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The Latin Version of lbn Mucādh's Treatise “On Twilight and the Rising of Clouds”*

Published online by Cambridge University Press:  24 October 2008

A. Mark Smith
College of Arts and Science, Department of History, 101 Read Hall, University of Missouri-Columbia, Columbia, Missouri 65211, USA


Written by the 11th-century Spanish Arab, Abū ʿAbd Allāh Muhammad ibn Mucādh al-Jayyānī, “On Twilight and the Rising of Clouds” represents a unique attempt to determine the height of the atmosphere on the basis of the first tinging of its upper reaches by dawn light. In fact, Ibn Mucādh's value of around 52 miles remained standard until the 17th century, when it was revised sharply downward in consideration of atmospheric refraction and barometric studies. The treatise itself survives in a single Hebrew exemplar, 25 Latin exemplars, and an Italian exemplar derived from the Latin. At the heart of this present study is a critical text based on a fullscale comparative transcription of 22 of the Latin manuscripts, ranging in date from the 13th to the 17th century.

Composé par Abū ʿAbd Allāh Muhammad ibn Mucādh al-Jayyānī, auteur du XIe siècle de l'Espagne arabe, “Du crepuscule et de l'ascension des nuages” représente une tentative, unique en son genre, de déterminer la hauteur de l'atmosphère en considérant le premier éclairement de ses confins supérieurs par la lueur de l'aurore. De fait, la valeur d'environ 52 milles (82.5 km), calculée par Ibn Mucādh, demeura la valeur admise jusqu'au XVIIe siècle; elle fut révisée à cette époque, en prenant en compte la réfraction atmosphérique et les études barométriques. Le traité en question a survécu dans un manuscrit hébreu, dans 25 manuscrits latins et un manuscrit italien qui dérive du latin. Le noyau de la présente étude est une édition critique du texte, appuyée sur une transcription comparative complète de 22 manuscrits latins, allant du XIIIe au XVIIe siècle.

Research Article
Copyright © Cambridge University Press 1992

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page 83 note 1 Among those works are the Tabulae Jahen (translated into Latin by Gerard of Cremona in the late 12th century), “On the Total Solar Eclipse” (translated into Hebrew by Samuel ben Jehuda in the 14th century), “Projection of the Rays of the Stars,” “Determinations of the Magnitudes of the Arcs on the Surface of a Sphere,” and On Ratio; for details and descriptions see Dold-Samplonius, Y. and Hermelink, H., “Al-Jayyānī,” in Gillispie, C. C. (ed.), Dictionary of Scientific Biography (New York, 1973), vol.7, pp. 8283.Google ScholarHill, Donald R., “A Treatise on machines by Ibn Mucādh Abū cAbd Allāh al-Jayyānī,” Journal for the History of Arabic Science (henceforth JHAS) I, 1 (1977): 3346,Google Scholar attributes an Arabic treatise on mechanical devices (“The Book of Secrets about the Results of Thoughts”) to Mucādh., IbnCf., however, Sabra, A.I., “A note on Codex Biblioteca Medicea Laurenziana Or. 152,” JHAS I, 2 (1977): 276–83,Google Scholar and Villuendas, Maria V., “A further note on a mechanical treatise contained in Codex Medicea Laurenziana Or. 152.,” JHAS II, 2 (1978): 395–96, both of whom deny Ibn Mucādh's authorship of the work in question.Google Scholar

page 84 note 2 For the details of this biographical reconstruction, see Dold-Samplonius and Hermelink, “Al-Jayyānī,” as well as Sabra, A.I., “The authorship of the Liber de crepusculis, an 11th-century work on atmospheric refraction,” Isis, 58 (1967): 7785.Google Scholar

page 84 note 3 See Sabra, “The authorship of the Liber de crepusculis,” for a detailed account. As he maintains, there is only the flimsiest support among the Latin manuscripts of “On Twilight” for an attribution to Ibn al-Haytham (Latin=Alhazen). Perhaps the error stems from the fact that at least seven manuscript versions of the treatise are bound with Ibn al-Haytham's monumental optical treatise, De aspectibus (Arabic=Kitāb al-manāzir). While this could easily lead to a conflation of authors, I know of only one case in which the conflation is explicitly made. Furthermore, armed with hindsight, we find clear indications within the Latin tradition of Ibn Mucādh's authorship. For instance, of the 23 manuscript versions I have consulted, 10 explicitly ascribe the tract to “Abhomadi” (in various orthographic permutations). This can be plausibly construed as a transliteration of “Abū… Mucādh” or even of “Ibn Mucādh.” Also, as Sabra points out, the mysterious words “malfegeyr” and “safac” found in the incipit to several of the Latin manuscripts (Liber Abhomady malfegeyr id est in crepusculo matutino et saffac id est in crepusculo matutino et saffac id est in vespertino crepusculo) make perfect sense as transliterations of Mā al-fajr (“What is dawn?”) and shafaq (“evening twilight”); hence the suggested reconstruction of the original Arabic title as: Mā al-fajr wa al-shafaq (“What is Dawn and Twilight?”). As far as I know, explicit reference to Gerard of Cremona as the Latin translator occurs only once in the manuscript tradition (in a 14th-century version), as well as in the first printed edition (1542).

page 84 note 4 At present we know of only one manuscript exemplar of this version (Paris, Bibliothéque Nationale, MS Hebr. 1036), which also contains Samuel ben Jehuda's translation of “On the Total Solar Eclipse.” See Goldstein, Bernard R., “Ibn Mucādh's treatise on Twilight and the height of the atmosphere,” Archive for History of Exact Sciences, 17 (1977): 97118, for an English translation and scholarly commentary. I have relied extensively upon this work not only in developing my own commentary but also in unravelling certain interpretive snags in the Latin text.CrossRefGoogle Scholar

page 85 note 5 This Italian version, which can be found in Vatican, Biblioteca Apostolica, MS Vat. Lat. 4595, is appended as an integral concluding part to Ibn al-Haytham's De aspectibus (also in Italian translation).

page 85 note 6 Ibn Mucādh also mentions an identity of “form” between dawn and dusk, although the illumination displayed during the former generally tends toward whiteness while that displayed by the latter tends toward redness. It is presumably on the basis of such overall identity that Ibn Mucādh restricts his subsequent technical analysis to dawn.

page 85 note 7 Evidently harking back to the ancient Pythagoreans, the notion that the earth casts a conical shadow when it blocks the sun depends upon assuming that the sun is larger than the earth.

page 85 note 8 See n. 4, “Commentary,” for a discussion of stellar luminosity and its causes according to Ibn Mucādh and his contemporaries.

page 86 note 9 Goldstein, “Ibn Mucādh's treatise,” p. 111, mentions reflection as the means by which Ibn Mucādh supposes that the atmospheric vapors convey sunlight to us at twilight. Actually, this is not quite correct if, as passages in the text indicate, Ibn Mucādh was an advocate of the visual-ray theory of sight championed by Euclid, Ptolemy and al-Kindī (see n. 6, “Commentary”). In that case, he would have supposed that the vapors merely absorb solar illumination to become secondarily luminous and, therefore, actually visible to the visual flux issuing from the eyes along the line of sight.

page 86 note 10 See n. 14, “Commentary,” for a brief account of the provenance of these figures. The estimate of 19° for the solar depression at first light is a bit high by modern standards, the accepted value being 18° for “astronomical” twilight. Before fastening on to 19° as his computational standard, however, Ibn Mucādh does mention the figure of 18°. The definition of twilight is, of course, arbitrary, depending upon what criteria of illumination and/or visibility are used: the Glossary of Meteorology, ed. Huschke, Ralph E. (Boston, 1959) offers three distinct variations:Google Scholar astronomical twilight, which begins/ends when the sun lies 18° below the horizon; nautical twilight, which begins/ends when the sun lies 12° below the horizon; and civil twilight, which begins/ends when the sun lies 6° below the horizon (note that dawn and dusk are treated throughout as symmetrical). For current accounts of twilight, see Minnaert, M., Light and Colour in the Open Air, trans. Kremer-Priest, H. M. (London, 1940),Google ScholarRozenberg, G. V., Twilight: A Study in Atmospheric Optics, trans. Rodman, R. B. (New York, 1966),CrossRefGoogle ScholarMeinel, A. and Meinel, M., Sunsets, twilights, and evening skies (Cambridge, 1983).Google Scholar

page 88 note 11 Cf., e.g., Dold-Samplonius and Hermelink, “Al-Jayyānī,” p. 82, who credit Ibn Mucādh with “obtaining [a] reasonably accurate value” for that angle. Determining the duration of twilight is crucial for defining the periods of evening and morning prayer in Islam, but obviously Ibn Mucādh's motivation for writing “On Twilight” had nothing to do with this problem.

page 88 note 12 Dold-Samplonius and Hermelink, ibid., and Sabra, “The authorship of the Liber de crepusculis,” p. 77, both cite 18° rather than 19° as the value that Ibn Mucādh uses in the treatise.

page 88 note 13 That atmospheric refraction has a significant displacement-effect on observations made at or near the horizon was known from at least the time of Ptolemy; see, e.g., Almagest, VIII, 6 and Optics, V, 23–30. Any attempt to take atmospheric refraction into account, however, would have vitiated Ibn Mucādh's method for determining the height of the rising vapors responsible for twilight- which may well explain his otherwise puzzling silence on the matter.

page 89 note 14 The lack of any ostensible practical application for either the method or the result as presented in “On Twilight” doubtless prompted Ibn Mucādh's gratuitous defense at the end of his introduction of those who promulgate new knowledge (see n.15, “Commentary”).

page 89 note 15 The relatively large number of extant manuscripts of the Latin version of “On Twilight” (at least 25 complete copies, ranging in date from the 13th to 16th century) attests to its continuing popularity in the Latin West throughout the Middle Ages and Renaissance. The absence of any known Arabic exemplar, on the other hand, does not necessarily imply a lack of interest in the treatise in the Islamic East; our knowledge of text-transmission in the Arab world is still too scanty to support such a conclusion.

page 89 note 16 There is no evidence that anyone before Ibn Mucādh attempted to derive the height of the atmosphere on the basis of a rigorous geometrical method, and we know that his determination of 52 miles, or thereabouts, remained standard until the 17th century. Perhaps it is Ibn Mucādh's method that is at play in Salimbene's amusing (and doubtless apocryphal) tale of Emperor Frederick II's attempt to cozen his court astrologer, Michael Scot, into miscalculating the height of the sky: “[Frederick] once asked… the distance of his palace from heaven. And after Michael gave the answer that seemed correct to him, the Emperor took him away for several months, … commanding his architects and stone masons in the meantime to lower that room of his palace in such a way that no one could detect it… When the Emperor returned to his palace with his astrologer, he asked him again how far distant the palace was from heaven. And after he had completed his calculations, Michael Scot answered that either the heavens had risen or the earth had sunk,” The Chronicle of Salimbene de Adam, trans. Baird, J. L., Baglivi, G. and Kane, J. R., Medieval Renaissance Texts and Studies 40, (Binghamton, N.Y., 1986), pp. 355–56.Google Scholar

page 90 note 17 See, e.g., Kepler, Johan, Ad Vitellionem paralipomena: quibus astronomiae pars optica traditur (Frankfurt, 1604), IV, sections 12,Google Scholar section 6, props. IX-XI, and section 9. Kepler informs us that, according to “physicists,” the twilight-producing atmosphere is “12 German miles” high (=89km =c. 55 miles, according to Chevalley, Catherine, Les fondements de l'optique moderne [Paris, 1982], p. 441, n. 8). Drawing upon his own analysis of atmospheric refraction, however, Kepler eventually revises this estimate downward to half a German mile (=c. 2 miles).Google Scholar

page 90 note 18 See Goldstein, Bernard R., “Refraction, twilight, and the height of the atmosphere,”Vistas in Astronomy, 20 (1976): 105–7, for an account of the fate of Ibn Mucādh's estimate during the 17th century. By the end of that century, in fact, his estimate has been completely superseded and much reduced on the basis of different methods, such as the use of barometry, of deducing the atmosphere's altitude.CrossRefGoogle Scholar

page 90 note 19 The three manuscripts I have not included are all Parisian: Bibliothéque Nationale, MSS Lat. 7319 (which I have consulted but not transcribed) and Lat. 16207; and Bibliothéque de la Sorbonne, MS 1037; my thanks to Dr. Richard Lorch, Institut für Geschichte der Naturwissenschaften der Universität München, for calling these manuscripts to my attention. In addition, I have also failed to include Oxford, Bodleian, MS Digby 104, because it contains only the first half or so of the treatise.

page 92 note 20 Autolycus de Pitane. Histoire du texte suivie de l'édition critique des Traités de la sphère en mouvement et des levers et couchers, Recueil de travaux d'histoire et de philologie de l'Université de Louvain 3, 37 (Louvain, 1950). For an instructive adaptation of this method, see Lejeune's, Albert introduction to L'Optique de Claude Ptolémée dans la version latine d'apr`s l'arabe de l'émir Eugéne de Sicile (1957; reprint, Leiden, 1989).Google Scholar

page 93 note 21 It is also possible, of course, that O and N represent two separate branches from the same root. The inordinately large number of variants that N contains in comparison to its nearest predecessor O indicates that its author, Pedro Nunes, took considerable liberties with the text.

page 93 note 22 This particular cluster is far more complex than the preceding one. For one thing, the detailed scheme of transmission involves several branches. Also, in the V-P4 combination, P4 actually precedes V chronologically. This would indicate that both manuscripts have a common ancestor of which V represents the more direct and faithful (albeit later) version. Therefore, the list as I have given it not only in this case, but in succeeding cases as well, indicates the direction of transmission only in the most general way.

page 94 note 23 A comparison of gross characteristics among the manuscripts tends not only to confirm these groupings but also to suggest cross-linkages among them. For instance, L, P4, E, P, S, and V are connected at least superficially by the fact that they are all appended to Ibn al-Haytham's De aspectibus. So too, certain manuscripts can be grouped according to shared incipits and/or explicits (group 1 = Val, Va2, V1, P1, P3, V2; group 2 = P2, D, Es, B, C; group 3=S, P, U, V, P4, E). At a somewhat less superficial level, F, P3, V1, Va1, Va2, B, O, D form a distinct group insofar as they alone contain the last paragraph of the treatise (lines 419–26 in the critical edition). Conversely, P2 and Es define a sub-group by lacking a particular, significant swath of text (lines 128–210 of the critical edition), as indeed do P1 and G by lacking the last several lines of text (386–426 in the critical edition), L and F, finally, form a distinct pair inasmuch as they share verbatim the same extensive glosses.

page 117 note 1 This title represents a composite, with some revision, of three titles consistently encountered either in the incipits or explicits to the manuscripts: 1) Liber Abhomady malfegeyr de crepusculis, 2) Liber de crepusculis, and 3) Liber de ascensionibus nubium.

page 117 note 2 The Latin term crepusculum, which I have translated as “twilight,” specifically denotes “dusk,” the term for “dawn” being diluculum. From the context, however, it is clear that the author (or the Latin translator) intends crepusculum in a general way to cover both dusk and dawn – hence the plural crepusculis in the title. Still, as will be clear later on in the text, Ibn Mucādh bases his analysis solely on dawn – presumably under the supposition that what holds for it also holds for dusk.

page 117 note 3 Both radius and its plural radii are used in the Latin text to describe the propagation of sunlight. In the singular form it connotes “radiation” in general, whereas in its plural form it connotes “rays” as vehicles of radiation. For the most part, I have rendered both Latin terms as “radiation,” since the arguments developed in the text do not depend on drawing a punctilious distinction between the two meanings.

page 118 note 4 In medieval Latin, the term piramis is almost always used to denote “cone,” the term conus being fairly rare. That the earth casts a conical shadow with its base on a great circle perpendicular to the line joining the centers of earth and sun was a commonplace from Greek antiquity, when it was used to explain lunar eclipses.

page 118 note 5 Among “stars" Ibn Mucadh seems to have included both planets and fixed stars. The dispute to which he adverts involves the source of stellar and planetary luminosity. Ibn al-Haytham cites the same dispute in his “The Light of the Stars” (c. 1030?); see cArafat, W. and Winter, H.J.J., “The light of the stars: a short discourse by Ibn al-Haytham,” British Journal for the History of Science, 5 (1971): 282–88.CrossRefGoogle Scholar The problem, according to Ibn al-Haytham, is that, having become convinced (rightly) that the moon's illumination is due solely to the sun, many “who profess philosophy” assume by extension (wrongly) that all celestial bodies owe their luminosity to the sun. Perhaps it is in view of this point that Ibn Mucadh claims universal agreement that the moon receives its primary light from the sun (cf., however, Galileo's discussion of the moon's secondary illumination at total eclipse in the “First Day” of the Dialogue Concerning the Two Great World Systems, trans. Drake, Stillman [Berkeley, 1970], pp. 6798).Google Scholar

page 118 note 6 In this, and the previous sentence, the Latin term visus denotes “visual powers” or, perhaps, “visual rays" rather than simply “sight.” In both passages, moreover, those visual powers/rays are understood to “reach out to” or “attain to” (consequuntur) the light in heavenly objects – all of which implies that, in contravention to Ibn al-Haytham, Ibn Mucādh subscribed to the extramissionist visual theory of Euclid, Ptolemy, and al-Kindī; for details about this theory and Ibn al-Haytham's intromissionist alternative, see Lindberg, David C., Theories of Vision from al-Kindi to Kepler (Chicago, 1976), pp. 1186.Google Scholar Ibn Mucādh's appeal to “density” as a prerequisite for stellar luminosity, in tandem with his previous assertion about sunlight “absorbed by the high stellar bodies,” suggests strongly that, again in contravention to Ibn al-Haytham, he assumed all celestial bodies to be illuminated by the sun.

page 119 note 7 Aer and ether are obviously being conflated here, as well as in other places in the text. Following Aristotle, Ibn Mucādh would presumably have admitted a sharp distinction between the two in essence, “air” being sublunar and “ether” supralunar. However, as far as Ibn Mucādh seems to be concerned, both are identical in terms of their pure transparency, which is clearly the primary concern here.

page 119 note 8 The point of this discussion is to establish that, when we see the first light of dawn or the last light of dusk, we are not seeing the sun itself or any direct radiation from it. Nor, for that matter, is the appearance of such twilight luminosity due in any way to air or ether since, being perfectly transparent, neither can become illuminated by the sun. Hence, neither can present light to us. As the next sentence makes clear, the point of all this is to establish that some intermediary, denser than air or ether, is needed to absorb the sunlight and present it back to us at twilight.

page 120 note 9 Ibn Mucādh's aim in this paragraph is to establish that it is not the horizon itself that we see illuminated at the moment of dawn – in short, that the necessary intermediate cause is not the earth or its outer fringes. Goldstein, “Ibn Mucādh's treatise,” pp. 111–12, explains how Ibn Mucādh's would have arrived at the stated parameters of 3 and 250 miles, assuming that an average man is around 6 feet tall. According to Goldstein, Ibn Mucādh's claim that 8 miles is the limit for the tallest possible mountain cannot be traced to any known source.

page 120 note 10 The argument to this point can be summarized as follows: 1) the sun orbits the earth in 24 hours; 2) the earth's circumference is 24,000 miles, which constitutes the projection of the sun's daily orbit upon the earth's surface at the equator (Ibn Mucādh in fact cites this figure later on in the treatise; see note 1/4; below); 3) thus, since the sun's motion projects over 1,000 miles of surface in an hour at the equator, it must project over only 1/4 of that (i.e., 250 miles) in 15 minutes (these distances will of course decrease with an increase in latitude either north or south). The guiding assumption for the rest of the argument seems to be that, by the time the sun's light has pervaded the entire 250 miles of space between horizon and elevated viewer, the sun itself will begin to peep over the horizon. Thus, from first light at the horizon to the total pervasion of the intervening space (and, thus, the first appearance of the sun over the horizon) fifteen minutes will elapse. By the same reasoning, presumably, the lower the viewer's elevation, the shorter the interval between first light, total pervasion of the intervening space, and first appearance of the sun – hence the author's qualification of “a quarter of an hour or less.”

page 120 note 11 Here we have a clear indication that Ibn Mucādh ignores, or at least discounts, the effect of atmospheric refraction on horizon-phenomena, since he assumes that, at the very instant the sun appears over the horizon, its cusp and the viewpoint will be connected by a straight line tangent to the horizon.

page 121 note 12 Having excluded all other possibilities, Ibn Mucādh finally cites vapors (i.e., moisture) rising into the air from the earth's surface as the requisite intermediary cause of twilight (see n. 8 above). That such moist vapors are in fact the true cause of twilight is presumably borne out by the tinging of the sky at dawn and dusk, a tinging that finds its counterpart in the rainbow, for instance. The ascription of such meteorological phenomena to vapors (both moist and dry) rising from the earth is, of course, a commonplace harking back at least to Aristotle's Meteorologica (see, e.g., 1.4).

page 121 note 13 This is a fairly clear articulation of the principle of logical economy commonly described as “Ockham's Razor” – frustra fit per plura quod per pauciora fieri potest.

page 122 note 14 As Goldstein, “Ibn Mucādh treatise,” p. 113, points out, the parameters mentioned by Ibn Mucādh were, with one exception, standard for his time. The estimate of 24,000 miles for the earth's circumference undoubtedly finds its basis in Ptolemy's figure of 180,000 stades (Geography VII, 5) converted according to the figure of 7.5 stades/mile. The claim that the sun's radius is 5.5 that of the earth comes directly from Ptolemy's Almagest V, 16, and the figures of 18 and 19 degrees for the depression of the sun below the horizon at dawn was perfectly consonant with the values accepted in Ibn Mucādh's day. The only problematical figure among the ones used by Ibn Mucādh is that of 1,110 for the mean distance between sun and earth. The standard value, given by Ptolemy in Almagest V, 15, is 1,210. It is of course possible, as Goldstein suggests, that the alternative of 1,110 came from a faulty text used by Ibn Mucādh, but it is at least equally likely that he got it from al-Battānī's figure of 1,108 ( = “1,100 et circiter 10”) for the mean solar distance; for details see Swerdlow, Noel, “Al-Battani's determination of the solar distance,” Centaurus, 17 (1972): 97105,CrossRefGoogle Scholar and for a general overview, see van Helden, Albert, Measuring the Universe (Chicago, 1985), pp. 433.Google Scholar

page 122 note 15 This excursus seems to have been intended by Ibn Mucādh as a defence against certain detractors (religiously inspired?), who were evidently opposed to his wasting time on such matters as those addressed in this particular tract. Certainly, by Ibn Mucādh's time, there was a growing backlash among conservative Islamic thinkers against what they saw as the unbridled rationalism of the falsafa tradition, a backlash that finds a culmination of sorts in al-Ghazālā's Tahāfut al-falāsifa (“Destruction of the Philosophers”). The same self-justificatory theme is taken up more vehemently by Ibn Mucādh in the epilogue, which was purposely omitted by the Latin translator but was retained in the Hebrew version.

page 122 note 16 As Goldstein, “Ibn Mucādh's treatise,” p. 113, points out, this theorem and the one following it are similar to the first two theorems of Aristarchus of Samos's On the Sizes of the Sun and the Moon, trans. Heath, T. L. in Aristarchus of Samos (Oxford, 1913), pp. 355–61.Google Scholar This treatise would have been readily available to Ibn Mucādh through Qusṭā ibn Lūqā's 10th-century Arabic version; see also al-Kindī, De aspectibus, ed. Bjornbo, Axel and Vogl, Sebastian, “Alkindi, Tideus und Pseudo-Euklid. Drei optische Werke,” Abhandlungen zur Geschichte der mathematischen Wissenschaften, 26 (1912), pp. 56.Google Scholar

page 123 note 17 Particularly apposite in this case is Elements VI, 12 where Euclid explains how to derive a fourth proportional to any three given straight lines.

page 126 note 18 This muddled theorem purports to demonstrate that, when a luminous body shines upon a non-luminous one, it illuminates only that segment delineated by the tangents connecting both bodies. The underlying principle here – as in the previous two propositions – is that light (as well as visual flux) radiates in perfectly straight lines. Accordingly, this proposition would seem to be logically misplaced, since what it purports to demonstrate (i.e., that the connecting tangents strictly limit the size of illuminated segment in relation to size of illuminating segment) is fundamental to what is proved in the preceding two theorems. Given the self-evidence of the point Ibn Mucādh is at pains to establish here, he would have done better to postulate it rather than attempt to prove it.

page 127 note 19 In this, as in subsequent calculations, Ibn Mucādh is working within the sexagesimal (base-60) system on the basis, essentially, of Ptolemy's analysis of chords in Almagest I, 11. Accordingly, he is operating under the assumption that the diameter of a circle is subdivided into 120 parts of which the radius thus contains 60. Given these assumptions, the calculations in this passage are as follows:

1) AZ = 1,110 units, and KZ = 4.5.

2) Sine angle KAZ = ZK/AZ = 4.5/1,110 = .004054 = 00; 14 + 3/5.

3) Hence, angle KAZ = 13'56" = 13 + 14/15 minutes.

4) Therefore, 2 × angle KAZ = 27 + 13/15 minutes = 27' 52".

5) Therefore, arc BGDE = 180° 27' 52".

According to Doncel, M. G., “Quadratic interpolations in Ibn Mucādh,” Archives internationales d'histoire des sciences, 32 (1982): 6877,Google Scholar Ibn Mucādh derived his values for trigonometric functions from al-Khwārizmī-Maslama's tables. Although I have not yet been able to consult it, I ought in this context to mention Villuendas, M. V., La trigonometria europeana en el siglo XI (Barcelona, 1979).Google Scholar

page 128 note 20 Elements III, 16.

page 130 note 21 Suffice it to say that, although Ibn Mucādh fails to stipulate it explicitly, line BHD must be constructed perpendicular to line AEH in order for the specified conditions to hold.

page 130 note 22 To treat EQK and BHD as if they actually coincided is legitimate within the framework of the celestial sphere to which, according to Ptolemy, Almagest 1,6, the earth bears the ratio of a point. Within the framework of the solar orbit (represented by ABGD), however, the ratio is 1:1,110 according to the parameters offered by Ibn Mucādh. Although this ratio is not large, it is nonetheless significant enough to force Ibn Mucādh to admit that “what we say here is not quite accurate.”

page 131 note 23 That line QH bisects angle EHL depends upon proving the equality of sections LQ and EQ of intersecting tangents TLM and EQK. The basis for such a proof is Elements III, 17, whence it is an easy matter to demonstrate that any two tangents dropped from single point (e.g., LQ and EQ dropped from point Q) are equal; see, for example, al-Nayrīzī's 10th-century commentary on Elements III, 17, in Curtze, Maximilian (ed.), Anaritii in decem libros priores Elementorum Euclidis commentarii (Leipzig, 1899), p. 130.Google Scholar

page 131 note 24 Although the Latin reads “chord” (corda), it is clear from the context that “sine” is intended.

page 131 note 25 The basic train of reason that Ibn Mucadh follows in this passage can be summarized thus:

1) The illuminated portion of the earth, represented by arc LFN is, as previously determined, 180° 27' 56".

2) Angle LHF = 1/2 angle LFN = 90° 13' 56", and angle BHF = 19° by construction.

3) Therefore, angle LHB = angle LHF − angle BHF = 90° 13' 56" − 19° = 71° 13' 56".

4) Therefore, angle EHL = angle EHB − angle LHB = 90° − 71° 13' 56" = 18° 46' 4".

5) Angle QHE = 1/2 angle EHL = 9° 23' 2".

6) Therefore, angle HQE = angle KEH − angle QHE = 90° − 9° 23' 2" = 80° 36' 58".

7) Sine angle HQE = EH/QH = .986618 = 59; 11,49 (text has 59; 11,48).

8) Since ZH = EH (both being radii of circle EFN), then ZH/QH = .986618 = 59; 11,49.

9) Therefore, QH = ZH/.986618 = ZH/59; 11,49.

10) Normalizing ZH to 1 (decimal) or 60 (sexagesimal), we get QH = 1/.986618 or 60/59; 11,49 = 1.013564 or 60; 48,50.

11) Therefore, QZ = QHZH = 00; 48,50 or .813889 parts out of 60.

12) Given that the earth has a circumference of 24,000 miles and assuming a value of 22/7 for n, then we can calculate ZH, the earth's radius, according to the formulation r = 24,000/2JI = 3818.181818 or 3818; 10,55 miles.

13) Divided into 60 radial parts, that value yields 63.636363 or 63: 38,11 miles per part.

14) Therefore, QZ = 63.636367 × .813889 or 63; 38,11 × 00; 48,50 = 51.792737 or 51; 47, 34 + 6/11 miles, which is the maximum distance above the earth's surface that rising vapors can reach into the air.

page 132 note 26 See Goldstein, “Ibn Mucādh's treatise,“ p. 110, for an English translation of the text omitted here by the Latin translator.

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