Al-Zarqalluh’s Graphical Method for Finding Lunar Distances

by

ROSER PUIC*

1. Introductory Remarks

Towards the end of the eleventh century, when the Toledan astrono- mer al-Zarqilluh (Azarquiel)‘ devised the instrument known as the safiba zarqcifiyya he introduced on its back some elements which do not feature on any astrolabes or other related instruments. The first, a representation in three quadrants of a system of celestial coordinates in meridian orthographic projection, I have treated elsewhere.’ The second, a small circle called “the circle of the Moon”, forms the subject of the present paper.3 It will be shown that the circle is used to construct a scale representation of the Ptolemaic lunar model, from which lunar parallax components are determined by an Indian tech- nique.

This study is based upon the following texts: two Arabic copies of al-Zarqalluh’s treatise on the sufiba zatqdfiyya, Mss Escorial 962 and Pans 4824 (Suppl. 2669); and the Castilian translation, ordered by King Alfonso X, preserved in the Libros del Saber de Astronomia,‘ which is virtually a literal translation of the Arabic original.’

2.1. The Construction of the Circle of the Moon

The instructions on the layout of this circle are contained in a few lines in the part of al-Zarqiilluh’s treatise dealing with the construction of

‘Univtrsidad de Barcelona, Facultad de filologia, Departamento de Arabe, Gran Via 585. 08007 Barcelona. Spain.

Ccniaurw 1989 vol. 32 pp 294-309

Al-Zarqahh and Lunar Distances 295

the instrument. This part is preserved in the Pans manuscript (see Appendix I) as well as in the Alfonsine translation.

The centre is on the lower radius of the “base circle” (al-dd’ira a/-shabiha, literally “similar circle”, from its analogy to the graduated circle on the face) which is divised into sixty equal parts by the lines of a sexagesimal sine quadrant, as shown in the representation of the back that appears in Figure 1. The centre is to coincide with the thirteenth division below the centre of the ,safi(za and its radius is to be

Figure 1: The back of the $am zarqdiyya. MS Exorid h.I.1, fol. 141v”

296 Roser Puig

equal to six of these divisions. A concentric circle, with a smaller radius, is also inscribed and the ring between the two circles is divided into twenty-four equal parts. The text gives no indication of the function of the circle or even what it is supposed to represent.

2 . 2 . The Purpose of the Circle of the Moon

Among the hundred chapters dealing with the use of the safiha there is only one (Chapter 68) in which the circle of the moon is mentioned. The title of this chapter is On the determination of the moon’sparallax (ikhtilaf manzar al-qamar) in the altitude circle, in longitude and lat- itude, with this jafiha. In this chapter, the critical edition of which appears in Appendix 2, two parts can be distinguished: a mechanical procedure in which the use of the circle of the moon together with the peculiar ruler on the back of the safiba zarqctliyya is described, and a numerical one where the operations can easily be reduced to formu- lae. In the first part we obtain the earth-moon distance which is needed to find the lunar parallax; and, in the second part, we deter- mine the parallax in the altitude circle and its latitudinal and longitudi- nal components.

The description of the construction of the ruler which is used on the back is rather obscure in the texts. There are no remaining examples of it , although an illustration appears in the Alfonsine text6 which I reproduce in Figure 2. In fact, it consists of two perpendicular rulers, one of which is the alidade proper, provided with two sights and attached to the centre of the sapha so that it can be rotated around the outer scale of the instrument like the alidade of a standard astrolabe. The second, called af-rnu‘tari(ia, the “transversal”, slides under the alidade. The length of the transversal is slightly more than the diame- ter of the base circle and one of its edges is graduated in the same divisions as the lower radius of that circle.

3. The Problem of Lunar Distance

Let us consider Figure 3. It represents the Ptolemaic lunar model which, as we shall see, is relevant:’ the deferent circle MCN is eccen-

Al-ZarqciNuh and Lunar Disrances 297

Figure 2: The ruler on the back of the zarqdiyya. A redrawing of the Alfonsine illustration

tric to E , the centre of the earth, and its radius is R. Its centre D describes, rotating in a retrograde sense, the small circle DD,, with centre E and radius e. Its rotation takes place in such a way that angle D E b is always equal to the angular distance between the mean sun and the mean moon, that is the mean elongation (ij). Thus, angle CED = 2q. Point C which rotates in the direct sense along the deferent circle is the centre of the epicycle AP, with radius f . The moon moves in the opposite sense on the epicycle with its anomalistic motion. The straight line D,C determines, on the epicycle, point A (mean epicyclic apogee) from which 6 , the mean motion in anomaly of the moon, is measured in order to determine the moon’s position P on the epicycle. Segment PE is the earth-moon distance.

The basic parameters of the model are stated by Ptolemy in the Almagest: e = 10;19, r = 5;15, and R = 49;41. However, the param- eters mentioned in the Canobic Inscription and in the Planetary Hy- potheses, works written by Ptolemy before and after the Almagest respectively, are the equivalents ones: e = 12;30, r = 6;20, and R =

298 Roscr Puig

M

N Figure 3: The Ptolernaic lunar model.

These figures are very close to those given by al-Zarq8lluh in his treatise on the construction of the jufi!za: 13, 6, and 60 respectively. It seems that al-Zarqdluh rounds them due perhaps to the difficulty of engraving the instrument with such precision.

4.1. Al-Zarqalluh’s Instructions to Find the Distance

I now represent in modem notation the instructions on the use of the circle of the moon as they are presented in Chapter 68. When drawing the accompanying figures I have taken into account only the elements involved in the solution of the problem, that is, the base circle and the

Al-Zarqrilluh and Lunar Distances 299

circle of the moon. The other lines represent the consecutive positions of the transversal and the alidade:

a) First (Figure 4a), we mark Son the left of the base circle, which is graduated from 0" to 180". the double elongation 2fi, twice the difference between the mean longitudes of the moon and the sun. If 2 i > lW, we subtract this amount from 360" and operate with the remainder in the same manner. Then, with the transversal passing through the centre of the jofiba 0, we rotate the alidade so that the edge of transversal falls on point S.

C)

Figure 4.

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Afterwards. we displace that ruler under the alidade, maintaining the position of the latter so that the ruler passes through the centre of the circle of :he moon C. The two successive positions of the transversal SO and S, C are parallel. The intersection of the transversal, in this second position. and the base circle determine the "first mark" S,.

b) After that, we join (Figure 4b) S, to the twenty-sixth division of the lower radius 0, by means of the transversal. Then we displace the latter again, so that it passes through the centre of the circle of the moon and determines on it the position of S, which will be the "second mark", placed at the farthest intersection from 5, of the transversal and the circle.

c) Then (Figure 4c) we mark S, so that the distance S2 S, equals the mean lunar anomaly a obtained for the time in question from a set of astronomical tables. We count S, .S3 clockwise for 2~7 < 180" and counter clockwise for 2fi > 180".

d) Finally (Figure 4). we join S, to 5, using the transversal, and measure the distance between the two points on the divisions of the ruler. The text does not specify the nature of the result thereby obtained, although it does indicate that we must divide seventy-two by the measured distance to obtain "the maximum lunar parallax in the altitude circle for any distance from the earth, according to Ptolemy'.

4.2. Commentary and Justification of al-Zarqdlluh's Procedures

If we consider Figure 3 drawn to scale according to al-Zarqalluh's parameters and Figure 5 which summarizes the four steps of the instructions in Figure 4 we will verify, by superposing both figures, that:

a) The centre of the rufrha 0 (Figure 5 ) coincides with the centre of the deferent circle D (Figure 3) and this circle coincides with the base circle on which we have marked S. When we displace the transversal 50 that it passes through the centre C of the circle of the moon we are transporting the vertex of angle 21 from 0 to C, the centre of the earth & in the Ptolemaic model, so that <OCS, = 21. Thus, S, of the base circle coincides with the point C of the deferent, which is the centre of the epicycle.

b) and c) When we join the division 26, 0,, D, in the Ptolemaic model. to S, we are trying to determine the position of the mean epicyclic apogee of the moon A. To do this we need an epicycle, with centre at S,, placed on the base circle. Since such a device is not available in the rafiba, al-Zarqilluh reverses the role of points S, and C of line S, C, so that S, represents the centre of the earth E and C the centre of the epicycle C. Thus, the circle of the moon represents, now, an epicycle that can be used for all values of the

Al-Zarqdlluh and Lunar Distances 30 1

double elongation. When we displace the transversal ruler, parallel to S, O,, so that it passes through the centre of the circle of the moon and determines S,, this latter point represents A, the mean epicyclic apogee, from which the anomaly is measured to obtain S3.

d) This does not give the lunar true, position. However, al-Zarqilluh does not need the lunar true longitude. He wants EP, the earth-moon distance, and since triangles ECP (in Figure 3) and S, CS, (in Figure 5 ) are congruent, the earth-moon distance is obtained by measuring S3 S, = d as prescribed above.

Figure 5 .

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I t is clear, therefore, that the circle of the moon represents the lunar epicycle. It is to be used together with the base circle as an in- strumental adaptation of the Ptolemaic lunar model to obtain graphi- cally the earth-moon distance. Since the radius of the base circle, or deferent, is divided into sixty divisions, al-Zarqalluh takes R = 60.

The geometric procedures described are closely related to those used by geometric equatoria. These instruments were developed by the Andalusian tradition including al-Zarqalluh himself,' although the purpose of these instruments, the determination of true planetary longitudes, is essentially different from the goal here.

5.1. A I-Zarqalluh's Instructions for Finding the Parallax

The numerical part of the chapter starts with the indication (quoted before, section 4.1) of dividing seventy-two by the measured earth- moon distance ( d ) . With this operation we obtain horP, the horizontal parallax of the moon in the altitude circle for a given distance d. Al-Zarqilluh ascribes to Ptolemy the expression horP = 7Zd.

Having determined horP we compute the parallax in the altitude circle for intermediate values of the zenith distance, h, from 0" to go", as well as the longitudinal and latitudinal components. The circle of the moon is not mentioned again; the sexagesimal sine quadrant is used to calculate sines of arcs. No symbols are used in the text and the trigonometric functions are defined on the basis of a circle with R = 60. The instructions can be reduced to the following formulae, ex- pressed in modem notation:

a) for the parallax in altitude P(h) = horP sin &

where h is the altitude, b) for the latitude component

P, = horP sin [900-(a + g)] where a is the angle between the ecliptic and the horizon and moon's latitude; and

is the

c) for the longitude component PL = horP sin a sin y

where y is the distance of the moon from the highest point of the ecliptic.

Al-Zarqalluh and Lunar Distances 303

5.2. Commentary and Justification of al-Zarqalluh 's Procedures

In fact, the Almagest expression for horizontal parallax in the altitude circle, expressed in modern symbols, is

horP = arcsin (rJd),

where r, is the radius of the earth and d is the earth-moon distance measured in the same units. For mean distance at syzygies Ptolemy takes d = 59 earth radii, '" whence

horP = arcsin (1/59) = 0;58,16".

Al-Zarqalluh's rule puts

horP = 72/d

that is, parallax is made inversely proportional to distance. For him, mean distance at syzygies is 60+13. Hence his corresponding

horP = 72/73 = 0;59,11",

which is close to Ptolemy, but by no means identical.

to rationalize the constant of proportionality, 72:

to designate the medieval sine function,

Following is an attempt to reconcile the two rules, and in particular,

In al-Zarq8lluh's units re = 73/59 = 1.2. Using the customary capital

(1) Sin(horP) = 60 sin(horP) = 60 (rJd) = 60 (1.Ud) = 7Ud. For small x , (2) Sin xo = 60 sin xo = 60 sin ( n ~ / 1 8 0 ) ~ ~ ~ = 60 (m/180) -- n(x/3) = 1.0472~ = x.

Application of expression (2) above to (1) gives al-Zarqilluh's rule. He was, of course, ignorant of radian measure. But he may have noticed that in a sexagesirnal sine table the first entries are almost equal to the corresponding arguments.

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Regarding the formulae, they are attested in the Indian tradition of the SLirya Siddhrtntha,” although there the formulae are given for the adjusted lunar parallax Pm (difference between lunar and solar paral- lax, Po, = P,,,-P,) and in the computation of the latitude component the Sirya-Siddhuntha does not take into consideration the lunar lat- itude. It is interesting to observe that al-Khw8rizmi’s zij’ uses the same expressions as the Sarya-Siddhrtnta to compute the parallax in altitude and in latitude but employs a different method to calculate the longi- tude component. Therefore, it seems that al-Zarqilluh was ac- quainted with a source within the Indian astronomical tradition that was other than al-Khw2rizmi’s z i j as it is known to us.”

6. Concluding Remarks

To conclude, al-Zarqilluh developed an original and ingenious device to adapt a geometric lunar model in order to find the earth-moon distance. His subsequent instructions for finding the lunar parallax are an illustration of the coexistence in Andalusian astronomy of Hellenis- tic and Oriental traditions.

Ackno wledgement

A first draft of this paper was presented at the “Congreso Interna- cional al-Andalus” held in C6rdoba (Spain) in January of 1987. In the course of the research a number of helpful suggestions were made by Professor E. S . Kennedy, to whom I express my sincere gratitude. I also thank Professor D. A. King for his useful comments on the penultimate version of this paper.

NOTES

1 . On this astronomer and his work see M i l k Vallicrosa and the article by Vernet in DSE. 2. On the use of that orthographic projection see Puig 1987a. pp. 125-138. 3 . There is an unsuccessful attempt to explain the meaning of this circle in Woepcke. 4. See Rico y Sinobas. 111, pp. 1 3 f 2 3 7 .

Al-Zarqalluh and Lunar Distances 305

5 . For a systematic comparison between them see Puig 1987b. 6 . See the edition by Rico y Sinobas, 111, p. 145. 7. On the third Ptolemaic lunar model see a brief hut adequate description in ESA by

8. See Hamilton. Swerdlow & Toomer, p. 65. 9. The earliest known treatises on the equatoria are. in chronological order: the treatises of Ibn

al-Samh (c. 1025) and al-Zarqilluh on the construction of the instrument, translated in Rico y Sinobas. 111, pp. 245-271 and 272-284 respectively; the treatise of al-Zarqilluh on the use of the instrument, translated and published by Millk Vallicrosa, pp. 457-479 and the treatise of Abu Salt (c. 11 10) studied by Kennedy 1970. Another treatise is that of al-Kashi (15th century). the astronomer of Samarqand. on whose equatorium we can see Kennedy 1960. All these texts have been taken into account by Poulle in his comprehensive study of the West equatoria, pp. 189-200, although he states that the Arabic material is not the purpose of his study. On Ibn al-Samh’s equatorium see also the more recent articles of Samsd and Mancha, and see Millas Vendrell.1983 for al-Zarqdluh’s equatorium. Finally. Goldstein 1987 has studied several descriptions in Hebrew of equatoria of the late 14th century.

Neugebauer 1957.

10. See Almagest V, 13; in Toomer’s translation p. 251. Also Pedersen, p. 207. 11. See Kennedy 1956. p. 178. 12. On al-Khwsrizmi’s z i j see Suter. Neugebauer 1962, Milks Vendrell and Goldstein 1967.

BIBLIOGRAPHY

Goldstein. B. R. 1967: Ibn al-Muthanna’s Commentary on the Astronomical Tables of al-Khwarizmi. Two

Hebrew Versions, Edited and Translaed with an Astronomical Commentary, New Haven and London.

1987: “Descriptions of Astronomical Instruments in Hebrew”, in From Deferent to Equanrr A Volume of Studies in the History of Science in the Ancient and Medieval Near East in

Honor of €. S. Kennedy, edited by D. A. King and G. Saliba, New York. pp. 105-1.11.

1987: “The Canobic Inscription: Rolemy’s Earliest Work”, in From Ancient Omens 10 Sfafrsri- cal Mechanics. Essays on the Exact Sciences Presented to Asger Aaboe. edited by J . L. Berggren and B. R. Goldstein, Copenhagen, pp. 55-73.

Hamilton. N. T., Swerdlow, N. M. & Toomer, G. J .

Kennedy, E. S. 1956: “Parallax Theory in Islamic Astronomy”, in fsis 47. pp. 33-52, reprinted in Studies in

1960: The Planetary Equatorium of Jamshid Ghijath al-Dh al-Kiishi, Princeton. 1970: “The Equatorium of AbO Salt”, fhysis 12, pp. 73-81, reprinted in Studies in the Islamic

the Islamic Exact Sciences, Beirut 1983, pp. 164-184.

Exact Sciences, Beirut 1983, pp. 481-489.

20 Cenlaurus X y x t I

306 Roser Puig

Mancha, J . L. 1987: "Sobre la version alfonsi del ecuatorio de Ibn al-Samb". in De Asrronomia Alphonsi

Regis. Barcelona, pp. 117-123. Millis Vallicrosa, J . M.

Millis Vendrell. E. I Y4>50: Esrudios sobre Azarquiel, Madrid-Granada.

1963: El comenrorto de Ibn al-Mu!anna 4 lar Tablm Astronomicas de al-Jwarizmi. Estudio y edicidn cririca del rexro lurino. en la version de Hugo Sancrallenslr. Madrid-Barcelona.

1983. "La5 posiciones de 10s apogeos planetarios en el ecuatorio de Azarquiel", in Nuevos Esrudios sobre Asrronomia Espunola en el Siglo de Alfonso X. edited by J . Vernet. Barcelona. pp. 119-123.

Neugebauer. 0. IYS7: (ESAJ The Exacr Sciences in Aniiyuiry. Providence, Rhode Island. 1962: The Asrroiiomical Tables of al- Khwurizmi. Transhiion with Commentaries of rhe Latin

Version Edited by H. Surer. Supplemenred by Corpus Chritri College MS 283, Ksben- havn.

Pedersen. 0.

Poulle, E. 1974: A Survey of rhe Almagesr. Odense.

1980: Les insrruments de la fheone des planefes selon Ptolemee: Equaroires er horlogerie planiraire du XIII' au XVI' siecle, 2 vols., Gencve-Paris.

Pulg, R. 1987a: "La proyeccion ortografica en el Libro de la acafeha alfonsi", in De Asironom'a

1987b: Lor Tratados de Construccidn y Us0 de la azafea de Azarquiel, Madrid.

1864: Libros del Saber de Asrronomia del rey Alfonso X de Casrilla, Madrid.

1983: "Notas sobre el ecuatorio de Ibn at-Samb". in Nuevos Esrudios sobre Asironomin

Alphonsi Regis, Barcelona, pp. 125-138.

Rico y Sinobas. M.

Samso. J.

fipanola en el Siglo X, edited by J. Vernet, Barcelona, pp. 105-118. Suter. H.

1914: Die Asrronomischen Tafeln &s M&ammed ibn Mtini al-Khwarizmi in der Bearbeitung des Maslama ibn Ahmed al-Madjn7i und &r Latein Ueberserzung des Athelard von Barh, Ksbenhavn.

Toomer. G. J.

Vernet. 1. 1984: Ptolemy's Almagesr. New York, Berlin. Heidelberg and Tokyo.

1976: (DSB) article "al-Zarq5li". in Dictionary ofscienrific Biography, vol. XIV, New York, pp. 592-595.

Woepcke, F. 1864: "Uber ein in der Kaiserlichen Bibliothek zu Paris befindliches arabisches Astrolabium".

in Bulletin de I'Academie ImpCriaIe dcr Sciences de SI . Pttersbourg 4, cols. 220-227, reprinted in E Woepcke. Enrdct sw les mathLmadques arabo-islamiques, 2 vols., Frankfurt: IGAIW 1986, pp. 535-540.

AI-Zarqrilluh and Lunar Distances 307

APPENDIX 1: ms. Paris, fol. 6b.

APPENDIX 2: ms. Esc.( I), fols. 62a-64a; ms. Pans ( u), fols. 60a-61b; Castilian translation ( j), pp. 216-218.

20'

308 Roser Puig

Al-Zarqalluh and Lunar Distances 309