1 Introduction

Calculus related concepts are to be found in Indian Siddhāntic texts, from Laghumānasa of Muñjāla (932 CE) onwards (Datta et al., 1984; Sriram, 2014; LM, 1944; Ramasubramanian & Srinivas, 2010). They are in the context of the rates of motion of the planets. Due to the eccentricity of the orbit of a planet, an ‘equation of centre’ correction should be apllied to the mean planet, \(\theta _0\) (which moves uniformly with time) to obtain the ‘true’ planet, \(\theta _t\). In many texts, the expression for the true planet, \(\theta _t\) is of the form

$$\begin{aligned} \theta _t = \theta _0 - \frac{r_0}{R} f (M) \sin M \end{aligned}$$

as such, or in an approximation. Here, \(M=\theta _0 - \theta _A\) is the manda-kendra (anomaly), where \(\theta _A\) is the ‘apogee’. Here, \(\frac{r_0}{R}\) is the ratio of the radius of the manda-epicycle and the radius of the mean planet’s orbit. Also, \(f(M)\approx 1\)is a function of M. The second term in the above equation is the mandaphala or the ‘equation of centre’.

In earlier texts, the rate of motion of the planet was found by just computing the true planet, \(\theta _t\) at the mean sunrise on two successive days. The difference between them was considered the true rate of motion through out the intervening day.

It is in Laghumānasa that the rate of motion is treated very differently. In this text, \(\theta _t\) has the form (LM, 1944, pp. 38–49; Shukla, 1990, pp. 125–127)

$$\begin{aligned} \theta _t = \theta _0 - \frac{r_0}{R} \times \frac{\sin M}{1 + \frac{r_0}{2R} \cos M}, \end{aligned}$$

and the true rate of motion is given as

$$\begin{aligned} \frac{\Delta \theta _t}{\Delta t} = \frac{\Delta \theta _0}{\Delta t} - \frac{r_0}{R} \times \frac{\cos M}{1 + \frac{r_0}{2R} \cos M} \times \frac{\Delta M}{\Delta t}, \end{aligned}$$

where the first term in the RHS is the madhyamagati or the ‘mean rate of motion’ and the second term is the gatiphala (result of correction to the mean rate of motion). This is the rate of motion at any instant or ‘instantaneous velocity’, though it is not stated explicitly in the text Laghumānasa. Here, it is clear that \(\frac{\Delta \sin M}{\Delta t}\) is taken as \(\cos M \times \frac{\Delta M}{\Delta t}\), and the variation due to the factor \(\frac{1}{1 + \frac{r_0}{2R} \cos M}\) is not taken into account. Clearly it is recognised that the derivative of \(\sin M\) is \(\cos M\), \(\left( \frac{\Delta \sin M}{\Delta t} = \frac{\Delta \sin M}{\Delta M} \times \frac{\Delta M}{\Delta t} = \cos M \times \frac{\Delta M}{\Delta t} \right)\), though not stated as such.

In the Mahāsiddhānta of Āryabhaṭa-II (tenth century CE) (MS, 1910, p. 58), the manda-sphuṭa-graha is given by

$$\begin{aligned} \theta _t = \theta _0 - \frac{r_0}{R}\times \sin M, \end{aligned}$$

and the rate of motion is given as

$$\begin{aligned} \frac{\Delta \theta _t}{\Delta t} = \frac{\Delta \theta _0}{\Delta t} - \frac{r_0}{R} \times \cos M \times \frac{\Delta M}{\Delta t}. \end{aligned}$$

Here also, the derivative of sine is recognised as the cosine.

In the Grahagaṇitādhyāya part of Siddhāntaśiromaṇi, in the chapter on Spaṣṭādhikāra (SS, 2005, chapter 2, verse 30, p. 50), Bhāskara’s expression for \(\theta _t\) is

$$\begin{aligned} \theta _t = \theta _0 - \left( \frac{r_{0}}{R} \times \sin M \right) . \end{aligned}$$

Then, the rate of motion would be:

$$\begin{aligned} \frac{\Delta \theta _t}{\Delta t} = \frac{\Delta \theta _0}{\Delta t} - \frac{r_0}{R} \times \cos M \times \frac{\Delta M}{\Delta t}, \end{aligned}$$

which is the same as in Mahāsiddhānta. It is stated in verse 37 of this chapter (SS, 2005, chapter 2, p. 52), as follows:

कोटिफलघ्नी मृदुकेन्द्रभुक्तिस्त्रिज्योद्धृता कर्किमृगादिकेन्द्रे ।

तया युतोना ग्रहमध्यभुक्ति तात्कालिकी मन्दपरिस्फुटा स्यात् ।।

koṭiphalaghnī mṛdukendrabhuktistrijyoddhṛtā karkimṛgādikendre ।

tayā yutonā grahamadhyabhukti tātkālikī mandaparisphuṭā syāt ।।

The daily motion of the mandakendra (mean anomaly) being multiplied by the koṭiphala and divided by the radius, and the result being added to or subtracted from the mean motion depending upon whether the anomaly is in karkyādi or mṛgādi gives the true instantaneous [rate of motion] of manda-sphuṭa.

In the next verse, Bhāskara stresses the need for using the instantaneous rate of motion in the case of the Moon whose rate of motion of anomaly is large:

समीपतिथ्यन्तसमीपचालनं विधोस्तु तत्कालजयैव युज्यते ।

samīpatithyantasamīpacālanaṃ vidhostu tatkālajayaiva yujyate |

In the case of the Moon, the ending moment or the beginning time of a tithi which is near at hand is to be computed using the instantaneous (tatkāla) rate of motion only.

This is explained in far greater detail in the vāsanā for the verses. Here, it is pointed out that the earlier computation of the rate of motion (by just finding the difference between the true longitudes at successive sunrises) is only approximate, and a more precise instantaneous rate of motion has to be computed.

The actual planets, Mars, Mercury, Jupiter, Venus and Saturn have one more correction, namely, Śīghra. Finding their exact rates of motion is challenging and Bhāskara solves this by adopting a novel approach, in which only the derivative of the sine function is involved (SS, 2005, pp. 54–58).

In Tantrasaṅgraha of Nīlakaṇṭha Somayājī [Ramsubramanian & Sriram (2011), chapter 2, p. 76, p. 90 and pp. 114–116], the manda-correction (mandaphala) for the mean planet to obtain the true planet is of the form \(- \sin ^{-1} (\frac{r_0}{R} sin M)\). Nīlakaṇṭha gives the exact expression for the correction to the rate of motion of the planet due to this mandaphala as

$$\begin{aligned} -\frac{\frac{r_0}{R} \cos M \times \frac{\Delta M}{\Delta t}}{\sqrt{\left( 1- \frac{r_0^2}{R^2} \sin ^2 M\right) }}. \end{aligned}$$

So, the derivative of the inverse sine function is calculated correctly in this text.

In his Sphuṭanirṇayatantra (late sixteenth century) (SNT, 1974, chapter 3, verses 17–18 p. 20), Acyuta Piṣāraṭi essentially considers a mandaphala of the form:

$$\begin{aligned} \frac{ - \frac{r_0}{R} \sin M}{(1 + \frac{r_0}{R} \cos M)} \end{aligned}$$

also, as in Laghumānasa. Acyuta gives the correct expression for the correction to the rate of motion due to this mandaphala which is a ratio of two functions, \(- \frac{r_0}{R} \sin M\) and \((1 + \frac{r_0}{R} \cos M)\), as

$$\begin{aligned} \frac{ -\left[ \frac{r_0}{R} \cos M + \frac{\left( \frac{r_0}{R} \sin M\right) ^2}{\left( 1 + \frac{r_0}{R} \cos M\right) } \right] }{\left( 1 + \frac{r_0}{R} \cos M\right) }\frac{\Delta M}{\Delta t} \end{aligned}$$

(SNT, 1974, chapter 3, verses 19–20, pp. 20–21; Ramasubramanian & Srinivas, 2010, pp. 279–280).

All these are in the context of the rates of motion of planets. However, a recent study of two Kerala texts, namely Karaṇottama (KTM, 1964) of Acyuta Piṣāraṭi, and Dṛkkaraṇa [DK1, DK2, Venketeswara and Sriram (2019)] by us has revealed that the calculus concepts (essentially the derivative of the sine function) are used in another context. This is in the context of finding the instant of vyatīpāta or vaidhṛta, when the magnitudes of the declinations of the Sun and the Moon are equal, whereas their rates of change are opposite (with one increasing and the other, decreasing). The computation involves the rates of change of the declinations of the Sun and the Moon, wherein use is made of \(\frac{d}{dt}\sin \lambda = \cos \lambda \frac{\Delta \lambda }{\Delta t}\), where \(\lambda\) is the longitude of the Sun or the Moon. In this paper, we elaborate the use of the derivative concept in finding the instant of vyatīpāta or vaidhṛta in the two texts.

2 Phenomena of vyatīpāta and vaidhṛta

Vyatīpāta or lāṭa and vaidhṛta occur when the magnitudes of the declinations of the Sun and the Moon are equal, and their rates of change are opposite, that is, one of them is increasing, while the other is decreasing.

For lāṭ­a or vyatīpāta, the ayanas of the Sun and the Moon should be different, that is, one is moving northwards, whereas the other is moving southwards. In the case of vaidhṛta, the ayanas of both are the same.

Fig. 1
figure 1

Vyatīpāta and vaidhṛta

Full size image

These are illustrated in Fig. 1. When the Sun is at \(S_1\) or \(S_2\) it is lāṭa when the Moon is at \(M_1\) and \(M_2\) respectively, where \(|\delta _s|= |\delta _m|\), but the two objects have different ayanas. For the same two positions of the Sun, it is vaidhṛta when the Moon is at \(M_{1}'\) and \(M_{2}'\) respectively, where \(|\delta _s|= |\delta _m|\), but the two objects have the same ayanas. Similarly, one can consider lāṭa and vaidhṛta, when the Sun is in the third or fourth quadrants.

2.1 Computation of vyatīpāta and vaidhṛta

To be specific, we consider the text Dṛkkaraṇa first. The text DṛkkaraṇaFootnote 1 (c. 1608 CE) is a comprehensive text on astronomy which was composed based on observational data [DK1, DK2, (Venketeswara & Sriram, 2019)].

The author declares right at the beginning of the text that he is going to expound a karaṇa based on observations, to enable young students to understand the mathematical methods of astronomy. He also emphasises that he is going to explain this in the [popular] language which he calls as Bhāṣā. In practice, the Bhāṣā is a highly Sanskritised version of Malayāḷaṃ, called Maṇipravāḷaṃ. A study of Dṛkkaraṇa reveals that it is actually a Tantra type of text which gives all the algorithms associated with the traditional topics in a typical Indian text in more than 400 verses spread over 10 chapters. These include the computations of the mean longitudes, true longitudes, tripraśna problems related to time and shadow, corrections associated with the terrestrial longitude and latitude of a location, detailed discussions of lunar and solar eclipses, vyatīpāta, heliacal rising and setting of planets, computations of the ascendant (lagna) at a given time, dimensions of the orbits of the Sun, Moon and planets, Vākya system and so on (Venketeswara & Sriram, 2019).

In particular, the seventh chapter is dedicated to the algorithms pertaining to the vyatīpāta and vaidhṛta. This chapter gives the details of the computation related to vyatīpāta. These include the expressions for the declination of the Moon including its latitude, for the ‘middle’ of the vyatīpāta, the procedure for finding the sparśakāla (beginning of the vyatīpāta) and the mōkṣakāla (end of the vyatīpāta), and the special case when the Sun and the Moon are close to their ayanasaṅkramas. Verses 1 and 2 in this chapter are as follows [DK1, DK2]:

figure a
figure b
figure c
figure d

व्यतीपातं गणिक्कुन्न प्रकारङ्ङळ् पऱञ्ञिटां |

अयनांशमिरट्टिच्चु संस्क्करिच्चुळळ सूर्यनॆ ||१||

आऱुराशियिल् वाङ्ङीट्टु मण्डलत्तिन्नुमङ्ङिनॆ |

तुल्यं चन्द्रनितिन्नोटु वन्नीटुं नाळिलोर्क्कणं ||२||

vyatīpātaṃ gaṇikkunna prakāraṅṅaḷ paṟaññiṭāṃ |

ayanāṃśamiraṭṭiccu saṃskkariccuḷḷa sūryan ||1||

āṟurāśiyil vāṅṅīṭṭu maṇḍalattinnumaṅṅine |

tulyaṃ candranitinnōṭu vannīṭuṃ nāḷilōrkkaṇaṃ ||2||

The methods for computing the vyatīpāta are being told. [The longitude of] the Sun which has been corrected by the twice the ayanāṃśa has to be subtracted from the six rāśis or twelve rāśis (maṇḍala). Then, it becomes equal to the [longitude of the] Moon. The day [on which this occurs] is to be noted down (ōrkkaṇaṃ).

The author states that different methods for computing vyatīpāta or vaidhṛta would be told. Now, at the vyatīpāta or vaidhṛta, the declinations of the Sun and the Moon should be equal and their rates of change should be opposite. Now, for a celestial object on the ecliptic,

$$\begin{aligned} \sin \delta = \sin \epsilon \sin \lambda , \end{aligned}$$

where \(\lambda\) is the tropical or the sāyana longitude of the object. Hence, if the latitude of the Moon is ignored (to begin with), then the equality of the magnitudes of the declinations of the Sun and the Moon implies thatFootnote 2

$$\begin{aligned} \sin \lambda _m = \sin \lambda _s, \end{aligned}$$

where \(\lambda _m\) and \(\lambda _s\) are the sāyana longitudes of the Moon and the Sun. This implies that

$$\begin{aligned} \lambda _m = 180^\circ - \lambda _s \end{aligned}$$

if the Sun and the Moon have opposite ayanas, or

$$\begin{aligned} \lambda _m = 360^\circ - \lambda _s \end{aligned}$$

when they have the same ayana. Now, \(\lambda _m = (\lambda _m)_n + a\) and \(\lambda _s = (\lambda _s)_n + a\), where \((\lambda _m)_n\) and \((\lambda _s)_n\) are the nirayaṇa longitude,Footnote 3 of the Moon and the Sun respectively. Hence, for the computation of vyatīpāta or vaidhṛta, first find the instant at which

$$\begin{aligned} (\lambda _m)_n= & {} 180^\circ - \left( (\lambda _s)_n + 2a \right) \\ \text{ or }\qquad \qquad (\lambda _m)_n= & {} 360^\circ - \left( (\lambda _s)_n + 2a \right) , \end{aligned}$$

respectively, as stated in the verses.

2.2 Lāṭ­avaidhṛtadoṣas

figure e
figure f
figure g

लाटवैधृतदोषङ्ङळ् रविचन्द्रौ च पातनुं |

गणिच्चिट्टयनांशत्तॆ संस्क्करिच्चङ्ङु वॆक्कणं ||३||

चन्द्रार्क्कन्मारॆ वॆच्चिट्टु क्रान्तिज्यावङ्ङु कॊळळुक |

lāṭavaidhṛtadōṣaṅṅaḷravicandrau ca pātanuṃ |

gaṇicciṭṭayanāṃśatte saṃskkariccaṅṅu vekkaṇaṃ ||3||

candrārkkanmāre vecciṭṭu krāntijyāvaṅṅu koḷḷuka |

[For obtaining the lāṭ­a and vaidhṛta-doṣas], place [the longitudes of] the Sun, the Moon and the node (pāta) which have been computed and corrected by the ayanāṃśa.Footnote 4 Then, find the Rsine of declination corresponding to the Sun and the Moon.

For obtaining the declination of the Sun, it is sufficient to know its sāyana longitude, that is the nirayaṇa longitude corrected by the ayanāṃśa. As the Moon’s orbit is inclined to the ecliptic, it is necessary to find its latitude also, to obtain its declination. For this, it is necessary to obtain its node (pāta) as well. The procedure to obtain the declination of the Moon, taking into account its latitude is described elsewhere in the text.

3 Use of the derivative of the sine function in Karaṇottama and Dṛkkaraṇa

For finding the instant of vyatīpāta or vaidhṛta, the law of propotions and an iterative procedure was prescribed in the earlier texts such as Brāhmasphuṭa-siddhānta (BSS, 1966, vol. 3, chapter 14, verses 39–40, pp. 1023–1025), Karaṇaratna of Devācārya [KR (1979), chapter 1, verses 54–57, pp. 37–38], Śiṣyadhīvṛddhida of Lalla (SVT, 1981, part 1, chapter 12, verses 6–9, pp. 171–173), and also later texts. This method has also been discussed in some recent articles (Plofker, 2014, pp. 1–11; Venketeswara Pai et al., 2015, pp. 69–89).

The same procedure is described in the Pātādhikāra of the Grahagaṇita part of Siddhāntaśiromaṇi. We summarise this procedure which is described elsewhere in detail (Venketeswara Pai et al., 2015).

Let \(t_1\) be a suitable instant at which the declinations of the Sun and the Moon are \(\delta _s\) and \(\delta _m\) respectively (including the sign). Now, finding their difference, we have \(\Delta _1 = \delta _s - \delta _m\). Now, again find the difference in declinations, \(\Delta _1 = \delta _s - \delta _m\), at some other instant \(t_2\). Then, the instant of vyatīpata is found by the law of proportion. If the difference in the declinations of the Sun and the moon changes by an amount equal to \(\Delta _1 - \Delta _2\) in the time interval, \(t_2 - t_1\), what is the instant T, when it has changed by an amount \(\Delta _1\), making the declinations equal, that is, when \(\Delta (T) = 0\). This is given by

$$\begin{aligned} T-t_1 = \frac{t_2 - t_1}{\Delta _1 - \Delta _2} \times \Delta _1. \end{aligned}$$

This formula is in terms of \(\Delta\)s including the sign.

Now, at instant T, \(\delta _s\) and \(\delta _m\) are found again. In general, they would not be equal. Hence, \(\delta _s - \delta _m\) is computed at T, and some other nearby instant, and the process is iterated, till an ‘invariable’ quantity is obtained, when the values of the instants of vyatīpāta in the successive stages of iteration are equal.

In the texts Karaṇottama and Dṛkkaraṇa, a different strategy is used for finding the instant of vyatīpāta implicitly, using the derivative of the declination.

Karaṇottama is an important karaṇa text composed by Acyuta Piṣāraṭi (1550–1621 CE). The author himself has written a commentary on the work. It consists of 119 verses divided into five chapters, which deal with the standard topic in a Siddhānta text. This includes the computations related to vyatīpāta/vaidhṛta in the fifth chapter.

Both Karaṇottama and Dṛkkaraṇa describe the procedure for obtaining the longitudes of the Sun and the Moon at the middle of the vyatīpāta. The algorithms given in both the texts are similar and an intermediate term referred to as krāntigati/gatikrānti (translated as rate of motion of the declination) is used by the authors to arrive at the true longitudes at the middle of the vyatīpāta. In the expressions for krāntigatis, we find the application of the differential calculus. In the following subsections, we would explain the procedure for krāntigatis as described in the texts Karaṇottama [KTM (1964), p. 41] and Dṛkkaraṇa [DK1, DK2] respectively.

3.1 The krāntigati of the Sun in Karaṇottama [KTM (1964), p. 41]

तत्रार्कस्य क्रान्तिगत्यायनमाह–

tatrārkasya krāntigatyāyanamāha–

There, the procedure for obtaining the rate of motion of Sun’s declination is being told.

कोटिक्रान्ते रवेर्दिग्घ्न्यास्त्रिशैलेषु हृता गतिः ।।५।।

kōṭikrānte raverdigghnyāstriśaileṣu hṛtā gatiḥ ||5||

The kōṭikrānti of the Sun when multiplied by 10 (dik) and divided by 573 (tri-śaila-iṣu), the gati is obtained.

रविकोटिज्यायाः क्रान्तिमानीय तां‌ दशभिर्हत्वा गोसमेन हृत्वा सूर्यस्यापक्रमगतिरिति ।।

ravikōṭijyāyāḥ krāntimānīya tāṃ daśabhirhatvā gōsamena hṛtvā sūryasyāpakramagatiriti ||

Having obtained the declination from the Rcosine of the longitude of the Sun and multiplying that by 10 (daśa) and divided by 573 (gōsama), the gati of the declination [of the Sun is obtained].

Let \(\delta _s (t)\) be the declination of the Sun at any instant t, then the krānti-gati (\(g_s\)) of the Sun is given as

$$\begin{aligned} g_s= &\, {} k \bar{o}{} {{ {t}}} ikr\bar{a}{} nti \, \text{ of } \text{ the } \text{ Sun }\; \times \frac{10}{573}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{10}{573}, \end{aligned}$$
(1)

where \(\lambda _s\) is the longitude of the Sun.

The rationale for the expression (1) can be understood as follows:

Let the declination of the Sun be \(\delta _s (t)\) at any instant t, then the krāntigati of the Sun (\(g_s\)) can be expressed as

$$\begin{aligned} g_s=\frac{\text {d}{(R\sin \delta _s (t))}}{\text {d}{t}}= &\, {} \frac{\text {d}{(R\sin \epsilon \sin \lambda _s)}}{\text {d}{t}}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{\text {d}{\lambda _s}}{\text {d}{t}}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{\text {d}{\left[ \frac{(R\lambda _s)}{R}\right] }}{\text {d}{t}}. \end{aligned}$$
(2)

Here, the term \(R\sin \epsilon \cos \lambda _s\) is referred to as the kōṭikrānti in the text. Also, \(R\lambda _s\) is the longitude of the Sun in minutes and

$$\begin{aligned} \frac{\text {d}{(R\lambda _s)}}{\text {d}{t}} \approx 60'/\text{day }. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\text {d}{(R\sin \delta _s (t))}}{\text {d}{t}}= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{\text {d}{\left[ \frac{(R\lambda _s)}{R}\right] }}{\text {d}{t}}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{60}{R}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{60}{3438}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{10}{573}, \end{aligned}$$
(3)

which is the same as the expression (1).

3.2 The ‘krāntigati’ of the Moon in Karaṇottama [KTM (1964), p. 41]

इन्दोर्गत्यायनमाह–

indōrgatyāyanamāha –

[Now, the procedure for] obtaining the rate of motion of Moon’s declination is being told.

कोटिक्रान्तिः पृथक्स्थेन्दोर्वर्गिता सहितोनिता ।

क्रान्तियुत्यान्तरघ्न्या स्वदोःक्रान्त्याधिक्यकार्श्ययोः ।।६।।

तत्पदाढ्या पृथक्स्थेषु हताग्न्यब्धिहृता गतिः ।

koṭikrāntiḥ pṛthaksthendorvargitā sahitonitā |

krāntiyutyāntaraghnyā svadoḥkrāntyādhikyakārśyayoḥ ||6||

tatpadāḍhyā pṛthakstheṣu hatāgnyabdhihṛtā gatiḥ |

Having kept the kōṭikrānti of the Moon separately, the product of the sum and difference of [the Rsines of] the declinations of the Sun and the Moon has to be added to or subtracted from the square of that [Rcosine of the declination of the Moon] depending upon whether the Rsine of the declination of the Moon is larger or smaller respectively. The square-root of this [result] is to be added to the quantity kept separately and that has to be multiplied by 5 (iṣu) and divided by 43 (agnyabdhi). [The result obtained] would be the gati [of the krānti of the Moon].

इन्दोः‌ कोटिक्रान्तिं पृथक् विन्यस्य वर्गीकृत्यास्यामर्केन्दुभुजाक्रान्त्योर्योगान्तरहतिं संस्कुर्यात् । तत्प्रकारस्तु इन्दुक्रान्तेराधिक्ये सति योजयेत् । अल्पत्वे वियोजयेदिति । एवं संस्कृतस्य कोटिक्रान्तिवर्गस्य यन्मूलं तत्पूर्वं विन्यस्तायां कोटिक्रान्तौ संयोज्य तां पञ्चभिर्हत्वा त्रिचत्वारिंशताप्ता चन्द्रस्य क्रान्तिगतिः । ... ...।।

indoḥ koṭikrāntiṃ pṛthak vinyasya vargīkṛtyāsyāmarkendubhujākrāntyoryogāntarahatiṃ saṃskuryāt |

tatprakārastu indukrānterādhikye sati yojayet | alpatve viyojayediti | evaṃ saṃskṛtasya koṭikrāntivargasya

yanmūlaṃ tatpūrvaṃ vinyastāyāṃ koṭikrāntau saṃyojya tāṃ pañcabhirhatvā tricatvāriṃśatāptā candrasya krāntigatiḥ | ... ... ||

Having kept the kōṭikrānti of the Moon separately and squaring it, that [square] has to be corrected by the product of the sum and difference of [the Rsines of] the declinations of the Sun and the Moon. The nature of correction is indeed additive if the [Rsine of the] declination of the Moon is larger. If it is smaller, then the subtraction has to be performed [as the correction]. Like this, having found the square-root of the corrected kōṭikrāntivarga, it has to be added to the Rcosine of the declination which has been kept separately before. The obtained quantity has to be multiplied by 5 (pañca) and divided by 43 (tricatvāriṃśat). [The result obtained] would be the krāntigati of the Moon.

The verse 6 and half of the verse 7 of the Karaṇottama give the procedure to obtain the krāntigati of the Moon. Let \(\delta _s (t)\) and \(\delta _m (t)\) are the declinations of the Sun and the Moon at any instant t respectively, then the algorithm for finding the krāntigati is as follows:

  • The kōṭikrānti of the Moon (\(R\sin \epsilon \cos \lambda _m\)) has to be kept at two places separately. Here, \(\lambda _m\) is the longitude of the Moon respectively. That is,

    $$\begin{aligned} \text{ Place } \text{(A) } \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{ Place } \text{(B) }\\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ R\sin \epsilon \cos \lambda _m \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m \end{aligned}$$

  • Find the square of \(R\sin \epsilon \cos \lambda _m\). That is, find \(R^2\sin ^2\epsilon \cos ^2\lambda _m\) and at Place (A), we have

    $$\begin{aligned} \text{ Place } \text{(A) } \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{ Place } \text{(B) }\\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ R\sin \epsilon \cos \lambda _m \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m\\ \downarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow \\ R^2\sin ^2\epsilon \cos ^2\lambda _m \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m \end{aligned}$$

  • Find the Sum (S) and difference (D) of the Rsines of the declinations of the Sun and the Moon. That is, we have

    $$\begin{aligned} S= & {} \left| R\sin \delta _s + R\sin \delta _m\right| \\ \hbox {and}\qquad \qquad D= & {} \left| R\sin \delta _s - R\sin \delta _m\right| . \end{aligned}$$

    Here, \(R\sin \delta _s\) and \(R\sin \delta _m\) are understood to be the magnitudes of the Rsines of \(\delta _s\) and \(\delta _m\). Also, the product of this sum and difference is given as

    $$\begin{aligned} \text{ Product }\;\text{(S,D) }= &\, {} S\times D\\= &\, {} \left( R\sin \delta _m + R\sin \delta _s\right) \times \left( R\sin \delta _m - R\sin \delta _s\right) \\= &\, {} \left( R^2\sin ^2\delta _m - R^2\sin ^2\delta _s\right) \;\; (\text{ if }\; \delta _m> \delta _s)\\ \text{ and }\qquad \qquad \text{ Product }\;\text{(S, } \text{ D) }= &\, {} S\times D\\= \,& {} \left( R\sin \delta _s + R\sin \delta _m\right) \times \left( R\sin \delta _s - R\sin \delta _m\right) \\= &\, {} \left( R^2\sin ^2\delta _s - R^2\sin ^2\delta _m\right) \;\; (\text{ if }\; \delta _s > \delta _m). \end{aligned}$$

  • Apply the product of the above Sum and the difference to the square of the koṭikrānti of the Moon. That is, we have

    $$\begin{aligned} R^2\sin ^2\epsilon \cos ^2\lambda _m +\text{ Product }\;\text{(S, } \text{ D) }= & {} R^2\sin ^2\epsilon \cos ^2\lambda _m + S\times D\;\; (\text{ if }\; \delta _m > \delta _s)\\= & {} R^2\sin ^2\epsilon \cos ^2\lambda _m + \left( R^2\sin ^2\delta _m - R^2\sin ^2\delta _s\right) \\= & {} R^2\sin ^2\epsilon \cos ^2\lambda _m + R^2\sin ^2\delta _m - R^2\sin ^2\delta _s\\= & {} R^2 - R^2\sin ^2\delta _s\\= & {} R^2\sin ^2\epsilon \cos ^2\lambda _s, \end{aligned}$$

    where \(\lambda _s\) is the longitude of the Sun. Similary,

    $$\begin{aligned} R^2\sin ^2\epsilon \cos ^2\lambda _m - \text{ Product }\;\text{(S, } \text{ D) }= & {} R^2\sin ^2\epsilon \cos ^2\lambda _m - S\times D\;\; (\text{ if }\; \delta _m < \delta _s)\\= & {} R^2\sin ^2\epsilon \cos ^2\lambda _m- \left( R^2\sin ^2\delta _s - R^2\sin ^2\delta _m\right) \\= & {} R^2\sin ^2\epsilon \cos ^2\lambda _m- R^2\sin ^2\delta _s + R^2\sin ^2\delta _m\\= & {} R^2 - R^2\sin ^2\delta _s\\= & {} R^2\sin ^2\epsilon \cos ^2\lambda _s. \end{aligned}$$

    Therefore,

    $$\begin{aligned} R^2\sin ^2\epsilon \cos ^2\lambda _m \pm \text{ Product of the Sum and the difference} \; = R^2\sin ^2\epsilon \cos ^2\lambda _s. \end{aligned}$$

    The above term is referred to as saṃskṛta-krānti-kōṭivarga. The square-root of this is \(R\sin \epsilon \cos \lambda _s\), which is the koṭikrānti of the Sun. It is not clear why this is stated in such a round-about manner.

  • This (\(R\sin \epsilon \cos \lambda _s\)) has to be added to \(R\sin \epsilon \cos \lambda _m\) which has been kept separately (at Place (B)). This sum has to be multiplied by 5 and divided by 43 to obtain the krāntigati of the Moon (denoted as \(g_m\)). Therefore,

    $$\begin{aligned} g_m= & {} \left( R\sin \epsilon \cos \lambda _m + R\sin \epsilon \cos \lambda _s\right) \times \frac{5}{43}. \end{aligned}$$
    (4)

    Therefore,

    $$\begin{aligned} \text{ Place } \text{(A) } \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{ Place } \text{(B) }\\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ R\sin \epsilon \cos \lambda _m \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m\\ \downarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow \\ R^2\sin ^2\epsilon \cos ^2\lambda _m\;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m\\ \downarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow \\ R^2\sin ^2\epsilon \cos ^2\lambda _s \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m\\ \downarrow \;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow \\ R\sin \epsilon \cos \lambda _s \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;R\sin \epsilon \cos \lambda _m\\ \downarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow \\ \;\;\;\;\;\;\;\;\;\;\;\rightarrow \;\;\;\rightarrow \;\;\;\;\rightarrow \;\;\;+ & {} \;\;\;\;\leftarrow \;\;\;\leftarrow \;\;\;\leftarrow \;\;\;\;\;\;\;\;\;\;\; \\&\downarrow&\\&\left( R\sin \epsilon \cos \lambda _s + R\sin \epsilon \cos \lambda _m\right)&\\&\downarrow&\\&\left( R\sin \epsilon \cos \lambda _s + R\sin \epsilon \cos \lambda _m\right) \times \frac{5}{43}&\\&\Downarrow&\\&{{\varvec{kr}}}{\bar{{\varvec{a}}}}{{\varvec{ntigati}}}\;\textbf{of the Moon}&\end{aligned}$$

The rationale for the expression (4) could be understood as follows: The krāntigati (\(g_m\)) of the Moon is obtained by finding the derivative of \(R\sin \delta '_m\), where \(\delta '_m\) is the longitude of a point on the ecliptic which has the same longitude as the Moon (essentially the declination of the Moon ignoring its latitude). Therefore,

$$\begin{aligned} g_m= &\, {} \frac{\text {d}{(R\sin \delta '_m (t))}}{\text {d}{t}} \end{aligned}$$
(5)
$$\begin{aligned}= &\, {} \frac{\text {d}{(R\sin \epsilon \sin \lambda _m)}}{\text {d}{t}}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _m \times \frac{\text {d}{\lambda _m}}{\text {d}{t}}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _m \times \frac{\text {d}{\left[ \frac{(R\lambda _m)}{R}\right] }}{\text {d}{t}}, \end{aligned}$$
(6)

where \(R\lambda _m\) is the longitude of the Moon in minutes and

$$\begin{aligned} \frac{\text {d}{(R\lambda _m)}}{\text {d}{t}} \approx 800'/\text{day }, \end{aligned}$$

which is the rate of change of Moon’s longitude in minutes. Therefore,

$$\begin{aligned} \frac{\text {d}{(R\sin \delta '_m (t))}}{\text {d}{t}}= &\, {} R\sin \epsilon \cos \lambda _m \times \frac{1}{R}\times \frac{\text {d}{(R\lambda _m)}}{\text {d}{t}}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _m \times \frac{800}{R}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _m \times \frac{800}{3438}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _m \times \frac{1}{4.2975}\nonumber \\\approx &\, {} R\sin \epsilon \cos \lambda _m \times \frac{1}{4.3}\nonumber \\= &\, {} R\sin \epsilon \cos \lambda _m \times \frac{10}{43}. \end{aligned}$$
(7)

Now, near vyatīpāta, \(\left| \cos \lambda _m\right| \approx \left| \cos \lambda _s\right|\), as

$$\begin{aligned} \lambda _m \approx 180^\circ - \lambda _s \approx 360^\circ - \lambda _s \end{aligned}$$

at vyatīpāta. Therefore,

$$\begin{aligned} R\sin \epsilon \cos \lambda _m= & {} \frac{1}{2}\left( 2 R\sin \epsilon \cos \lambda _m \right) \nonumber \\\approx & {} \frac{1}{2}\left( R\sin \epsilon \cos \lambda _m + R\sin \epsilon \cos \lambda _s\right) . \end{aligned}$$
(8)

Applying (8) in (7), we have

$$\begin{aligned} \frac{\text {d}{(R\sin \delta '_m (t))}}{\text {d}{t}}= & {} \frac{1}{2}\left( R\sin \epsilon \cos \lambda _m + R\sin \epsilon \cos \lambda _s\right) \times \frac{10}{43}\nonumber \\= & {} \left( R\sin \epsilon \cos \lambda _m + R\sin \epsilon \cos \lambda _s\right) \times \frac{5}{43}, \end{aligned}$$
(9)

which is the same as the expression (4) for the krāntigati of the Moon given in the text Karaṇōttama.

Noting that \(\frac{5}{43}\times \frac{1}{2}\times \frac{800}{3438},\) and \(\frac{10}{573} = \frac{60}{3438}\), the sum of the krāntigatis (\(g_{sum}\)) of the Sun and the Moon can be expressed as

$$\begin{aligned} g_{sum}\;\; {(Kara {} {{\d {n}}} {} ottama )}= & {} \frac{\text {d}{(R\sin \delta '_m (t))}}{\text {d}{t}} + \frac{\text {d}{(R\sin \delta _s (t))}}{\text {d}{t}}\end{aligned}$$
(10)
$$\begin{aligned}= & {} \frac{1}{2}\left( R\sin \epsilon \cos \lambda _m + R\sin \epsilon \cos \lambda _s\right) \times \frac{800}{3438} \nonumber \\{} & {} + R\sin \epsilon \cos \lambda _s \times \frac{60}{3438} \nonumber \\= & {} \frac{800}{3438}\times R\sin \epsilon \left[ \frac{1}{2}\cos \lambda _m +\frac{1}{2} \cos \lambda _s + \cos \lambda _s \times \frac{60}{800} \right] \nonumber \\= & {} \frac{800}{R}\times R\sin \epsilon \left[ \frac{1}{2}\cos \lambda _m + \cos \lambda _s \left( \frac{1}{2}+ \frac{60}{800}\right) \right] \nonumber \\= & {} \frac{800}{R}\times \left[ \frac{1}{2}R\sin \epsilon \cos \lambda _m + R\sin \epsilon \cos \lambda _s \left( \frac{1}{2}+ \frac{60}{800}\right) \right] \end{aligned}$$
(11)

Now,

$$\begin{aligned} \frac{1}{2}+ \frac{60}{800}= & {} \frac{1}{2}\left( 1 + \frac{60}{400}\right) \nonumber \\= & {} \frac{1}{2}\times \frac{23}{20}. \end{aligned}$$
(12)

Applying (12) in (11), we have

$$\begin{aligned} g_{sum} = \frac{800}{R}\times \left[ \frac{1}{2} R\sin \epsilon \cos \lambda _m + \left( \frac{1}{2} \times \frac{23}{20}\right) R\sin \epsilon \cos \lambda _s \right] . \end{aligned}$$
(13)

We shall see later that this sum (\(g_{sum}\)) is used to obtain the longitudes of the Sun and the Moon at the middle of the vyatīpāta.

3.3 The gatikrānti (krāntigati) in Dṛkkaraṇa [DK1, DK2]

The author of Dṛkkaraṇa uses the term gatikrānti for the rate of change of declination instead of krāntigati, as in Karaṇottama. This intermediate term is used to obtain the correction term by applying which one can obtain the longitude of the Sun and the Moon at the middle of the vyatīpāta. Now, we shall explain the algorithm to obtain the gatikrānti as described in Dṛkkaraṇa in verses 15-\(-\)17.5 in chapter 7.

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चन्द्रन्ऱॆ कोटिजक्रान्ति वेऱॆयॊन्नङ्ङुवॆच्चतु |

वर्ग्गिच्चिट्टतिलुं पिन्नॆ रवीन्द्वोः क्रान्ति तङ्ङळिल् ||१५||

कूट्टित्तयोरन्तरत्ताल् पॆरुक्किस्संस्क्करिक्कणं |

चन्द्रक्रान्ति कुऱञ्ञीटिल् कळवू कूट्टुकन्यथा ||१६||

अतु मूलिच्चु कूट्टीट्टु कोटिजक्रान्तिलिप्तयेत् |

अर्क्कस्य कोटिजक्रान्तिं गारघ्नं नरभाजितं ||१७||

फलवुं‌ कूट्टियर्द्धिच्चाल् गतिक्रान्तियताय्वरुं |

candranṟe kōṭijakrānti vēṟeyonnaṅṅuvccatu |

varggicciṭṭatiluṃ pinne ravīndvōḥ krānti taṅṅaḷil ||15||

kūṭṭittayōrantarattāl perukkissaṃskkarikkaṇaṃ |

candrakrānti kuṟaññīṭil kaḷavū kūṭṭukanyathā ||16||

atu mūliccu kūṭṭīṭṭu kōṭijakrāntiliptayēt |

arkkasya kōṭijakrāntiṃ gāraghnaṃ narabhājitaṃ ||17||

phalavuṃ‌ kūṭṭiyarddhiccāl gatikrāntiyatāyvaruṃ |

Having kept the kōṭ­ijakrānti of the Moon separately, find the square of it. To this [square of the kōṭ­ijakrānti, apply the product of the sum and the difference of the declinations of the Sun and the Moon. If the declination of the Moon is lesser [than that of the Sun], then that [product] has to be subtracted from [the square of the kōṭ­ijakrānti], otherwise it has to be added. Then, having found the square-root of this [quantity] and having added this [square-root] to the kōṭ­ijakrānti [of the Moon], [the obtained quantity] has to be converted into minutes. When the sum–of thisFootnote 5 and the result obtained by multipying the kōṭ­ijakrānti of the Sun by 23 (gāra) and divided by 20 (nara)–is halved, then the result obtained would be the gatikrānti (krāntigati).

These verses give the procedure to find the gatikrānti. The method is the same as in Karaṇottama, with the gatikrānti here differing by a factor compared to the ‘krāntigati’ of Karaṇottama. We summarise the procedure in the following.

  • The kōṭijakrānti of the Moon \(R\sin \epsilon \cos \lambda _m\) (where \(\lambda _m\) is the longitude of the Moon) has to be placed at two places. That is,

    $$\begin{aligned} \text{ Place } \text{(A) } \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{ Place } \text{(B) }\\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ R\sin \epsilon \cos \lambda _m \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m \end{aligned}$$

  • Find the square of \(R\cos \delta _m\). That is, find \(R^2\cos ^2\delta _m\) and at Place (A), we have

    $$\begin{aligned} \text{ Place } \text{(A) } \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{ Place } \text{(B) }\\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ R\sin \epsilon \cos \lambda _m \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R\sin \epsilon \cos \lambda _m\\ \downarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow \\ R^2\cos ^2\epsilon \cos ^2\lambda _m \;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; R^2\cos ^2\epsilon \cos ^2\lambda _m \end{aligned}$$

  • Find the Sum (S) and difference (D) of the Rsines of the declinations of the Sun and the Moon. That is, we have

    $$\begin{aligned} S= &\, {} \left| R\sin \epsilon \sin \lambda _s + R\sin \epsilon \sin \lambda _m\right| \\ \hbox {and}\qquad \qquad D= &\, {} \left| R\sin \epsilon \sin \lambda _s - R\sin \epsilon \sin \lambda _s\right| . \end{aligned}$$

  • Apply the product of the above Sum and the difference to the square of the kōṭijakrānti of the Moon. There are two cases depending upon whether the declination of the Moon is smaller or larger.

    $$\begin{aligned} R^2\sin ^2\epsilon \cos ^2\lambda _m + S \times D, \;\;\; \text{ for }\; \delta _m > \delta _s\\ \text{ and }\qquad \qquad R^2\sin ^2\epsilon \cos ^2\lambda _m - S \times D, \;\;\; \text{ for }\; \delta _m < \delta _s. \end{aligned}$$

    In either case,

    $$\begin{aligned} R^2\sin ^2\epsilon \cos ^2\lambda _m \pm S \times D = R^2 \sin ^2\epsilon \cos ^2\lambda _s. \end{aligned}$$

    The square-root of this is \(R\sin \epsilon \cos \lambda _s\) and is referred to as the kōṭijakrānti of the Sun.

  • Now, the sum of the above result (\(R\sin \epsilon \cos \lambda _s\)) and the kōṭijakrānti of the Moon (\(R\sin \epsilon \cos \lambda _m\)) is to be found. It is not clear whether this sum is the ‘phala’ referred to in the half-verse following the verse 17.

  • Now, the the kōṭijakrānti of the Sun has to be multiplied by 23 (gāra) and divided by 20 (nara). That is, we have a new quantity

    $$\begin{aligned} Y= &\, {} {k\bar{o}{} {{ {t}}} {} ijakr \bar{a}{} nti\, of\, the\, Sun}\, \times \frac{{{g\bar{a}ra}}}{{nara }}\\= &\, {} R\sin \epsilon \cos \lambda _s \times \frac{23}{20}. \end{aligned}$$

  • This new quantity (Y) has to be added to the phala (X). The half of this is known as gatikrānti (denoted as g (Dṛkkaraṇa)). If the ‘phala’ (X) here is interpreted as the sum of the kōṭijakrāntis of the Sun and the Moon, it will not lead to anything meaningful. However, if the ‘phala’ is interpreted as the kōṭijakrānti of the Moon only, we obtain a result which is in accordance with the procedure in Karaṇottama, which gives the sum of the Rsines of the declinations of the Sum and the Moon. Hence, we adopt the later interpretation. Then,

    $$\begin{aligned}g \;{(D {} {{ \d{r}}} {} kkara {} {{ \d{n}}} {} a )}= &\, {} \frac{X + Y}{2}\nonumber \\= &\, {} \frac{\left( R\sin \epsilon \cos \lambda _m + R\sin \epsilon \cos \lambda _s \times \frac{23}{20}\right) }{2}\nonumber \\= &\, {} \frac{1}{2}\left( R\sin \epsilon \cos \lambda _m\right) +\frac{1}{2}\times \frac{23}{20}\times \left( R\sin \epsilon \cos \lambda _s\right) . \end{aligned}$$
    (14)

    Comparing the expressions for the sum of the krāntigatis of the Sun and the Moon, \(g_{sum}\) (Karaṇottama) as defined in Karaṇottama, and the ‘gatikrānti’, g (Dṛkkaraṇa) as defined in Dṛkkaraṇa, we find that

    $$\begin{aligned} g \; {(D {} {{ \d{r}}} {} kkara {} {{ \d{n}}} {} a )} = \frac{R}{800}\times g_{sum} \; {(Kara {} {{ \d{n}}} {} ottama )}. \end{aligned}$$

    It can be recollected that \(g_{sum}\) (Karaṇottama) is the sum of the rates of changes of the Rsines of the declinations of the Sun and the Moon (ignoring its latitude). In Appendix 2, the folio corresponding to the verses describing the ‘gatikrānti’ in Dṛkkaraṇa is presented.

4 Instant of vyatīpāta/vaidhṛta and the corrections to the longitudes of the Sun and the Moon

Let the instant corresponding to Vyatīpāta be T units of time (day or nāḍikā) after the instant when \(\delta '_m = \delta _s\) (when \(\lambda _m = 180^\circ - \lambda _s\) or \(360^\circ - \lambda _s\); where t is taken as 0). Then,

$$\begin{aligned} R\sin \delta _m (T) - R\sin \delta _s (T) = 0\approx & {} R\sin \delta _m (0) - R\sin \delta _s (0)\\{} & {} + \left( \frac{\text {d}{\left[ R\sin \delta _m(t) - R\sin \delta _s (t)\right] }}{\text {d}{t}}\right) \times T. \end{aligned}$$

Hence,

$$\begin{aligned} T= \frac{R\sin \delta _s (0) - R\sin \delta _m (0)}{\frac{\text {d}{\left[ R\sin \delta _m(t) - R\sin \delta _s (t)\right] }}{\text {d}{t}} }. \end{aligned}$$
(15)

Now, when the Moon has a latitude, \(\beta\),

$$\begin{aligned} R\sin \delta _m (t)= &\, {} R\sin \epsilon \sin \lambda _m \cos \beta + R\sin \beta \cos \epsilon \\= &\, {} R\sin \delta '_m (t) \cos \beta + R\cos \epsilon \sin \beta \\\approx &\, {} R\sin \delta '_m (t) + R\beta \cos \epsilon , \end{aligned}$$

ignoring terms of \(O (\beta ^2)\). Hence,

$$\begin{aligned} R\sin \delta _s (0) - R\sin \delta _m (0)= &\, {} R\sin \delta _s (0) - R\sin \delta '_m (0) - R\beta \cos \epsilon \nonumber \\= &\, {} - R\beta \cos \epsilon , \end{aligned}$$
(16)

as \(R\sin \delta '_m (0) = R\sin \delta _s (0)\).

Also, near Vyatīpāta

$$\begin{aligned} \frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}} \approx -\frac{\text {d}{R\sin \delta _m (t)}}{\text {d}{t}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\text {d}{(R\sin \delta _m (t) - R\sin \delta _s (t))}}{\text {d}{t}}= & {} \pm \left[ \left| \frac{\text {d}{R\sin \delta _m (t)}}{\text {d}{t}} \right| + \left| \frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}} \right| \right] \nonumber \\= & {} \pm \left[ \left| \frac{\text {d}{R\sin \delta '_m (t)}}{\text {d}{t}} + R\cos \epsilon \frac{\text {d}{\beta }}{\text {d}{t}} \right| \right] + \left| \frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}} \right| . \end{aligned}$$
(17)

Here, the ‘+’ sign is applicable if \(\frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}}\) is negative (Sun in even quadrant) and ‘-’ sign is applicable if \(\frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}}\) is positive (Sun in odd quadrant).

Now, applying (17) and (16) in (15), we have

$$\begin{aligned} T= & {} \frac{R\sin \delta _s (0) - R\sin \delta _m (0)}{\frac{\text {d}{\left[ R\sin \delta _m(t) - R\sin \delta _s (t)\right] }}{\text {d}{t}}}\\= & {} \pm \left[ \frac{R\sin \delta _s (0) - R\sin \delta _m (0)}{\left| \frac{\text {d}{R\sin \delta '_m (t)}}{\text {d}{t}} + R\cos \epsilon \frac{\text {d}{\beta }}{\text {d}{t}} \right| + \left| \frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}} \right| }\right] . \end{aligned}$$

Now,

$$\begin{aligned} R\sin \delta _s (0) - R\sin \delta _m (0) \approx - R\beta \cos \epsilon = O(\beta ), \end{aligned}$$

already. Hence, \(\frac{\text {d}{\beta }}{\text {d}{t}}\) term in the denominator, can be neglected if T is being computed to \(O(\beta )\). Hence,

$$\begin{aligned} T\approx \pm\, \left[ \frac{R\sin \delta _s (0) - R\sin \delta _m (0)}{\left| \frac{\text {d}{R\sin \delta '_m (t)}}{\text {d}{t}} \right| + \left| \frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}} \right| }\right] , \end{aligned}$$

where the ‘\(+\)’ sign is applicable when the Sun is in the even quadrant and the ‘−’ sign, when it is in the odd quadrant. In fact, it can be seen that

$$\begin{aligned} T\approx \pm\, \left[ \frac{R\sin \delta _s (0) - R\sin \delta _m (0)}{\left| \frac{\text {d}{R\sin \delta '_m (t)}}{\text {d}{t}} \right| + \left| \frac{\text {d}{R\sin \delta _s (t)}}{\text {d}{t}} \right| }\right] , \end{aligned}$$

where, ‘−’ sign is applicable when the object in the odd quadrant has a greater declination which means that the vyatīpāta/vaidhṛta has elapsed, and ‘\(+\)’ sign is applicable when the object in the odd quadrant has a lesser declination.

Using the expression for the rate of change of the sum of the Rsines of the declinations of the Sun and the Moon in the expression for T, we have

$$\begin{aligned} T\approx \pm\, \left[ \frac{\left| R\sin \delta _s (0) - R\sin \delta _m (0)\right| }{\left( \frac{800}{R} \right) \left[ \frac{1}{2} R\sin \epsilon \cos \lambda _m + \left( \frac{1}{2}\times \frac{23}{20} \right) R\sin \epsilon \cos \lambda _s\right] }\right] . \end{aligned}$$

This result, as such, is not stated in the two texts. However, the changes in the longitudes of the Sun and the Moon, during the interval between the instants when the Rsines of the declinations of the Sun and the Moon have a given difference and the middle of the vyatīpāta, when it is zero can be readily computed from T. Let these changes be \(\Delta \lambda _s\) and \(\Delta \lambda _m\) respectively.

$$\begin{aligned} \Delta \lambda _s\; \text{(mins.) }= &\, {} T\; \text{(in } \text{ days) } \times \frac{\text {d}{\lambda _s\; \text{(mins./day) }}}{\text {d}{t}}\nonumber \\= &\, {} T\times 60\nonumber \\= &\, {} \frac{\pm \left( R\sin \delta _m \sim R\sin \delta _s\right) \times 60}{\frac{800}{R}\left[ \frac{1}{2} R\sin \epsilon \cos \lambda _m + \left( \frac{1}{2} \times \frac{23}{20}\right) R\sin \epsilon \cos \lambda _s \right] }\nonumber \\= & {} \pm \left[ \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 3438}{\left[ \frac{1}{2} R\sin \epsilon \cos \lambda _m + \left( \frac{1}{2} \times \frac{23}{20}\right) R\sin \epsilon \cos \lambda _s \right] }\times \frac{3}{40}\right] , \end{aligned}$$
(18)

and

$$\begin{aligned} \Delta \lambda _m\; \text{(mins.) }= &\, {} T\; \text{(in } \text{ days) } \times \frac{\text {d}{\lambda _m\; \text{(mins./day) }}}{\text {d}{t}}\nonumber \\= &\, {} T\times 800\nonumber \\= & {} \frac{\pm \left( R\sin \delta _m \sim R\sin \delta _s\right) \times 800}{\frac{800}{R}\left[ \frac{1}{2} R\sin \epsilon \cos \lambda _m + \left( \frac{1}{2} \times \frac{23}{20}\right) R\sin \epsilon \cos \lambda _s \right] }\nonumber \\= &\, {} \pm \left[ \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 3438}{\left[ \frac{1}{2} R\sin \epsilon \cos \lambda _m + \left( \frac{1}{2} \times \frac{23}{20}\right) R\sin \epsilon \cos \lambda _s \right] }\right] . \end{aligned}$$
(19)

It is readily seen that

$$\begin{aligned} \Delta \lambda _s\; \text{(mins.) } = \Delta \lambda _m\; \text{(mins.) } \times \frac{3}{40}. \end{aligned}$$
(20)

The expressions for \(\Delta \lambda _s\) and \(\Delta \lambda _m\) which are to be subtracted from or added to the longitudes of the Sun and the Moon respectively at the instant (with a given value of (\(R\sin \delta _m \sim R\sin \delta _s\))) to obtain the longitudes at the middle of the vyatīpāta/vaidhṛta. We have already noted that ‘−’ sign is applicable when the object in the odd quadrant has a greater declination, and ‘\(+\)’ sign is applicable when the object in the odd quadrant has a lesser declination. These are explicitly stated in both the texts, Karaṇottama [KTM (1964), pp. 41–42] and Dṛkkaraṇa [DK1, DK2]. The verses with the translations are presented in Appendix 1. The folio describing the changes in longitudes of the Sun and the Moon in Dṛkkaraṇa is presented in Appendix 3.

5 Concluding remarks

It is well known that the derivative of the sine function is used for computing the instantaneous velocity (tātkālikagati) of planets in Indian astronomical texts from Laghumānasa onwards. In this paper, we have reported the use of the derivative for finding the rates of change of the declinations of the Sun and the Moon in two Kerala texts of the late sixteenth and early seventeenth century, namely Karaṇottama of Acyuta Piṣāraṭi in Sanskrit, and a Malayāḷam text, Dḳkaraṇa. This is used for the computation of the instant of vyatīpāta/vaidhṛta implicitly, which used to be computed using proportionality arguments earlier. It would be interesting to investigate whether the concept of derivative is used in other contexts also, in later Kerala texts.