Use of the concept of derivative in the computation of vyatīpāta in two Kerala texts

Abstract

It is well known that the concept of derivative was used in finding the rates of motion of planets in Indian astronomy texts beginning with Laghumānasa (c. 932 CE). In his Vāsanābhāṣya of his own work, Siddhāntaśiromaṇi (c.1150 CE), Bhāskarācārya explains the necessity of using the concept of tātkālikagati (instantaneous rates of motion) of planets, which involves using the derivative of the sine function, and discusses the retrograde motion of planets also, using the concept. Later, Kerala texts like Tantrasaṅgraha also discuss this concept. In two Kerala texts, Karaṇottama of Acyuta Piṣāraṭi (late sixteenth century) and Dṛkkaraṇa (1608 CE), the use of the concept of derivative is used in a very different context, namely, computations pertaining to vyatīpāta. In this paper, we describe the algorithms involving the ‘krāntigati’ or the rate of change of the declinations of the Sun and the Moon involving the derivative conept, in these two texts.

Appendices

Appendix 1

We present the algorithms, given in karaṇottama and dṛkkaraṇa, for finding the longitudes of the Sun and the Moon at the middle of the vyatīpāta.

1.1 Obtaining the longitudes of the Sun and the Moon at the middle of the vyatīpāta as per karṇaōttama

द्विष्ठात् क्रान्त्यन्तरात् षष्ट्या खखनागैश्च ताडितात् ।

गतियुत्याप्तलिप्ताः स्वं क्रमादर्कशशाङ्कयोः ।

अल्पाचेदोजगाक्रान्तिर्महती चेदृणं तयोः ।।७।।

dviṣṭhāt krāntyantarāt ṣaṣṭyā khakhanāgaiśca tāḍitāt |

gatiyutyāptaliptāḥ svaṃ kramādarkaśaśāṅkayoḥ |

alpācedojagākrāntirmahatī cedṛṇaṃ tayoḥ ||7||

The difference in [Rsines of] the declinations [of the Sun and the Moon] which have been kept separately at two places have to be multiplied by 60 (ṣaṣṭi) and 800 (khakhanāga) respectively and divided by the sum of the gatis of their declinations (sum of the gatikrāntis of the Sun and the Moon). The obtained results, in minutes, have to be added to the longitude of the Sun and the Moon respectively when the declination of the object (Sun/Moon) situated at the odd-quadrant is lesser than that of the other one. If the declination is larger, then those [results] have to be subtracted.

सूर्येन्दुक्रान्त्योरन्तरं द्वयोः स्थानयोर्निधायैकं षष्ट्यान्यं शताष्टकेन च ताडयेत् । क्रान्तिगत्योर्योगेन विभजेच्च । तत्र प्रथमं फलं लिप्ताल्पकमर्के संस्कार्यम् । द्वितीयं फलं‌ चन्द्रे संस्कार्यम् । संस्कारप्रकारस्तु अर्केन्द्वोर्मध्ये य ओजपदगतस्तस्य क्रान्तिरल्पा चेद् धनं महती चेदृणमिति । ... ...

sūryendukrāntyorantaraṃ dvayoḥ sthānayornidhāyaikaṃ ṣaṣṭyānyaṃ śatāṣṭakena ca tāḍayet | krāntigatyoryogena vibhajecca | tatra prathamaṃ phalaṃ liptālpakamarke saṃskāryam | dvitīyaṃ phalaṃ candre saṃskāryam | saṃskāraprakārastu arkendvormadhye ya ojapadagatastasya krāntiralpā ced dhanaṃ mahatī cedṛṇamiti | ... ...

Having kept the difference in [Rsines of] the declinations of the Sun and the Moon at two places, multipy [the term at] the first place by 60 (ṣaṣṭi) and [the term at] the other (second) place by 800 (śatāṣṭaka). Also, divide [both the quantities] by the sum of the rates of motion (krāntigatiyōga/(gatikrāntiyōga)) [of the Sun and the Moon]. There, the first result in the form of minutes has to be applied to [the longitude of] the Sun. The second result has to be applied to the Moon. The nature of correction is like this. Among the Sun and the Moon, if the declination of the one which is situated at the odd quadrant is smaller [than that of the other one], then addition is to be performed. If it is larger, then the subtraction [is to be performed].

Verse 7 of Karaṇōttama gives an algorithm to obtain the longitudes of the Sun and the Moon at the middle of the vyatīpāta. This is as follows:

• Place the difference in Rsines of the declinations of the Sun and the Moon at two places. That is, we have

  $$\begin{aligned} \text{ Place } \text{(A) } \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{ Place } \text{(B) }\\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ R\sin \delta _m \sim R\sin \delta _s \;\;\;\;\;\;{} & {} \;\;\;\;\;\; R\sin \delta _m \sim R\sin \delta _s \end{aligned}$$

• Multiply by 60 at one place and by 800 at the second place. That is,

  $$\begin{aligned} \text{ Place } \text{(A) } \;\;\;\;\;\;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\; \text{ Place } \text{(B) }\\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ R\sin \delta _m \sim R\sin \delta _s \;\;\;\;\;\;{} & {} \;\;\;\;\;\; R\sin \delta _m \sim R\sin \delta _s \\ \Updownarrow \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Updownarrow \\ \left( R\sin \delta _m \sim R\sin \delta _s\right) \times 60 \;\;\;\;\;\;{} & {} \;\;\;\;\;\; \left( R\sin \delta _m \sim R\sin \delta _s\right) \times 800 \end{aligned}$$

• Divide both the results by the sum of the rates of motion (gatikrāntiyōga) of the Sun and the Moon. These are the corrections to be applied to the longitudes of the Sun and the Moon respectively. Therefore, correction to the Sun’s longitude is given by

  $$\begin{aligned} \Delta \lambda _s= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 60}{{{gatikr\bar{a}ntiy\bar{o}ga}}}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 60}{R\cos \delta _s \times \frac{10}{573} + \left( R\cos \delta _m + R\cos \delta _s\right) \times \frac{5}{43}}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 60}{R\cos \delta _s \times \frac{10}{573} + \left( R\cos \delta _m + R\cos \delta _s\right) \times \frac{5}{43}}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 60}{R\sin \epsilon \left[ R\cos \lambda _s \times \frac{10}{573} + \left( R\cos \lambda _m + R\cos \lambda _s\right) \times \frac{5}{43}\right] }. \end{aligned}$$

  (21)

• Similarly, the correction to the Moon’s longitude is given by

  $$\begin{aligned} \Delta \lambda _m= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 800}{{{gatikr\bar{a}ntiy\bar{o}ga}}}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 800}{R\cos \delta _s \times \frac{10}{573} + \left( R\cos \delta _m + R\cos \delta _s\right) \times \frac{5}{43}}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 800}{R\cos \delta _s \times \frac{10}{573} + \left( R\cos \delta _m + R\cos \delta _s\right) \times \frac{5}{43}}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 800}{R\sin \epsilon \left[ R\cos \lambda _s \times \frac{10}{573} + \left( R\cos \lambda _m + R\cos \lambda _s\right) \times \frac{5}{43}\right] }. \end{aligned}$$

  (22)

• Now, corrections given by the expressions (21) and (22) have to be applied to the longitudes of the Sun and the Moon respectively. Therefore,

  $$\begin{aligned} \lambda _s (T)= & {} \lambda _s (0) \pm \Delta \lambda _s\\ \text{ and } \qquad \qquad \lambda _m (T)= & {} \lambda _m (0) \pm \Delta \lambda _m, \end{aligned}$$

  where ‘$+$’has to be used if the declination of the object which is situated at the odd quadrant is smaller than that of the other one. Otherwise, ‘−’ sign has to be used.

1.2 Obtaining the longitudes of the Sun and the Moon at the middle of the vyatīpāta in Dṛkkaraṇa [DK1, DK2]

क्रान्त्यन्तरं‌ जालभोगैः पॆरुक्कीट्टु हरिक्कणं ||१८||

गतिक्रान्त्या फलं वन्नाल् चन्द्रनिल् संस्क्करिक्कणं |

अतु वेऱॊन्नु वॆच्चिट्टु गानं कॊण्टु पॆरुक्कियाल् ||१९||

नाभिकॊण्टु हरिच्चिट्टङ्ङर्क्कन्ऱॆ लिप्तयिल् तदा |

ओजपादग्रहत्तिन्ऱॆ क्रान्तियेऱिल् कळञ्ञिटू ||२०||

कुऱकिल् कूट्टि वॆक्केणमविटॆ क्रान्तिकॊण्टुटन् |

क्षेपवुं संस्क्करिच्चिट्टु क्रान्तिसाम्यं वरुत्तुक ||२१||

krāntyantaraṃ‌ jālabhōgaiḥ perukkīṭṭu harikkaṇaṃ ||18||

gatikrāntyā phalaṃ vannāl candranil saṃskkarikkaṇaṃ |

atu vēṟonnu vecciṭṭu gānaṃ koṇṭu perukkiyāl ||19||

nābhikoṇṭu haricciṭṭaṅṅarkkanṟe liptayil tadā |

ōjapādagrahattinṟe krāntiyēṟil kaḷaññiṭū ||20||

kuṟakil kūṭṭi vekkēṇamaviṭe krāntikoṇṭuṭan |

kṣēpavuṃ saṃskkaricciṭṭu krāntisāmyaṃ varuttuka ||21||

Now, having multiplied the difference in declinations (krāntyantara) by 3438 (jālabhōga), divide it by the gatikrānti; the result obtained is to be applied to [the longitude of the] the Moon. Having kept this aside, multiply this by 03 (gāna) and divide by 40 (nābhi). Both these results in minutes have to be subtracted from [their respective longitudes] if the declination of the odd-quadrant-planet is larger, if it is smaller they have to be added. Thereby the equality in declinations is to be obtained by correcting this by the latitude as well.

• Multiply the difference in Rsines of the declinations of the Sun and the Moon by 3438 (jālabhōga) and divided by the gatikrānti. The result is the correction ($\Delta \lambda _m$) applied to the Moon’s longitude. Therefore,

  $$\begin{aligned} \Delta \lambda _m= & {} \frac{{{kr\bar{a}ntyantara}} \times {{j\bar{a}labh\bar{o}ga}}}{{{gatikr\bar{a}nti}}}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 3438}{\frac{1}{2}\left( R\sin \epsilon \cos \lambda _m\right) +\frac{1}{2}\times \frac{23}{20}\times \left( R\sin \epsilon \cos \lambda _s\right) }. \end{aligned}$$

  (23)

• The correction ($\Delta \lambda _s$) applied to the longitude of the Sun is obtained by multiplying $\Delta \lambda _m$ by 03 (gāna) and divided by 40 (nābhi). That is,

  $$\begin{aligned} \Delta \lambda _s= & {} \Delta \lambda _m \times \frac{3}{40}\nonumber \\= & {} \frac{\left( R\sin \delta _m \sim R\sin \delta _s\right) \times 3438}{\frac{1}{2}\left( R\sin \epsilon \cos \lambda _m\right) +\frac{1}{2}\times \frac{23}{20}\times \left( R\sin \epsilon \cos \lambda _s\right) }\times \frac{3}{40}. \end{aligned}$$

  (24)

• Now the longitudes of the Sun and the Moon at the instant of the middle of the vyatīpāta are given by

  $$\begin{aligned} \lambda _s (T)= & {} \lambda _s (0) \pm \Delta \lambda _s\\ \text{ and } \qquad \qquad \lambda _m (T)= & {} \lambda _m (0) \pm \Delta \lambda _m, \end{aligned}$$

  where ‘$+$’ has to be used if the declination of the object which is situated at the odd quadrant is smaller than that of the other one. Otherwise, ‘−’ sign has to be used.

• From $\lambda _s (T)$ and $\lambda _m (T)$, the declinations of the Sun and the Moon can be obtained respectively.

• The true declination of the Moon can be found by applying the correction due to the latitude. The true declination of the Moon is expressed as

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

  taking the latitude of the Moon into account.

Appendix 2

See Fig. 2.

Fig. 2

Folio corresponding to gatikrānti in Dṛkkaraṇa, Trav. c. 7c., Kerala University Oriental Research Institute and Manuscript Library, Trivandrum

Appendix 3

See Fig. 3.

Fig. 3

Folio corresponding to the longitudes of the Sun and the moon at the middle of the vyatīpāta, Trav. c. 7c., Kerala University Oriental Research Institute and Manuscript Library, Trivandrum