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 c8: Chapter III. Sun's and Moon's Anomaly and Actual Moment of New Moon
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 c8
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Chapter III. Sun's and Moon's Anomaly and Actual Moment of New Moon
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9. We computed the time of occurrence of the first mean new moon in K.Y. 5011, and we arrived at the result, 7 ghaṭikas and 17 palas after mean sunrise on 9th May 1910. If now we look into a Pañchāngam, computed from the English Nautical Almanac, like Mr. Srauti's Tiruvâdi Pañchāngam or Messrs. Râghava Châri & Son's Nungumbaukam Pañchāngam, we shall find the time of occurrence of the new moon in May 1910 to be 9th May, 13 ghaṭikas and 2 palas after sunrise. It is encouraging that we are so near the Pañchāngam result, but we are not near enough. The Pañchāngams in question are calculated for the latitude and longitude of Kumbakonam and Madras, respectively, whereas our calculation is for mean sunrise (6 a.m.) at the Equator on the meridian of Ujjain, which is accepted by Hindu astronomers as a starting point in the same way as Greenwich is accepted by modern English astronomers. It is known that Madras time is 18 minutes or 45 palas in advance of Ujjain time, and, further, sunrise at Madras on the 9th May 1910 is noted in the Panchangam as occurring 12 minutes or 30 palas before 6 a.m. These two circumstances account for a difference of 75 palas or 1 ghaṭika and 15 palas between our mean time and the Panchangam time, but there is still a difference of 4 ghaṭikas and 30 palas or ·0750 of a day to be accounted for.
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10. We account for it thus. Although mean new moon occurs every 29·53059 days, the actual new moon does not recur at exactly this interval. About the time when the moon is due to become new, she may be fast, or she may be slow, and we have to take this circumstance into consideration. Hindu astronomers have devised a table of the moon's anomaly which enables us to calculate exactly by how many degrees the moon is in advance of, or behind, her mean position at new moon or at any other time. The original tables of the moon's anomaly and equation of the centre, as these constants are called, will be found in Professor Jacobi's standard article on the subject in Vol. I of the *Epigraphia Indica*. For the sake of practical convenience, we may convert the degrees into time and further calculate the anomaly for each ·001 of the equation, and the result will be our Table IX. This table shows us exactly what time should be added to or deducted from mean time in order to arrive at the actual moment of occurrence of new moon.
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11. To use this table, you should know the moon's anomaly at the time you are dealing with, and the moon's anomaly is determined for centuries and odd years in the same way as the mean moment of the first new moon is determined for every solar year. The moon completes an anomalistic month, that is, returns to a particular pace round the earth in 27·5546 days, and in a solar year the moon's anomaly increases by 365·25875 less 13 × 27·5546 = 7·04896 days as shown in Table VII, Column 4. In 100 years the anomaly increases by 16·03078 days and we calculate, as before, the anomaly for the commencement of the solar year K.Y. 3101 (B.C. 1).
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| | | | | | | Days. |
| For 3000 years | ... | ... | ... | ... | ... | 12·49527 |
| For 100 years | ... | ... | ... | ... | ... | 16·03078 |
| For 1 year | ... | ... | ... | ... | ... | 7·04896 |
| For 3101 K.Y. | ... | ... | ... | ... | ... | 35·57501 |
| | Deduct one completed anomalistic month | ... | 27·5546 |
| | | | | | | 8·02041 |
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Add the moon's anomaly at the first instant of K.Y. 0, which, according to the Sûrya Siddhânta, was 90° from perigee or 6·8886 days, and deduct Sodhya | 2·1707 days, | 4·7179 |
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| | | 4·7179 | 12·73831 days. |
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12. The result, 12·7383 days, is just what we find entered in Table VI as the moon's anomaly at the commencement of K.Y. 3101 (B.C. 1). From this point we go on, year by year and century by century, till we reach 1600 A.D., when a correction, or *bija*, introduced by the Hindu Astronomer Ganesa Daivajna, comes into operation. The effect of this *bija* is to raise the annual increase of the moon's anom. from 7·04896 d. to 7·04898 d. or by 2 seconds of time every year, and also to diminish the length of the anomalistic month from 27·5546 to 27·55459797 or ·000002 d., *i.e.*, ⅙ of a second every month. As a result of this correction, the anomaly for the commencement of K.Y. 5011 (1910-11 A.D.) is 2·1759 days. As the first new moon in the same year occurs (*vide* Section 8 supra) on the 25·9134th day after the commencement of the year, the anomaly for the time of the first new moon is 25·9134 + 2·1759 ————— 28·0893 Deduct completed anomalistic month—27·5546 ————— ·5347 of a day.
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13. As new moon is the moment when the moon has gained exactly 360° over the sun, we have, in determining this moment as well as the moon's anomaly at this moment, to take account of the sun's pace as well as the moon's. The sun's pace depends on his position in the anomalistic year, and that we know (Sec. 4) is the Hindu solar year. We also know that at the time of occurrence of the first new moon in K.Y. 5011 the sun had advanced 25·9134 days in the anomalistic year. We simply look for the corresponding equation in the table of sun's anomaly (Table IX-c) and we find it to be midway between +·145 and +·144. We put down +·1445 as the equation we require and add thereto ·5347, the moon's anomaly already found. Net moon's anomaly, ·6792.
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We look in the table of moon's anomaly (Table IX-a) against ·679 of a day and we find the equation to be between —·059 and —·060. The actual anomaly entered in the table is ·677, and as our anomaly is ·679, we take a suitable equation by proportional parts, *i.e.*, ·0592.
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We now *add* the total of sun's and moon's equations (+·1445 —·0592), or +·0853 to the mean time already found, 9·1213* d., and obtain 9·2066, *i.e.*, 12 ghaṭikas 24 palas after mean sunrise on 9th May 1910. For the present we might be satisfied with this result, which agrees with sufficient closeness with that (13 ghaṭikas 2 palas) of the Nautical Almanac and the Panchângams based thereon; for if we add the effect of longitude and local sunrise, already adverted to in Section 9 supra, our local time for the ending moment of the new moon tithi will be 13 ghaṭikas 39 palas after sunrise.
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14. The reasons for the various processes whereby we arrive at the actual ending moment of any tithi will be clear to us when we come to Chapter XXIII “Theory of anomalies and equations,” but one word on the subject may perhaps be usefully said before we close this chapter. The reader will observe that each anomaly table in our Table IX is divided into four equal portions, two having additive equations and two subtractive. Each equation, as given in the table, consists of three decimal places, the first two being given in vertical columns and the third in a horizontal column. The horizontal column is not necessary when all we want to know is how many hours or how many ghaṭikas after sunrise a new moon or other tithi occurred, and consequently, the horizontal place of the equation, namely, its third decimal place, is omitted in the Eye-table at the end of the book. On the other hand, when such a course is necessary for very accurate work, a fourth figure may be added to the third shown in the horizontal column by noting the difference between two successive anomalies and taking a proportional part of the difference between the corresponding equations. Thus, supposing we want the equation for a ☾'s anomaly of 7·25 days: the anomalies and equations next to those we want are— ☾'s An. 7·077 d. ... ☾'s An. 7·275 d. ... Diff. ·198 d. Eqn. ·413 d. ... Eq. ·414 d. ... Diff. ·001 d.
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∗ The reader should note that two fractions of the day are usually quoted for the moment of occurrence of any astronomical event, *tithi*, *nakshatra*, *yoga*, *karana*, etc. The first fraction (in this case ·9134) is part of a period of time reckoned from the beginning of the solar year, while the second fraction (in this case ·1213) is the time of the day, reckoned from mean Lanka sunrise, 6 a.m. The difference between the two fractions is (as will be seen from Sec. 8 supra) the fraction of day marking the commencement of the solar year, one of the data entered in Table X.
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Inasmuch as the difference in anomaly between 7·077 and 7·25 is ·173 which is 173/198 = ·9 of the tabular difference, we can add 9 to the lower equation and take ·4139 as the equation corresponding to ☾'s anomaly, 7·25 days.
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We should note that sometimes the lesser anomaly has the higher equation. Thus, supposing we require the equation for ☾'s An. 20·375 d. We note the following as the nearest anomalies: ☾'s An. 20·210 d. ... ☾'s An. 20·577 d. ... Diff. ·297 d. Eqn. +·414 d. ... Eqn. +·413 d. ... Diff. ·001 d.
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Our anomaly is lower than 20·577 by ·202, which is 202/297 = ·7 of the difference. We therefore add 7 as a fourth figure to the equation +·413 and take +·4137 as the equation we require.
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In all cases we should add to, or subtract from, the equation corresponding to the anomaly from which we took the difference. In this case, if we had taken the difference from 20·280, it would have been ·095 and the proportional part 095/297 = ·3, which we should deduct from the fourth place of ·414. Thus our equation would be +·414 minus ·0003 = +·4137, the same as before.
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The same observations apply to solar anomalies.
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15. For ordinary results, we do not require a fourth place in the equation, and we can take the nearest three place equation that we can find in Table IX and thus, by means of a simple sum in addition or subtraction, discover, without a moment's trouble, the actual from the mean moment of a tithi or nakshatra. Even then we shall be far more accurate than by any of the rough-and-ready methods now in use.