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View metadata, citation and similar papers at core.ac.ukbrought to you byCOREprovided by Elsevier - Publisher ConnectorHistoriaMathematics7 (1980)134-140SOLARAND LUNAR VELOCITIES IN THEALFONSINE TABLESBY BERNARD R. GOLDSTEINUNIVERSITY OF PITTSBURGH, PITTSBURGH, PA,AND INSTITUTE FOR ADVANCED STUDY, PRINCETON. NJSUMMARIESAlthoughnot previouslynoticed,the solarandlunar velocitytablesin the 1483 editionof the dlfonsineTables differfrom the correspondingtablesin the 1518 edition(Venice)which are identicaltothose that appear in the Toledan Tables and the zijof al-Battzny.The velocitytablesin the unpublishedHebrew versionof the dlfonsineTables,composed in1460 (dvignon),agree with the 1483 edition,and theHebrew translatortellsus that these tables(togetherwith some others)were added to the AlfonsineTablesby Christianscholarsafterthe time of Alfonso.Adiscussion.of the underlyingparametersis also presented.Personnen'a observeque l'&ditionde 1483 destablesalphonsinescontientune table des v&locit&solaireset lunairesqui differede la table correspondantede l'editionde 1518 (Venise)qui est identique 2 cellesdes tablesde Tolkde et 2 celledu zIjd'al-BattZn?.La table des velocitksde la versionh&bralquedes tablesalphonsines,composee en 1460(dvignon),s'accordeavec l'editionde 1483, etl'auteurde la versionhebrai'quenous dit que dessavantschrgtienspost&ieursa l'&poqued'dlphonseont ajoutgcettetable(et d'autresaussi)aux tablesalphonsines.Cet articleprkenteen plus une discussiondes &l&mentsfondamentauxdu calculde cesvelocitk.In order to compute the durationof eclipsesand the timefrom mean syzygy to true syzygy,most medieval sets of astronomical tables(Arabic:zzj)includedvelocitytables[ll.there is no indicationof the procedureused to conUsually,Althoughsome printededitionsof the Alfonsinestructthem.Tables containthe hourlysolarand lunar velocitytablesthatappear in the z?j of al-Battan?(a. 929) [21, via an inter0315-0860/80/020134-07 02.00/OCopyright 0 1980 by Academic Press, Inc.All rights of reproduction in any form reserved.134

HM7Solarand LunarVelocitiesinthe (deg/h)AlfonsineTables(deg/h)LunarLunarminimum velocitymaximun velocity0;30,180;36, 40;30,210;36,25SolarSolarminimum velocitymaximum velocity0; 2,230; 2,330; 2,230; 2.34mediary(1483)text (e.g.,the Toledan Tables), the 1483 edition[33 has(folios[g6lr-Ig73r)that is quite differentand of unknown origin(see Table 1).This table is constructedfor degreeintervalsfrom 1' to 180“ of argument (solar or lunar anomaly)with entriesfor the hourly solar and lunar velocities.The solarvelocityhas a minimum value, 0;2,23O/h,that agrees withal-BattZni'sminimum, and a maximum value, 0;2,34'/h,that exceeds his maximum; this may be due to the use of a slightlygreater eccentricity.For example, an eccentricityof 2;14 yieldsthe maximum and minimum values that appear here (cf. Goldstein[1974, 1101) .The lunar velocityalso has maximum values that differslightlyfrom those of al-BattZni.The line-by-linedifferencesare always 0, 1, 2, or 3 seconds (but for an isolatedprintingerror at 123"); 77 to 128O is the longest stretch where the lineby-linedifferencesare constant(3").For the most part thesevelocitiesare based on a lunar eccentricityof 5;lO in a simpleeccentricmodel (see Table 2) [4], whereas Ptolemy and al-Battaniused 5;15 for the lunar eccentricity.However, the last severalentries depart seriouslyfrom the expected values and require adifferentexplanation.The entry in the text for 180' suggestsa lunar eccentricityof 5;47, but this parameter does not producecloseagreementelsewhere in the table.Moreover, the line-byline differencesin the text near the end of the table are incompatible with any plausiblemodel for lunar motion.The onlyexplanationI can offer is that some error was committed incalculatingthe last 20 (approximately)entries and that theauthor was unaware that he had exceeded the maximum value forthe velocitybased on the eccentricity5;lO.The line-by-linedifferencesare all smooth in this region of the table,and thismay mean that he had alternativesets of line-by-linedifferencesPresumably, only a small numberand applied the wrong set here.of entries were computed directly,and the rest were found by interTable 3 displays a reconstructionof the end of thepolation.a table

Bernard136R. GoldsteinHM7table with a simple set of line-by-linedifferencesthat agreeswith the recomputationto within1".The maximum lunar equationof centerin the AlfonsineTablesis 4;56',which is also found in the KhwarizmlTables and isbased on an Indian tradition(cf. Neugebauer[1962, 961); thecorrespondingeccentricityis 5;lO ( 60 sin 4;56O).This leadsme to believethat al-Battk?'svelocitytableswere replacedbysomeone who feltthis tableshould be consistentwith the equation table.The solareccentricitymentionedabove, 2;14, mayalso be connectedwith the Indian traditionrepresentedin theKhwarizmiTables,becausethe maximum solarequationthereis 2;14O(cf. Neugebauer[1962, 951); strictly,this would requirea solareccentricityof 2;20 ( 60 sin 2;14O).The maximum solarequation in the AlfonsineTables is 2;10, correspondingto an eccentricityof 2;16.The differencesin the resultantentriesarenot large enough to decide with confidencebetweenthese eccentricities.The Hebrew translationof the Latin versionof the AlfonsineTables with the introductionof John of Saxony was made by et)Text(deg/h)Comp.theAlfonsine(e 5;lO)(deg/h)TablesT - C(set)(1483) Comp.2591118200;30,2130,2230,2830,3730,5031, 4235, 535,26302823212233,4234,1234,4135, 735,29221-2-335,4916017018035,4836, 936,25211635,4635,5736, 12122436, III(Text-Comp.).0100-20(e 5;47)(deg/h)0;30,2130,2330,2830,3730,4831, 601020304050ColumnLunar2in column0;30,I(Text).4T-C:

HM7Solarand LunarVelocitiesin the AlfonsineTableA 36,192222235,5835,5835,5935,5936, 1Of interesthere isben Abraham de Nimes in Avignon in 1460.that the solarand lunar velocitiestablesin the Hebrew versionagree exactlywith the Latintext of 1483, but for the absenceof the printer'serrornoted above.Moreover,some notes addedby the Hebrew translatorsupport.theclaim that the solarandlunar velocitytablesin the 1483 Latin editionwere added byaIscholarafterhave consultedSteinschneider(in translation)theAlfonsineTablesMs. Munich,Hebrew,11895, 7811, and thein the d.126 copied in 1496 (cf.relevantpassagesare includedMS. MUNICH HEB. 126Folio"Thewerethis69btabletogetherwiththe eight

Bernard-38HM7R. Goldsteintablesthat followafterit do not belong to Alfonso,were added to fulfilla need and to be ableof ycontentsand ties(-Ratdoltin 1/60th of a day (R)in 1/60thof a day (R)(R)for the planets(not in R)Folio"End of the supplementary73b(tables),Foliopraiseto God."114b"The translatorsaid:here end the tablesof King Alfonsowithall the supplementarytablesthat the Christiansadded to make themcompleteso that the astronomerwillnot need any other tables.I also added some tablesfor obliqueascensionfor latitudes400, 410, 42' which in my view were necessaryfor one who wishesMay he whoto compute (?: lahqoq)accuratelyand completely.grantsperfectionto his creaturesbe blessedand exaltedforever,amen."CommentaryThe eight tablesmentionedon fol.69b presumablyrefertothe five that come afterit plus the threetablesof obliqueascensionmentionedon fol.114b that are found on folios103b108b.The table of lunar velocitiesin 1/60thof a day has a minimumvalue of 0;12,g0/d'and a maximum value of 0;14,25/d'(both inthe 1483 editionand the Hebrew text),where d' is 1/60thof aday.With an eccentricityof 5;10, I find the minimum lunarvelocityto be 0;12,8'/d'and the maximumto be 0;14,240/d'.In the tablefor hourlylunar velocitiesthe maximum was 0;36,25'/h,which correspondsto 0;14,34'/d';it followsthat thistable wasnot computed from it.With an eccentricityof 5;15, the minimumand maximum velocitiesare 0;12,7O/d'and 0;14,26'/d'.Thus itis hard to decide with confidencethe exact parameterthat was usedto constructthistable.AcknowledgmentsThis study was supportedresearchgrant.ProfessorsUniversity,and ProfessorA.paper and I am gratefulforby a NationalScience FoundationD. Pingreeand G. J. Toomer of BrownAaboe of Yale Universityread thistheirsuggestions.

SolarHM7and LunarVelocitiesinthe AlfonsineTables139NOTES1. For some examplesdiscussedrecently,see Toomer [196882 ff.];Neugebrauer[1962, 105 ff];as-Saleh[1970];Goldstein[1974, 108 ff].2.See, for example,Venice 1518 editionof AlfonsineTables(Liechtenstein).The originalform of the AlfonsineTables andthe transmissionof the textin Latin has been the subjectofmuch discussion:for a recentstudy,see North[1977].Themanuscriptshave not been systematicallyexamined,and I aminformedby ProfessorG. J. Toomer that Ms. Paris,BibliothequeNationale,Lat 7295 A, a copy of the AlfonsineTables,containsthree sets of velocitytables:I.ii.iii.The usual Battan?table at lo intervals(folio140r).A lunar velocitytable added into the originalmanuscript:the entriesare given only to minutesforargumentsat lo intervals.The minimum entryis 30'and the maximum is 36'A set of solarand lunar velocitiesat 6' intervalsto3 30 /h0;29,37,13O/hMaximum0;2,33,40 '/h0;36,58,54'/h3. TabulaeastronomicaeAlfontii,E. Ratdolt of Augsburg,1483.4. My calculationswere performed on a Hewlett-Packardcalculator(HP-97), based on modern formulas for the velocityin aneccentricmodel.This procedure produces very good agreementwith al-BattZni'stable (see Goldstein[1974, 1131).The basicformula isv,(a) 2’ ;acos c v m - v a'where v,(a)is the hourly lunar velocityas a functionof thelunar anomaly, CI; OM is the distance from the observer to themoon as it travelson an eccentriccirclewhere the radius ofthat circleis 60; Va is the mean hourly motion in anomaly; c isthe correctionangle correspondingto the anomaly, 1; and Vm isthe mean hourly motion in longitude.REFERENCESas-Saleh, J. A. 1970. Solar and lunar distances and apparentvelocitiesin the astronomicaltables of Habash al-Hasib.al-Abhath23,129-177B. R. 1974.The astronomicalGoldstein,Hamden, CT: Archon Books.tablesofLevibenGerson.

140BernardR. GoldsteinHM7Neugebauer,0. 1962.The astronomicaltables of al-Khw%i&.Copenhagen:Munksgaard.North,J. D. 1977.The AlfonsineTablesin England.In Prismata:FestschriftfEr Willy Hartner.pp. r,M. 1895.Die hebraeischen HandschriftenderK. Hof- und Staatsbibliothekin Muenchen. Munich.Toomer,G. J. 1968. A survey of the Toledan Tables. In Osiris15; pp. 5-174.

tables that follow after it do not belong to Alfonso, but they were added to fulfill a need and to be useful." Folios 69b-73b: contents 1. table of hourly solar and lunar velocities (-Ratdolt 1483) 2. table of lunar velocities in 1/60th a day (R) 3. table of solar velocities in 1/60th of a day (R) 4.