Part 4 - Problem 4: The Altazimuth Equation

Indian Method Let a denote the azimuth of the Sun from the south. In the same triangle SKL in the same figure, we have, or, LK SL R sin : R cos : = 'Sankutala' : 'Sanku' =R sin': R cos 'Sankutala'= R cos ZXR sin R cos Now 'Sankutala' is made up of two parts, namely, 'Bahu' and 'Agra, of which the former is the distance of L from the observer's East-West line; the 'Agra' has been already found. R sin ZXR cos a Here 'Bahu'=. and 'Agra'= R R sin XR R cos or 'Sankutala' 'Bahu'+'Agra' R cos ZXR sin _R sin ZXR cos a R sin 8 * R or R sind= R cos R +. R cos R cos R cos ZXR sin o R sin ZXR cos R R cos which is easily seen to be equivalent to sin &=cos Z sin - sin Z cos p, cos a Greek Method PH Ptolemy has also a method of finding the Sun's altitude at any hour of the day. His method is as follows:H S ME Z Fig. 9 N (i) He would find by means of his tables for the times of risings of the signs of the zodiac, the orient ecliptic point. (ii) He would then find the culminating point of the ecliptic. (iii) He would finally apply Menelaus's theorem in spherics thus:Let ASC be any position of the ecliptic,(Fig. 9)NZC the 1. The equivalent of this. in a particular case, is first found in Brahma. sphutasiddhanta, Ch. III, 54-56 Cf. Suryasiddhanta, III, 28-31, also Bhaskara Grahaganita, IX. 50-52. 2. Manitius, ibid, pp, 118, 19.

meridian, NAMH the horizon, Z, the zenith and S the Sun. Here the celestial longitudes of C, S and A are taken to be known; hence ZC and CH are also known. Now take ZCS for the triangle and HMA to be the transversal; we then have by Menelaus's theorem. sin ZH sin CA sin SM ☑ ☑ sin HC sin AS sin MZ or sin SM= cos CZxsin AS sin CA 1 It is thus clear that Ptolemy had no direct method for connecting the Sun's altitude and the hour-angle. This method is workable for the problem "given time, find the altitude" but is not workable in the converse problem; besides, the calculation of the longitudes of A and C is very cumbrous. Again, when EA has been found out, taking ZHM for the triangle and CSA for the transversal, we get, sin HA sin MS sin ZC ☑ CH sin Am sin SZ sin CH 1, whence and thence HM, the azimuth can be found. The method is here also cumbrous, there being no direct connection between altitude and azimuth ; besides the time-element is not avoided. The Analemma of Ptolemy and the Indian Method. When the Sun's declination is zero and his hour-angle, is H. Zeuthen following the method of the 'Analemma' of Ptolemy, as explained by Braunmuhl² has deduced the following equations: (1) cos Z=cos H. cos (2) tan a= tan H sin To these two, Heath following Braunmuhl, adds (3) ³tanZQ tan H cos 1. Heath, Greek Mathematics, Vol. II. pp. 290-91. Zeauthen, Bibliotheca Mathematica, 13, 1900, pp. 23-27. 2. Braunmuhl, ibid, pp. 12-13, 3. The Indian form of this equatiom is R Sin ZQ = Bhaskara's, Goladhyaya, Com. on VIII, 67. R Sin HX R √ R² - R 2 cos 2 HXR² Sin²O R

where Z is the zenith and Q is the point of intersection of the prime vertical and its secondary passing through the Sun and the north-south points. Zeuthen¹ points out that later in the same treatise Ptolemy finds the arc 28 described above the horizon by a star of given declination &' by a procedure equivalent to the formula. (4) cos B=tan d'tan o. With regard to the 'Analemma' of Ptolemy. it may be noted, as Heath 2 says, that "the procedure amounts to a method of graphically constructing the arcs required as parts of an auxiliary circle in one plane." Many things may be, in practice, done graphically far more easily than by the theoretical method. Besides, no theoretical calculations occur in the 'Analemma'. Zeuthen², following the method of this work, has deduced in the general case, the two equations. (5) cos Z=(cos 8, cos H+sin 8. tan () cos 0. (6) tan a= sin 8 coso cos 8.sin H +(cos 8.cos H+sin 8.tan ) sin These equations are suggested to a modern reader from a study of the figures in the 'Analemma.' But neither in this work nor in the 'Syntaxis' are they to be found. With regard to the first four formulae, it is possible that they were recognised by Ptolemy. With regard to the last two, Zeuthen³ remarks "mais le texte nen contient rien,' and they were certainly not recognised by Ptolemy. Besides the tangent function is wholly absent in Greek trigonometry. They are also different in form from those arrived at by the Indian method as explained before. Thus, it is clear that the Indian methods are in no way connected with the method of the 'Analemma." Even taking for granted that the Indians followed a method of projection much allied to the method of the Analemma' there is no adequate reason for assuming that their method is derived from any Greek source. Analogy and precedence do not necessarily constitute originality-there is still the chance of a remoter origin from which both the systems drew their inspiration. The method of the 'Analemma,' as has been already stated, presents a 1, 2, Bjornbo, loc. cit. p. 86. 3. Zeuthen, loc. cit. p. 27.

graphical method for constructing the Sun's altitude and azimuth from the hour angle when the Sun's declination is zero but such a graphical method is generally complex as compared with the elegant Indian method. An astronomer who constructs and uses an armillary sphere to arrive at his equations in spherical astronomy and who has not a well-developed spherical astronomy at his command must have to draw perpendiculars from the positions of the heavenly body, not only on the meridian plane, the horizon or on the prime vertical, as the occasion arises, but also on the line of intersection of the diurnal circle with the horizon. Hence Braunmuhl's statement that the Indian methods of spherical astronomy have their origin in the 'Analemma's, in spite of his admitting that Indians were first to utilise its methods, is rather far-fetched and tends to take away the honour from the great Indian astronomers, who devised the beautiful methods. The 'Analemma' as it now exists is a Latin translation from an Arabic version of the original Greek¹. We may reasonably doubt that the Arabic version was greatly influenced by the ancient Indian system. We now pass on to the consideration of other allied or similar problems in the two systems of astronomy.