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].

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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.

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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. we have 푀푅 = 90, which is the relation stated in (2). The above verses (in the anuṣṭubh metre) essentially in- It may be noted that since the kālalagna is the same troduce the concept of madhyakāla, which is the right as- for all observers on a given longitude,10 the rāśikūṭa- cension of the point at the intersection of the equator and lagna and the madhyalagna are the same too for these the prime meridian, represented by point 푇 in Figure 3. observers. They also present expressions detailing the relationship between the madhyakāla and the kālalagna. 2.2 Computation of the madhyakāla In Figure 3, let the point 푀 represent the madhyalagna, which has been discussed in the previous verse. Taking 훼 मले पुनः कुय配 푡 as the right ascension of 푇, and 휆 as the longitude of 푀, नजप्राणकलार륍 । 푚 the relation given in the verse for the madhyakāla can be मकालो भवेोऽयं expressed as: त्रभां काललक륍 ॥३२॥ काललं त्रराूनं madhyakāla = madhyalagna ± nija-prāṇakalāntara मकालः प्रकत셍तः । or, 훼푡 = 휆푚 ± |휆푚 − 훼푡|. (3) madhyalagne punaḥ kuryāt nijaprāṇakalāntaram | Having defined the madhyakāla thus, starting with the madhyakālo bhavet so’yaṃ latter half of verse 32, the author describes the connection tribhāḍhyaṃ kālalagnakam ||32|| between the kālalagna and the madhyakāla through the kālalagnaṃ trirāśyūnaṃ following relations: madhyakālaḥ prakīrtitaḥ | kālalagna = madhyakāla + tribha

One should again apply the nija-prāṇa-kalāntara or, 훼푒 = 훼푡 + 90, (4) 8푀퐸 = 90 as 퐸 is the pole for any point on the prime meridian. and 푀퐾 = 90 as 퐾 is the pole of any point on the ecliptic. 9Except when the points are separated by 180 degrees, two points are madhyakāla = kālalagna − trirāśi sufficient to define a unique great circle on asphere. 10See verse 30 in [7]. or, 훼푡 = 훼푒 − 90. (5)

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푍

푃 푀

ecliptic 휆 푚 푇 훼 푡

equator

푊 Γ

푁 horizon Ω 퐸

Figure 3 Determining the madhyalagna and the madhyakāla.

The relation between the madhyakāla and the madhya- मलमवाते । lagna given in (3) can be easily understood with the help madhyakāle punastasmin of Figure 3. Here, both the points 푇 and 푀 lie on the prime koṭiprāṇakalāntaram ||33|| meridian, which immediately indicates that 훼푡 is the right vyastaṃ kuryāt tadā vātra ascension of a body at 푀. Thus, if the madhyalagna (휆푚) madhyalagnamavāpyate | is already known, then the madhyakāla (훼 ) can be read- 푡 Again, one should apply the koṭi-prāṇa-kalāntara ily found by applying the prāṇakalāntara to it.11 That is, to that madhyakāla reversely. Then also the merid- ian ecliptic point (madhyalagna) is obtained here. 훼푡 = 휆푚 ± |휆푚 − 훼푡|, which is the first relation given in the verse. The latter half of verse 33 and the first half of verse As the east cardinal point (퐸) is the pole for any point 34 (both in the anuṣṭubh metre) together present the fol- on the prime meridian, it is also evident from the figure lowing expression to determine the madhyalagna or the that the kālalagna (훼푒) is given by meridian ecliptic point from the madhyakāla:

훼푒 = 훼푡 + 90, madhyalagna = madhyakāla ∓ koṭi-prāṇakalāntara which explains the relations (4) and (5) stated in the verse. or, 휆푚 = 훼푡 ∓ |휆푚 − 훼푡|. (6)

It is easily seen that the above relation is a corollary of 2.3 An alternate way of obtaining the (3). The koṭi-prāṇakalāntara is applied here as we want madhyalagna to convert the meridian ecliptic point’s right ascension 12 मकाले पुनꅍ into its longitude. Furthermore, it must be noted that कोटप्राणकलार륍 ॥३३॥ the prāṇakalāntara is applied reversely here for the same ं कुय配 तदा वात्र 12Refer to our discussion on nija-prāṇakalāntara and koṭi- 11Note that for a point on the ecliptic, there is only one kind of prāṇa- prāṇakalāntara in verse 31. The points 푇 and 푀 here are analogous kalāntara. to points 퐵 and 퐶 in Figure 1.

308 IJHS | VOL 54.3 | SEPTEMBER 2019 ARTICLES reason, as we want to convert right ascension into longi- two quantities, as well as the Rsine and Rcosine of the lat- tude.13 itude (휙), is stated as follows:

madhyaguṇa = (akṣajyā × doḥkrāntikoṭi ± 2.4 Determining the madhyajyā akṣakoṭijyā × doḥkrāntijīvā) ÷ trijyā मला配 पुना配 दोःकोोः क्रामानये配 ॥३४॥ or, दोःक्राकोाक्षममु (푅 sin 휙 × 푅 cos 훿 ± 푅 cos 휙 × 푅 sin 훿 ) 14 푅 sin 푧 = 푚 푚 . कोा दोःक्राजीव車 च नह भूयः । 푚 푅 तोगभेदा配 समभदे (7) त्रातं मगुणं वद ॥३५॥ The verses further state that the sign in the above re- lation has to be taken as positive or negative depend- madhyalagnāt punastasmāt ing upon whether the latitude and declination are in the doḥkoṭyoḥ krāntimānayet ||34|| ‘same’ or ‘opposite’ directions. This remark, as well as the doḥkrāntikoṭyākṣamamuṣya validity of the above expression, can be understood from koṭyā doḥkrāntijīvāṃ ca nihatya bhūyaḥ | Figure 4. The figure depicts the madhyalagna (푀) when tadyogabhedāt samabhinnadiktve it has northern as well as southern declination. The figure trijyāhṛtaṃ madhyaguṇaṃ vadanti ||35|| also depicts its declination (훿푚), as well as zenith distance Again, from that meridian ecliptic point (madhya- (푧푚), in these two cases. As can be seen from the figure, lagna), one should derive the Rsine and Rcosine the zenith distance of 푀 is equal to 휙+훿푚 when the eclip- of its declination. Having multiplied (i) the [Rsine tic is in the same direction with respect to both the zenith

of the] latitude (akṣa) with the Rcosine of the dec- and the equator, and 휙 − 훿푚 when the equator and the lination corresponding to the longitude (doḥkrān- zenith are on either side of the ecliptic.16 We therefore tikoṭi) [of the madhyalagna], and also (ii) the Rsine have of the declination corresponding to the longitude (doḥkrāntijīvā) [of the madhyalagna] with the sin 푧푚 = sin(휙 ± 훿푚) = sin 휙 cos 훿푚 ± cos 휙 sin 훿푚, Rcosine of this [latitude], their sum or difference— or, depending upon [whether the equator and the zenith are in the] same or opposite direction [with (푅 sin 휙 × 푅 cos 훿 ± 푅 cos 휙 × 푅 sin 훿 ) 푅 sin 푧 = 푚 푚 respect to the ecliptic]—divided by the radius (tri- 푚 푅 jyā), is stated to be the madhyaguṇa. which is the same as (7). Now, a brief note on the advantage of the choice of form In one and a half verses (half in the anuṣṭubh and full in of the rule prescribed by (7). Naively, it may appear that the indravajrā metres) Mādhava gives the relation for cal- expanding sin(휙 ± 훿 ) and determining the sum or dif- culating the madhyajyā (i.e. madhyaguṇa) or the Rsine 푚 ference of products of the sine and cosine functions could of the zenith distance (푧 ) of the meridian ecliptic point 푚 be more cumbersome than directly determining the sine (madhyalagna). To this end, the verses instruct that first of the total quantity. However, this need not be the case the Rsine and Rcosine of the declination (훿 ) correspond- 푚 if the constituent terms of the expansion were already ing to the longitude of the madhyalagna should be calcu- known to the practitioners, who could then easily deter- lated.15 The relation for the madhyajyā in terms of these mine the result by the simple summation of two products. 13The prāṇakalāntara as defined in the first chapter ofthe Lagna- For instance, the sine of the declination of the madhya- prakaraṇa is for converting the longitude of an ecliptic point to the lagna (sin 훿푚) can be derived directly using the relation corresponding right ascension. Thus, it is prescribed to be applied ‘re- versely’ here. declination can be determined easily using the well known relation 14 The available manuscripts give the reading as न. This however sin 훿 = sin 휆 sin 휖. This would be the doḥkrāntijīvā. Its cosine would appears to be a transcribing error as the correct relation requires mul- be the doḥkrāntikoṭi. 16 tiplication and not division. Here, 훿푚 represents only the magnitude of the declination of the 15 Knowing the longitude 휆푚 of the madhyalagna, the sine of its madhyalagna.

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휙

푧푚 푧푚 휙 푍 푍 푀 훿푚

ecliptic 푇 푇 푃 푃 훿푚 equator equator

푀 ecliptic

Γ Ω 푁 푁 horizon horizon

퐸 퐿 퐿 퐸

(a) Same direction. (b) Opposite directions.

Figure 4 The direction of the equator and the zenith with respect to the ecliptic for determining the madhyaguṇa. sin 훿 = sin 휆 sin 휖. Its cosine too can be easily determined The dṛkkṣepajyā, or simply the dṛkkṣepa, is the Rsine using basic arithmetic, while the sine and cosine of the of the zenith distance of the dṛkkṣepalagna or the nona- latitude would be readily available for a given location. gesimal.17 This verse (in the śālinī metre) gives an expres-

On the other hand, to directly determine sin(휙 ± 훿푚), one sion for determining the dṛkkṣepa in terms of the madhya- would have to first calculate the inverse sine of the quan- jyā and another intermediary quantity called the bāhu, 18 tity sin 훿푚 to obtain 훿푚, add or subtract this to the latitude, which is defined as follows: and then determine the sine of the composite quantity. madhyajyā × koṭikrānti bāhu = The complex and time consuming task of determining the bāhukrāntikoṭi inverse sine can be avoided by following the prescribed 푅 sin 푧푚 × 푅 cos 휆푚 sin 휖 procedure. = . (8) 푅 cos 훿푚 The quantity madhyajyā thus determined is now used to determine the dṛkkṣepajyā in the next verse. Now, the dṛkkṣepajyā is defined as: (dṛkkṣepajyā)2 = (madhyajyā)2 − (bāhu)2 2 2 2 2.5 Determining the dṛkkṣepajyā or, (푅 sin 푧푑) = (푅 sin 푧푚) − (bāhu) , (9)

कोटक्राेम셍जीवाहताया where 푧푑 corresponds to the zenith distance of the लं बाक्राकोा तु बाः । dṛkkṣepalagna. माया वग셍तो बावग ा शं ा 饃ेपवग셍ः ॥३६॥ Rationale behind the expression for dṛkkṣepajyā koṭikrāntermadhyajīvāhatāyā The expression for the dṛkkṣepajyā given in (9) can be labdhaṃ bāhukrāntikoṭyā tu bāhuḥ | arrived at as follows. In Figure 5, 퐷 represents the madhyajyāyā vargato bāhuvargaṃ drkkṣepalagna or the nonagesimal, which corresponds tyaktvā śiṣṭaṃ syācca dṛkkṣepavargaḥ ||36|| to the highest point of the ecliptic lying above the hori- The result obtained from the koṭikrānti, which is zon. In other words, its zenith distance 푍퐷 (which is the multiplied by the madhyajīvā (madhyajyā), and di- 17The nonagesimal is a point on the ecliptic above the horizon which vided by the bāhukrāntikoṭi is bāhu. The residue is ninety degrees from the rising ecliptic point, and is also the highest obtained after the subtraction of the square of the point of the ecliptic. 18The standard relation for krānti or declination is sin 휆 sin 휖. The bāhu from the square of the Rsine of the madhya- term koṭikrānti is to be instead understood to be cos 휆 sin 휖. The term jyā, would be the square of [the Rsine of] the bāhukrāntikoṭi here is to be understood to mean the cosine of the dec-

dṛkkṣepa. lination of the madhyalagna, i.e., cos 훿푚.

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푍 휙 퐽

훿푚 푇 퐾 푃 훿푚

푀 휙 90 − 휇 6 o’clock circle

퐷 Γ 푆 푁 푅

휇 horizon 퐶 퐸 퐿 ecliptic

equator

Figure 5 Visualising the dṛkkṣepa. dṛkkṣepa), is the least compared to any other point on the the dṛkkṣepa and the zenith distance of the madhyalagna ecliptic. This is only possible when 푍퐷 is perpendicular to respectively. Taking 푍퐷 = 푧푑 and 푍푀 = 푧푚, we have the tangent to the ecliptic at 퐷.19 This in turn implies that ′ ′ the secondary 퐾퐷 from the pole of the ecliptic (퐾), which 푍퐷 = 푅 sin 푧푑, and 푍푀 = 푅 sin 푧푚. is perpendicular to the ecliptic at 퐷, passes through the We also have zenith (푍).20 ′ ′ Also, in the figure, the ecliptic points 푀 and 푅 corre- 푂퐷 = 푅 cos 푧푑, and 푂푀 = 푅 cos 푧푚. spond to the madhyalagna and the rāśikūṭalagna respec- tively, while the equatorial point 푇 corresponds to the Now, as 퐾푍퐷 is perpendicular to the ecliptic, the planar ′ ′ ′ madhyakāla. As 퐸 is the pole for any point on the prime triangle 푍푀 퐷 is right-angled at 퐷 , and lies in a plane meridian, we have 퐸퐽 = 90. Now, let 퐸푅 = 휇. Then, obvi- perpendicular to the ecliptic. Therefore, we have ously 푅퐽 = 90 − 휇. As 푀 is the pole of the great circle arc (푅 sin 푧 )2 = (푅 sin 푧 )2 − (푀′퐷′)2, 퐾퐽푅퐸,21 the arc 푅퐽 also corresponds to the angle between 푑 푚 the prime meridian and the ecliptic. which is the relation given in (9), where the side 푀′퐷′ has From the form of (9), it is evident that the author vi- been called bāhu. sualised a right-angled triangle, with the madhyajyā as the hypotenuse, and the dṛkkṣepajyā and bāhu as sides. Rationale behind the expression for bāhu To help visualise this triangle, the celestial sphere is de- picted from the point of view of the ecliptic plane in Fig- The relation for the bāhu given by (8) can be understood ′ ′ ure 6a. Here, 푍퐷′ and 푍푀′ correspond to the Rsines of from the same right-angled triangle 푍푀 퐷 . In this trian- gle, the angle ′ ′ 19The arcs corresponding to the zenith distances of other points on 푍푀̂ 퐷 = 90 − 휇 the ecliptic will not be perpendicular to it, implying that they will be longer than 푍퐷. corresponds to the angle between the prime meridian and 20The great circle containing the arc 퐾푍퐷 is sometimes referred to as the ecliptic. Using simple trigonometry, we have the dṛkkṣepavṛtta, or the great circle corresponding to the dṛkkṣepa. 21Shown in our discussion of verse 31. 푀′퐷′ = 푅 sin 푍푀 cos(90 − 휇),

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퐾 휇 퐽

푃

prime

meridian 90 − 휇

푍 equator 퐿

푂 푅

휇 푀′ 퐷′ ecliptic 퐹 Γ 퐷 퐸 푀

(a) Visualising the dṛkkṣepa from the point of view of the ecliptic plane.

푍

퐷

푚 푑 푧 푧 ′ sin 퐷

sin 푅 푅 푅 푀퐷 푧 푑

푅 cos sin 푅 bāhu 90 − 휇

′ ′ ′ 푀 bāhu 퐷 푂 푅 cos 푧푚 푀 퐹 푅 cos 푀퐷

(b) Planar triangles used to determine the dṛkkṣepajyā and the dṛkkṣepalagna.

Figure 6 Determining the dṛkkṣepajyā and the dṛkkṣepalagna.

312 IJHS | VOL 54.3 | SEPTEMBER 2019 ARTICLES or When the square of the dṛkkṣepa[jyā] is subtracted

bāhu = 푅 sin 푧푚 sin 휇. (10) from the square of the radius (trijyā), [people] state its (the difference’s) square-root to be the para- The expression for sin 휇 can be determined from the śaṅku. The bāhu [stated in the previous verse] spherical triangle Γ퐸푅 in Figure 5, where we have 퐸푅 = 휇, multiplied by the radius (trijyā) divided by this 푅Γ퐸̂ = 휖, and Γ퐸푅̂ = 90 − 훿 .22 Applying the sine rule in 푚 (paraśaṅku) is converted to arc and applied posi- this triangle, we have tively and negatively in order to the meridian eclip- sin Γ푅 × sin 휖 tic point (madhyalagna) depending on [whether sin 휇 = . (11) sin(90 − 훿푚) the madhyalagna is in the six signs] Capricorn (mṛgādi) etc., or Cancer (karkaṭādi) etc. It (the However, as the rāśikūṭalagna is ninety degrees from the positive or negative application of the arc to the madhyalagna, we have madhyalagna) is reversed when the madhyajyā is northward. Then, the nonagesimal (dṛkkṣepa- Γ푅 = 90 − 푀Γ = 90 − (360 − 휆푚) = 휆푚 − 270. vilagna) would be [obtained]. That added by three Using the above expression for Γ푅 in (11), we have signs is stated to be the rising ecliptic point (udaya- lagna). cos 휆 sin 휖 푅 cos 휆 sin 휖 sin 휇 = 푚 = 푚 . cos 훿푚 푅 cos 훿푚 The two verses above (in the indravajrā and upajāti metres respectively) are essentially meant for providing Substituting the above expression in (10), we have an expression for the ascendant or the udayalagna. The 푅 sin 푧 × 푅 cos 휆 sin 휖 expression for the udayalagna is given in terms of the bāhu = 푚 푚 , 푅 cos 훿푚 dṛkkṣepalagna, which in turn is defined in terms of the paraśaṅku. Hence, the set of verses above first give a re- which is the same as (8). lation for the paraśaṅku or the gnomon corresponding to the nonagesimal, then for the dṛkkṣepalagna or the longi- 2.6 Determining paraśaṅku, dṛkkṣepalagna, and tude of the nonagesimal, and finally for the udayalagna udayalagna or the rising ecliptic point. The relation for the paraśaṅku is given as: 饃ेपवग त्रगुण वग配 ेऽ मूलं परशमाः । ु paraśaṅku = √(triguṇa)2 − (dṛkkṣepajyā)2 त्राहतं बामनेन भं 2 2 चापीकृतं मवलकेऽꅍ ॥३७॥ or, 푅 cos 푧푑 = √푅 − (푅 sin 푧푑) . (12) क्रमानण मृगककटाोः The expression for dṛkkṣepalagna is given to be: ं च तगुणे तु सौे । तदा तु 饃ेपवलकं ा配 bāhu × trijyā dṛkkṣepalagna = madhyalagna ± cāpa ( ) तत्रभं तूदयलमाः ॥३८॥ paraśaṅku dṛkkṣepavarge triguṇasya vargāt or, bāhu × 푅 tyakte’sya mūlaṃ paraśaṅkumāhuḥ | 휆 = 휆 ± 푅 sin−1 ( ) , (13) 푑 푚 푅 cos 푧 trijyāhataṃ bāhumanena bhaktaṃ 푑 where bāhu is given by (8), and paraśaṅku by (12). cāpīkṛtaṃ madhyavilagnake’smin ||37|| Now, the expression for the udayalagna stated in terms kramāddhanarṇaṃ mṛgakarkaṭādyoḥ of the dṛkkṣepalagna in the last quarter of verse 38 is: vyastaṃ ca tanmadhyaguṇe tu saumye | tadā tu dṛkkṣepavilagnakaṃ syāt udayalagna = dṛkkṣepalagna + tribha tatsatribhaṃ tūdayalagnamāhuḥ ||38|| or, 휆푙 = 휆푑 + 90. (14) 22 As 푃 is the pole of the equator, and 푀 is the pole of the great cir- We now provide the rationale behind the above expres- cle arc 퐾퐽푅퐸, we have 푃푇 = 퐽푀 = 90, and 푃퐽 = 푇푀 = 훿푚. ̂ ̂ ̂ Therefore, Γ퐸푅 = 푇퐸푃 − 퐽퐸푃 = 90 − 훿푚. sions.

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Expression for the paraśaṅku where the dṛkkṣepa is to the south of the zenith, the pro- cedure of applying the arc 푀퐷 to the madhyalagna is re- The paraśaṅku is the gnomon dropped from the dṛkkṣepa- verse in the case when the dṛkkṣepajyā is northwards. (point 퐷 in Figure 5) to the horizon. In later verses, lagna The length of the arc 푀퐷 can be determined by consid- this quantity is also referred to as the dṛkkṣepakoṭikā or ering the triangles 퐷퐹푂 and 퐷′푀′푂 shown in Figure 6b. the rāśikūṭaprabhā. The length of this gnomon would be The triangle 퐷퐹푂 lies on the plane of the ecliptic, in which equal to the Rsine of the arc 퐶퐷. As 퐶퐷 = 90 − 푧 , we 푑 푂퐷 = 푅, and 퐷퐹 is the perpendicular dropped on the ra- have dius 푂푀 from 퐷. Thus,

paraśaṅku = 푅 sin(90 − 푧푑) = 푅 cos 푧푑, 퐷퐹 = 푅 sin 푀퐷, and 푂퐹 = 푅 cos 푀퐷.

The triangle 퐷′푀′푂 also lies on the plane of the ecliptic, which is equivalent to (12). where we have already shown in our discussion of the ′ ′ ′ previous verse that 푀 퐷 = bāhu, 푂푀 = 푅 cos 푧푚, and ′ Expression for the dṛkkṣepalagna 푂퐷 = 푅 cos 푧푑. Using (8), it can be shown that

For observers in the northern hemisphere, the dṛkkṣepa- (푂퐷′)2 = (푂푀′)2 + (푀′퐷′)2, lagna is generally south of the zenith, and can be either ′ ′ ′ in the eastern hemisphere or the western hemisphere, de- which implies that triangle 퐷 푀 푂 is right-angled at 푀 . ′ ′ pending upon the longitude of the madhyalagna. When Thus, the triangles 퐷퐹푂 and 퐷 푀 푂 are similar, as they the longitude of the madhyalagna is in the range of 270 are both right-angled, and also share the common angle degrees to 90 degrees (mṛgādi), the dṛkkṣepalagna is in at 푂. Applying the rule of proportionality of the sides of the eastern hemisphere, and when the longitude of the similar triangles, we have madhyalagna is in the range of 90 degrees to 270 degrees bāhu × 푅 푅 sin 푀퐷 = . (16) (karkaṭādi), the dṛkkṣepalagna is in the western hemi- 푅 cos 푧푑 sphere. In Figure 5, where the longitude of the madhya- Substituting this result in (15), we have lagna is mṛgādi, it can be seen that the dṛkkṣepalagna is in the eastern hemisphere, and that its longitude is equal −1 bāhu × 푅 휆푑 = 휆푚 ± 푅 sin ( ) , to the sum of the longitude of the madhyalagna (푀) and 푅 cos 푧푑 the arc 푀퐷. Alternatively, when the dṛkkṣepalagna is in which is the same as (13). the western hemisphere, this arc would have to be sub- tracted from the longitude of the madhyalagna to obtain Expression for the udayalagna the dṛkkṣepalagna. Thus, we have In Figure 5, one observes that the udayalagna (퐿) is 90 de- grees from the pole of the ecliptic (퐾), as well as the zenith 휆푑 = 휆푚 ± 푀퐷. (15) (푍). Therefore, 퐿 is the pole of the great circle 퐾푍퐷퐶, which means 퐿퐷 = 90. Therefore, In some cases, for observers at lower latitudes (휙 < 휖), the dṛkkṣepalagna and the dṛkkṣepajyā can be north of 휆푙 = 휆푑 + 90, the zenith (for instance, see Figure 7b). In these cases, the dṛkkṣepalagna is in the eastern hemisphere when the lon- which is the same as (14). gitude of the madhyalagna is in the range of 90 degrees to From the fact that 퐿 is at 90 degrees from both 퐶 and 퐷, 270 degrees (karkaṭādi), and in the western hemisphere the angle 퐶퐿퐷̂ between the ecliptic and the horizon will when the longitude of the madhyalagna is in the range be equal to the measure of the arc 퐶퐷, and therefore of 270 degrees to 90 degrees (mṛgādi). Therefore, for ob- ′ 퐶퐿퐷̂ = 푧 = 90 − 푧푑. (17) taining the dṛkkṣepalagna in this case, the arc 푀퐷 has 푑 to be subtracted from the longitude of the madhyalagna Though the above result is not stated in the above verses, when it is mṛgādi, and added to it when the madhyalagna we make a note of it here as it is essential for later discus- is karkaṭādi. Therefore, when compared to the situation sions which will be brought out as a sequel to this article.

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훿푚 휙 휙 훿푚 푇 푍 푇 푍 푀 푀 퐷

퐷 Γ Ω

푃 푃 ecliptic equator equator ecliptic

푁 푁 horizon horizon 퐶 퐶 퐸 퐿 퐿 퐸

(a) Southern hemisphere. (b) Northern hemisphere.

Figure 7 Direction of the dṛkkṣepa at lower latitudes.

3 Conclusion Kerala astronomical tradition. In subsequent papers, we plan to present other tech- The Lagnaprakaraṇa is a unique text focusing on a sin- niques of determining the ascendant described by Mā- gle problem in astronomy, namely the determination of dhava in the Lagnaprakaraṇa. the ascendant. This is indeed a non-trivial problem that seems to have attracted the attention of astronomers in Acknowledgements India and around the world. While the problem has been attempted to be solved using a variety of approximations We would like to place on record our sincere gratitude by various civilisations at different points of time, in our to MHRD for the generous support extended to carry understanding, precise formulations appear in this work out research activities on Indian science and technology of Mādhava for the first time in the annals of Indian and by way of initiating the Science and Heritage Initiative world astronomy. Indeed, Mādhava seems to have ap- (SandHI) at IIT Bombay. We pay our obeisance to the late proached this problem from the viewpoint of a pure math- Prof. K. V. Sarma, who saved the Lagnaprakaraṇa for fu- ematician and employs a variety of techniques to present ture generations by painstakingly copying the crumbling multiple precise relations for the computation of the as- manuscripts of the text. We are also extremely grateful to cendant. the Prof. K. V. Sarma Research Foundation, Chennai, for In this paper, we have discussed the first technique de- preserving and sharing the copies of these manuscripts, scribed by Mādhava in the Lagnaprakaraṇa. From our which has enabled us to study this interesting and impor- discussion it is evident that Mādhava seems to have been tant work. Finally, we would also like to profusely thank exceptionally good at visualising the motion of celestial the anonymous referee for thoroughly reviewing our pa- objects in the celestial sphere and the projection of various per and making constructive suggestions. points on it in several planes. This mastery enabled him to precisely derive the udayalagna through a series of fairly complex mathematical steps involving the determina- Bibliography tion of several quantities such as rāśikūṭalagna, madhya- [1] Āryabhaṭa. Āryabhaṭīya. Ed., trans., and comm., lagna, madhyakāla, madhyajyā, dṛkkṣepajyā, paraśaṅku, with an introd., by Kripa Shankar Shukla and dṛkkṣepalagna, etc. This complexity perhaps explains K. V. Sarma. New Delhi: Indian National Science why previous astronomers did not give precise relations Academy, 1976. for the ascendant, and attests to Mādhava’s reputation as the Golavid, or the knower of the celestial sphere, in the

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[2] Bhāskara. Siddhāntaśiromaṇi. With a comment. [12] Mādhava. “Lagnaprakaraṇa”. KVS Manuscript by Satyadeva Sharma. Varanasi: Chaukhamba No. 37a. Prof. K. V. Sarma Research Foundation, Surabharati Prakashan, 2007. Chennai.

[3] Brahmagupta. Brāhmasphuṭasiddhānta. Ed. and [13] Mādhava. “Lagnaprakaraṇa”. KVS Manuscript comm. by Sudhakara Dvivedi. Reprint from The No. 37b. Prof. K. V. Sarma Research Foundation, Pandit. Benares: Printed at the Medical Hall Press, Chennai. 1902. [14] Nīlakaṇṭha Somayājī. Tantrasaṅgraha. Trans. and [4] Śrī Kṛṣṇa Candra Dvivedī, ed. Sūryasiddhānta. comm. by K. Ramasubramanian and M. S. Sriram. With the commentary Sudhāvarṣiṇī of Sud- Culture and History of Mathematics 6. New Delhi: hākara Dvivedī. Sudhākara Dvivedī Granthamālā. Hindustan Book Agency, 2011. Varanasi: Sampurnanand Sanskrit University, [15] Putumana Somayājī. Karaṇapaddhati. Trans. and 1987. comm. by Venketeswara Pai et al. Culture and His- [5] Jñanarāja. Siddhāntasundara. Trans. and comm. tory of Mathematics 9. New Delhi: Hindustan Book by Toke Lindegaard Knudsen. Baltimore: John Agency, 2017. Hopkins University Press, 2014. [16] Śrīpati. Siddhāntaśekhara. Part 1. Ed. and comm. [6] Jyeṣṭhadeva. Gaṇita-yukti-bhāṣā. Ed. and trans. by by Babuāji Miśra. Calcutta: Calcutta University K. V. Sarma. With a comment. by K. Ramasubra- Press, 1932. manian, M. D. Srinivas, and M. S. Sriram. 2 vols. [17] Śrīpati. Siddhāntaśekhara. Part 2. Ed. and comm. Culture and History of Mathematics 4. New Delhi: by Babuāji Miśra. Calcutta: University of Calcutta, Hindustan Book Agency, 2008. 1947. [7] Aditya Kolachana, K. Mahesh, and K. Ramasub- [18] Vaṭeśvara. Vaṭeśvarasiddhānta and Gola. Part 2. ramanian. “Determination of kālalagna in the Trans. and comm. by Kripa Shankar Shukla. New Lagnaprakaraṇa”. In: Indian Journal of History of Delhi: Indian National Science Academy, 1985. Science 54.1 (2019), pp. 1–12. [19] Vaṭeśvara. Vaṭeśvarasiddhānta and Gola. Part 1. [8] Aditya Kolachana, K. Mahesh, and K. Ramasubra- Ed. by Kripa Shankar Shukla. New Delhi: Indian manian. “Mādhava’s multi-pronged approach for National Science Academy, 1986. obtaining the prāṇakalāntara”. In: Indian Journal of History of Science 53.1 (2018), pp. 1–15.

[9] Aditya Kolachana et al. “Determination of ascen- sional difference in the Lagnaprakaraṇa”. In: In- dian Journal of History of Science 53.3 (2018), pp. 302–316.

[10] Lalla. Śiṣyadhīvṛddhidatantra. With the commen- tary of Mallikārjuna Sūri. Ed., with an introd., by Bina Chatterjee. Vol. 1. 2 vols. New Delhi: Indian National Science Academy, 1981.

[11] Mādhava. “Lagnaprakaraṇa”. Manuscript 414B, ff. 53–84. Kerala University Oriental Research Insti- tute and Manuscripts Library, Thiruvananthapu- ram.

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