Indian Chronology (Solar, Lunar and Planetary): A Practical Guide to the Interpretation and Verification of Tithis, Nakshatras, Horoscopes and other Indian Time-records, B.C. 1 to A.D. 2000 c22: Chapter XVII. Modern Equations for Lunar Inequalities
Indian Chronology (Solar, Lunar and Planetary): A Practical Guide to the Interpretation and Verification of Tithis, Nakshatras, Horoscopes and other Indian Time-records, B.C. 1 to A.D. 2000 c22
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Chapter XVII. Modern Equations for Lunar Inequalities
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116. In compliance with a desire expressed by many persons interested in the *Drig-ganita* system, it has been thought advisable to include in the present work three very simple tables (XVI-A, XVI-B and XVI-C) for correcting the ending moments of *tithis* worked out according to the *Sûrya-Siddhânta*, so as to bring them within 2 ghaṭikas of *Drig-ganita* or *Nautical Almanac* results, further approximation not being possible within the limits of a popular work. Seeing that Siddhânta and *Drig-ganita* *panchângams* occasionally exhibit a divergence of 17 ghaṭikas (7 hours) in regard to *tithis* intermediate between new-moon and new-moon, the approximations now made possible will go a great way towards leading to a better understanding between the followers of the two systems.
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117. Modern Astronomers use five principal equations in order to determine the moon's actual, from her mean, elongation (or *tithi*). These are (1) the sun's equation of the centre, (2) the moon's equation of the centre, (3) the moon's evection, (4) the moon's variation, (5) the moon's annual equation. Of these (5) is already included in the sun's equation and both are given together in Tables IX-*c* and IX-*h*, (4) depends only on the *tithi* and is given in the last column of Table XVI-A, while (2) and (3) are given respectively in Tables XVI-B and XVI-C.
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118. The example given below will show clearly how Table XVI is to be used, but it will be useful to bear in mind the following rules:—
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Rule (1) Table IX gives *equated anomalies* (as explained in sec. 251) for mean *tithis*, i.e., each anomaly given in Table IX-*a*, *b* and *g*, for the moon, contains the equation which has to be separated before applying the equation to the mean *tithi*, whereas the anomaly given in Tables XVI-B and XVI-C is a *mean anomaly* for the actual moment of the *tithi*. The principle of the method is to work until the equation obtained from the *equated anomaly* of Tables IX-*a*, *b* and *g*, is equal to the equation obtained from the *mean anomaly* in Tables XVI-B and XVI-C. Accordingly, Table XVI enables us to ascertain an ending moment *not per saltum*, but gradually and by successive approximations, using the *equated anomalies* of Tables IX *a*, *b* and *g*, as our base.
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Rule (2). Tables XVI-B and XVI-C apply, as they stand, only to the first quarters (0 d. 0 gh. 0 p. to 6 d. 53 gh. 20 p.) when the equations are *negative*. For the 2nd quarter (6 d. 53 gh. 20 p. to 13 d. 46 gh. 40 p.) the equation is *negative*, and the argument should be obtained by deducting the given anomaly from 13 days 46 ghat. 40 palas. For the third quarter (13 d. 46 gh. 40 p. to 20 d. 40 gh. 0 p.) the equation is *positive* and the argument should be obtained by deducting 13 d. 46 ghat. 40 palas from the given anomaly. For the fourth quarter (20 d. 40 gh. 0 p. to 27 d. 33 gh. 17 p.) the equation is *positive* and the argument is obtained by deducting the given anomaly from 27 days 33 ghat. 17 palas.
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Rule (3). On account of differences between Indian and European astronomical constants, 45 palas have to be added to the moon's mean elongation or *tithi* and 14 ghaṭikas to the moon's mean anomaly. These corrections are applicable from the year A.D. 1800 to the year A.D. 2000.
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*Example.*
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119. To find the ending moment of *bahula Shashthi* (21st tithi), Nija Sravana A.D. 1909.
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| | d. | gh. | pal. | ☉'s Anomaly. d. gh. pal. | ☽'s Anomaly. d. gh. pal. |
| According to the Ephemeris for A.D. 1909 (Table XII), | | | | | |
| Mean new-moon of Nija Sravana was on Aug. ... | 16 | 20 | 46 | 125 23 49 | 10 18 19 |
| Add equivalent of 21 tithis (*bahula shashthi*) ... | 20 | 40 | 17 | 20 40 17 | 20 40 17 |
| Add (for diff. between Ind. and Eur. constants)... | | | + 45 | | + 14 |
| | | | | | |
| | 37 | 1 | 48 | 146 4 6 | 31 12 36 |
| ☉'s Eqn. for 146 d. 4 gh. (Table IX-*h*) | -9gh. 40p. | | | Ded. 1 an. month. | 27 33 17 |
| ☽'s Variation for Shashthi (Table XVI-A) | -1gh. 43p. | | | | 3 39 19 |
| | | | | | - 11 23 |
| | -11gh. 23p. (**A**) | | | | 3 27 56 |
| ☽'s Anom. for 3 days 27gh. 56p. by Table IX-*g* is | -16gh. 40p. (**B**) | | | | - 16 40 |
| | | | | (**C**) | 3 11 16 |
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The argument for Table XVI-B is the given moon's anomaly (C) :— 3 11 16 Do. Table XVI-C is 2 (☽'s—☉'s long.)—☽'s Anom. Now, 2 (☽'s—☉'s long.) for 6 tithis is given in Table XVI-A as either 11d. 7gh. 19p. or 38d. 34gh 36p. 11 7 19 The given ☽'s anom. has to be deducted from either of these data 3 11 16 7 50 3 Since the last anomaly is in the second quarter, we deduct it from 13 46 40 [*vide Rule (2) supra.*] 5 56 37 (**D**)
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This is the argument for Table XVI-C. The equation acc. to Table XVI-B. for 3d. 11gh. 16p. is —20gh. 33p. Do. do. Table XVI-C. for 5d. 56gh. 37p. is — 6gh. 27p.
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Total... —27gh. 0p. (**E**)
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We compare the last equation (E) with (B) found *supra* from Table IX-*g* (*i.e.*, 16gh. 40p). Clearly, our Sûrya Siddhânta equation is too small and we must try again with—27gh. 0p. (E) in lieu of 16gh. 40p. (B). That is, we must diminish the argument for Table XVI-B by 10gh.20p and increase the argument for Table XVI-C by the same amount.
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Our new arguments for Table XVI-B will be 3d. 0gh. 56p. and its eqn.—19gh. 38palas. Do. do. Table XVI-C will be 6d. 6gh. 57p. and its eqn. — 6gh. 30palas.
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Total... —26gh 8p. (**F**)
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This time we began with —27 gh. (E) and ended with —26 gh. 8 p. (F) which shows that our last argument for Table XVI-B must be increased by 1 gh. and our last argument for Table XVI-C correspondingly diminished.
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Our new argument for Table XVI-B is 3d. 1gh. 56p. and its equation —19gh. 43p. Do. do. Table XVI-C is 6d. 5gh. 57p. — 6gh. 30p.
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—26gh. 13p. (**G**)
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We began with - 26gh. 8p. and ended with - 26gh. 13p. (G). We may be satisfied with this result. Our final equation will be - 26gh. 13p.
(1) Sun's equation of the centre and ☾'s annual eqn. + Variation - 11gh. 23p.
(2) Moon's do. do. and ☾'s evection - 26gh. 13p.
- 37gh. 36p.
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This being deducted from our mean tithi, we have mean tithi or elongation Aug. 37. 1gh. 48p.
- 37gh. 36p. Aug. 36. 24gh. 12p. To which add for local time (Table XIII); 1651" = 1gh. 9p. (Lat. 13°, 146th day).
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Aug. 36 or Sep. 5, 25gh. 21p. which is about 2 ghaṭikas in excess of the time given in Drig-ganita panchângams for the year Saumya (A.D. 1909—10), whereas the Siddhânta panchângams give 12 or 13 ghaṭikas more as the ending moment of bahula Shashṭhi in Nija Śrâvana of that year.
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120. Ordinarily, the greatest difference between Siddhânta and Drig-ganita pan-chângams occurs from the 6th to the 10th tithis. For the reason of this divergence, the reader is referred to the article on the Moon in the Encyclopedia Britannica, where we are told that ancient astronomers were unacquainted with the evection and variation, as per-turbing elements of the lunar theory.
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121. If the reader wishes to know how Drig-ganita panchângams obtain their tithis from the Nautical Almanac, we may state the process in a very few lines :— Taking the above example, we look out in the Nautical Almanac the ☉'s and ☾'s longitude at Greenwich mean noon on two or three neighbouring days, say 4th, 5th and 6th September 1909.
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| (1) September 1909. | (2) ☾'s long. at noon. | (3) ☾'s long. at noon. | (4) ☾'s elongation. | (5) ☾'s elongation for 24 hrs. |
| | ° ' " | ° ' " | ° ' " | ° ' " |
| 4 | 41 14 39.3 | 161 17 19.1 | 240 57 20.2 | |
| 5 | 55 25 16.2 | 162 15 29.9 | 253 9 46.3 | 12 12 26.1 |
| 6 | 69 11 33.6 | 163 13 42.8 | 265 57 50.8 | 12 48 4.5 |
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Columns 2 and 3 of this table are extracted from the Nautical Almanac for 1909. Column 4, Moon's elongation, is of course column 2 less column 3, and the difference between the elongations of two successive days is entered in column 5. The tithi we want is Bahula Shashṭhi, i.e., the 21st tithi which ends when the Moon's elongation is 21 × 12° = 252°. This, we find, was sometime before Greenwich noon on 5 September 1909. Since the Moon's exact elongation at Greenwich noon on 5th September 1909 was 253° 9' 46.3" which is 1° 9' 46.3" (= 1.1628°, by Table XIX) more than we require, we have to deduct the time taken by the Moon to effect this elongation. This last time may be determined by working out a proportion either to the elongation of the previous 24 hours (12° 12' 26.1" = 12.2072° by Table XIX) or to the elongation of the next 24 hours (12° 48' 4.5" = 12.8011° by Table XIX). The result is (1) 1.1628/12.2072 = 0953 day, or (2) 1.1628/12.8011 = 0908 day.
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If we deduct the first result from Greenwich noon, we have September 4·9048, to which we add ·4605 day for the difference between Greenwich astronomical and Ujjain Lankâ time, and again add 1651 seconds or 27½ minutes or ·0190 day for the difference between Ujjain time and Madras time (Table XIII; lat. 13°, 146th day). Net result, September 4·9048 + ·4605 + ·0190 = September 5·3843 or 23 ghat. 4 palas on 5th September 1909.
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On the other hand, if we deduct the second result, ·0908 day from Greenwich noon, 5th September 1909, we have September 4·9092 to which we add, as before, + ·4605 + ·0190 and also (for Madras standard time) 9 minutes + ·0062, and we have, as the final result, September 5·3949 or (by Table XIX) 23 ghaṭikas 40 palas, which differs by only 13 palas from the result given in the Nungumbaukam Panchângam.
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Our working shows that a difference of about 15 palas might result according as we took the proportional part from the elongation of the previous or of the following 24 hours.
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122. This is very much in favour of the Drig-gaṇita method, even if exact astronomical accuracy is wanted, and that method only postulates the possession of a Nautical Almanac. In the absence of a Nautical Almanac, or for years long past or a long way ahead, we can obtain a very fair approximation to Nautical Almanac results by means of the modern equations in our Table XVI.