GROWTH AND FORM CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANAGER London: FETTER LANE, E.C. Edinburgh: 100 PRINCES STREET [Illustration] New York: G. P. PUTNAM’S SONS Bombay, Calcutta and Madras: MACMILLAN AND Co., LTD. Toronto: J. M. DENT AND SONS, LTD. Tokyo: THE MARUZEN-KABUSHIKI-KAISHA _All rights reserved_ ON GROWTH AND FORM BY D’ARCY WENTWORTH THOMPSON Cambridge: at the University Press 1917 “The reasonings about the wonderful and intricate operations of nature are so full of uncertainty, that, as the Wise-man truly observes, _hardly do we guess aright at the things that are upon earth, and with labour do we find the things that are before us_.” Stephen Hales, _Vegetable Staticks_ (1727), p. 318, 1738. PREFATORY NOTE This book of mine has little need of preface, for indeed it is “all preface” from beginning to end. I have written it as an easy introduction to the study of organic Form, by methods which are the common-places of physical science, which are by no means novel in their application to natural history, but which nevertheless naturalists are little accustomed to employ. It is not the biologist with an inkling of mathematics, but the skilled and learned mathematician who must ultimately deal with such problems as are merely sketched and adumbrated here. I pretend to no mathematical skill, but I have made what use I could of what tools I had; I have dealt with simple cases, and the mathematical methods which I have introduced are of the easiest and simplest kind. Elementary as they are, my book has not been written without the help—the indispensable help—of many friends. Like Mr Pope translating Homer, when I felt myself deficient I sought assistance! And the experience which Johnson attributed to Pope has been mine also, that men of learning did not refuse to help me. My debts are many, and I will not try to proclaim them all: but I beg to record my particular obligations to Professor Claxton Fidler, Sir George Greenhill, Sir Joseph Larmor, and Professor A. McKenzie; to a much younger but very helpful friend, Mr John Marshall, Scholar of Trinity; lastly, and (if I may say so) most of all, to my colleague Professor William Peddie, whose advice has made many useful additions to my book and whose criticism has spared me many a fault and blunder. I am under obligations also to the authors and publishers of many books from which illustrations have been borrowed, and especially to the following:― To the Controller of H.M. Stationery Office, for leave to reproduce a number of figures, chiefly of Foraminifera and of Radiolaria, from the Reports of the Challenger Expedition. {vi} To the Council of the Royal Society of Edinburgh, and to that of the Zoological Society of London:—the former for letting me reprint from their _Transactions_ the greater part of the text and illustrations of my concluding chapter, the latter for the use of a number of figures for my chapter on Horns.