American Mathematical Society

London Mathematical Society

History of Mathematics Volume 22

BRUCE C. BERNDTROBERT A. RANKINEDITORS

RAMANUJAN:

Essays and Surveys

Selected Titles in This Series

22 Bruce C. Berndt and Robert A. Rankin, Editors, Ramanujan: Essays and surveys,2001

21 Armand Borel, Essays in the history of Lie groups and algebraic groups, 2001

20 Kolmogorov in perspective, 2000

19 Hermann Grassmann, Extension theory, 2000

18 Joe Albree, David C. Arney, and V. Frederick Rickey, A station favorable to thepursuits of science: Primary materials in the history of mathematics at the United StatesMilitary Academy, 2000

17 Jacques Hadamard (Jeremy J. Gray and Abe Shenitzer, Editors), Non-Euclideangeometry in the theory of automorphic functions, 1999

16 P. G. L. Dirichlet (with Supplements by R. Dedekind), Lectures on number theory,

1999

15 Charles W. Curtis, Pioneers of representation theory: Frobenius, Burnside, Schur, andBrauer, 1999

14 Vladimir Maz′ya and Tatyana Shaposhnikova, Jacques Hadamard, a universalmathematician, 1998

13 Lars Garding, Mathematics and mathematicians: Mathematics in Sweden before 1950,1998

12 Walter Rudin, The way I remember it, 1997

11 June Barrow-Green, Poincare and the three body problem, 1997

10 John Stillwell, Sources of hyperbolic geometry, 1996

9 Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and commentary, 1995

8 Karen Hunger Parshall and David E. Rowe, The emergence of the Americanmathematical research community, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore,1994

7 Henk J. M. Bos, Lectures in the history of mathematics, 1993

6 Smilka Zdravkovska and Peter L. Duren, Editors, Golden years of Moscowmathematics, 1993

5 George W. Mackey, The scope and history of commutative and noncommutativeharmonic analysis, 1992

4 Charles W. McArthur, Operations analysis in the U.S. Army Eighth Air Force in WorldWar II, 1990

3 Peter L. Duren et al., Editors, A century of mathematics in America, part III, 1989

2 Peter L. Duren et al., Editors, A century of mathematics in America, part II, 1989

1 Peter L. Duren et al., Editors, A century of mathematics in America, part I, 1988

Ramanujan:Essays and Surveys

Bruce C. BerndtRobert A. RankinEditors

History of Mathematics

Volume 22

American Mathematical Society

London Mathematical Society

https://doi.org/10.1090/hmath/022

Editorial Board

American Mathematical SocietyGeorge E. AndrewsJoseph W. DaubenKaren Parshall, ChairMichael I. Rosen

London Mathematical SocietyDavid Fowler, ChairJeremy J. GrayTom Korner

Abstract. This book contains essays on Ramanujan and his work written especially for this vol-ume, as well as important previously published survey articles in areas influenced by Ramanujan’smathematics. Most of the articles are nontechnical, but those that are more technical containsubstantial sections which general readers will find engaging. Some of the previously publishedarticles have heretofore been difficult to locate. The book complements the authors’ earlier volume,Ramanujan: Letters and Commentary.

The book begins with the four extant photographs of Ramanujan and historical accounts ofthem, including information about others appearing in the photographs. Included in this bookare biographies of Ramanujan’s wife, S. Janaki, and S. Narayana Iyer, the Chief Accountant ofthe Madras Port Trust Office where Ramanujan worked. A previously unpublished cryptic familyhistory written by Ramanujan’s younger brother has been deciphered. Articles on Ramanujan’sillness by the second named author; the British physician, D. A. B. Young; and S. Chandrasekharare included, with the latter being published for the first time. Survey and historical articles onRamanujan’s notebooks and lost notebook by the authors and George Andrews are also reprinted.Jonathan and Peter Borwein’s captivating article on Ramanujan and pi is included. Fascinatinglectures given on the centenary of Ramanujan’s birth by Chandrasekhar, Freeman Dyson, AtleSelberg, and Richard Askey are contained herein, along with several further articles as well.

2000 Mathematics Subject Classification. Primary 01A61; Secondary 11–03, 11A99,11P82, 11P83, 33–03, 33C05, 33C20, 33D15.

Library of Congress Cataloging-in-Publication Data

Berndt, Bruce C., 1939–Ramanujan : essays and surveys / Bruce C. Berndt and Robert A. Rankin.

p. cm. — (History of mathematics, ISSN 0899-2428 ; v. 22)Includes bibliographical references.ISBN 0-8218-2624-7 (alk. paper)1. Ramanujan Aiyangar, Srinivasa, 1887–1920. 2. Mathematicians—India—Biography.

3. Number theory. I. Rankin, Robert A. (Robert Alexander), 1915–2001. II. Title.III. Series.

QA29.R3B47 2001510′.92—dc21 2001045097

c© 2001 by the American Mathematical Society. All rights reserved.Printed in the United States of America.

The American Mathematical Society retains all rightsexcept those granted to the United States Government.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

The London Mathematical Society is incorporated under Royal Charterand is registered with the Charity Commissioners.

Visit the AMS home page at URL: http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 06 05 04 03 02 01

In Memory of

Srinivasa Ramanujan

S. Janaki

S. Narayana Iyer

Contents

Preface ixPhotograph Credits xiiiCitations for Previously Published Papers xv

Part I

The Life of Ramanujan

1. The Four Photographs of Ramanujan, by Bruce C. Berndt 32. The Books Studied by Ramanujan in India, by Bruce C. Berndt

and Robert A. Rankin 93. The Influence of Carr’s Synopsis on Ramanujan, by Bruce C.

Berndt 174. The Notebooks of Srinivasa Ramanujan, by Bruce C. Berndt 215. A Recently Discovered Letter Giving Ramanujan’s Examination

Scores 236. On Ramanujan, by S. Chandrasekhar 257. The Ramanujan Family Record, by Robert A. Rankin 29

Part II

Ramanujan’s Illness

1. Ramanujan As a Patient, by R. A. Rankin 412. Ramanujan’s Illness, by D. A. B. Young 653. An Incident in the Life of S. Ramanujan, F.R.S.: Conversations

with G. H. Hardy, F.R.S. and J. E. Littlewood, F.R.S. and TheirSequel, by S. Chandrasekhar 77

Part III

S. Janaki

1. S. Janaki Ammal (Mrs. Ramanujan), by Bruce C. Berndt 832. Conversation “I Didn’t Understand His Work, but I Knew His

Worth,” by P. Nandy 89

Part IV

S. Narayana Iyer

1. A Short Biography of S. Narayana Iyer, by Bruce C. Berndt 972. The Distribution of Primes, by S. Narayana Aiyar 993. Some Theorems in Summation, by S. Narayana Aiyar 101

vii

viii CONTENTS

Part V

E. H. Neville

1. Srinivasa Ramanujan, by E. H. Neville 1072. University Lectures in Madras 113

Part VI

Ramanujan’s Manuscripts and Notebooks

1. Ramanujan’s Manuscripts and Notebooks, by R. A. Rankin 1172. Ramanujan’s Manuscripts and Notebooks, II, by R. A. Rankin 1293. An Overview of Ramanujan’s Notebooks, by Bruce C. Berndt 1434. An Introduction to Ramanujan’s “Lost” Notebook, by George E.

Andrews 165

Part VII

Nontechnical Articles on Ramanujan’s Work

1. Ramanujan and Pi, by Jonathan M. Borwein and Peter B. Borwein 1872. π Related Developments Since 1988, by Bruce C. Berndt 2013. Reflections around the Ramanujan Centenary, by Atle Selberg 2034. The Problems Submitted by Ramanujan to the Journal of the

Indian Mathematical Society, by Bruce C. Berndt, Youn–Seo Choi,and Soon–Yi Kang 215

Part VIII

Somewhat More Technical Articles on Ramanujan’s Work

1. A Walk through Ramanujan’s Garden, by Freeman J. Dyson 2612. Ramanujan and Hypergeometric and Basic Hypergeometric Series,

by R. Askey 2773. The Final Problem: An Account of the Mock Theta Functions, by

G. N. Watson 325

PREFACE

It is with deep sadness that I begin this preface by informing readers that mycoauthor, Robert A. Rankin, died on January 27, 2001, after a steadily weakeningillness. Only a few days before he died, except for this Preface, we had completedthe manuscript for the book.

In the preface to our earlier book, Ramanujan: Letters and Commentary [3],we indicated our fervent desire to preserve for future generations as many lettersto, from, and about Ramanujan as possible and to help readers appreciate boththe mathematical and cultural contents of the letters. In particular, Ramanujan’sletters from India to G. H. Hardy contain several of Ramanujan’s mathematicaldiscoveries, which spawned considerable research in the decade after their publica-tion in Ramanujan’s Collected Papers [6] and continue to do so today. It was ourgoal to communicate as much of this activity as possible.

A few years after the publication of [3], we again experienced a historical call.In the next two paragraphs, we offer some of our motivation for writing this secondvolume.

Although we love and admire R. Kanigel’s biography [5] of Ramanujan, we feltthat a few historical tasks remained unfinished. In particular, we were in possessionof a very cryptic historical family record sketched by Ramanujan’s younger brother,S. Lakshmi Narasimhan. However, we could not decipher most of it. S. Janaki,Ramanujan’s wife, is one of the most famous spouses in mathematical history, butwe knew little about her life and strongly believed that those inspired by Ramanu-jan’s life would be enthused to read about Mrs. Ramanujan as well. Althoughmany biographers have written about the strong influence of G. S. Carr’s Synopsis[4] on Ramanujan’s development, no one had carefully examined Carr’s book todetermine the extent of this influence. The Chief Accountant at the Madras PortTrust Office, S. Narayana Iyer, is recognized by name to those familiar with theelements of Ramanujan’s life, but very little is known about him. Furthermore,Narayana Iyer published two papers in the Journal of the Indian MathematicalSociety communicating some of Ramanujan’s theorems, especially in the theory ofprime numbers. We reproduce these little-known papers in this volume.

Many articles have been written about Ramanujan’s life and his work. Someof these are not particularly well known. Our second goal has been to collect thebest of these together so that they might have wider readership. In particular,an English physician, D. A. B. Young, has carefully studied the symptoms of Ra-manujan’s illness and has made a convincing diagnosis of the cause of Ramanujan’sdeath, but his paper is not known to most people. Rankin’s thorough expositionof Ramanujan’s life as a patient in English sanitariums and nursing homes is alsoreprinted here. Many surveys have been written about Ramanujan’s work. Read-ers should particularly consult Ramanujan Revisited [2], providing over 30 surveysof Ramanujan’s work at the time of the centenary of Ramanujan’s birth. Twoarticles from that book are reproduced here. One is S. Chandrasekhar’s [2, pp. 1–6] banquet speech delineating the influence of Ramanujan’s life on him and other

ix

x PREFACE

Indian scientists. The other is F. J. Dyson’s [2, pp. 7–28] delightful survey on cur-rent research in partitions emanating from Ramanujan’s famous congruences forthe partition function. At centenary celebrations in India, A. Selberg spoke aboutthe influence of Ramanujan’s life on his mathematical career. Since this speech hadbeen published only in Selberg’s Collected Papers, we thought that it merited widercirculation. Thus, much of this volume contains articles about Ramanujan and hiswork published elsewhere, and we are grateful to the original publishers of thesepapers for permission to reproduce them here.

We have divided the articles into eight categories, which we now briefly describe.Part I contains nontechnical articles centering on Ramanujan’s life. To the best of

our knowledge, only four photographs of Ramanujan exist. Two of these are groupphotographs. V. Viswanathan has successfully identified all of Ramanujan’s friendsin one of the photographs, and with Viswanathan’s help we have written biographiesof all these gentlemen. We next try to answer the frequently asked question aboutthe mathematics books that Ramanujan might have studied in India before comingto England. We then attempt to assess the influence of Carr’s book on Ramanujan’snotebooks and published papers. A one-page description of the notebooks is givennext. C. A. Reddi obtained for us a letter (which we produce here) from theTamil Nadu Archives giving Ramanujan’s examination scores when he attemptedthe F. A. Examination at Pachaiyappa College in 1907. This is followed by thecentenary lecture of Chandrasekhar mentioned above. Lastly, we present an accountof the family history as recorded by Ramanujan’s brother, S. Lakshmi Narasimhan.

Part II focuses on Ramanujan’s illness and begins with the aforementioned arti-cles by Rankin and Young. Although Chandrasekhar’s account of his conversationwith Hardy about Ramanujan’s attempt to take his own life has been widely dis-seminated, the full text of Chandrasekhar’s recollections has not been heretoforepublished.

Our biography of S. Janaki and a conversation of P. Nandy with Mrs. Ramanujanform the content of Part III.

Part IV comprises our biography of S. Narayana Iyer and the two papers he com-municated to the Journal of the Indian Mathematical Society about Ramanujan’swork.

Part V is devoted to E. H. Neville, who was instrumental in Ramanujan’s sojournto England. We reproduce a talk given by Neville on All India Radio in 1941. Onemight question our decision to select this article, published in Nature, since itcomprises mostly a biography of Ramanujan, the details of which are well known,and since it contains some factual errors. However, the article contains detailsconnected with Ramanujan’s trip to England that are generally not found elsewhere.We also reproduce a page from the Journal of the Indian Mathematical Societyannouncing Neville’s course of lectures in Madras in early 1914.

Part VI is devoted to Ramanujan’s manuscripts and notebooks. Rankin’s twogeneral articles from the Bulletin of the London Mathematical Society are presentedfirst. Berndt’s survey of Ramanujan’s notebooks, published in 1997, follows. Thesection ends with G. E. Andrews’s article on the lost notebook written for theAmerican Mathematical Monthly two years after he rediscovered Ramanujan’s lostnotebook in the library at Trinity College, Cambridge.

The final two sections are devoted to survey articles on certain primary themesin Ramanujan’s mathematics. The line between “technical” and “nontechnical”is somewhat blurred, but we have nonetheless divided the articles accordingly in

PREFACE xi

two such categories. We begin with a charming article by J. M. Borwein andP. B. Borwein entitled, Ramanujan and Pi, which we follow by a short update onRamanujan-type series for 1/π and recent calculations of the digits of π. This isfollowed by Selberg’s lecture cited earlier. Ramanujan published 58 problems inthe Journal of the Indian Mathematical Society, many of which have had enormousinfluence on mathematical research. The last article in this first grouping of surveyarticles is an updated account of all these problems written by Berndt, Y.–S. Choi,and S.–Y. Kang. Commencing the second set is Dyson’s article mentioned aboveand perhaps more appropriate for the first grouping than the second. Next is a his-torical survey on hypergeometric series and basic hypergeometric series emphasizingthe contributions of Ramanujan, written by R. A. Askey at the time of the celebra-tion of the 100th anniversary of Ramanujan’s birth. We end with G. N. Watson’saccount of perhaps Ramanujan’s last discovery, his mock theta functions. This ar-ticle is more technical than the others here, but it is important. Those readers notdesiring to wade through the technical details might nonetheless read the beginningand end of the article. A general survey of the work of Ramanujan and others onmock theta functions up to the centenary of Ramanujan’s birth can be found inAndrews’s paper [1].

We thank the writers and publishers of the articles included in this volume forpermission to republish their articles. We have attempted to reproduce the articleswith complete accuracy, and we apologize if any misprints have crept into theirworks.

Many people have helped the authors in preparing articles for this volume. Anattempt here to record the names of all of them would inevitably lead to unintendedomissions. Some are thanked in the articles themselves. However, four people espe-cially deserve our sincere thanks. In February 1999, Mrs. Ramanujan’s adopted son,W. Narayanan, and the first author conversed for three afternoons about Ramanu-jan and Janaki. These conversations have been invaluable to our understandingof both of them. K. Srinivasa Rao, from the Institute of Mathematical Sciencesin Chennai, helped make numerous arrangements and offered expert advice andcritical comments on our writing. C. A. Reddi (and his family) hosted the firstauthor on his visit to Chennai in September 2000, transcribed the Ramanujan fam-ily history, unearthed the letter mentioned above, and gave much historical advice.Lastly, V. Viswanathan provided considerable information about his grandfather,S. Narayana Iyer. Without Viswanathan’s help, the friends of Ramanujan in thegroup photograph would not have been identified. Moreover, through his persis-tent efforts, relatives of Ramanujan’s friends in the photograph were contacted andinformation about them was obtained.

The first author is enormously grateful to the Number Theory Foundation and, inparticular, its founder, John Selfridge, for a generous grant which provided releasetime from other duties so that this book could be prepared.

Finally, we express our sincere gratitude to Tori Corkery, who scanned many ofthe articles reproduced here. Since mathematical formulas cannot be accuratelyscanned, Jaebum Sohn and David Gepner took the scanned papers and accuratelyset them in AMS-TeX to restore them. For all their labors, we are very grateful.

Urbana, IllinoisApril 2001

xii PREFACE

References

1. G. E. Andrews, Mock theta functions, Theta Functions Bowdoin, Part 2, Proceedings ofSymposia in Pure Mathematics, vol. 49, American Mathematical Society, Providence, RI,1989, pp. 283–298.

2. G. E. Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan, and R. A. Rankin (eds.),Ramanujan Revisited, Academic Press, Boston, 1988.

3. B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, American Math-ematical Society, Providence, RI, 1995; London Mathematical Society, London, 1995.

4. G. S. Carr, A Synopsis of Elementary Results in Pure Mathematics, C. F. Hodgson andSon, London, 1880, 1886; reprinted by Chelsea, New York, 1970, under the title Formulasand Theorems in Pure Mathematics.

5. R. Kanigel, The Man Who Knew Infinity, Scribner’s, New York, 1991.6. S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927; reprinted

by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence,RI, 2000.

PHOTOGRAPH CREDITS

The American Mathematical Society gratefully acknowledges the kindness ofthese individuals and institutions in granting the following permissions:

Mrs. S. Chandrasekhar

Passport photo of S. Ramanujan, following Part I title page, courtesy ofMrs. S. Chandrasekhar

V. S. Ramachandran, Curator, Visvesvaraya Industrial and Technologi-cal Museum, Bangalore, India

Photo of S. Ramanujan alone, sitting in chair, following Part I title page,courtesy of V. S. Ramachandran

Photo of S. Ramanujan with friends at Trinity College, following Part I titlepage, courtesy of V. S. Ramachandran

Photo of S. Ramanujan with fellow graduates at Cambridge, following Part Ititle page, courtesy of V. S. Ramachandran

Photo of Mrs. Ramanujan with sewing machine, following Part III title page,courtesy of V. S. Ramachandran

Photo of S. Narayana Aiyar, following Part IV title page, courtesy of V. S.Ramachandran

Photo of E. H. Neville, following Part V title page, courtesy of V. S. Ra-machandran

Sincere thanks to S. Janaki Ammal, wife of Ramanujan, for information andsupport.

Indian Mathematical Society

Photo of participants at Second Conference of the Indian Mathematical Soci-ety, January 11–13, 1919, Bombay, India; frontispiece in J. Indian Math. Soc. 11(1919); following p. 27, courtesy of the Indian Mathematical Society

P. K. Srinivasan

Photo of Mrs. Ramanujan (upper left), following Part III title page, reprintedwith permission of P. K. Srinivasan

W. Narayanan

Photo of Mrs. Ramanujan (bottom right), following Part III title page, reprintedwith permission of W. Narayanan

Richard Askey

Photo of bust of S. Ramanujan sculpted by Paul Granlund, following p. 93and on cover, reprinted with permission of Richard Askey

All other photos were supplied by the editors.

xiii

CITATIONS FOR PREVIOUSLY PUBLISHED PAPERS

Part I

The Life of Ramanujan

2. B. C. Berndt and R. A. Rankin, The books studied by Ramanujan in India,Amer. Math. Monthly 107 (2000), 595–601.

6. S. Chandrasekhar, On Ramanujan, Ramanujan Revisited (G. E. Andrews, R. A.Askey, B. C. Berndt, K. G. Ramanathan, and R. A. Rankin, eds.), AcademicPress, Boston, 1988, pp. 1–6.

Part II

Ramanujan’s Illness

1. R. A. Rankin, Ramanujan as a patient, Proc. Indian Acad. Sci. (Math. Sci.) 93(1984), 79–100.

2. D. A. B. Young, Ramanujan’s illness, Notes Rec. Royal Soc. London 48 (1994),107–119.

Part III

S. Janaki

2. P. Nandy, Conversation “I didn’t understand his work, but I knew his worth,”Sunday (Ananda Bazar Patrika Ltd., Calcutta) 9 (7 February 1982), 28–29.

Part IV

S. Narayana Iyer

2. S. Narayana Aiyar, The distribution of primes, J. Indian Math. Soc. 5 (1913),60–61.

3. S. Narayana Aiyar, Some theorems in summation, J. Indian Math. Soc. 5 (1913),183–186.

Part V

E. H. Neville

1. E. H. Neville, Srinivasa Ramanujan, reprinted by permission from Nature 149,pp. 292–295, copyright c© 1942 Macmillan Magazines Ltd.

2. University Lectures in Madras, J. Indian Math. Soc. 6 (1914), 2.

Part VI

Ramanujan’s Manuscripts and Notebooks

1. R. A. Rankin, Ramanujan’s manuscripts and notebooks, Bull. London Math. Soc.14 (1982), 81–97.

xv

xvi CITATIONS FOR PREVIOUSLY PUBLISHED PAPERS

2. R. A. Rankin, Ramanujan’s manuscripts and notebooks, II, Bull. London Math.Soc. 21 (1989), 351–365.

3. B. C. Berndt, An overview of Ramanujan’s notebooks, Charlemagne and HisHeritage: 1200 Years of Civilization and Science in Europe, Vol. 2: Mathemat-ical Arts (P. L. Butzer, H. Th. Jongen, and W. Oberschelp, eds.), c© BrepolsPublishers, Turnhout, 1998, pp. 119–146.

4. G. E. Andrews, An introduction to Ramanujan’s lost notebook, Amer. Math.Monthly 86 (1979), 89–108.

Part VII

Nontechnical Articles on Ramanujan’s Work

1. J. M. Borwein and P. B. Borwein, Ramanujan and pi, Sci. Amer. 256, No. 2(1988), 112–117. Reprinted with permission. Copyright c© 1988 by ScientificAmerican, Inc. All rights reserved.

3. A. Selberg, Reflections around the Ramanujan centenary, Collected Papers, Vol. 1,c© Springer-Verlag, Berlin, 1989, pp. 695–706.

4. B. C. Berndt, Y.-S. Choi, and S.-Y. Kang, The problems submitted by Ramanu-jan to the Journal of the Indian Mathematical Society, Continued Fractions:From Analytic Number Theory to Constructive Approximation (B. C. Berndtand F. Gesztesy, eds.), Contemp. Math., vol. 236, American Mathematical So-ciety, Providence, RI, 1999, pp. 15–56.

Part VIII

Somewhat More Technical Articles on Ramanujan’s Work

1. F. J. Dyson, A walk through Ramanujan’s garden, Ramanujan Revisited (G. E.Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan, and R. A. Rankin,eds.), Academic Press, Boston, 1988, pp. 7–28.

3. R. A. Askey, Ramanujan and hypergeometric series and basic hypergeometricseries, Russian Math. Surveys 45 (1990), 37–86; Ramanujan International Sym-posium on Analysis (Pune, 1987) (N. K. Thakare, ed.), Macmillan of India, NewDelhi, 1989, pp. 1–83.

4. G. N. Watson, The final problem: An account of the mock theta functions,J. London Math. Soc. 11 (1936), 55–80.

AMS on the Web www.ams.org

RAMANUJAN:

Essays and Surveys

HMATH/22

BRUCE C. BERNDT AND ROBERT A. RANKIN, EDITORS

This book contains essays on Ramanujan and his work that were written especially for this volume. It also includes important survey articles in areas influenced by Ramanujan’s mathematics. Most of the articles in the book are nontechnical, but even those that are more technical contain substantial sections that will engage the general reader.

The book opens with the only four existing photographs of Ramanujan, presenting historical accounts of them and information about other people in the photos. This section includes an account of a cryptic family history written by his younger brother, S. Lakshmi Narasimhan. Following are articles on Ramanujan’s illness by R. A. Rankin, the British physician D. A. B. Young, and Nobel laureate S. Chandrasekhar. They present a study of his symptoms, a convincing diagnosis of the cause of his death, and a thorough exposition of Ramanujan’s life as a patient in English sanitariums and nursing homes.

Following this are biographies of S. Janaki (Mrs. Ramanujan) and S. Narayana Iyer, Chief Accountant of the Madras Port Trust Office, who first communicated Ramanujan’s work to the Journal of the Indian Mathematical Society. The last half of the book begins with a section on “Ramanujan’s Manuscripts and Notebooks”. Included is an important article by G. E. Andrews on Ramanujan’s lost notebook.

The final two sections feature both nontechnical articles, such as Jonathan and Peter Borwein’s “Ramanujan and pi”, and more technical articles by Freeman Dyson, Atle Selberg, Richard Askey, and G. N. Watson.

This volume complements the book Ramanujan: Letters and Commentary, Volume 9, in the AMS series History of Mathematics. For more on Ramanujan, see these AMS publications: Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Volume 136.H, and Collected Papers of Srinivasa Ramanujan, Volume 159.H, in the AMS Chelsea Publishing series.