Sources and Studies in the History of Mathematics and Physical Sciences ManagingEditor J.Z.Buchwald AssociateEditors J.L.Berggren and J.Lu¨tzen AdvisoryBoard C. Fraser, T.Sauer, A. Shapiro Forfurthervolumes: http://www.springer.com/series/4142 Michael N. Fried Edmond Halley’s Reconstruction of the Lost Book of Apollonius’s Conics Translation and Commentary ABC MichaelN.Fried ProgramforScienceandTechnologyEducation Ben-GurionUniversityoftheNegev MarcusFamilyCampus Beer-Sheva84105 Israel [email protected] ISBN978-1-4614-0145-2 e-ISBN978-1-4614-0146-9 DOI10.1007/978-1-4614-0146-9 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2011933227 ©SpringerScience+BusinessMedia,LLC2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerScience+BusinessMedia(www.springer.com) FormyparentsDanand Thelma Preface The present translation of Halley’s reconstruction of Book VIII of Apollonius’s Conicswascarriedoutoverthecourseofseveralyears.IttookthatlongbecauseI wearanotherhat;namely,thatofalecturerinaprogramforscienceandtechnology education.And,inthatfield,translatingandcommentingonearly18thcenturyre- constructionsofancientGreekmathematicalworks,unfortunately,hasasomewhat lowpriority. Still, though my interest in Halley’s reconstruction of Conics VIII grew origi- nally outof previouswork I haddoneon ApolloniusofPerga,the moreI worked onthetranslationandthoughtaboutHalley,themoreIbegantoseethattheproject actuallyspoketomyeducationalinterestsaswell.Ihavelongbeenconcernedwith therolethathistoryofmathematicscanplayinmathematicseducation.Ihavehadto askmyselfinthisconnectionwhatexactlydoesastudentofmathematicsgainfrom history?ThiswasthequestionthatpreoccupiedmemoreandmoreaboutEdmond Halley. Thoughhe was a master of the modern and increasingly powerfulmathe- maticsofhistime,Halleytreatedtheancientmathematicianswithgreatseriousness. Why? What did he see himself learning fromengagingwith mathematicianssuch asApollonius? This really became for me the main question in the backgroundof the present work.OfcourseoneshouldalsoaskhowfarHalley’sreconstructionofBookVIII succeededinreproducingApollonius’sownthoughtandthislostbookoftheCon- ics.Afterall,thatwasHalley’simmediategoal,andwhatheproducedinthisregard washardlytrivial.Indeed,asapersonwhohasspentmanyyearsstudyingtheCon- ics, I am tremendously impressed by the profundity of Halley’s understanding of Apollonius.Ultimately,Halley’sreconstructionofBookVIIIismoreaboutHalley than about Apollonius. It is, in a way, the portrait of this man’s deep relationship withthepast. AsIremarkedattheoutset,myownhistoricalworkhasoftenhadtotakeaback- seat to otherinstitutionalobligations.Duringa periodwhen I feltthis particularly acutelyanddespairingly,IwasurgedtotakeuptheHalleyprojectbyadearfriend and, by all accounts, a brilliant anthropologist, Tania Forte. Tania persuaded me thatwithoutsuchaprojectIwoulddriftinexorably,asahistorianofmathematics, vii viii Preface toward the River of Lethe. So, to Tania I must offer my very first thanks. Sadly, Tania will never read this: she died suddenly and tragically in 2005, just as she wasbeginningherownpromisingcareerandjustasIwasmakingmyfirststepsin thistranslationandcommentary.AfterTania,ImustthanknextJohnNeufromthe historyofsciencelibraryattheUniversityofWisconsinwhogenerouslymadethe text of Halley’s reconstruction available to me when I was working on my Ph.D. some ten yearsago. I am still touched by the kindnesshe showed me then. I also want to thank Gideon Freudenthal, who managed to find for me Halley’s preface totheentire1710editionoftheConics,whichIcouldnotfindinthebestlibraries hereinIsrael.MayerGoldberghelpedmeobtainpapersonHalley’sactuarialwork and,moreimportantly,providedinvaluablehelpinallthingsdigital,nottospeakof wonderfullonghoursofconversationovercoffee.ThroughoutmyworkonHalley’s Book VIII, Sabetai Unguruwas my constant advisor:he was always the first per- sonIturnedtowhenIhadquestions,doubtsorideasthatneedingworkingthrough andheneverfailedme.Iowetohimdeepthanks,notonlyforthis,butalsoforhis encouragementandwarmfriendship. ThanksarealsoduetoMarinusTaisbakwhoneverturnedmeawaywhenIcame to him with Latin problems. I also want to thank my friend and fellow historian of Greek mathematics, Alain Bernard, with whom I discussed the reconstruction onseveraloccasions.Theintroductionbelowowesmuchtomyconversationswith Alain. LennartBerggren,who read the manuscriptwith a fine-toothedcomb, also deservesthanks:hismanycommentsandsuggestionshavemadethisamuchbetter workthanitwaswhenhefirstreceivedit. Finally,Iamgratefultomywife,Yifat,whotrulysupportedmeinthisproject, saying often, “I think you should put everything else away today and work on Halley.” Although the work might have been begun without her, without her, it wouldneverhavebeencompleted. Beer-Sheva MichaelN.Fried April,2011 Contents PartI Introduction 1 EdmondHalley:AncientandModern ............................ 3 2 Apollonius’sConics............................................. 7 3 ThePathtoHalley ............................................. 13 4 Halley’sGeneralStrategyforReconstructingConics, BookVIII .... 17 5 Halley’sDialoguewiththePast .................................. 23 6 ANoteontheTranslation....................................... 33 PartII ApolloniusofPerga’sOnConics:BookEightRestored APOLLONIUS OF PERGA’S ON CONICS: BOOK EIGHT RESTOREDORTHEBOOKONDETERMINATEPROBLEMS CONJECTURED................................................... 37 PartIII SynopsisandAppendices SynopsisoftheContentsofHalley’sConics,BookVIII .................117 PropositionsI-IV:TheParabolaandInitialPropositionsfortheEllipse andHyperbola .............................................117 PropositionsV-XVIII:ConjugateDiameters.........................117 ScholiumSpecialCasesofthe“ApplicationofAreas”.................119 PropositionsXIX-XXXIII:DiametersandtheirLateraRecta...........119 ix