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Precise Determination of the Ascendant in the Lagnaprakaraṇa -I

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Precise Determination of the Ascendant in the Lagnaprakaraṇa -I ARTICLES IJHS | VOL 54.3 | SEPTEMBER 2019 Precise Determination of the Ascendant in the Lagnaprakaraṇa -I Aditya Kolachana∗, K. Mahesh, K. Ramasubramanian Indian Institute of Technology Bombay (Received 25 March 2019; revised 09 September 2019) Abstract The determination of the ascendant (udayalagna) or the rising point of the ecliptic is an important problem in Indian astronomy, both for its astronomical as well as socio-religious applications. Thus, as- tronomical works such as the Sūryasiddhānta, the Brāhmasphuṭasiddhānta, the Śiṣyadhīvṛddhidatantra, etc., describe a standard procedure for determining this quantity, which involves a certain approximation. However, Mādhava (c. 14th century) in his Lagnaprakaraṇa employs innovative analytic-geometric approaches to outline several procedures to precisely determine the ascendant. This paper discusses the first method described by Mādhava inthe Lagnaprakaraṇa. Key words: Ascendant, Dṛkkṣepajyā, Dṛkkṣepalagna, Lagna, Lagnaprakaraṇa, Madhyakāla, Madhya- lagna, Mādhava, Paraśaṅku, Rāśikūṭalagna, Udayalagna. 1 Introduction In the first chapter of the Lagnaprakaraṇa, Mādhava discusses several procedures (many of them novel) to determine astronomical quantities such as the prāṇa- As the name implies, the Lagnaprakaraṇa (Treatise for kalāntara (difference between the longitude and right as- the Computation of the Ascendant) is a text exclusively cension of a body), cara (ascensional difference of a body), written to outline procedures for the determination of the and kālalagna (the time interval between the rise of the ascendant (udayalagna) or the rising point of the ecliptic. vernal equinox and a desired later instant). The physi- To our knowledge, it is the first text to give multiple pre- cal significance of these quantities, the crucial role they cise relations for finding this quantity. We have defined play in the computation of the ascendant, as well as the the udayalagna and discussed its significance in an earlier import of Mādhava’s procedures in their determination paper.1 In the same paper, we have briefly noted the state have been discussed in earlier papers.3 It may be briefly of udayalagna computations in Indian astronomy prior noted here that the kālalagna is an ingenious and novel to Mādhava, and also remarked upon the approximations concept, apparently first introduced by Mādhava in the involved therein.2 Lagnaprakaraṇa, which greatly facilitates precise deter- mination of a number of astronomical quantities, includ- DOI: 10.16943/ijhs/2019/v54i3/49741 ing the udayalagna. *Corresponding author: [email protected] 1See the introduction to [7]. Tripraśnādhikāra of Śiṣyadhīvṛddhidatantra. For a detailed discus- 2The standard procedure adopted by Indian astronomers prior to Mā- sion of this technique, see [10, pp. 61–69]. dhava to determine the udayalagna is perhaps best described in the 3See [8], [9], and [7] respectively. IJHS | VOL 54.3 | SEPTEMBER 2019 ARTICLES From the second chapter onwards, the Lagna- kuryāt kālavilagnake | prakaraṇa describes several techniques of precisely rāśikūṭavilagnaṃ tat determining the udayalagna. These techniques are tribhonaṃ madhyalagnakam ||31|| fairly involved, spread over many verses, and require One should apply the nija-prāṇakalābheda (nija- the calculation of numerous intermediary quantities. prāṇakalāntara) to the kālavilagna (kālalagna). The current paper focuses only on the first technique That is the rāśikūṭavilagna. That decreased by of determining the udayalagna described in the second three signs is the meridian ecliptic point (mad- chapter of the Lagnaprakaraṇa. hyalagnaka).4 Besides this introduction, this paper consists of two more sections. In Section 2, which consists of several This verse (in the anuṣṭubh metre) shows the method subsections, we provide the relevant verses of the Lagna- to determine the madhyalagna, or the meridian eclip- prakaraṇa which describe the first method of the compu- tic point, in degrees. Towards this end, the verse first tation of the udayalagna, along with their translation and gives the following relation to determine the rāśikūṭa- detailed mathematical notes. In the third and last section, lagna, which is a point on the ecliptic ninety degrees from we make a few concluding remarks. the meridian ecliptic point: rāśikūṭalagna = kālalagna ± nija-prāṇakalāntara 2 Precise determination of the or, 휆푟 = 훼푒 ± |휆푟 − 훼푒|, (1) ascendant where 훼푒 and 휆푟 represent the kālalagna and the longi- tude of the rāśikūṭalagna respectively. Then, the verse The second chapter of the Lagnaprakaraṇa commences notes that the madhyalagna can be simply determined as with the definition of two quantities known as the rāśi- follows: kūṭalagna and madhyalagna (meridian ecliptic point). Through eight verses (31 to 38), Mādhava successively madhyalagna = rāśikūṭalagna − tribha and systematically defines several quantities such as or, 휆푚 = 휆푟 − 90. (2) the madhyakāla, madhyajyā, dṛkkṣepajyā, paraśaṅku, dṛkkṣepalagna, and finally the udayalagna. As we go through the verses, one cannot but help conclude that Mā- Note on nija-prāṇakalāntara and koṭi-prāṇakalāntara dhava’s approach is quite meticulous and methodical. The verse states that the rāśikūṭalagna can be determined As a prelude to our discussion, it may be mentioned by applying the nija-prāṇakalāntara (lit. own prāṇa- that in the following discussion we employ the symbols kalāntara) to the kālalagna. The nija-prāṇakalāntara 휆, 훼, 훿, and 푧 to respectively refer to the longitude, right refers to the magnitude of the difference of the longitude ascension, declination, and zenith distance of a celestial and right ascension of any body,5 not necessarily lying body. The kālalagna, the latitude of the observer, and the on the ecliptic. The term nija-prāṇakalāntara is used in obliquity of the ecliptic are denoted by the symbols 훼 , 휙, 푒 contrast to the term koṭi-prāṇakalāntara, which appears and 휖 respectively. It may also be mentioned that all the in later verses, to differentiate between the two possible figures in this section depict the celestial sphere foranob- prāṇakalāntaras for a point 퐵 on the equator shown in server having a northerly latitude 휙. Figure 1. Here, we have 2.1 Obtaining the rāśikūṭalagna and the nija-prāṇakalāntara = |휆푓 − 훼푏|, madhyalagna 4The term madhyalagnaka employed in the verse refers to the madhyalagna only. In other words, the suffix ka is not meant to mod- नजप्राणकलाभेदं ify the meaning of the noun here (ाथ). कुय配 कालवलके । 5Though the definition is generic, and is quite inclusive to be appli- राशकूटवलं त配 cable to celestial bodies that are off the ecliptic, it may be noted that the term prāṇakalāntara discussed in the first chapter of the Lagna- त्रभोनं मलक륍 ॥३१॥ prakaraṇa assumed the body (typically the Sun) to lie on the ecliptic. nijaprāṇakalābhedaṃ See [8]. 305 ARTICLES IJHS | VOL 54.3 | SEPTEMBER 2019 푃 퐾 푋 Ω 퐹 퐶 equator 휖 퐵 훼푏 ecliptic Γ Figure 1 The significance of nija-prāṇakalāntara and koṭi-prāṇakalāntara. which gives the difference in terms of point 퐵’s ‘own’ lon- From the above two relations, we have gitude and right ascension, while the −1 cos 훼푏 휆푓 = cos ( −1 ) cos[sin (sin 훼푏 sin 휖)] koṭi-prāṇakalāntara = |휆푐 − 훼푏|, −1 cos 훼푏 or, 90 − 휆푓 = sin ( −1 ) . gives the difference in the longitude and right ascension cos[sin (sin 훼푏 sin 휖)] of point 퐶 on the ecliptic whose right ascension (훼푏) corre- The above relations can be used to determine the nija- 6 sponds to that of point 퐵. Thus, only one kind of prāṇa- prāṇakalāntara in the form of kalāntara is applicable for a point on the ecliptic, while the two kinds discussed above are possible for a point on 휆푓 − 훼푏, the equator. where 훼푏 is already known. The same relations can also Perhaps assuming the procedure to be straightforward be arrived at using planar geometry, and would surely (though it doesn’t seem to be so), the text does not describe have been known to the author of the text. how to determine the nija-prāṇakalāntara. Hence, for the convenience of the readers, here we outline how to ob- Deriving the expressions for rāśikūṭalagna and tain this quantity using modern spherical trigonometrical madhyalagna results. Applying the cosine rule of spherical trigonome- try in the spherical triangle 퐹Γ퐵, where The verse states that the nija-prāṇakalāntara is to be ap- plied to the kālalagna to obtain the rāśikūṭalagna. This 퐵퐹Γ̂ = 90, 퐵Γ퐹̂ = 휖, Γ퐵 = 훼푏, Γ퐹 = 휆푓, can be understood from Figure 2, where the great circle arc 퐾푅퐸 is the secondary7 from the pole of the ecliptic yields (퐾) to the ecliptic, and also passes through the east car- cos 훼푏 = cos 휆푓 cos 퐵퐹. dinal point (퐸). Let the right ascension of 퐸 be 훼푒. This secondary meets the ecliptic at the rāśikūṭalagna (푅). Let Applying the sine rule in the same triangle, we get the longitude of 푅 be 휆푟. The points 퐸 and 푅 are analo- gous to points 퐵 and 퐹 in Figure 1. Therefore, applying sin 퐵퐹 = sin 훼푏 sin 휖. the nija-prāṇakalāntara to 훼푒 gives 휆푟, or 6The term koṭi-prāṇakalāntara may have been employed as it refers 휆푟 = 훼푒 ± nija-prāṇakalāntara = 훼푒 ± |휆푟 − 훼푒|, to the prāṇakalāntara corresponding to the koṭi or the upright 퐶퐵 of the triangle 퐶Γ퐵, which is right angled at 퐵. 7The secondary would of course be perpendicular to the ecliptic. 306 IJHS | VOL 54.3 | SEPTEMBER 2019 ARTICLES 푍 퐽 푇 푃 퐾 equator 휙 푀 ecliptic Γ 푆 푁 푅 horizon 퐸 퐿 Figure 2 Determining the madhyalagna from the kālalagna and the rāśikūṭalagna. which is the relation given by (1). to the meridian ecliptic point (madhyalagna). Now, the madhyalagna is the longitude of the point 푀 That would be the madhyakāla. That increased at the intersection of the ecliptic and prime meridian in by three signs would be the kālalagna. The kāla- Figure 2. As the arcs 푀퐸 = 푀퐾 = 90,8 one can conclude lagna diminished by three signs is stated to be the that 푀 is the pole of the great circle arc 퐾푅퐸.9 Therefore, madhyakāla.
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