantor, who treats of the history of Greek mathematics in about 400 pages of vol. i. While no one would wish to disparage so great a monument of indefatigable research, it was inevitable that a book on such a scale would in time prove to be inadequate, and to need correction in details; and the later editions have unfortunately failed to take sufficient account of the new materials which have become available since the first edition saw the light. The best history of Greek mathematics which exists at present is undoubtedly that of Gino Loria under the title. Le scienze esatte nell' antica Grecia (second edition 1914, |