Determination of Ascensional Difference in the Lagnaprakaraṇa

Indian Journal of History of Science, 53.3 (2018) 302-316 DOI:10.16943/ijhs/2018/v53i3/49462

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 determining 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 The text Lagnaprakaraṇa—a hitherto unpublished manuscript attributed to Mādhava, the founder of the Kerala school of astronomy and mathematics—however discusses not one, but a number of techniques for the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 interesting 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. of [the Rsine of] the amplitude of the Sun (bhāskarāgra) and the Rsine of the latitude Therefore, we have (palamaurvikā), or of the Rsine (dorguṇa) [of sin 퐴′ sin 훿 the Sun’s longitude] and the last earth-sine = (antimamahīguṇa), by the day-radius (dina- sin 90 sin(90 − 휙) 푅 sin 훿 × 푅 maurvī). The quotient indeed would be the or, 푅 sin 퐴′ = , Rsine of the ascensional difference (carajīvā). 푅 cos 휙

The above two verses, composed in the indra- which is the same as (5). Using the well known vajrā and svāgatā metres respectively, together relation connecting the longitude and declination present two expressions for the cara. As these of the Sun expressions involve the Rsine of the Sun’s ampli- sin 훿 = sin 휆 sin 휖, tude6 (arkāgraguṇa) at the time of rising or set- the above relation can be written as ting, the verses first present two relations to deter- 푅 sin 휆 × 푅 sin 휖 mine the same, and then describe a right-angled 푅 sin 퐴′ = , 푅 cos 휙 triangle having the mahīguṇa and the Rsine of the declination as sides, and the arkāgraguṇa as the which is the same as (4). 6Amplitude (퐴′) refers to the angular distance of separation from the east or west point of the horizon. It is related to the azimuth (퐴) by the relation 퐴 = 90 ± 퐴′, with the positive sign employed when the arkāgraguṇa is south, and negative sign employed when it is north. 306 INDIAN JOURNAL OF HISTORY OF SCIENCE

Expression for mahīguṇa 푋퐺푋′ is the right-angled triangle which the author alludes to in this verse. It may also be noted Having defined arkāgraguṇa in the first three that upon substituting (5) in the above expression quarters of verse 19, the author proceeds to iden- for the mahīguṇa (퐺푋′), the expression reduces to tify what mahīguṇa or bhūjyā is geometrically in (1). the last quarter of the verse. It is said that the right-angled triangle having arkāgraguṇa as its hy- Expressions for cara potenuse, has the mahīguṇa and the Rsine of the declination as its sides. This can be understood In verse 20, the author gives the following rela- from Fig. 2a, which is the same as Fig. 1, but ro- tions to determine the Rsine of the ascensional dif- tated about its axis for ease of visualisation. This ference, in terms of the relations described above: figure is depicted in greater detail in Fig. 2b, where bhāskarāgraguṇa × palamaurvikā ′ carajīvā = , 푋 is the projection of the point 푋 on the equato- dinamaurvī ′ rial plane. The planar right-angled triangle 푋푂푋 푅 sin 퐴′ × 푅 sin 휙 푅 sin Δ훼 = , (6) lies in the meridian plane passing through 푋 along 푅 cos 훿 which the declination is measured. Therefore, we and, have 푋푂푋̂ ′ = 훿, and hence dorguṇa × antimamahīguṇa carajīvā = 푋푋′ = 푅 sin 훿, 푂푋′ = 푅 cos 훿. dinamaurvī 푅 sin 휆 × 푅 sin 푀 i.e., 푅 sin Δ훼 = 푙 , (7) The right-angled7 triangle 푂퐺푋 lies on the hori- 푅 cos 훿 ′ zon, and 퐺푂푋̂ = 퐴 . Therefore, where 푅 sin 푀푙 is the last earth-sine, or the earth- sine when the declination of the Sun is maximum 푋퐺 = 푅 sin 퐴′, 푂퐺 = 푅 cos 퐴′. and equal to the obliquity of the ecliptic. There- fore, substituting 훿 = 휖 in (1), we have ′ The right-angled triangle 푋퐺푋 lies in a plane per- 푅 sin 휖 × 푅 sin 휙 pendicular to the equator, and parallel to the plane 푅 sin 푀 = . (8) 푙 푅 cos 휙 of the prime meridian. Here, we have already de- The rationale behind (6) can be understood with termined 푋퐺 and 푋푋′ as the Rsines of the ampli- the help of Fig. 3. This figure is the same as Fig. 2b, tude and declination respectively. As in this trian- only modified to highlight the similar triangles gle 푂퐺푋′ and 푂푄퐹 . From Fig. 2a, we know that 퐸퐹 퐺̂ = 휙′ = 90 − 휙, is the arc corresponding to the ascensional differ- we have ence, which means that in the planar right-angled 퐺푋′ = 푅 sin 퐴′ sin 휙. triangle 푂푄퐹 we have the angle 푄푂퐹̂ = Δ훼. As the side 푂퐹 = 푅 in this triangle, we obtain This third side 퐺푋′ is nothing but the earth-sine or the mahīguṇa. This can be understood as fol- 퐹 푄 = 푅 sin Δ훼. lows. Consider the diurnal circle passing through We have already shown that 퐺푋′ = 푅 sin 퐴′ sin 휙, 푋 in Fig. 2a. A perpendicular dropped from 푋 and 푂푋′ = 푅 cos 훿. Therefore, considering the onto the radius of this circle, which is parallel to similar triangles 푂퐺푋′and 푂푄퐹 , we have ′ the line 푂퐸, will be the earth-sine. 퐺푋 is the im- 푅 sin Δ훼 푅 sin 퐴′ sin 휙 age of this line on the equatorial plane. Therefore, = , 푅 푅 cos 훿 7The triangle does not appear right-angled in the figure due to the difficulty in depicting the three-dimensional celestial sphere on a two-dimensional surface. DETERMINATION OF ASCENSIONAL DIFFERENCE IN THE LAGNAPRAKARAṆA 307

푃 푍 Δ훼

푁

90 − 훿 diurnal 6 o’clock circle 푆 (Sun) circle 푌 푋 equator 휙 훿 Ω 퐸 Δ훼 퐹 ecliptic horizon

(a) Rotated celestial sphere.

푃

푍

푁 푊

′ 푂 휙 푉 푈 훿 푋 퐴′ equator 휙′ 퐺 푋′

퐸 horizon (b) Planar right-angled triangle formed by the amplitude, the declination, and the earth-sine.

Fig. 2. Visualising the triangle which has the amplitude, the declination and the earth-sine as its sides. 308 INDIAN JOURNAL OF HISTORY OF SCIENCE or, koṭi of the desired declination multiplied by the 푅 sin 퐴′ × 푅 sin 휙 maximum ascensional difference (caramacara) 푅 sin Δ훼 = , 푅 cos 훿 and divided by the day-radius (dyujyā) is the which is the same as (6). Note that (6) can also Rsine of the ascensional difference (carajyā). be validated by simply substituting the relation for The above verse is composed in the long 푅 sin 퐴′ from (5) into it, upon which the given re- sragdharā metre, consisting of twenty-one sylla- lation reduces to (3). Similarly, (7) also reduces to bles per quarter. The choice of the long metre for (3) upon substituting for 푅 sin 푀푙 using (8) in it. this verse stems from the fact that the author gives several relations here. The first two relations are METHOD 3 meant for defining a quantity called the koṭi. Next, he defines a relation for determining the radius of दोवग भुजापमकृतरहता the diurnal circle (dyujīvā) in terms of koṭi. This is मूलमैव कोटः followed by an expression for the Rsine of the as- दोजीवाु भगुणवता censional difference (carajyā) in terms of koṭi and dyujīvā. The expressions for koṭi along with their वापमा कोटः । equivalent in mathematical notation are:8 तगद कोटीगुणकृतसहता मूलमाुजीवा koṭi = √(dorjyā)2 − (bhujāpakramajyā)2 इापाकोटरमचरहता 2 2 यााु चरा ॥२१॥ = √(푅 sin 휆) − (푅 sin 훿) , (9) dorjyāvargād bhujāpakramakṛtirahitāt or, mūlamasyaiva koṭiḥ dorjyā × antyadyujīvā koṭi = dorjyāghnyantyadyujīvā tribhaguṇavihṛtā tribhaguṇa vāpamasyāsya koṭiḥ | 푅 sin 휆 × 푅 cos 휖 = . (10) tadvargādatra koṭīguṇakṛtisahitāt 푅 mūlamāhurdyujīvām The dyujīvā and carajyā are given as iṣṭāpakrāntikoṭiścaramacarahatā dyujyayāptā carajyā ||21|| dyujīvā = √(koṭi)2 + (koṭīguṇa)2 The square-root taken from—the square of the or, 푅 cos 훿 = √(koṭi)2 + (푅 cos 휆)2, (11) Rsine (dorjyā) [of the Sun’s longitude] de- and creased by the square of [the Rsine of] the dec- koṭi × caramacarajyā lination corresponding to the longitude (bhujā- carajyā = pakrama)—is the koṭi of this [declination] it- dyujyā koṭi × 푅 sin Δ훼 self. Or, the last day-radius (antyadyujīvā) mul- or, 푅 sin Δ훼 = 푚 , (12) tiplied by the Rsine (dorjyā) [of the Sun’s lon- 푅 cos 훿 gitude] and divided by the radius (tribhaguṇa) where Δ훼푚 is the maximum ascensional difference is the koṭi corresponding to this declination which occurs at the maximum declination of the (apama). Here, [scholars] state the square root Sun. Therefore, substituting 훿 = 휖 in (1) and (2), taken from—its (koṭi) square increased by the from the two equations we have square of the Rcosine (koṭīguṇa) [of the Sun’s 푅 sin 휖 × 푅 sin 휙 푅 푅 sin Δ훼 = × . (13) longitude]—to be the day-radius (dyujīvā). The 푚 푅 cos 휙 푅 cos 휖 8The phrase ‘koṭi corresponding to this declination’ in the verse is not to be confused with the Rcosine of the declination here. Instead, it is to be taken as a technical term denoting the given mathematical expressions. DETERMINATION OF ASCENSIONAL DIFFERENCE IN THE LAGNAPRAKARAṆA 309

푃

푍

푁 푊

′ 푂 휙 푉 푈 훿 푋 Δ훼 ′ equator 휙′ 푋 퐺 푄 퐹 퐸 horizon

Fig. 3. Determining the relation for the cara.

Rationale for the expressions for koṭi and dyujīvā figure. In this triangle, we have the hypotenuse 푂푆′ = 푅 cos 훿, and the sides 푂푇 = 푅 cos 휆, and We now show that the expression for the koṭi de- 푇 푆′ = koṭi. Since 푂푆′2 = 푇 푆′2+푂푇 2, we obtain scribed in this verse corresponds to the line seg- (11). ment 푇 푆′ shown in Fig. 4, which depicts the celestial sphere from the point of view of the equatorial plane. Here, let 푆 be the Sun, having longitude 휆 and declination 훿. In the right-angled triangle Rationale for the expression for the carajyā 푆푇 푆′, we have the side 푆푆′ = 푅 sin 훿, and the hypotenuse 푇 푆 = 푅 sin 휆. As 푆푇̂ 푆′ is nothing The given expression for the carajyā or the Rsine but the angle between the equator and the ecliptic, of the ascensional difference can be understood we have 푆푇̂ 푆′ = 휖. Therefore, with the help of Fig. 3 as well as Fig. 4. In Fig. 3, noting that 푂푁 = 푅, and that 퐺푋′ is 9 ′ 푇 푆′ = 푅 sin 휆 cos 휖. the mahīguṇa, from similar triangles 푋퐺푋 and 푁푂푈, we have This is equivalent to the expression for koṭi given ′ in (10). In the right-angled triangle 푆푇 푆 , we also 푅 sin 휙′ 푅 sin 훿 = mahīguṇa × . have 푅 cos 휙′ 푇 푆′2 = 푇 푆2 − 푆푆′2. Noting ′ koṭi, and substituting the values of ′ 푇 푆 = In Fig. 4, noting that 푂푆1 = 푅 and 푇 푆 is the koṭi, the other two sides of the triangle in this expres- ′ from similar triangles 푆푇 푆 and 푆1푂퐷, we have sion yields (9). Similarly, (11) can be easily derived by consid- 푅 sin 휖 ering the right-angled triangle ′ in the same 푅 sin 훿 = koṭi × . 푂푇 푆 푅 cos 휖 9See Method 2. 310 INDIAN JOURNAL OF HISTORY OF SCIENCE

푃 퐾

푆1 Ω

푂 휖 퐻 퐷 훿 푆 휆 equator 휖 푇 푆′

Γ ecliptic

푆 푆

푅 훿 푅 sin 휆 훿 sin sin 푅 푅 훿 휖 푂 푅 cos 훿 푆′ 푇 푅 sin 휆 cos 휖 푆′ 푂 푂

휆

휆 푅 휆 푅 cos 훿 cos cos 푅 푅

푇 푅 sin 휆 푆 푇 푅 sin 휆 cos 휖 푆′ Fig. 4. Understanding the term koṭi. DETERMINATION OF ASCENSIONAL DIFFERENCE IN THE LAGNAPRAKARAṆA 311

Equating the above two expressions, noting 휙′ = The cara is then given in terms of the kālajīvā as: 90 − 휙, and rearranging, we have the mahīguṇa kālajīvā × paramacarajyā koṭi × 푅 sin 휖 × 푅 sin 휙 carajyā = 푅 sin 푀 = tribhajīvā 푅 cos 휖 × 푅 cos 휙 푅 sin 훼 × 푅 sin Δ훼푚 koṭi × 푅 sin Δ훼푚 or, 푅 sin Δ훼 = . (15) = . [using (13)] 푅 푅 Now, multiplying both sides by 푅 , and using The expression for the right ascension given by 푅 cos 훿 (2), we have (14) can be understood using Fig. 5. This figure is the same as Fig. 4, only modified to highlight the koṭi × 푅 sin Δ훼 푅 sin Δ훼 = 푚 , similar triangles 푂푇 푆′ and 푂퐶퐵. Here, the arc 푅 cos 훿 Γ퐵 represents the right ascension of the Sun, and which is the required expression (12). therefore 퐵푂퐶̂ = 훼. As 푂퐵 = 푅, we have the semi-chord METHOD 4 퐶퐵 = 푅 sin 훼. 10 परमगुणाहताु भुजाा We also already know that वतेगुणेनु कालजीवा । ′ ′ परमेण चरेण ताडता सा 푇 푆 = 푅 sin 휆 cos 휖, and 푂푆 = 푅 cos 훿.

भजीवापताथवा चरा ॥२२॥ ′ paramadyuguṇāhatā bhujājyā Now, from similar triangles 푂푇 푆 and 푂퐶퐵, we vihṛteṣṭadyuguṇena kālajīvā | have parameṇa careṇa tāḍitā sā 푅 sin 훼 푅 sin 휆 cos 휖 = tribhajīvāpahṛtāthavā carajyā ||22|| 푅 푅 cos 훿 푅 sin 휆 × 푅 cos 휖 Or, the Rsine (bhujājyā) [of the Sun’s lon- or, 푅 sin 훼 = , gitude] multiplied by the extreme (last) day- 푅 cos 훿 radius (paramadyuguṇa) and divided by the which is the same (14). desired day-radius (iṣṭadyuguṇa) is the Rsine From (10) and (14), we have of the right ascension (kālajīvā). That (kāla- jīvā) multiplied by [the Rsine of] the maximum koṭi 푅 sin 훼 = . ascensional difference (parama cara), and di- 푅 cos 훿 푅 vided by the radius (tribhajīvā), is the Rsine of the ascensional difference (carajyā). One can easily see that substituting the above expression in (12) yields (15). The above verse composed in the mālabhāriṇī metre prescribes the formula for obtaining cara in METHOD 5 terms of the kālajīvā or the Rsine of the Sun’s right ascension. For this purpose, the verse first gives an ाघातलशगुणता ामौवका । expression for the kālajīvā as follows: ााु चरजीवा ा तापं ह चरासवः ॥२३॥ bhujājyā × paramadyuguṇa kālajīvā = trijyākṣaghātalambāṃśa- iṣṭadyuguṇa guṇitā krāntimaurvikā | 푅 sin 휆 × 푅 cos 휖 dyujyāptā carajīvā syāt or, 푅 sin 훼 = . (14) 푅 cos 훿 taccāpaṃ hi carāsavaḥ ||23|| 10See Method 3. 312 INDIAN JOURNAL OF HISTORY OF SCIENCE

푃 퐾

푆1 Ω

푂 휖 퐻 퐷 훿 푆 훼 equator 휖 푆′ 푇 퐶 퐵 Γ ecliptic

Fig. 5. Derivation of kālajīvā or right ascension.

The Rsine of the declination (krāntimaurvikā) 3. APPLICATION OF THE ASCENSIONAL multiplied by one-lambath11 part of the prod- DIFFERENCE uct of the radius (trijyā) and [the Rsine of] the latitude (akṣa) and divided by the day-radius ातरं वणगजादवशानण (dyujyā) would be the Rsine of the ascensional अेऽथा न तदहनशयोु मे । difference (carajīvā). Indeed, its arc would काय पमलवादषु चानुपाता be the ascensional difference in arc-minutes नैवानुपातवधर घटीसारे ॥२४॥ (carāsavaḥ). prātaścaraṃ vaṇigajādivaśāddhanarṇam The above verse in the anuṣṭubh metre gives the aste’nyathā na tadaharniśayostu madhye | following relations: kāryaṃ tripañcamalavādiṣu cānupātāt trijyā×akṣajyā × krāntimaurvikā naivānupātavidhirasya ghaṭīprasāre ||24|| lambajyā carajīvā = dyujyā At sunrise, the ascensional difference is posi- 푅×푅 sin 휙 × 푅 sin 훿 tive or negative depending on [the Sun’s posi- 푅 cos 휙 or, 푅 sin Δ훼 = , (16) tion with regard to] Libra or Aries. While set- 푅 cos 훿 ting, it is otherwise. That should not be applied and, during midst of the day and night. And in case carāsu = cāpa(carajīvā) of third, fifth etc. portions [of the day, the cara] should be calculated [afresh] successively. The or, Δ훼 = 푅 sin−1(푅 sin Δ훼). (17) rule of proportion for this (cara) will not be The above expressions are a restatement of (2), applicable in [setting] the motion of the clock upon substituting for the mahīguṇa from (1). (ghaṭīprasāra).

11 The term lamba refers to the Rcosine of the latitude. The compound word लश is to be understood here as लेन वभाः अंशः। This is an unusual way to express division by a variable. DETERMINATION OF ASCENSIONAL DIFFERENCE IN THE LAGNAPRAKARAṆA 313

This verse composed in the vasantatilakā me- mine 푡휙 is summarised in Table 1. tre prescribes rules for the application of the cara depending upon the time of the day, and the Sun’s 3.2. Application of cara at noon and midnight position on the ecliptic for an observer in the northern hemisphere. The verse also briefly comments It is clearly stated in the verse that the correction on the nature of the cara function. due to ascensional difference is zero at the midst of the day and the night. In other words, the magnitude of the cara at the prime meridian is zero at 3.1. Application of cara at sunrise and sunset mid-noon and mid-night when the Sun reaches the When the Sun is in the northern hemisphere (i.e. prime meridian either during the day or the night, its longitude is in the range of 0∘ to 180∘, referred irrespective of the latitude of the observer. This to as ajādi or ‘beginning with Aries’ in the verse), is another way of stating that the Sun reaches the the cara is to be subtracted in the morning, and prime meridian at the same instant for all observers added in the evening to obtain the time of the local on a given longitude. sunrise and sunset respectively. This is because, After discussing the application of cara at sun- during this period, the Sun rises earlier and sets rise, sunset, noon, and midnight, the author makes later than at the equator, where sunrise and sunset noteworthy remarks about the very nature of the occur at six o’clock throughout the year. cara function itself, not usually found in other as- Fig. 6a depicts the cara at the time of sunrise tronomical works. when the Sun has northern declination. Here, it can be seen that the Sun reaches the horizon at 3.3. The nature of the cara function point 푋 before crossing the six o’clock circle at point 푌 . Therefore, the Sun rises before 6 a.m. In the latter half of the verse, the author makes the The cara gives the magnitude of this difference following remarks regarding the nature of the cara and needs to be subtracted from six o’clock to de- function from a mathematical viewpoint: termine the local time of sunrise. Fig. 6b depicts (i) the cara has to be computed successively the cara at the time of sunset when the Sun has during every third part, or fifth part etc. of northern declination. Here, it can be seen that the the day, and Sun first reaches the six o’clock circle at point 푌 , before it sets at point 푋. Therefore, the Sun sets (ii) the variation in cara cannot be taken to be after 6 p.m., and the cara needs to be added to de- proportional to the change in time for setting termine the time of local sunset. the clock. On the other hand, when the Sun is in the southern hemisphere, the cara is to be applied positively It seems to us that the intent of the author here is to in the morning, and negatively in the evening to caution against assuming a linear variation in cara obtain the time of local sunrise and sunset respec- from sunrise to noon, and also from noon to sunset. tively. This is because, during this period, the Sun This prescription to determine cara successively at rises later and sets earlier than for observers at the various intervals during the day, and not to apply equator, and the duration of the day is shorter for the method of proportion while setting the clock observers in the northern hemisphere. Figures 6c is quite interesting and gives us a clue to the fact and 6d depict this situation. If 푡푒 is the time of that the day—irrespective of its duration— was sunrise or sunset at the equator, and 푡휙 is the time divided into a fixed number of units, for certain of sunrise or sunset at a desired latitude (휙), then purposes. Clearly, the duration of each such unit application of cara in different situations to deter- would vary over the course of the year along with 314 INDIAN JOURNAL OF HISTORY OF SCIENCE

푍 푍

푆

diurnal circle 푃 푃

equator 6 o’clock circle 휙 휙 Δ훼

ecliptic Δ훼 diurnal circle Γ 6 o’clock circle 푁 푁 푌 푌 horizon horizon

퐸 푋 푋 푊 퐹 퐹 푆 Γ equator ecliptic

(a) (b)

푍 푍

ecliptic

equator 푃 푃

diurnal circle Δ훼 휙 휙 Δ훼 6 o’clock circle

푁 푁 퐹 horizon 퐹 푆 6 o’clock circle horizon 푊 푋 푋 퐸 Ω 푌 Ω 푌 ecliptic

equator 푆 diurnal circle

Fig. 6. Application of cara at sunrise and sunset for an observer in the northern hemisphere: (a) and (b) Sun having northern declination, (c) and (d) Sun having southern declination.

Table 1. Application of cara in determining the time of local sunrise and sunset for an observer in the northern hemisphere.

Sun’s declination Sunrise Sunset

North 푡휙 = 푡푒 − Δ훼 푡휙 = 푡푒 + Δ훼 South 푡휙 = 푡푒 + Δ훼 푡휙 = 푡푒 − Δ훼 DETERMINATION OF ASCENSIONAL DIFFERENCE IN THE LAGNAPRAKARAṆA 315 day length, as indicated by the following verse of be familiar to most students of Indian astronomy Śrīpati:12 as that which is discussed in standard astronomical texts like the Siddhāntaśiromaṇi.13 The sub- दन यः पदशो वभागो sequent methods are more interesting and involve राेथा त मुतमान । determining the cara using parameters like the am- dinasya yaḥ pañcadaśo vibhāgo plitude of the Sun, the maximum earth-sine, the rātrestathā taddhi muhūrtamānam | maximum cara, and so on. Methods 2, 3 and 4 in Muhūrta is that [span of time] which is [equal particular showcase a great dexterity and artistry to] one fifteenth of a day; and also of the night in conceiving of various planar triangles and pro- similarly. jections that lead to a variety of formulae and approaches for arriving at the same result. This in- The verse clearly states a muhūrta is equal to one- deed reveals a strong grasp of mathematical and fifteenth of a day, indicating that it is not a fixed astronomical concepts. time unit, but rather varies with the length of the While method 5 is a modification of method 1, day (and night) over the course of the year. the subsequent discussion on the application of the Now, in the Lagnaprakaraṇa, the author seems cara at different times of the year and during dif- to be indicating that the variation in the cara is to ferent times of the day is important to help students be considered even while measuring time during understand the practical application of an other- a day, and that this variation cannot be considered wise abstract astronomical concept. The author to be proportional to the change in time. In any deserves praise for succinctly conveying several case, it is quite interesting and noteworthy that the mathematical results in a single verse, and in one author presents such discussions in the context of particular case, all the information contained in Ta- application of cara. Obviously, building any such ble 1 in less than two quarters of a verse. clock which takes into account the non-linear vari- The variety of approaches presented here to ation of cara would be quite complicated. How- compute the same physical quantity suggests that ever, it may be noted that in the calculation of the purpose of this work was more than just provid- the śaṅku and the dṛggati in verses 53–56 of the ing users with a way to compute the cara. It was Lagnaprakaraṇa, the author makes use of the cara fundamentally about showcasing the number of which is determined using the declination of the ways in which the same phenomenon could be de- Sun (or an ecliptic point ninety degrees behind it) scribed mathematically, and thus demonstrating a at a desired instant, rather than at sunrise. There- certain mathematical ingenuity and facility. Taken fore, it is clear that the author definitely had a the- in conjunction with the thorough exposition on oretical use for this quantity, whether or not it had the prāṇakalāntara in this same text,14 the above any practical application. discussion certainly reveals a deftness and robust- ness of approach towards solving problems which 4. CONCLUSION are signs of a pure mathematician. Being privy to subsequent discussions in the Lagnaprakaraṇa, Of the various approaches for determining the we are confident that our future investigations will cara described in the article, the first method will only further strengthen this claim. 12The given verse is attributed to Śrīpati by the commentator of the Muhūrtakalpadruma. See Viṭṭhala Dı̄ kṣita (Muhūrta- kalpadruma, p. 43). The exact origin is unclear. Similar verses are found in many muhūrta texts. 13See Bhāskarācārya (Siddhāntaśiromaṇi, p. 114). 14See Kolachana, Mahesh, and Ramasubramanian (“Mādhava’s multi-pronged approach for obtaining the prāṇa- kalāntara”). 316 INDIAN JOURNAL OF HISTORY OF SCIENCE

ACKNOWLEDGEMENTS Jñānarāja. Siddhāntasundara. Trans. and comm. by Toke Lindegaard Knudsen. Baltimore: John Hopkins Univer- sity Press, 2014. The authors would like to place on record their sincere gratitude to MHRD for the generous sup- Jyeṣṭhadeva. Gaṇita-yukti-bhāṣā. Ed. and trans. by K. V. Sarma. With a comment. by K. Ramasubramanian, M. D. port extended to them to carry out research ac- Srinivas, and M. S. Sriram. 2 vols. Culture and History tivities on Indian science and technology by way of Mathematics 4. New Delhi: Hindustan Book Agency, of initiating the Science and Heritage Initiative 2008. (SandHI) at IIT Bombay. They also would like to Kolachana, Aditya, K. Mahesh, and K. Ramasubramanian. pay their obeisance to the late Prof. K. V. Sarma, “Mādhava’s multi-pronged approach for obtaining the who saved the Lagnaprakaraṇa for future gen- prāṇakalāntara”. In: Indian Journal of History of Science erations by painstakingly copying the crumbling 53.1 (2018), pp. 1–15. manuscripts of the text. Finally, they also would Lalla. Śiṣyadhı̄ vṛddhidatantra. Ed., trans., and comm. by like to acknowledge the K. V. Sarma Research Bina Chatterjee. 2 vols. New Delhi: Indian National Sci- Foundation, Chennai, for sharing the manuscripts, ence Academy, 1981. which enabled them to study this interesting and Mādhava. “Lagnaprakaraṇa”. Manuscript 414 B, ff. 53– important work. 84. Kerala University Oriental Research Institute and Manuscripts Library, Thiruvananthapuram. –“Lagnaprakaraṇa”. KVS Manuscript No. 37a. Prof. K. V. Sarma Research Foundation, Chennai. BIBLIOGRAPHY –“Lagnaprakaraṇa”. KVS Manuscript No. 37b. Prof. K. V. Sarma Research Foundation, Chennai. Āryabhaṭa. Āryabhaṭı̄ ya. Ed., trans., and comm. by K. S. Shukla and K. V. Sarma. 3 vols. New Delhi: Indian Na- Nı̄ lakaṇṭha Somayājı̄ . Tantrasaṅgraha. Trans. and comm. tional Science Academy, 1976. by K. Ramasubramanian and M. S. Sriram. Culture and History of Mathematics 6. New Delhi: Hindustan Book Bhāskarācārya. Siddhāntaśiromaṇi. With a comment. by Agency, 2011. Satyadeva Sharma. Varanasi: Chaukhamba Surabharati Prakashan, 2007. Putumana Somayājı̄ . Karaṇapaddhati. Trans. and comm. by Venkateswara Pai et al. Culture and History of Mathemat- Brahmagupta. Brāhmasphuṭasiddhānta. Ed. and comm. by ics 9. New Delhi: Hindustan Book Agency, 2017. Sudhakara Dvivedi. Reprint from The Pandit. Varanasi (formerly Benares): Printed at the Medical Hall Press, Vaṭeśvara. Vaṭeśvarasiddhānta. Ed., trans., and comm. by 1902. Kripa Shankar Shukla. 2 vols. New Delhi: Indian Na- tional Science Academy, 1986. Gangooly, Phanindralal, ed. Sūryasiddhānta. Trans. and comm. by Ebenezer Burgess. Delhi: Motilal Banarasidass Viṭṭhala Dı̄ kṣita. Muhūrtakalpadruma. Ed. by Sureshchan- Publishers Private Limited, 2000. dra Mishra. New Delhi: Ranjan Publications, 2005.