Indian Journal of History of Science, 53.3 (2018) 302-316 DOI:10.16943/ijhs/2018/v53i3/49462 Determination of Ascensional Difference in the Lagnaprakaraṇa Aditya Kolachana∗ , K Mahesh∗ , Clemency Montelle∗∗ and K Ramasubramanian∗ (Received 31 January 2018) Abstract The ascensional difference or the cara is a fundamental astronomical concept that is crucial in de- termining the durations of day and night, which are a function of the observer’s latitude and the time of the year. Due to its importance, almost all astronomical texts prescribe a certain procedure for the determination of this element. The text Lagnaprakaraṇa—a hitherto unpublished manuscript attributed to Mādhava, the founder of the Kerala school of astronomy and mathematics—however discusses not one, but a number of techniques for the determination of the cara that are both inter- esting and innovative. The present paper aims to discuss these techniques. Key words: Arkāgraguṇa, Ascensional difference, Cara, Carajyā, Carāsu, Dyuguṇa, Dyujyā, Earth-sine, Kujyā, Lagnaprakaraṇa, Mādhava, Mahīguṇa 1. INTRODUCTION of the rising times of the different zodiacal signs (rāśis) at a given latitude. The ascensional difference (cara henceforth) is an The Lagnaprakaraṇa1 (Treatise for the Compu- important astronomical element that is involved in tation of the Ascendant) is a work comprised of a variety of computations related to diurnal prob- eight chapters, dedicated to the determination of lems. It is essentially the difference between the the ascendant (udayalagna or orient ecliptic point), right ascension and the oblique ascension of a and discusses numerous techniques for the same. body measured in time units. At the time of rising, However, as a necessary precursor to determin- the cara gives the time interval taken by a body to ing the ascendant, the text first discusses various traverse between the horizon and the six o’ clock methods to obtain the prāṇakalāntara,2 as well as circle or vice-versa depending upon whether the the cara. While the procedure for determining the declination of the body is positive or negative. It is cara is discussed in other astronomical works, no most commonly determined for the Sun because it text treats it and related phenomena as thoroughly is this quantity that helps one in finding the time of and comprehensively as the Lagnaprakaraṇa. To sunrise and sunset at a given location, for a given demonstrate this, in this paper we excerpt verses time of the year. It is also used in the determination from the first chapter of this text which deal with ∗Cell for Indian Science and Technology in Sanskrit, Indian Institute of Technology Bombay, Powai, Mumbai - 400076; Email: [email protected] ∗∗School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand 1The authors obtained two manuscripts of the Lagnaprakaraṇa from the Prof. K. V.Sarma Research Foundation, Chennai. 2The prāṇakalāntara (lit. difference between kalās and prāṇas) is the difference between the longitude and corresponding right ascension of a body. DETERMINATION OF ASCENSIONAL DIFFERENCE IN THE LAGNAPRAKARAṆA 303 five different ways to compute the cara (verses 18– by the Rcosine of the latitude (avalambaka) is 24), and explain the technical content of the verses the earth-sine (mahīguṇa). They (i.e. schol- along with their rationale. ars) know [the result] from the radius (tribha- maurvikā) multiplied earth-sine divided by the 2. DETERMINATION OF CARA IN THE day-radius (dyuguṇa) converted to an arc to be LAGNAPRAKARAṆA the ascensional difference (cara). The above verse composed in the mañjubhāṣiṇī The Lagnaprakaraṇa presents five different meth- metre gives the relations to determine (i) the earth- ods for obtaining the cara of the Sun at any lat- sine (mahīguṇa), and (ii) the ascensional differ- itude, and also discusses how to apply this quan- ence (cara). The given relations can be expressed tity depending upon the position of the Sun and as follows: the time of the day. The first method below is the apakramajyā × palamaurvikā standard approach to determining the cara given mahīguṇa = avalambakajyā in a number of Indian astronomical texts, while the 푅 sin 훿 × 푅 sin 휙 subsequent methods appear to be unique.3 or, 푅 sin 푀 = (1) cos In the following discussion, we employ the sym- 푅 휙 bols 휆, 훼, and 훿 to denote the longitude, right as- and, cension, and declination of the Sun respectively. mahīguṇa × tribhamaurvikā cara = dhanuṣ The symbols 휙 and 휖 are employed to denote the ( dyuguṇa ) latitude of the observer and the obliquity of the or, ecliptic respectively. The radius of the diurnal cir- 푅 sin 푀 × 푅 Δ훼 = 푅 sin−1 . (2) cle, which measures 푅 cos 훿, is generally referred ( 푅 cos 훿 ) to by the word dyujyā. However, in the following Using (1) in (2), we have verses, the author employs the term dyuguṇa more 푅 sin 휙 × 푅 sin 훿 × 푅 frequently. Δ훼 = 푅 sin−1 . (3) ( 푅 cos 휙 × 푅 cos 훿 ) METHOD 1 To derive the cara, we refer to Fig. 1, where the Sun has northern declination and therefore rises पलमौवकाभनहतादपमा earlier for an observer in the northern hemisphere अवलकोतफलंृ महीगुणः । compared to an equatorial observer. The measure भमौवकावनहताहीगुणा of this time difference, i.e. cara, is given by the गुणातंु कृतधनुरं वः ॥१८॥ great circle arc 퐹 퐸, which corresponds to time taken by the Sun to reach the six o’clock circle palamaurvikābhinihatādapakramāt (point ) after sunrise (point ).4 The relation for avalambakoddhṛtaphalaṃ mahīguṇaḥ | 푌 푋 the cara given in the verse can be derived from tribhamaurvikāvinihatānmahīguṇāt spherical triangle , where is the vertical dyuguṇāhṛtaṃ kṛtadhanuścaraṃ viduḥ ||18|| 푃 푍푋 푍푋 through 푋. In this triangle, we have The quotient obtained from dividing the prod- 푃̂ = 90 + Δ훼, 푃 푋 = 90 − 훿, uct [of the Rsine] of the declination (apakrama) and the Rsine of the latitude (palamaurvikā) 푍푋 = 90, 푃 푍 = 90 − 휙. 3In the Kerala school, Nı̄ lakaṇṭha Somayājı̄ (Tantrasaṅgraha, pp. 76–80) gives the same formula as Method 1 for calcu- lating the cara. The Gaṇita-yukti-bhāṣā does not explicitly show how to calculate the cara. However, methods 4 and 5 below are discussed by Putumana Somayājı̄ (Karaṇapaddhati, pp. 252–256). 4푃 퐸 and 푃 퐹 are meridian circles perpendicular to the equator passing through 푌 and 푋 respectively. 304 INDIAN JOURNAL OF HISTORY OF SCIENCE 푍 90 − 휙 푃 Δ훼 휙 푆 (Sun) 6 o’clockcircle 90 − 훿 푌 푁 Ω 휙 푋 horizon 퐸 훿 diurnal Δ훼 circle 퐹 equator ecliptic Fig. 1. Determining the earth-sine and the ascensional difference. Applying the cosine rule of a spherical triangle, we It may be noted that the verse prescribes to first have: determine the mahīguṇa, and then the cara using this quantity. However, for convenience, here we cos 90 = cos(90 − 휙) cos(90 − 훿) first derived the cara using spherical trigonome- + sin(90 − 휙) sin(90 − 훿) cos(90 + Δ훼), try, and then the mahīguṇa from it. In contrast to the technique shown here, Indian astronomers or, typically first derived the mahīguṇa using planar sin 휙 sin 훿 Δ훼 = sin−1 , geometry, and then determined the cara there- ( ) cos 휙 cos 훿 from, using the fact that they are semi-chords of which is the same as (3).5 corresponding arcs of the diurnal circle and the The mahīguṇa and the Rsine of the cara are equator respectively. The method of deriving the the semi-chords of corresponding arcs of the di- mahīguṇa using planar geometry is shown in our urnal circle (푋푌 ) and the equator (퐹 퐸) respec- discussion of Method 2. tively, and are therefore proportional to their re- spective radii 푅 cos 훿 and 푅. Therefore, we have METHOD 2 the mahīguṇa 푅 cos 훿 × 푅 sin Δ훼 अापयानाहतदोगुणाा 푅 sin 푀 = 푅 ाहतादभुजापमाा । 푅 sin 훿 × 푅 sin 휙 लातोऽकगुणः स कणः = , 푅 cos 휙 ा कोटीह भुजा तु भूा ॥१९॥ भारापलमौवकयोव which is the same as (1). दोगुणाममहीगुणयोव । 5The relation for the cara can also be derived from the spherical triangle 퐸퐹 푋 in Fig. 1. Here, we have 퐸̂ = 90 − 휙, 퐹 퐸 = Δ훼, 퐹̂ = 90, and 푋퐹 = 훿. Applying the cotangent four-part formula using these values also gives (3). DETERMINATION OF ASCENSIONAL DIFFERENCE IN THE LAGNAPRAKARAṆA 305 घातम वभजे दनमौ hypotenuse. तलं ख भवेरजीवा ॥२०॥ antyāpayānāhatadorguṇādvā Expressions for arkāgraguṇa trijyāhatādiṣṭabhujāpamādvā | Denoting the Sun’s amplitude at the moment of ris- lambāhṛto’rkāgraguṇaḥ sa karṇaḥ ing or setting as 퐴′, the two expressions for arkā- krāntiśca koṭīha bhujā tu bhūjyā ||19|| graguṇa given in the verse are: bhāskarāgrapalamaurvikayorvā dorguṇa × antyāpayānajyā dorguṇāntimamahīguṇayorvā | arkāgraguṇa = ghātamatra vibhajed dinamaurvyā lambajyā 푅 sin 휆 × 푅 sin 휖 tatphalaṃ khalu bhaveccarajīvā ||20|| i.e., 푅 sin 퐴′ = (4) 푅 cos 휙 The quotient of either the Rsine (dorguṇa) [of the Sun’s longitude] multiplied by [the Rsine and, of] the last (maximum) declination (apayāna), iṣṭabhujāpamajyā × trijyā or [the Rsine of] the declination (apama) corre- arkāgraguṇa = sponding to the desired longitude multiplied by lambajyā 푅 sin 훿 × 푅 the radius (trijyā), divided by the Rcosine of the i.e., 푅 sin 퐴′ = . (5) latitude (lamba), is the Rsine of the Sun’s ampli- 푅 cos 휙 tude (arkāgraguṇa). That [Rsine of the Sun’s The above relations can be derived by apply- amplitude] is the hypotenuse (karṇa). [The ing the sine rule to the spherical triangle 퐸푋퐹 in Rsine of] the declination (krānti) is the upright Fig. 1. In this triangle, we have (koṭi) here, and indeed the earth-sine (bhūjyā) is the lateral (bhujā). 퐸푋 = 퐴′, 푋퐸퐹̂ = 90 − 휙, Here, one should divide either the product 푋퐹 = 훿, 퐸퐹̂ 푋 = 90.