Ibn a1-Ha'im's Trepidation Mode1
Merce Comes l
Table of contents
I Introduction 293
J. Generaliries. 293 2. Aim 01 rhe paper, 297 3. Structure o/lhe papero 298 3.1 Edition ofrlle lexl. 298 3.2. Commentary. 298 3.3. Sectiolls 01 lile book edited Qnd commetlted. 299
[[ Commentary 300
J. Ibn al-Ha 'im '5 sources on rrepidarion and obliquity 01 (he ecliptic: lhe ~Uber de Motu" and Azarquiel's "Book Qn che Fixed Stars ", 300 2. On lile errors detected by Ibn al-Ha 'im in Ibn 01-
Kammad's "al-Kawr (ald al-Dawr" and "al-Muqtabas lO, 306 2. J. A jurther commentary on lhe errors thar Ibn al-Hii'j", altributes to Ibn aI-Kammad. 315
I This paper was written as pan of a research prograrnme. entitled Astronolllkol Theory and rabies in a[~Andalus and al-Maghn'b between Ihe twelfth and founeenrh centuries, funded by the "Dírección General de Investigación Científica y Técnica~ of the Spanísh "Ministerio de Educación y Ciencia~.
Suhayl 2 (2001) 292 M.Com<$
3. On rhe errors deleeted by Ibn al-Ha 'im in "0/-211 al·Muntakhab ". 318 4. On rhe construction ojcorreCl rabies for (!lis malion. 323 5. 011 rhe description 01 ¡he accession and recession and obliquity models. 329 6. On the paramelers o/the malion o/¡he Head o/Aries and rhe Pole 01 che &liptic. 337 7. On ¡he impossibi/ity ojconslTUeting on everlasring cable 10 detem'úle t:.A. 344 8. On how 10 detennine E using che corresponding lables. 346 9. On how lO detemzine dA using (he corresponding lab/es. 347 JO.OIl how LO determine the mblimum obliquity from on observed ob/iquity. 348 JJ.On how 10 determine the obliquity (E) corresponding lO a given momelll. 351 12.0n the knowledge o/ ·al·;qbdI al--al1lWClr. 352 13.0n the knowledge o/ "al-iqlxll a/-lhanr. 356 14.0n the knowledge o/ "al-;qbál al-llulnf" between two g;ven pos;lions ojlhe Head o/Aries. 359
III Conc1usions 360 l. General conclusions. 360 2. lbn al-Hd'im and AzarquieJ. 361 3. lbn aJ-Hd'im and lbn aJ-Kanmlád. 362 4. lbn aJ-Hl1'im and lhe anonynwus vers;on ojIbn IslJ4q al-TtiIlisr's "zIj". 363 5. lbn aJ-Hl1';m and lbn al-Raqql1m. 364
IV Appendix 366 j. Edilion 01 lhe seclions ollhe lext dealing with lrepidation 01 lhe equinoxes and obliquiry 01 lhe ecJiplic. 366
V References 399
~ 2(2001) 1M ai-HIJ'i11l's TrqJidalion Motkl 293
I Introduction
1. Generalities.
The aim of trepidation rn 1 A full account of Ihe masl recent bibliography on this subject is to be found in J. Samsó [1992:219·2251 and [1994:VIII.l-31]. J. Ragep (1996:267-298). R. Mercicr [1996:299 3471, and M. Comes [1996:349-3641. J See A. Tihon [1978:236-2371319]. See J. Ragep (1996:271·2951. StQyll (XX)l) 294 M. Comes To/edan Tables\ and the work oC AzarquielÓ and his followers, from 11th 7 eentury onwards . The maio difference is Ihat while in Ole Kittib lfaraklil al-shams and in al-zO al-SaJá 'il]. we find just purely geomelrical demonstrations without parameters, in the Western authors there is a complete set oC parameters as well as tables to compute the positían oC Ihe Head oC Aries and the obliquity of rhe ecliptic ror any desired moment. Tú this, il should be added that in (he trepidatían theories of (he Ancients, the solstitial po"inls are (he fixed reference. The same occurs in 8 9 the aforementioned works oC SiDan b. Thabit and al-Khazin • In Zarqallian and Andalusian models, the reference system is defined by (he equinoctial points. Andalusian models were probably reelaborations ofthese earlier Eastern models with the attribution ofparameters that accounted for the differences found by the observatians. In fact there is a paragraph in Azarquiel's introduction to his Book on che Fixed Scars where there seems to be an indirect reference to Thabit. Azarquiel lO states that. the previous authors did not pursue [he subject of the motion of the fixed stars further because reliable observations of earlier astronomers were not available to them. According to J. Ragep, presumably "one ofThabit's reasans far writing Is~aqll was to asceflain if he knew of any observatioll of the sun between Sec G.J. Toomer 11968: [18-1221. ~ Following J.M. Millás [1943-1950], I use lhe Spanish form Azarqlliel, deriving from Ihe Al1Ibic walad al-ZIIrqiya/. documente Sce chapler 1 of lhe following cornrncmary. , sce Sa Ms. Asrinagar. 314. We owc a photocopy of Ihe manuscripl lO the kindncss of Professors Ansari and King. This lJj (ll-!iof(1'i~1 is mcntioncd by al-Brn1nT in his Kitáb a/-Atllár al Oaqiyo 11 :3261 as comaining a good dcscriplion of an acccssion and reccssion model. 10 J.M. Millás [1943-1950:2781. 11 He is referring lO a scicmific cpisllc Ihal Thabit wrole tO Is~¡¡q b. J:lunayn. the well known physieian amI lranslalor from Greek inlo Arabie. kepl in Ibn Yünus's u/.ZiJ 01 ~/(jk¡II1[. Suh>.y1 2 (2()J1) Ibn al-Hd'jm 's Trepidatioll Modef 295 Plolemy and their own lime" in order lO lesl whether Irepidation could account for "lhe differences between the MumfQ~JaIl results and lhose of Plolemy lo a generalized phenomenon affecling aH the stars"12. Furlhermore, Thlibil ends the leHer saying Ihat he has not cOllununicaled his studies on that respecl to anyone "even if many have asked me for it, parlicularly because they are nol found on something precise, bu! I eSlablish now Ihe poinl to which Ihe queslion has evolved, from Plolemy lo Ihis time epoch which is ours"13. Azarquiel also crilicizes lhe efforts ofprevious Andalusian astronomers, especially Ibn al-SarnJ:I (d. 1035)14. Indeed he seems to feel a particular animosity towards Ibn al-SarnJ:l, because similar negalive judgemen!s are also found in an indirecl reference in his book on Ihe use of Ihe equalorium 'S . However, the facl {hat we find here a slatement affirming tha! lbn al~ Sarnl) had reached an underslanding of the motion ofthe fixed slars, albeit incomplete, raises Ihe possibilily that he was Ihe aUlhor ofthe model oftlle Liber de Motu, on which Azarquiel based his own models. Moreover, in l6 ~acid's TabaqtU , Ibn al-Sanll:J is described as a specialist in mathemalics and geometryl7, as well as in Ihe motion of the slars and lhe configuration of the celestial spheres. Another possibilily is ~licid himself for in his Tabaqtit lS he states rhat afier reading Ibn al-Ádami's treatise Qn trepidation, he was able lO understand trepidatioo mOlioo and to expound it in his book on !he motion 12 J. Ragep 11996:282-283J. See also R. Morelon [1994:131-132J. 13 R. Morelon [1994:132]. II 1.M. Mill IS See M. Comes [1990:225]. 16 Abo I'Qasim ~¡¡ 17 lbn al-Sa~'s skill as mathematidan and geometer, especially as far as his study on!he sections of (he cylinder IS concemed, has been confinned by R.Rashed's outs¡anding study [1996]. 11 ~¡¡ SIl1IyI 2 (2001) 296 M. Comes of the celestial boclies. This might explain (he presence in me Toledan Tabies, also attributed to ~acid and bis group, afthe same trepidation rabIes we find in several manuscripts of the Liber. Finally. othee possibilities are: lbn BarghUth, whose observations ofthe st3r Qalb al-Asad (44IH/I049-1050AD) are quoted by AzarquieJl9; or, 20 beuer still. aJ-Istijji , who according to lbn al-Ha'im wrote a Risa/al aJ iqbt1/ wa-l-idbar (c. 1050) and seems to have devised his own trepidarian model. There is no doubt, however, that Azarquiel remains one of the most Iikely candidates as author of (his book. OC cDurse, the raer (hat lbriihlm b. SiRan does not attribute any observations oc theoretical work on that subject to his grandfather is ror me conclusive evidence that the Liber was not due to Thabit b. Qurra. Be that as it may, il is beyond question thar the l Liber is intimately linked with the Toledan Tablei2 • Sorne other authors describe a combination of constant precession and variable trepidation, a combination well known in al-Andalus. In fact, the authors of the Alfonsine Tables used a procedure in which a constant precession is combined with a trepidation model, placing themselves in the Andalusian tradition, which is probably derived from al-Battanf's ll reinterpretation of Theon's statement in this respect . 2J J. Ragep, in his notable study of Na~Jr al-DIn al-Tl1sf's Tadhkira , shows that al-Tl1sI evokes this hybrid model in his book and adds that ShirazT and other commentators raised objections. Al~Tl1sT, as we shall see, presents other features which are basically Andalus¡'24. 19 J.M. Millás [1943-1950;309]. !ll Qti4i ~¡¡'¡d al-Andalusi in his '[abaqlil al-UlIIlIIII also states that Aba Marwan al-Istljji had sludied the subject of lrepidation. See H. Bífalwán's edition (~a'id al~Andalusr [3: 196-7]). JI On lhe aulhorship of lhe Liber de Motu. see the account in J. Samsó 11992:225/1994a:2- 31: J. Ragep [1993400403] and R. Mercier [1996:321-325J. II See J. Samsó [1987a:IXI/175-183J, and J. Ragep [1996:267·298]. !J J. Ragep [[993]. !< See lhe commenlary inlroduced after 11.4.(51. Sufu,yl 2 (2001) lb" al-HIJ 'jm's TrqJidatiOf/ Modd 297 We also have me writings ofsome Maghribi authors ofthe 15th and 17th centuries, which provide infonnalion aboul sorne aS1fonomical activity in al-Maghrib between the 12th and Ihe 14th centuries. [n the Iighl of the disagreernents belween computation and observation, (hey consider Ihat trepidation Iheories are no ¡onger applicable or at ¡east that another motion 25 should be added to accession and recession • Even Copernicus was to propose superimposed motions of this kind in Book 111 of De Revolutio"ibus26. 2. Aim of the papel". The aim of this paper is to study the trepidation models described in al-Zij al~KdJ1lj[ F 'I~Tcrdtrm by Abü Mu~ammad tAba al-J:laqq al-GatiqT al IshbTIl, known as Ibn al~Ha'im27. The only copy of Ihis zt) is MS: Arab 285 (Marsh 618), kept in the Bodleian Library. It consists of 170 pages and appears to have no tables, only the canons divided into seven books (maqdldt), after quite a long introduction. AI-2J) al-Kámil was composed ca. year 601 Hijra (1204~5), probably in al-Maghrib or al·Andalus, and was dedicated to lhe Almohad caliph Abü tAbd Allah Mu~ammad al-Na~ir (1199~1213). In this book, Ibn al-Ha'im seems to describe aH he knows of the trepidation and obliquilY of the eclipüc models developed in al-Andalus, especially Azarquiel's third model. He provides nol only the description of the model, the geometrical demonstrations, and the use of the tables, already found in Azarquiel's Book on t!le Fi:ced SlQrs, bUI also the spherical trigonometrical formulae involved. n See J. Samsó (1998) and M. Comes (1997]. 111; See in the respect W. Hartner [1984:277J. On Copemicus's trepidation and its relationship with \he Arabic tradition see Swerdlow and Neugebauer [1984:1:42-43:61, 72·74 and 127·1721 n On this author see: E.S. Kennedy [1956:I32(n.48»); J. Samsó (1992:320-325]; M. Abdulrahman [l996a:365·3811; E. Calvo 11998:51·111 and 11997]; and R. Puig 12lXXJ:71·72J. ~ 2(2001) 298 M. Comes 3. Structure oC the papero 3.1 Edition 01 the tuJ. An edition of the sections of lile (ex[ dealing with lhe trepidation of lile equinoxes and the obliquity of [he ecliptic appears as an appendix al me end ofthe paper. Each chapter has been divided into paragraphs, numbered in square brackets. The change of folio is noted in parentheses. As the manuscript is a unieum, there are no varian! readings. I have standardized the spelling of hamza and added the corresponding shadda and punctuation marks. Mistakes have been corrected and the manuscript readings indicated in the critical apparatus. The ¡astlines ofalllhe folios have beeo badly damaged by humidity. Sorne arthem are almost completely erased, while others have beeo repaired with patches which hide the last two Of three lines of the page completely. The guessed words appear in me edition belween angle brackels. Most of me reconslructed texl has becn taken from similar senlences in omer parts of me book; from me Book on the Fixed Stars by Azarquiel, 00 which lbn al· Ha'im based sorne seclions of his book; or from Ibo al·Raqqam's al-a) al· 2I Shdmil, which reproduces verbatim sorne parts of our reXI • In Ihe lasl case, I have used square brackets to distioguish memo The source is indicated in me critical apparatus. TItree dOls between angle brackels indicate a missing or unreadable passage and belween square brackets a passage dealing with unrelated subjects. 3.2. Commenlary. The conunentary of Ihe tex! is organized under headings dealing with lhe different subjects. Each conunenl is numbered as follows: Roman numeral of lhe maqiíla (lhe introduclion is numbered O); number of rhe biib; aod a square bracketed number of the paragraph corresponding to the annexed editioo. 11 Ibn aJ-Raqqam and Ibn al·H:I'im'$ ¡ex! are the only soutceS in Arabic. SlIlIyt Z lZllOl) /1)// ol-HI1'im 's TrepidoJiOfl Mockl 299 The figures of me manuscript have been reproduced and extra figures have been added: Below me reproduced figures, me number identifying the folio is specified inside parentheses. As the lettering in text and figures does nol always coincide, 1 have chosen the first ones for .the reworked figures and me commenlary; me figures ofme lexl are incomplete and the Jeuering is ofien misJeading. The mathematical symbols use .6,}.. lncrease of longitude due to the trepidation motion Pmv; Maximum accession or recession value ,. Angle of rotation around me equatorial epicycle o Declination of a point of the equatorial epicycle R Radius of the great circle r Radius of the epicyc1es E Obliquity of the ecliptic f llllll Minimum obliquity of the ecliptic Eman Mean obiquity of the ecliptic E... Maximum obliquity of the ecliptic j Angle of rotation around the polar epicycle 3.3. Sections ol/he book ediJed and commented. Muqaddima (fols. 3v-Sv and 8v-9v), mainly on the criticisms oftwo of Ibn al-Karnmad's books: al-Kawr "ald al-dawr and al-Muqtabas. Maqtlla U Báb l (fols. 23v-25v), on the accession and recession mode1. Btlb 2 (fols. 26r-27r), on Ihe paramelers of lhe different motions. Báb 3 (fols. 27r-28r), on the impossibility of constructing an everlasting table for bOlh motioos. Btlb 4 (fols. 28r~29v), on the errors found in Ibn al-Ka~d's trepidation modelo $ltIIyI 2 (2001) 300 M. Comes Maqdla III Bah 1 (fol. 35v), 00 how to determine e with the use afthe tables. Bah 2 (fols. 35v~36r), on how lO determine AA with the use of the rabies. Maqala VII Bah 1 (fo1s. SOr-8Ir), 00 how lO determine the minimum obliquity oC the ecliptic from an observed obliquity. Bab 2 (fols. 81r-81 v), 00 how to determine (he obliquity afthe ecliptic from the already known maximum and minimum obliquities. Bab 3 (fols. 82r-82v), 00 how lO determine the distance between the Head oC Aries and the spring equinoctial point, that is the first accession (al-ma~sas). Bah 4 (fo1s. 82v-83v), 00 how to determine the right ascension oC the degrees oC (he equatorial epicycle with respecl to the meridian, that is the second accession (al-ma"qal). 11 Commentary 1. lbn al-Ha'im's sOllrces 00 trepidation and obliquity of the ecliptic: ll U the ltLiber de Motu and Azarquiel's "Book 00 the Fixed Stars To understand lbn al-Ha'im's criticisms and models, it seemed to me worthwhile to start with a brief description of the most imponant Andalusian rnodels previous to Ibn al-Ha'irn. The tables found in the Toledan Table$29, as well as in sorne of the rnanuscripts of the Uber de motu octave spereYJ, are the first documentation we have of the Toledan aslronomers' work on this subjecL 1'1 In lhis regard, see GJ. Toomcr [1968:118-122]. JO On ¡he Liber de Motu, see the Latín tex! edited by J.M. Millás [1943-195O:496-509J: an English ¡ranSlalion and commelltary by O. Ncugebauer (l962-290-299J: as well as the foJlowing Sludies: B.R. GoldSlein [1964:232-2471; J. Dobrzycki [1%5:347]: J.D. North (l967:73-83J and 11976:155-158): and R. Mercier [1976: 197-200J and (1977:33-711· Suhayl 2 (2001) lbn al·116'im's Tnpidation Modef 301 The model of the liber, whose authorship is unknown although, as we have seen, it seems to have an Andalusian origin, is well known and has been Sludied in deplb by R. Mereier [19761 and [1997]. o , o ¡ • ,/ Fig. 1 In Ibis model (Fig. 1): QQ. Equator, P Pole of !he equator, E Intersection between the Equator and the mean Ecliptic (mean Equinox), B,BI'~' etc Moving Head of Aries. Pe Section of a meridian circle, the pole of which is E (Ee = m, so that the distance between e and the equator will correspond lO the mean obliquity of the ecliptic (,~.,). Section of the Mean Ecliptic. Ecliptic, which is the great circle detennined by point e and points B, Bh B2, ete. reaching lhe Equalor al points E, El' Et, etc. El' Ez, etc Equinoctial points, where the Ecliptic cuts the Equator. SInyl 2 (2OOl) 302 M. C<>m EIB h EzB2• etc Ares of the Ecliptic, which determine the ¡ncrease or decrease in longitude due lO the accession and recession motion (aA). The mOOel worlcs as follows: 00 the small equatorial epicycle BBlBz. whose centre is E, tbe Head of Aries (B, BI • Bz. etc.) moves with a unifonn motiao producing an angle with the equator called i. When point B rotales, it carries with it a moving ecliplic which always passes by point C. The intersections of this moving ecliptic with the equator are the equinoctial points (E, Eh E:z. elc.). The ¡ncrease or decrease in longitude proouced by the accession and recession movemeDl are the corresponding ares (BE) between the different positions of the Head of Aries (8, B\. Bz. etc.) and !he corresponding equinoxes (E, El. E:z. etc.). The objeclive is lO compute are BE (AA) for any vaJue of angle i. Obviously. there is another equatorial epicycle at 180" around which !he moving Head ofLibra rotates. The Toledan Tables and the Liber de Motu present three tables lO account for the different values of precession: a mean motion table allowing ealculation of!he value ofangle ; for a spCcific momenl; and two lables which allow calculation of !he ¡necease of longitude (dX) corresponding to a particular i by lwO differenl procedures. The firsl gives directly !he inerease of longilude as a function of ; while !he second calculales the declination ofpoint B, as a funclion of i and of the radius of the small epicycle, and should be used in combinalion wilh a declination table. Goldstein [1964] explains both procedures. Azarquiel, for his part, devised three models, but concluded !hat the correct Olle was the !hird. This third model (Fig. 2) is basically similar lO !hat of!he liber although, from a practical point of view, it introduces sorne important changes. First of all, the parameters are different. Second, in lhe model of the Liber there is a fixed point, placed in a meridian al9(f from lhe centre of the small epicycle carrying the moving Head of Aries, through which a1l the different ecliptics will pass. In contrast, in Azarquiel's third model the 90" are to be found between !he moving Heads of Aries and Libra and lhe also moving pole of lhe ecliptic. And finally, the mosl imponam difference is lhal in his model lbn nl-HI1·im·.f TupidnJion Mcxkl 303 Azarquiel considers lhe Illovements of accession and recession and obliquity of lhe ecliplic to be independent, lhough related. This involves lwO different kinematic models, one for obliquity and anolher for accession and recession (Figs. 3 and 9). , / Q Q , Fig.2 Azarquiel gives three tables: lhe firsl for the mean motion of Ihe Head of Aries (1); the second for the second accession (ZG in Fig. 9)31; and the third for the declinalion of the Head of Aries (6). In order to calculate the increase or decrease in longitude due to the accession and recession motion (..1.X), Azarquiel does nol use atable Iike the liber, but an indirecl method described in his 800k on rhe Fixed Srars and established by B. Goldstein32 • 11 Called by Azarquiel the Requalion of !he diameterR and by lbo al-Banm' thc "accession of the perpendicular of the diameter-. On !he second accessior. Stt B. Goldslein (1964:242-244) as wdl as chapters 12, 13 and 14 of lhis commentary. lZ Goldslein (19641. SIN)'! 2 (2001) 304 M. Comes EID'IJ e... ,-,1-'e f -'-* , c... f /Y , . I " I·~ ••••__ D O· -'0_.. B Emn Fig.3 Once the value of angle i has been deterrnined, using lhe first table, similar to lhat in me Toledoll Tab/es, a second table aUows calculation of the declination oC poiot (B, Fig. 9); men, (he following formula must be used: sin .6.'" = sin Ó I sin E This means that in arder to calculate the precession, the value of the obliquity must be determined beforehand. Azarquiel's model for the obliquity of the ecliptic (fig. 3) seems to have been influenced by a model proposed by Jbráhím b. Sinan, involving 3D ecliptic, controled by aD epicycle. The airo of Ibn Sinan's model was to aCCOUlll (or (he decrease in lhe value oflhe obliquity and me ¡ncrease in me s.t.yI 2 (2001) lbn al·HtJ 'im 's Trepidarion Model 305 lJ longirude of the apogee of lhe Sun . Azarquiel's model (Fig. 2) consists of a deferent circle, the centre of which is the pole of (he equator (Pq) with radius 23;43°, corresponding lo lhe mean obliquity (€mea,J; and a small epicycle with radius O; l(Y, (he centre of which is placed on the deferent circle, around which the pole of the ecliptic (PJ moves. Aceording to Goldstein, the cenlre ofthe epicyc1e moves along a limited arc of its deferent. Neither Azarquiel nor Ibn al-Ha'im saya single word aboul tllis specific motion, unless we consider that tlley are talking about it when saying that NThe orb advances 8° and then retreats tbe same amount and tbe potes oC (he ecliptic orb elevate and depress 8° alternatively "34. At the same time (Fig. 3), the pole of the ecliptic (O), rotates about its centre with a uniform motion (j). In addition, in (he model of Azarquiel and Ibn al-Ha'im, the pole of the ecliptic and the Head of Aries keep a constant distance of 90°. Since the radii are 23;43° and O; lOO, the obliquity will vary between 23;33° for Azarquiel's time and 23;53° for a moment before Ptolemy's time. So in this model (Fig. 3) the obliquity is easily determine lJ R. Mercier [1987: 1061 . .\ol See 1I.I.P]. )S On the procedure for calculating me obliqulty according to !he parameters of the Liber, see R. Mercier [1976:212 & 1977:391; See also 1. Samsó (1987b:367-377 & 1992:224 2251 and K.P. Moesgaard (1975:97]. SlN~1 2 (1001) 306 M. Comes Later astronomers in the Iberian península and the Maghrib. 8uch as Ibn al-Kanunad, Ibu Is~aq al-TunisT, Ibn al-Banna', Abü 'l-J:lasan al MarriUcushI, Ibn al-Raqqam, Abü 'I-J:lasan al-Qusan~ni, lbu cAzzuz a1 Qusan~¡n¡, Of the astronomers of King Peter the Ceremonious, followed Azarquiel's third model. However, aH afthem introduced a table, similar to that of the Toledan Tables and the Liber de Motu. giving the accession or recession value (dA) corresponding to any value of angle i. The use of this kind of table is eriticized by Ibn al-Ha'im. The model presented by Ibn al-Ha'im when dealing with lbu al KalTlITl3:d's errors is described in Báb 4 of Maqála II and analysed in chapter 2 of (his conunentary. Although the lines immediately before the description correspond to a damaged end ofpage and are iIIegible, there is no doubt that the model depicled by lbn al-Ha'im is Ibn al-Karnmad's model. The model followed by lbn al-Ha'im is depicted in Báb 1 of Maqála II and analysed in chapler 5 of this cornmentary. 2. On the errors detected by Ibn al-Ha'im in Ibn al-Kammad's nal_ Kawr tala al-dawrn and tlal-Muqtabas" Within the rhetorical introduction, which goes from fol. 2r lo fol. IOr, the author makes his first criticisms of Ibn al-Karnmad. In fols. 3v lo 4r the errors found in Ibn al~Karnmad's trepidation model are expounded. From fol 5r [O 8r, Ibn al~H¡¡'im describes up to 25 errors he detected in two books by lbn al-Karnmad, al-Kawr talO. al-dawr and al-Muqtabas, introduced by the formula wa-min dhiilika. The errors are ordered and numbered at the edge of (he page. Amongst them, Ihere is the error referring to Ibn al-Karnmad's model for the 36 trepidation motion . The model is compared by the author with what he calls Ihe "Toledan observations" (al-ar~iid al-fulayfuliyya) and the works .11> see in this respect E. Calvo [1997]; 1. Chabás & B.R. Goldstein [1994] and 119961; and J.L. Mancha [1998:1·111. ~yl 2 (2001) /bn al-Ha 'im 's Trepida/ion Motief 307 31 J8 on the subject by Abo Is~aq al-Zarqalla , Abo Marwan al-IstijjT and J9 Abo tAbd Allah b, Barghiith • As the subjecl is also deah with in Bab 4 of Maqala n, here 1 will cornment on the two chapters together. 0.[1] As. far as the introcluction is concerned, after a short foreword on the differences found between observations and calculations of the various )1 The llame of Ihis author is spelt al-ZarqiIla in the manuscripl, written in MaghribT script. Aceording to J. Samsó, mis seems lO be a c1assicised Eastem fonn, not documenled in Andalusian sources. This information is lO appear. under lhe entry al-Zar~liIT by J. Samsó, in a forthcoming volume of the Encycfopédie de /'Islam. However, al-Zarqala is also the spelJing in Ibn aJ-Baqqar's Kirdb af-AdwdrftTasyrr af·Anwár (Escorial, ms. 418. foI238r). )f Aba Marwa"n 'Abd Allah b. Khalaf al-lstii¡l, according to fbn al-Baqqar's KiUlb al-Adwár ft Tasyrr al-Anwdr (Escorial, ms. 418, f. 242r. See also Samsó, Bemni [1999]). Aba Marwan 'Abd Al1lih (or 'Ubayd Allah) b. Jalaf al-lstijT, according lO $:r'id's 'faJxu¡tll, although L. Cheikho read "al-Asluhi". Consequently. I have adopled ¡he spelling al-/slijjr in spite of me faet thal in the manuscript this llame appears as a/-/stijibr. J9 This name is found as al-FaqTh al-Qli¡p- Aba 'Abd Allah b. Barghat in fol. 4r and al-FaqTh Aba 'Abd AJJah b. Barghüth in 9v. The spelling also varies in Ihe different edilions or translalions of $:r'id's 'fabaqdt. l;Iaylit Ba 'Alwan in $:r'id al-Andalusi [3: 173-6) calls him Mu~ammad b. 'Umar b. Muhammad (b. 'Umar in some mss.) known as Ibn BarghiJt, reading accepted by Maíllo withoul any explanation ($li'id al·Andalusi [5:130], while according to R. Blachere's lranslation ($:r'id al-AndalusT (2: 135-51), based on Cheikho's edilion ($li'id al-Andalusi [1]), his llame is lbn Barghuth. The mosl recenl edilion of Tehran (l?a'id al-Andalusi [41) points again to Ibn Barghulh. placing Ba 'Alwlin's reading amongsl Ihe various errors that Ihe author of Tehran's edition found in Ba 'Alwan's edilion. According lO Suter [1986:n. 221J lhe name is Mu~ b. 'Umar b. MUJ:1ammad, AbO Subayl 2 ('lOO1) 30S M. Comes motioos. lbn al-Ha'im starts bis criticisms oC lbn al-Karnmid's theory of trepidation found in his two books, al-Kawr'ald al-dawr and a/-Muqtabas. particularly the first. Ibn al-Kanunid's errors loo people to criticise and reject the Toledan observalions after having accepted mem. A group oC astronomers, however. were interested in this book (al-Kawr) and they paid no attention lo i15 mistakes. Several coincidences suggest that me model attributed by Ibn al-Ha'im to lbn al-Kamm.ad was either Azarquiel's first model or Azarquiel's second model. First oC all, according to Chabás and Goldstein O.{2] lbn al-Ha'im goes on to say that he has seen a copy of the book (a/lKawr) with a note wriuen in one of the aforementioned astronomers own hand, in whieh he proves his ignorance by praising Ibn aJ-Karrunad's trepidation table with the argument lhat it is Azarquiel's table, although in faet il is ROl. lbn al-Ha'im marveles at the ignorance showed by Ihis mano O.{3] In faet, we find here a quotation from Azarquiel's Book on the Fixed Stars about (he impossibility of eonstrueting atable that gives 00 J. ChaMs & B. Goldstein 11994:241 . • 1 See M. Comes [1991). <1 M. Comes 11990:88-92). ~ 2{lOO1) Ibn al-Ha 'im's Trepidrltion Model 309 directly the value foc AA because of the changing obliquity. The quotation eorresponds word for word with the last paragraph of Azarquiel's 8111 43 chapter • Ibn al-Ha'im expands his crilicisms on the use of such atable in 0.[13J and 1l.3.[4]. 0.[4] Aeeording to Ibn al-Ha'im, an otherwise unlrnown astronomer, a contemporary of his, prepared a zlj named al-Z1.1 al-Muntakhab, which reproduced Abü Marwan al-IstijjI's mean motion tables and al-Battani's equation tables. In this zlj, Ibn al-Ha'im also finds the bulk of errors he criticized in Ibn al-Karnmad's al-Kawr ~alii al-dawr. However, as we will 44 see , the paramelers implied in al-Kawr ~altí iil-dawr and in the Muntakhab zlj, at least as far as the trepidation tables are concerned, seem lo be unrelated. This zlj appears also mentioned in 0.[11]. Unfortunately, we have no references lO the aUlhor of this zt} nor to the zt} itself. Following Ihis, there is an excursus, nOl edited here, on the errors found in the Muntakhab zlj related lo the molion of the sun. In il we find another quotation on the solar motion from Abü Marwan's Risalal al-Iqbal wa-l idbar, which has nol survived. Since Ihis book is lost, and given the exactitude oflhe previous quolation ofAzarquiel, I think thallhis quotalion deserves to be studied wilhin the framework ofthe Andalusian models for 4S the molion of the sun • 0.[5] The next step will be to discuss Ihe errors in Ihe trepidatioo molion: firslly, Ibn al-Kammad's errors in al-Kawr rala al-dawr; and secondly, the mistakes found in lhe Mumakllab zlj'46. 0.[6] Ibn al-Kammad's enors 00 Ihis subject, appearing in fols. Sno 8r, are divided ioto four aspeets (jillat). o J.M. Millás [1943-1950:3351. .. See lhe commenlary lO O.IIOJ . •s On this see E. Calvo [19981. <06 The errors in al-'ZJ.] al~MIII1/(/k}Ulb will he discussed in chapler 3 uf lhis COl11mcmary. ~yl 2 (2001) 310 M.Comes Firsl: Ibn al-Kammad's premises are two: tha! (he moliao of Ihe pole of (he ecliPlic slarts from one of ilS mean values. while me motioo of (he Head oC Aries staTl:S rrom [he cqualor. and thal lhe two mOlioos are equal. See Fig. 4 described in 11.4.[2]. , , , , , (O G G Fig.4 0.[7] Second: lbn al~Kammiddevised atable for the two motioos, which he considered valid for any time, and strucrured as an equalion table for argumems between 10 and 90°. According to Ibn al·H1i'im mis was nol correet, because ror mis range oC arguments oC the motion oC the pele oC the ecliptic around Ihe polar epicycle the paJe does nol achieve all of ils obliquities. To go rrom tbe maximum to the minimum obliquity. the pole needs DOI only to cover 180° bUI also lO meet the condilions explained in 0.[8]. WII,t 2 ami) l/m nl-Ha'jm's Trepjdt:Jtjon Model 311 0.[8] Third: The problem posed by lbn al-Ha'im in this rather obscure paragraph is that even if the table was syrnmetrical for 180°, the problem remains. The pole of the ecliptic should not be placed at I:ioew because in this case, from one to the other I:"",.n' the pole would not complete all its possible obliquities, from maximum to minimum. To achieve this, the pole should be placed at its minimum or maximum distance from the pole ofthe equator (I:,n;" or Emu)' but, in this case, the Head of Aries would be at the Northern or Southern limits of the equatorial epicycle and not at the intersection with the equator, where AA is 0° and the Head of Aries coincides with the vernal equinox, as proposed by lbn al-Kammad. 0.[9] Fourth: Finally, lbn al-Ha'im's last criticism is tIlat the values auributed to trepidation by Ibn al-Kammad do not coincide with the values observed, which are lhe starting point (a~f) used by the Toledan school to determine the parameters of the trepidation model. Table 1 Azarquiel lbn al-Ha'irn's u~ül Hipparchus Table I summarizes the values of the AA and the argument for the Head of Aries (i) for lhe times of Hipparchus and Ptolemy, as reported by Azarquiel in his Book 011 lhe Fixed Stars, and lhe values mentioned by lbn al-Ha'im. As we can see in this table, Ibn al-Ha'irn's values are very clase to those of Azarquiel's third rnodel. Entering wirh these values of i Ibn al-Kamrnad's equation table, Ibn al Ha'im obtains the following results: dif. with the u~111 Hipparchus 0;100 Ptolemy O; 15 0 Suh:I~1 1 (2001) 312 M. Comes Tha! is lO say, a difference of up lo fifreen minutes between Ihe slatedH values (o:fl) and the anes ohtained using Ibn al-Kammad's tables in his z¡J 8 al-Kawr rala al-dawr. Using Ihe tables in lhe MUQtabas nt , lobtained lhe same values. that is 9; 19° for Hipparchus and 6;57049 for Ptolerny. 11.4.[1] Bab 4 of lIIaqaLa 11 is also devaled lo Ibn al-Kammad's errors 011 this subjecl. In lhe firsl paragraph -hard to read because it coincides with lhe damaged end of lhe page- Ibn al-Ha'im jusI sta!es his two main criticisl1ls of Ibn a\-Kammad's trepidation model: lhe similarity of [he two mo¡jons, and lhe errors in lhe construclion of Ihe tableo 11.4. [2] The model is depicled in fol. 29v and reproduced in Fig. 4. Ibn al-Kanunad's model corresponds roughly to Azarquiel's third model and thal of the Liber de Mo/u~. The deseriprion in Ihe texl is as follows: AB are of the equator (90°) AG are of rhe eeliplic (90°) AHK equatorial epicycle O 01 In the lext, lhese values are quoted as "obscrved". " Biblioteca Nacional de Madrid, ms. n. 10023. '9 In lhe manuscripl lhe value is 7;57°. However. Ihis value should be correcled lO 6;57° becausc 7° has to end al lO' 18" and nol al lO' 20° as il appears in !he manuscript due lO a displacement in the column of lhe degrees. In fact displacement crrnrs are fairly common in lhis manuscript. This value is found corrected by samsó in [1997:109], as it appears in lhe MUWáfiq liJ. which offers Ihe same table. In the table reproduced by Chabás & Goldslein [1994:25], Ihe value is nO! corrected, although the correet value appears in \he variant readings taken from Ihe Bareelollo rabIes (Ms. Ripoll 21, fol. 137r). lO See chapler I of Ihis cornrnenlary. 11 Points T and E are not mentioned in lhe descriplion of lhe model, although \hey are used in lhe rest of Ihe tex!. see U.4.{4]. SUhayl 2 (2001) lbn al-Ha 'im's Trepidntion Model 313 N pole of the equator D pole of the ecliplie In this model, the figure of 90°, whieh has ereated many problems of inlerprelation from the very beginning, is to be taken on the equator and Ihe ecliptic, between the Head of Aries and Libra at the beginning of their motions and rhe colure. In both Azarquiel's third model and Ibn al Ha'im's, Ihe 90° are taken berween the moving Head of Aries and Ihe moving Pole of Ihe ecliptic. In the Liber de Motu they are considered to be between Ihe centre ofthe equatorial epicycle (mean first point ofAries) and a fixed point in thecolure, which corresponds with the intersection between 52 the moving and fixed ecliptics. This was Ihe assumplion made by North , 51 followed by Sams6 , and afterwards accepted by Mercier as the best one. Regiomonlanus' model, however, coincides wilh Ibnal-Ha'im'sdescription 54 of Ibn al~Kammad's model • 11.4.[3] We find here the repetition of Ibn al-Kammad's opinion. The Head of Aries and the pole of the ecliptic would cover equal ares in equal times. The Head of Aries s!arts from Ihe point corresponding to Ihe inlerseclion belween the ecliptic and the equator and the pole oflhe ecliptic slarts from i!s mean dislance from Ihe pole of lhe equator (E nIC3,J. 11.4.[4] This paragraph aims lO show once again Iha! atable for 90° would be enough for the motion ofthe Head of Aries, because Ihe variable used to compute Ihe lable is the declination of the Head ofAries (o), which 52 See 1. Nonh [1%7:71-831 and [1976:155-1581. In Ihe fjrsl papero Nonh presenls lhe conslraim. afterwards acceple SJ 5ee 1. Samsó 11992:222-2241. so See R. Mercier's account [1996;305-6] and 1. Samsó (1992:222]. SUlayl l (20011 314 M. Comes goes rrom a mínimum value ror i=O° to a maximum value for i=9005S . However, to compute atable for obliquities, we nce I1.4.[5] Theo, lhe Head of Aries will go back lo the place from which lhe mOlioo was initialed and reach poim K, while lhe pole of lhe ecliptic will have moved to point E, which corresponds to lhe minimum obliquily of lhe ecliptic (EmiJ. II.4.l6+7] To deal with tllis problem Ibn al-Ha'im suggesls a solution lhat he had already proposed in tbe introduction: lO place lhe beginning of the mOlioo of lhe Head of Aries at its maximum southern declinalion and, therefore, lhe beginning of the molioo of lhe pole of lhe ecliptic at its Emill, Le. lO compule atable eilher from E to T and from K to H or from T lo E and H lo K (Fig. 4). However, lhis would be correcl only if the two mOlioos were lhe same, which, according lO Ibn al-Ha:'im is nol the case, as he has already showo io n.1 lo 3. SI The lenninology is. apparently, eonfusing. The leXI says mal after having moved Ihrough an are of 90°, the Head of Aries will nave altained "all ils maximum declinations from !he equalor" (jam{ muyüfi+hi al-klllliyya can m¡faddil al-nahtlr). AI-nla)'l al-kullf is the obliquity of Ihe ecliptie bUI in this case 1 assume it lO mean Ihe declination of Ihe Head of Aries. There is also lhe possibility that lbn al-Ha'im was trying \O say that in faet, according lo Ihe relationship he wiJI show in IU, during Ihese 90° lhe ecliplic would have attained al! ilS values for lhe obliquity. Cenainly the ratio between Ihe revolulion period of the pole of the ecliptie and that of Ihe Head of Aries is nol 2/1 (1 ;54,4()PW) bUI Ihe difference as far as lhe table is coneemed will be negligible. Suhayl 2 (2001) IIHI aJ-HI2 'im's TrqtidatiOfl Modtl 315 2.1. A rurtber commeotary 00 tbe errors tbat Ibn al-Hli'im altribules lo Ibo al-Kammad The eriticisms we have just seen need a more extensive eommenlary. We know of a reference to the faet that Ibn al-Ha'im had deteeted various errors in lbn al-Karnmad's Irepidalion models. This reference is found in the Natd 'jj al-ajkiirft shar~ Rawt:!at al-az/uir, one of the commenlaries on the Rmw!at al-az/lar ft Ci/m waqt al-layl wa-(+nahdr (794/1391·92), me urjüZa on limekeeping wriuen in Fes by Abü Zayd cAbd al-R~min al Lakhmr al·Ja.diri (777-818/1375·1416)S6_ There are two anonyrnous copies 51 dated 1183/1770, although the annus praesens is 920/1515 , and another copy, undated but probably wriuen around the end of lhe 16th century, in which the name of the author appears as Abü Zayd cAbd al-Ra~man al Janatl al-NafawP'. Most of the data in it are expanded on in the Kant. al asrar wa-Nard'lj al-afkiir ft shar~l Rawtfar al-azhdr by Abü'l-cAbbas al Mawasl al-Fasl (d. 911 H liSOS). However, [he reference to Ibn al-Ha'im and Ibn al-Karnmad is not found in the lalter work. These commentaries conlain sorne interesting remarks which indieate Ihat Ihe trepidation Iheory was no longer appl icab]e~9. Unli] recently, we had no documenlalion Ihal confirmed {he error auribuled lO Ibn al-KalllITllid by Ibn al-Ha'im. However, in a manuscript in Ihe Cathedral of SegoviaOO there is a medieval Caslilian translation of a ehapter by lbn al-Karrunad dealing with lrepidalion, which gives a full ,. Ms. 80 in tht' Maktab31 al-Z1wlyya al-f:!amzawiyya (Ayt Ayache). fols. 203-220. This manuscript is partially described lO M. Mannfini 1I9631. n. 157. A full descriptlOfl of It IS tO he found In A. Alkuwaifi & M. Rius 119981. )J 00 ms. Cairo K 4311. see D. Kiog r1981];119861. See also ms. Loodon Brilish Library Or411. SI Maktaba I~amzawjy)'a SO. 228-334. " See J. Samst"l (19981 aod M. Comes [19971 . ... Segovla. BlbhOlcca de la Catedral ms 115 fols. 2ISvb·22Ovb. "Yucaf Benac01Tlfillibm sohre l;lrcunfereocia de moto". See J.L. Mancha 119981, where lhe edl!ed lext appears 10 pp. 8·10 "\dayl 2 Inlll 316 M.Comes explanation of the identity of me two tnOlions. lts title, Libro sobre ~ircuflferencia de mOfO sacado portiempo seculo, seems to be a translation of al-Kawr ca/ti al-dawr and/or al-Amad ralá al-abad. The idea, slated in the Cathedral of Segovia manuscript, is [he one we have seen crilicized by lbn al·Há'im. Tbe motiao of!.he pele of lhe ecliptic around ilS polar epicycle is equal to the motian afthe Head ofAries around its equatorial epicycle. Furthermore, the former will s13rt al its mean value (E_ = 23;43°), Ihat is when the pole of the ecliplic is either on M or O, between its nearest distance m and farthest distance (E) lO the pole of lhe equator. and the lauer will 5tart when the Head of Aries is al lhe equator (A) and (he declinarían is O°. Hence, accession will result from Ihe molioo ofthe Head of Aries in Dile half of (he epicycle, while recession would occur in the other half and, al the same lime, (he obliquity of the ecliplie will inerease as its pole rotates from (he midpoint ofone halfofits epicycle to the other midpoim, and will decrease in the other half. When laIking about the ehanging obliquity ofthe ecliptic and the mation of the equinox.es, in his Tadhkira, AI·TüsT introduces a difficult (O undersland paragraph, that and has been edited and translated by J. Ragep61 as follows: "Dne [sorne?] of them carne to be satisfied with one mover for both divergencies. This mover would cause the ecliplic orb to mDve in such a way tIlat every point on it moves about a smal1 circle. Accession would then result from the motion in one ofits halves, while recession would occur in the other half. [In ~ 1. Rigep [1993:D.4(5)]. Sliayl 2 (2001) Itm o/-Hd 'im"s Tr~pidDJiOfl ModLl 317 addition], tbere would occur a decrease io the obliquity during the rootioo from the midpoint of one of these halves to the midpoim of lhe other half, while there would be an increase during the motioo in the other half" , Hawever ifwe replace a single word, nuq{a (~) ~point" with qu{b (~) "pole", two words that are easily confused in Arabic, and of course reading the verb as masculine instead of ferneoioe, we have exactly the same idea found in Ibn al-Kamrnad and criticized by lbn al~H¡fim, It is theo c1ear that al-Tus! is describing Azarquiel's model, but in Ibn al Kammad's version. Al-TUS! alsa specifies that this is not his own apinion, but the opinion of other people. This is usual in the Tadhkira, where al Tus! orten introduces ideas and opinions thal he does nOI share in an attempt to present everything he koows on a particular subject. There are sorne other indications that al-Tus! knew about Azarquiel's trepidation and obliquity of the ecliptic models, and mat he had access to materials caming fioro Ibn al-K~d. In a recent communicatian in Tehran62 1 showed me existence oC rnany points of contaet between tbe Maragha astronorners and those oC the court of Alfonso X. amongst Ibero those related to Azarquiel. 1 will surrunarize them here: 1) J. Ragep63 notes that in Theon's theory of trepidatioo the solstitial points are c1aimed to move, while al-Tús! modifies the theory as reported by Theon in order to make il compatible with his general cosmological perspective. in which it is tbe vernal equinox that defines the reference system. To this we should add that, once again, al-Tüs! attributes this modification to "sorne of the practitioners oC this discipline". Unlike EasIem trepidation models, such as the models of Ibn Sinan or al 6t Khazin , in tbe Andalus! trepidation models derived traro Azarquiel's models, the vernal equinox is precisely the reference paint. 61 see M. Comes [1998]. u J. Ragep [1993:Commentary 11.4 [tll. .. See InlrodUClion 1. ~ 1(2001) 318 M.Clxn<> 2) In bis Tadhkirtf6. al-TUsi refers lO a "maximum and minimum" value Cor tbe obliquity lbat inunediately recalls Azarquiel's model, especially ifwe take ioto account lbat according lo al-Tusi, the maximum should be less than 24° and the miniDlUm nol less than 23;33° and Azarquiel's model gives a maximum of23;53° and a minirnum of23;33° precisely. Other authors defend a decreasing valuc. but Azarquiel is the first ane lO give a model with a maximum and minimum parameters. 3) J. Ragep also suggeslS the possibility lba! al-Tüsi was aware oC me work: oC Azarquiel when he menti~os an apparenl recognition in the 66 Tadhkira mal lhe solar apogee may have its own motion • In Caer lbn al ·Sha!ir (Damascus, d. 1375) maintained that according to his observations the solar apogee moved al a differenl rate from precession, aD apioion that coincides with Azarquiel, and is at variance with his own contemporaries who thought that the solar apogee moved at the same speed as precession. He also demonstrates his knowledge of the trepidatio'n theory and the corresponding models although he dismisses them because they do not 67 accord with the observations • 3. 00 the errors detected by Ibn al-Ha'im iu the "Muntak.hab zij" 0.[10] Ibu al-Ha'im will show now how to test the errors found in the Mwuakhab zIi, in which, according to him, Abú Marwan al-Istijji's mean motions were used sorne 150 years later by the aforementioned contemporary of Ibn al·Ha'im. Neither the Munrakhab 'lfj nor Abü Marwan's V) are extant. However, in his Risá/al ft a/-Tasyfrát wa-Ma!iiri~ a/-ShlftYtil" Abü Marwan suggests the use oC a cert'ain 'liju-nii Cour 'lCj"), which could be either a 'll) ofhis own, and the source for the abovementioned Mutltakhab 'llj, or the W J. Ragep 11993:11.41111. 66 J. Ragep 11993:Comrrtentary 11.6[1]]. • 7 See G. Satiba [1994:2351 and M. Cornes f19971 . .. Ms. Escorial n. 939, fol. 12r. See J. Sarnsó & H. Berrani 11999:296-2981. Solh>yI 2 (1001) lbu al-Hli '¡m 's Trepidllliou MoJel 319 Toledan Tables, However, as we will see in this chapler, Abü Marwan's dala, as quoted by lbn al-Ha'im, do not agree wirh lhe trepidalion parameters in the Toledan Tables. Ibn al-Ha'im uses the following procedure to determine thal the trepidation lables in the MUllfakhab z¡j are Ilot correcL First, he gives the data on which Abü Marwan based his trepidation model. Although he does not say so explicitly, I assume thal Ibn al-Hinm is referring here to lhe model described by Abü Marwan in his Risa/m al lqbiil wa-l-Idbiir, previously mentioned by Ibn al-Ha'imoo , and which is not extant either, Abü Marwan's values are different from Azarquiel's and Ibn al Kammad's and it seems mat he probably used his own model, which differs from all of Azarquiel's three models: neither ax nor i coincide wilh the lO data given by Azarquiel for his first and second models • The data are reproduced in Table 2. Table 2 Azarquiel11 Abü Marwan Hipparchus ax 9;28,30' 9;38,40' i 9' 22;32,12' 95 23;42° Ptolemy ax 6;42,45' 6;50,40' i 10' 19;01,30' 10' 19;23' Entering with i in the tables in !he Muntakhab z¡j, lbn al~Ha'im obtains a difference ofabout five minutes between Ahü Marwan's precession value and the ones derived fram lhe aforementioned z[j. The values and differences are the following: 69 See O,[4J. 70 On these pamneters see J. Samsó [1994:VllII9-27]. 11 In Azarquiel's third model. SlIIIyl 2 arol) 320 M. Comes Muntakhab Abli Marwan dif. Hipparchus 9;45,300 9;38,40' 0;6,50' Ptolemy 6;55,15° 6;50,40' 0;4,35° This information requires a few cornments: 1) The values obtained by Ibn al~H¡¡'im using the tables in (he Muntakhab zt] are clase lo the anes of (he second approach in Azarquiel's 2nd model: Azarquiel MUlIlakhab dif. Hipparchus 9;45° 9;45,30° 0;0,30° Ptolerny 6;57° 6;55,15' 0;1,45° As Ibn al-Ha'im stales that lhe MuntakJ¡ab zE] contained the bulk of 12 errors found in Ibu al-Kamrnad's uJ-Kawr cala al-Dawr , 1 have al50 recalculare 2) As regards the ¡ncrease of precession, determined by the estimated difference between the two positions of QallJ al-Asad for Hipparchus and Ptolemy's times, we have the following differences: Az. s.p & 1B 73 Az. 2874 Aba Marwan Mumakhab 2;48' 2;50,25' J! See Ihe paragraph 0.14]. JJ The difference is based on Ihe determinalion of the 10ngilude of Qalb al-Asud used by A7..arquiel as slaning poinl (s. p.) for !lis sludy of precession. 1B slands for Az.arquiel first model. second variant. See Samsó [1994:7, 26]. 1< 2B slands for A7.arquiel second model, second variant. See Samsó (1994:26) Suhayl 2 (2001) IblJ a/-f/li ';111 '5 TrepidariQIl Model 321 75 According lO J. Sams6 , lhe facl lhal for Aba Marwan the increase in lhe value of precession belween Hipparchus' and Ptolemy's limes was 2;48°, very c10se to the difference between Ihe corresponding longitudes of the star Qalb al-Asad (2;47°) delermined by Azarquiel, also seems to poim 10 a relalionship between lhe {wo authors. And, in fact, as we can see, ¡he difference between Hipparchus' precession and Ptolemy's in the 0 second approach ofthe second madel gives us exactly 2;48 . However, the difference derived from the values obtained using the tables in the 0 Muntak/¡ab zi] is 2;50,25 , 3) As Samsó & Berrani76 suggesled Ihal Abü Marwan could be one of the authors of the Toledan Tables, I have tried 10 emer ilS trepidation tables with i as argument. The result gives precession values which do not agree with the ones that, according [O Ibn al-Ha'im, proceed from Abü Marwan. ~>.. Toledan Tables Abii Marwan dif. Hipparchus 9;49,30' 9;38,40' 0;7,520 Ptolemy 6;58,33' 6;50,40' 0;10,500 4) As far as Pmu is concerned and to recompute the lable which supplies the increase or decrease of precession in the Toledan Tables, 1 have used the T7 following approximation : sin Pmax = sin ~>"I sin i From Abii Marwan's parameters, 1derived a Pmax of 10;32,53° (from Plolerny's values) and 10;32,32° (from Hipparchus'). These values are a150 different from Pmax = 10;45°, which appears in the Toledan Tables and the Liber de Motu. 1S See 1, Samsó [1994a:28J. 16 See 1, Sams6 & H. Berrani [1999:296-298]. 11 See 1. Sams6 {1994a:4]. SuhIIyl 2 (2001) 322 M.Comc:s From the vaJues Ibn al-Ha'im obtained using the rabies in the Munrakhab vl. I derived the following maximum ¡ncrease of precession: P_ ~ 1O;4O,0,37'(P) /1O;4O,1,54'(H). Surprisingly enough, 10;40° is the Pmu found in lbn al-Raqqam's al-Zll ll a/-Qawrm and in Abü '1·J:lasan al-Qusan!int . Using Ibn al-Raqqam's table, me values obtained and [he difference between [hese values and the values obtained by Ibn al-Ha'im using the Muntakhab Zl} are as follows: AA Ibn al-Raqqám .6.>" Muntakhab dif. Hipparchus 9;45,30' 9;45,30' 0;0,0' Ptolemy 6;55,56'" 6;55,15° 0;0,41 0 As a cooc1usian, we can say 1hat Abü Marwan's paramerers differ from the Olher known parameters, although his determination oC the ¡ncrease of precession between Ptolemy and Hipparchus coincides wirh Azarquiel's. Furthermore, his parameters are nol related to lhose in the Toledan Tab/es or Ihe liber de Motu, allhough this does nol rule OUI his intervention in the preparation of the Toledo" Tab/es. The parameters for Hipparchus' and Ptolemy's precession in Ibe M/lnIakhab zí], however, are very similar (O Ibose of Azarquiel's second approach in Ibe second model. Furtherrnore, it seems that the table for determining the increase or decrease of precession in Ibis zí] may have coincided wi(h Ihe table found in Ibn al-Raqqam's a/-ZIj al-Qawrm and in Abü 'I-J:lasan al-Qusan!,inT, which also differ from the tables found in the resl ofAzarquiel 's followers. 11 Sec M. Comes 11996:360]. ... It seelllS that there is a misreading here. either between 56" and 15" or previously in lhe yalue of i. lbn al-HI1'im's Trepida/ion Model 323 4. 00 the construction of correet tables for this motiooSO 0.[11] Ibn al-Ha'im computed [planetary] positions using al-Battani's equation tables and the value ofprecession calculated using the ~ables in the Muntakliab zij. However, the observations ofmeridian rransits (al-majdztit al-istiwa 'iyya) did not agree with the positions calculated. There are major differences between computation and observations in the periods of time berween {Wo meridian transits. 5uch differences cannot be attributed to observational errors. This is the kind of error one finds in modern books on the subject. Ibn al-Ha'im states that somebody, whom 1cannot identify, but who according to him was aman who tried to reach truth through both theory and practice, warned him of this. In fact, Ibn al-Ha'im states that he has checked this trepidation motion by using it in combination with al-Battanl's planetary equations. This appears to be a reference to the Muntakhab zij which, according to 0.[4], contained al-lstijjT's mean motion tables and al-BattanT's planetary equations. It is logical to imagine that al~lstijjT's mean motions were sidereal and that the computation gave sidereal longitudes to which precession should be added, using trepidation tables of sorne kind. The only information we have on these trepidation tables appears in 0.[10], where, as we have seen, Ibn al-Ha'im states that the values oftrepidation obtained with the tables of the Muntakhab zO did not agree with the ~al used by al·lstijjI. We may wonder whether the mean motion of the Head of Aries in rhe Muntakhab zO was eomputed with a mean motion table copied from al-lstijjI. On the other hand, in my cornmentary to 0.[10] l have also suggested that the table to calculate the equation oftrepidation in the Muntakhab zO might be the one we find in Ibn al-Raqqam's Qawfm z,j. 0.[12] Taking aH this into aceount, Ibn al-Ha'im decided to devise his own tables for the trepidation motion, after having studied what the followers of this motion had said before him, as well as the errors made. He creates atable caUed Jadwal al-juyub li-mayl Ra's al-Ifamal. The variables used are the following: 1Il On (he possibility that the KamilZij had tables. see M. Abdulrahman [l9%a]. SJhayl 1: (2001) 3~ M.C~ i Angle of rotation around the epicycle. 6 Declination of a point of the epicycle from the equalor. r Radius of the epicycle. t:.A Accession or recession. f Obliquity of the ecliptic. This table seems lO be similar lo the table found in Azarquiel, which gives ó. However, the wordjuyiUJ appears explicitly in the IWe and hence this table, which is not extam, would give ·sin Ó"II instead of "6". So, Ibn al-Hii'im would have atable giving directly Ihe sine of the declination (sin ó). Thís table is also found in lbn al-Raqqiim's al-Zij al Sliámif'l, composed lo provide tables for Ibn al-Ha'irn's book, as state O.{ 13] Once he has delermined sin Ó, he devises a table lO calculate ah direclly, but only for!he accession motion, which QCCurs in the northem side orthe equator and between ()i and 6". lbn al-Ha'im explicitly states that me table can be used for Ihe seeond halfof the equatorial epicycle, that is, ror recession, because in Ihat case the errors would amount only to seconds of arco However, as this table will not be useful ror a second revolution, he recornmends devising a new table using Ihe sines orthe new obliquities. The eXplanalion given by lbn al-Ha'im, based on {he relationship belween Ihe revolution periods of the two motions, the motion of the Pole 1I This is oonfimK:d by VII.3.[ 11. u see E.S. Kennedy [1997:561. u M. Comes 11996:349-364). SublI)'l 2 (2001) 100 al·HlJ'i",'s T1?pidarion Motkl 325 and mat of me Head oC Aries, is found in 11.3.[1-3]". This passage is, then, only apparently contradietory: lhe motioo of the Head oC Aries through an are of 180° willlake a long period of lime (e_ 2000 years) during which there will be importanl ehanges in the obliquity of lhe ecliptie. Tbe onJy explanation is that Ibn al-Ha'im's equation lable takes into aecount the value of E for eaeh mean position of the Head of Aries on lhe equarorial epicyele and, hence, usiog a different E for eaeh value of i and applying an expression oC the type: sin aX = r sin i I sin E In fact, he will establish below (11.3.[1-3]) lhat while lhe Head of Aries moves through aD are of 180°, the pole will have moved through 180° x 1;54,40°, lhat is 344°. This implies that lhe values for argumems between 180° and 360° will not be syrnmetriealto those between 0° and 180°. The error will be small, but it will inerease eonsiderably when the Head of Aries has eompleted a revolution, while the pole of lhe ecliptie will have moved 688°: equivalent to lwo revolutions minus 32° (see below 1I.[3]. 0.[14] Here there is a rhelorical paragraph, ool edited, in which the author acknowledges the indebtedness of his work on this motiOD to Azarquiel's mOOels. He outlines his exhaustive investigations and praises the skill of Azarquiel and the excellence of bis mOOels and books. lbn al-Ha'im talks here about al-lJarakaJ al-mustadraka talá al-qudanuJ'. The term al-mustadraka, whieh also appears when dealing with the lunar modelu, seems to have been used in al-Andalus to refer to the objections made to different assertiODS of the ancients, mainly Ptolemy. In fael, Saliba16 has followed the use of this terro in al-Andalus. He mentioos a lost lreatise entitled Kitdb al-lslidrtik talli Ba.{lamiylls, by an unknown eontemporary oC Azarquiel. This book seems lo be devoled to expounding .. See ctLapter 7 of this commenlary. ., Af-tJytJda al-lTWSladraka "a/d al qudaml1'. See R. Puig (200:1:76,911 . • For me implication of lhis tenn, see G. SaJiba [1996:83-86 &. 1999:3-25). SlmyIl{200l) 326 M. """'" the various problems thal Ptolemy's astronomica! theories posed. Ibn al-Ha'im then introduces the observations of different astronomers ofthe position of Qalb al-Asad, the sta! most used roc this purpose, due to its proximity 10 the ecliptic. Table 3 Positiom oC QaJb al-Asad Ibn al-Ha'im Azarquiell Azarquiel2 Hipparchus 31 29;50° 31 29;39°11 3' 29;50° Ptolerny <41 >2;300 41 2;26°88 41 2;35°89 al-Batwij" 41 14;00° 41 13;58°90 _._------Ibn Barghülh 41 16;20091 ------41 16;20°92 Table 3 shows the values found in this paragraph and Azarquiel 's values fram his Book on the Fixed Stars. In this table, AzarquieIJ corresponds lO Azarquiel's caJculations based, according toAzarquieJ himself, on Thabít b. Qurra's ·observed"9J values, n Probably Ibis value should real:! 3' 29;49°. and lhe error comes from taking 3;300 inslead of 3;400 as thc basis (or calculation for (he solar apogcc posilion related 10 me posi(ion of QaJb al-Asad. bc;(ween Hipparchus' and Thabil's epochs. In (aet 3' 29;50° is a well knownvalue. .. 1ñe value in the Almagnt 2;30°. see Toomer (1984:367] and Kunítzseh [1986:94 95;266-267]. 19 This posilion is approximate and comes from deducling fom Azarquicl's posilion (c. 136;35) a difference eslablished between Azarquíel and Ptolemy of 14°. 90 The value in al-BauitnT's slar lable is 14°. See Nallino [1899:2581. 91 160 in words and 16;20° in abjad. 91 lbn Yunus (1032) detennined a value of 4' 16;19°. n If Azarquiel (ook Ihis value from Thabit's Trua on the Solar Yt'ar, we are dealing here wilh Ma'muni's observalions, as stated by Morelon [1994:132-1331 and Samsó (1994:Vm,8j. However lhe source is not explicitly stated in Azarqulel"s 8ook. on the SW)1 2 r.ml) Ibn al-lid '/In 's Tr~pidoliOll Mod~1 327 9l and laking into accounl the posilion of Ihe solar apogee . Azarquiel2 corresponds to the supposed observations of Ihe differenl aslronomers as iS qUOled by Azarquiel • Table 4 shows Ihe inerease in lime between the various attested OT calculated observations. Table 4 .4. oC time between the various atlested positions oC Qalb al-Asad .6.t ace. Ibn al-Ha'im .ó.1 ace. Azarquiel Hipparchus ------Ptolemy 287 from Hipparchus 285 id. al-Baltan! 741 96 from Ptolemy 744 id. Ibn Barghüth 171 from al-Battan! 167 id. In Ihis table, the increase in lime between Ihe supposed observations quoted by Azarquiel in his Book on lile Fixed Srars is not explicilly staled 97 and has been deduced by researchers • According to Azarquiel, the dates ofthe observation were the following: Hipparchus 150 Be; Ptolemy 137 AD; al-Battan! 883 AD; Ibn Barghüth 1049-1050 AD. However. Azarquiel is confusing. He gives the data in Arabie years hut Ihe inerease in time between obsérvalions in Julian years, Fiud Suu:r, and Azarquiel nol ooly adds a paragraph praising ThIbics skill and mcthod as an OOserver bul also gives a value (4' 13; 13°) that is different from lile value fauOO in Th1bit's Traa (4' 13;2°) and from that of J:labash and al-Fargtt:inT (4' 13;15°) (Girke [1988J). AlIlhis suggeslS that Azarquiel may nave bren using a different source. 901 J.M. MiIJ~ (1943-1950:297-8]. 1" J.M. MiIIis [1943-195O:305,309-314J. 96 In !he lext lile value is 941, which most probably oorresponds lO a confusion between 700 aOO 900, which is very usual in Arabic scripl. f1 1 nave used J. Samsó (1992:236J. R. Mercier 11996:328) caleulates !he following observation limes: Hipparchus 145BC; Ptolemy 140AD and al-BananJ 884AD, which gives intervals of 285 yean: and 744 yean between" me aforementioned asuonomers. ~ 2 (2lXl1) 328 M. Comes although this is not always stated in the texto To devise the third model in the Book on "che Fixed Stars9&, Azarquiel used lhe positions ofQalb al~Asad fOf different times, deduced froro Thabit 99 b. Qurra's values , taking into aceDunt the motion of me solar apogee, as was studied in chaptee ane of his lost solar treatise. According to him, these deduced values are more reliable that the observed anes, because he hold TIüibit to be more reliable than aoyone cisc. As we can see, lbn al-Ha'irn's values are exactly the values attributed by Azarquiel lO thedifferent astronomers, except fOf al-BattanT's, which is not in Azarquiel2 bul i~ a rounding offofAzarquielJ. The small differences in (he intervals of time probably correspond to lhe data from which the calculations had been deduced. At the end ofchapter 5, Azarquiel determines that the star Qalb al-Asad was at Leo 9;8° at the moment when .6.>" = O°. This is the value found in all the following Andalusi and Maghribi star tables entitled jadwal al kawtikib min al-mabda' al-dhiítr and calculated for thal moment. The difference between this position (Leo 9;8°) and the position of Qalb al Asad determioed by Ptolemy (Leo 2;30°) will give us the iocrease of6;38° 1oo 00 Ptolemy's star positions found in all these tables . 0.[15] The motion affecling this star is what is called molion of accession. The positions of the star indicate that the orb was in accession (muqbila) at Ihese times. Finally, Ihere is a long excursus, not edited in the appendix, on Ihe difficuIties Ihe other astronomers had found in underslanding Ihis motion correctly, which ends wilh me explanations of the differenl reasons why the author entitled his book al-Zij al-Kámil. 93 J.M. MiJlás [1943.-1950:295-300). 99 00 lhis see J. samsó [1994:7-8]. 100 GoldSlein & Chabás disagree wilh this; Ihey mainlaio Ihal Ihese lables are ca1culaled for lhe beginning of Hijra and are nol relate Suh.~1 2 (2(XI1) lbl/ af-Hti '¡m 's Trepida/iol/ Model 329 s. 00 tbe descriptioo of the accessioo aod recessioo aod obliquity models The first báb of the second maqala deals with the description of lhe model for lhe morion oC the Heads of Aries and Libra in their equatorial epicycles as well as that oC the Pole of the ecliptic in its polar epicycle. ll.l.[l] The description begins wilh an introduction in which it is slated that Ihe group of Toledan astronomers (al-jamlfa al-!ulay!uliyya) reached the conclusion - based on the information available to them about anciem observations - that lhe difference between tropical year and sidereal year (calculaled using a zt) based on the Indian astronomical tradition) corresponds to the precession ofthe stars. There follows an ancien( version of the lheory of trepidalion, aUributed to lhe Babylonians as in al-BírünI's lol Taflll-'n . 102 In the imroduction lO his Book on lhe Fixed Srars , Azarquiel auributes Ihis lheory to Hermes and his followers, and differentiales it from Theon's theory, which he takes as a combination of precession and trepidation, as did al-Banan! in his al-ZJl al-$ábi '103. However, in chapter 104 6 , Azarquiel attributes the theory to the "Hatelesmat" and says that this was stated by Hermes in a book called Book on lhe Longirude. In chapter 4105 he slates that the opinion of the authors of the "Hind"l06 was closer lO the truth lhan the opinion maintained by Hermes and the authors of the "Halelesmat". The last two lines of the page are badly damaged and impossible to read. The parts oí lhe missing sentence that can be read are lhe beginning; "Tlle 1\11 On this version see J. Ragep [1993:397-398. 4041 11)2 J.M. Millas [1943-1950:277]. IlIJ See J. Ragep [1996:271-272]. 104 J.M. Millás [1943-1950:320]. j03 J.M. Millás [1943-1950:3041. 106 see Pingree [1972]. SlNyl 2 (2001) 330 M. Comes 0 orb advances 8 ... "; and the end "... the ecliptic orb elevates and depresses approximately the same quantity in this lime". The whole sentence appears in Azarquiel's Book on the Fixed Stars (p. 277) as follows: "The orb advances 80 and tbeo goes backwards tbe same amount and the potes uf lhe ecliptic orb elevate and depress 8 0 allernatively"I07. It is worth stating hefe that Ibn al-Ha'im is using the verbs inafaca wa inkhafatf.a, also used by al-Hashimi (c.890) in bis KiUib ftcIlal al-ZJ.ldt. This confirms the apioion of E.S. Kennedy and D. Pingree that al-Hashimi is referring to trepidation, although, as J. Ragep states. he caUDal be referring to lhe model of lhe Liber de Motu 'fY3 • Of course, al-Hashimi, Iike Azarquiel and lbn al-Ha'im, is talking about a simpler and older moclel in which Ihe 8 0 apply not only for the trepidation motion but also for the lO9 motion of LIle pole of the ecliptic • 11.1.[2] The group of Toledan aslronomers, through the differences found and the observations made, conjectured that the position of the ecliplic related to the equator was changing and that lhe beginning of the signs (mabda' al-burüj, i.e, Ihe Head ofAries) moved around the equator, sometimes north of it and sometimes soulh of il, wilh equal motions in equaltimes. This, according to the author, explains the changes observed in the velocity of Ihe precession, which affecls the fixed stars and the different lenglhs of the tropical year, which in turo produce time differences in lhe passage ofthe Sun through the equinoxes. After working in {his direction together and reaching complete agreemenr, lhe group of Toledan astronomers devised three different models, although aH of them 107 AJso on pages 280 ("lhe fixed stars advance 8° and ¡hen relreat the same quantity ... and the same quantity corresponds tO Ihe elevation and depression of the pales of lhe ecliplic orb~) and 321 ("The poles of lhe ecliptic orb move up and down 8° and the beginning of Aries advances 8° and retreat 8°"). Maybe this couJd be related lO lhe fact lhat ahhough neilher Ibn Sinán nor aJ-Khazin give any parameter for their modeJs, a 4° radius for the palh of the ecliptic paJe appears quoled by lhe commentators of aJ-Tü~rs Ttrdhkira, SCC J. Ragep 11993:4041. 101 Ragep [1996:2791. 109 AJ-IUshimi [l:f.97r and p.2251. Suhayl 2 (2001) Ibll al-Ha '¡m's Trepidatio/1 Model 331 agree that only one offers a perfeet tit for the conditions provided by the observations. 1lO In the introduction to his Book on file Fixed Sfars , Azarquiel confirms Ibn al-Ha'im's reference to the group of Toledan astronomers lll working together on this subjeet. In chapter 6 oC the same book , pe also affirms that the third hypothesis offers allthe conditions required for (he motion stated aboye. IJ.I.[31 Therefore, (he description ofthe mo<1el devised and accepted by the group ofToledan astronomers starts with lhe celestial sphere as it was sorne 50 years befare Ihe Hijra, coinciding, according to lbn al-Ha'im, with the birth of Ihe Prophet. At this time, lhe Head of Aries was on Ihe equator and tropical and sidereallongitudes were equal; so Ihe IwO schools, ll2 thal of al-Hind and al-Mu"UaJ,zan, coincide . ll3 Ibn al-Ha'im is Ihus correcting Azarquiel , aceording lO whom the dale of the birth of the Prophet, in which .6.>" = D°, would be sorne 40 1l4 years before the Hijra, a date also maintained by al-Marralcushf • Here begins the description of the rnodel properly speaking. The radius of the equatorial epicycle will be approximately 4;8°, a rounding off of Azarquiel's third mo<1el radius (4;7,58° 113), which appears in the tables of the Book on lile Fixed Stars Cor the second accession and for the declination of the Head of Aries (Ó)116. In fact, it is the related angle 110 1.M. Mil1&s 11943-1950:278]. 111 1.M. Millás [1943-1950:321]. 112 This paragraph is a summary of Azarquiel's beginning of section 2. see Millás [1943 1950:338]. IIJ 1.M. Millás LI943.1950:338]. 11. On the different opinions about the presumed date of the binh of the Prophet, in which .0.), = 0°, er. J. Samsó [1997: 108 -109] and M. Comes LI997]. According to the Toledan Tables.o.), = 0° for year 604. see Mercier [1996:306-307]. U$ In Ibn al-Raqqim's al-'lJ] al-SMmil the value for the radius is 4;7,57°. 116 J.M. Mill&s (1943-1950:336]. 5uNyl 2 (1001) 332 M.Comes 0Ill arcsin oC D;4,19,26 , a value a150 found in the corpus oC 800k on the lll Fixed Stars , attributed lo Azarquiel by Ibn al-Ha'im in [1.2.[1] (as 4;19.26.8011~, and used in VIl.4.[l] (0;4,19,26 as r/6fJ). This parameter is similar lO the ane used in the liber de MOlU (4; 18,43°). The mode!, as ir was sorne 50 lunar years befare tbe Hijra, is depicted in fol. 26r and Fig. 5. 11 consists of211: Fig. S ART Ecliptic. AHT Equator. O Centre oC the Universe. 111 sin 4;7,58° '= 0;4,19.26,42°, r = 4;19,26° for R = 60 in lhe Hebrew tex! edited by MiIIlls. 111 According lO R. Mcrcier 11996:328], this is lhe value as quOted in ¡ex!. 1.M. Millás 11943-1950:3181. stales4;19.26ll on lhe basis of R ... 60. This is atoo lhe "alue found in lhe Hebrew manuscript. 11. Aresin 0:4,19,26.8° = 4;7,57,27°. 1/10 For lhe 111lnscription of ¡he Arabic letters I have use SlmyI 2 (lOOl) 11m al-HIl';", '$ TrepidmiOfl MOtkI 333 A Poiot al which rhe Head of Aries and the equ3tor coincide. E A poiot at c. 4;08° from A, centre of the Aries equatorial epicycle. AE Radius of tbe equatorial epicycle. ABG4> Aries equarorial epicycle. TKLM Libra equatorial epicycle. GL& TA Straight Iines crossing al the centre of the Universe. N Pole of the equator. HRNS Solstitial colure. HR Angle corresponding 10 the obliquity of lhe ecliptic al thar moment. SFQ Polar epicyc1e called in the [ext ~Circle of lhe differences of the obliquity" l2l_ From this initial position, at which me Head ofAries coincides with the equator and me Pole of me ecliptic is at its medium distance from the pote ofthe equator (E-J in direction to the minirnum distanee (E->, the mation in \he equatorial epicycle will take place around the equator: flf'St, on me North side, eastwards (with regard to \he equinox), and then, on the South m lt seems lhat a number of lioes are missing heTe. because lhere is no mention of me polar epicycle. nor of me value of ilS radius. It looks like a -saut du mtme l mbne-. A StlggeslCd 101 appears inside angle brackets in me edilion of me Arabic lOt. ID The letIering and values appearing in me following ¡=.ragraphs of me lexl bu! missing in this sea.ion are shown inskle angle brackelS. S\Nyl 2 (2001) 334 M. Comes side, westwards (with regard to the equinox)I23, The East is identified in the text with the place fram where the Qabl1L wind blows, and the West l24 from where the Dabür wind blowS . Obviously, he uses the names of the winds in connection with the names oC the motions aJ-iqbal and al idlJiír. The description of the moliao coincides with Azarquiel'sl2S. A similar reference can a150 be faund in maqala 5 bah 1, were Ibn al Ha'im states mal when {he sphere is mut1bir, the equinox goes forward towards the East, and when (he sphere is muqbil, the equinox goes l26 backward towards the West • 11.1.[4] Let us now consider {hat the two equatorial epicycles move carrying the Head of Aries and Libra respectively. The Head of Aries in are AEcJl is 00 the northern side oC the equator and to the east of rhe equinoctial point, and the Head of Libra in TZK is on the southem side of the equator and to the west of (he equinoctial point. The pole ofthe ecliptic will also have moved eastwards in the are se. Then the different positions ehange, and the situation will be the foUowing (Fig. 6): E Head of Aries Z Head of Libra C Pole oC the ecliptie at thar moment NC Obliquity of the ecliptie at that moment, corresponding to f min HNC Solstitial colure e Spring equinox l1J This interpretation is hased on what the author says below in n.I.[4) and 11.1.[7]. 110 On the Cardinal Winds, see M. Forcada [1994J and D.A. King [1989]. I~ Sce 1.M. MiIlás [1943-195O:289-294J. 1101 See E. Calvo, ll998:[29]66}. The use of "accession" (muqbil) and "recession" (mudbir) both for Ihe equino); and !he Head of Aries is also found in olher sources related 10 Azarquiel: for instance, in the Kirt1b al-Adwdr/fTasyrr al-AnWl:1r by Ibn al Baqqár. Ms. Escorial, 418. fol. 237. A paragraph lhat seems to be Ibn al-Baqq~r's source is found in Ibn Istmq's ZJj. Ms. Hyderabad Andra Pradesh State Library 298. fol. 92. Suhayl 2 (2001) lbn al-H4'im's Trepidation Model 335 X Autumnal equinox1t7 &121 Summer solstice at that time, Fig.6 n.1.[5] As the motion continues, the positions change. The Head of Aries moves in (he circle AE~ up to lhe point m At the equinoxes. the points e and 1, on lhe ecliptic, and U and X, on me equalor, mentioned here for the firsl time, coincide. 121 & is used here 10 express !.he combinalion l4m-alij not describe SuIlayl 2 (2001) 336 M. Comes v Spring equinox W Autumnal equinox J Summer solstice ,.• ·t·_._------._--_ Fig.7 n.I.[6] The motion continues in the next q'uarter of the equatorial epicycles (4(; and KL), and the Heads oC Aries and Libra again reach the equator at points G and L. The al-Hind and aJ-MumtalJan schools coincide again al a moment in which mere is no accession nor recession, while the pole comes near its first E_. n.1.[7] 1be motion continues in the same way in both halves oC the equatorial epicycles (GBA and LMT) as the pole rises to its Emin . At that moment, the Head ofAries is in the southem halC oC the equatorial epicycle and to the west oC the equinox, while the Head oC Libra is in the northem halfand to the east of the equinox. Exactly the opposite of me situation at the beginning of the motion. n.1.[8] The motion in the two epicycles will be completed once the Heads oC Aries and Libra have again reached points A and T. sw.yI 2 (2001) /bn al-Ha'im 's Trepj(JaJjoll Model 337 The Heads ofAries and Libra, hence, will once again be in the position in which they were sorne 50 years befare Hijra when the trepidation value was O°. According, then, to Ibn al-Ha'im, the molion of the Heads of Aries and Libra takes the solstitial colure with it and this implies that there should be a connection between the two motions which justify trepidation and the variation of the obliquiry of the ecliptic, for the solstitial colure passes by the different poJes of the ecliptic moving around the polar epicyeles. However, if these two motions have to be connected, both cannot be circular, unless the centre of the polar epicyele moves along a limite 11. l.[9] The molions of the equatorial and polar pairs of epicycles each produce a pair of equal and opposed cones of a cylinder, whose bases are the cireles created by these molions, connecte 6. 00 the parameters oC the motioo oC the Head oC Aries and the Pole oC the Ediptic 8ab 2 ofmaqala 11 is entitled "On the quantity of the molioos of the Head of Aries, Ihe Pole and the Centre,in lheir circles and the times of their returo to the ioitial points". We will exclude rhe references to lhe centre of the solar eccentric, whose values, with the exception of rhe parameter of its motion for Julian years (0;6,27,42,33), are confirmed in Maqala I1I, báb 3 edited by E. Calvo in her paper on the model of the Sun in Ibn al H¡,¡'im's Z,-yI3lJ, aHhough she does not edit or comment 00 lhe references to the Sun found in the present chapler. 11.2. [1] There is no general agreemenl among lhe astronomers who established the parameters and periods of revolution. Azarquiel fixed the 119 J.M. Millás [1943-1950:289]. 1)0 See E. Calvo [1998:56-58/89-90]. 338 M. Comes radius of the equatorial epicyc1e at 4;19,26,8°. This parameter ¡s, according to Ibn al-Ha'im, Ihe mast accurate of (hose established by Azarquiel. The value, qUOled by Ibn al-Hii'im, coincides with the radius stated by Azarquiel himself in his Book 011 (he Fixed Stars (4; 19,26°), and 13l is even more precise • As foc lhe res! of Azarquiel's data Ihat have reached him, Ibn al-Ha'im considers tha! they show a variety of imperfections. Furthermore, Ibn al-Ha'jm states Ihe following: "When weexamined the queslion in Ihe pages (SU~IUf) of him (Azarquiel) Ihal were available to us related to this question [...] we found that they disagreed with the foundations (u~al) which were the basis of oue knowledge. This eompels us to review lhe whole question in arder to eorreet him, especially in relation lO the motion ofthe pole. The results caleulated by him coneerning the values of the obliquity for the times of the observations are differeot from those acrually observed by him and by others. The differenee is oot small. Therefore, we cannot build our resuits on this basis, not eveo in ao approximate way, for ir produces important ehanges in the distance between the Head of Aries and the equinoctial point". This paragraph seems to suggest Ihatlbn al-Ha'im was working 00 Ihe basis ofsorne papers wriuen in Azarquiel 's own hand, whose contents may be slight1y different from the ones in the Book on lhe Fixed Stars. This would explain tbe faet that the parameters that Ibn al-Ha'im quotes as Azarquiel's are sometirnes more exaet or aceurate than the parameters of the Book. Furtherrnore, in Ibn al-Ha'im's 21) Roser Puig has also found a reference to the faet that our author saw sorne writings on Azarquiel's 132 observations of the motioo of the Moan in Azarquiel's own hand • 11.2.[2] Ibo al-Ha'im's own values are then stated. First of al! he determines the ratio (nisba) concerning the time elapsed between the observations of Ptolerny, al-Battan'ill3 and Azarquiel, as follows: IJI See n.1 [3). 132 SUhayl 2 (2001) lbn aJ-HIl'im's Trepidmion Model 339 Ptolerny-BattanI/BattanI-Azarquiel = 3;50,291'/11' The proportion requires a few cornments: in his Bookon the Fixed Stars Azarquiel also states this relationship, and extends it to the relationship between the observations of Hipparchus, Ptolemy and al-BattanI. l34 According to Millás , tbe ratios are the following: Hipparchus-Ptolerny/Ptolemy-BattanT= 11'12;36,37,53,401' BauanI·AzarquieI/Ptolerny-Bauani = [1 ]1'/[2] ;40,28,47,2()1'll5 The first oC Azarquiel's ratios presents no problems at al!. If we adopt the dates of the "observations" oC Qalb al-Asad calculated by Mercier l36 (Hipparchus 17/2/-145; Ptolemy 4/3/140; al-BaltanI 19/3/884), we have the following ratio: 11'/2;36,37,53,411'. As the fourths are stated in the text as 2/3 instead oC 40, we have a very good approximation. The only correction required is the one stated by MilIás for the degrees. However, the Hebrew manuscript is confusing in relaríon to the second ratio lJ7 : lO begin wilh, Ihe integer parts do not appear in the text. MiIlás corrected the integer parls to [1] and [2], furthermore, Ihe 40 minutes appear only in the margin of the manuscript; the seconds are corrected by Millás from 25" to 28"; the 47 thirds appear in the manuscript as fourths and the 20 fourths appear as fifths, without any mention of the thirds. MilIás's correction (I P/[2];[40],28,47,2Q1') would give a difference of 279 years between al-BaltanI and Azarquiel, which would place Azarquiel 's observations ofQalb al-Asad around 1165, i.e. almost one century after his lime. Azarquiel uses Julian years when talking of time differences and this should also be applied [O the relationships. Considering thal Azarquiel himselfplaces his observatiolls in 1074-1075 1:14 See 1.M. Mi1lás [1943-1950:315). m Neilher of Ihe inleger parts are stated in the lex!. Ilb See R. Mercier [1996:3281. 131 1 owe lhe readings of the Hebrew, manuscript lO T. Martinez. Suhayl 2 (2001) 340 M. Comes and Mercier's calculation places them in 1076, we have lO accepl a correclion ror lhis parameter. The firSI suggestion would be lO correet Azarquiel's parameter on me basis of Ibn al-Ha'irn's. Ifwe suppose the integer parts lO be 3P instead oC 2P, and keep the 28". corrected by MilIás, we would have a difTerence of about 203 years. meaning that AzarquieJ's observations should be placed around 1087. If we also correet the 40' to 50'. and forget about the fact Ihat 47 are fourths in (he manuscript, this would give us a difference belween al·Battaní's time and Azarquiel's oC 193 yeaes, which places Azarquiel's observations around 1077, fitting (he known data fairly well. Furthennore. 3;50,28,47,2()P seems 10 correspond to lbn al-Ha'im's rouRded off value [3;50,29P]. The problem is that Ihe aforementioned correclion involves a great number of changes. There is a second possibility, whieh is closer to the manuscript reading. If we accept that between Ptolemy and al-Hattani the difference of time is, as staled in the aboye ratio, 744 Julian years, and the difference between Azarquiel and al-Battani 196 Julian years, we have a ratio of l P/O;IS,47,2QP. This difference would place Azarquiel's observations around 1080. This may also be a valid reading, a1though it involves severa! changes: to apply a correction from 25" to 25'138 and from 25' to IS'; and forgel the 40' in the margin as well as the faet thal47 and 20 may be thirds, fourths or fifths. which is nol c1ear in the manuscript. The third possibility would be to suppose the integer parts to be 3P and maintain Ihe resl of Ihe data in the manuscript. We will then have 3;40,25,O,47,2QP which will give us a difference of 202 years between al· Battani and Azarquiel, placing Azarquiel's observations in 1086, exactly 10 years after the data calculated by Mercier, and llar 12 years after the data stated by Azarquiel himself. It seems that Ihe first possibilily is more reasonable, mainly if we take into aceount Ihe exaetitude of Ibn al-Hii'im when quoting Azarquiel's words and parameters. 1. 1.M. MillAs [1943-1950:315) corrects 28~ to 25', although in (he manuscript it appears as 25". &Q:yI 2. (lOO1) /brr al-Ha 'im 's TrqJidation MOtkI 341 n.2.[3] Afterwards, Ibn al·Ha'im determines the revolution periods1lll: a) Head of Aries: (Julian years) = 3874 y. and 3 and ca. 112 months. From this revolution period, which is approximate, we can derive a daily value of 0;0,0,54,57,3,5,42° very c10se to the parameter given by lbn al Ha'im himself (0;0,0,54,57,3°) and quoted in 11.2.[4]. The parameter derived trom the mean motion values (0;0,0,54,57,2,41°) also tits the value Slated. The value giveo by Azarquiel is 3874 Julian years. As always, Ibn al Ha'im follows Azarquiel but is more precise in the parameters. From Azarquiel's revolution pararneter, we oblaio a daily value of 0;0,0,54,57,17,38,5°, very c10se to the parameter derived fiom mean molioos io Persian and Julian years (0;0,0,54,57,17.38,4°). The parameter for Persian years agrees to the sevenths with the parameter for Julian years, wheoever we use me value s18tOO for l Persian year; if we correct this value 00 the basis of the values for 10 aod 100 Persian years we obtain 0;0,0,54,57,7,46°. Arabic years show a different parameler, a problem l40 which has beeo sludied in depth by Mielgo • b) Pole of Ihe ecliptic: (Persian years) = 2032 y. and c. 29 days. This "alue is once again approximate and gives a daily parameter of 0;0, I,44,50,20°, very c10se to the parameler given by the same Ibn al Ha'im (0;0,1,44,49°) and also quoted in n.2.(4]. The value given by Azarquiel is 1850 Julian years. Here Ibn a1-Ha'im is correcting Azarquiel. Similar correclioos in the paramelers of the obliquity of the ecliplic rnodel will appear in almost all of Azarquiel's followers, because after Azarquiel's time, the obliquity was sliII decreasing '1'1 The values are very close to (hose of Azarquiel and the rest of his followers. although the basic parameters seem lO have bttn sligh(ly modified. Azaquiel t.ables are calculaled for Arabic. PersWl and Julian Years. '4 See H. M~lgo (19961. 5lm)'l 2 (2001) 342 M, Comes and the paramerers had to be modified in arder to tit the observations'~'. According to Ibn Isl).aq and Ibn al-Raqqam, to make a complete revolution the pole of the ecliptic needs 2698 Julian years. 11.2.[4] From the revolution parameters he obtaios the following motioos: a) Head of Aries: (lulian year) ~ e. 0;5,34,30,45,40' (Persian year) = c. 0;5,34 142,17,1,24° (Arabie year) ~ 0;5,24,32,43' (Daily) ~ 0;0,0,54,57,3' The values stated by Azarquiel Coc the mean motioo of the Head of Aries are: (lulian year) ~ 0;5,34,32,16,36' (Persian year)~ 0;5,34,18,32,16,35' (A rabie year) ~ 0;5,24,32,23,21,40' (Daily) = no1513tOO. b) Pole of the ecliptic (Julian year) = e. 0;10,38,4,43,3' (Persian year) = 0;10,37,38,30,45,48' (Arabie year) ~ e. 0;10,19,3,56,16,48' (Daily) = 0;0,1,44,49' Ibn al-Há'im's daily motion of the Head oC Aries, recalculated froro Persian and Julian years, is slightly different froro the daily motioo 1<1 See M. Comes 11996). Sul~yJ 2 (2001) Ibn al-Htl. 'im 's Trepida/ion Model 343 recalculated trom Arabic years. The difference starts in (he sixths. (Daily motion from Persian year) = 0;0,0,54,57,2,41,48" (Daily motion from Julian year) = 0;0,0,54,57,2,41,45" (Daily motioo from Arabic year) = 0;0,0,54,57,2,43,33" This aloo happens with the motion of the Pole of (he ecliptic. The difference also starts io the sixths. (Daily motioo from Persiao year) = 0;0,1,44,49,4,14,17" (Daily motion from Julian year) = 0;0,1,44,49,4,14,6" (Daily motion from Arabic year) = 0;0,1,44,49,4,6 0 Similar differences are also found when recalculatiog Azarquiel's daily value for the motion of the Head of Aries143, although in this case the differeoce starts in tbe fourths. (Daily motioo from Persian year) = 0;0,0,54,57,17,38,4,1 0144 (Daily motioo from Julian year) = 0;0,0,54,57,17,38,4,11" (Daily motion from Arabic year) = 0;0,0,54,56,59,24,2,500 l45 However, in Azarquiel's motion of the Pole of the ecliptic , the difference does not start uotil the sevenths. 1.) On the differences between the daily parameter derived from Azarquiel's Arabic and PersianlJulian years, see H. Mielgo [1996: 172-178]; and M. Comes [1996:357-359). Mielgo worked on !he assumption tIlat MiIlás's edilion was correet; though Ihere are several misreadiogs, the differences do nol change lile majn idea of Mielgo's paper. 1... As conected by Mielgo [1996:174-178]. l.' 1bere is a mistake in MiIlás's translalioo of lile Hebrew texto 1 have used !he correet value which is !he one tIlat appears in the tables edited by Millás (1943-1950:329] (0;11,19.40°), different from lhe vaJue stated in !he translation for 1 Arabic year, p. 327 (0;10,19,59,39,40°). From the values for 10 and 100 years io the text!he daily value recalculated is O; 11, 19,39,59,39,40°, which corresponds exactly 10 !he vaJue stated in lhe Hebrew text, rounded off in the labre 10 0;11,19,40° SlilayI 2 (2001) 344 M. Comes (Daily molioo froro Persian year) = 0;0,1,55,4,42,42,5,44° (Daily motian froro Julian year) = 0;0, I ,55,4,42,42,5.35° (Daily motioo traro Arabic year) = 0;0,1,55,4,42,42,4,17° As we have seen, Ibn al-Ha'irn's parameters do nol coincide with Azarquiel's Dor with the values found in lhe rest of Azarquiel's 1ol6 followers • However, this is nol surprising; Ibn al-Ha'im himself states thal he deteImined new parameters 00 finding substantial differences amongst his predecessors. Ibn al-Ha'im is JUSI accepting sorne of the old parameters, and in the case of the trepidation model he considers ooly Azarquiel's radius of (he equatorial epicycle as correcto Ibn al_Raqqam '47 will al50 introduce slight corrections lO Ibn al-Ha'im's parame(ers. 7. 00 lhe impossibility oC constructing 3D everlasting table lO determine .n. Báb 3 of maqdfa II deals with the relationship between the revolution periods of the Head of Aries and the pale of the eclíptic. 0.3.[1] According to lbn al-Ha'im, the ratio (~) of the velocities is approximately the following: Rev. PolelRev. Aries = 1;54,40/1. Then, the revolution period of the pole of the ecliptic is almost twice the revolution períod of the Head of Aries, the difference being only 0;5,20. 1J.3.[2] From this difference, Ibn al-Ha'im calculates the relationship between the recurrences of both motions, using differenc procedures. First of all taking a revolulion to be 60, he has: I1 revolutions and 1/4 for the Head of Aries. t46 00 lhe parameters used by lhe different astrooomers dealing with trepidation. see M. Comes [1996:357-3631. See also J. Samsó &. E. Mi11b 11998:2621 . •<1 M. Abl!ulrahman 11996b:511. Slayl 2 (2lXllJ Ibll al-fifí '¡m 's TrepidO/io" Model 345 21 \48 revolutions and 1/2 for the Pole of the ecliptic. In effecl, 60/5;20 ~ 11;15 and 11;15 x 1;54,40 ~ 21;30. l49 Then he uses the term "bsr , which means that he reduces the aforementioned values to a conunon denominator, obtaining: 45 revolutions for the Head of Aries. 86 revolutions for the Pole of the eclíptic. We can see mal 11 1/4 = (11 x4)+ 1J/4 ~ 45/4 and 21 112 ~ [(21 X2)+ 1]/2"" 43/2. Then, multiplying both fractions by 4, we llave 45 and 86, which are the minimum value of entire numbers which keep the same ratio as that of the two velocities. He then repeats the operation giving the revolution the value of 3600 p instead of 60 • Therefore the ratio 5;20/60 becomes 5;20/60 X 360° "" 32°, which means that the pole of the ecliptic will miss completing two revolutions by 32° (360° x 2 - 32° = 688°), while the Head of Aries performs one revolution. Furthermore, 360° / 32 = 11;15° and he repeats here that 11;15 revolutions of rhe Head of Aries would correspond to 21 ;30 revolutions of the Pole. He then uses again the term "bsr' , meaning to reduce an emire number and a fraction to a cornmon denominator, and he obtains: 180 revolutions for the Head of Aries 344 revolutions for the Pole of the ecliptic In effecl, 1;54,40 ~ 1 + 54/60 + 40/3600 ~ 6880/3600 ~ 344/180. However, 180 and 344 are not the mínimum entire numbers in lhe aforementioned ratio of velocities, although if we simplify the ratio !la 22 in the manuscript: this is obviously incorrect because tbe ratio would be more lhan double; furthennore. 21 is explicitly stated at the eOO of this very same chapter. 109 lbn al-H'im uses "bs!", which. according to Souissi [1968:90-92}, in a mathematical context means to reduce an entire number and a fraction 10 a corrunon denominator. Suhayl 2 (2001) 346 M. Comes 180/344, we obtain 45/86 as before. OC course, the 344 revolutions can also be deduced froro the previous step: 360 - 32/2 oc 688/2. This is the proof he needed to demonstrate that it is impossible to construct aD everlasting table to determine A)." The Teason is that the table will only agaio be useful when the Head of Aries has completed the 45 revolutions, which, bearing in mind that the Head of Aries completes a revolution every 3874 years, gives us 174330 years, a period too long lO be considered. 11.3.[3] According to hirn, Azarquiel, Abü Marwan and the Qt'ü!f AbO 'l-Qasim ~acid understood this impossibility and this is why there are no tables of rhis kind in their works. However, the trepidatían tables in lhe Liber de Motu and in the Toledan rabIes give directly [he precession value. This suggests two possibilities: that neither Aba Marwan nor ~acid or Azarquiel were the authors of the Toledan Tables or that the trepidation tables were a Jaler addilion. This Jauer option, however, conflicls with Mercier's proved opinion that (hese tables were designed to work onJy wilh Ihe Toledan Tables1'JO. 11.3.[4] Ibn al-Ha'im states that more recent authors, amongst them Ibn al-Kammad and some contemporaries of Ibn aJ-Ha'im, were unaware of this and mistakenly prepared everlasting tables for ~>..15l and believed 152 lhat the two motions coincide • 8. 00 how to determine f using the corresponding tables Bab 1 of maqiila 111 is devoled to determining the obliquity of the ecliptic using the corresponding tables. 11O Mercier [1996:299]. 111 lbn al.H:i.'im himself prepared tablcs for the molÍon of the Head of Aries and ¡he Pole of the ecliptic. but to be used for a limited period of time. see 0.[13]. 111 see 0.16-7]. S\IIIll~1 2 (2lXI1) /bn af-Hd 'jm 's TnpidJJJion Mode/ 347 m.l.[l] It involves the use oftwo tables: 1) a mean motion table for the motion of the Pole of the ecliptie around its epicycle, lhen giving anglej; and 2) atable for me differenl values of obliquily in which the argument ¡sj. The procedure suggests the use ofseconds, which do nol appear.in all the obliquity tables -al-Marrakushi'sl~3 for instanee- and corresponds exactly with Azarquiel's model. 9. 00 how lo determine the 4>-' using the corresponding tables Báb 2 of maqala In deals witb the use of tbe tables te detennine the distance between the Head of Aries and the spring equinox, that is the accession and recession molion. UI.2.[t] Ibn al-Ha'im uses two tables (See Fig. 9): a) atable for Ihe mean motioo of the Head of Aries (aogle 1). b) atable giviog directly 8/.., using angle i as argument. The firsl table calculales Ihe motioo ofthe Head of Aries in its epicycle, while the second one computes the distance between the moving Head of Aries and the equinox, which corresponds to the ¡ncrease or decrease ofthe longitude due 10 accession and recession motion. The second table is calculated to the precision of seconds. The result should be added to or subtracted from the sidereal longitude of the star or planet, depending on whether the equation is positive (iqbál) or negalive (idbár). Although he is describing Azarquiel's third model, he does not use the tables and the procedure given by Azarquiel but the ones found in the Liber de MOlU and the Toledan Tables, and used by mosl of Azarquiel's followers sueh as Ibn al-Karnrnad, Ibn Is1:).aq al~Tünisi, lbo al-Banna', Abü 'l-l:Iasan al-Murrakushi, Abü 'l-l:Iasan al-Qusan~ini, lbn CAzzüz al 1Yo Qusan~ini, lbn al-Raqqam and the autbors of the Barcelona Tables • The main difference is that Azarquiel does not have atable giving .6./.. 111 See M. Comes (1996:362-3]. 1)4 See allhis respect M. Comes [1996:356-362). ~ 2(2001) 348 M. Comes directly, bul uses an indirect procedure involving a Rumber oC calculationsus. As we have seen, the use oC atable that gives dA directly seems to be precisely what he has been criticizing in Ibn al·Kammad and sorne oC the lS6 astronomers oC his own time • Howeveas 1 understand ir, what he is criticizing is nol the use oC the table for a short period of time whenever great accuracy is nol required, bUI the use oC an evertasting tableo 1tI.2.[2] In (act, he is merely explaining how lO use atable ofthis kind bUl warning the user thal when the mean molioo of the Head of Aries is greater than 6 signs the approximation obtained is worse. We have to lake ioto account that the tables are calculated using a fixed obliquity, while the calculation with the fonnula allows the use oC the obliquity corresponding lO any moment. In maqdla 7. bábs 3 and 4 he will explain how lO detennine the J:1)o., exactly. As we will see, in the corresponding commemaryl57, he will use the same procedure and trigonometrical formula as Azarquiel. 10. 00 how to determioe the minimum obliquity rrom aD observed obliquity In maqlJ/a VII, bah 1, Ibn al-Ha:'im uses a procedure of spheric trigonometry lO delermine E_ from a given observation ofthe obliquity of the ecliptic. The model for the obliquity of lhe ecliplic described by Ibn al·Ha'im is l5l Azarquiel's. It is depicled in fol. 81r , panially erased by moisture; il is also shown in Fig. 3, and described in VII.I.[4]. lIS Sce J. Samsó 11994:22-251. 116 St:e E. Calvo [1997] and J.L. Mancha (1998:6]. "1 Sc:e VII.). [141. ISI In lhe figure appearing in lhe manuscript, Itlters B and G are inlen:hanged. 1 have oonttted Ihis misuke followlOg the description of lhe model found in VII.I.[4j. ~ 2(2001) lbn al-Ha 'im's Trepida/ion Model 349 VIL 1.[1] COnSiS1S of a sel of instruetions for obtaining ZA. Followiog Azaquiel, lbn al-Ha'im uses here the eosine lheorem formulated in al Andalus by lbn Mucadh in his KiUib Majhúlát Qisr al-Kural59 and by Jabir b. AfJa~ in his lslii~ al-Majis{l'4oo. First, Ibn al-Ha'im obtains ZD, using the angle of rotation around the polar epicycle at the momenl of the observation, the radius of the polar epicycle, and the observed obliquity. The procedure is as follows: Sin BG = 60 x sin BG; 0;0,10,20° x Sin BG = O; 10,20° X sin BG In order to obtain the are of great circle GD, he operates: GD ~ Sino' (O; 10,20· x sin BG) He then obtaios Cos GD, whieh will be a divisor (imiim) in the final formula: Cos ZD = Cos ZG X 60 I Cos GD At lhis point in the lext there seems lO be a mistake or a eopyist's error, for it states that we should oblain the arcsine (inslead of the arecosine) of ZD_ Then, Ibn al-Ha'im gives the ¡ostruetions for obtaining AD, again using lhe motioo of lhe pole (Cordj for j < 180°) or (Cord (360° -J) for 180° < j < 360°) and the radius of the polar epicycle. Cord AG ~ 60 x cord AG 0;0,10,20· X Cord AG = O; 10,20· cord AG In arder to obtain the are of great circ1e AG, he operates: 1"" On Ibn Mtflidh's lrigonometry see M.V. Villuendas [1979] and J. Samsó [1980:60-68]. 160 See R. Lorch [1975:38] and J. Samsó [198O:64J. SUhayl 2 (;'0)1) 350 M. Comes AG ~ Cord-' (O; 10,20' X CQrd AG) He theo obtains Cas AG, which will be used in lhe final formula: Cas AD = Cas AG x 60 I Cas GD As aboye, the arcsine oC AD is used instead oC the arccosine. Finally he obtaios ZA. which is the desired minimum obliquity, (har is to say .the minimum distance between the pote oC the ecliptic and the pole of the equator, through a simple subtraction (ZA = ZD - AD). 0 161 The Tadius given by Ibn al-Hii'im is O; 10,20 (0;0, tO,20 X 60 ) which does not correspond exactly to Azarquiel's r = O; 10°. However, as we have seen in I1.2.[1], lbn al-Hii'im does Dot consider Azarquiel's radius afthe polarepicycle as ane ofthe correct values, so it is nol strange lO find it correcled. VIl.1.[2] By the usual formula wa-l-cilLajl dhálika, he introduces (he description oC the figure as follows: AGB Polar epicycle BAZ Arc of a great circle passing through the centre of the polar epieyele, the pole of the equator and point A Z Pole of the equator A PaJe of the eeliptic at its minimum distaneel62 from the pole of the equator (AZ = €miJ B Pole of the eeliptie at irs maximum distanee from the pole of the equator (BZ = €'na~) G Pole of the ecliptie at a given moment ZG Are of a great eirele from G to lhe pole of the equator (Z), that is the distanee between lhe two poles al the given moment = observed €. 161 80lh Azarquiel and Ibn al-Ha·im lnSlsl repealcdly that they use R=60. Sec VI1.4.(lI. 162 The text States merely "distallce from Ihe pole uf the equalor". S..lIoyl 2 (2001) lb" (ll-Ha ';III'S Trepidmioll Molle! 351 AZ Minimum obliquity of the ecliptic (E""u) VII.1.[3] Furthermore, he determines that if we know an observed obliquity, the desired minimum obliquity, that is ZA, is also known. He describes the triangles he needs to sol ve and adds the following: AG Arc of a great cirele, represented in the drawing by the chord AG GO Arc of a great circle perpendicular from point G to the are of great eircle AB, represented in the drawing as a stright Hne vn.l.[4] The Icnown data are the following: BG (derived from AG); Sin BG = DO; ZO (known by observation); R = 60P and r (radius ofthe polar epicycle = O; 1O,2QP). VII. 1.[5] He shows here how to solve the right angled triangle ZOO, made of arcs of great eireles, in whieh O is the right angle. He uses the eosine theorem to determine ZO: eos GZ/eosZO = eos GO/sin 90° (the fonnula used in VII.l.[IJ). VIL 1. [6] He then salves the right angled triangle AGO, using the known data, in arder to determine AO. The procedure used here, as elsewhere, is extremely careful: are AG of the equatorial epicycle is known (for r = 0;,10,20), we can thus ealculate chord AG (for R = 60) and obtain, from atable of chords, tbe arc AG of a great circle of the sphere. VII. 1.[7] From these known data, he again uses the eosille theorem to determine AO: eos AG/eos GD = cos AD/sin 90° (the formula used in VII.l.[IJ). Then, he only needs to subtract AO from ZO to obtain ZA, the desired minimum obliquity. 11. On bow to determine the obliquity (f) corresponding to a given moment Maqála VII, bah 2, shows how lO determine the obliquity of the ecliptie for a given momento ~ 2(2001) 352 M. Comes VII.2.[I}-[4] The procedure is lhe same as befare bUl in the reverse arder. Firsl of al1 he uses the cosine theorem to salve the right angled triangle AGD. The known data are as befare: AG, and consequently BG, which implies that DG, beiog the sine of BG, is also known. With this he determines DA, using lhe same procedure as in VIl.l.[I],[6]. Afterwards, he uses right angled triangle DZG. Here, (he difference is Ihar instead of knowing the hypotenuse and ane side and then determining the other side he knows lhe two sides and has to determine the hypotenuse. The known data are: ZD, which is ZA (emiJ plus DA, {he value aboye determined and DG, beiog (he sine of BG. To determine za, Ihal ¡s. (he hypolenuse. he uses the relationship mentioned in VJI.I.[5], based on the cosine theorem. ZG will correspond to the desired obliquity for a given moment. l63 This is exactly what Azarquiel does in his Book 0/1 (he Fixed Srars , where he also solves the same two right angled spherical triangles '64 to determine the obliquily of the ecliptic, using as known data the maximum and minimum obliquily and delermining the obliquity for a given moment. He uses (he same theorems as lbn al-Ha'im, but more than a century before, more or less al (he same time as or slightly after Ibn Muca:dh, withouI specifying the procedure and the relations as Ibn al-Ha'im does. Azarquiel is jusI showing that in right angled spherical triangles, once one of Ihe sides and Ihe hypotenuse are known lhe other side is known; and once the two sides are known the hypotenuse is known, which implies Ihe use of spherical trigonometry as in Ibn al-Ha:'im's tex!. 12. 00 lhe knowledge of "al-iqbal al-awwal" Báb 3 of maqála VII deals with lhe determination of the ¡qMI al-awwal, which computes AA., thal is Ihe positive or negative amount of precession l',) J.M. MllIa~ 11943-1950:3301. 1". Aeeordmg IU Ibn al~H:rim "min qisiy l/a\l'(¡ 'ir 'i?!im" and following Millás' IlOlllslalion of Azarquiel'~ lh:hrcw lex! "Arcos de eirculn grande" (Ares of great eircles). Sullayt 2 (200tl Ibn al-Hil'/ltl's Trrpidarion Modd 353 for a given momenl, which is not found tabulaled in Azarquiel. The title introduces for the firsl time lhe notion of accession and recession "perceptible by the senses" (ma~lSlls) in contraSI wilh Ihe title of bab 4, dealing wilh al-iqbcU aL-rlltil/[, in which lhe accession and recession is "perceptible by Ihe intellecl" (mcfqtU). Azarquiel also refers lO al-iqbál al-owwal as perceptible by lhe senses; however, he considers a/-iqbtí/ OI-IIIolI[as Ihe "lrue" iqbtilwa-idbtir of the Head of Aries, Ihat is the mation of lhe Head of Aries not in its equatorial epicycle, and hence in tbe eclíplic, but projecled anta the equator. The astronomers wbo follow Azarquiel's Irepidation moclel consider that the outermast orb (al-aq$ii) has IWO mOlions, one tropical and perceptible by the senses (!ab¡<[ ~lissO and the otber sidereal and perceptible by lhe inteIlect (dht1tr'"aqli)lM. VU.3.[ 1] This paragraph corresponds almosl verbalim 10 a paragrapb in chapter 8 of Azarquiel's Book ol/Ihe Fixed Stars l66 and explains the use of tbe table for the declinalion of the Head of Aries. Should we wam to calculate the accession or recession for a given moment, we willtake as argument (he mean mol ion of the Head of Aries and use illo enler the rabie of "declinations" in Azarquiel, or "sine of lhe declination" for Ibn al~Ha'imI61. Ifthe argument is between 1° and 180°, 1) will be northero; on the otber hand, between 180° and 360°, it will be southem. TIten, we should apply the formula: sin.6.). = sin 1) x 60 I sin E Ifi is belween 1° and 90° or between 270° and 360°, the Head of Aries is moving towards (muqbil) lhe nOrlh, while in the rest of the epicyc1e it is 163 This terminology is found in Ibn al.RaqqiIm's al-Zl} al-Shtlmi/ /f Tohdhfb al-KIJmil. chapters 72 and 73, laken from Ibn al-HI'im, but also in his al-lij al-Muslal\fl'" (chapler 17) and other sources such as Ibn Isl)Jq's zJ1 (Hyderabad. fol. 92) and Ibn al-Baqqlr's Kirdb al-AdwtJr (fol. 139v). 166 see 1.M. Mi1lis [1943-1950:335-337). 1" see in this regard 0.(12]. &D)'l 2 (2001) 354 M."""'" moving lowards (muqbil) the soulh. Ir the declination is northem. lhe accession and recession (iqbdl and id/x}r) -one should understand the position of (he Head oC Aries· are to the east oC (he spring equinox and the Head of Aries will be forward (mutaqaddima) towards (he East, bUI if the declination is southern, !he accession and recession (iqbtíl and idbtir) will be lO the west of the equinox while lhe Head oC Aries will be backward l68 (mura'akhkllira) towards the West . Ifwe follow Azarquiel's and Ibn al·Ha'im's instructions for calculating ,6,).. step by step, the ooly difference we find is thal Azarquiel uses one table for the declination and another one for sines, while Ibn al-Ha'irn's table, as we have seen, gives Ihe sine oC the declination direclly. VlI.3.[2] Introduced as is customary by me fonnuJa wa-(Jilla jf dhdlikiJ, this section presents the trigonometricaJ explanation. Te begin with, we find the description of the model, which corresponds to 169 Azarquiel's geometrical description of chapter 8 . The lettering in the figure that appears in the manuscript (81 v) does not correspond to the descriptlon of the text. 1 have reproduced lE as Fig. 8, although 1 llave followed the lettering in lhe tex!: ABO Equatorial epicycle Aa Diameter of ABO B Head of Aries at a given moment BZ Are of a great circle perpendicular to AG and (, at that mament DB Section of the ecliptic and 8A perceptible by the senses at that moment 161 TIx' possible confusion arise5 from twO facts: lhe firsl Is lhal place were lhe "accesslon or reccssion" (in Azarquicl's lemlinology) of [he Heads of Aries and Libra slans (i "" O" or 180") is nol lhe place werc lhe "accession or recession" of the equinoclia\ poims Star1S (i - 90" or 270°). and lhe second is [hat ncilher Azarquiel nor Ibn al-HA'lm sate explicilly whether they are lalklllg about the dlrection of lhe Heads of Anes and Libra, lhelr po5ltlons with regard lO [he equinoxes or lhe forward and backward mollon of [he cqUlllOxlal potnlS. HIt MlIlis 11943-1950:3)3-3341. WDyl 2 (DIII Ibn a1-Htrim 's Tr~pidarion Mod~1 355 AGD Section of Ihe equator O Inrerseclion belween equator and ecliplic and hence Ihe spring equinox at lhal momenr A EZ G , D B Fig.8 This implies one ofthe following possibilities: eilher 1) Azarquiel knew Ibn Mucadh's trigonometry, which apparenLly was unknown in Toledo in lhe time of $acid al-Andalusi or 2) The new Eastem trigonometry reached al-Andalus nol only lhrough Ibn MuCadh but via other channels as well. Vll.3.[3] Ibn al-Ha'im lhen detennines the relationship foc solving the spherical triangle oza (Fig. 8), whicb does not appear in Azarquiel, although it is implicit in his geometrical resolution ofthe spberical lciangle. Ibn al-Ha'im uses me sine theorem here, formulated also in al-Andalus by Ibn MuCadh in his Kitc'ib Majhaldl Qisf al-Kural70 and Jabir b. Af1a~ in his isla!}. al-Majis!i-HI. Sin aZ/Sin BD = Sin D/Sin Z no On Ibn Mlfidh's trigonomeuy see M.V. ViIluendas (1979] and J. Sa.msó {I980:60-68I. III See R. Lorch (1973:38J and J. Sa.msó (1980:641. SlmyI 2 (XlII) 356 M. """'" VII.3.[4}lbn al~H¡¡'im tries lO determine BD = .6.A, from the known data, Ihat is: BZ = ó; D = E; Z = 90°, of a right angled spherical lriangle. Theo, according to the given relation, the formula to be applied has to be: sin.1.).. = sin ó X sin 90°/ sin E. We find exactly the same in Azarquiel's text. 13. 00 the knowledge of tbe "iqbil al-thini" As we have seco, biib 4 of fIlaqtUa VII deals with the determination of the iqbtilol-lhtillrn , which calculates what Azarquiel calls the "equation of the diameter~ (ZG in Fig. 9). As usual, the figure in the text (fol. 82v) is ¡ncomplete. The iqbl11 a/-lhállf is the difference between lhe right ascensions of the Hcad of Aries (B) and of point (G). In Ibn aJ-Ha'im's words iqbtíl al-ramlldft '/-qu!r. A , o B Fig.9 1Jl On tll-iqbal a/-IMm Stt B. Goldstein [1964:242-2441; and J. Samsó 11994b:X; 151. SlNyl 2 (1001) 1bn al-Ha 'im 's Trepidarían Model 357 D A HEZ G B Fig. 10 The terminology used needs sorne eommentary, The iqbiil al-lhiillfis an are on the equator. while the iqbál al-awwal is an are 00 the ecliptie. The iqbtil al-1Julnfis defined in the title ofthe ehapter as ir it were the ascension ofeaeh degree ofthe equatorial epieycle with respect to the meridiano The same kind of expression appears agaio io [3] and is c1arified in [4], in which the words mií ya!llf Ji da 'irat nis] al-naJuir beeome mti yajuz ft da'irat nis]al-nahtir. This seems to refer to the are of the equator between the meridian transits of two points of the equatorial epicycle. In bab 17 ofmaqtila VI, Ibn al-Ha'im uses the iqbal al-tJuinflo caJeulate the equation of time. The equation of time is tbe differenee between two passages of the sun through {he meridian, the passage of the real sun traveling aloog the ecliptie and the hypotbetical passage of the mean sun traveling aJoog the equator. He uses {he iqbal al-(htinr to determine the precession vaJue projected onto the equator, that is lhe right ascensioo of LlA m, The same use is mentiooed io ehaplers 6 and 17 of Ibn al-Banna's l14 Kittib Minhiij al-'[tilib li-tddrt al-Kawtikib , l1l On this. see A. Meslres (1999:1.29-31] m See Vernct {1952:28.82;53.117). SItlayl 2 (mI) 358 M. Comes VII.4. [1-2] Paragraph one partially corresponds lO a paragraph in 17S chaplee 8 oC Azaquiel's Book 01/ the Fixed Srars • In il we find lhe instruclion Cor calculating al-iqbál al-lhanf with lhe corresponding table using the mean motian oC the Head of Aries as argument. However, lbn al-Ha:'im here presents a fonnula nol faund in Azarquiel to calculate the al-iqbtil al-lhiint for lhe positions of lhe Head of Aries taking into aceDun! if i < 1800 oc i > 1800. In lhe first case, he uses: Cord ix 0;4,19,26(1 "" 60 cord ¡x 0;4,19,26P = cord ¡x 4; 19,26P Sin (180 - i) x 0;4,19,26' ~ 60 ,in (180 - ¡) x 0;4,19,26' ~ sin (180 -1) x 4;19,261' The developrnent oC the formula is as follows: Cord-¡ (cord i x 4;19,26P) = ....BG (are oC geeat circle) Sin- l (sin (180 - i x 4;19,26") = ~BZ (are of great circle) He, then, determines ZG, the desired iqbtil al-thiinf, using the relationship stated in VIl.4.[4], where we find the geometrical explanation, corresponding to Azarquiel'sI76: Co, ZG ~ Co, BG x 60 I Co, BZ; For the second case, the procedure is the same, using: 173 J.M. MillAs [1943-1950:336-337]. 17(i J.M. MillAs [1943-1950:333-334]. Suhayl 2 (2001) Ibn al-HIJ.'jm's Trepidmion Model 359 Cord (360 - 1); Sin (180 -(360 -1)) ~ Sin (i - 180) The radius oC the equatorial epicycle is 4;19,2& (0;4,19,26P x 60). As in the case oC the radius oC the polar epicycle, Ibn al-Ha'im uses two values, tbe radius given in the text and the vaiue of the radius divided by 60, used in the geoinetrical description. The description found in VII.3.[2] is also repeated, with the following additions: E Centre oC the epicycle BG Are oC great circle and straight line connecting the two points VIl.4.[3] Now. 'he known data are: GB (i); BA (180" -1) and ZB (o). The geometrical explanation is as follows: given the triangle GBZ, made oC ares of great eircles, of which Z is the right angle, the relationship, aeeording to lbn al-Ha'im but nOl stated by Azarquiel, will be: cos BG I cos GZ = cos BZ I sin 90° As BG and BZ are known, we also know GZ, that ¡s, al-iqbdl al-thánt. So Ibn al-Ha'im is using the eosine theorem here. Azarquiel uses the same triangle and Ihe same data so that, implicitly, he is using the same relalionship. 14. 00 the knowledge oC the ltiqbal al-thani" between two given positions oC the Head oC Aries VII.4.[4] Then, Ibn al-Ha'im adds Ihe procedure to be used 10 determine the "second aceession" between two different positions of the Head of Aries. Lel us 1l0W suppose lhat at anOlher momen{ lhe Head of Aries is at point l) and draw a perpendicular lO AG, in order to oblain lhe value of () at that mamen!. We also draw an are of a great circle between l) and G and join Ihe t\Vo points with the straighlline cJlG (Figs. fol. 83v. and 10). 360 M. Comes VI1.4.[S] Then, Jet us consider, as befare, lhe rigill angled triangle 111. Conclusions l. General couc1usioDS This manuscript is interesling because il affers, fOf lhe first time, a detailed description of lhe Andalusian theory of trepidation in an Arabic tex! aud not in a Latin Of Hebrew translation of ao Arabic text. In lhe Kámil zij, accession and recession, Of trepidation, is a conslant throughout the book because i[ has to be taken into accounl for most of the calculalions and procedures lhat the book presents: from the equation of time and the explanation of the different kinds of years to the knowledge of the procedure used [O calculate the tropical longitudes of the heavenly boclies referred to the ecliptic. lbn al-Ha'im is not keen on tables; however, he suggests chat they may be used, whenever the user is aware of the fact that, in general, tables have a lime ¡imit ofno more than 40 years. In particular, trepidation tables have to be adapted to the changing obliquity. In fac[ lbn al-Ha'im's equation table takes into account the value of f. for each mean position of the Head of Aries on !he equatorial epicycle (See 0.13). However, tite fact that the period of revolution of the Pole of the ecliptic is not exactly twice the period of revolution of tite Head of Aries implies that tite values for arguments between 180 0 and 3600 in the table will not be symmetrical to 0 0 0 those between 0 and 180 • Even if the table is prepared for 360 , it will be val id just for one revolution of tite Head of Aries. lbn al-Ha'im also offers other ¡nformalion on Andalusian sources: for instance, the parameters used by Abü Marwan al-lstijji, which suggest that he wrote a z(j omer than the Toiedan Tabies. In fact, Abü Marwan's Suhayl 2 (X01) Ibn al-HlJ 'im 's Trepida/io" Model 361 parameters differ from the known parameters, almough his determination of the increase of precession between Hipparchus and Ptolemy coincides wim Azarquiel's. Furthermore, his parameters seem not to be related to those of me Toledan Tables or the Liber de Motu. Ibn al-Ha'im also introduces us a certain Muntakhab zlj, by a comemporary of his, whose values for precession corresponding to Hipparchus and Ptolemy's time are very close to the values in Azarquiel's second model. And the table for the ¡ncrease or decrease of precession in mis zlj seems to have coincided with the one in Ibn al-Raqqám's al-zfj al-Qawfm and Abü 'I-J:lasan al Qusan~iní's zt), which differ frorn the tables found in the rest ofAzarquiel's followers. To determine Ibn al-Ha'im's role in the spreading of the trepidation models, 1 will analyze his relationship with his forerunners, Azarquiel (III.2) and lbn al-Kammad (1II.3), and with his followers, the author of me Hyderabad recension of Ibn Is~aq's zfj (lIlA) and Ibn al-Raqqam (lII.S). 2. Ibn al-Ha'im and Azarquiel As we have seen, in me Kiimil z'J, Ibn al-Ha'im relies heavily on the models in Azarquiel's 800k Off elle Fixed Scars. He uses Azarquiel's third lrepidation model as well as his model for the obliquity of the ecliptic. Allhough it has always been considered that in Azarquiel's third model the obliquity of the ecliptic model was independem of the trepidation model, Ibn al-Ha'im, following Azarquiel, connects the two motions and models. In them, the pole of the ecliptic carries the solstitial colure and maintains a constant distance of 90° from the Heads of Aries and Libra. Ibn al-Ha'im is, however, more precise in the parameters given, perhaps indicating that he is noc using the Book Off rhe Fixed Srars, but sorne sheets ofpaper written in Azarquiel's own hand; in fact, Ibn al-Ha'im states this in a couple of places of this book. As far as the geometrical resolution is concerned. both used spherical trigonometry. In the Kámil zlj, Ibn al-Ha'im adds to Azarquiel's text me delailed description and use of spherical lrigonometry. as regards to formulae and relationships between sides and angles of righl angled spherical triangles, showing a good knowledge of lhe subject (See VII. 1 and Vll.2) 5..lIayl 2 (2001) 362 M. Comes As fae as lhe use of (he rabIes is concerned, as we llave seen, tllere are (wo basic differences between Ibn al-Ha'im and Azarquiel: 1) lbn .al Ha'im, like all Azarquiel's followers, devises atable thal gives the ¡nerease oc decrease of trepidation directly. Nevertheless. he recornmends Azarquiel's procedure for calculating Ihis value due to lhe faet 1hat standard trepidation tables, based on a fixed value for the obliquity of the ecliptic, are only valid for a shoet period aftime, due to which, as we have seco, he calculates atable taking ioto account the different values of the obliquity corresponding fa every position of Ihe Head of Aries; 2) instead of Azarquiel's table of dec1inations of the Head of Aries, lbn al-Ha'im gives atable for the "sine of the declination", because in fact this is the value needed to apply the formula sin.6.A = sin ó I sin E. Furthermore, we should note that sorne paragraphs of Ibn al-Ha'im's text are excerpts, often word for word, from Azarquiel's book. Examples can be found in lI.I.{l,3,9]; VIL3 (I,2] and VIJ.4.{l]. However, Ibn al Ha'im ofien modifies the parameters to agree with the observations. 3. Ibn al-Ha'im aud Ibn al-Kammad Ibn al-Ha'im considered Ibn al-Karnrnad an ignorant disciple of Azarquiel who had distorted his master's theory. He states that Ibn al-Karnmad's trepidatioo model is based 00 two fundamental errors: 1) Ibn al-Karnmad believes that the motion ofthe Head of Aries and the motioo oC the Pole oC the ecliptic are the same and, therefore, devises a single rabie for both motioos; 2) he thioks that this table is valid for any time. According to lbn al-Ha'im, the ratio between the two motions is approximately 112m and Ibn al-Kamrnad's table cannot be valid for both motioos or for aoy time, because ofthe obliquity ofthe ecliptic used io the formulae underlying the table chaoges. To this it must be added that lbn al-Ha'im states in different pans ofthe 171 111;54,40° in n.3.[I}. Suh¡yJ 2 (2001) Ibn al-Hl1'im 's Trepida/ion ModeJ 363 book mat zijes have a limiled duration oC no more than 40 years 118, One oC lbn al-Karnmad's errors was precisely lo consider his zij as everlasting (al-Amad cala al-abaá). Surprisingly enough, in the Hyderabad's version oC Ibn Is~aq's zlj, we can read that in al-Kawr 'ala al-Dawr l79 lbn al Kanunad slated lhat lhe duration of lhe zijes was Iimited. 4. Ibn al-Ha 'im and the Hyderabad version oC Ibn Is1}.aq al-Tñnisi's "zij"UlO As the author of this version of Ibn ls~aq's zlj mentions Ibn al-Ha'im as one of his sources, which also inelude Azarquiel and Ibn al-Karnmad, it seemed worthwhile to study the trepidalion chapters in this zlj. Il was, however, disappointing to find ~at in the lheoretical discussion of lrepidation, the author relies only on Ibn al-Karruna:d's al-Kawr cala al Dawr. In his tables, lhe aUlhor follows Azarquiel, direclly or through Ibn al Karnmad, although he introduces corrects slight corrections io sorne of lhe parameters. For inslance, io [he daily parameter for lhe motioo of [he Head of Aries in its equatorial epycicle, lbn Is~aq as well as Ibn al-Banna', Ibu al-Raqqam aud lbu cAzzüz al-Qusun~ini follow Azarquiel's value for Persian and Juliao years, although adapted to Arabic years, while Ibn al Karrunad, al-Marrakushi aod Ibn al-Ha'im follow Azarquiel's table for 17! In raCt, in different parts of the book Ibn al-Ha'im affirms lhat his tables will be useless after a period of 40 years fmm ¡he dale of eomposition, which, according to him, corresponds lO the beginning of "'Fe, Hijra (13"'c. AD). See for instance fols. 36r, 70v and 71v, or when he speaks of other iteros such as lhe delermination of (he true longitudes of the sun by means of the corresponding lables. On this see E. Calvo 11998:571. In the Hyderabad revision of Ibn Ist:Uiq's ,[j, il is stated thal the 40 years limit duration of a v7 is due to the changing obliquity. see the Arabic edition of the text in A, Mestres [1999:11,296]. 11'l F. 92. see eonclusions ¡>oim 4. leo Hyderabad Andra Pradesh State Library rns. 298. On trepidation and obJiquilY of tlle ediptic in this MS see M. Comes ll992: 147-1591 and (1996:355-364). A general account of Ihe eontems of the l[j is {O be found in A. Mestres [1996:383-4441 and an edition of the main part of the ll} in A. Mestres [1999]. Suhayl 2 (::mI) 364 M. Comes Arabic years, which as we have seen is differenl rrom bis lables for Persian and Julian years. However, he does nOI propose Azarquiel's method ror calculating the accession oc recession value ror a given moment bUI atable giving the value directly. as do all the rest oC Azarquiel's followers. Be thal as il may. in the context oC trepidation. Ibn al-Ha'im is nol mentioned al all. It appears that me anonymous author was following Ibn Is~aq direcLly here, and as Ibn Isl:1aq was a eontemporary oC Ibn al-Ha'im, he did nol probably even Icnow oC him. 5. Ibn alMHa'im and Ibu al-Raqqam. As is already known, lbn al-Raqqam has three differenl zijes: al-ZI) aLM Sllámil Ji Tahdhib al-Kámil, al-ZlJ al-Qawi"m Ji Funun al-Tcfdil wa-l Taqwrm l81 and al-ZlJ al-Mustawjt. These zijes, especially Ihe first one, rely heavily on Ibn al·Ha'im, although the parameters do nol coincide'12 • According to Ibn al·Raqqam himself. a/-2J.l al-Shtimil jf TaMluo a/ Kdmil. as ils name indicates, was expressly ¡ntended lO provide tables for Ibn al-Ha'im's al-Zij al-Kdmil, and ror this reason I have used it to fill sorne oC the blanks in the manuscript. For the arguments in favour of or againsl the possibilily that Ibn aI llJ Ha'im's l.l} had ilS own tables • 1 would suggest that he may have devised sorne tables and Ibn aI-Raqqam was rnerely trying to complete the book with the ones that were missing. Seven chapters oC this zfj deal with trepidation: - Blib 17, "On the quantities oC the motion oC the Head oC Aries, the Pole and the Centre in their circles". It corresponds to Ibo al-Ha'im's paragraph 1I.2.[3J. • Bdb 25, "00 the knowledge oC the obliquity oC the ecliptic using the tables". It correspoods, almost word for word, to Ibn al-Ha'im's m.l. - Bdb 26, "00 the koowledge oC the distance between the Head oC Aries 1'1 Cf. E.S. Kennedy {l997:3S-72] 10 See Abdulrahman {1996b]. 11) See poilll 3 of the comrnentary. ~2amll Ibn al-f/(l'jm·s Trepidation Model 365 and Ihe spring equinox using the lables·. It is a surnmary of Ibn al-Ha'im's 11I.2. - Btib 70, "00 Ihe k:oowledge of Ihe rninirnurn obliquily and Ihe distance belween Ihe two poles (ecliplic and equator) from an observed obliquity·. It corresponds, almosl vcrbatim, to Ibn aI-Ha'im's Vil 1.[1-3]. As in me nexl chapler. he uses the descriplioo of Ihe formula. but avoids Ihe Irigonometrical explanation. - Btib 71, "00 Ihe koowledge of Ihe value of the obliquity and the distance between Ihe two poles al any moment". It corresponds, almosl word for word, to Ibn al-Ha'irn's VU.2.[1]. As in me chapter aboye, he gives me description of me fonnula, but avoids Ihe trigooometrical explanation, + Báb 72, "00 Ihe Icnowledge of the distance between the Head of Aries aod me spring equinox at any moment, which is Ihe firsl accession and recession, perceptible by the senses·. It corresponds lO Ibn al-Ha'irn's VIL3.[1], which io its turo corresponds to a paragraph in chapter 8 of Azarquiel's Book on lile Fixed Slars. Ibn al-Raqqam avoids the geornetric descriprion. - Bab 73, "On the Icnowledge ofme ascension degrees in the equalor ofthe accession and recession circle degrees, which is me second accession, perceptible by me intellecl·. Itcorresponds lo Ibn al·Ha'irn's VU.4.[I]. lbn al-Raqqarn here has avoided lhe geometrical description as well as Ihe trigonomelrical explanation. In the Qawfm zJj only one chapter refers to trepidation: • &lb 9, "On the accession and recession molion·. It corresponds lO bdb 26 in Ibo al·Raqqarn's Shámil. The Mustawfi zfj also devotes two chapters (16 aod 17) to trepidation. In chapler 16, he deaIs with the iqMl al-lhdnf in his treatment of the equation oftime and in chapter 17, after a long and detailed description on the use of lhe corresponding lables, there is a short paragraph on the differences belween lropical and sidereal motions and a brief mention of !he use of the formula (sin) ,d}., = sin o/sin E to achieve me increase or decrease of precession ror a given momento 5UtQyl 2 amI) 366 M. Comes IV. Appendix 1. EditiOD of the sectiODS oC tbe text dealing witb trepidatiOD oC the equinoxes snd obliquity of tbe ediptlc. (ÚL;JI.:H'í ';"WI ~)JI .,;wt J....oll, .J4..~1, J!.,l1' 45_ ....·~l .;,,» uJ;" J" ---J .1:.0111 ..,.t:,..JI 41 ·,,1 .~! (...] (.Ji>Y) [\] J,..n ..u. 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J'~YI ¡;a",¡;, 1~ .."...... 1.1 1Jo't~•• ...' ,.. ú"~.••.... -.-.. _185 .~I.J ~ ~ ·~~I·.;..1S~-I..:- J.,Ju-1I.,.JI86 .",.,)JI L'" yA L.SJI ~1-L;a;¡J1 ~ ~~.J~ (~Y ....,.) ~'ll ,JJ-1I18'7 .L.1II Uf1~ la SDJ!1(D)1) 368 M. Comes ¡Pi• ..,.. ~ :<5.,-, ~ J4l~1 ¡pi• ..,.. J.-JI .,.1. :<5_ ·ui ;¡~ ~l~ U-.:: i ~.t ~'.J ~~,. ~L..j .~JÍJ A.J~I ...u.. u~ ~ ~ .,..~I .:,...... :",...... :J :<5.,-n .:..J ~.= J- .::.I ...~ ....,.n [V] ~• .::.1 (J"J..,) ul.ojYI ~ ...~ ~l.u llo.>.Y l. J4l~I ..... pl• .."u. ~...... :.JI ~I ~ ~ • ." :o::...... c.... I..,... ~ •." :o::.... .~ I ..,.. ..,.. ..,.Jo.>JI u ...... rJ'::'~ :<5.,-n.:....,.. JL-o..u... ..¡,.u:oJI J.I= ~ ~ .4..:.WI_ .u.Jo=.'o4 ~ ¡O' .:,.. J •...,.n .,.l• ..,.. J.-JI .,.l. :<5_i..... J=.::.I ¡:JUJI ....,.n [Al ,Ji ...J.aw.¡~ o~ ~ uU: ~1~.i1 ~ C,Ji ~j) ..\iJ JI~ 'il ¡.PI.) ~I"' I (W) l·;,.,u,,:.W. ;;':'LoJJ-~I.:" ~ J•...,.n f.o" .,.l• .!J;.,.., 1 u,s, ui -y! ¡,,:ID! oJ",.. (~U1.o.;JI ~.. ..,.. LA, i u,s, ~~ t:..oJ,"',..,.n t:..o! 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'1:. t t' t c·~ -~f'., '. .:i r~.1 ~."-"t- f, ,"i :; ~ 1._ oC.... t'.'_",t... _ ..... 1".. Ibn al-Ha'irn's Trtpidation Mothl 371 .....JillJ;,JI ..t-.I..,.,1 ,1.-) ~ ... W"~ L..::.¡J L.~, 'Lo»JI .,.a .....J ..,u..>,) ".J! [...) (~~) [...) ~ J:.-i ..,: .~J .~ 1ll6J 1.:-' 195~ 19o1~ ~WJ ~.J ~L.. ~ ~ .~,J ~ ü~_Lui J ~~!J 197.¡:.~ Ü"' ~ .~ .v=a~ ~.t-'~ l.J.! -:JII ~ ,Hi ~ .~J "rJ ~tl ~ ~J.:I ~ 19l1~ .~ ~ .~~ ~ ~J ~~JJ ~ ~L.. Ü"' ~ ..w~ ~ '~.Jo! ~tl U--199~J..) ¡~ ~~".Ji -!IJ", JL,j1' AS.-~ I;J"' oo·'..,: .J,il ...411 (~Yl') ·J4..~I, JL,.i~1 .pl.. ..,J J-l1 UOiJ '-S_ ~ ..,J .J.~~;I95 ." """"tl,j.e- ü~ uf ~ J.,Jw....J1 ~ t.~1 ~1..,.....,..196 . J.~ ~ -4.:IL..a.-¡"197 .J.~~;l" .J.~ ~ .~~ u.~" ..,.....,..199 ~ 2(2001) 372 M. Comes l..t~ ~j ~ J." ~J ;¡_lh_,lhI14~ w.A.i, t:..J, ['l ...I.>.....YI, ..,) lUoJ1 "'" ~ r;,J, Lo""1", •Lo..:¡)1 .L-) "'"~ ~, ~~ lJ~ ~~ ~~~IJ1.)",.s ~l~ ~tspv-iJ~uL.j~ ..,) ~ L"..,..:. "..JI l!':H =o'~ I:'! .."I.....YI .,.....:.Jl ""1)1.>,...... 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