Centaurus Volume 28 issue 1 1985 [doi 10.1111%2Fj.1600-0498.1985.tb00799.x] J. L. Berggren -- The Origins of al-Bīrūnī‘s “Method of the Zijes” in The Theory of Sundials

8/13/2019 Centaurus Volume 28 issue 1 1985 [doi 10.1111%2Fj.1600-0498.1985.tb00799.x] J. L. Berggren -- The Origins of

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The Origins of al-BirZlnis Method ofthe Zves in The Theory of SundialsJ. L. BERGGREN*

In his great work on mathematical geography the Central Asianscholar Abu 1-Rayhgn al-Biriini fl. 1010) described a method for find-ing the azimuth of one locality relative to another and called it themethod of the zijes. Zij is a Persian word that Islamic astronomersapplied to astronomical handbooks, and in a previous pape? we por-trayed the development of this method in the zijes of Habash al-Hgsibfl. 850 , Abu 1-Waf2 al-BiizjZini fl. 980) and Jamshid al-Kgshi fl.1400) as well as in writings of al-Birijni fl. ca. 1010) and Abu 1-Wafgscontemporary, Abii Sahl al-Kiihi. The present paper continues ourstudy with an account of the versions of the method found in an anon-ymous treatise in AS 4830 and the following works: al-Zijal-Shdmil, awork by an unknown astronomer who bases it on a z i j of Abii I-Wafs,al-Zij ul-Hdkimiby the 10th Century Egyptian astronomer Ibn Yiinus,and al-Zij ul-Jdmz by Ktishyar ibn LabbBn, a contemporary of Abu 1-Waf2 and Ibn Yfmus. This study sheds additional light on the rela-tionships both between the zijes and between their authors, and itcontains evidence to suggest that the origin of the method of thezijes lies in the theory of sundials.The problem the method solves, that of finding the azimuth of oneplace relative to another, is important to the Muslim world when thefirst place is Mecca and the second is ones own locality, for the azi-muth is then the qiblu, which Muslims must face to perform their fivedaily prayers. During the centuries from Habash al-Hasib to Jamshidal-Kiishi, Muslim astronomers discovered a variety of solutions to theproblem, solutions whose study presents the historian of exact sci-ences with a great diversity of methods, and this paper, with its prede-

Department of Mathematics, Simon Fraser University, Burnaby, B.C. Canada VSA 1S6.Centourus 1985: vol. 2 8 pp, 1-161 Centaurus XXVIII


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2 1.L.Berggren

Fig. 1

cessor, s an attempt to tell the history of one of these methods over aspan of six centuries. (Recent work by D. King has shown that in addi-tion to methods based on scientific principles there are a great numberbased on folk-astronomy and that the latter were often used in orien-ting mosque^.^)To make this study as self-contained as possible, we begin with abrief account of the method of the zijes as Al-Kiihi explains it in histreatise.Figure 1 shows the celestial sphere seen from the outside, whereABGD represents the horizon of our locality and DEB its meridian,passing through the south ( D ) and north (B) points of our horizon aswell as through the zenith of our locality 9 nd the north celestialpole H). The circleAZG is the celestial equator whileN s the zenithof Mecca.In the following account, X Y or X Y Z will always denote the arc of agreat circle on the sphere, and Sin X Y denotes the medieval sine func-tion r sin XY where sin is the modem function and r is the radius

of the sphere. Following E. S. Kennedy, we write S h ( x U = WZ)


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al-Birmrs Method of the Zqes 3

Fig 2

to mean the sine of the arc XY hich arc is also equal to the arcWZ . n addition the reader may find helpful the circle of latitudesshown in figure 2, which portrays the meridian of some locality, inwhich Z is the zenith of the horizon, P the ole of the equator, 9 helocal latitude and @J = 90 - 9. n general, XYwill denote 90 -XY.We may now identify various arcs in figure 1.HN s the complementof the latitude of Mecca (qM) nd TZ is the positive difference betweenthe longitude of our locality and that of Mecca A).he circle ES,called the altitude circle of Mecca, cuts from our horizon the arc of theinhirdf DS, hich measures the angle through which a worshipper fat-ing south must turn to face Mecca.To find DS he methods of the zijes is the following: Let GNOA bea great semicircle passing through the east (A) and west G) points, sothat the angles at 0 are right angles. By the rule of four quantities6ap-plied to the spherical right triangles ONH and ZTH,

Sin(NH = eM)Sin(TZ = A n )Sin(TH = 90 )Sin NO =

I


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4 J . L.Berggrenand a table of Sines yields a value for NO from that of Sin NO. Thesame rule, this time applied to the right triangles GOZ, GNT, yields

Sin(NT= rpM)Sin(NG = NO)/Sin(GO = 90)Sin 02 =

and from 02 we calculate EO = Q,- 02. The complement of the dis-tance from our zenith to that of Mecca, measured in degrees, (a, sgiven bySin No. Sin EOSin(G0= 90Sin(NS = 2) = 3)

when the rule of four quantities is applied to the right triangles GSN,GDO. Finally the relationshipSin NO

Sin(NE = d)/(Sin ES = 90Sin SD = (4)obtained from the triangles DSE and ONE, yields S D , the inhiraf ofMecca. The one drawback to this method is that, since using Sine ta-bles to produceXY rom Sin X Y yields only angles between 0 and 90,the method will not, without further instructions, give a correct an-swer when, e.g., dl > 90. Indeed, in the previous paper one of ourconcerns was to trace the elaboration of this method up to the work ofJamshid al-KBshi, where one finds the method as the core of a pro-cedure valid for every locality on earth, providing an example of whatD. King has called universal solutions in Islamic astronomy.Before turning to the z i j literature, where the examples we cite inthis work are all from the late-tenth century onwards, we begin with amuch earlier treatise outside the corpus of zijes.In a forthcoming paper on early methods for determining the qiblaD. King has called attention to a short treatise on the subject found inAS 4830.8The method in the second section of this treatise is of con-siderable relevance to our study since it calculates, just as 1) abovedoes, Sin NO (which it calls the first quantity), then Sin m ( t h esecond quantity), then, just as 2), specifies Sin 02 nd, from 02,


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al-BirLinis Method of the ZGes 5

E O = q - 02 the third quantity). The next step has been garbledby the omission of a line but, as the text is emended in Kings paper, itspecifies the use of Sines as lines inside the sphere together with thePythagorean theorem to calculate Crd E N and then Sin EN =Sin(2 arc Sin Crd EN) . Finally, after this tortuous bypath, the authorreturns to the main track and specifies the calculation of what he callsthe distance of the qibla from the meridian, i.e. the inhirdJ exactlyas in (4).We clearly have here a variation of the method of the zijes, andthe awkward and inconsistent way the calculation of O E is handled,together with the terminology first quantity, etc. suggests a treatiseindependent of the z i j of Habash. According to D. King, The styleof the Arabic and clumsiness of the mathematical methods identifythe treatise as early Abbasid and, in our view, this early date and itsevident independence from Habashs treatment support the statementwe made in our previous study that it seems unlikely that his [Ha-bashs] zij marks the first appearance of the method ....We now turn to the zijes, beginning with al-Zij al-Shdmil. In his pi-oneering study of medieval Islamic zijes Kennedy classifies thisanonymous work, extant in a 131hcentury copy, as one based on themean motions established by Abu 1-Wafs and his colleagues, and Sez-gin describes this z i j as a revision (Bearbeitung) of a z i j of Abu 1Waffi. Thus, although the extant work is no mere copy, it is connec-ted with a z i j by Abu 1-Waffi, and we shall see evidence supportingthe supposition that such a z i j is the source of the qibla method inal-Zij al-Shdmil. If this is so we have an alternate treatment of theqibla problem by Abu I-Waffi and one that differs markedly from thatof his al-Zij al-Maji-sfi. In this latter work the rule of tangents forspherical triangles is the basic tool for deriving the various arcs, whilein al-Zij ul-Shdmil the author follows exactly the steps we have givenas (1)-(4); but, instead of referring e.g. to Sin AllSin 90, he speaks ofthe Sine of what is between the two longitudes depressed (munhuf-.tan, i.e. divided by 60 which, since the radius is 60, comes to thesame thing and differs only in terminology. Further, in the latter partof al-Zijal-Shdmil it is explained when to measure the inhirdffrom thesouth point and when to measure it from the north point. If this pas-sage is indeed by Abu 1-Waffi it appears to be the first in which thedistinction is made. The following chart summarizes that text, show-


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6 1 L.Berggrening the two subcases of each of the two main cases. (Corrected lati-tude refers to EO.)

When the and the direction of Mecca the inhiraf liescorrected latitude isis

northerly east of the meridian northeastnortherly west of the meridian northwestsoutherly east of the meridian southeastsoutherly west of the meridian southwestHere, to say that the corrected latitude is northerly or southerlymeans that O in figure 1 lies to the north or south of the zenith E,which would appear in the calculations by comparing Q and 02. hewhole procedure is Abu 1-Wafiis substitute for negative numbers.The terminology employed in this section of the z i j is at least consis-tent with the supposition that Abu l-WafZi wrote it, for the de-scriptions of NO as correction of longitude fcfdd Z-MZ , of 20 scorrection of latitude (fcfdl lal-ard) and of EO as corrected lati-tude (al-ard al-muaddal) are precisely those used in al-Zij &Ma-jisJi.The assumption that the z i j of Abu 1-Wafii on which al-Shdmil isbased contains the above description of the qibla method also helpsexplain al-Bixiinis criticism of Abu 1-Wafii in his Maqdlid where hewrites If, on the contrary, his pretentions rest on the determinationof the azimuth of the qibla, this does not justify them either becausehis method offers nothing original not already found in the z i j of Ha-bash al-Hiisib. He (only) contents himself with dividing into sectionsthe steps of the calculation and with changing the term correctedlongitude (al-tul al-muaddal) to correction of longitude and cor-rected latitude to correction of latitude. If this judgement onAbu 1-Wafii is taken to apply to the work al-Majis6 which is the onlyone we know of where all the arcs are calculated by the tangent law, itis hard to understand why al-Biriini would have judged Abu 1-WafZiso severely, for there is surely much there that is quite different fromHabashs treatment. On the other hand, al-Bixiinis words apply wellto the treatment given in al-Zij al-Shdmil, where the treatment, pre-


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al-Bininis Method of the Z ?S 7sented in four steps with no proofs, goes beyond that of Yabash onlyin the concluding discussion summarized in the table, and this al-Biriini indeed may have considered no great matter since it is readilyapparent from the diagram. The point here is not to vindicate Abu 1-Wafii at the expense of al-Bixfini but only to say that of the two texts,al-Zij al-Shdmil and al-MujkK it seems that al-Biriinis criticisms ap-ply more obviously to the former.Al-Biriinis criticism of Abu 1-Wafiifor having introduced the termscorrection of longitude and correction of latitude only to makehis work appear original makes it plain al-Biriini did not know of AbiiSahls work, which uses precisely the same terminology. Abii Sahl wasthe director of a group of scholars who made astronomical observa-tions under the Biiyid king Sharal al-Dawla, and one member of thisgroup was Abu 1-Wafii.14 Since the method of the zqes, in commonwith many other solutions of the qiblu problem, involved the celestialsphere it is natural that these two astronomers would have discussedthe problem and may have agreed to adopt a common terminology.The two other zijes were composed at about the same time, early inthe eleventh century. AZ-Zij aZ-Jdmicof KiishyBr b. LabbBn, writtenaround the year 1010, exists in two similar, but by no means identical,versions, one in Istanbul and the other in Leiden.I5 The first two stepsof the Istanbul copy are exactly (1) and (2) and, beyond noting that,like Abu 1-WaW and Abii Sahl, Kiishyiir calls NO the correction oflongitude and 02 the correction of latitude these steps need notdetain us here. The next step is precisely that of Abu 1-Wafii, i.e.from the comparison of RZ and Z E to form RE - RO = EO the cor-rected latitude of the locality, southern, when RE > RO or ZO ZE= EO, the corrected latitude of the locality, northern, when Z E< 20 Similarly Kiishyiir states the azimuth is on the east-west linewhen ZE = EO. Finally the computationsof (3) and (4) follow, just asin Abu 1-Wafii and Abii Sahl.In the Leiden version of Kiishyiirs z i j however (ff. 19, 20) he cal-culates first the Sine of an unnamed arc X according to (1) above and,since no arc exceeds 90, = NO. The text then calls for the additionof @ to X and if the result

< 90 the sum is the second arc(q,) southerly,= 90 the azimuth is on the east-west line,> 90. set the second arc ( q2 )= 180- (sum), northerly.is {


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8 1. L. BerggrenThis makes no sense, however, for NO s an arc of a great circle per-pendicular to the local meridian and its addition to @, a piece of thatmeridian, produces nothing useful. A scribe has evidently left out aline that prescribes the formation of Sin 20 from Sin NO, as in (2)above. Since J = ZD, ZO + J = OD and, indeed, if this is less than90 the qiblu is southerly, if it is equal to 90 the qiblu is on the east-west line, and if it is greater than 90 hen alls between E and B,and OB, the supplement of OD, is what we want.The next step in KiishyBrs procedure requires the calculation ofCos q , Sin &Sin W here q1 has not been previously mentionedbut is now referred to as the first arc. If we take q 1= NO, which isthe first arc Kiishyiir calculates, then this expression is exactly (3) andthe result of the calculation is, as Kiishyiir says, the Sine of the altituded ) . (In the case when 02 + @ > 90, s in figure 3, so that q2= OB,we obtain the above expression for Sin d from the rule of four quanti-ties applied to the two spherical right triangles GNS, GOB.)Finally the fourth step is precisely 4), and thus, with the singleemendation we have suggested, i.e. inserting the calculation of 02 sin 2) above, and the understanding that the first arc refers to NO,the first arc calculated, the procedure in the Leiden copy of KiishyBrsz i j is again the method we are investigating.Consequently, there are two differences between sections of theLeiden and Istanbul versions of Kiishy2rs z i j dealing with the qibla.First, the Leiden version reflects a different tradition in its vocabularywith its use of the first arc rather than correction of longitude.

Second, the Leiden version handles the case E Z < 02 (the case of anortherly qiblu) by calculating what is called the second arc as 180-OD = OB), where the Istanbul version calculates its complement,OE the corrected latitude, by finding 02-EZ. The differences bothin terminology and mathematical minutiae that appear in the two ver-sions of KiishyBrs z i j in the sections on the qibla, show that the twoversions are to a certain extent independent of each other.To judge by the above criteria the next zij belongs to the tradition ofAbu 1-WafB, al-Kiihi and the Istanbul version of KiishyBrs zij. It is al-Zjal-fdkimi,written by the Egyptian astronomer Ibn Yiinus in 1009,a few years before his death.16 The method in question appears fourtimes in this zi j namely in Chapters 15,20,26 nd 28, and since threeof these appearances are in closely related problems we shall discuss


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al-Bininls Method of the Z es 9

Fig. 3

them first. In Chapters 15 and 20 steps 1)- 4) are used to computethe azimuth of the sun or a star given the local latitude, the declinationof the body and its hour-angle. (If instead of a celestial body it werethe celestial point that is the zenith of Mecca then the latter two ofthese quantities would be called the latitude of Mecca (qM) nd thedifference in longitude respectively, so that the input is that ofthe qiblu problem.) The only difference between the procedures ofthese two chapters is that in Chapter 15 (see King I, pp. .153-55), as inthe Leiden version of Kiishyfirszij, Ibn Yiinus follows step 2) with adefinition of the quantity @ + ZO, which he calls the inclination arc(it being the inclination of the great circle GNA to the horizon),whereas in Chapter 20 at this point (King I, pp. 191-92) he computesZE - Z O as most other writers do and calls it the corrected latitude.Thus Chapter 15 follows the procedure in the Leiden copy of Kiish-yfirs zij, while Chapter 20 follows that of the Istanbul copy and suchwriters as Abii Sahl al-Kiihi. It does not appear that in these two sec-tions Ibn Yiinus is interested in determining whether the angle of theazimuth should be measured to the north or the south of the primevertical AEG.


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10 J . L.BergpenIt is in Chapter 28 that Ibn Yiinus applies the method to the prob-

lem of finding the qibZa,18and the steps of the method are identicalwith those of Chapter 15. Also that Ibn Yiinus designates 20 inChapter 20 and 28) and EO (in Chapter 20) just as did Al-Kiihi andAbu 1-Wafft, so, contrary to what, al-Biriini said, Abu 1-Waffts ter-minology is not just an idiosyncracy.Before we discuss the remaining appearance of the method in IbnYllnus zij in Chapter 26, it will be useful to make some general re-marks about the method and the mathematical nature of both theproblem and the solution it offers. This problem, like most in medi-eval spherical astronomy, asks for what we should now call a changein spherical coordinates. The equator, with its poles, and the meridianof our locality provide two orthogonal reference circles with respect towhich we can measure two coordinates: (1)the height, q, f a celestialobject relative to the equator, and 2) the distance (AA) from thesouth point of the equator to the foot of the great circle measuring theheight. We may call this the equator-meridian system. The problemis to find the analogous coordinates in a horizon-meridian, system,where the horizon circle plays the role of the equator and its zenithplays the role of the north pole. Thus we seek: (1) he height relativeto the horizon, i.e. the altitude (a ) , and (2) the distance from thesouth point of the horizon to the foot of the great circle measuring theheight, i.e. the inbinif (See figure 4).The elegance of the solution consists in introducing an intermediatecoordinate system, that of the local meridian and equator, whichshares a great circle with both systems, namely the local meridian,with its poles. he E- and W-points. The method shows how to use thedata to describe height relative to the meridian @) and distance rela-tive to the equator ( A x ) , i.e. in the meridian-equator system. Thisis why the great circle through the west-point and the zenith of Meccais introduced, for it is the arc NO of this circle that measures theheight relative to the meridian, and it is from the foot of this circle onthe meridian, the point 0 (figure l , hat the distance 02 s meas-ured. This distance immediately supplies us with another, namelyOD, which, mathematically, involves changing the second referencecircle of the new system from the equator to the local horizon. The el-egance of the method lies in the fact that the same rules that transformcoordinates from the equator-meridian system to the system me-


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al-Birzinis Method of he ZGes 11

Fig. 4

ridian-equator and then to meridian-horizon will effect the changefrom this latter system to that of horizon-meridian. This is so becausethe geometric relationship of the first to the second system is the sameas that of the second to the third, and this geometric fact is reflected inthe formal identity of (1) and 2) with (3) and (4) respectively. Onlythe names of the variables differ.19The Islamic authors perception ofthis mathematical relationship is reflected in their use of the termi-nology of corrected latitude for cj5 (the analog of q ) and correctedlongitude for (the analog of A n ) , and to this extent al-Biriini iscorrect in saying that the phrases correction of latitude and correc-tion of longitude are derivative from the more correct participial con-structions.The crucial insight of the solution appears in the literature in an-other, earlier context as well, namely in the construction of the so-called meridian sundial, whose receiving plane is parallel to that of thelocal meridian and whose gnomon points eastward (see figure 9,westward, or both. Then the length of the shadow that the gnomonAB casts on a vertical plane E A D depends only on the length of thegnomon and the measure of the arc NG = NO), the direction of the


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12 J L.Berggren

Fig. 5

shadow depends only on the arc DO =m).hus the transformationof coordinates would have been of great use to the ancient specialistsin the design of sundials. In fact, in the Analemma, Ptolemy employsthe great circle through the E-W points and the sun, a circle he claimsto have introduced into gnomonics. In this work, the arc NO isnamed the hektEmoros and is one of the six quantities of which anytwo determine the others. The remaining are: meridianus (OD), de-censivus ( E N ) , horizontalis SG),and two horarius and verticalis)which do not appear in figure 5 because they involve the prime ver-tical.Ptolemy explains how, by rotating circles on the sphere into a work-ing plane, a procedure he calls an analemma, the student may de-termine the six arcs for a given local latitude q, declination 6 , and thehour angle t of the sun. P. Luckey gives a translation of Ptolemys pro-cedures into trigonometric terms, from which it transpires that thedescription of NO is precisely that of l ) , hough this is not the casewith other quantities.According to Sezgin, Ptolemys treatise was not translated intoArabic, but an earlier example of an,analemma, that of Diodoros(first century B.C.), was cited by Ibriihim b. SinBn, the grandson ofThabit b. Qurra, and it is no surprise to see Thiibit b. Qurra, who waswell-schooled in Hellenistic mathematics, make use of the hektt3noro.s

(which he calls by no special name) in his treatise on ~undials.~hiibit


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al-Biralnis Method of he ZGes 13gives the same description of the arcs NO and OR as that found in (1)and ( 2 ) ,so it is clear that both basic transformations were familiar tothe ninth century writers on gnomonics and that the key idea is foundeven earlier in Ptolemy.We conjecture, therefore, that the origins of the qibla method thatconcerns us today lie in the theory of sundials. It may not be coin-cidental that the writer in whose z i j we first find the method, Habashal-Hiisib, also wrote on sundials, though no copy of his book On Sun-dials and Gnomons25s known today.Another piece of evidence for the close connection between this qi-bla method and the theory of sundials is Chapter 26 of Ibn Yiinus z i jwhich is devoted to sundials. This Egyptian astronomer gives exactlythe same method as does habit for finding the arcs NO nd 02 andthen finds OD = @ + OZ. e claims the method as his own inventionand, evidently ignorant of Thabits work, he may have genuinely be-lieved it was; however, since Ibn Yiinus gives the same qibla methodas Abu I-Wafii and others, and he does not claim that as his own in-vention, the method of the zijes may be an example of a technique forvertical sundials inspiring - during the 9th century - a qibla methodwhich, 150years later, inspires the original method for sundials. In ad-dition, however ignorant Ibn Yiinus may have been of ThBbits worki t is clear that he saw the mathematical core of the method, for heheads Chapter 26 On finding the altitude and azimuth with respect tothe meridian, and there can be no doubt that he saw that the threepairs of coordinates involved in the method were in essence all meas-uring the same pairs of quantities and differed only in the choice of thereference circles.Finally, to conclude our survey, we note an interesting transforma-tion of the mathematical reasoning behind the method in al-Biriiniszi j ul-Qdniin al-M&zidi. Since this source is clearly explained inKings readily-available survey,26 nd the steps in the computation ofthe inbirdfare precisely those of the method of the zijes, we shall notrepeat them here; rather we limit ourselves to four noteworthy pointsin Biriinis treatment. The first is that the horizon of Mecca plays abasic role, and the great circle GNOA of figure 1 is defined as onewhose poles are the intersections of the local meridian with the hori-zon of Mecca. This approach seems to be unique to al-Biriini. Second,the law of sines for spherical triangles is used in addition to the rule of


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14 J L. Berggrenfour quantities. Third, al-Biriini explains, using the same test as in thework al-Zij a l-Shdmil, how to tell if the qibla is north or south of theprime vertical and east or west of the local meridian. Finally al-Biriinichooses the odd terminology al-bud fil-maddr (the distance onthe ma&r) for the arc NO n figure 1. In his K. l-tafhim (p. 56) al-Binjni defines al-rnaddrrit al-yaum iyyat as day-circles and mad-drat al-ar# as circles parallel to the ecliptic, so the basic meaningof mad& seems to be a small circle parallel to a great circle. Thus itseems to be a strange word to use to designate a great circle.We have, in al-Binjnis treatment, what we would expect in theeleventh century, namely an elaboration of the method to cover awider variety of cases and the use of spherical trigonometry with theemployment of spherical angles. The use of the horizon of Mecca wasa variation on what everyone else had done, while the odd name forthe arc NO eminds us of a careless slip we have previously observedin the companion treatment of the qibla problem by an analemma inthe next section of this zij.We therefore conclude that the roots of the method of the zijesfor finding the qibla lie in Ptolemys treatise The Analemma, wherethe Alexandrian astronomer solves the problem of drawing the hourlines on a vertical dial. In the third century Hijra (9th A.D.), at leastby the time of Yiibash al-Hasib, some Islamic writer saw that the sun-dial transformations, applied twice, would produce a solution to theqibla problem. This solution became extremely popular during thefourth century Hijra (10th A.D.), occumng at least six times either asa solution to the qibla problem or the more general problem of findingthe altitude and azimuth of any celestial object given q, the declinationand the hour-angle. In the various solutions, the terminology em-ployed reflects two different traditions, and the justifications offeredshare in the progress of spherical trigonometry from the rule of fourquantities to theorems involving the angles of spherical triangles.2sThe other characteristic of the fourth-century Hijra treatment of theproblem, necessitated by the fact that the medieval sine function as-sumed only positive values, was an elaboration of the method to dealwith a wider variety of cases of the position of Mecca relative to the lo-cality in question. This elaboration reached its culmination by thetime of Jamshid al-Kiishi in the latter part of the 9* Hijra century(early 15th century A.D.) who, in his Zij-i-Khdqani, presents an ex-


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al-Birrinis Methodof the Zces 15panded version of the method to deal with any location on the earthssurface and thus provides a universal solution to the qibfa problem.AcknowledgementsI wish to thank E. S. Kennedy for sending me notes based on micro-film copies of the work af -Zi j al-Jdmi by Kiishygr b. Labban andD. A. King and R. Lorch for helpful comments on a preliminary ver-sion of this paper. I also thank the Natural Sciences and EngineeringResearch Council of Canada for its Grant A3486 in support of the re-search on which this paper is based.

BIBLIOGRAPHYBerggren I J. L. Berggren, A Comparison of Four Analemmas for Determining the AzimuthBerggren 2: J . L. Berggren, On at-Birunis Method of the Zijes for the Qibla. Proc. of theBirLini: Abu I-Raybin at-Biriini, On the Determination of he Coordinatesof Cities (tr. J Ali),Kennedy I : E. S. Kennedy, ,A Survey of Islamic Astronomical Tables, Transactions of theKennedy 2: E. S. Kennedy,A Commentary Upon Birrinis Kitcib Tabdid al-Amrikin Beirut 1973.Kennedy 3: E . S. Kennedy, The History of Trigonometry, an Overview, 31sf Yearbookof heKing I : D. A. King, The Astronomical Works of Ibn Ycinus (Ph.D. diss., Yale Univ., 1972).King2: D. A. King, Art. Kibla, Encyclopaediaof Islam, 2nd ed., 4 vols. to date, Leiden 1960-

present.King 3: D. A. King, Some Early Islamic Approximate Methods for Determining the Qibla, to

appear.King 4: D. A. King, The World about the Kkba: A Study of he Sacred Direction in Medieval Is

lam to be published by Islamic Art.Publications, S.p.A.Luckey P., Das Analemma von Ptolemaus, Astronomische Nachrichren 23 1927) No. 498,

cols. 1 7 4 .Neugebauer: 0. Neugebauer, A History of Ancient Mathematical Astronomy Parts 1-3. New

York 1975.Sezgin: F. Sezgin, Geschichte des arabirchen Schrifttums Band V: Mathematik and Band VI:

Astronomie Leiden 1974 and 1978, respectively.Thabit:Thibit ibn Qurra, K. fiilit al-siCit, ed., trans., and comm. by K. Garbers as Ein Werk

Tibit b. QurrL iiber ebene Sonnenuhren), Quellen und Sncdicn zur Cesch. derMath., s*.und Physik: Abt. A 4 (1936), pp. 1-80.

of the Qibla, Journalfor the History ofArabic Science 4 (1980). pp. 6 W .16th International Congress of the History of Science (Bucharest, 1981), C.pp. 23745.Beirut 1967.American Philosophical Society N.S. 6, Part. 2 (1956).

National Council of TeachersofMathematics (Washington, D.C. %9), pp. 333-59.


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16 1.L. BerggrenABBREVIATION

AS Aya Sofia, Istanbul.NOTES

1. See Binini, pp. 253-55.2. Berggren 2.3. These methods were of several types ranging from exact or approximate mathematicalmethods to those based on folk astronomy. The former are discussed in King 2 nd 3 whilethose of the latter type are found in King 4.4. This is from the MSMeshhed, Ridn 5412, p. 43-50.5. See Kennedy 2,p. xvii.6. The rule says that in two spherical right triangles ABC and AB'C' with a common acute an-gle at A and right angles at C and C', Sin B'C'ISin B A = Sin BClSin BA. See Kennedy 3 or

a discussion of this rule in Islamic mathematics.7. in a talk delivered to the 16th International Congress of the History of Science held in Bu-charest in 1981.8. See King 3.9. See King 3.10. In Berggren 2, p. 241.11. See Kennedy 1, p. 129.12. In Sezgin, V p. 324.13. We are quoting this from the unpublished Ph.D. thesisof Dr. M.-Th. Debarnot who kindlysent us extracts from her thesis relevant to our research. T h e quotation is from p. 110.14. See Sezgin V, .V. al-KOhi.15. See Sezgin VI,pp. 246-8.Our account of the two versionsof this text is based on notes supplied to us by E. S.Kennedy from microfilms of Leiden, Or. 8 and Istanbul, Fatih 3418.There are several other manuscripts of this zi j listed by Sezgin.16. See Kennedy 1 p. 126.For a detailed study of much of Ibn Yunus' z i j see King 1.17. King 1, pp. 153-5 and p. 191.18. King 1, p.265.19. The substitutions NT NO, TR .--* OD produce (3) from l), and when NS in (3)has beencalculated its substitution for NG in (2). along with the above substitutions, produces (4)from (2).20. See the discussion in Neugebawr, pp. 848-852.21. For some remarks on the history of this method see Ntugebauer, pp. 839-56.On the use of22. In Luckey, col's. 31-32, the results of J. Drecker are reported.23. See Sezgin, V, p. 170.24. See Thabir, . 47.25. Cited by Sezgin, from the Fihrisr of Ibn al-Nadim, in V, . 276.24. See K i n g 2 .27. See Berggren 1, p. 74.28. For an outline of the history of trigonometry see Kennedy 3.

analemmas tb determine the qibla see Berggren 1.

Centaurus Volume 28 issue 1 1985 [doi 10.1111%2Fj.1600-0498.1985.tb00799.x] J. L. Berggren -- The Origins of al-Bīrūnī‘s “Method of the Zijes” in The Theory of Sundials

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