jyothisha bashings and eclipses -2

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Lunar Eclipses Observed by Parame´svara

 

 

 

 

 

 

No:

 

Kalidina

 

Date and LMT of Full Moon

 

Beginning of phase

 

Maximum

 

End of phase

 

 

Total

 

Partial

 

Total

 

Partial

 

 

1

 

1655647

 

17.01.1942, 21:22

 

20:50

 

19:38

 

21:26

 

22:02

 

23:13

 

 

2

 

1655293

 

28.01.1431, 19:14

 

---

 

17:03

 

19:27

 

---

 

21:51

 

 

3

 

1652694

 

17.12.1423, 20:03

 

---

 

19:51

 

20:13

 

---

 

20:36

 

 

4

 

1654614

 

21.03.1429, 04:00

 

03:35

 

02:16

 

03:56

 

04:16

 

05:35

 

 

5

 

1653403

 

25.11.1425, 02:29

 

01:53

 

00:44

 

02:25

 

02:57

 

04:06

 

Parame´svara's account of lunar eclipses has the mention of Kalidina only and the modern astronomical data of the lunar eclipses mentioned by him are given in the above Table.

 

8. Mean Luni-Solar Positions Given by Parame´svara

 

Parame´svara's account of eclipses ends with the accurate mean positions of Sun, Moon, Moon's apogee and Node for Kalidina = 1651700: Saturday, March 29, 1421AD 05:47 Ujjain sunrise. JD (TDT) = 2240165.53472291.

 

 

 

 

 

 

1

 

2

 

3

 

5

 

6

 

 

Mean

l of

 

Parame´svara's

 

 

Col.2+ 14046'

 

 

Modern l

 

Difference

Col.3- 4

 

 

Sun

 

00015'

 

15001'

 

14051'

 

10'

 

 

Moon

 

304006'

 

318052'

 

318037'

 

15'

 

 

Apogee

 

279057'

 

294043'

 

294036'

 

7'

 

 

Node

 

143055'

 

158041'

 

158027'

 

14'

 

 

When compared with the modern values the differences average to be only 11.5 minutes of arc. Truly speaking the mean ls of the Siddhantic tradition cannot be compared with modern mean ls as the equation of center and other corrections based on which the mean longitude is arrived at from the true observed positions are different. As for example, here in the case of Sun the difference is (+) 10 minutes of arc and this error gets nullified when true ls are compared as the siddhantic equation of center is deficient by 11.8 minutes for anomaly of nearly (-) 780. These accurate positions were probably obtained by applying additional correction to the method outlined in Grahanamandana as can be noted from the Drgganita II.47-50:

"Additional corrections required for the mean positions of Grahanamandana, which was not stated there-in, I shall mention now: One second should be subtracted for every two hundred years from the Sun of Grahanamandana while for Moon addition of one second is required for every 41 years. For apogee the correction is (+) one second for every 135 years and in the case of Rahu one minute needs to be subtracted for every three years."

 

If we go by the Kalikhandam of 1648157, Grahanamandana's formulation took place in 1411 AD – quite early in the career of Parame´svara. Kalikhandam given above in the correction mentioned falls after 10 years and hence these corrections may be the fruits of his constant observation of astronomical phenomena like eclipses.

 

9. Epoch of Drgganita

A discussion on Parame´svara shall remain incomplete without a reference to his magnum opus – Drgganitam. According to the information available in the text (´Sloka 46) the composition took place in ´Saka 1353 corresponding to AD 1431. The epoch as such will be the expiry of the Kali year 4532: Kalidina = 1655352.34: JD[TDT] =2243817.54490972; Thursday, March 29, 1431, 05:55 LMT[ Ujjain Sunrise].

 

Comparison of the Mean Elements

 

Kalidina: 1655352: 29thMarch 1431AD, Thursday, Ujjain sunrise 05:55

 

 

 

 

 

 

 

 

Drk l+A

 

Tropical l

 

 

Sun

 

14036'

 

14027'

 

 

Moon

 

199004'

 

199012'

 

 

Apogee(Moon)

 

350008'

 

217044'

 

 

Rahu

 

329047'

 

325004'

 

 

Mars

 

211031'

 

211030'

 

 

Mercury

 

118048'

 

113011'

 

 

Jupiter

 

45005'

 

46006'

 

 

Venus

 

08011'

 

07035'

 

 

Saturn

 

292050'

 

291040'

 

Derivation of the mean longitudes from observed positions calls into play both the observational skills as well as the mathematical genius of an astronomer. Strictly speaking, the modern mean longitudes and the siddhantic madhyamams are not entirely the same even if the precession is accounted for. True positions decremented by the equation of center formed the siddhantic madhyamams while in modern astronomy the mean positions are arrived at after scores of other corrections, which represent the perturbations to the mean orbit. Obviously equality cannot be looked for while comparing the epochal mean co-ordinates. But even then the contrast, siddhantic versus modern, of the different epochs have been employed by earlier scholars to gain a prima facie understanding of the merits of the siddhantic works. True picture can be gained only by comparing the true longitudes as per Drgganita with those of modern algorithms. Those who are adept in the Siddhantic computational process may attempt a comparison of the Drgganita output with the modern true longitudes provided below:

 

 

 

 

 

 

Planet

 

True l

 

b

 

dl per day

 

db per day

 

 

Sun

 

16°21'

 

-00°00'

 

00°58'41"

 

-00'00".08

 

 

Moon

 

195°44'

 

-03°53'

 

12°06'45"

 

-42'19".53

 

 

Mercury

 

35°52'

 

02°37'

 

01°06'11"

 

06'27".13

 

 

Venus

 

12°22'

 

-01°17'

 

01°14'13"

 

01'04".14

 

 

Mars

 

212°09'

 

01°54'

 

-00°17'56"

 

-01'37".83

 

 

Jupiter

 

44°10'

 

-00°48'

 

00°13'19"

 

00'08".16

 

 

Saturn

 

294°39'

 

-00°01'

 

00°02'44"

 

-00'04".82

 

 

The Kalikhanda of 4532 years (of 365.2586806 days) constitute 1655352.34 days. To remove the fraction 0.34 of a day Parame´svara introduced the Kali-dhruva of 20'22" for mean sun, for the first time in the history of siddhantic astronomy. At no time earlier has there been a correction applied to Sun in terms of Kali-dhruva. Perhaps, Parame´svara is the first astronomer of the Aryabhatan tradition of Kerala, who has postulated zero correction for the planets at the beginning of Kaliyuga. In Drgganitam he has given: Sun: 20'22"; Moon: +3015'2"; Mars: +3'; Mercury:(-) 3044"; Jupiter:(-) 2059'; Venus: (-) 40;Saturn: + 408'; and Node: 182036'.[1] Also for Suryasiddhanta: Sun: +5'35"; Moon: (-) 2'37"; Apogee: + 4017'13"; Mars: (-) 5'; Mercury: (-) 40;Jupiter: (-) 3045';Venus:(-) 4024';Saturn:+407';Rahu:+2035'. Parame´svara could disregard the traditional belief of zero mean longitudes at Yugadi only because of his scientific outlook and the knowledge he has gained about planetary motions through observations of the sky.

 

10. Conclusions

 

In the above is presented a succinct account of the golden period of medieval Indian as well as Kerala astronomical tradition authored by the genius of Parame´svara. His record of astronomical observations, the refinement of astronomical parameters and the accompanying work of his disciple Nilakantha Somayaji, following the same footsteps to maximize the accuracy of planetary longitudes etc., are reflective of the past glory of Indian Science. The accuracy that we see in the Sun and Moon of Parame´svara and Nilkantha can be inferred as the result of eclipse observations as well as experimentation with the computational techniques. Planetary longitudes of Kujadis have relatively more deviation when compared to the results of modern algorithms, obviously due to the constraints of the pre-telescopic Age in the matter of observation.

 

 

 

 

[1] Indian Astronomy – A Source Book: ed. B.V. Subbarayappa and K.V.Sarma, p.153; Nehru Centre, Bombay