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Author’s Note and Acknowledgements
In writing the life of Ramanujan, I faced the barriers of two foreign cultures, a challenging discipline, and a distant time. As I am expert in none of these, I owe a debt of gratitude to the many persons who have helped me surmount those barriers—who have consented to interviews, spent hours explaining recondite areas of mathematics or Indian cultural life, guided me to out of the way documents in libraries and archives, read and criticized early drafts, befriended me in England and India—and, back in Baltimore, offered a supportive hand or word of advice. I am abashed at how much space I require to simply say thank you, but it is an apt measure of my debt.
First, thanks are due Judy, who has borne more husbandly changes of mood than any person should have to bear, and to whom I owe much in the way of nurturing, encouragement, and support. Those intangibles are easy to take for granted when you have them, but almost impossible to get along with when you don’t.
To Judy and my sister Rachele, and their refusal to tell me what I wished to hear at a crucial point, I owe an approach to writing about the mathematics to which otherwise I might not have turned.
Thanks go to Davy, whose Daddy was gone in England and India for three months and then, for the year and a half it took to write this book, was too much in the office and not enough with him. “So how’s the Ramanujan book?” he asked me one day when he was five. I hope that one day he’ll read it and be able to answer for himself.
To my parents, Bea and Charles Kanigel, for leaving me fascinated equally by words, numbers, people, and ideas.
To Michael, Kevin, and Jonathan, to Harry and Rachele, and to Elise and Liz—all of them, each in their own way, irreplaceable parts of my life.
To Bill Stump for encouraging a young writer twenty years ago.
To V. Viswanathan, of Madras, who made room for a confused American in an already overstuffed auto-rickshaw. To him, his brother V. Meenakshisundaram of the Madras Port Trust, to S. Sankara Narayanan and V. Subramanyam, and to the many other members of his family who befriended me in Madras, I owe a debt of kindness I cannot possibly repay.
To Sambandam, Vijaya, Mahalingam, and all their family in Kumbakonam, for their boundless hospitality.
To the “Gang of Three”—three American mathematicians, admirers and students of Ramanujan, who have helped me to understand his work, and who have read the manuscript along the way, invariably peppering their excellent advice with dollops of needed encouragement: George Andrews, Pennsylvania State University; Richard Askey, University of Wisconsin; and Bruce Berndt, University of Illinois. Without the help of these three men this book could not have been written.
To Robert Rankin of Glasgow, Scotland, whose research into the lives of Ramanujan and Hardy has materially contributed to this book, and who has been unfailingly patient in responding to my trans-Atlantic queries.
To Freeman Dyson, for taking such interest in the book, rounding up old letters, reading the manuscript, and making important suggestions.
To Barbara Grossman, whose idea this book was, and who has placed the full force of her personality behind it. To Joy Smith for her diligence and unfailing good nature. To Erich Hobbing, David Frost, and others at Scribner’s for their help and talents. To Zoë English Kharpertian for her fine job of copyediting.
To Vicky Bijur, my agent, for the important role she played in bringing this book into being, and for the almost frightening efficiency with which she has acted in my behalf along the way.
To Jane Alexander, who sent me to India the first time and has been a friend since.
So many have helped, in small ways and large, to make this book possible that it seems scarcely possible to remember them all. I apologize to any I may have inadvertently omitted.
In America: Sudarshan Bhatia and Asha Rijhsinghani. V. Anantharaman and Malini. Ashvin Rajan. Ranjan Roy, Beloit College. Henry S. Tropp, Humboldt State University. S. Chandrasekhar, University of Chicago. Arthur Magida. Gary Leventhal. Ann Finkbeiner and Cal Walker. Ken Gershman. Lee and Phyllis Jaslow. Mildred Foster. Carolyne, Kathy, and everyone at the Red Balloon. William Dyal and Thomas Slakey, St. John’s College, Annapolis. Warren Kornberg, Mosaic magazine. Jacek Mostwin, Johns Hopkins Hospital. Steve Fisher. Alan Sea. Adrianne Pierce, Johns Hopkins University. John Halperin, Vanderbilt University. Maurice St. Pierre, Morgan State University. Suzanne Holland, Harvard magazine. George W. Comstock, Johns Hopkins University School of Hygiene and Public Health. S. Bhargava, University of Illinois, Urbana. Wayne Markert, University of Baltimore.
In Britain: Emma and Jonathan Leigh, Cranleigh School. Kevin Gray, Trinity College. Béla Bollobás, Trinity College. Charles Burkill, Cambridge. Mary Cartwright, Cambridge. Rajiv Krishnan, Christ’s College. Roger David Hugh Custance, Winchester College. Guy Newcombe, Trinity College. Vince Darley, Trinity College. John Vickers, St. John’s College. R. Robson, Trinity College. Constance Willis, Cambridge. Deborah and Bryan J. B. Gauld, Putney. Theodor Schuchat and Louise Harper, London. Paul and Clare Friedman, London. S.J. Mann, Cranleigh School. Susan M. Oakes, London Mathematical Society. J. D. Webb, Cambridge City Council. Tom Doig, Cambridge Folk Museum. Pat Kattenhorn, India Office Library.
In India: Janaki Ammal, the widow of Ramanujan, and her son, W. Narayanan, and his family, Triplicane, Madras. T. V. Rangaswami (“Ragami”), Triplicane, Madras. A. P. Victor, Kuala Lumpur, Malaysia. P. K. Srinivasan, Association of Mathematics Teachers of India, Madras. T. M. Srinivasan and T. M. Kasturirangan, Madras. John Herbet Anand, Mylapore, Madras. R. Janarthanan, Kumbakonam. R. Viswanathan, headmaster, and V. Vaidynathan, assistant headmaster, Town Higher Secondary School, Kumbakonam. S. Subbarathinan, Erode. A. Nazimuddin and H. Sharmila, Madras. K. Elangovan, Kodumudi. R. Chandrasekhar, Kodumudi. A. Sanguttuvan, Erode. L. Rajagopolan, S. Elango and D. G. Ramamurthy, Kumbakonam. A. Saranathan and his family, Kumbakonam. S. Govindaraja Battachariar and P. Vasunathan, Uppiliapan Koil, Thirunageswaram. T. U. Bhanumurthy and Kalyanalakshmi, Madras. M. Vinnanasan, Kumbakonam. K. S. Padmanabhan, Ramanujan Institute, Madras. K. Narayanan and A. V. Chandrasekhar, The Hindu. A. Ranganathan, Madras. K. Rajamani, Kumbakonam. T. C. Krishna and his family, Madras. Bhama Srinivasan, University of Illinois, Chicago. P. P. Kulkarni, Nagpur. And, of course, the unforgettable Hari, of Madras.
• • •
I wish also to express my gratitude to the staffs of the many libraries in the United States, England, and India who have helped me in the quiet, faceless, but unfailingly competent way we all expect of them. That, of course, is the problem: it’s so easy to take the library for granted when it has the book or document you need, and to grumble when it doesn’t. But the very concept of the library, as a place to store, preserve, and give access to books, was thrown into sharp focus during my five weeks in India. There, libraries cannot always treat their treasures with the expensive care Western libraries can lavish on theirs. In one, I found books and journals set out on the floor, piled this way and that, the pages of even recent books crumbling, dusty and mildew
ed. And yet never have I seen libraries so intensively used, books so hungrily devoured. In one small library in Erode, I saw every seat at every table taken, and many people standing in the aisles to read.
My appreciation, then, goes to librarians, archivists, and other staff at the following institutions:
Baltimore: The Milton S. Eisenhower Library of Johns Hopkins University. William H. Welch Medical Library. Enoch Pratt Free Library—central library, St. Paul Street branch, and telephone reference service.
Washington, D.C.: Library of Congress.
Cambridge, Mass.: Harvard University Archives.
Princeton, N.J.: Seely G. Mudd Manuscript Library.
Cambridge, England: University Library of Cambridge University, and the libraries of these Cambridge colleges: Trinity, St. John’s, Gonville & Caius. Scientific Periodicals Library. Cambridgeshire Collection. Cambridge Folk Museum.
London: Royal Society Library. India Office Library. University of London. National Maritime Museum, Greenwich.
Elsewhere in England: Guildford Public Library (Local Studies Collection). Cranleigh School, library and archives. Cranleigh Library. St. Catherine’s School Library, Bramley. New College Library, Oxford. University of Reading.
Erode: Public Library.
Madras: Connemara Library, University of Madras. Archives of The Hindu. British Embassy Library. Fort St. George Museum. Madras Museum.
Index
Abel, Niels Henrik, 147, 150, 169–70, 366
Acta Mathematica (journal), 363, 370
Additive theory of numbers, 303
Advisory Committee for Indian Students, 174, 184
Agincourt, Battle of, 163
Algebra, 41–44, 46, 53, 60, 86, 119, 149, 161, 205, 366
computer, 349
genetics and, 146
Algebraic geometry, 344
Allen, G. C., 126
Allied Powers, 212
Alternative Sciences (Nandy), 335
American Mathematical Monthly, 336
American Physical Society, 348
Analytical Club, 85
Analytical number theory, 249, 340
Ananda Rao, 230
Anantharaman, 55, 72, 190, 282, 290, 318, 332–33
Andrews, George, 6, 252, 325, 326, 344–46, 349, 350–52
Anna University, 351–52
Apostles. See Cambridge Apostles
Applied mathematics, 347–49
Archimedes, 209
Are Conjectandi (Bernoulli), 90
Aristotle, 287
Arithmetical functions, 343
Arnold, Thomas, 112, 121
Arybhata, 86
Askey, Richard, 280, 350–52
Athens, University of, 366
Atomic research, 348
Atshara Abishekam (ritual practice), 13
Australian National University, 349
Bach, Johann Sebastian, 344, 350, 351
Bacon, Francis, 239
Bailey, W. N., 202, 231, 371
Baker, H. F., 106, 170–72, 291, 358
Balakrishna Iyer, S., 78, 321
Ballistics, 228
Barnes, E. W., 151, 233, 243, 261, 277, 290
Bauer (mathematician), 167
Baxter, R. J., 349
Bayeux tapestries, 133
Beethoven, Ludwig van, 346, 350
Belgium
in Great War, 211–14, 227, 260, 277
tuberculosis in, 267
Bell, E. T., 65, 81, 256, 280, 287
Below the Magic Mountain (Bryder), 270
Benedictine order, 126
Bergson, Henri, 157
Berlin, University of, 365
Berndt, Bruce, 183, 204–5, 280, 344, 346, 350, 351
Bernoulli, Jacob, 89
Bernoulli numbers, 89–92, 105, 161, 203
Berry, Arthur, 201–2
Bertrand Russell and Trinity (Hardy), 277
Besant, Annie, 314
Bhagavad-Gita, 30, 179
Bhaskara, 86
Bhavaniswami Rao, 106
Bieberbach, Professor, 366
Birmingham, University of, 204, 366
Birth of Britain, The (Churchill), 133
Bloomsbury literary movement, 111, 137, 141
Bohr, Niels, 365
Bollobás, Béla, 253
Bolshevik Revolution, 306
Borel (mathematician), 153
Borwein, Jonathan M., 208
Borwein, Peter B., 208
Bose, J. C., 335, 336
Bounded errors, 251–52
Bourne, A. G., 103
Boyles, Robert, 291
Brahmagupta, 86, 209
Brahmins, 20–26, 29–33, 36, 46, 51, 54, 77, 81–82, 112, 174, 210, 354
dietary regulations of, 240–41, 298
foreign travel forbidden for, 174, 185–86, 198, 240, 316, 329
morning ritual of, 244
Brighton Railway, 113
British East India Company, 82, 99
British India Lines, 195, 311
Bromwich, T.J. I’A., 104, 105, 145, 176, 245, 284, 291
Brouwer, Luitzen, 340
Bryder, Linda, 270, 290
Budapest, University of, 338
Buddhism, 32
Bulstrode, H. Timbrell, 266–67, 270
Burkill, J. C., 254
Burma Oil Company, 314
Butler, Henry, 213
Byron, George Gordon, Lord, 2, 345
Calculus, 46, 89, 148–50, 153, 161, 249
differential, 56
integral, 166, 183
pi in, 209
California Institute of Technology, 364
Cambridge Apostles, 137–41, 147, 157, 172, 228, 276
Cambridge Magazine, 260, 277
Cambridge Philosophical Society, 130, 245, 276, 295, 301, 303, 306
Cambridge Review, 152, 307
Cambridge Shakespeare Society, 136, 137, 142
Cambridge Tracts in Mathematics and Mathematical Physics, 160
Cambridge University, 1–4, 6, 40, 66, 100, 105–7, 109–11, 124, 126–33, 145, 154, 157, 163, 170, 226, 253, 285, 332, 344, 345, 361–65, 367, 368, 372
arrangements to bring Ramanujan to, 172, 174, 190–94
debating societies at, 137
and Great War, 212–13, 227–29, 244, 246, 259–61, 311
homosexuals at, 140–44
Newton’s influence at, 148–50
Ramanujan at, 198, 200–207, 210–15, 219–20, 228–31, 233–35, 237–46, 252–59, 289–91, 315, 330, 351, 357
Tripos system at, see Tripos system
See also specific colleges
Cambridge University Press, 337, 341
Camford Observed (Rose and Ziman), 361
Campbell, Islay Makimmon, 259
Cancer research, 348
Cardus, Neville, 123
Carmichael, Robert, 337
Carnot’s Cycle, 131
Carpenter, Edward, 140
Carr, George Shoobridge, 39–46, 57, 58, 72, 90, 92, 105, 164, 173, 203, 253, 356
Carstairs, G. Morris, 23
Cartwright, Mary, 156, 171, 364, 369, 372
Case of the Philosopher’s Ring, The (Collins), 109
Caste system, 20–23, 112, 314
Cauchy, Augustine Louis, 150, 287
Cauchy’s theorem, 249
Central College (Bangalore), 91
Central Powers, 212
Chandler, Laura, 118
Chandrasekhar, Subrahmanyan, 329–30, 333, 335, 352, 354, 372
Chandrasekar, P. S., 321, 322, 330
Charley’s Aunt (play), 231
Chatterji, G. C., 230–31, 237–39, 275
Chattopadhyaha, Mrilani, 237–38
Chaudhuri, Nirad C., 75, 243
Chengalvarayan, K., 315
Chettiar, Alagappa, 355
Chicago, University of, 364
Chola Empire, 15
Churchill, Winston, 133
Circle method, 249, 252, 324
Clarke,
Eustace Thomas, 116
Class system, British, 112, 113
Colinette House, 311–12, 319
Collins, Randall, 109
Collins, Robert, 261
Colorado, University of, at Boulder, 340
Combinatorics, 250
Combinatory Analysis (MacMahon), 305
Commercial Printing Press, 341
Composite numbers, 231–32, 234, 337
Comptes Rendus, 252
Compton, Herbert, 55, 101, 102, 272
Computers, 349
Congruences, 302–3, 306, 313, 339, 370
Connemara Library (Madras), 180
Continued fractions, 57, 215
Continuous functions, 134
Continuous quantities, 249
Convergent series, 88
Conversazione Society, 137
Coombs, “Iron Man,” 364
Corpus Christi College, Cambridge, 212, 256
Cotterell, C. B., 193
Course in Pure Mathematics, A (Hardy), 153
Cours d’analyse de I’Ecole Polytechnique (Jordan), 134, 149, 172
Cranleigh School, 113–20, 123, 126, 136, 142, 171, 172
Crystallography, 348
Cubic equations, 27
Cubism, 157
Curzon, Lord, 99
Darling (mathematician), 168
Davenport (mathematician), 344
Davies, Arthur, 174, 184, 190
Davies, P. D. O., 267–68
Decemviri debating society, 137
Dedekind (mathematician), 150
Definite integrals, 90, 182–83
Delhi, University of, 348
Deligne, Pierre, 343, 344
Denmark, tuberculosis in, 267
Denton, Eliza, 118
Derby, Lord, 228
Descartes, René, 287
Deshmukh, C. D., 230
Dewsbury, Francis, 192–93, 233, 234, 307, 309–12, 317–18, 322, 337, 345, 362
Diaghilev, Serge, 157
Dickinson, Goldsworthy Lowes, 138, 228
Discovery of India (Nehru), 354
Discrete quantities, 249
Divergent series, 80, 174, 203
Dougall-Ramanujan Identity, 167
Doyle, Arthur Conan, 109, 325
Dravidian peoples, 32, 33
Dryden, John, 239
Dubos, Jean, 270
Dubos, René, 270
Dudley Smith, Charles Jervoise, 260
Dyson, Freeman, 339–41, 346, 349, 368
Ecole Polytechnique, 285
Eddington, Sir Arthur, 372
Edinburgh, University of, 366
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