Paper - (Suhayl 01) the Theory of the Moon in the Al-Zij Al-Kamil Fi-l-Ta'Alim of Ibn Al-Ha'Im - Roser Puig - 2000

The Theory of the Moon in the Al-Zij al-Kiimilfi-l-Ta'iilrm of Ibn al-Hii'im

(ca. 1205)

Roser Puig

Introduction.The Zfj and its Author!

At the beginning of the thirteenth century (60tH I 1204-1205) Abu Mu~ammad cAbd al-J:laqq al-Ghafiql al-lshbTIi known as Ibn al-Ha'im composed his work entit led a/-Zij al-Kamil jr-I-Ttflitrm in honour of the Caliph Abu ~Abd Allah Muqammad al-Na~ir (who reigned from 1199-1213). All we know of Ibn aJ -Ha'im's life is that he came frolll Sev ille and that he appears to have worked in North Africa.

Ibn al-Ha' im's 21) is included in the MS Oxford Bodleian 285 (Marsh 618) . It is qui le a long text, with an introduction and seven books (maqdld/). Each book is divided imo several chapters, of which there are eighty altogether. The text can be considered as a 'i.E} on the basis of its structure and its contenlS, although it does not include astronomical tables;

This paper is part of the complete study of the I II of Ibn al-Ha'im which we started in Barcelona some years ago as pan of a research program entitled "Astronomical Theory and Tables in al-AndaJus and the Maghrib between the 12th and 14th Centuries", sponsored by the Direcci6n General de Investigaci6n Cientifica y Tecnica of the Spanish Ministry of Education and Culture, Emilia Calvo ( 1998) has recently published a paper on the astronomical theories related to the Sun in the same vland Merce Comes is preparig a paper on Ibn al-Ha 'im's trepidation model.

72 R. Puig

it may in fact have had some tables in its origina l version2, The text offers ca lculati ng procedures and g ives geometrical proofs for the rules proposed .

Ibn al-I-Hi'im describes the astronomy practised in al-Andalus and the Maghrib al the beginning of the Ihinecmh century and informs us of the activities of the Andalusian astronomer Ibn al-Za rqalluh (died 11(0) and the ToJedan astronomers (al:Jamtla al-!ulay!iiliyya) who worked under the patronage of qaqI ~atid in the eleventh century.

In this paper I shall deal with the theory of the Moon in the zij, which is of considerable hislOrical interest. The ZI} deals with two aspects of the theory of the Moon: the computation of its longitude, and the compUlalion of its latitude It does nOt contain specific chapters on eclipses, the visibility of the new moon or the parallax, which are typ ical of similar z!)es. The Arabic text of the chapters that I will comment on appears as Appendix I at the end of the paper .

Determination of the Moon's Longitude

This question is discussed in chapter 4 (fols. 36v-37r, pp. 72-73) and 9 (fols. 4Iv-43v, pp . 82-86) of the third book . In chapter 4, ent itled A" Delermilling lite Distance of the Lunar Position ill the Ecliptic from the "Beginning of Aries" alld "(he Vemal Equinox", the Moon's siderea l and tropical longitudes are computed by means of a set of tables; the difference between them is the amount of precession calculated according to the theory of trepidation. In chapter 9, entitled all Fillding the Variatiolls (ikhtilarat) oj the LUllar Epicycle Centre Due to the Displacement oj (lie Poim of Aligllmem (markaz al-mu~aQat)Jrom rhe Cellfre o/the Ecliptk we find Ibn al-Ha 'im's explanation of the theory set out in chapler 4 .

Ibn al-Ha'im's instructions for calcu lating the longitude of the Moon in chapter four are as follows: [I ) Find the mean longitude of the Moon, its mean anomaly, and the mean longitude of the Sun, all for the place and the time desired. [2] Then correct the mean longitude of the Moon by mUltiplying the si ne

l See M. Abdulrahman (l996a).

Suhayl I (2(0))

11le 111eory of the Mooll in rhe Al-Zlj al-Krmlil ft-I-Ttf(firm ... 73

of the distance between the Moon and the corrected solar apogee3 by 0 ;0,2404. [3J Then add this produclto the mean longitude if the Moon is between the solar apogee and the solar perigee, or subtract it from the mean longitude if the Moon is between the solar perigee and the apogee. [4J Subtract the mean longitude of the Sun from this corrected mean longitude of the Moon in order to obtain the elongation (1]) and hence the double elongation (211). Then, enter with the double elongation as argument in the Lunar Equation Tables. [5]-[8] The rest of the steps are standard and the only point of interest is the terminology used by Ibn al-Ha 'im: the equat ion of the cent re is ca ll ed

i"~iraf al-qli(' (A lmagesf C3: Equarion f or Mean to True Apogee); the interpolation function corresponds to the daqa'iq al-nisba (Aim. C6 : Sixtieths); the tcfdfl a/-bucd a/-aqrab is the difference between the epicyclic equation at the perigee and at the apogee (AIm. Cs: Increment in Epicyclic Equation); the equation of anomaly at the apogee is the taCdz1 al-~i~~a (Aim. C4 : Epicyclic Equation). The final result (equat ion of anomaly for a particular double elongation and true anomaly) is called al-Iacdil al-murakkab. We must add it to or subtract it from the mean lunar longitude 10 obtain the sidereal longitude.

This "corrected solar apogee" (al-awj al-mrladdal) is defined in the same manuscript, Maqtila II, Chapter 10, fol. 34v: "The mean motion of the apogee for the momCll[ and [he era we wish plus [he radix position for [he beginning of the era. The result will be the corrected position of the apogee on the ecliptic, that is the distance from the point of the Head of Aries for that moment". This means thai the corrected solar apogee is a sidereal apogee corrected with the apogee's own motion. See OJ . Toomer (1969) and (1987). See also 1. Sams6 and E. Millas (1994).

4 He is using 0;24' * R sin (Lm~A,) so, for (L",-A,) = 90 then , the correction will be 60 0;24' = 0;24 as stated in [26)

S See, for instance, OJ . Toomer (1 984), p. 238. On the Ptolemaic model, see O. Neugebauer ( 1969), Appendix I, pp. 191-207.

74 R. Puig

[9] In order to obtain the tropical long itude, we must add or subtract the amount of precession calculated with the trepidation tables .

T he most interesting thing in this chapte r is the correction of [2] and [3], which will modify the Moon's initial mean longitude obtained with the tabl es and consequently the double elongation as argument . We find Ibn al-Hii ' im's justification in chapter!)6:

Ibn al-Ha ' im begins ([10]-[23]) by explaining the computation of the lunar equation of the centre using Ptolemy's model. For that purpose, he begins by explaining the computation procedure ([IOH17]) and then adds a geometrical justification (tilla) of the fannula employed ([18]-[23]). There is nothing new in Ihis part of the text , but one should note that Ibn

al~H a' im is not copy ing the Almagest (V, 9). As in the rest of the book , he is providing a clear explanation of the methods used to calculate tables of equations: something which is implicit in Ptolemy 's work , and which our author wishes 10 develop.

The computation of the equation of the centre appears as follows: [IO~II] He calcu lates Sin 211 (al-jayb al-awwal, the first sine) and Cos 211 (al-jayb al~tMn{, [he second sine) [ 12] He then determines:

dJ = 10; 191' x Sin 21/160 (al~qif a/~awwal,the first side) f ~ 2 x 10; 19' x Cos 2~ / 60 (a/{ad/a, difference) [13~ 14] dz = V(49;4 12 d/) + !(alqif a/~IJuj"r, the second side)

b = v(49;4 12 ~ d,z) + f/2 (bifd al~markaz al-mar'ft the distance between the centre of the lunar epicycle and the centre of the ecl iptic) [1 5J He next gives an alternative approx imate procedure to calculate b. He enters the table of the interpolation function (daqtJ'iq lIisbat al-khiirij af-markaz) with the value for the double elongation, obtains a value m, and establishes a proportion:

x / 20;38 ~ m / 60 where x ~ 60 - b - fo r 211 = 0 , then m = 0, x = 0 and b = 60 - for 21/ = 180, then m = 60, x = 20;38 and b = 39;22 = R - e , e

6 A description and delai led analysis of this chapter is in R. Puig (1992), slill unpublished. See a fi rsl approximation 10 the subjecl in J.Sams6 (1992), pp. 218-219.

The Theory o/the Moon in (he AI-Zlj al-Ktltnilft-I-Td'lllim _.. 7S

being the eccentricity Finally. he determines [16] w = v(d/ + d/) (al-watar, the hypotenuse) and [l7J q = 5in" (60Iw x dl ), the equation of the centre, which he uses to calculate the true anomaly .

The-geometrical justification for the computation procedure begins in [1 8J which is described in fig. 1: ABG is the lunar deferent with centre E. o is the centre of the ecliptic and T the proslleusis point, diametrically opposite to E. Angle AOB is the double elongation (21/) . B is the centre of the epicycle HM .

A

E

T

G

Figure 1

[19] He joins Band E. B and 0 , Band T. TB and DB will intersect the epicycle at points H and M. The Moon will be at 5, which he joins with B. Point H will be the mean apogee of the epicycle (ai-burd ai-abcad al-wasa(i), while M will be the true apogee (ai-buCd al-abcad ai-mar 'i). [20] < TBO (=

76 R. Puig

between < HBS (mean anomaly) and < MBS (true anomaly). [211 < HBM (equation of the centre) is known. From points E and T he traces EZ and TL, perpendicular to DB, which they intersect al points Z and L. T riang les DEZ and DTL are congruent (the text says shabih) , for TO = DE, TL = EZ, and LD = DZ. In the triangle DEZ, < 0 is known (21/) and < Z is a right angle. TD = 10; 19P is the eccentricity.

This explains the two computations of [12] : TL = EZ = 10;19P x Sin 271 1 60 = d, LD = DZ = IO;19P X Cos 21/ 160

LZ~2x DZ ~f [22) EZB is a right angled triangle, < Z being the right angle. EZ and EB (deferent radius = 49;41 P) are known, as is side BZ . This imp lies that he is using Pythagoras' theorem, and we have:

BZ ~ v{49;41' -EZ') (as in [13]) and BL is also known, for

BL = BZ + ZL (which should be identified with dz) BD = BZ + ZLI2 (b ll'd al-markaz al-mar'f) (as in [1 4]) . The explanation is obvious although it is not given explicitly in [23],

where the text merely states that side TL and < LBT are known: IV ~ BT ~ v(BL' + LT') (as in [16]) < LBT ~ sin-'{LT/BT) ~ Sino' (60/BT X LT) (as in [17])

[24J-(26Jlbn al-Ha'im goes on to explain the aforementioned correct ion (see [21-(31) as an add ition wh ich fill s an empty space or corrects an error of the Ancients (al-ziyllda al-musradraka calii.-l-qlldallu'iY. He ascribes it to Ibn al-Zarqall uh , as a result of some 37 years of lunar observations. This is a new and extremely va luable reference to Ibn al-Zarqalluh's work on the Moon, which has not been preserved . Ibn aJ-Ha'im says that he found in Ibn al-Zarqalluh's own hand that the results of his observations of the

G. Saliba has pointed out to me that there is a . tradition for the terms mustadraka. istidrdk, etc. to mean "objections to Ptolemy and his discovery of a reference to the earliest Andalusian criticism of Ptolemaic astronomy (eleventh century) in a lost anonymous work entitled al-Istidrdk ~ali't Ba{lt1mYlls. gives valuable support to this theory. See G. Saliba (1994). p. 83, and (1999).

StWyl I (1Xl)

The Theory oj Ilze Moon if! fhe A/-Z.} a/-KtimilJT-/-Ta'lilim . 77

Moon's motion in longitude, latitude, and variations (ikllli/lifiit) during the eclipses were in abso lute ag reement with the Ancient astronomers except fo r the mean motion in longitude. In his own observations of eclipses Ibn al-Zarqalluh found the Moon in the middle of them, either delayed or advanced with respect to its mean motion obtained from the tables. The maximum amount of the variat ion was about 24 minutes at 90 from the Sun's apogee. 1271 The conclusion was that the Moon's mean motion, as tabulated by the Ancient astronomers, was not around the centre of the Earth, as they thought, but around another centre displaced from it on the Sun's apsidal line in the direction of the apogee.

rbn al-Ha'im gives us a figure (Fig. 2)8 to show this: [28] Let us consider circle ABD with centre L as the deferent circle and diameter BO as the deferent's apsidalline. E is the centre of the Ecliptic. The figure corresponds to the situation in [mean] conjunctions and oppositions in which E is placed between Land 0 (lunar perigee) so that the geocemric distance of the centre of the lunar epicycle (B) is R +e. The text adds a confusing remark according to wh ich points Land D "are placed on this side of the figure, I mean between the apogee and the perigee of the Sun, in the direction of the signs. This is point E". The sentence apparently describes the side of the figure corresponding to points E and D. f29] Diameter AG is the Sun's apsidalline, A being on the direct ion of the apogee. [30] B is the centre of the epicycle of the Moon. Angle AEB corresponds to the distance observed between the cent re of the ep icycle of the Moon and the so lar apogee. [B is the Moon's -mean position at conjunction or opposition and at 90 from the Sun's apogee]. [3 1] According to observation, B is delayed with respect to its pOSition obtained from the tables. So we will look for another centre Z displaced from E in the direction of the solar apogee. The new angle AZB will

! In order to make clearer the relationshIp between figure t and figure 2, I have rotated figure 2 and put letters in brackets which correspond with telters in fig. I.

78 R. Puig

correspond to the mean distance between the centre of the epicycle of the Moon and the solar apogee obtained from the tables.

B_,

L G A

DO' Figure 2

[32] Angle ZBE will be the maximum difference, which reaches about 0;24 for this position of B. [From this max imum value we will find a partial value which will correspond to any possible posit ion of the centre of the epicycle]: [33] T is now the cenlre of the epicycle. Ibn al-Ha'imjoins T and Z, T and E with lines ZT and ET. In this case, as before, angle AET corresponds to the mean observed distance between the centre of the lunar epicycle and the solar apogee, while angle AZT is (he mean distance obtained from the tables. The end of the paragraph (almost two lines) is difficult to reconstruct due to the blanks in the manuscript. [34]-[35] He drops perpendicular EQ from E to the prolongation of ZT. Angles Band T, being small enough , are considered by Ibn al-Ha' im as equivalent [0 segments EZ and EQ subtended by these two angles (01-khu{u! al-mustaqfma a/-muwouora bihd).

SoWoyl 1 (2000)

The n,eory oj Ihe Moon in the Al-Zl} al-Kt1miljr-I-Td(J{[m ... 79

[36] He solves right angled triangle EQZ in which Q is the right angle, EZ is approximately equivalent to the 24' of the max imum variat ion and Z equals ang le AZT, the mean lunar longitude measured from the solar apogee as obtained from the tables. Then,

< T = QE = EZ/60 x Sin < Z = 0;24'/60 x Sin < AZT

Therefore , the final fo rmula (see above [2] and [3]) to correct the mean longitude obtained with the tables is :

where L'm is the corrected mean longitude , Lm is the mean longitude obtained with the tables, and

Lm~As is the distance between the mean position of the Moon obtained with the tables and the corrected solar apogee.

We find the same correction in zljes of later Andalusian and Maghribf astronomers such as:

Al-Zij al-Muqrabas by Ibn al-Kammad (fl . Cordova 12th century)9. His procedure fo r calculating the longitude of the Moon is Ptolemaic, and he uses the correction in the case of the calcu lation of eclipses. In fact, the canons of Ibn al-Kammad state the following rule: ~ Quod sf volueris locum Lune vemm pro eclipsi minue augem Solis a cemro et quod remanseritJac illud cordam er multipliea eam in duobus quintis lmillS minuri et duo quinri unius min uti sum 24 secunda et divide quod i!lde colligetur per 60". Ibn al-Kamrnad 's words deserve some comments . First, the use of the term corda for the sine, following al-BattanL This use is confirmed in chapter 17 of the canons in which he explains how to use the table of cords, cords of the complement (= cosine), and sagiuae (versed sines). Second: there seems to be an error in his formulation for , accord ing to him, the correction is:

9 I have used the Latin manuscript 10023 of Biblioteca Nacional de Madrid , Chapter (pona) 13. See a general commemary on the tables of this;,O in J. ChabAs and B.R. Goldste in (1994) .

80 R. Puig

(0;0,24 X 60 sin (L.-AJ) / 60

The division by 60 does not seem !O have much sense and il is probably the result of a confusion with a formula like:

(0;24 x 60 sin (Lm-A, / 60

The zfj of Ibn Ist)aq al-Tunisi (fl. Tunis and Marrakesh ca. 1193-1222), as preserved in the Hyderabad manuscript Andra Pradesh State Library 298 1, and the Al-Minhtij al-{tJlib fi tifdrl al-kawttkib, by Ibn al-Banna' al-Marrakushi (1256-\321 )11. They add 10 their procedure fo r calcul ati ng the long itude of the Moon another correction which appeared in Eastern Islamic astronomy in the ninth cent ury: as Ihey obtain the longitude of the Moon on its orbital plane, they must ~ reduce" it to the ecl iptic by means of a table ca lled Jadwa/ ((fdrt al-falak a/-mti'iI by Ibn I s~aq and Jadwa/ tcfdfl falak al -qamar al-ml1'U by Ibn al -Banna'. The manuscript of Ibn Isl.laq comains (Wo versions of (he same tablell which we also find in the zt) of Ibn al-Banna '(MS Escoria l 909, fol. 25v) beside the lunar latitude (able , with a maximum ofY. In both authors, (he arguments are comprised between 00 and 1800 with a maximum of 0;6,39" fo r argument 45. We can find the same table in (he ast ronomical handbook known as Mum/a~all ilJ (MS Escoria t Arabic 927 , fols. 20v-23r) , in the C, of the lunar equation table of Yal.lya ibn Abi Man~Or (fl. ca. 830) calcul ated for a max imum

10 l owe this information to Angel Meslres who is preparing the edition of this zrj. To date, he has published a delailed survey of the contents of Ihe manuscript in Mcstres (1996).

11 The canons oflhis zl} were ediled by 1. Vernet (1952) . Chapters on the Moon are on pp. 3 1-32 (Arabic lex!) ~nd 8789 (Spanish translalion). On the computation of planetary longirudes in the V}, including the Moon, see a recent paper by J. Sams6 and E. Millas (1998).

11 See Mestres (1996), p. 415 .

~~I ] (2000)

The Theory o/fhe Moon ill the AI-ZIj al-Kami{/f-I-Ttf(1(fm . 81

inclination of the lunar orbit of 4;46013 , The "reduction to the ecl iptic" as a correction to the Ptolemaic longi mde is pointed ou t by Pedersenl4 who has calculated a maximum correction of about 0;70 fo r a longitude of 45;3 and for a maximum latitude of SO.

In addition to this adjustment, both Ibn IsJ:taq and Ibn al-Banna' recommend using the correction explained by Ibn al-Ha'im when a high degree of accuracy is needed , i.e. for the calculat ion of eclipses and new moon . Ibn al-Banna' also incl udes lhe computed values of the second correct ion in a table called Table o/rhe ~arf~.

AI-il] al-ShtimilJr whdhfb al-Kamif by Ibn al- Raqqam (1245-1315). He follows Ibn al-Ha'im word for word but he does not include the geometrica l just ifications; nor does he mention Ibn al-Zarqalluh I6 .

Finall y, the recent publication by Mancha17 of a Provencal version of the tables for eclipses of Levi ben Gerson (1288- 1344) shows a possible introduct ion of the Za rqalJi an correction in the work of the Jewish astronomer: Lev i says that in lunar conjunctions and oppositions one fi nds a correction (divercitat) in the lunar equation (the computat ion of luna r longitude ?) which could reach an amount of up to 29 minutes . I wonder whether these 29 minutes are the result of a copy ing error and they correspond (0 the 24 minutes of Ibn al-Zarqalluh or whether Levi ben Gerson made a new est imation of the maximum vallie of the ZarqlHlian correct ion .

I) See H. Salam and E.S. Kennedy (1967), reprinted in E.S. Kennedy (1983), pp. 109-113.

It See O. Pedersen (1974). p. 200. IS The table of $aif(MS Escorial909. fo!. 25v) is only described. not edited, in a

footnote by 1. Vernet (1952). p. 88, note 177. 16 I have used the MS Istanbul Kandilli 249, fo l. l4r (Chapter 28) and foJ. lSv

(Chapter 33), as well as the partial edition of this vJ presented by Mu~ammad "Abel a l -Ra~man as a doctoral dissertation (l996b).

11 See Mancha (1998) .

82 R. Puig

Detennination of the Moon's Latitude

As in the case of longitude, this quest ion is discussed in two chapters. Chapter 2 of the fourth book (fa Is. 50r~50v . pp. 99- 100) and chapler 6 of the seventh book (fols. 84r-84v, pp. 165- 166). tn chapter 2 of the fourth book, ent itled On Determining the Lunar Lmirude 10 the South and North of the Ecliptic, rlie Argument of Latitude (~i~~a l al_Car~) and its Verification. we have Ibn aJ -Ha'im's instructions for us ing a Table of {he Differences of the Arguments of Latitude (JadwaJ tara~ul ~i~a~ carc.! al-qamar) which does nOI appear in the standard layout of the Ptolemaic lunar tables . In chapter 6 of the seventh book, entitled 011 Determining rhe Inclinations a/the Degrees o/the Lunar Orbit (muyUl ajza ' falak alqamar) from the EcliptiC and the Differences of the Argumems of Latitude, we find the general formula for the computat ion of a table of latitude and another one for the aforementioned differences.

In Chapter 2 of the fourth book, the instructions to calculate the latitude are as follows: [37] Take (he distance between the Moon's true position (mawtfl alqamar almllqawwam) and the neare r of the two nodes. Call it the first argument

(al-lJi~~a al-Ula) or the difference (al jaqla). [38J Emer with it in the Table of the Differences of the Argumellis of Latitude, tabulated in minutes and seconds, and always add them to the first argument in order to obtain the true argument of latitude Vli~~at al-Carq a/-~/Qqfqiyya). [As the argument used is true argument of latitude, these tables must be calculated from 0" to 9O"J. [39] Enter with it in the Lunar Latitude Tab/e to obtain the latitude. [40] and (41) We must take into account four possible positions of the Moon with respect to both the ascending and descending nodes according to the distance obtained in [37] , in order to determine if the latitude is northern or southern and if the Moon is ascendi ng or descending on its orbit. [42] Ibn al-Ha ' im gives a figure (fig . 3) to just ify all this : Circle AG O is the lunar o rbit (aljalak a/-nul 'if) and circle ABO is the parecliptic (a/ja/ok al-muwafiq) . A is the ascending node and 0 the descending one. G and Z

Somayl I (2IXKl)

Th~ Theory of(h~ Moon in (h~ Af-Zij af-Ktlmifjr-I-Ttftlffm ... 83

are , respectively, the northern and southern positions of the Moon on its orbital plane.

A

B

G

o

Figure 3

[43J We drop perpendiculars ZL and GS on the plane of the ecliptic: Sand L will be the positions of the Moon in longitude. [44] Sand L being right angles. arcs AZ and AG, on the orbital plane, are longer than the corresponding arcs AL and AB, on the ecliptic . This produces a difference, which. according to [bn al -Ha ' im, will reach a maximum of about 0 ;7. He then explains that the value of the latitude in the tables is given as a funct ion of the argument of latitude measured on the Moon's orbit. Thus , latitude ZL corresponds to arc AZ . But , if we enter the latirude table with the arc AL of the ecliptic as argument instead of arc AZ, the latitude corresponding to AL will be approximate. [451 At the limits of ec lipses Ibn al-Ha' im gives a difference between both

Suhayl I (2lKXl)

84 R. Puig

arcs of about 0;311, The end of the paragraph is difficu lt to understand due to the blanks in the manuscript , but there is a reference to Ibn a l ~ Ha'im's treatment of eclipses in another part of his work, which I have been unable to find in the extant manuscripl. [46] Finally, he repealS the four lunar positions detailed in [40} and [41] .

Chapter 6 of the seventh book explains the problem in more detail: [47] First, Ibn al-Hii'im says, following Ptolemy, that to calculate the latitude we muSt proceed as in the case of declinations, but he has a number of remarks to make. [48] Then. he explains how to calcu late the latitude. We obta in the lunar distance from the node, measured on the lunar orbit (al-fa/ak ai-nul 'it): this will be the true argument of iatimde (~i-Hat al-rarc! al-~aqrqiyya, AG in fig . 3). The formula is standa rd :

{3 = Sin ., (Sin AG x Sin 5) I 60

He is using the Ptolemaic parameter SO for the maximum inclination of the lunar orbit. (49] He then gives us (he formula to find the differences . He calculates AB which he calls ~i~~at al-(a/ (a rgument of longilUde):

AS = sin" (sin AG cos 5 I cos (3)

and the difference (fae//) will be:

AG - AB

(50) Ibn al-Ha'im gives a figure (fig. 4) saying that AG is the true argument of latitude, measured on the lunar orbit, and AS the argument of longitude, measured on the ecliptic. SG is the latitude of point G.

II This figu re is correct according to the Ptolemaic values of 5" for the maximum inclination and 15:12~ for a lunar eclipse limit (Almagest VI, 5, see GJ . Toomer (1984) , p. 287 and O. Pedersen (1974), p. 230).

Suha)'l I (lOX)

Thl! Thl!ory of Ihl! Moon ill Ihl! AI-ZIj of-KlJmil jr-f-TtftJlfm . 85

G

Figure 4

[5 1] In the figure he justifies the formula explained in [48], applying the sine law to spherical triangle ABG:

sin BG I sin < GAB = sin AG I sin 9

86 R. Puig

The conclusion we may draw from Ibn al*Hii ' im's treatment of lunar latitude is that it contains important informat ion about his determination of the lunar longitude. It seems clear that he interprets his corrected lunar position (maw4l: al-qamar al-11Iuqawwam) as corresponding to a lunar longilUde measured on the ecliptic (AB in fig . 4) and not on its own orbit (AG) , in spite of the fact that the instructions given by the amhor to calcuJate longitudes do not include any reference to the "reduction to the ecliptic" and that the Zarqiillian co rrection , which is a function of the mean lunar longitude and of the longitude of the solar apogee, has nOlhing to do with Ihis . As the argument for lunar latitude is not AB but AG, Ibn a1-Hii'im uses his table of differences in order to obtain an amoum d, and [hen AS + d = AG, which he uses to obtain the lunar latitude. The procedure is related to, although to some extent in conn ict with , the tradition of the Mllmta~lOli ilj, preserved by Ibn l s~aq and Ibn al-BaIIDa': there , Ya~ya ibn AbT Ma~ijr considers that, when he calculates the lunar longirude using his lunar equation tables derived from the Halldy Tables, he obtains AG which he uses directly in order to obtain the lunar latitude. On the other hand , to compute the longitude correctly, he must obtain AB and, for that purpose, he uses his own table of differences mentioned above . Since his table is calculated for arguments between 1 and 1800, the correction can be positive or negative.

Ibn al- Ha' im's nOll-extant table uses the Ptolemaic 5 for (jmn and calculates his entr ies for arguments between 1 and 900: as he subtracts the lunar position from the nearer node (and not from the ascending node as

Ya~ya does) , his corrections are always positive . A table probably similar to that used by Ibn al-Ha'im is preserved by Ibn al-Raqqam in his SIlilmil il)-1O , which is also computed for arguments between 1 and 90" and for

(j~x = SO (see Appendix 2) .

Concluding Remarks

Ibn al-Ha'im proposes two corrections to the standard Ptolemaic theory of the Moon. The first is an attempt to correct the theory of lunar longitude .

HI E.S. Kennedy (1997).

~yI 1(2000)

TIle Theory of lhe Moon in lhe Ai-Zfj a/-Klimii /f-I-Ta'lUim ... 87

The correction is ascribed to a lost astronomical work of Ibn al-Zarqalluh which Ibn al-l-Ia'im had read in an autograph manuscript wriuen by the Toledan astronomer himself. The mod ification consists of correcting the Moon's mean 10ngilUde and, consequently, the double elongat ion, used as initial arguments to enter the tables. Ibn al-Ha'im imerprets this correction as a result of the displacement of the centre of the lunar mean motion to a point on the straight line joining the cemre of the Universe and the solar apogee, and at a distance of O;24P from the cemre of the Universe. This seems to imply the existence of a lunar equam point which will rotate with the mot ion of the solar apogee. We do not know to what extent Ihis generalization of the correction of the Ptolemaic lunar model is due to Ibn al-Zarqalluh himself or is the result of Ibn al-IH ' im's interpretation of Ibn al-Zarqalluh's work. Whatever the case, this model met with some success, for we find the same correction in later Andaiusl and MaghribI zljes although restricted to the calculation of eclipses and new moon.

The second is a peculiar correction to the computation of (he lunar latitude lhat has a direct relation with a standard practice of Muslim astronomers since the MumtafJan Zij in the calcu lation of longitudes. However, it implies a change of altitude in their respect: Ibn al-Hifim believes that his lunar model gives ecliptic longitudes and that, consequently, YaJ:tya's reduction to the ecliptic is unnecessary fo r the computation of long itudes and that an inverse reduction to the lunar orb it has to be operated for calculating latitudes . Ibn al- Ha'im's final result , however, in the calculation of latitudes is necessarily different from Ptolemy's, for he has introduced a double correction (in the "argument of longitude" and in the reduction to the lunar orbit) . lt is also obviously different from the result obtained by YaJ:tya b. AbI Man~i.lr and his followers.

88 R. Puig

APPENDIX I

The manuscript is badly damaged and the last lines of each page are vcry difficult 10 read . I have used square brackets to mark the holes and the words added or reconstructed. In most oflhe cases, [0 reconstruct the lex!. the vi of Ibn al -Raqqam (d.131S) has been useful since he follows Ibn al -Ha'im textually in some passages. J have also used brackets 10 indicate the MS page number as well as sequential paragraph numbers which are not pari of the original lex!. This paragraph numbering is referred 10 in my English commentary . A few orthographic and morphological corrections have been made without comment.

'JSJ L."i [..,....:JI] .h..JJ =J.-'-":;JI.h..J (~U djj ""') Iji [I] _I" ~ L.. ~ ~y (,f:JI .ll,J1J w:i..,lJ d!J

,-"-,",,,IJ ~ ~...-'" W .,...;JI J...J U-- J.u..J1 ..,....:JI () .b:i...1 'r' [2] .oI.....b..i..::u~, W 4+>~u.J~J t":') ..!l..l~J 4o.:"J ~ ~ ~ L. ':'; ~I.,:JI ~ u;" ;, , IIJ (J~I ~.,...;JI ulS U>' -,""I 'r' [3] ~I.,:JI .j)l;. ~ (J~IJ u;', ;" II ~ .,...;JI ulS uiJ .,...;JI .h..J ~

.~u J.u..J1 J...~I -*'...-"') ~I W """" ~U .a..J1 . - 1 w......."u - W..,....:JI.h.. """" -- 1 [4]

. .,...~ ,..ru J ~r" . ~I .,...;JI J..>.>.O JJ-"> ~ .., ~U ...ll La.J1 ,":-,~'il .u..,Jl [.....i\~\) ~~u..l~.J..rb1Jl .....il~1 rY>.u~ l...o i.:...,J. [5]

.dJj ~IJ ?,.JJU l...o ~ ~U.J\ J.lLJ ~ .....iIl~"Jl .lj-i (J.>:' ~ u-o "Jji ~L.;,...Jl -Lt...,J1 uLS ul-i.;-lOu1 of> [6] W ~\ u-o .....jl~)'\ ~l...i (J..r.' ~ u-o JiSi 0LS ub ~1 JJ~I ~ 4-, ~u .. i.u..J1 ~I ....... ~I U-- ...-'" ) ~I [L...] ~I [J..>.>.O u--] J ,-,,-"~I .a..J1 J..>.>.O U-- 4-I~ L. :c.J J.,s:W1

.dJj ..l.>.O ~ [J,.>WI 'jJ ,-,,-"~I] .a..J1 J..>.>.O ~ ~I ,-".>-"1 'r' [7]

' [J

171e Theory of the Moon in the Ai-ll] al-Ktimiijr-i-Ttftili"m ... 89

J,.uOJ1 Ji' [J-" ~ .:,. .,osi ' .. (.u...J1 ~I w:.lS U>, pi ",.,. [8] ~L.; [J-" ~.:,. "j>i w:.lS ulJ [.,...o..ll] J.u...J1 .L.,JI ~ .,D.,..JI .a.,. ~ dl~ .a.,. ~ ) ~I WI J.a....JI ~, .h....J U-

90 R. Puig

jS.,..J1 EoWI

11Ie 11Ieory of the Moon in the A!-Zlj ai-Komi! jr-I-Ttfd/im ... 91

u. J'.J' ,.,s:.:. L. ~ -oU)I.o.;.~1 ,'-" u-- oL..,;,J1 ~ '-SJ.,:.....JI ',,,jll L.:.iJ [24] . ~L.:; UJI J.p,., -'-'-!

-oL,.,jll ~ ..... J-" I...::.! JI,,.JI ..... '",.,.11 ~I .b...J 'J d]'J [25] (J.,.ul dli pyo ~ ~I ..... ~~I '-S~I .; ~! d],~, ,.,...aJ L., ~ wLi.,.....sJl .kL..,J1 0-' ~ .;-a-A-Jl .l.....J 1.1,>1 ~ uLS ......:.1 d.!j,J [26J ~! ~I () u-- ipL.. ~I ul$ ~ L.:.i J....-lIJ ~jll " ;"", L. L.:.i J ~lyJl "';)1..;. ~! JJI.w1 u-- (~I " i.r>-w I, ;" ;"

o '1

The Theory of the Moon in the Al-Zlj al-Klimiljr-I-Ttfl1/fm ,. . 93

~ i...l..:>.IJ .0.:9..10..0 i~ ~ wj .,t . .,...!1 ~JI .; 0-=-:'J ~L,J\ ~JI.; V"YJ 4 ,,",,~I ~ L:.) ~ ~:L_n ......j)L...:,..)'\ uh tSl.; ~lA 0-> ~i ~lA ......j\.j -u)tJ [35] i...l..:>.IJ .0.:9..1 ~ 2-jii _)'J t-4=)' 4-i 4 t..:.hl l~! ~LbJ\ ~JI.; V"~ 0-=-:'J ..L...c:..u ~ ~~J ~I 0-=-:' ......j)L...:,..),1 u,.... ~)~ liA uL.$ L.J "'"""~ Jo'..IL.:a...... ~ i:J, ~?,JI ,:..u~)'1 ~iA ~J\'; Jo'..l~ "" ~ .)..J..iI 'v=-J\

.,",,~L, 4J ,';:;yJl '.,,-.. 1\ .b~1 "" ul~ .,i U~JL...... ......j\.j ~lAJ 1$\'; ~LA ~ 0-> iJ! ..l..:>.IJ -jS..i [36] 4.i.o ~1.;J1 ~Jlj ~lj......j!.J ~LA c..:.t....., -u-j J .. LbJ1 J ~L,JI ~J\j 0..0.u o';:'.,...Jl '-" ..? [(J 51 J 50)1.,"-.,JI 4 OJL;,...JI] -;2-; ? oJ 0 ~I ~ J c:: J.r.JI dU GJ.:.., u.c- '" y..,.JI J J~I

o.i'..iA.. ,-;-,.;:ii.., r-yuJ\ .~I ~Y' 0-=-:' l!i.I\ ~I h.i dJ~.:...I) lJ! [37] ~I ~ uLS W l.,Jt.:;) 4J yUJl uLS L. .ili.o [.l;o . ] 11 \I u.,..J1

l E

c l' ~ t 1 ~. '::, ~'i.. ';; tf 'i ft: 1 lA ' f 1, 't ~ t:- -c." 'i.. 'i.. r t- l ~:~ ~t r::: [. ~ 11 't r~ t 1:' t q= t ~ ~ [r't c ' - 'i, Ii" .:..., 1 ~ 1 \ f ll'l l'~' E {f,;;; b C=. { 1,; ,"' \d: 1 ~,'i:!~, 1 b f, ~ 1 ,,-' - ';; c,=

171e 1heory 0/ /he Moon in /he Ai-Zij af-Ktimil ji-f-Trft1irm . 95

yUJl "'-li ol:r.J J""" U->,-- '" [(.lO 84-J 84) 6 ",L,JI 7 -,-" WJ) Ii! [48] w 1-1,>i ~0.1,j d~i F~.J ;;.1L...l1 U-o i.1.o-:>J:.Wl d1.iJ1 ~ ~

~ ~ ",-:,..,....ou ~ .:s....s d. -.:;,.:;,.." 11_ u-'>.r--J1 ~ ~ I>jl~i ..:...:.LS [~w ~ ~ '--"u ~I W ~I -l...J1 "'J i~i .Ije-i ~ ~jL "'J ",~I -l...J1 ...... ""yJl .:.:.[5 W ",~I """~ ..... yo..;

. [J~I dli.J< ~I ~I ;;: -.:;,.:;,.." 11 ~I ~ '-:-'--,"",,=,U uo'>.r--J1 ~J..OL..;,;.)~WJ) 01-i [49] [~ W ~j>-ll ~w ~ ~ ~I L "'-:;IJ US -l...J1 ~w ~ '" c,. 4L--L J.,hll ~ ...... ""yJl .:.:.[5 W ",~I """~ ..... -y..; .L.llL,..1 ~u u.:-, - 0 .." II ~ L.. j....:....i .J-4-i ~ W ;;-': ;'1;"" II 4..:....o..:.J1

MsJl c,. ~ J r:"" ...:.Ji ~ j:.WI dJ.;J1 c,. ~ ~ .ill, '" LWI J [50] , u~ [J~I dli c,. .L,...:.Ji ",,->, ~ iJY"' .L i'+"- ",,->, F' "'J [J~I dli c,. .LlI .;-aJ d-_, ;, .: ;, ,> II ~1 ~.1 ;;.1L...ll U-o

' -ulu [5 1] ~ ~ 0~ ~L,....:aJi U"'~~.J 4..1..:>..! ~ ..1-,""",", "L,. ~ U"'~.1 ..:.J'11 [ ~] ~! i'+"- L, " ->' [~] o..,....:.s ; .;.Lul I::! J ~ ~! i'+"- ...:.Ji "->'

96 R. Puig

Th~ Theory of Ihe MOOfI in Ihe AI-ZlJ al-KtlmjJ p.l.rulllrm ... 97

APPENDIX II

Ibn al -Raqqam. Shtimil ZlJ. MS. Kandilli 249. rol. 73r Table of the difference between the two arguments

(Jadwal/041 mli bayn al-~i.r.ra/ayn)

[Entries are in minutes. The differences in seconds between Ibn al -Raqqii m's values and the recomputed values of the table appear in brackels)

,;"' ,+

I

I .t 2;:

-"" I

15 ;16 145 3; 16 16 ;27 146 34 +1 05

;39 14e 33 + 1 53(+1)

" ;5U ! 4H 31 : 7H I (+ I)

19 ;01 !49 29 179 28 2U ;12 50 27 180 15 21 ;23 24 (- I) 18 1 02 22 ;34 .+ 1 21 (- I) I H2 49 23 1;44 +: 53 IH 183 36 24 ;54 .+: 54 (+ I) I H4 23 + 1

" JJ +:

" IU H5 IU + 1

2O '2 +: )0 i .0 )0 + 1 ;2U + . UU , .. 42 ;2' .+;

'"-+ 1 , HH ,. (+ I)

';'0 "

4> +, : "'

14 (- I)

So.duyl I (2(0))

98 R. Puig

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