Developments in trigonometry

The last great representative of the House of Wisdom in Baghdad was the tenth___13

century astronomer and mathematician Abu al-Wafa. He made significant

11 See van Helden (1985), p. 32. Copernicus later based his own determination on parameters adapted from al-Battani (see Swerdlow (1973)).

In his L' histoire de l'astronomie du moyen age (1819), Delambre devotes fifty-three pages to a thorough analysis of al-Battani's zij, but the best modern work is held widely to be the Latin translation and commentary written by C. A. Nallino between 1899 and 1907. Abu al-Wafa wrote a major astronomical work that was modelled on the Almagest but which did not introduce anything essentially new to theoretical astronomy. He was, however, credited by the nineteenth-century French scholar L. Sedillot, with discovering the so-called variation of the Moon, though this view has subsequently been shown to be false and Tycho Brahe has been reinstated as the true discoverer of this phenomenon.

Fig. 4.2. The rule of four quantities.

achievements in the development of spherical trigonometry and the construction of trigonometric tables that were more accurate than those of Ptolemy. One result that facilitated calculations with right spherical triangles was what has become known as the 'rule of four quantities', illustrated in Figure 4.2. If ACB and AED are right-angled spherical triangles with a common angle at A and right angles at C and E, then sin BC sin DE sin AC sin AE A result of perhaps greater significance was the sine theorem for spherical triangles, which states that in any spherical triangle ABC, sin a sin b sin c sin A sin B sin C This result (which interestingly predates the sine theorem for plane triangles) simplified greatly the solution of many problems involving oblique spherical triangles, because it removed the need for decomposing them into a number of smaller right-angled ones.

The sine formula for spherical triangles was used to good effect by the famous Islamic scholar al-Biruni with his solution to the qibla problem, this being to determine the direction in which Mecca was closest from a given location on the Earth, i.e. along a great circle. Thus in Figure 4.3, in which P represents one's own position, M is Mecca, and PN the north pole of the Earth, the required angle is 0.

One of al-Biruni's solutions to this problem was as follows.14 The latitude and longitude of one's own position (a, y) and of Mecca (fi, S) are assumed known. We then have PNP = 90° - a, PNM = 90° - fi and /PPNM = S - y. The sine formula is not immediately applicable to the spherical triangle PPNM, but al-BirunTdevised a solution procedure that involved a sequence of triangles for which the sine formula could be used. The technique is illustrated in Figure 4.4, which is a view looking down on the Earth from directly above the position

See Katz (1998), p. 279 (see also Kennedy (1984) and references cited therein).

Fig. 4.3. The qibla. The observer is at P, M is Mecca, PN is the north pole.
Fig. 4.4. Al-Biruni's solution to the qibla problem.

P. Thus, ANIDSE is the horizon circle for which P is the pole. The horizon circle for which M is the pole is ABHCGD. Two further great circles are shown both passing through M. First, MFG is the horizon circle for which B is the pole and, second, MPNH is the great circle passing through Mecca and the north pole. The solution procedure involves three applications of the sine formula and one application of the rule of four quantities.

We begin by applying the sine formula to the triangle PNFM, noting that /MPnF = 5 - y , PNM = 90° - p and /MFPN = 90°. Thus, sinMF = sin(5 — y) cos p.

Hence, arc MF is known, and then so is FG = 90° — MF. Since FG is part of the horizon circle the pole of which is B, we know that / PN BH = / FBG = FG. This then permits the application of the sine formula to the right-angled triangle PNBH (in which PNH = p):

sin FG

Thus, PNB is now determined, and since PNN = a, immediately we get BN = a — PNB and BP = 90° — BN. Next, we use the rule of four quantities on the triangles BPC and BFG, noting that BF = 90°, sin PC = sin PB- sin FG.

The quantities on the right-hand side are all known, so PC and then CI = 90° — PC can be determined. Finally, we use the sine formula on triangle BAN. As A is the pole of EMPCI, we have /BAN = LCAI = CI, and so sin AN sin BN

sin ZABN sin CI Now ZABN = ZPNBH and the right-hand side is known, so AN can be determined. The qibla can then be calculated since 0 = NE = 90° + AN.

Al-Biruni's interests ranged over virtually all the branches of science known in his time. He studied Aristotle closely at an early age and engaged in a fairly acrimonious exchange of letters with the philosopher Ibn Sina (known in Latin as Avicenna), arguably the most famous of all Islamic scientists. Al-BTrumsent Ibn Sina a series of questions attacking Aristotelian natural philosophy, pointing out that many of its tenets had scant justification. As just one example, al-BTrunT wrote: 'There is nothing wrong in imagining the forms of Heavens as elliptic. Aristotle's reason for making them spherical is hardly convincing.' How true!

Al-Biruni wrote eight major astronomical works, the most comprehensive being his Canon, which was one of the most important astronomical encyclopedias, dealing with such subjects as cartography as well as theoretical astronomy. His models for the heavenly bodies essentially were Ptolemaic, though many of his parameters were derived independently. The Canon was not translated into Latin, and al-BTrunT's work remained unknown in the medieval West.

Another great Muslim astronomer from this period, whose works remained largely unknown in Europe during the Renaissance, was Ibn Yunus. He lived and worked in Cairo and is reported to have been an eccentric figure who devoted

15 Said and Khan (1981).

his time to astronomy, astrology and poetry. He wrote a major astronomical handbook (the Hakemite Tables) that, unlike those of many of his contemporaries, contained a large number of observations of eclipses and conjunctions. This work was used widely in the Islamic world but only became known in the West in the eighteenth century.16 Ibn Yunus also made significant contributions to trigonometry and introduced the idea of prosthaphaeresis, which was used by astronomers to shorten calculations before the invention of logarithms in the seventeenth century. He realized that the recently discovered identity

could be used in conjunction with trigonometric tables to simplify the process of multiplication by converting it into one of addition and dividing by 2. This method was one of the reasons why tables were often computed to over twelve significant figures.

Islamic religious observances presented quite a few problems in mathematical astronomy, the qibla being just one example, and this was one of the factors that encouraged the study of such problems. This influence is very evident in Ibn Yunus's work. He produced a set of tables for time-keeping by the Sun and regulating the astronomically defined times of Muslim prayer and, up until the nineteenth century, virtually all Egyptian Muslim prayer tables were based on his work.17 He also investigated the problem of determining just when the lunar crescent became visible in the evening sky following a conjunction of the Sun and Moon. This latter problem was of great importance because the beginning of each month was determined by the sighting of the crescent, and it is a complex problem involving, among other things, the relative positions of the Sun, Moon and horizon.

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