K. Ramasubramanian Editor Gan. itānanda Selected Works of Radha Charan Gupta on History of Mathematics ṇ ā Ga it nanda R.C.GuptareceivestheKennethO.MayPrize(2009) K. Ramasubramanian Editor ṇ ā Ga it nanda Selected Works of Radha Charan Gupta on History of Mathematics 123 Editor K.Ramasubramanian Cell for Indian ScienceandTechnology in Sanskrit, Department of Humanities andSocial Sciences Indian Institute of Technology Bombay Mumbai,India ISBN978-981-13-1228-1 ISBN978-981-13-1229-8 (eBook) https://doi.org/10.1007/978-981-13-1229-8 LibraryofCongressControlNumber:2018945878 ©SpringerNatureSingaporePteLtd.2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721, Singapore Many hitherto hidden facts about [Indian] mathematics were brought to light by R. C. Gupta through the light of his brilliant expositions. May the sun[light]oftheseexpositionsblossomthelotusminds of scholars. Having compiled these expositions, we present them to the scholarly world with immense pleasure and a deep sense of gratitude. May this offering be gracefully accepted. Foreword It is a great pleasure as well as a privilege to be invited to write these few words aboutthiscollectionoftheselectedworksofProf.RadhaCharanGupta,oneofthe great Indian scholars on the history of Indic mathematics. Dr. Gupta began his career as a mathematics teacher in a college and was inspired by Datta & Singh’s book on the History of Hindu Mathematics, which he came across in 1963, to pursue the history of Indic mathematics. He went on to complete his Ph.D. thesis under the guidance of Prof. T. A. Sarasvati Amma, whose book on Geometry in Ancient and Medieval India represents a landmark in Indic historical studies of mathematics. Dr. Gupta dedicated the rest of his life to carrying out a whole series of meticulous studies on Indic works in mathematics—studies which have stood the testoftimeandwonhimthe7thKennethO.MayPrizeintheyear2009,awarded bytheInternationalCommissionontheHistoryofMathematics—thefirstIndianto get the Prize. Among his most notable contributions are the analyses of Parameśvara’s third-order series approximations for the sine, and the methods of Govindasvāmin for interpolating in sine tables. He has contributed articles to StudiesinHistoryofExactSciences,volumeinhonourofDavidPingree,editedby Burnettetal.,andWritingtheHistoryofMathematics:ItsHistoricalDevelopment, edited by Dauben and Scriba. He has over 500 publications and has served the world of scholarship in many different ways. Hispersonalqualitiesofsimplicity,modesty,frugalityandgenerositymatchthe passion and extraordinary care with which he has pursued the history of mathe- matics. He is at once both an outstanding scholar and a sage, in the classical Indic tradition.IcongratulatetheeditorofthisvolumeProf.K.Ramasubramanianforhis effort in producing a collection of selected works of Dr. Gupta. The volume is appropriately calledGaṇitānanda (roughly theJoyofMathematics)—andIcannot help recalling how the great Bhāskara promised that his algorithms will bring ānanda to all gaṇakas. I am sure this book will do the same. vii viii Foreword We need more of the kind of authentic papers that Dr. Gupta has written with suchdedication.Suchpapersarerare,andaregenerallyscatteredoverthescholarly literature, so it is convenient to collect them into a volume and make them easily available. Prof. Ramasubramanian deserves our gratitude for his effort. October 2015 Roddam Narasimha Preface to the Second Edition In the year 2015, the Annual Conference of the Indian Society for History of Mathematics (ISHM) was held at IIT Bombay. As the 80th birthday of Prof.R.C.Guptaalsohappenedtobeinthesameyear,therewasasuggestionfrom the then Administrative Secretary S. L. Singh1 and the President of the Society Prof.S.G.DanithatwecouldtaketheopportunitytofelicitateProf.R.C.Guptaon thisoccasionfortheimmensecontributionshehasmadetothestudiesinthehistory ofmathematicsaswellastothesociety.Fromthisgerminatedtheideaofbringing out a volume containing selected works of Prof. Gupta. Since the time interval between the emergence of this idea and the conference was hardly two months, it was not possible to approach any professional publisher to bring out this volume. Hence, the volume was brought out as a publication of ISHM, and only a few copies were printed, primarily for the distribution among the participants of the conference and a few others. Sincethiswasalimitededition,subsequentlyitwasfeltthatitwouldbehighly desirable to bring out the volume through a reputed publisher and make it com- mercially available for various institutions and individuals to purchase and get benefitedfromitscontents.ProfessorDanitooktheleadinthisdirectionandgotin touch with Springer. They readily agreed to publish thevolume andrequested that I, as an editor of the volume, should consider adding a few more articles to thevolume.However,itoccurredtome,Ithoughtthatitwouldbemoreappropriate to seek inputs from Prof. Gupta himself in this regard. When I approached him, hegavealistofarticlesthatcouldbeconsideredtobeincludedinthisrevisedand enlarged edition. From this list, ten articles have been typeset and included in this volume.Theinclusionshavebeendoneinsuchawaythatwemayfindonearticle addedperpartofthevolumeeitheratthebeginningorattheend,exceptforPartI and Part VIII of the previous edition. 1ProfessorShyamLalSingh,bornonJanuary20,1942,tookhislastbreathattheageof75on October2,2017.HisdedicationtothepromotionofstudiesinthehistoryofmathematicsinIndia andhisselflessservicetothesocietycanneverbeoverstated.Formoredetails,seeGaṇitaBhāratī 39.1(2017). ix x PrefacetotheSecondEdition WhilePartIofthepreviouseditionremainsunchanged,PartVIIIisincludedwith two very interesting articles, one on Prof. P. C. Sengupta—a doyen among the historians of Indian mathematics and astronomy—and the other on Prof. Augustus DeMorgan,whowasabrilliantteacher,aprolificwriterandascholarparexcellence onmathematics,logic,andphilosophy.Incidentally,duringoneoftheconversations withProf.Gupta,IcametoknowthathetravelledallthewaytoCalcuttatomeetthe directheirsofProf.Senguptainordertogetanauthenticbiographicalsketchabout their father,whichhe has nicely written up inthe form of ashort article. Another valuableaddition that hasbeenmade tothis volumeistheinclusionof anup-to-datebibliographyofProf.Gupta.In1996,Prof.TakaoHayashiprepareda bibliography of the writings of Prof. Gupta which was published in Historia Scientiarum. Then again in 2011, he compiled an updated version of it which was publishedinGaṇitaBhāratī.Thosetwoyearsincidentallyhappenedtobethe60th and 75th birth anniversaries of Prof. Gupta. It was so kind of Prof. Gupta that he gave us the necessary materials to further update and present an up-to-date bibliography of his writings while working on this revised edition. This bibliography is provided at the end of this volume. While preparing this edition, as done in the earlier edition, much attention was paidtoeliminatethetyposthathadinadvertentlycreptintotheearlierpublications, particularly in the Sanskrit passages quoted by the author as well as in the transliteration of Sanskrit terms. It is worth mentioning the suggestion from Prof. Gupta that this volume be supplemented with an exhaustive index. Efforts weretakeninthisdirection,andsignificantprogresswasalsomade.Butconsidering the current trend of both the individuals and the institutions preferring to procure digital versions of the texts, it was suggested by the publisher that efforts towards preparinganindexmaynotbeworthwhile.Hence,theideaofaddinganindextothe volume was dropped. Finally,Iwouldliketoacknowledgeallthosewhocontributedinvariouswaysto the production of this revised and enlarged edition of Gaṇitānanda. First, I would like to thank Prof. Dani of ISHM and Mr. Shamim Ahmad of Springer for having takeninterestinbringingoutthisrevisededition.WorkingwithShamimhasbeena great pleasure as he has been very gentle, highly encouraging and extremely cooperative. Secondly, I would like to thank all my project staff, namely Ms.SushamaSonak,Ms.SreelekshmyRanjit,Ms.LalitaHotkar,Dr.DineshMohan Joshi and Mr. G. Periasamy, for their editorial assistance. In particular, Sushama needsaspecialmentionforbeingkindenoughtoworkquitelateintheeveningsona few days inorder tomeet the deadline for thesubmission ofthe manuscript. All these project staff are supported under the auspices of Science and Heritage Initiative (SandHI) at IIT Bombay. But for the financial support made available, I cannot imagine venturing into the task of producing this volume, as well as others that were produced in the past, or are in the pipeline. In this connection, I would like to express my sincere gratitude to Ms. Amita Sharma (former Additional Secretary, MHRD) and Prof. Devang Khakhar (Director, IIT Bombay) for getting thisprojectsanctionedandlettingmeworkwithitwithinthenormsofthesystemin an unfettered system.