Precise determination of the ascendant in the Lagnaprakaraṇa-IV

Abstract

Mādhava’s Lagnaprakaraṇa is an important astronomical text which discusses numerous innovative techniques of precisely determining the udayalagna or the ascendant. In previous papers, we have detailed the various methods of determining the ascendant described in the second and third chapters of this text. In this paper, we discuss a technique described in the fifth chapter, which makes use of two quantities known as the viṣuvannara and the ayanāntaśaṅku.

1 Introduction

In a series of earlier papers, we have discussed the various contributions of Mādhava to astronomy by way of developing ingenious computational techniques in the focused treatise Lagnaprakaraṇa. The first three papers presented techniques of determining astronomical quantities known as prāṇakalāntara, cara and kālalagna,¹ that are employed in determining the ascendant. These techniques are described in the first thirty verses of this text, corresponding to its first chapter. Three subsequent papers discussed the various techniques of precisely determining the udayalagna or the ascendant, described in verses 31–61 of the Lagnaprakaraṇa, corresponding to its second and third chapters.² In this paper, we discuss further methods of determining the ascendant described in verses 80–87, corresponding to the fifth chapter of this text.³

The following verses, belonging to the fifth chapter of the Lagnaprakaraṇa, first describe the techniques to determine the gnomons—viṣuvannara and ayanāntaśaṅku—corresponding to the equinoctial and solstitial ecliptic points, and then show how to determine the ascendant therefrom. As the equinoctial and solstitial ecliptic points are ninety degrees apart, the verses in this chapter can be considered as addressing a special case of the procedures laid out in our previous paper.⁴

In order to have a better appreciation and full comprehension of the contents of this paper, it should be read in conjunction with our earlier papers, as various physical and mathematical quantities described therein are employed here as well. In this regards, it may be reiterated that we employ the symbols $\lambda$, $\alpha$, $\delta$, and z to respectively refer to the longitude, right ascension, declination, and zenith distance of a celestial object. The kālalagna, the latitude of the observer, and the obliquity of the ecliptic are denoted by the symbols $\alpha _e$, $\phi$, and $\epsilon$ respectively. It may also be mentioned that all the figures in this paper depict the celestial sphere for an observer having a northerly latitude $\phi$. In these figures, N, S, E, and W denote the cardinal directions north, south, east, and west, while P and K denote the poles of the celestial equator and the ecliptic respectively.

2 Obtaining the viṣuvannara and the ayanāntaśaṅku

लम्बाहता कालविलग्नदोर्ज्या

व्यासार्धभक्ता विषुवन्नरं स्यात् ।

अन्त्यद्युजीवाहतकाललग्न-

कोटीगुणाद्यत् त्रिभमौर्विकाप्तम् ॥८०॥

तल्लम्बघातं पलमौर्विकान्त्य-

क्रान्त्योर्वधे कर्किमृगादिकत्वात् ।

स्वर्णं च कृत्वा त्रिगुणेन हृत्वा

लब्धं तु विद्यादयनान्तशङ्कुम् ॥८१॥

स्वल्पे विदिक्कोटिजलम्बघाते

कर्क्यादिकत्वं च भवेदमुष्य ।

lambāhatā kālavilagnadorjyā

vyāsārdhabhaktā viṣuvannaraṃ syāt |

antyadyujīvāhatakālalagna-

koṭīguṇādyat tribhamaurvikāptam ||80||

tallambaghātaṃ palamaurvikāntya-

krāntyorvadhe karkimṛgādikatvāt |

svarṇaṃ ca kṛtvā triguṇena hṛtvā

labdhaṃ tu vidyādayanāntaśaṅkum ||81||

svalpe vidikkoṭijalambaghāte

karkyādikatvaṃ ca bhavedamuṣya |

The Rsine of the kālalagna multiplied by the Rcosine of the latitude (lamba) and divided by the semidiameter (vyāsārdha) would be the gnomon of the equinoctial ecliptic point (viṣuvannara). One should know that product of the Rcosine of the latitude (lamba) and that which is obtained by dividing the product of the last day-radius (antyadyujīvā) and the Rcosine of the kālalagna by the radius (tribhamaurvikā), applied positively or negatively to the product of the Rsine of the latitude (palamaurvikā) and [the Rsine of] the last declination (antyakrānti)—depending on [if the kālalagna is in the six signs] Cancer (karki) etc., or Capricorn (mṛga) etc.,—and [the result] divided by the radius (triguṇa), to be the gnomon of the solstitial ecliptic point (ayanāntaśaṅku).

[When the kālalagna is in Capricorn etc.,] if the product of the vidikkoṭija and the Rcosine of the latitude (lamba) is smaller [than the product of the Rsines of the latitude and the maximum declination], this [ayanāntaśaṅku] would be in Cancer (karki) etc.

These verses, in the indravajrā metre, give the following relation for the gnomon corresponding to the equinoctial point of the ecliptic:

(1)

They also give the following relation for the gnomon corresponding to the solstitial point of the ecliptic:

(2)

The above relations can be simply derived by employing (1) of Kolachana et al. (2020b) to determine the śaṅku of the Sun when it is at these positions.

2.1 Obtaining the viṣuvannara

Fig. 1

Determining the viṣuvannara

When the Sun is at either equinoctial point, its declination ($\delta$) and ascensional difference ($\Delta \alpha _s$) are zero, while its right ascension ($\alpha$) can be 0 or 180 degrees. Substituting these values in (1) of Kolachana et al. (2020b) yields (1) of this paper.

This relation can also be derived geometrically as shown in Fig. 1. In this figure, $\Gamma$ is the vernal equinoctial point, and Y is its projection on the horizon. The planar right-angled triangle $\Gamma XY$ lies in a plane perpendicular to the horizon and parallel to the prime meridian. Its side $\Gamma Y$ represents the viṣuvannara or the gnomon dropped from the vernal equinoctial point. Its hypotenuse $\Gamma X$ is equal to the Rsine of the arc $\Gamma E$, which is nothing but the kālalagna ($\alpha _e$). Finally, the angle $\Gamma \hat{X}Y$ represents the angle between the planes of the equator and the horizon. Thus,

The planar right-angled triangle $TOT'$ lies in the plane of the prime meridian, which is perpendicular to the horizon. Here, T represents the intersection of the equator and the prime meridian, and $T'$ is its projection on the horizon. Thus, the side $TT'$ is perpendicular to the horizon, the hypotenuse OT is equal to the radius of the celestial sphere, and the angle $TOT'$ represents the angle between the planes of the equator and the horizon. Therefore, we have

$$\begin{aligned} TT' = R\cos \phi , \quad OT = R, \quad {\hbox {and}} \quad T{\hat{O}}T' = \phi '= 90-\phi . \end{aligned}$$

Fig. 2

Determining the ayanāntaśaṅku when the kālalagna is karkyādi

It can be clearly seen that the two right-angled triangles $\Gamma XY$ and $TOT'$ are similar. Applying the rule of proportionality to the sides of these two triangles, we have

which is the required relation.⁵ It may be noted that the same relation can be obtained in a similar manner when the autumnal equinoctial point is above the horizon.

2.2 Obtaining the ayanāntaśaṅku

When the Sun is at the solstitial points, its declination is $\pm \, \epsilon$, and its right ascension is 90 or 270 degrees. Its ‘instantaneous’ ascensional difference, calculated using (16) of Kolachana (2018a), would be

$$\begin{aligned} R\sin \Delta \alpha _s = \frac{R\times R\sin \phi \times R\sin \epsilon }{R\cos \phi \times R\cos \epsilon }. \end{aligned}$$

It can be seen that substituting these values in (1) of Kolachana et al. (2020b) yields (2) of this paper.

The geometric derivation for the ayanāntaśaṅku is somewhat involved, but very interesting. This derivation can be carried out for three different scenarios where (i) the kālalagna is in the range of 90 to 270 degrees (i.e. Cancer etc.) and the summer solstitial point is above the horizon, (ii) the kālalagna is in the range of 270 to $270\,+\,\Delta \alpha _m$ degrees,⁶ or $90{\,-\,}\Delta \alpha _m$ to 90 degrees, and the summer solstitial point is above the horizon, and (iii) the kālalagna is in the range of $270\,+\,\Delta \alpha _m$ to $90-\Delta \alpha _m$ degrees and the winter solstitial point is above the horizon. The latter two cases together constitute the mṛgādi (Capricorn etc.) period of the kālalagna.

The construction required for determining the ayanāntaśaṅku when the kālalagna is in the range of 90 to 270 degrees is shown in Fig. 2. In this figure, $S_1$ is the summer solstitial point, and V is its projection on the horizon. Thus, $S_1 V$ corresponds to the ayanāntaśaṅku. The equatorial point $S_1'$ corresponds to the right ascension of $S_1$, and U is its projection on the horizon. Here, $OS_1$ and $OS_1'$ are the radii of the celestial sphere, and $S_1 B$ is the perpendicular from $S_1$ dropped on $OS_1'$. This perpendicular corresponds to the Rsine of the arc $S_1 S_1'$, which is nothing but the declination ($\epsilon$) of $S_1$. Thus, we have

$$\begin{aligned} S_1 B=R\sin \epsilon , \quad OB=R\cos \epsilon , \quad {\hbox {and}} \quad BS_1'=R\, \mathrm{versin}\,\epsilon . \end{aligned}$$

Now, let BF and BC be the perpendiculars dropped from B onto $S_1 V$ and $S_1'U$ respectively, which makes them parallel to the horizon. Then, it can be seen from the figure that the measure of the gnomon $S_1 V$ is given by

$$\begin{aligned} \quad S_1 V = S_1 F + S_1'U - S_1'C. \end{aligned}$$

(3)

The derivation of each of the quantities in the right hand side of the above expression is discussed below.

2.3 Obtaining S ₁F

$S_1 F$ can be obtained by considering Fig. 3a. This figure depicts an instant when the kālalagna is exactly 90 degrees. Thus, the vernal equinoctial point ($\Gamma$) is on the prime meridian, and $S_1$ and $S_1'$ are located on the six o’ clock circle, the latter coinciding with the east cardinal point (E). In this case, the radius $OS_1'$ is located on the horizon, and the perpendicular $S_1 B$ dropped from $S_1$ onto $OS_1'$ (in the plane of the six o’ clock circle) is equal to the Rsine of the arc $S_1 S_1'$, whose measure is $\epsilon$. That is, $S_1 B = R\sin \epsilon$. As B lies on the horizon, the perpendicular BF from B meets the gnomon of $S_1$ at its foot. Thus, F is the projection of $S_1$ on the horizon.

Now, it can be seen that the planar right-angled triangle $S_1 BF$ is perpendicular to the horizon, and parallel to the plane of the prime meridian. In this triangle, the angle $S_1 \hat{B}F$ is equal to the angle between the planes of the six o’ clock circle and the horizon, which is equal to the latitude ($\phi$) of the observer. Therefore, we have

$$\begin{aligned} \quad \quad \quad S_1 F = S_1 B\times \sin \phi = R\sin \epsilon \sin \phi . \end{aligned}$$

(4)

As the relative positions of $S_1$ and $S_1'$ are fixed, the measure of $S_1 F$ is also fixed, irrespective of the location of $S_1$ on the celestial sphere. Thus, $S_1 F$ will have the same measure in Fig. 2 as well.

2.4 Obtaining S $_\mathbf{1}^{\prime }$ U

$S_1'U$ can be determined from Fig. 3b. The celestial sphere depicted here is the same as that depicted in Fig. 2. In this figure, the planar right-angled triangle $S_1'UH$ is perpendicular to the horizon, and parallel to the plane of the prime meridian. Its hypotenuse $S_1'H$ is equal to the Rsine of the arc $S_1'E$. As $\Gamma S_1'=90$, and $\Gamma E = \alpha _e$, we have $S_1'E=\alpha _e-90$. Thus, $S_1'H=R\cos \alpha _e$. The angle $S_1'\hat{H}U = \phi ' = 90-\phi$ gives the measure of the angle between the planes of the equator and the horizon. Therefore, we have

$$\begin{aligned} \quad S_1'U = S_1'H\times \sin (90-\phi ) = R\cos \alpha _e\cos \phi . \end{aligned}$$

(5)

2.5 Obtaining S $_\mathbf{1}^{\prime }$ C

$S_1'C$ can be obtained by considering the similar triangles $S_1'CB$ and $S_1'UO$ in Fig. 2. In the right-angled triangle $S_1'CB$, we have

$$\begin{aligned} BS_1' = R\,\mathrm{versin}\,\epsilon , \qquad B{\hat{S}}_1' C=\theta '=90-\theta , \end{aligned}$$

and thus,

$$\begin{aligned} S_1'C=R\,\mathrm{versin}\,\epsilon \sin \,\theta . \end{aligned}$$

However, from the right-angled triangle $S_1'UO$, we have $\sin \theta =\cos \alpha _e\cos \phi$. Therefore,

$$\begin{aligned} \quad \quad\qquad S_1'C=R\cos \alpha _e\cos \phi \,\mathrm{versin}\,\epsilon . \end{aligned}$$

(6)

Fig. 3

Obtaining $S_1 F$ and $S_1'U$ for determining the ayanāntaśaṅku

Substituting (4), (5), and (6) in (3), we obtain the relation for the gnomon corresponding to the summer solstitial point when the kālalagna is in the range of 90 to 270 degrees:

$$\begin{aligned} \quad \quad S_1 V = R\sin \epsilon \sin \phi + R\cos \alpha _e\cos \epsilon \cos \phi . \end{aligned}$$

(7)

Fig. 4

Instances of the summer solstitial point above the horizon when the kālalagna is mṛgādi

Fig. 5

Instance of the winter solstitial point above the horizon when the kālalagna is mṛgādi

For an observer in the northern hemisphere, the summer solstitial point does not set immediately when the kālalagna goes beyond 270 degrees, but continues to be above the horizon for a time period corresponding to the maximum ascensional difference ($\Delta \alpha _m$) at that latitude. Similarly, the summer solstitial point rises earlier by a time interval of $\Delta \alpha _m$ before the kālalagna reaches 90 degrees. The former case, when the kālalagna is in the range 270 to $270 + \Delta \alpha _m$ degrees, is shown in Fig. 4a. The latter case, when the kālalagna is in the range of $90-\Delta \alpha _m$ to 90 degrees, is shown in Fig. 4b. In both these cases, $S_1$ is above the horizon, but $S_1'$ is below it. In either scenario, employing a similar construction as shown in Fig. 2, the gnomon of the summer solstitial point can be shown to be equal to

$$\begin{aligned} \quad \quad \quad \quad S_1 V&= S_1 F - S_1'U + S_1'C \nonumber \\&= R\sin \epsilon \sin \phi - R\cos \alpha _e\cos \epsilon \cos \phi . \end{aligned}$$

(8)

Finally, when the kālalagna is in the range of $270+\Delta \alpha _m$ to $90-\Delta \alpha _m$ degrees, the winter solstitial point ($S_2$) is above the horizon, as shown in Fig. 5. Employing a similar construction as shown in Fig. 2, the gnomon corresponding to $S_2$ can be shown to be equal to

$$\begin{aligned} \quad \quad \qquad S_2 V' = R\cos \alpha _e\cos \epsilon \cos \phi - R\sin \epsilon \sin \phi . \end{aligned}$$

(9)

Taken together, (7), (8), and (9) yield (2), and also satisfy the conditions for addition and subtraction of the constituent terms as stated in the verse.

2.6 Distinguishing between the summer and winter solstitial points when the kālalagna is mṛgādi

We have seen that either the summer or winter solstitial points can be above the horizon when the kālalagna is in the range of 270 to 90 degrees. Their corresponding gnomons are given by the relations (8) and (9) respectively. To obtain positive values for these gnomons, it can be seen that the expression $R\cos \alpha _e\cos \epsilon \cos \phi$ has to be smaller than $R\sin \epsilon \sin \phi$ in the former instance, and greater in the latter instance. Therefore, depending upon the relative magnitudes of these two expressions, one can determine which solstitial point is above the horizon. The first half of verse 82 thus states that, when the kālalagna is mṛgādi, the gnomon calculated using (2) would correspond to the summer solstitial point (karki or the solstitial point at a longitude of 90 degrees) when the quantity⁷

$$\begin{aligned} \frac{R\cos \alpha _e\times R\cos \epsilon }{R}\times R\cos \phi \end{aligned}$$

in the numerator is smaller than the product $R\sin \phi \times R\sin \epsilon$.

3 Relationship between viṣuvannara, ayanāntaśaṅku, and dṛkkṣepakoṭikā

शङ्क्वोस्तयोरत्र तु वर्गयोगात्

मूलं च दृक्क्षेपगुणस्य कोटिः ॥८२॥

śaṅkvostayoratra tu vargayogāt

mūlaṃ ca dṛkkṣepaguṇasya koṭiḥ ||82||

The square root of the sum of the squares of those two gnomons [described in the previous verse] is the Rcosine of the dṛkkṣepa.

This half-verse, in the indravajrā metre, gives the following relation:

(10)

The above relation is a special case of (7) in Kolachana et al. (2020b), where the viṣuvannara and the ayanāntaśaṅku (which are gnomons dropped from points ninety degrees apart on the ecliptic) replace the śaṅku and dṛggati (which too are gnomons dropped from points ninety degrees apart on the ecliptic). That is, if the śaṅku is measured at either of the equinoctial points, then the dṛggati will be measured at the corresponding solstitial points and vice-versa. Therefore, the above relation is just a special case of (7) in Kolachana et al. (2020b).

4 Determining the udayalagna from the viṣuvannara

व्यासार्धनिघ्नात् विषुवन्नराद्यत्

दृक्क्षेपकोट्याप्तफलस्य चापम् ।

तदेव तत्रोदयलग्नमाहुः

आद्ये पदे कालविलग्नकस्य ॥८३॥

तस्य द्वितीये तु पदे धनुस्तत्

चक्रार्धतः शुद्धमुशन्ति लग्नम् ।

चक्रार्धयुक्तं च पदे तृतीये

संशोधितं मण्डलतश्चतुर्थे ॥८४॥

vyāsārdhanighnāt viṣuvannarādyat

dṛkkṣepakoṭyāptaphalasya cāpam |

tadeva tatrodayalagnamāhuḥ

ādye pade kālavilagnakasya ||83||

tasya dvitīye tu pade dhanustat

cakrārdhataḥ śuddhamuśanti lagnam |

cakrārdhayuktaṃ ca pade tṛtīye

saṃśodhitaṃ maṇḍalataścaturthe ||84||

That arc, which is of the quotient obtained from the division of the semi-diameter (vyāsārdha) multiplied viṣuvannara by the Rcosine of the dṛkkṣepa (dṛkkṣepakoṭi), itself is stated to be the rising ecliptic point (udayalagna) there, in the first quadrant of the kālalagna. Indeed in the second, third, and fourth quadrants of that (kālalagna), that arc [respectively] subtracted from a semi-circle (cakrārdha), added by a semi-circle, and subtracted from the full circle (maṇḍala), is stated to be the [rising] ecliptic point (lagna).

Fig. 6

Determining the udayalagna from the viṣuvannara and the ayanāntaśaṅku

These two verses, in the upajāti and indravajrā metres respectively, give the following procedure to determine the udayalagna using the viṣuvannara and the dṛkkṣepakoṭi, which have been defined earlier in this chapter:

Taking $\lambda _l$ as the longitude of the udayalagna, and denoting the dṛkkṣepakoṭi as $R\cos z_d$, the above relations can be expressed in mathematical notation as follows:

(11)

(12)

(13)

(14)

Here, the viṣuvannara is obtained using (1).

The viṣuvannara is the gnomon corresponding to the equinoctial points. Verses 58–60⁸ detail the general procedure for obtaining the udayalagna given the position of the Sun, and the measure of its gnomon (śaṅku). Therefore, the relations for udayalagna given in these two verses can be obtained by considering the Sun to be at either equinoctial point, taking the śaṅku to be equal to the viṣuvannara, and applying (12)–(15) of Kolachana et al. (2020b). However, to demonstrate the physical significance of the relations given in these two verses, here we derive (11)–(14) of this paper with the help of Figs. 6a–6d, which show the kālalagna in different quadrants.

4.1 Obtaining the ecliptic arc from the equinoctial point to the horizon

As the longitudes of the equinoctial points, as well as the separation of the rising and setting ecliptic points are known, the udayalagna can be determined by obtaining the arcs $\Gamma L$, $L'\Gamma$, $\Omega L$, and $L'\Omega$ in each of the Figs. 6a–6d respectively. We have shown that (16) of Kolachana et al. (2020b) gives the measure of the arc from the Sun to the rising ecliptic point (SL) when the Sun is in the eastern hemisphere, and the measure of the arc from the setting ecliptic point to the Sun ($L'S$) when it is in the western hemisphere. Thus, the arcs $\Gamma L$ or $L' \Gamma$ can be obtained by simply considering the Sun to be present at the vernal equinoctial point ($\Gamma$), and employing (16) of Kolachana et al. (2020b). If the Sun were at the vernal equinoctial point, its śaṅku would be equal to the viṣuvannara, given by (1), and thus (16) of Kolachana et al. (2020b) would reduce to

(15)

Similarly, the arcs $\Omega L$ and $L' \Omega$ can be obtained by considering the Sun to be present at the autumnal equinoctial point. In this case too, the śaṅku of the Sun would be equal to the viṣuvannara, given by (1),⁹ and (16) of Kolachana et al. (2020b) would reduce to

(16)

4.2 Obtaining the udayalagna

When the kālalagna is in the first quadrant, as shown in Fig. 6a, the vernal equinoctial point is above the horizon and in the eastern hemisphere. Here, the longitude of the udaya-lagna is given by the arc $\Gamma L$ which is directly obtained from (15). This is the result stated in (11).

When the kālalagna is in the second quadrant, the vernal equinoctial point is above the horizon and in the western hemisphere, as shown in Fig. 6b. Here, the longitude of the udayalagna is given by the arc $\Gamma L = L'L - L'\Gamma$, where $L'L=180$, and $L'\Gamma$ is obtained using (15). Thus, we get (12).

When the kālalagna is in the third quadrant, the autumnal equinoctial point is above the horizon and in the eastern hemisphere, as shown in Fig. 6c. Here, the longitude of the udayalagna is given by the arc $\Gamma L = \Gamma \Omega + \Omega L$, where $\Gamma \Omega =180$, and $\Omega L$ is obtained using (16). Thus, we obtain (13).

Finally, when the kālalagna is in the fourth quadrant, as shown in Fig. 6d, the autumnal equinoctial point is above the horizon and in the western hemisphere. Here, the longitude of the udayalagna is given by the arc $\Gamma L = \Gamma \Omega - L'\Omega + L'L$, where $\Gamma \Omega =L'L=180$, and $L'\Omega$ is obtained using (16). Thus, we obtain the result stated in (14).

5 Determining the udayalagna from the ayanāntaśaṅku

व्यासार्धनिघ्नादयनान्तशङ्कोः

दृक्क्षेपकोट्याप्तधनुर्धनर्णम् ।

निजायनान्ते युगयुक्पदत्वात्

कृत्वा भवेदौदयिकं विलग्नम् ॥८५॥

मृगादियाते सति काललग्ने

तत्कोटिजीवाफललम्बघाते ¹⁰ ।

स्वल्पे परक्रान्तिगुणाक्षघातात्

पदान्यता तस्य च कल्पनीया ॥८६॥

vyāsārdhanighnādayanāntaśaṅkoḥ

dṛkkṣepakoṭyāptadhanurdhanarṇam |

nijāyanānte yugayukpadatvāt

kṛtvā bhavedaudayikaṃ vilagnam ||85||

mṛgādiyāte sati kālalagne

tatkoṭijīvāphalalambaghāte |

svalpe parakrāntiguṇākṣaghātāt

padānyatā tasya ca kalpanīyā ||86||

The result obtained by applying the arc of the quotient obtained from the division of the product of the ayanāntaśaṅku and the semi-diameter (vyāsārdha) by the Rcosine of the dṛkkṣepa (dṛkkṣepakoṭi) to [the longitude of] the own solstitial point (nijāyanānta) positively or negatively, depending on even or odd quadrants [of the kālalagna], would be the rising ecliptic point (audayikaṃ vilagnam).

When the kālalagna is in Capricorn etc. (mṛgādi), if the product of the result obtained from its Rcosine (koṭijīvā) and the Rcosine of the latitude (lamba) is smaller than the product of the Rsine of the maximum declination (parakrāntiguṇa) and [the Rsine of] the latitude (akṣa), the quadrant of that [kālalagna] should be considered to be otherwise (padānyatā) [i.e., odd as even, and even as odd, for the purpose of deciding whether to add or subtract the obtained arc to the nijāyanānta, in determining the udayalagna].

Verses 85, in the upajāti metre, describes the following procedure to determine the udayalagna using the ayanāntaśaṅku and the dṛkkṣepakoṭi, which have been defined earlier in this chapter:

Taking $\lambda _l$ as the longitude of the udayalagna, denoting the dṛkkṣepakoṭi as $R\cos z_d$, and noting that the longitudes of the summer ($S_1$) and winter ($S_2$) solstitial points are 90 and 270 degrees respectively, the above relations can be expressed in mathematical notation as follows:

(17)

(18)

(19)

(20)

Here, the ayanāntaśaṅku is obtained using (2), taking care of the quadrant of the kālalagna in which it is sought to be applied. It may be noted that the ‘own solstitial point’ (nijāyanānta) of the kālalagna refers to $S_1$ when the kālalagna is in the first two quadrants, and $S_2$ when the kālalagna is in the third and fourth quadrants. This is only used to represent the quantities 90 and 270 degrees in the above relations, and does not indicate which solstitial point is above the horizon in each of these cases.

The above relations can be directly derived by considering the Sun to be present at either solstitial point, taking its śaṅku to be the appropriate ayanāntaśaṅku, and employing (12)–(15) of Kolachana et al. (2020b). However, to demonstrate the physical significance of the relations given here, we derive (17)–(20) of this paper with the help of Figs. 6a–6d, which depict the kālalagna in different quadrants.

As the longitudes of the solstitial points, as well as the separation of the rising and setting ecliptic points are known, the udayalagna can be determined by obtaining the arcs $L'S_2$, $S_1 L$, $L'S_1$, and $S_2 L$ in each of the Figs. 6a–6d respectively. These arcs can be obtained by considering the Sun to be present at $S_1$ or $S_2$ and applying (16) of Kolachana et al. (2020b). Here, however, we would have to replace the śaṅku of the Sun with the respective ayanāntaśaṅkus of $S_1$ and $S_2$. In our discussion of verses 80–81, we have shown that the relation for the ayanāntaśaṅku varies depending upon the quadrant of the kālalagna. Therefore, these different cases are dealt separately below.

5.1 Obtaining the udayalagna when the kālalagna is karkyādi

When the kālalagna is in the second and third quadrants, the summer solstitial point is above the horizon and the ayanāntaśaṅku is given by (7). Substituting this quantity in (16) of Kolachana et al. (2020b), we have

(21)

When the kālalagna is in the second quadrant, as shown in Fig. 6b, the summer solstitial point lies above the horizon in the eastern hemisphere. Here, the longitude of the udayalagna is given by the arc $\Gamma L = \Gamma S_1 + S_1 L$, where $\Gamma S_1= 90$, and $S_1 L$ is given by (21). Thus, we obtain (18).

When the kālalagna is in the third quadrant, as shown in Fig. 6c, the summer solstitial point lies above the horizon in the western hemisphere. Here, the longitude of the udayalagna is given by the arc $\Gamma L = \Gamma S_1 - L'S_1 + L'L$, where $\Gamma S_1 = 90$, $L'L=180$, and $L'S_1$ is given by (21). Thus, we have $\Gamma L = 270 - L'S_1$, which is the same as (19).

5.2 Obtaining the udayalagna when the kālalagna is mṛgādi and the winter solstitial point is above the horizon

We have shown in our discussion of verses 80–81 that when the kālalagna is in the range of $270+\Delta \alpha _m$ to $90-\Delta \alpha _m$ degrees,¹¹ the winter solstitial point is above the horizon, and the ayanāntaśaṅku is given by (9). Substituting this quantity in (16) of Kolachana et al. (2020b), we have

(22)

Figure 6a depicts a situation where the kālalagna is in the first quadrant, and the winter solstitial point ($S_2$) lies above the horizon in the western hemisphere. In this case, we have $\Gamma L + 360= \Gamma S_2 - L'S_2 + L'L$, where $\Gamma S_2=270$, $L'L=180$, and $L'S_2$ is given by (22). Thus, the longitude of the udayalagna is given by the arc $\Gamma L = 90-L'S_2$, which is the same as (17).

Figure 6d depicts a situation where the kālalagna is in the fourth quadrant, and the winter solstitial point lies above the horizon in the eastern hemisphere. In this case, the longitude of the udayalagna is given by the arc $\Gamma L = \Gamma S_2 + S_2L$, where $\Gamma S_2=270$, and $S_2L$ is obtained using (22). Thus, we obtain (20).

5.3 Obtaining the udayalagna when the kālalagna is mṛgādi and the summer solstitial point is above the horizon

We have shown in our discussion of verses 80–81 that when the kālalagna is in the range of $90-\Delta \alpha _m$ to 90 or 270 to $270+\Delta \alpha _m$ degrees, the summer solstitial point is above the horizon, and the ayanāntaśaṅku is given by (8). Substituting this quantity in (16) of Kolachana et al. (2020b), we have

(23)

Figure 4b depicts a scenario where the kālalagna is in the first quadrant, and the summer solstitial point is above the horizon in the eastern hemisphere. Here, the longitude of the udayalagna is given by the arc $\Gamma L = \Gamma S_1 + S_1 L$, where $\Gamma S_1 = 90$, and $S_1 L$ is given by (23). Therefore, we have

(24)

Figure 4a depicts a scenario where the kālalagna is in the fourth quadrant, and the summer solstitial point is above the horizon in the western hemisphere. In this case, the longitude of the udayalagna is given by the arc $\Gamma L = \Gamma S_1 - L'S_1 + L'L$, where $\Gamma S_1 = 90$, $L'L=180$, and $L'S_1$ is given by (23). Thus, we obtain

(25)

Comparing (24) with (17), it is seen that the arc has to be added in the former case, and subtracted in the latter, though both relations pertain to the first quadrant of the kālalagna. Similarly, comparing (25) and (20), it is seen that the arc has to be subtracted in the former case, and added in the latter, though both relations pertain to the fourth quadrant of the kālalagna. Thus, we find that (24) and (25) are an exception to the rule given in verse 85, which states that the appropriate arc is to be added to or subtracted from the nijāyanānta in the even and odd quadrants respectively. This exception is addressed in verse 86 (in the upajāti metre), which states that the even quadrant of the kālalagna has to be considered as odd, and the odd as even, for the purpose of determining between the addition or subtraction of the arc to the nijāyanānta in these two cases.¹² This is equivalent to stating that the appropriate arc has to be added to the nijāyanānta in the odd quadrant, and subtracted from it in the even quadrant, when the kālalagna is mṛgādi and the summer solstitial point is above the horizon. As can be seen, this exception to the rule satisfies both (24) and (25).

6 Determining the lagnakarṅa

लम्बाप्तो लग्नकर्णः स्यात् त्रिज्यापरनराहतेः ।

तेन वा लग्नमानेयं स्फुटीकरणवर्त्मना ॥८७॥

lambāpto lagnakarṇaḥ syāt

trijyāparanarāhateḥ |

tena vā lagnamāneyaṃ

sphuṭīkaraṇavartmanā ||87||

The lagnakarṇa would be the quotient obtained from the division of the product of the radius (trijyā) and the paranara (i.e. paraśaṅku) by the Rcosine of the latitude (lamba). The rising ecliptic point (lagna) can also be computed from it by means of the sphuṭīkaraṇa process.

This verse, in the anuṣṭubh metre, defines a quantity known as the lagnakarṇa in terms of the paranara or the paraśaṅku ($R\cos z_d$), and the lambajyā ($R\cos \phi$) as follows:

(26)

The verse further states that the udayalagna can be computed from this quantity by means of a process known as sphuṭīkaraṇa. This process is described later in the Lagnaprakaraṇa. We only present this verse here for the sake of completeness, and intend to discuss this procedure in detail in a forthcoming paper.

7 Discussion

In our previous paper,¹³ we disc ussed the procedure of determining the ascendant described in verses 53–61, constituting the third chapter of the Lagnaprakaraṇa. There, the author presents two quantities, known as śaṅku and dṛggati, which are the gnomons corresponding to the Sun and an ecliptic point ninety degrees behind the Sun. In that paper, we have shown how these two quantities, along with a third quantity known as dṛkkṣepakoṭi, have been manipulated to precisely calculate the ascendant.

Similarly, in the verses discussed in this paper, the author defines two gnomons corresponding respectively to the equinoctial and solstitial ecliptic points, and together with the dṛkkṣepakoṭi, again precisely determines the ascendant. The relations for the gnomons in terms of the kālalagna, just as in the case of the śaṅku and dṛggati, are quite innovative, and capture the variation in these two quantities accurately. While the variation in the viṣuvannara is fairly straightforward, the variation of the ayanāntaśaṅku is more complex, having different relations in different quadrants of the kālalagna. The text captures the variation in the measure of the ayanāntaśaṅku in different scenarios particularly well, revealing a deep study and strong comprehension of this topic by the author. Later verses discuss the means to precisely determine the ascendant using the viṣuvannara and the ayanāntaśaṅku, based on various possible values of the kālalagna. Exceptions to the stated rules are made abundantly clear, showing that the author has carefully considered all possible scenarios. Thus, Mādhava once again lives up to the epithet of “golavid”, bestowed upon him by later scholars.

8 Conclusion

In this paper, we discussed the fifth chapter of Lagnaprakaraṇa, which describes yet another sophisticated technique to precisely determine the ascendant. The chapter ends with a tantalising verse, describing a quantity known as lagnakarṇa, and hinting at a process known as sphuṭīkaraṇa for determining the ascendant. We intend to discuss this fascinating procedure and other contributions of the Lagnaprakaraṇa in forthcoming papers.