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Annals of Science Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713692742 What We Can Learn from a Diagram: The Case of Aristarchus's On The Sizes and Distances of the Sun and Moon Nathan Sidoli a a Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada

Online Publication Date: 01 October 2007 To cite this Article: Sidoli, Nathan (2007) 'What We Can Learn from a Diagram: The Case of Aristarchus's On The Sizes and Distances of the Sun and Moon', Annals of Science, 64:4, 525 - 547 To link to this article: DOI: 10.1080/00033790701336841 URL: http://dx.doi.org/10.1080/00033790701336841

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Malzbender, T. the Hockley, Seiradakis, P. ‘Decoding Crawley, Mechanism’, J.H. A. Edmunds Allen, M.G. Moussas, M. Ramsey, and A. X. Bate, D. Bitsakis, Hadland, R. Y. Freeth, T. B.C., c.80 from et ihafwecpin,i sol eetyta h igasi h medieval the the as scrutiny in critical diagrams of the level same that the recently received only have itself. sciences is text exact it the exceptions, for few sources a With text. key text. the the form of always are must edition tradition removed Latin, manuscripts an manuscript the far of the of or social study basis thus Nevertheless, a the are the and Arabic, composition. texts, out, as ancient original are the carried well for was the source sources as work of this copies, best that which medieval from are in our these contexts they cases, making intellectual originals for and some the reasons In after The translations. preserve. centuries many to compiled presumed manuscripts Greek usually Fakuta an 01,122 2001), Main, am (Frankfurt Ptolemy’s eoro srnm NwYr,19) 569 1993), York, (New Astronomy on Memoir efn ntemnsrp ore.Mnsrp igas oee,aemr than more are however, diagrams, Manuscript what sources. than text manuscript the figures the in printed ideas in these mathematical cases, find many understanding for image In we reader. what helpful more the of fact, benefit idea editor’s in and are, the text on the based support diagrams best revised will print to been has recently, 4 3 2 Coniques .Netz, R. exact medieval of MSS Latin in diagrams of studies valuable of number a made has Eastwood B.S. 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Vat. ubro aucit nbt re n rbc hs,te,aeoreiec for evidence our are then, These, Arabic. of and text Greek the both in manuscripts of number srnmebdc.40H. 430 ca. bid Astronomie etr)hdwitno the on written had century) ahmtclcneto h raieadt nueta hsudrtnigntbe not understanding this that ensure in the to understand and when to reader Nevertheless, attempt treatise serious lost. the the some a scholia of that making aid shows which these content were presence transmission mathematical of their to scholia, Greek nonetheless, value the negligible; mathematical meant the in often readers perspective, and of were best mathematical at number a mediocre and From is fair text. diagrams a the auxiliary interpreting contains include codex sometimes the time, same aucitbssfrHahsciia dto was edition critical Heath’s for basis manuscript nthe on the on work 95,18. 1965), etof text oismd rmoeo oeBzniemnsrpswihaenwlost. now are which manuscripts Byzantine more or one from made copies fot nBgdd ehv oemnin ftecleto n fa es one least at of and collection the of it. mentions on some written have commentary we Arabic Baghdad, in efforts ahmtclperspective mathematical 25 24 23 22 21 20 ` ı o oto htw oko ftehsoyof history the of know do we what of most For h letsuc o h re eti elkonnnhcnuymanuscript, century ninth well-known a is text Greek the for source oldest The ap npriua,tefgrsfor figures the particular, In ancient late and medieval early Heath, for evidence considered be can that MSS Vatican five are There 35). and century 34 twelfth (notes the In below discussed are editions Hultsch, Latin Renaissance two The ’ ideBooks Middle ea ob rsne n edotietecneto hscleto ftexts of collection this of context the outside read and presented be to began ostrh 0.I otisanme ftetsso the of treatises of number a contains It 204. a.gr. Vat. Aristarch ideBooks Middle Aristarchus, nSizes On ap collectionis, Pappi ma ´ tvn a.gr. Vat. ideBooks. Middle hl oTha to v ’ nSms nescugnzrU zur Untersuchungen Samos: on 0 sgo,tedarm r oeie ofsn rma from confusing sometimes are diagrams the good, is 204 `o ı ideBooks Middle .36(oe6.H lomd s fWli’ciia dto,wihicue an includes which edition, critical Wallis’ of use made also He 6). (note 326 p. 9,12 0,23 n 0.Tetx itr of history text The 204. and 203, 202, 192, 191, yka itself. Lie,17) 65 1978), (Leiden, o xml Abu example for , ¯ CE i b ur;seF Rosenthal, F. see Qurra; ibn bit ideBooks; Middle .TeTasiso fteTreatise the of Transmission The 4. lSmwa ie h il facmetr htQus that commentary a of title the gives al-Samaw’al , ` ı nSizes On * solh Elements .44f.(oe16). (note ff. 474 p. nSizes On atta sntcerfo et’ edition. Heath’s from clear not is that fact a iteAstronomy Little ´ nhw h itadodro h raie eentfxdbut fixed not were treatises the of order and list The 22 Webdn 1992). (Wiesbaden, n ya es h hrenhcnuy tincluded it century, thirteenth the least at by and , ¯ e .Sezgin, F. see a rnltdit rbc ntennhcentury, ninth the in Arabic, into translated was ,1 n 4(f 1v 1rad116v). and 116r 113v, (ff. 14 and 13 7, Sa‘ı n Ptolemy’s and oevr rmtemdea eid ehv a have we period, medieval the from Moreover, 6 hr r te,msl ae,mnin ftetranslation the of mentions later, mostly other, are There 66. ¯ al-S d Conics ˙ afadı ¨ hl ntemdea rbctradition Arabic medieval the in while , eleeuggshct e Schrift der berlieferungsgeschichte ecihedsaaice crftm:Bn VI, Band Schrifttums: arabischen des Geschichte ¯ h lsia eiaei Islam in Heritage Classical The hr r e te important other few a are There . futet etr)atiue dtra work editorial attributes century) (fourteenth a.gr. Vat. Almagest. nSizes, On 21 iteAstronomy Little o h rbctranslation Arabic the For 204. nSizes On nSizes On 24 nlt niut,this antiquity, late In lhuhteGreek the Although twsicue na in included was it ˙ ta ¯ a ensuidby studied been has b Lu ibn a generally was Bree,CA, (Berkeley, ¯ por qa swl as well as 25 ¯ lt ninth (late ` ı mogouv tthe At 23 The On . 20 ˜ n Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 htmd yNas by made that h adi la,adatog hr r arnme fcmo rosi h text, the in errors common of fine. number fair very a are are diagrams there the although and clear, is hand The lost a to reference a is Baghdad Qus in by made known collection was the treatises on of commentary group this that evidence Greek the of sense make to work considerable required it sources. sixteenth, the in Latin, and ubro re S htcnandsm eeto fthe of selection some contained that MSS Greek of number edition the included which sciences, exact Arabic and Books Greek in works foundational al-T lived, he which through turmoil military o h otpr,T part, most the For ocs,eiiae eeiin n neesr lmnso ro tutr n only and structure proof of elements unnecessary and T repetitions idiom. eliminated concise, technical standardized more new of series a was work this of core The preservation. cultural ieta u aletGekM ftetx a en oidi yatu,Tha Byzantium, in copied being was text the of MS Greek earliest our that time b ur opsda rbceiini aha.Ti a oea a as done was This Baghdad. in edition Arabic an composed Qurra ibn rnltosbtti a o eesrl iiino aorbtenaphilologist a between labour Tha of example, previous division of for Arabic a basis mathematician; original necessarily the a on wrote not and out was also carried this texts often but were Greek translations Revisions of topics. related revisions on produced treatises who the Those of subject. argumentation. idea mathematical be author’s out fleshed Arabic redrawn, to by and the characterized figures held often relabelled, on are sakeand texts was course, mathematical the of of for what Revisions sacrificed content. based, original preserve was preservation, preservation to textual conceptual strict poorly of but words, purposes, other distorted our text In for corruption, figures. significantly textual drawn the most transmission: and, of change translations vagaries strange the argumentation, to from the content not essential for its was mathematicians revision Baghdad a the by used sources. Greek commonly of preservation composition of genre aeb Qus by made hs rnlto Tha translation whose Mu a isl rnltr esol di h osblt hth aeterevision the made he Tha text. that Greek possibility a the of admit basis the should on we directly translator, a himself was rmasnl Smd oadteedo h hrenhcnuy rbbyi Cairo. in probably century, thirteenth the of end the toward made MS single a from 27 26 ¯ h otcmo rbcvrinof version Arabic common most The encountered certainly almost researchers scientific Arabic These nodrt netk eiin coa a ohv oi omn fthe of command solid a have to had scholar a revision, a undertake to order In A sa .Kheirandish, E. Sezgin, F. ¯ re . nterrvso fteltrboso Apollonius’ of books later the of revision their in a tlsial lsrt t rbcsuc hna than source Arabic its to closer stylistically was v ision Aristarchus’s editions ˙ ecihedsaaice Schrifttums, arabischen des Geschichte ta ¯ a orcinaogtertcl rmteaia,lns h olof goal The lines. mathematical, or theoretical, along correction a was b Lu ibn h rbcVrino Euclid’s of Version Arabic The ˙ ¯ ı eeanwgnefrtepeetto fcnnclsucs An sources. canonical of presentation the for genre new a were ˙ al-Dı r u ¯ ¯ ¯ sı ¯ i sda h oreo i dto of edition his of source the as used bit qa seiin eemr ocs hnhssucsadwitni a in written and sources his than concise more were editions ’s nTeSzsadDsacso h u n Moon and Sun the of Distances and Sizes The On ¯ hl ehmefpoue h rnlto sdb h Banu the by used translation the produced himself he while ¯ al-T n ˙ u ¯ sı ¯ (1201 ˙ u ¯ ¯ ¯ sı i eie h rnlto fthe of translation the revised bit i’ eso of version bit’s nSizes, On ¯ pis os NwYr,19) xxvii. 1999), York, (New vols. 2 Optics, ˙ eokdidvda hsst emore be to phases individual reworked 24.I h is ftepltcland political the of midst the In 1274). .6 nt 22). (note 66 p. ta ˙ ¯ u ¯ b Lu ibn sı ¯ are u eakbepoetof project remarkable a out carried nedo l the all of indeed ¯ qa ¯ nSizes On d 912). (d. re v ision Conics. ideBooks. Middle nSizes On nSizes. On 26 oisGeksource Greek its to scretyknown currently is rudtesame the Around ideBooks, Middle ti o known not is It editions navr small very a in ic Tha Since re Elements h first The v ision, Middle fthe of ¯ ¯ 531 bit bit 27 is a ¯ . Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 3 .Sidoli N. 532 uao mrsu eeisprSmnmPpesmdcu eiaum 48 i lioSeptembris. ultimo Die 1498. Beuilaquam. dictum Papiensem Simonem per Venetiis Impressum humano, aeiitoutru,Glnsd ucdni,AeadrAhoiesd assfbim slsde Pselus febrium, causis de Aphrodiseus Alexander de presagio, succidaneis, de Galenus Galenus distemperantia, de presagitura, theologicis de inequali Galenus Galenus quibusdam de humani, introductorium, Galenus corporis de confirmatione Galeni pestilentia, Pamphili de de Galenus Eusebii habitu, Rhazes corporis poetica, Aristotelis musica, bono cello, ars de Cleonidis Aristotelis Aristotelis resurrectione, mundo, ethica, de magna de philosophi Athenagorae et mundo, Timeus magnitudinibus de de lunae, Cleomedes ambiguitatibus, Samii et Aristarchi astrolabo, elementorum, solis de decimus Nicephorus quartus distantiis Euclidis, Euclidis argumentis, libri de eiusdem libellus interpretatio Valle Hypsiclis Georgii logica, Nicephori continentur hec ubro e emsadrde ayo h igast elc i understanding his reflect to diagrams the mathematics. of the many a redrew of supplying and arguments lemmas key mathematical of the number all through carefully worked Commandino Tha the of those on based are diagrams the all corruption. particular, textual In suffered he places, had some In that argument. arguments an corrects in steps also or words clarifying introduced occasionally eemd,nn fteehsbe found. been has these of none made, were itne fteSnadMoon: 213 and Sun the of Distances xlctoiu ubsa.AFdrc omnioi aiu ouru,a omnaisillustratus commentarijs Commandini Federici ac cum conuersus, editus Latinum in Commandino Federico 1572), (Pesaro, A quibusdam. explicationibus sie rsnstetets ntesm otx stemnsrp rdto.I 48and 1488 these In of tradition. manuscript Neither the ( sources. Valla as Giorgio Greek scholar context humanist same the of 1498, the in basis again in the treatise the on presents made translations Renaissance a enpitdi h yeaa dto fT of edition Hyderabad the in printed been has T The lettered. identically rdcdtefrtciia dto fteGektx ae nGekMSand MSS Greek on based text Greek the apparatus. of critical edition a with critical supplied first the produced osntse ohv fetdteltrtaiino edn h text. the reading of tradition (1506 later the affected have to seem not does Sizes, 98,77 1998), tteaprts e ..Ree Jh als dtro re ahmtcltxs nGW ot ed., Most, G.W. in texts, mathematical Greek of editor Wallis, innovations edieren, useful ‘John Texte of Reeve, number Text, M.D. a included Editing see and apparatus; criticism text the of standards it contemporary highest the on based nacleto htwsmsl aeu ftasain fGektetss nthis In medical, treatises. philosophical, Greek of of array texts. translations dizzying mathematical a of and with up together mystical, made appears of mostly treatise version our was Latin book, that a published collection manuscripts, a Greek in of library his for scholars ftemnsrps arzNtoa irr S38,hsbe rne yJA Chavoshı J.A. by printed been has 3484, MS Library National Tabriz manuscripts, the of ore,poue atn nlec nteErpa coasi ntetreatise. the on scholarship European the in influence lasting a produced sources, i tTs:Tah at-Tusi: Din `cle 34 33 32 31 30 29 28 onWli (1616 Wallis John lhuhteei oeeiec htmdea ai rnltosof translations Latin medieval that evidence some is there Although nteohrhn,teeiino h etpbihdb rdrc Commandino Frederico by published text the of edition the hand, other the On 254. .Wallis, J. Commandino, F. follows: as reads edition 1498 the of page title The Noack, ‘Aristarchus’s Sidoli, and Berggren Al-T in shown is This Rashed, R. see practice, textual his of examples For os Lno,19) 12 1996), (London, vols. 2 , tcnb hw htT that shown be can it 55 oadteedo ogcre ftasaigGekmathematical Greek translating of career long a of end the toward 1575) ˙ u ¯ 93. sı ¯ , Aristarch Kita rsaciSmid antdnbs&dsati oi ua,lbr ucpiu Graece primum Nunc liber. lunae, & solis distantiis & magnitudinibus de Samii Aristarchi ˙ rı ¯ ¯ - Mutawassı r-e Arist b rsaciD antdnbs tdsati oi,e ua,lbrcmPpiAlexandrini Pappi cum liber lunae, et solis, distantiis et magnitudinibus, De Aristarchi v ˙ nSamos, on rhsfı arkhus prmt:Kiih tde u hlgeecihe ad2(Go 2 Band Philogiegeschichte, zur Studien Kritiche Aporemata: 73 aego s fCmadn’ okwe he when book Commandino’s of use good made 1703) ˙ ¯ ˙ t u re n rbcTexts’, Arabic and Greek ¯ ¯ v a ¯ rin aia oiqeilu editoris & illius notisque Latina, ersione imya-aianwa-bu‘dayhima al-nayirayn jirmay sı t ¯ ˙ p 45 pp. Tha,2005). (Tehran, 7adJL egrnadN ioi ‘Aristarchus’s Sidoli, N. and Berggren J.L. and 27 u ¯ eso of version sı ¯ aehseiino h ai fTha of basis the on edition his made 32 7(oe25). (note 47 34 al’ rnlto a enltl tde and studied little been has translation Valla’s et,i un sdWli’aprtsa his as apparatus Wallis’ used turn, in Heath, nSizes On 31 h aletetn ai dtosaetwo are editions Latin extant earliest The eri al lcnioItrrt o in hoc Interprete Placentino Valla Georgio e mathe Les nteSizes’ the On Archi ˙ u ¯ sı setn nanme fMS and MSS, of number a in extant is ¯ c v ’s o itr fEatScience, Exact of History for e .1430 ¯ ideBooks Middle mtqe infinite ´matiques Hdrbd 90.Afciieo one of facsimile A 1940). (Hyderabad, nt 28). (note 49,nwbs nw to known best now 1499), Ofr,18) als etwas text Wallis’ 1688). (Oxford, 28 ¯ i eso n are and version bit ntecs of case the In smlsd X uXIe au IXe du ´simales . 30 ¯ i’ revision. bit’s nteSzsand Sizes the On ¯ intro., , nSizes On nSizes On 1(2007), 61 v ¨ ai al- Nasir olumine ttingen, v On ictu 29 33 Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 qiiru fPlanes, of Equilibrium eut o xml,tefrtsxpooiin r eddt show to needed are propositions six first the the or example, For earth using result. conveyed from are luminaries sizes of the Ratios of Earth. volumes. and and distances Moon diameters relative Sun, both the the of pairs of of ratio sizes relative a on bounds as rvdsbud eaigteslrdsac otelnrdsac,whereas distance, lunar the to distance solar 12 the relating bounds provides Phenomena, 80,wihgv i cest oeraig fMSi ai htdUbnhdnoted. had d’Urban that Paris in MSS of readings some to access him gave which 1810), swrhntn htAciee’mcaia treatises, mechanical Archimedes’ that noting worth is BCE against check primary Autolycus’ tep ostaeteewti nitrrtv rmwr.Teeaepoal a probably text. are the of There goals framework. the interpretive understand to an ways within valid of these number situate to attempt diameter. terrestrial the nSizes On with similarities computational te ad Archimedes’ hand, other Astronomy Mathematical Ancient of History A antb eti twa on,pirt h is century, first the we to Although prior world. point, the what in at things certain the be mathematical about in established cannot statements these, through numerical proceeds, and derive then diagrams It to geometric computations. methods, in in depicted used be be to to world turn, physical the of aspect some h tl fddciemteais tsosta fw tr ihafwsimple, few Earth. their a the and of with distances that and start we sizes to then relative we relative luminaries, their sizes if about the statements of that certain phenomena demonstrate shows visible can the It about mathematics. assumptions crude, deductive perhaps of style the luisPtolemy Claudius oistasiso yteBzniesrbs ewl ou ntedarm to diagrams the on focus will We scribes. opposed Byzantine as the translation, Sizes of by periods transmission during read its was to treatise this way the in differences subsequent all text. for Greek point an the starting the were of formed which readings have diagrams, scholarship, his his of and aspect mathematics important the of understanding Commandino’s h paetszso h u n Moon,’’ and sun the of sizes apparent the egrnadSdl,‘Aristarchus’s Sidoli, and Berggren 37 36 35 ubro rpstosaeesnilylma edn pt nimportant an to up leading lemmas essentially are propositions of number A serve that numbers between ratios two generate to is text the of strategy basic The h raiea tsad ssml eiso ahmtclsaeet;i ae no makes it statements; mathematical of series a simply is stands it as treatise The nSizes On ecnuetedarm nteGek rbc n ai ore oexamine to sources Latin and Arabic, Greek, the in diagrams the use can We . 4aelma for lemmas are 14 tsol entd oee,ta ftee only these, of that however, noted, be should It Neugebauer, O. example, For text. the of approach and aim the of discussions of number a are d’Urban, There Fortia A. of work the consulted also Heath 37 13. ti eeal eivdta h te re raie fti om uhas such form, this of treatises Greek other the that believed generally is It and Aristarchus’s adReckoner Sand naMo a On eeas opsddrn hsperiod. this during composed also were sawr fsrcua omlg ratooy hc speetdin presented is which astronomy, or cosmology structural of work a is Blioe D 97,171 1977), MD, (Baltimore, esreto Circle, a of Measurment v n Sphere ing nSizes On hc r ftesm ye eewitni h hr century third the in written were type, same the of are which nTeSzsadDsacso h u n Moon and Sun the of Distances and Sizes The On . a.gr. Vat. nSizes. On nteSizes the On .TeTets Itself Treatise The 5. 5 hc onstertoo h oa imtrto diameter solar the of ratio the bounds which 15, n,a led oe,teeaecnieal iiaiisbetween similarities considerable are there noted, already as And, 0,wihWli i o consult. not did Wallis which 204, and nieNtrisncatudir Rezeption ihre und Naturwissenschaft Antike 36 ec,i eiswt yohss hc allow which hypotheses, with begins it Hence, 7ad389 and 77 iig n Settings and Risings NwYr,17) 634 1975), York, (New nt 28). (note ’ lhuhpr ahmtc,ehbt ubro key of number a exhibits mathematics, pure although laigBodies Floating itiedAitru eSamos de d’Aristarque Histoire 4 .Pnhno ‘rsacu fSmson Samos of ‘‘Aristarchus Panchenko, D. 94, laigBodies Floating otisnmrclrsls nthe On results. numerical contains nSizes On 3 ..Newton, R.R. 43, n Euclid’s and nSizes On a opsd it composed, was 35 1(01,23 (2001), 11 nti way, this In os(Paris, vols 2 , and Optics h rm of Crime The nSizes On ,which 7, nthe On and 533 On 29, Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 3 .Sidoli N. 534 efreog wyt evr uhlre hnteSnappears. Sun could bright the so than is larger which object much no very that be claim the to and away Sun enough the trans. far of and brightness be ed. the Hicks, R.D. concerned appears.’’ this it for argument as us, to relative much, Li as just is stars remaining a htteSnilmntsteMo,()ta hyapa ob h aesz,(c) size, same the be to appear small they think is that we (b) span if Moon, angular is, the their That illuminates that apparent. Sun immediately the be that themselves (a) not may that implications hdwi h w onsFadE h ieF a ecle the called be may FE line The E. and F points two the in divides shadow that the circle called the be the of may call line diameter This the shadow. its is from DE line. Moon line the of The side Moon. illuminated the the C circle and Earth, which in context intellectual provides mathematics that case general the heavens. the making of phenomena as the examining taken for tools be it. powerful than may Sun smaller It the the is that relevant. than Moon and away us, the further from while distance much Earth its and the to larger relative than much small larger fact, quite is is in Moon is, the Sun that the Moon, another that one shown from be angles can right at nearly are they h aesz steshr nwihteSnmvs Heiberg, moves. Sun the which in sphere the as size same the ocnrcshr oe ftecosmos. the of model concentric-sphere ecpini rdcn nweg bu h ie n itne fastronomical of distances and sizes the sense about of knowledge objects. function the producing about school in his perception and Epicurus by raised problems philosophical twswitnfrasalgopo edr h ol easmdt aehda had have to its assumed of be relevance the could of who appreciation an readers and of uses it group results. methods small the of a knowledge working for written was it hyaeraiyaprn hnoecnrnsteeiec ftemanuscript the of evidence the confronts one although when science, of apparent demands. historians diagrams. readily a by proposition draw noticed are to the fact, been possible In recently configuration they actually Moon. only not the have the is difficulties of of it These orbit and diagram true, the coherent strictly to are relation mathematically assumptions its these and of shadow none Earth’s the of geometry treatise. the in demonstrating theorems for important necessary most are the they of require, they work the worth to prefatory v 41 40 39 38 so mnn Philosophers, Eminent of es tabsclvl tage htcam bu perne aemathematical have appearances about claims that argues it level, basic a At nFgr ,tecrl bu eteAi h u,tecrl bu eteBthe B centre about circle the Sun, the is A centre about circle the 1, Figure In specific the know to claim cannot we that however, case, the remains It been have would work this which in contexts different of think to possible is It nSizes On with along proposition, thirteenth The egrnadSdl,‘Aristarchus’s Sidoli, and Berggren his in Archimedes, by told are We his in Epicurus, that us of tells goal Laertius the Diogenes as this asserts Neugebauer rsacu el sta h n-onso Esepotacrl,wihw may we which circle, a out sweep DE of end-points the that us tells Aristarchus 39 npitcircle endpoint 41 ial,i a drs omlgclise htwr eeatt a to relevant were that issues cosmological address may it Finally, nSizes On 3poed ymkn ubro mlctsmlfctost the to simplifications implicit of number a making by proceeds 13 n htacodo hscrl,ln E nescsteEarth’s the intersects FE, line circle, this of chord a that and , 5 hti,wieisrslsmyntb hmevsinherently themselves be not may results its while is, That 15. os(abig,M,17) o.2 1.Acrigt igns the Diogenes, to According 618. 2, vol. 1972), MA, (Cambridge, vols 2 nSizes On adReckoner, Sand * nteSizes’ the On 6. a,a ot2 most at say, nSizes On nSizes On 40 a rdcd eetees ti la that clear is it Nevertheless, produced. was nNature, On in htms srnmr osdrdtecso obe to cosmos the considered astronomers most that nt 28). (note nSizes On HAMA, 13 * 8 d hthl onocr when occurs Moon half that (d) , Archimedis ra ot a,87 say, most, at or ttdta,‘Tesz ftesnadthe and sun the of size ‘‘The that, stated .63(oe38) (note 643 p. 2&1,myb ae as taken be may 14, & 12 o.2 1 nt 10). (note 218 2, vol. , 38 tmyadesspecific address may It . nSizes On npitchord. endpoint ignsLaertius: Diogenes 8 pr,te it then apart, di 5 one 15, v iding Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 ersra hdw htB stnett h on n httepito agny E, tangency, of point the that and Moon, the to tangent is BE that shadow, terrestrial shadow. terrestrial the of w uvs h rcs hp fwihdpn nteata ie n oeet of movements and sizes the actual as the out congruent on sweep Moon. depend remain actually and which will not Sun of DE the shape will of precise endpoints DEB the the triangle curves, Indeed, two Hence, Earth. the Earth. however, relative around fact, the position moves Moon’s In Moon not the B. by Sun, around determined is rotates the DE, it to line, that dividing the the so of moves to of orientation Moon the reasonable motion the fairly as the fact, congruent view in remain is, to it which day, against a shadow degree a terrestrial Moon. roughly configuration while stationary hour, move an the of a about to motion is in assume the seen eclipse to lunar shadow is relative total still average Sun Earth’s is an Sun the of the duration the that the of imagined Since be Moon. triangle must the it the 1, Figure the intersect in depicted when to moment EDF first the circle at set shadow. Earth’s be the objects enters the fully that Moon requires proof the Moreover, h emti tpltoso h hoe eur htB steai fthe of axis the is BD that require theorem the of stipulations geometric The nodrfrteedonso Et we u ice h rageBEmust BDE triangle the circle, a out sweep to DE of endpoints the for order In the for order In assumptions. hidden of number a contain statements few These Aristarchus’s iue1 osbercntutddarmfor diagram reconstructed Possible 1. Figure nSizes On nTeSzsadDsacso h u n Moon and Sun the of Distances and Sizes The On J 3apoiae hs uvswt icei h vicinity the in circle a with curves these approximates 13 F B D G A C E H nSizes On 13. 535 Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 3 .Sidoli N. 536 otedaee fteMo,EF: Moon, the of diameter the to Sizes ua imtrpredclrt h iejiigtecnr fteSnwt h centre the with grey. Sun the in of Moon’s shown centre the the Moon, the to by joining equivalent the perceptibly line determined the of is the line not to of dividing perpendicular is the proof diameter fact, line lunar In the dividing Earth. the the time, to relative of same position shadow. orientation Earth’s the the the above, At within stated entirely shadow. be terrestrial should Moon the the of that demands edge proposition the on falls npitcod hs r sdin used are These chord. endpoint t itnefo h at.Ti,i un emt i odtrietesz fthe of size the determine in Earth to the him of permits to size Moon turn, the the in of to size This, relative the him Earth. relating Sun allow for point the assumptions object contact from These theoretical the key Earth. distance E, a the its as at of line GH centre endpoint line the the from the use Moon meets shadow.to the it the to that from tangent and protrude a As chord shadow. of will endpoint terrestrial Moon an the is the there GH, of that on edge is the E GH, when line 1, on Figure Moon in the shown touch not will Moon onsfrtertoo h npitcodt h imtro h u,EF: Sun, the of diameter the to chord endpoint the of ratio the for bounds xatdarm iltl sago elaothwtetx a edi h medieval the in the read of was text study the a how about Nevertheless, periods. deal modern text. good reproducing early a his as diagrams and us taken accompany the tell be will can in to diagrams sources variety drew extant our the in Aristarchus diagrams of which the consideration of that fact, we none diagrams that consult In likely same we edition. When it the original. makes contains published the them the to of get in none can that find we as find close we however, as manuscripts, is that the diagram a them. with text around the constructed are proof his and and objects, a coherent diagram of perfectly depiction his as accurate both them strictly treats a they Aristarchus is, in Nevertheless, That objects eclipse. theoretical. geometric lunar purely well-defined are as line, taken endpoint be the cannot and circle endpoint the objects, eipeeo h onfcn wyfo h u,sneteSni uhgreater much is Sun the since Sun, the from away Moon. the facing than Moon the of hemisphere oteegso h oeo h ersra hdw FJ,i ie oe bound. lower a given is EF:JH, shadow, terrestrial diameter solar the follows. the of as of are cone extension bounds the the These to of chord edges endpoint the the to of ratio the and provided; 43 42 lal,i sntpsil o l fteetig ob rea h aetm.As time. same the at true be to things these of all for possible not is it Clearly, h ups of purpose The eetees nodrt banhscmuainlgas rsacu ilassume will Aristarchus goals, computational his obtain to order in Nevertheless, nodrt drs h atclr fAitrhssapoc,w hudcompare should we approach, Aristarchus’s of particulars the address to order In of features interesting most the of One nSizes On Sizes On rvsta h u smc agrta h Moon. the than larger much is Sun the that proves 9 eosrtsta agrshr luiae oeta afo mle pee while sphere, smaller a of half than more illuminates sphere larger a that demonstrates 2 hw httedvdn icei o ecpil ifrn rmagetcircle. great a from different perceptibly not is circle dividing the that shows 4 nSizes On nSizes On nSizes On 43 nete ae agn rw rmtecnr fteErht the to Earth the of centre the from drawn tangent a case, either In nSizes On 13a 3:(0 13e: 13c 3i odvlpfv ueia ttmnsivligthe involving statements numerical five develop to is 13 ; ; 42 :(0 d: :(2 b: paigasltl,tedvdn iewl ei the in be will line dividing the absolutely, Speaking : 10 d nSizes On : m 11 sbuddaoeadblw pe n lower and upper below; and above bounded is , 2:1 ) 979:10 ) nSizes On )1:9 B nSizes On 4&1.Tertoo h npitchord endpoint the of ratio The 15. & 14 EF: B ; 125 EF: d m 15. B B d s 3i httetokygeometric key two the that is 13 EF:JH 84 ( 88:45 225( 22:225 1 : 0 96) : 10) d s are , On Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 ahmtclyncsay hscnb otatdwt ae hr oeta one than in more example, for where as, proof, cases the for with required contrasted actually is be figure can This necessary. mathematically h is iga,o iejoining line a or diagram, first the f 2] loteeyoeo h le re S ae ifrn praht the to approach than different better a is takes none MSS but Greek Moon, older the the in of lines one every Almost 62r]. [f. thirteenth good generally a ie nteSnta r eddfrtepof u tdosteMo niey nfact, In entirely. Moon the drops it but the proof, the contains for diagram needed This line manuscripts. are however, that Greek Sun in the practice in common lines the is as text, the rti nt ,p 2) htvrtecs,ti sntoeo h l S hc ocr shere. us concern which MSS old the of one not is this case, the Whatever James, likely 325). was M.R. p. it theorem. in 5, me Cambridge the (note tells listed & College, he Britain of not correspondence, Trinity understanding In is figure. Library of peculiar It this Library saw a Cambridge. Vatican he College, had where Trinity have say The to at must able Reborn: figure not is this Netz ‘‘Rome Unfortunately, drew Whoever exhibit features. Netz (http://www.loc.gov/exhibits/vatican/acknow.html). interesting the Grafton of forSizes A. http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/ figures part different by significantly at prints is edited online for Culture,’’ image diagram first Renaissance seen the This of image full-colour be A Greek_math2.html. 25). (note can XXV and A 204 XIII B, XII B, XI on hr otedaee fteMo,aecridotuigFgr .This 2. Figure using out carried are diameter lunar Moon, that except the end- this, the of for relating sufficient theorem, diameter basically the is the of diagram parts to two first chord The point grounds. visual purely on made otnainof continuation irr,Vat. Library, propositions. igas n o h aeo qa pee n nte o h aeo unequal of case the for another and in spheres equal fact, manuscript. of in the case Indeed, spheres. the in for diagrams one two diagrams, has that proposition ohv enraie ymdea coas ssonb h ait fmedieval two of variety in the Moon by shown the as intersect scholars, drawing medieval of must by impossibility appears GEH proposition The realized diagrams. the line. of Sun, been features different dividing have geometric a the the true to all while depicts the another from that Moon, diagram meet be the accurate E to an cannot to fails DE on tangent figure Hence, is the falls points. that BE means line GH, This Since E. chord, line, proof. point, endpoint the shadow, The single lunar of the E. a of requirement and in edge the D meet and line, all DE, dividing line, GH, dividing the is the EDF, of DE, circle, endpoints endpoint line, the the dividing EF, at The proof. it by the been touch shown has of Moon as it requirements diameter, section, the lunar previous of a number nearly the a in meet described to As drawn evidence. manuscript any on based 45 44 hsi aecs fapooiinuigmr hnoedarmweei snot is it where diagram one than more using proposition a of case rare a is This o h atrprsof parts latter the For In h letGekMScnantodarm for diagrams two contain MSS Greek oldest The of discussion foregoing the accompanies which 1, Figure hsi reo l fteGekmnsrpsta aese,teods S,ali h Vatican the in all MSS, oldest the seen, have I that for manuscripts diagrams the Greek of the number A of all of true is This 3i ige obndfgr n,fo ahmtclprpcie tcnan ubrof number a contains it perspective, mathematical a from and, figure combined single, a is 13 nSizes On a.gr. Vat. Aristarchus’s gr. bn 9,12 0,23 0,adanme fterdescendants. their of number a and 204, 203, 203, 192, 191, jl 3 oee,tecoc oicuetodarm em ohv been have to seems diagrams two include to choice the however, 13, sas sdi h ae at,s htterae utrfrbc to back refer must reader the that so parts, later the in used also is hra h etseiista tb aallto parallel be it that specifies text the whereas , 0,tetoprsof parts two the 204, nSizes On nTeSzsadDsacso h u n Moon and Sun the of Distances and Sizes The On .TeDarm for Diagrams The 7. nSizes On nSizes On orenhcnuyM,teeaen ie nteMoon the in lines no are there MS, century fourteenth ,tesm iga antsrefrbt ae.In cases. both for serve cannot diagram same the 1, nSizes On Cmrde 94,adHahke fn rsacu S in MSS Aristarchus no of knew Heath and 1904), (Cambridge, b 3aerpoue nNoack, in reproduced are 13 13, and nSizes On 3&1 in 14 & 13 a.gr. Vat. n nSizes On a.gr. Vat. a oemissing. gone has .Tetnet rw rmBt the to B from drawn tangents The 4. nSizes On 0 otisFgr ,cmn after coming 3, Figure contains 204 Shaping, aaou fWsenMnsrpsi the in Manuscripts Western of Catalogue 204. nSizes On r ubrda separate as numbered are 1 p 1ad4 nt ) htfor That 5). (note 42 and 41 pp. 45 13 nSizes On nSizes On Aristarch 13. nSizes On 44 l jn nSizes On hsi h second the is This v per ob a be to appears o nSamos on ranme of number a or 1 .In lohstwo has also 1 a.gr. Vat. 3in 13 3 snot is 13, ,Taf.VIA, a.gr. Vat. 191, 537 On Downloaded By: [Sidoli, Nathan] At: 06:29 11 October 2007 3 .Sidoli N. 538 ml oe etfrte hntetx a copied. was text the Cassino, when Monte them Sinai, for Mount left Catherine, boxes St. small of Monasteries the of Libraries the Nanantola from Lorsch, Centuries 17th the to 28). uhdul igasadteriacrc a ru gis aigte as them taking against for argue possible,diagram anything. may not prove not is inaccuracy do It gone features their text. these has and of but the rarity line original, The diagrams with one. only A or inconsistent double diagrams, text. two are originally such were the others there if of accompanied determine to couple however, originally a that and those missing, from diagrams the hoesApollonius’ theorems o i eiin hc a,i un h ai o Tu for basis the turn, in was, which revision, his for eas twssl ytebo elrH .Kraus. consistent, P. 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Science These Bennett this National for the the by necessary supported Sciences works at work obscure upon Yuen Computational the at Vera of and place number University. a a Mathematical Fraser locate me Simon the giving referees by The at the in research trivial. and Center my Research as Bowen overlooked supported Alan Interdisciplinary had has of I Archibald the readings Tom that careful journal. details the this of from for number benefited a was has workshop consider this diagrams itself to at paper participants on led Osaka, other was in the material Saito I with Ken discussions that The by during initially organized Berggren. was diagrams It Len manuscript Japan. on with workshop a collaboration for prepared in out carried Aristarchus’s on research My when of prudence author exercise the that by to intended centuries and accompany. was diagrams the they care us these text with of in encourage the feature diagrams should times particular manuscript any realization various that the This claiming at study composition. altered to original other been their both figures name from have the can that us may We admit separate evidence. generally texts must documentary during canonical we out no but of carried editors, or individual been little have or have also junctures may possible we sort appropriation. which this cultural for of of periods revisions periods that during recognize revised must considerably We was the diagram of a views their which on light shed to and preservation. activity cultural diagrams their new study of the to project and use texts improved can these We an study. for for be drew available to work competent they the considered mathematically of and they version what Baghdad by intelligible making ninth-century more redrawn in both interested in were were that who diagrams significant scholars is the It Italy, used. and sixteenth-century computational read was of text of sake the features the interesting for most the assumptions of implicit some and are objects results theoretical of use ial,ti td hudsrea atoaytl.W aese pcfccssin cases specific seen have We tale. cautionary a as serve should study this Finally, which in way the of indications are diagrams the perspective, historical a From Aristarchus’s nTeSzsadDsacso h u n Moon and Sun the of Distances and Sizes The On nteSzsadDsacso h u n Moon and Sun the of Distances and Sizes the On Acknowledgements nSizes. On was 547