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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

Sep 29, 2015



Theory of the Moon in the Al-Zij Al-Kamil

  • 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 .





    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