J×bir b. Afla¬ on lunar eclipses1

José Bellver Key Words: Astronomy, Greek Astronomy, Medieval Astronomy, al-Andalus, J×bir b. Afla¬, Ptolemy, Almagest, I½l׬ al-MajisÐ÷, Eclipse Theory, Lunar Eclipses, Criticisms of Ptolemy’s Almagest Abstract In his most important work, the I½l׬ al-MajisÐ÷ or Improvement of the Almagest, the Andalusian mathematician and theoretical astronomer, J×bir b. Afla¬, presents a list of criticisms of Ptolemy’s Almagest, mainly of a mathematical nature. One of these is devoted to the computation of the magnitude and phases of lunar eclipses. Ptolemy uses plane trigonometry and some approximations that J×bir b. Afla¬ contests. Ptolemy obtains the magnitude and phases for two particular cases – when the Moon is at its apogee and when it is at its perigee – and computes a table of interpolation for any other lunar anomaly. J×bir b. Afla¬ avoids the need for tables of interpolation providing a slightly different method for computing the magnitude and phases of a lunar eclipse. In addition, he claims to have found an error in Ptolemy’s method of interpolation. However, thanks to aquotation of the Almagest appearing in the I½l׬ al-MajisÐ÷, we conclude that J×bir b. Afla¬’s criticism is due to the fact that there is a section

1 The present paper was prepared as part of the research programme “The evolution of Science

in al-Andalus’ society from the Early Middle Ages to the Pre-Renaissance”, sponsored by the Spanish Ministry of Science and Innovation (FFI2008-00234/FILO) and the European Regional Development Fund (ERDF).

Suhayl 8 (2008) pp. 47-91

48 J. Bellver Jabir b. Afla/:z on lunar eclipses 49

missing in his manuscript of the Almagest, and not to an error committedby Ptolemy, nor to a deficient translation of the Almagest.

Prolegomena

Jabir b. Aflal) al-IshbIlI/ known as Geber filius Afflay Hispalensis inmedieval Western Europe, was a mathematician and theoreticalastronomer who most probably flourished in Seville in the first quarter ofthe 12th century.3 Jabir b. Aflal) is a leading figure in medieval astronomythanks to his most important work, the I$lab al-Majis!f, which was

2 For a general introduction on Jabir b. Aflab, see R.P. Lorch [1975], "The Astronomy ofJabir b. Aflab", Centaurus, VoI. 19, pp. 85-107 - reprinted in R.P. Lorch [1995a],Arabic Mathematical Sciences: Instruments, Text, Transmission, Aldershot, VI - anabridgement of his doctoral thesis: Jabir ibn Afla/:z and his Influence in the West(Manchester, 1971); and forthcoming J. Bellver [2009],"Ellugar del 1o;la/:z al-MayistfdeYabir b. Aflab en la lIamada «rebeli6n andalusi contra la astronomia ptolemaica»", al­Qantara, VoI. 30, fasc. I (2009). Other papers by Lorch on the work of Jabir b. Aflab,are R.P. Lorch [1976], "The Astronomical Instruments of Jabir ibn Aflab and theTorquetum", Centaurus, VoI. 20, pp. 11-34 [reprinted in R.P. Lorch [1995a], XVI]; R.P.Lorch [1995c], "Jabir ibn Aflab and the Establishment of Trigonometry in the West" inLorch (1995a), VlII; R.P. Lorch [1995b], "The Manuscripts of Jabir's Treatise" in Lorch(1995a), VII; R.P. Lorch [2001], Thabit ibn Qurra, On the Sector-Figure and RelatedTexts. Edited with Translation and Commentary, Frankfurt am Main, pp. 387-90. Otherscholars have studied aspects of Jabir b. Aflab's work, such as N.M. Swerdlow [1987],"Jabir ibn Aflab's interesting method for finding the eccentricities and direction of theapsidalline of superior planets" in D.A. King and G. Saliba (eds.) [1987], From Deferentto Equant. A Volume of Studies in the History of Science in the Ancient and MedievalNear East in Honour ofE.S. Kennedy, New York, pp. 501-12; H. H. Hugonnard-Roche[1987], "La theorie astronomique selon Jabir ibn Aflab", in G. Swarup, A.K. Bag andK.S. Shukla (1987), History of Oriental Astronomy. Proceedings of an InternationalAstronomical Union Colloquium n° 91 (1985), Cambridge, pp. 207-8; J. Sams6 [2001],"Ibn al-Haytham and Jabir b. Aflab's Criticism of Ptolemy's Determination of theParameters of Mercury", Suhayl, VoI. 2 (2001), pp. 199-225 - reprinted in J. Sams6[2007], Astronomy and Astrology in al-Andalus and the Maghrib, Aldershot - Burlington,VII -; J. Bellver, "Jabir b. Aflab on the four-eclipse method for finding the lunar periodin anomaly", Suhayl, VoI. 6 (2006), pp. 159-248; 1. Bellver [2007], "Yabir b. Aflab entomo a la inclinaci6n de los eclipses en el horizonte", Archives Internationales d'Histoiredes Sciences, VoI. 57, Fasc. 158 (2007), pp. 3-25 and the forthcoming J. Bellver [2008­9], "Jabir b. Aflab on the lunar eccentricity and prosneusis at syzygies", Zeitschrift flirGeschichte del' Arabisch-Islamischen Wissenschaften.

3 See Lorch [1975], pp. 85-6.

translated into Latin and Hebrew.4 The I$lab al-Majis!f or Improvement ofthe Almagest is a rewriting of the Almagest in which he simplifies itsmathematical structure by avoiding Menelaus's theorem and ignoring itsmore practical elements. He also presents some criticisms of the Almagest,mainly of a mathematical nature.

Some of Jabir b. Aflal)'s criticisms are aimed at Ptolemy's procedurefor computing the magnitude and phases of eclipses. Mainly, thecriticisms are centred on certain statements by Ptolemy dealing with theproblem of parallax while considering solar eclipses. However, one of hiscriticisms refers to the fact that Ptolemy uses a table of interpolation toaccount for the effect of the lunar anomaly in both solar and lunareclipses. In order to avoid the use of this table, Jabir b. Aflal) provides anew method for computing eclipses which is slightly different fromPtolemy's. This paper describes this method and Jabir b. Aflal)'s criticismof this point.

1. Introduction

The aim of the method discussed is to compute the magnitude and phasesof both solar and lunar eclipses. In the case of solar eclipses, in addition tothe method provided, the lunar parallax must also be taken into account.So the method discussed, as it is, is suitable for lunar eclipses, while it isalso a part of a broader method for dealing with solar eclipses.

Ptolemy devotes chapters VI.7 to VI.9 of the Almagest to the study ofthe magnitude and the phases of both solar and lunar eclipses and presentssome tables for obtain these values.5

The magnitude of an eclipse Cm) is the number of digits of the diameterof the eclipsed body obscured at the mid-eclipse - a digit being the twelfthpart of a diameter. To compute the magnitude, the value of the immersion

4 In this research I have mainly considered the only three extant Arabic manuscripts inArabic script: Mss. Escorial 910 [henceforth referred to as Es1], Escorial 930 [henceforthreferred to as Es2

] and Berlin 5653 [henceforth referred to as 8.). In some sections of theIsla/:z al-Majistf, Mss. Escorial 910 and Berlin 5653 differ, while Ms. Escorial 930follows one or the other. On the Isla/:z al-Majistfs manuscripts, see Lorch, R.P. [I 995b).

5 See G.J. Toomer [1984], Ptolemy's Almagest, London [henceforth referred to as PtA],pp. 294-310; O. Pedersen [1974], A Survey of the Almagest, Odense, pp. 231-5; O.Neugebauer [1975], A History of Ancient Mathematical Astronomy, 3. vols., Berlin ­Heidelberg - New York [henceforth referred to as HAMA], pp. 134-9 and 1240.

50 J. Bellver Jiihir b. Ajlal; on lunar eclipses 51

(;1) is needed. This is the angular distance between the rim of the eclipsingbody and the point of the rim of the eclipsed body nearest the centre of theeclipsing body at the mid-eclipse.6 See Figure I for an example of thevalue of immersion.

Figure I. Example of immersion of the Moon

B'

Figure 2. Phases of a lunar eclipse

Knowing from (1.) thatfl = m r) / 6 ,the half of totality, GD, for a givenmagnitude, m, is

To compute the magnitude, when the immersion of the eclipsed body iscomplete, the value of immersion - fl - is equal to the diameter - d - ofthe eclipsed body, and thus the magnitude of the eclipse - m - must be 12digits. Therefore, the value of the magnitude is

and the minutes of immersion, BG, for a given magnitude is

BG=BD-GD

(2.)

(3.)

m=12fl /d=6fl /r (1.)provided that

where rs is the radius of a section of the shadow cone of the Earth and ris the radius of the Moon. )

Ptolemy provides five tables in Almagest VI.8 to compute themagnitude and phases of both solar and lunar eclipses. The design of thesetables is explained in Almagest VI.7. Two tables are devoted to solareclipses, and the other two to lunar eclipses. Ptolemy considers two tablesfor each kind of eclipse since he takes two cases into account: when theMoon is in its apogee and when it is in its perigee. These tables aretabulated using as argument the true argument of latitude, and provide thevalues of the magnitude and the duration of the phases. Once these valuesare obtained for both cases - that is, when the Moon is located in itsapogee and when it is in its perigee, Ptolemy uses a fifth table to obtain a

where d is the diameter of the eclipsed body and r its radius.As for the phases of the lunar eclipse, Ptolemy considers two values:

the minutes of immersion and the half of totality. The minutes ofimmersion correspond to the arc from the beginning of the eclipse to thebeginning of totality, while the half of totality corresponds to the arcbetween the beginning of totality and the middle of the eclipse. So let usconsider Figure 2, where point A is the centre of the shadow cone, andpoints Z, E, D, G and B refer respectively to the beginning of the eclipse,the beginning of totality, the middle of the eclipse, the end of totality andthe end of the eclipse; and let us consider that the phase of immersion isequal to the phase of emersion.

6 See HAMA p. 135 and pp. 1241-2.

BD = [ ( r s + r) )2 - (rs + ( I - m/6) r) )2 ] 1/2 (4.)

52 J. Bellver Jiihir b. Ajla/;l on lunar eclipses 53

{ [77;48°, 79;12°], [100;48°, 102;12°], [257;48°,259;12°],[280;48°,282;12°] }

In this case, Ptolemy points out that in (5.) and (6.) rn. and 1'/. must be zeroand therefore the values of the magnitude and phases under theseconditions are:?

Figure 3. Ms. Es 1 62v.

B

Previous elementsJiibir b. Aflal) points out some previous elements needed to solve themagnitude and phases of an eclipse. In fact, it is in the consideration ofthese previous elements that Jiibir b. Aflal) improves Ptolemy's method.

E

2. Jiibir b. Aflal}'s method

Jiibir b. Aflal) follows Ptolemy's method for computing the value of themagnitude and the duration of the phases closely, except for a few smalldifferences that he introduces. The main one is that he provides a way toaccount for the effect of the anomaly on the lunar radius before obtainingthe value of the magnitude and the duration of the phases, whereasPtolemy, as we have seen, accounts for it after obtaining these values.

when there is no entry in table III for the maximum lunar distance.

(7.)(8.)

(6.)

(5.)

rn = rnp q(a)1'/ = I'/p q(a)

rn = rn. + ( rnp - rn. ) q(a)

where q(u) is the coefficient of interpolation obtained with Table V.In addition, Ptolemy considers what happens when some arguments of

latitude appear in table IV, but not in table III; that is, there are somearguments of latitude in which, when the Moon is at its perigee, an eclipsecan take place, but not when it is at its apogee. The intervals of argumentsof latitude in this situation are the following:

where q(a) is the coefficient of interpolation obtained with Table V.Similarly, let us consider the duration of a phase of an eclipse - 1'/ .

Suppose that 1'/. and I'/p are the values of the duration of an eclipse whenthe Moon is at its apogee and at its perigee. The duration of the phase - 1'/- for a given anomaly - a - is

coefficient of interpolation - q(a) - between the two cases according tothe value of the lunar anomaly (a) which ranges from 0° to 180°.

So let us consider the magnitude of an eclipse. Suppose that rn. and rnp

are the magnitudes of an eclipse when the Moon is in its apogee and in itsperigee. The magnitude - rn - for a given anomaly - a -is

7 Ptolemy points out: "If, however, it happens that the argument oflatitude falls within therange of the second table only, we take [as final result] the appropriate fraction(determined by the number ofsixtieths found [from the correction tableVofthe digits andminutes [of travel] corresponding to the argument of latitude in the second table alone."In this quotation, the second table refers to the one computed when the Moon is at itsperigee. See PtA p. 306.

Firstly, Jiibir b. Aflal) endeavours to obtain the arc of great circle betweenthe centre of the Moon and the centre of the shadow cone - for lunareclipses - or the centre of the Sun - for solar eclipses. Jiibir b. Aflal) reliesupon Figure 3 where point B is the lunar node, circle BH is the eclipticand circle BZ is the lunar inclined orbit, point G is the centre of the Sun(or of the shapow cone) at the true syzygy, point A is the centre of the

54 J. Bellver Jiihir b. Ajlal) on lunar eclipses 55

where i is the angle between the lunar inclined orbit and the ecliptic. Toobtain GD, Jabir b. Aflab applies the rule of the four quantities as follows:

'.' AB known

Sin AB / Sin AG = Sin BZ / Sin ZH with Sin BZ = 60P (if we

consider that he is applyingthe rule of four quantities)

... LZ=LD=90°

'.' LEAZ=LGAD

. '. Sin AG / Sin GD = Sin AE / Sin ZE

'.' Sin AG, Sin AE and Sin ZE known since AE = 90° - GA and ZE

= 900 -HZ

Moon at the true syzygy, point D is the centre of the Moon at the mid­eclipse and point E is the pole of the ecliptic. In addition, arc GD isperpendicular to the lunar incline orbit - circle AB - arc AG isperpendicular to the ecliptic - circle BG - and arc AZ is perpendicular tocircle EZ while circle EZ passes through the pole of the ecliptic and thepole of the lunar inclined orbit. Hence, point Z is the point of the inclinedorbit with maximum latitude which can be found at 90° from the node andJabir b. Aflab calls it "maximum inclination of the inclined orbit" (nihiiyatmayl al-falak al-mii'il).

Jabir b. Aflab obtains arc GD - which is the arc of the great circlebetween the centre of the Moon and the centre of the Sun (or the shadowcone) at the mid-eclipse - in order to obtain the magnitude of the eclipse ­which is a sector of arc GD. Therefore, he needs to know arc AG - whichis the latitude of the Moon at the true syzygy. Ptolemy assumes that thevalue of the two is approximately equal since the difference between AGand GD amounts to two minutes. Jabir b. Aflab considers that thisdifference amounts to four minutes, and criticises Ptolemy for hisassumption. This minor criticism is dealt with in the next section of thispaper.

Jabir b. Aflab describes the procedure for obtaining arc AG as follows:

As for the value of arc AG - which is the lunar latitude at the truesyzygy -, it is known since arc AB - which is the lunar nodaldistance in the inclined orbit - is known. And the ratio of its sine tosine of arc AG is equal to the ratio of the radius to the sine of themaximum inclination of the inclined orbit (nihiiyat mayl al-falakal-mii'il).8

or

Sin AB / Sin AG = Sin LG / Sin LB

... AGknown

.'. Sin GD known

with Sin LG = 60P (if weconsider that he is applyingthe sine law)

with AG < 90° since Sin ZH

= Sin i

with GD < 90°

Jabir b. Aflab applies the rule of four quantities or the sine law, sinceboth, in this case, are equivalent. By radius, he means the radius of ageneric circle; i.e. 60P• Since Sin BZ = 60P, the sine of the maximuminclination of the inclined orbit - i.e. the maximum latitude of the inclinedorbit - corresponds to the sine of arc ZH. Therefore, the procedure is asfollows:

8 ef. inji'a p. 85.

This is the procedure for obtaining the distance between the centre of theMoon and the centre of the Sun (or the shadow cone) at mid-eclipse.

In order to ascertain the value of the difference between AG and GD,let us consider, according to Ptolemy, that the inclination of the lunarinclined orbit relative to the ecliptic (i = LABG) is 5° and that themaximum value of BA - the maximum nodal distance along the inclinedorbit in which an eclipse can take place - is 12°. We know that

tan BG = tan BA cos itan BD = tan BG cos i

56 1. Bellver Jiibir b. Ajlab on lunar eclipses 57

and thus

BG = 11;57,20° and BD = 11;54,41°

so the difference we are looking for is

DA = BA - BD = 0;5,19°

its apparent diameter at the apogee. Line TM corresponds to the apparentdiameter of the Moon at point G. Jiibir b. Aflab wants to know the valueof line TM.

Firstly, he endeavours to obtain the distance between the centres of theMoon and the Earth, knowing that the distance between the centre of theepicycle and the centre of the Earth is 60P and that the diameter of theepicycle is lOP. Jiibir b. Aflab's procedure is as follows:

This difference is slightly greater than the one indicated by Jiibir b. Aflab.9

Later, Jiibir b. Aflab continues by obtaining the diameter of the Moon andthe shadow cone of the Earth for any lunar anomaly. IQ To do so, heprovides two figures which are described in Figure 4. The right-handfigure shows a plane epicycle where circle ABG around centre D is thelunar epicycle, point E is the centre of the Earth, point A is the lunarapogee and point B its perigee, point G is the centre of the Moon, line GZis perpendicular to line ADBE and distance EG is equal to distance EH.Jiibir b. Aflab wants to obtain the diameter of the Moon and of the shadowcone for distance EG.

... AB = lOP and ED = 60P

'.' LADGknown

.'. Sin LADG known

·'. AZ = Vers ADG known

.'. EZ = EA -ZA known

.'. EGknown

sInce LADG IS the gIven

anomaly

T

L AQ--<:b-'----lo-----4>-----------::=O'-oE

M

K

Figure 4. Ms. Es 1 63r.

The left-hand figure shows a scale of the diameters where TK correspondsto the apparent diameter of the Moon at the perigee and TL corresponds to

9 See PtA p. 298 n. 55 and HAMA p. 83 n. 5.

IQ See PtA pp. 283-5 for Pto)emy's method to obtain the maximum radius of the Moon andof the shadow cone of the Earth, and PtA pp. 252-4 for his method to obtain theminimum radius of the Moon.

EG is known since, using a modem function,

LZEG = arctan (GZ / ZE)EG = EZ / cos ( LZEG )

And finally, Jiibir b. Aflab ends this first part of the resolution of the lunardiameter for any lunar anomaly indicating that:

·'. EH = EG known

Once he has found the lunar distance to the centre of the Earth, Jiibir b.Aflab continues with the resolution of the lunar diameter for any lunaranomaly. He bases his resolution upon the left-hand figure of Figure 4.which shows the proportion of the lunar diameters as related to the lunaranomaly. Jiibir b.-Aflab's procedure is as follows:

·.' Lunar diameters at the apogee - TL - and at the perigee - TK ­

known.

58 J. Bellver Jiihir b. Ajla/:z on lunar eclipses 59

'.' LK=TK-TLknown

'.' AB =AE-BEknown

'.' AH=AE-HE known

... AH / AB == ML / LK

.'. LMknown

.'. TM = TL + LM known

So in order to obtain the lunar diameter relative to the anomaly, Jiibir b.Aflah considers that the lunar diameter decreases linearly with distance.This 'allows him to establish a proportionality between, on the one hand,the maximum and minimum diameters pertaining to the perigee and theapogee and, on the other, the diameter relative to any other lunar anomaly.Jiibir b. Aflab seems well aware of the fact that the epicyclic hypothesisdoes not explain the actual diameter of the Moon at its perigee. Hence heonly considers the observed values of the lunar diameter when the Moonis at its apogee and its perigee, not the values computed with thehypothesis. Therefore he bases the aforementioned proportionality upondifferences in the lunar diameter when the Moon is at its apogee and itsperigee, thus avoiding the use of distances related to the centre of theEarth. This is the main difference between Jiibir b. Aflab' s method andPtolemy's.

Jiibir b. Aflab uses the proportion of the difference between the lunardiameter at the apogee and the perigee - line LK - and the differencebetween the lunar distance at the perigee and the apogee - line AB. In thisway, he obtains the difference in the lunar diameter for a given anomalyrelative to its diameter at the apogee - ML - from the difference betweenthe lunar distance for the given anomaly and the lunar distance at theapogee - AH. By this trick, he avoids using the difference between theobserved lunar diameter at the perigee and the computed one.

Finally, Jiibir b. Aflab points out, first, that the method for computingthe diameter of the shadow cone is the same as the one shown for thelunar diameter and, second, that the solar diameter, which is slightlyaffected by the parallax, is equal to the lunar diameter when the Moon isat its apogee.

On the magnitudes and phases of lunar eclipses

After considering the procedures for finding the elongation between thelunar and solar centres at the mid-eclipse and the diameters of the Moonand of the shadow cone for any anomaly, Jiibir b. Aflab investigates themagnitude of lunar eclipses and the duration of their phases according toFigure 5, where arc AB is the ecliptic, arc GB is the lunar inclined orbitand point A is the centre of the Sun at the solar eclipse and the centre ofthe shadow cone at the lunar eclipse.

B

Figure 5. Ms. Es1 63v.

Jiibir b. Aflab considers three significant points which determine thephases of solar eclipses - the beginning, middle and end of the eclipse ­and five significant points which determine the phases of lunar eclipses ­the previous three points plus the beginning of totality and the end oftotality. The common significant points for solar and lunar eclipses arepoint G, which is the lunar centre at the beginning of the eclipse, point D,which is the lunar centre at the mid-eclipse and point E, which is the lunarcentre at the end of the eclipse. In addition, point Z corresponds to thebeginning of totality, and point H to the end of totality.

Jiibir b. Aflab establishes the next set of conditions. Firstly, he pointsout that arc AG is equal to the sum of the radius of the Sun and Moon atthe solar eclipse, or to the sum of the radius of the Moon and the shadowcone at the lunar eclipse. Moreover, he also establishes that arc AD isperpendicular to arc BG. And finally, he points out that arc AZ is equal tothe radius of the s-hadow cone.

Additionally, he considers that the different phases of an eclipse aresymmetrical as related to the mid-eclipse - to put it with modemterminology. So arc DE is equal to arc GD, arc GZ is approximately equalto arc HE and arc ZD is approximately equal to arc DH.

60 J. Bellver Jiihir b. Ajla/:l on lunar eclipses 61

Once he has described this figure, he tries to obtain the magnitude of theeclipse. The procedure is as follows.

'.' The lunar latitude at the mid-eclipse -fJ(])- is known

.'. AD known.

... dj) + ds -AD known

... magnitude of the eclipse known

where dj) is the diameter of the Moon and ds is the diameter of the shadowcone. In order to obtain the magnitude of the eclipse, he considers thedistances obtained in the previous section. The lunar magnitude at the truesyzygy corresponds to side AG of Figure 3, while arc AD corresponds tothe arc between the centre of the Moon and the centre of the shadow conei.e. arc GD in the same figure. Next he subtracts arc AD from the sum ofthe diameters obtained in the previous section and points out that themagnitude is therefore known.

from the sum of the diameters as Jabir b. Aflal). suggests. 11 Maybe this

lack of thoroughness is due to the fact that he is indicating the procedureof resolution - though this would be out of character in his case - or itmay be a textual error. The Arabic text is as follows:

.U . (. \\ \.. .. .L~I\ , I" •y= L>-a '9 11"0 .J~ ~ 0:u"""..... t~ L>-a4 h911,i9

12La)- u y.o&J\

"And we subtract it from the sum of both diameters andtherefore the magnitude is known"

In addition, I cannot find textual variants between the differentmanuscripts. Nor does the text of the Latin translation by Gerard ofCremona published by Petrus Apianus differ from the Arabic manuscripts:

Proijciam ergo ipsum de aggregatione duarum diametrorum etremanebit quantitas eclipsantis de quantitate eclipsati nota. 13

In any case, the immersion must be:

After obtaining the eclipse magnitude, Jabir b. Aflal). considers, withoutfurther indications, the value of the arcs related to the phases of the

11 In fact, when he considers this point for solar eclipses, he points out that, to obtain themagnitude of the eclipse, the distance between the solar and lunar centres must besubtracted from the sum ofthe solar and lunar radius.

r.

___.1....- --+-<>-1>-- .1-1')

Figure 6. Value of the immersion from the radius of the Moon and the shadow cone

lI=r +r -ADr )l s

Once the immersion is obtained, the magnitude is as

m = 6 11 / r)l = 6 ( r j) + rs - AD ) / r j)

(9.)

(10.)

But from Figure 6, the value of immersion is obtained by subtracting arcAD from the sum of the radius of the Moon and of the shadow cone, not

12 er. infra p. 79.

13 Petrus Apianus, Instrumentum primi mobilis. Accedunt iis Gebri filii Affla HispalensisAstronomi vetustissimi pariter et peritissimi, libri IX de astronomia, ante aliquot seculaArabice scripti, et per Giriardum Cremonensem latinitate donatio nunc vera omniumprimum in lucem editi, Nuremberg, 1534, p. 78.

62 J. Bellver Jiibir b. Afla/.l on lunar eclipses 63

eclipse. Firstly, he obtains arc GD after stating that it is approximatelyequal to arc DE (see Figure 6). The procedure is as follows:

... AGknown

... AD known

... LD = 90°

... GD is known.

GD is actually known since

since AG = rJ> + rs

as demonstrated in the introductory section

Jabir b. Aflal;t's method for obtaining the magnitude of the eclipse isslightly different from Ptolemy's. First, he avoids considering that the arcbetween the centre of the shadow cone and the centre of the Moon at themid-eclipse is equal to the latitude of the Moon at the apparent syzygy. Healso uses spherical trigonometry throughout his method. But the maindifference is that Jabir b. Aflal;t considers the lunar anomaly beforecomputing the magnitude and phases of the eclipse, while Ptolemy uses aninterpolation table for any lunar anomalies, thus considering the lunaranomaly after computing the magnitude and phases of lunar eclipses whilethe Moon is at the perigee and the apogee.

GD = arccos [cos AG . cos AD]3. Jabir b. Aflal;t's criticisms of Ptolemy

Lastly, Jabir b. Aflal;t seeks to solve the arcs related to the intermediatephases: arc GZ, which is the arc between the initial eclipse and thebeginning of totality, and arc ZD, which is the arc between the beginningof totality and the mid-eclipse.

So Jabir b. Aflal;t continues to assume that ZD is know and that

ZD = arccos [ cos AZ . cos AD ]

... AZknown

... AD known

... ZDknown

t

since, as above, we can apply

... GDknown

... GZknown

since AZ = rs

as demonstrated in the introductory section

since GZ = GD - ZD.

Jabir b. Aflal;t criticises two points of Ptolemy's approach to lunareclipses. The first criticism centres upon Ptolemy's premises that heconsiders to be approximate. Jabir b. Aflal;t states:

As to the procedure mentioned by Ptolemy, it is an approximateprocedure from two points of view. Firstly, he used straight linesinstead of arcs. And secondly, he considered that the arcbetween the centres of the two bodies - the eclipsing andeclipsed bodies - at the middle of the eclipse was equal to thetrue latitude of the Moon at the true syzygy. 14

That is, Jabir b. Aflal;t points out that (i.) Ptolemy uses plane trigonometryand that (ii.) he considers that the difference between the position of theMoon at the mid-eclipse and its position at the apparent syzygy isnegligible. However, it must be remembered that Ptolemy was well awarethat his procedure was approximate.

Jabir b. Aflal;t corrects the value of the error due to the approximation,stating:

In short, Jabir b. Aflal;t's procedure for obtaining the arcs relative to thephases is similar to Ptolemy's when he considers the situations when theMoon is at the apogee and the perigee, although Jabir b. Aflal;t appliesspherical trigonometry and Ptolemy the theorem of Pythagoras.

14 ef. infra p. 89.

64 J. Bellver Jabir b. Ajla/:l on lunar eclipses 65

He [i.e. Ptolemy] considered that [the difference between thecentres under these conditions] was of two minutes. IS However,[this difference] is greater than this value. 16

Jabir b. Aflal) considers this difference to be 4 minutes. 17 The cause of thisdifference is not that Ptolemy bases his computation on apparent syzygieswhile Jabir b. Aflal) uses true ones since the parallax does not affect lunareclipses, but the fact that Jabir b. Aflal) is using spherical trigonometry.This difference has been computed in the previous section of this paper.

The second point Jabir b. Aflal) criticises involves some difficultiesderived from the use tables computed for the lunar apogee and perigee andthe need to interpolate the results for a given anomaly.

Jabir b. Aflal) describes first the method of Ptolemy for computing themagnitudes and phases of the lunar eclipse. He then describes how to usethe table of interpolation. 18 He quotes the Almagest:

Enter the value of the [argument in] latitude in both tables andtake what you find in front of both [tables] and write it downseparately. Enter the degrees of anomaly in the row of the[corresponding] value in the table of minutes and take the valueof minutes that you find in front of it. Take a value from thedifference of the values obtained from both tables such that itsproportion to [this difference] is equal to the [proportion of theminutes obtained relative to 60']. This [proportion] is added tothe value obtained from the first table. The result in digitscorresponds to the eclipsed section of the lunar diameter. 19

This quotation describes equations (5.) and (6.) where Ptolemysummarizes his method of interpolation.

15 cr. PtA p. 297-8.

16 Cr. infra p. 89.

17 Cr. infra p. 86.

18 See the Table of Correction in PtA p. 308.

19 Cr. infra p. 90, and also PtA pp. 305-6.

The quotation of the Almagest in the l~liib al-Majisti continues:

In case the value of the [argument of] latitude does not exist inthe first table, but only in the second, take the digits that youfind opposite it. This [value] corresponds to the magnitude of thelunar eclipse (miqdiir al-munkasifmin qutr al-qamar),z°

In the last quotation, the following situation is considered: when, for thesame lunar argument of latitude, the lunar eclipse occurs when the Moonis at its perigee, but not when it is at its apogee. From this quotation of theAlmagest, as it appears in the l~liib al-Majisti, whenever a lunar argumentof latitude is not found in the table for the maximum lunar distance, butonly in the table for the minimum lunar distance, the table of interpolationmust not be considered. In this situation, from this quotation as it appearsin the l~liib al-Majisti, equations (7.) and (8.), should be

(11.)(12.)

Hence, whenever a lunar eclipse occurs when the Moon is at its apogee,but not when it is at its perigee, the magnitude and phases of the eclipseare independent of the anomaly and should be considered as if the Moonwere at the perigee. Jabir b. Aflal)'s criticism follows the above quotation.He states:

If the lunar [argument in] latitude were 79°, we will not find [anentry] in the first table. However, in the second table in front ofthe [argument of latitude of 79°], we will find a value greaterthan two digits.

[But] we said that the eclipsed section of the Moon is greaterthan two digits. And this is only so when the Moon is at theperigee of its epicycle. If the Moon were at the apogee or next toit, in such a way that the addition of the two diameters was equalto the lunar latitude or less than it, no part of the Moon will beeclipsed. But we have stated that it is eclipsed more than twodigits. And his pretension that the eclipsed section of the Moon

20 Cr. infra p. 90.

66 J. Bellver

is greater than a sixth is extremely clumsy for it is not eclipsed atal1. 21

Jiibir b. Aflab on lunar eclipses

the above correction will give us the magnitude of theobscuration, in twelfths of the lunar diameter, at mid-eclipse.24

67

The argument of latitude of the first row of the table for the maximumlunar distance is 79; 12°. Therefore there is no entry for an argument oflatitude of 79°. Instead, in the second table, the argument of latitude78;56° corresponds to a magnitude mp = 2d

• So, according to the quotationof the Almagest found in the I$lab al-Majistf, equation (11.) can beapplied, since there is no corresponding argument of latitude in the tablefor the maximum lunar distance. In this case, since the magnitude does notdepend on the anomaly, when the Moon is located in anomalies next tothe apogee, the eclipsed magnitude is m = 2d

; however, it must be Od asJabir b. Aflab states.

After noting this fact, Jabir b. Aflab goes on to make a fierce criticism22

based upon two facts. First, he considers that Ptolemy is inconsistentsince, on the one hand, he points out imperceptible minutiae - as when heconsiders that the immersion and emersion times of an eclipse aredifferenf3

- and, on the other, he does not account for a fact that providesa difference of two digits in the magnitude of an eclipse. The second pointcriticised by Jabir b. Aflab is based the fact that Ptolemy made thecomputation of these functions excessively complicated.

In any case, the origin of the error pointed out by Jabir b. Aflab must beexamined. Firstly, the quotation of the Almagest in the I$lab al-Majistfdiffers from Toomer's version. He translates:

If, however, it happens that the argument of latitude falls withinthe range of the second table only, we take [as final result] theappropriate fraction (determined by the number of sixtiethsfound [from the correction table]) of the digits and minutes [oftravel] corresponding [to the argument of latitude] in the secondtable alone. The number of digits which we find as a result of

21 Cr. infra p. 91.

22 Ibidem.

23 See the forthcoming Bellver [2009] for a summary of his criticism of this point.

That is, Ptolemy considers equations (7.) and (8.) which I copy here:

m = mp q(o.)'1 = 'lp q(o.)

in which a correction as a function of the anomaly is taken into account,and not equations (11.) and (12.) which have been derived from thequotation of the Almagest as it appears in the I$lab al-Majistf. So Jabir b.Aflab's criticism seems to be unjustified since it derives from a possibletextual error in the manuscript of the Almagest he is using. Let us considerthis point.

The text ofIsbaq b. !:Iunayn's translation is as follows:

t.... ~.J o..l.:l..J ~W\ J.J~\ ~ u.:::.yJ\ ~~ ~ ui ~\ uJ.J

L.>-" I.ll ~y.. W J.i1j,,)j\.J tj'l......a'J\ L.>-" ~~\ .clb ~\j~ o..l.:l..J ~ ~.J:!

~~ u~ tj'l......a'J\ ~ ~~ 25{~} 0J \.ili f':ui::J\ \~ L.>-" tj'l......a'J\

. ...! ..~1\ .Lt; . ~\ \. . ..b:...J r' , .,\. \. i-.r~~l.J-'l '1i":' __ ~....?'-~~l.J-'l~....?'-

26u..,....sJ\ L.>-".b...,..,1\ ut....jl\

While in al-!:Iajjaj's translation we read:

24 Cr. PtA p. 306.

25 In the margin.

26 Version of Isbiiq b. Bunayn according to Ms. BN Paris Ar. 2482 f. 123v.

68 J. Bellver Jahir b. Aflab on lunar eclipses 69

.clb ~~ -. I lll! -~-:II \~. ~"il' U1 . oU.l:o. LaLl, ~Y""'" U'" Cl U'" <:..?- . J

<.JA ~\ ~~ \"..,?- y;.c. ~I <.JA "I..,?-i ~~ u~ CI~"i\

27uyo.sJ\ <.JA b....J"i1 uLajl t.} ~I .;hi

Finally, the text of the quotation which appears in the I$liib al-Majisfl isthe following:

t.} ~J:l cfi'iJ J}-jl JJ~I t.} ~J:l 'i ~yJI ~~ u1.S uJJ

9,,,S;,,11 )~ ~m U1.S! CI~"i\ <.JA -u.a Aj\j~ La i:..i o.l:o.J ~~\

28o.fi~ ""ill J,..a..ll ~ \~ .~\ .;hi <.JA

From the collation of the previous texts, it can be concluded that Jabir b.Aflab is not quoting literally any of the translations of the Almagestavailable to him.29 Instead, he is adapting the text or quoting a previouslyadapted text. He is clearly adapting the terminology, since the longperiphrasis

27 Version of al-I;Iajjaj according to Ms. British Museum Add. 7474 f. 176v.

28 Cf. infra p. 82.

29 For the quotation of Jabir b. Aflab's in which he states that he had consulted Isbaq b.I;Iunayn and al-l;Iajjaj's translations of the Almagest, see Lorch [1975], pp. 96-7,especially n. 61. See my forthcoming Bellver [2008, forthcoming], for a discussion ofJabir b. Aflab's criticism in which he makes this comment and also Bellver [2009,forthcoming] for a discussion ofhis degree of acquaintance with both translations.

which is almost identical in both translations is rendered in the I$liib al­Majisfl as miqdiir al-munkasifmin qufr al-qamar, which can be translatedas 'magnitude of the lunar eclipse'.

Secondly, from an analysis of the texts, it is possible that the version ofthe Almagest used by Jabir b. Aflab is, as usual, Isbaq b. I:Iunayn'stranslation, since he uses certain expressions and terms that only appear inthis version; i.e., in the I$liib al-Majisfl we find the termjadwal, which isused in Isbaq b. I:Iunayn's version, but not in al-I:Iajjaj, who uses fa$linstead. Another case is the expression bi-'izii' (opposite) which in al­I:Iajjaj appears as aUatl tuqiibilu maw<;li'a-hu.

Here is a translation ofIsbaq b. I:Iunayn's version:

And if it happens that the argument of latitude only appears inthe second table, the digits and minutes found will be placedopposite [the argument of latitude]. [As for] the digits obtainedafter this correction ifa-mii kharaja lanii min al-a$iibi' min hiidhiial-taqwlm), we said: the number of these digits corresponds tothe number of the parts relative to twelve of the lunar diameterobscured at the mid-eclipse.

A comparison of this version of the Almagest with that of Jabir b. Aflabshows clearly that the author of the I$liib al-Majisfl did not take intoaccount the sentence "[As to] the digits obtained after this correction" ifa­mii kharaja la-nii min al-a$iibi' min hiidhii l-taqwlm). From this sentence,it can be concluded that the digits corresponding to the magnitude of thelunar eclipse, when an argument of latitude is found in the table for thelunar perigee and not in the table for the lunar apogee, have beencorrected. In addition, from the context, it must be assumed that thecorrection is performed using the table of interpolation. In this particularcase, the sense of al-I:Iajjaj's version does not differ from that of Isbaq b.I:Iunayn. In conclusion, despite the fact that the text is not completelyclear, equations (7.) and (8.) fit better the description found in the ArabicverSIOns.

Given that Jabir b. Aflab usually reads the Almagest with great care, itis unlikely that he forgot to mention the word taqwlm (equation,correction). In addition, it does not make sense to base his fierce criticismof Ptolemy on a careless reading. Therefore, the most likely hypothesis isthat the words min hiidhii l-taqwlm did not appear in the manuscript of theAlmagest Jabir b. Aflab used when he wrote the I$liib al-Majisfl.

70 J. Bellver Jiihir b. Afla/:l on lunar eclipses 71

However, the fact that Jiibir b. Aflab's criticism is not justified does notimply that Ptolemy's procedure is completely correct, assuming thepresuppositions ofthe ancient astronomy.

We know from equation (5.) that the general function of interpolationfor eclipses according to Ptolemy is:

m = ma + (m p - ma) q(a)

Ptolemy applies m = mp q(a) whenever ma is not greater than zero.However this implies that for some anomalies the resulting magnitude willbe greater than zero when in fact it is below zero if we were to extend theline resulting from slope mp - ma to such cases in which ma is actuallynegative. To be consistent with the rest of cases in which, for a givenargument of latitude, there exist both mp and ma, it would be necessary toconsider negative magnitudes (ma < 0) in the following intervals of theargument of latitude { [77;48°, 79;12°], [100;48°, 102;12°], [257;48°,259; 12°], [280;48°, 282; 12°] } in which an eclipse can take place when theMoon is at its perigee, but not when it is at its apogee. In this case,equation (7.) will be as follows:

ma = 21;36d -ld/0;30° I w - 90° I with 77;48° < w < 102;12°ma = 21;36d -l d/0;30° I w - 270° Iwith 257;48° < w < 282;12° (13.)

where w is the argument oflatitude. Graphically,

rn,

21;36t--~~ 28212'77.'48° 102~12°Od , ,

00 ~w79; 12° 90° 100;48°

259;12° 270° 280;48°

Figure 7. Lunar magnitude as function of the argument of latitude when the Moon is at itsapogee.

However, the use of negative numbers as such and of magnitudes notrelated to physical phenomena was not possible at Jiibir b. Aflab's time.

4. Conclusion

In this paper I have studied Jiibir b. Aflab's criticism of the computationof lunar eclipses in Ptolemy's Almagest. In this criticism, Jiibir b. Aflabclaims to have found an error in Ptolemy's computation of lunar eclipses.However, he reaches this conclusion only because there were somemissing sections in the manuscript of the Almagest he used when he wrotethe I$lab al-Majistf. In fact, this is one of the main reasons for Jiibir b.Aflab's criticisms of Ptolemy, although not the only one. In addition, heprovides a slightly different method for computing lunar eclipses. Thismethod is not a significant improvement from a computational point ofview, although, since it avoids interpolation tables, it is far more elegant.However, even though the I$lab al-Majistf is a close rewriting of theAlmagest, this method for calculating lunar eclipses shows that Jiibir b.Aflab deserves to be considered as a creative and original theoreticalastronomer.

5. On the edition

What follows is not a critical edition but a working one, based only on theArabic manuscripts extant in Arabic script. Consequently, we have notused the Arabic manuscripts in Hebrew script, or the Hebrew or Latinmanuscripts, although Apianus's Latin edition published in 1534 wasconsulted during the preparation of this study.

72 1. Bellver Jiihir b. Ajla/:l on lunar eclipses 73

6. Edition

[Es1

f. 62r, Es2

f, 74v and B. f. 64r]

lJ:1~\ ut!~I .}

rJ r...... \+i1a).J ut!~\ .J:l~tia ~I~I .}} ~~ ~} ~th.; r.......J, 30

~\ t....JA {ofi~

o

Figure 8. Ms. Est 62v.

-...ali (.)"..,9.J ~.J~I ~ 0""~ ..~ ~ (.)"..,9 L>Sili [1\ 0.J~ ~I] [1§]

u ~\ .} ~\ jS.JA ~ 4..bii ~.J y>ill JjW\ .illil\ 0""~ ..~

4....,.jl§ ~ -...ali (.)"..,9 ~.J .,;~I u ~\ .} J.l=J\ 0y\ ~ jS.JA.J ~\31 ,

. .J .. ~I\ '.< {.~} -...all 4..bii ' .<.... 4....,.jl§ lJ\ 'I~ I.J A~ I~I.,?..r- ..;-.JA I",? - LJ~ • .J.J cs- ~ .. . (.)".r cs-

31 Not in Ms. Es2•

32 "Jb {~} (.)"..,9 ~.J ~I 4.....:.:..JC ~ -...all (.)"..,9.J ~I JI.......:U~\

.;JI ~\ ~ Jb 4..bii ufi:i! 4....,.jl§ Y\.Jj .)c ..~ -...al1 (.)"..,9 .)c 4....,.jl§

33~ Jb} (.)"..,9 u§j.J u ~\ Ut... j k....J .} ~I jS.JA {~u.Al}

• 34U ~\ Ut... j k....J .} lJ:1~\ jS.JA {~ u.Al} .;Jl (.)"~l {~

• 35.,;~\ u~\ Ut...j k....J.} J1;JI oyb.J ~\ {jS.JA.J} ~\

36uy..&J\ .;hi 0"" ' .",S;~l\ ).li.c. ~ ~ Jb (.)"..,9 {).li.c. 0"".J} [2§]

J~~\ .} ~I tY:>.JC ~ .;JI ~ -...al1 (.)"..,9 ).li.c. r..t! lJ:1~1 0""

0"" ~I .Ai ~ .;J\ ..~ -...al1 (.)"..,9 J ~ 0"" ~~ u§j ~~ ~\

~ ~ -...al1 (.)"..,9 ~ ~} ~ ~.J ~~ JjW\ .illil\ .} o~139 38 37{~I} -.::..w\.5; W.J JjWI.illil\ {~} ~4-i {~}~}.;bill ~

41 40 '. ). .L.{~.J} {t...~} ~ -...all (.)".J9 ~ .JA.J &\ \ u.Al A.Aye- Y.J*

42 • 2 ,~~ {\~n ~ [Es f. 75r] oyb &.J 0"" ~\

32 Not in Ms. Es 1•

33 Ms. Es2 up:, 4-:!k.

34 Ms. Es2 • C .l.

37 In the margin in Ms. Es2.

38 In the margin in Ms. Es2. Not in Ms. B.

39 Mss. B., Es2 6~1.

40 Mss. Es 1, B. r).....

41 Ms. B. .Y".J.

74 J. Bellver Jiibir b. Ajla/.l on lunar eclipses 75

Figure 9. Ms. Es 1 63r.

~~ --¥ ~ J y. ~ ~~ ~J ~~ L-J [B. f. 64v] [5§]

.u u~ ,:?il\ yil\ ~~i <.J.4 ~ Js .} J,l:J\ oy\~ .;hiJ yil\ .;hi ..>.1~ti.a52 , ~.

{o..>.1Jii ~ J.;..I <.J.4} ~.J ~ \ jSy. <.J.4

jSy. J..? ~ ~~ '-.i1i oy\~ ~ ..>.1J.ii1\ ~~ ['\ 0.J~ ~\] [6§]

'-.i1i ~ U~ '-.i1i Jb ~~ ~l.A. ~J ~l.A. ~ ~.JY\ jSy. ~J Jb, - ~. ' 53Y o~ y.)1 ~~ ~J w'i~'i\ .} ~.J~I <.J.4 yil\ ~ ~\ {~}

54 &; ... ~ J: '" J:

{ \~I.J} ~\. I 0 ~.ul . - ~ ~I.J u.ll ..b:.. .&. \ _.~.. I .~l\ t.....4l\.J . .Y?- ~ ~. ..T"J ~~ J _• 55 •

yil\ .;hi ) ~ 4.9.Jl'-4 li1 f'~ ~J I~ j.;>.. U~ {Jb ~ l.A.} ..b:.. ~ ':?ill

~.PJ ~ l.A. ~~J] ~ l.A. '-.i1i ':?~ <.J.4 ~\ J JsJ J,l:JI 0yl ~ .;hi ) ~J

oJ

'-.i1i U"'~ ~JL.......a ~J Y e:al.... U"'~ 01! Jb ~ U"'...,! t..i J [3§], , 43

U~ La ~~ 0\ fi~J ~~ J\~ U"'...,! ~~ ~~ u.ll U"'...,! {~lES.J} ~~ , «{l.A.)~} 4.9.Jl'-4 ~J Jjli~ Cl) p.j~ ,:?il\J u~~ 4l::! {La}

• .&.L -:. .~- -oil 1~~ U..T".~ -.s-

y.J ~ ~~ U"'...,!~ yu ~ ~ '-.i1i U"'...,! li:ilii uJ Gi ~~J [4§],~ %

~~J] ~~ u.l\ {';yl~} ~ yu ~b ,:?\j ~l.A. U"'...,! us.::JJ {~l.A.~}

~ ,:?\ j u.li ~ l.A. ~J\ jJ Wli JbJ ,:?I j ~JI j <.J.4 o~1 J Js u~ ~, 48 , 1

u.l\ ~~~ {oli...~} L- U~ Jb u.l\ [Es f. 62v] ~ ~J\j

I' .\.~ 11 l.A. u.li .\.~ ~ J\~ .\.~ 1\':? .J c::--~ c.s', ~ c::--~ . ~ c::--~ c.s', ~

~} f'~ ~ l.A. ,:?\ j[J] ~ l.A. u.li[J] ~ u.li ~ y.J* <.J.4 ~\ J JsJ ~ l.A., 49 ,

oyb CI.J <.J.4 ~I Jb ~ U"'...,!J La~ Jb ~ U"'...,!~ U~ {Ul51 " 50 •

{u;.u U\ li~) La ~~J} 4.....~ {I~J} ~

44 Not in Ms. B.

46 Ms. Es2 j.

45 Mss. B., Es2 ~ ( L>".,! )~.

43 Ms. B. ..illil.

42 Ms. Es2 u~).

48 Mss. B., Es2 w..:i!.

49 Not in Ms. B.

50 Ms. Es2 u~).

53 Not in Ms. Es2.

54 Ms. B. )~I -..J[)].

55 Ms. Es2 • ~.

76 J. Bellver Jiihir b. Ajlal;l on lunar eclipses 77

oyb .,;b!J o.,;b! )~ ~ J ~yJ .J:U..liJ\ .ill! 0-" ~ ~ ~ ~\

~ wJ\ U"..,! J ~ u..J tj\j ~ ~.JAC c:y..ili ~ ~IA ~ Jhll-

y.J ~ (jfi: ~I Jt...-J'i\ w!J ~ ~'i\ U"..,! c/'J ~J~56 2

w!J ~ ~'i\ U"..,!} {-ill~J} L..)- [Es f. 75v] tjlj ~ ~.JAC

- , - 57-k - L.. _L - \. wJI k.lA _c:~ -11 1.00_ ·.C { .,._11 Jt...-J'il~ .Y-" tj ..J ..T"J U""""""" ~ U-"""':i ~

~J (")- ~}il j5.Y' 0-" ~I ~ y.J ~ ~ lA k:..! L..)- tj\ j ~ lA59 -, 58 -

{("'i ~lb k y.} ~IA w..l\ ~ ~\ .,;b! ~J .u YJL.......a ~h ~IA {k}- 62 - 61 60 -k {y.}~ ~IJ uts ~lb k {y.} ~IA ~~ {k} ~ o.,;b!J}- - 63k y. ~h ~IA k ~ y. tj~\ ~ ~IA ~ ~\ .,;b! )~J {uts ("'i

~\ .,;b! )~ y. ("'i ~lb k.. J ~ u..J o)~ ~ J ~YJ ~ ~lb65 64 - ,

Y.~ ~\J {~IA ~q ~ o)~ y. uts ~lb k J ~IA w..l\ ~

..w' I .. ;I\.lA ~\ h wJ\ k..~ '-l.J .~-:I\ 1- ..C:-. uts 'i k..tj . ~~ ..T" tj ~ .. _...J&"'l r..s- UJ"""' ("

56 Ms. B.~.

57 Not in Mss. Es I, B.

58 Not in Ms. Es2.

59 Mss. B., Es2 .b J 1u..

60 Not in Ms. B. Ms. Es2 ~.

61 Not in Mss. B., Es2.

62 Not in Mss. B., Es2.

63 In the margin in Ms. Es2.

64 Ar this point, there is a line partially deleted in Ms. B.

65 Ms. Es2 Y •.

, 66 , - ,~IA w..l\ {tj~}~} ~\ y. tj~1 ~~ w..l\ k ~J ~h ~IA[J] ~IA w..ll

68 67-0.l.;-biJ\ tj)~ 0:H L.. {~\ y. tj~1 {("'i~} k ~ ~IA ~~J]

- 70 ,69 .~I y. tj~\ uts ("'i k ~J {~h ~IA}[J] ~IA w..l\ tj~ {0.1:ill\}

~IA w..li} tj~ ~ L.. ~J ~IA ~~J] ~IA wJi tj~ 0.l.;-biJ\ tj)~ ~

~ 0.l.;-biJ\ ~ L.. ~J} 0Y"~ .J:U..liJ\ .ill! .,;b! y.J (")- ~ lA ~~J]1 , 72 71-

Es f.] ~IA w..l\ {tj~ ~ L..~ -ill~J {(")- uts ("'i k y.J- 73 , -

~ ~\ y.J ..cl J k J} {(")-} ~h w..ll k y.J ~IA ~hJ [63r- , 74

~ ~\ y.J ~ ("'i k (jfi: (j\ ~ {(")- J .b[J] ..cl ..b tj.,;b!

w..l\ ~ .;-biJ\ ) ~ ~J ~ L..)- ~h ~ lAJ ~ lA w..l\ tj~ 0.l.;-biJI

~ ~lb k.. y.J ~J.,;WI .A,Ill .;-biJ\ ) ~ (jfi: ("'i ~lb k.. ~\ ~ lA75

{-uk -- } L...L ... ~.J .Y-"

66 Not in Ms. Es l •

67 Ms. B.?, J.

68 Not in Ms. Es2.

69 Ms. B. Li>ll. Not in Ms. Es2.

70 Ms. B.. C.

71 Not in Ms. Es2•

72 Not in Ms. B. In the margin in Ms. Es2.

73 Ms. B. L..,Ja...

74 Not in Ms. Es l.

78 J. Bellver Jiibir b. Ajlal,i on lunar eclipses 79

"~IA f'* ~ J1;J\ oyb .,;b! )~ .} ~ ~ ~lb ~J [7§]

~J.,;iJI76 ,

jSY' <.J.4 {1A.:l'L..y\} ~ .} [B. f. 65r] ~\ .,;b! uts WJ [8§]" 77

WI ',< ' ~1.h;J\·:. _1 {:,:.-:11 .. ,<} ".- ':l . ~I<:.J ,J-Y' lY.:J ',? ~ ~ ~ ~ (..)o'='.J79 , 78

YJL......, {o~J} U~ o)~ J:j:J .:l! UtsJ <:J~\ ~ jSY'J ~ {jSyJ\}

~I .,;b! 80{~~} uts! ~':l~':l\ .} o.:l1.! ~ .} y>i1\ .,;b! ).:li.J

L.~, 81

~~ 1....£1\ U".,,! <:J~I ~ {<.J.4}~~.) I~).! [' • 0.J~ ~\] [9§]

u."...&1\ .} ~\ jSY' uli A.biiJ ~~ f'* U".,,! y>ill JjW\ -cllli\~J

.. ::11 ',< A.bii~ ..::1\ u .<11 ,.J "1\'11 0 'I.:l ',< i '''A'~'\\..r"'" ,J-Y' f'* ' J ',?..r"'" ."........ '-r U""'" Y ,J-Y' J c...r-

.,;b!J y>i1\ .,;b! ~i l>.l.,;bi1\ ~ t~ ~JL......, f'* uli U".,,!J" , 2

.} J1;JI 0yb .,;b!J y>i1\ .,;b! JI [Es f. 76r] ~\ u."...&11 .} ~I

Wli Y\Jj~ f'* ~~ U".,,!~ Wli Jb uli U".,,! ~J ',?y>i1\ u."...&1\82

JjWI -clllil <.J.4 Jb f'* U".,,!J u."...&11 UL.j h...J ~ Jb A.bii {U§:JJ}

U"."il ~JL......, ~ lA JI.:l U".,,! J,....:,.iiJ u."...&11 UL.j~~\ U"."ill ~

76 Ms. B. .~'---!\.

77 Not in Mss. Es I, B.

78 Not in Ms. B.

79 Ms. Es 1 "",.,.

80 Ms. B. ..illi5..

81 Not in Ms. Es I•

82 Ms. Es2 u§:i9.

, 83, '~\ .uL.'~ .b.:..:i ~\ ,~ {Jb lA} .ii' .<-:. Jbt>"J ..r .J ,_ _ '-.? - ~ U"..r u.,,-. f'*

u..~ L. 1~ L..L. '.<. •.•- \1 J~':ll ~ ' ..::1\, -, i ~ii.s-' .J--'" u."....; -.r::- J -r ..r"'" (..)0'='y:. u .

.) uy-&J\J ~lS..l\ ',?jS~ o~W\ ~ .;J\ Jb uli U".,,! ~i U§:J85 _ " 84

t~ <.J.4 {'4 bA,,,;,,} \.i;\ o~ L. ~ {~~} u."...&11 UL.j h...J

U1i k.. :) ~ <.J.4J L.~ uy-&JI .,;b! <.J.4 ' 9",S;JI )~ ~ U:Ubi1\, 86

Jb ~·:uIjJ f'~ JI.:l 1....£1\ U".,,!J {f'~} l>.l.,;bi11~ )~ JAJ f'*87 "

L.~ Jb f'* F ~I ~~.} {o~} r...... U~ Wli

Figure 10. Ms. Es 1 63v.

83 Ms. Es2 • ~.

84 Ms. B. L."J.....

85 In the margin in Ms. Es2~ t.

86 Ms. B. L."J.....

87 Ms. B.~.

80 J. Bellver Jabir b. Aflab on lunar eclipses 81

JI~i ~) ""y-ill u~ 0~ ~ ~I oyb uts W.J [lO§]< 88 <

u~1 J.JI l.c.A.J {""y-ilI.J ~I ~I} ~~:..s.~ ~ 0'il:JI< ~ -

I~J .JA.J ~~'il y..1 {l.c.A.J y-ill} 4 ..).;ii;l 04!41\ 0'il:JI.J ~y..I.J

- < 90 < -

J.b11 oYl..) l.JC ~.JyJY I~ I~J .JA.J ~~'il {J.JI.J}.us; y-ill u..,.&jl

~I J) .} ~JS.JA.J ""Ij~~I yJ .} y-ill JS.JA # [ll§]91

{.,ill~.J} ~lA ~6. U".,! ~ '-:-l:lfil .)c ""Ij ~ U".,! 0..P ~6. ~

< < I01J,;..I ()-c>.J ~6. Jb U"..,il '-:-l:lfil [Es f. 63v] .)c ~.JL...... Jb ""Ij U".,!

Jb un U".,!.J A..-a..,k..o 0~ ~I 0yb ..;hi u..-.l j ~ ""I j un U".,!

U".,!~ A..-a..,k..o~ Jb ~ U".,! uts ~.J A..-a..,k..o Jb ",,\ j U"..,i! A..-a..,k..o

""Ij ~ ~ ()-c> o..l.:o.l.J JS ""y-ill u~1 .) 0..P A..-a..,k..o ""Ij ~

Js ~ lA ~6.[.J] ~6. Jb~ ~.JL...... '-:-l:lfil .)c ~.J A..-a..,k..o Jb ""I j.J]

'+i~ U".,!

..;hi ()-c> , 9",<;01\ )~ [B. f. 65v] ~J ~ J.u.\1 I~ [l2§]< < 92

u~1 J.JI ()-c> ",,~I 01....)1 ~I u~1 {0L.j} )~.J uy-&JI

y..i ~J 94{y-il1 u..,...,s} J) ()-c> 93{",,~1 0L.)I.J} 4.b....J ~J ~I

88 Not in Ms. Es'.

89 Mss. B., Es2 t....~\ <i.;-oill .....~I.

90 Ms. B. Jjl ...,.;t:illj . Ms. Es2 Jjl jAj ...,.;t:ill j .

91 Ms. B.~j.

92 Ms. B. uL.).

94 Ms. B. <i.;-oill .....~\.

< 2u~1 0L.j .h......J ~J ~I y..1 ()-c> [Es f. 76v] ",,~I 0L.)I.J ~\

< 95 . < •

~J ~I J.JI ()-c> {",,~I.J} ~I J.JI ~J u~1 0L.j.h......J ()-c> ",,~I.J96 <

~ {'-:-l:l.)J} 0.J..) ~~'i\ y..198 97 <

~.J ()-c> {y~ J,..c.} ~ U"~ {~.fi~} ",,~I J.u.\1 L:..I.J [l3§]

.;JI U"..,ill ~ ~i ..,-iW1.J ~\ 0~ ~ 0)0-"tlo11 .b.,h:JI ~I J l.c.A..l.:o.i99 <

u~1 0L.j .h......J .} uy-&JI.J u.....t5..l1 {~I ~..r.JI} ""JS.JA l.J:!.!< 100 -\ _.:.. ~I -. 1 .(~ {;;,;;-II Jt...-:i'i\ .\ .,0_11 . -~II . ~

~ "" U J'"".J -.r:- ..rJ -.r:- ~ ~.JC- 101 - <

y-ill u..,...,s J1~ 'i.J~ ~.J {I~} ~ 4,jJ f.J .,ill~ ()-c> j61 .JA.J 0~..)

~ L..)c

()-c> ~'JI ~I .) ~I oYb.J y-ill ""..;hi~~ ~i .,ill~.J [l4§]

4-iW.J ~1j.;J 0~ ~~..,! =iliWI 0YI..lJI ()-c> .,ill~ ~ I.... ~.J .J:UiiJ1 &103 102 < <

\:i.;j1.J {I~..»} 0~.J ~ {~} ~ ()-c> .,ill~ .hi...,1j ~..) 0~'Lu\.J

.J."h.... ()-c> J.J'JI .,;h..JI .)~ 4..;.lWI ~~I ()-c> ~I ~.J ~..) o~

95 Ms. B. <iili.

96 Ms. 8. y:!jI:i!I.

97 Ms. Es2 ~.

99 Not in Mss. B., Es2.

100 Not in Mss. 8., Es2•

101 Ms. Es2 .M,.

102 Ms. Es2~.

103 Not in Ms. 8.

82 J. Bellver Jahir b. Aflab on lunar eclipses 83

104 "l,'~ 0:l..;bil\ {~} t~ ~I i:..\J ~~\ ~ ",,~I JJ~I

105\~~ {oy:;.c.} \:i.ijl ~~J ~WI oybll l,)A ~~ ~ L.. ~J ~Y\

0W J I~~ 0yt-,W>J ~ ~ ~ l,)A ~~ .bi...,U ~~ oy:;.c. \:i.ij\J

~ ""~\ ~tJ\ JJ~\ .J~ l,)A JJYI .,;h.JI ~ ~~ ~\j ~~ 0~)J

.llul\ I·~ ~ ~~ .: • \ .-: .. : -;\\ "tiill 'i ~ 1 - • YI• ~. • ~ ("j4 ~ c...r 0l J. ~J ~

",,~I .J:u.fl.ll -ill!...rh! ~l u~1 ~J ~ ~.JYI l,)A yJ.lI ~ ~ ~Y\

o' \ .llull· \.:.,. \. I·- ; '1 .li.~ J . r-- ~ U'-"""" .r

, 107 106~ [I~] ~yJ\ ~~ ~,fl 01 {w\j~\} J:1.llU ~ {Jlli} [l5§].. "~\ JS u.UJ \ .. :. ~\'(' . .J Qjl'w L.. i:..' '..1 ~I . ~\ JSJ . J --e-"'" J (,? <.,? .J, . ):U U::'J. ("j4 J

108 ,J ~ . .J {~I} ~~I .L .. .J ~'il I' W I:,fl"> ".:j~ 1_J. <.,? "" .J~ <.,? ~ ..». ~ ~ r.s-" I 109~ l,)A ~\.5. W [Es f. 64r] ~1j,,)j1 l,)A {~I j).~} L.. i:..YJ ~1j,,)j1

'\.5. W ·..1 ~I . i:..i L..' L.. 1 '-~ • ~\ ~ .,.... i:..i ~~U U::'J. ("j4 ~ ~ ("j4. U'"'" •

l. l, ' 110~ (:!L....::. ~ I l,)A 0\.5. W JJ ~1 JJ~\ l,)A i:..1 L.. ~ ~ {~~}

2 III~ [Es f. 77r] ~J;l 'i ~yJ\ ~~ 0\.5. {dJ} yJ.lI ...rh! l,)A , 9",siJI

104 Not in Ms. 8.

105 Ms. Es2 y.:.c.

107 Ms. Es2 u.,...s.Jl.

108 Not in Ms. Es I.

109 Mss. 8., Es2 .wljH.

110 Not in Mss. B., Es2.

III Ms.B.u}.!.

l. ' 112 l,(:!~~I l,)A .u... Qjlj~ L.. i:..1 o~J ~tJ\ ~ ~J;l US'i J {JJ~I} JJ~\

113{ 0 fi~} ""~I J...a.l\ .JA \1\ .yJ.lI ...rh! l,)A' 9, ,,' i 01\ ) ~ ~~ 0\.5.9

114JJ~I ~ l.A~ ~ I~~ ~J {~} ~yJ\ ~~ 0\.5. 0).9 [16§]

l,)A [B. f. 66r] ..;si (:!L....::.Y\ l,)A ~tJ\ JJ~\ ~ ~\j~ l.i~JJ JJY\

. \~

. \.5. I~I '.(, W\ 11\' \. ".<\.u...~ .. ~II ". I I.ili [17§]U ,U~ , J~ ("j4 ~ • ..r"'" U,

) ~YI o~ ~ ~ yJ.lI 0\.5. 0).9 o.J:U,fl -ill! l,)A ~.) y.)\ ~ yJ.lI

.u... . \ \ .. ~II . ~ I.... -cl~ . .L~I\ t ..<.~ .u... WJ ,Q~ J ..r"'" lJ""l'..,. U'"'" ~~ ~ u~ .. .....r

.>:<\ .u...~ .ul I.ili ~ . . ~I > .u...~ 'i~ .. ~I\ ·.<.4~ " (.)="-lJ . ~-.r' •• ..r"'" U~

~.l.ul' .>:<\.u...~ ..~II"·L ~ .\ ~~\ ~\..c. 01\' \.("j4 ~ • ..r"'" U. .J • U • J~ ("j4

112 In the margin in Ms. Es2•

113 Mss. B., Es2 P.

114 Ms. 8.~.

115 Ms. B. uL.YI.

116 Ms. Es2 "-:U:!.

117 Not in Ms. 8.

118 Not in Ms. Es2•

84 J. Bellver Jiibir b. Ajlab on lunar eclipses 85

7. Translation

[Jabir b. Aflal)'s method]

Introductory comments, not mentioned by [Ptolemy], needed to obtain themagnitudes (maqiidir) and phases (azmina) ofthe eclipses.

[§ I] [See Figure 8.] Let arc GB be a segment of the ecliptic and arc AB asegment of the lunar inclined orbit. Let point G be the centre of the Sun ata solar eclipse and the centre of the shadow cone at the lunar eclipse. Letarc AG be perpendicular to arc BG. Point A is the centre of the Moon atthe true syzygy while arc AG is its true latitude. Let arc GD beperpendicular to arc AB. Point D is the centre of the Moon at the mid­eclipse. The centre of the Sun and the centre of the Moon at the solar mid­eclipse, or the centre of the Moon and the centre of the shadow cone in thelunar mid-eclipse are on arc DG.

E

119 Not in Ms. B.

120 In the margin in Ms. Es2•

121 Mss. B., Es2 ()lA)1 U..o .-kl.

122 Mss. 8., Es2 <.j"y.:J.

123 Mss. 8., Es2 4-,.

124 Ms. B.~.

B

126 Ms. B. ....:;i.

127 Not in Ms. Es2•

128 Ms. Es 1 ul.a... Ms. Es2 ~I.a...

129 Ms. B. J:4-lI-,. Not in Ms. Es2.

Figure 8. Ms. Est 62v.

[§2] From the value of arc DG, the magnitude (miqdar al-munkasifminqutr al-maksuf) of solar and lunar eclipses is known. As for the value ofarc AG, which is the lunar latitude at the true syzygy, it is known since arcAB, which is the lunar nodal distance in the inclined orbit, is known. Andthe ratio of its sine to sine of arc AG is equal to the ratio of the radius tothe sine of the maximum inclination of the inclined orbit (nihayat mayl al-

86 J. Bellver Jiibir b. Aflab on lunar eclipses 87

falak al-mii'il). Since the three sines are known, the fourth, which is thesine of arc AG, is known and is less than a quarter of a circle. [Es2 f. 75r]Therefore, [the arc] is known.

[§3] As for arc GD, Ptolemy acted perfunctorily when he considered itto be equal to arc AG. In like manner, he considered arc AB to be equal toarc DB. He mentioned that the maximum difference between themamounts to two minutes, although the [actual] difference amounts to fourminutes. [In addition,] it is easy to know what its actual value is. 130

[§4] If we extend arc AG until it passes through the pole of arc BG ­i.e. point E - and we consider that arc EZH passes through the two polesof circles AB and BG, then angles Z and D are right angles and angleEAZ is equal to angle [Es I f. 62v] GAD. Hence, [it can be concluded]from what we have demonstrated previously that the ratio of the sine ofside AG to sine of side GD is equal to the ratio of sine of side AE to thesine of side ZE. Since the sines of arcs AG, AE and ZE are known,therefore the sine of arc GD is known. Arc GD is less than a quarter or acircle and, thus, it is known. And that is what we wanted to prove.

[§5] [B. f. Mv] [We must] also consider, in our introductory comments,the way of obtaining the values of the diameters of the Moon and of theshadow cone for all the possible distances of the Moon to the Earth [since]they vary due to the [motion of the Moon in its] epicycle.

T

L A!~:i>-L----";b---9----------===-OE

M

K

Figure 9. Ms. Es1 63r.

[§6] [See Figure 9] Let circle ABG around centre D be the epicycle of theMoon and let point E be the centre of the Earth. Let us join [points] E B DA in such a way that point A is the lunar apogee in the syzygies and point

130 er. PtA p. 297-8.

B its perigee in them. Previously we explained that the difference betweenthe two [points], i.e. line AB, is lOP whenever line ED is 60P. In addition,we knew the diameter of the shadow cone for distances AB and BE. Wewill consider as a given condition that the Moon is on point G of itsepicycle. We want to know the value of its diameter and the value of thediameter of the shadow cone for distance EG. Let us trace theperpendicular GZ. Since we consider as a given condition arc AG, whichis the arc of the anomaly at the true syzygy, [as point G is a givencondition], its sine - i.e. the perpendicular GZ - [Es2 f. 75v] is known. Inlike manner, the versed sine of the arc of the anomaly at the true syzygy ­i.e. line AZ - is known. [Hence], the remaining line EZ is known.Therefore, line EG, which is the distance of the Moon to the centre of theEarth, is known. Let line EH be equal to it, [i.e. to line EG]. Let thediameter of the Moon, for distance AE, be line TL, and its diameter fordistance BE be line TK. Therefore, the difference between them is lineLK. The lunar diameter for distance EG - which is equal to line EH - isline TM. We want to know its value. Since line TL is the value of thelunar diameter for distance AE, and line TK its value for distance BE andthe difference between them is line LK, the ratio of line AH - which is thedifference of distances AE and EH - to line AB - which is the differencebetween AB and BE - is approximately equal to the ratio of line ML ­which is the difference between the value of the diameters for distancesAB and EH - to line LK - which is the difference between the values ofthe diameters for distances AB and BE. The difference between distancesAE and BE is known and corresponds to the total diameter of the epicycle.And the difference between the diameters for both [distances] - i.e. lineLK - is known. In like manner, the difference between distances AE [Es I

f. 63r] and HE - i.e. line AH - is known; and line LK - which is thedifference between diameters TK and TL - is known. Therefore, line LM- which is the difference between the diameters for distances AE and EH- is necessarily known. Thus, [if] we add it [- i.e. line LM -] to the valueof the diameter for distance AE - i.e. line TL -, the value of the diameterfor the given distance - i.e. line TM - is known. End of the demonstration.

[§7] And exactly in like manner we can apply this method to obtain thediameter of the shadow cone for a given distance GE.

[§8] Since the solar diameter [B. f. 65r] for all [possible] distances tothe centre of the Earth does not change due to the small value of theeccentricity - [remember that Ptolemy] obtained its value when he found

88 J. Bellver Jiibir b. Aflab on lunar eclipses 89

that it was equal to the value of the lunar diameter when it is on its apogeeat the syzygies - [the solar diameter] for all these reasons, was known.

[§9] [See Figure 10] Let us consider as given conditions that arc AB isa segment of the ecliptic, that arc GB is a segment of the lunar inclinedorbit, and point A is the centre of the Sun at solar eclipses or the centre ofthe shadow cone at lunar eclipses. And let us consider that point G is thecentre of the Moon and arc AG is equal to the sum of both radius - i.e. theradius of the Moon and the radius of the Sun for solar eclipses, [Es2 f. 76r]or the radius of the Moon and the radius of the shadow cone for lunareclipses. Let arc AD be perpendicular to arc BG. Point D corresponds tothe mid-eclipse and the arc GD of the inclined orbit corresponds to thehalf duration of the eclipse. We take arc DE provided that it is equal to arcGD. Therefore, arc ED corresponds to the second half [of the eclipse].Since the lunar latitude at the true syzygy is known, as we have shownpreviously, arc AD - which passes through the centres of the eclipsingand eclipsed [bodies] (kfisif and maksiij) at the mid-eclipse - is known asexplained previously. We subtract it from the sum of the two diametersand, thus, the eclipse magnitude (miqdiir al-munkasifmin qutr al-maksiij)is known. Since [i.] line AG - which is the value of both radii - is known,[ii.] arc AD is known and [iii.] angle D is right, we can conclude fromwhat we explained for spherical triangles that arc GD is known.

A

B

Figure 10. Ms. Es' 63v.

[§10] Since the shadow cone is greater [than the lunar diameter], the lunareclipse goes through four different states (abwiil). Two of these states are

to be found in both kinds of eclipses - i.e. solar and lunar [eclipses] ­which are the beginning and the end of an eclipse. The two other statesapply only to the lunar [eclipse]. [These are] the beginning of totality(akhir al-imtila') - which is the time when the Moon [begins to be] totallyeclipsed - and the end of totality (awwal al-injila') - which is thebeginning of the reappearance of the Moon from the shadow [cone].

[§ 11] Let point Z be the centre of the Moon at the beginning of totalityand point H its centre at the end of totality. Arc GZ is approximately equalto arc HE. In like manner, arc ZD is [Es' f. 63v] approximately equal toarc DH. Since arc AZ - which subtends the radius of the shadow cone - isknown and arc AD is known, arc ZD is known. And given that the wholearc GD was known, the remaining arc GZ is known. Hence, for the lunareclipse, arcs GZ and ZD are known and are approximately equal to arcsDH and HE, and therefore each arc corresponds to its counterpart.

[§12] By this procedure we can obtain [B. f. 65v] the eclipse magnitude(miqdar al-munkasif min qutr al-maksiij) and the value of the phases(zaman, pt. azman) of the eclipse - i.e. the phase from the beginning of thesolar eclipse to the mid-eclipse, the phase from the beginning of the lunareclipse to the beginning of totality (akhir al-imtila'), the phase from [Es2 f.76v] the beginning of totality to the mid-eclipse (wasat zamiin al-kusiij),the phase from the mid-eclipse to the end of totality (awwal al-injila'), andthe phase from the end of totality to the end of emersion (iikhir al-injila')­despite the degree of approximation obtained.

[Jabir b. Aflab's criticism of Ptolemy]

[§13] As for the procedure mentioned by Ptolemy, it is an approximateprocedure, from two points of view. First, he used straight lines instead ofarcs. And second, he considered that the arc between the centres of thetwo bodies - the eclipsing and eclipsed bodies - at the middle of theeclipse was equal to the true latitude of the Moon at the true syzygy. Heconsidered that [the difference between the two centres under theseconditions] was of two minutes. However, [this difference] is greater thanthis value. After that, he tabulated a table to compute the lunar eclipsewhich I will summarize here:

[§14] He added the radius of the Moon and the shadow cone when theMoon is considered to be in the apogee of its epicycle and obtained thevalue for the inclined orbit. He found that this value was 10;48° andsubtracted it from 90°, obtaining 79; 12°. This value corresponds to the

90 J. Bellver Jiibir b. Afla/;l on lunar eclipses 91

distance to the Northern maximum [nodal] distance [in the inclined orbitin which an eclipse can occur]. He placed this value in the first row of thetable which is the maximum distance. In the same manner, he took, thesum of both radii [when the Moon is placed] at its minimum distance[from the Earth] and obtained this value for the inclined orbit, which was12; 12°. He subtracted this value from 90°, obtaining 77;48° and wrote it inthe first row of the second table devoted to the minimum distance [of theMoon to the Earth]. He tabulated a table in minutes in which the ratio ofthese values in minutes relative to 60' is equal to the ratio of thedifference between the maximum and minimum distances of the Moon tothe Earth at the eclipse relative to the diameter of the epicycle, which isthe difference between its maximum and minimum distances. 13I

[§ 15] On the computation of eclipses he mentioned:

Enter the value of the [argument in] latitude in both tablesand take what you find in front of both [tables] and write itdown separately. Enter the degrees of anomaly in the row ofthe [corresponding] value in the table of minutes and take thevalue of minutes that you find in front of it. [Es I f. 64r] Takea value from the difference of the values obtained from bothtables such that its proportion to [this difference] is equal tothe [proportion of the minutes obtained relative to 60']. This[proportion] is added to the value obtained from the firsttable. The result in digits corresponds to the eclipsed sectionof the lunar diameter. If this value of the [argument of]latitude does not exist [Es2 f. 77r] in the first table, but onlyin the second, take the digits that you find in front of it. This[value] corresponds to the magnitude of the lunar eclipse(miqdiir al-munkasifmin qutr al-qamar).132

131 Cf. PtA p. 296.

132 If it falls within the range of the numbers in the first two columns, we take the amountscorresponding to the argument of latitude in the columns for the [lunar] travel and thecolumn for the digits [of magnitude] in both tables, and write them down separately.Then, with the anomaly as argument, we enter into the correction table, and take thecorresponding number of sixtieths. We then take this fraction of the difference betweenthe [two sets of] digits, [derived from] the two tables, which we wrote down, and also ofthe difference between the [two sets of] minutes of travel, and add the results of theamounts derived from the first table. If, however, it happens that the argument oflatitude

This is the procedure that [Ptolemy] mentioned.[§16] If the lunar [argument in] latitude were 79°, we will not find [an

entry] in the fust table. However, in the second table opposite the[argument of latitude of 79°], we will find a value greater than [B. f. 66r]two digits.

[§ 17] [But] we said that the eclipsed section of the Moon is greater thantwo digits. And this is only so when the Moon is at the perigee of itsepicycle. If the Moon were at the apogee or next to it, in such a way thatthe addition of the two diameters was equal to the lunar latitude or lessthan it, no part of the Moon will be eclipsed. But we have stated that it iseclipsed more than two digits. And his pretension that the eclipsed sectionof the Moon is greater than a sixth is extremely clumsy for it is noteclipsed at all.

[§ 18] It is impossible that the authors of astronomical tables (zfjiit) ­in which there are no demonstrations - would have neglected this point.And how would Ptolemy have neglected it - a man who intended that allconcepts of this science be based upon correct demonstrations, to the pointthat he drew attention to [such minutiae] as that the time elapsed betweenthe beginning of the solar eclipse and its middle was different from thetime elapsed between the middle of the solar eclipse and its end. Heintended to compute such matters which are is not worthy of attentionsince, being so small, they are not perceived by the senses. How is itpossible for someone who investigates this value and claims attention tosuch unnecessary minuteness, which is not perceptible, to neglect an issuethat ends in a falsity when he pays attention to an issue that never takesplace - and is refuted byh the naked eye. In addition, he could haveverified this easily. Whoever claims to be just and does not want to enterinto disputes cannot doubt that he neglected this point and did not noticeit, particularly when we have found in his book easier questions (ma'iinin

)

that he had neglected. May the One whose perfection is unique beglorified, His nobility dignified and His issue enlarged.

falls within the range of the second table only, we take [as final result] the appropriatefraction (determined by the number of sixtieths found [from the correction table]) of thedigits and minutes [of travel] corresponding [to the argument of latitude] in the secondtable alone. The number of digits which we find as a result of the above correction willgive us the magnitude of the obscuration, in twelfths of the lunar diameter, at mid­eclipse. Cf. PtA pp. 305-6.