PREFACE 



IN this book I have attempted to give some account of 

 the chief results emerging from a study of the relative 

 growth of parts in animals which I have undertaken 

 during the last ten years. I have tried to correlate my own 

 findings and conclusions with those of other workers in the 

 same and related fields, but am well aware of the many gaps 

 that remain. However, it has not been my main intention 

 to produce an exhaustive survey of the subject, but rather 

 to set forth certain new facts and ideas and some of their 

 chief implications. 



There are, I think, four chief points in the book which are 

 more or less new. One is the quantitative formulation of 

 heterogonic growth (Chapters I and II) ; a second is the dis- 

 covery of the widespread existence of growth-gradients, and 

 their quantitative analysis (Chapters III and IV) ; a third is the 

 recognition that growth of logarithmic spiral type as seen in 

 Molluscan shells, etc., operates with the same growth- 

 mechanisms (growth-centres and growth-gradients) as does 

 growth of ordinary type as seen in a Crustacean antenna or 

 a sheep's leg (Chapter V) ; and the fourth is the application 

 of these results to certain evolutionary problems, as set forth 

 in the final chapter. 



I owe a great deal to previous work in this field : first and 

 foremost to D'Arcy Thompson's Growth and Form, but also 

 to the books and papers of Champy, Teissier, Schmalhausen , 

 and others too numerous to mention. 



I have to thank Professor L. T. Hogben and Dr. R. A. 

 Fisher, F.R.S., for reading the book in typescript and making 

 various useful suggestions ; and I owe a great deal to Pro- 

 fessor H. Levy for helping me with some of the mathematical 

 problems involved. My thanks are also due to Dr. C. F. A. 

 Pantin and Professor Selig Hecht, whose discussions with me 

 of various problems raised in this book have helped greatly 

 in clarifying my ideas. And especially I would like to thank 

 my pupils and co-workers, Mr. E. B. Ford, Mr. F. N. Rat- 

 cliff e, Miss M. Shaw (Mrs. White), Miss M. A. Tazelaar, Mr. 



ix 



