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 c26: Chapter XX. Use of Ephemeris or Table XII for Lunar Tithis
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 c26
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Contents
- c26.27 — 4 parts
Chapter XX. Use of Ephemeris or Table XII for Lunar Tithis
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150. Even in those parts of the Indian Continent where the Solar Calendar is used for *civil* purposes, lunar tithis are observed for religious purposes: elsewhere the civil as well as the religious calendar is *lunar* and is regulated by tithis, instead of by days of the month.
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151. Like the lunar calendar of the Jews, and unlike that of the Muhammadans, the Hindu calendar may be described as luni-solar since all its periods of time, though regulated by the movements of the moon, are made to fit in with divisions of the Solar Year. A lunar tithi is not counted at all unless the Sun rises upon it. A Tithi, Nakshatra, or Yoga may begin, or end, at any moment of the day, but the Tithi, Nakshatra, or Yoga pertaining to a day is that which is current at *sunrise*. A lunar month takes its name from the next following Solar *Sankrānti* : if there are two lunar months entitled to derive their name from a single Solar *Sankrānti*, they both receive the same name and one, the first, is called *adhika*. On the other hand, if a new moon is followed by two *Sankrāntis* before it is followed by another new moon, the lunar month which would ordinarily have been derived from the second of the two *Sankrāntis* is suppressed and is said to be *Kshaya*, that is, in defect. Lastly the lunar year, beginning in the month of Solar *Chaitra*, which is itself the last month of the Solar year, is named after the *next* Solar Year. These considerations suffice to prove that the Hindu lunar year is properly a luni-solar year.
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152. Lunar periods of time are characterized by an element of certainty or rather of palpable evidence, not found in solar periods, and in another sense the actual moments of lunar phases are marked by much greater uncertainty than Solar *Sankrāntis*. We will explain each of these propositions.
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153. *Palpable evidence of lunar periods.*—We cannot visibly perceive in the heavens the fact that the sun has completed any definite stage in his annual course: but new moon, when the moon's longitude is the same as the sun's, is a patent fact: so are full moon and each quarter of the lunar month. There cannot be a difference of a whole day between the moment of new moon in one part of the country, and the same moment in another part, just as we saw there might be in the commencement of a solar month. If an inscription states that a certain tithi fell on a Monday, and by calculating backwards we trace the tithi to a Saturday we may be quite certain that there is some error in the inscription.
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154. On the other hand, there are greater fluctuations in the moon's than in the sun's movements, in the sense that the actual time of new moon may be as much as 14 hours before or 14 hours after the time of mean new moon. Two calculations are therefore necessary to determine the time at which the moon reaches any particular stage of her course: we must first ascertain the *mean time*, which is simply the expected time, taking an average over very long periods; and then we must calculate the *actual time* by making a correction to the mean time according to the moon's and sun's *anomaly* at the particular moment. Both these operations can be performed very easily and very accurately, that is to say, in exact accordance with the Siddhântas, by means of the tables in the present work. The correction is called a correction for the sun's and moon's equation of the centre. There are also one or two other minor corrections recognized in modern astronomy, but the Siddhântas do not make them. (See sections 121 to 127 supra.)
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155. In the case of new moons from A.D. 1840 to A.D. 1920, it was thought best not to give the reader even the slight trouble of calculating the anomaly and the equation of the centre; and the exact moment of each new moon during these eighty years as well as the exact day of its occurrence, where it differs from the mean date, is entered in separate columns, in Table XII "General Ephemeris." Here, however, the day and fraction of a day marking the occurrence of mean new moon, and the sun's and moon's anomalies at the moment of mean new moon are also given, as they are necessary for the calculation of tithis between one new moon and another.
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156. Supposing now the reader wishes to ascertain the exact time of new moon in July 1910. He finds the entries:—
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(Tab. XII) A.D. 1910. Date and fraction of day of Ashadha new moon. | Actual moment of new moon. --- | ---
(5) July 7, 10 ghaṭikas 57 palas ; | July 6, 49 ghaṭikas 55 palas.
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*Moon's and Sun's anomaly in days, ghaṭikas and palas.*
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Sun's anomaly : 84 days 58 ghaṭikas 29 palas.
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Moon's anomaly : 4 days 29 ghaṭikas 13 palas.
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157. As regards the exact moment of occurrence of new moon in July 1910, there is no difficulty at all; for it is Wednesday, 6th July 1910 (the date of mean moon as well as the week-day has to be diminished by one, by reason of the anomalies) and the time of day is 49 ghaṭikas 55 palas.
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158. This, as explained already, is mean Laṅkâ time. If now you wish to know the true local time of the actual moment of new moon in July 1910, and if you happen to be at one of the 30 places named in Table XIII, all you have to do is to apply to Laṅkâ time the correction indicated in the column "Total correction" for the day of the solar year entered under the new moon date in Table XII.
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159. Supposing you are at Tanjore, the correction for the 84th day is + 2478 seconds of time, *i.e.*, 103 palas or + 1 ghaṭika 43 palas. Adding this to Laṅkā time, 49 ghaṭikas 55 palas, we have 51 ghaṭikas 38 palas, exact time at Tanjore according to Sūrya Siddhānta. If now you wish to know how far this time agrees with modern astronomical computation, you have merely to look at the time of new moon according to the Tiruvādi Almanac (Mr. Srauti's Pañchāngam) which is calculated from the English Nautical Almanac and is reduced to the latitude of Tanjore, and you find there "6th July 1910, 52 ghaṭikas 10 palas", showing a difference of 32 palas. This difference of 32 palas is due to lunar acceleration which has changed since the date of the Sūrya Siddhānta. (*Vide* Sec. 2.)
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160. If you are at Madras (lat. 13°) you will find the correction according to Table XIII to be + 2200 seconds of time, *i.e.*, + 92 palas or + 1 ghaṭika 32 palas. Adding this to mean Laṅkā time, you get (49 ghat. 57 palas + 1 ghat. 32 palas =) 51 ghat. 29 palas. The time according to the "College" Pañchāngam (Mr. Raghavachari's) is 52 ghaṭikas 33 palas, showing a difference of 1 ghaṭika 4 palas. The increased difference, as compared with Mr. Srauti's Pañchāngam, is mainly due to two facts, (1) The correction for Madras local time is less than that for Tanjore local time by 11 palas, (2) Madras standard time, which is shown in the College Pañchāngam, is nine minutes or 22½ palas in advance of Madras local time. There is still, however, an unexplained difference of a few palas between our tithi and that of the College Pañchāngam.
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161. Next, suppose the reader wishes to know the ending moment of the 8th tithi or *ashṭamī* in the bright fortnight following the new moon of 6th July 1910. For this we have to calculate first of all the *mean* ending moment of the 8th tithi, and then its *actual* ending moment. The mean ending moment is given by adding the collective duration of 8 tithis according to the Eye-table to the *mean* moment of new moon.
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(Tab. XII) Mean moment of new moon ... A.D. 1910. 7 July 10 ghat. 57 palas. *Add* collective duration of 8 tithis ... 7 days 52 ghat. 29 palas. according to the Eye-table. A.D. 1910 15 July 3 ghat. 26 palas.
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162. This, then, is the *mean* ending moment of the 8th tithi or *ashṭamī*. For the *actual* ending moment of the same tithi, we first of all add the collective duration of 8 tithis to the moon's and sun's anomaly noted in Table XII under the new moon of 7 July 1910.
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| | Moon's anomaly | | Sun's anomaly | |
| | 29 ghat. | 13 palas | 84 days | 58 ghat. | 29 palas. |
| *Add* collective | | | | | |
| duration of 8 tithis. | 7 days | 52 ghat. | 29 palas | 7 days | 52 ghat. | 29 palas. |
| | 12 days | 21 ghat. | 42 palas | 92 days | 50 ghat. | 58 palas. |
| *Deduct* ☉'s Eqn. | | -2 ghat. | 20 palas | | | |
| *Net* ☽'s anom. | 12 days | 19 ghat. | 22 palas | | | |
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163. According to the Eye-table we find the Sun's Equation for an anomaly of 92 days, 51 ghat to be minus 2 ghat. 20 palas, and we deduct this equation from the moon's anomaly already found. The Equation of the centre for the moon's net anomaly of 12 days 19 ghat. 22 palas is, by the Eye-table, minus 9 ghatikas.
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164. The sum of moon's and sun's Equations :
- 2 ghat. 20 palas.
- 9 ghat . . . . . 11 ghat. 20 palas.
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165. If 11 ghatikas be deducted from the mean ending moment of the 8th tithi, we obtain, as the *actual* ending moment of sukla ashtami, 14 July A.D. 1910, 52 ghatikas after sunrise.
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166. This is Lankā time. If now we wish to know the ending moment of the tithi in true local time, say at Madras (Lat. 13°), all we have to do is to apply the figure entered in the column "Total correction" under Madras (Lat. 13°) against the 92nd day of the Solar Year in Tab. XIII. The correction being + 2095 seconds or + 87 palas, that is + 1 ghat. 27 palas, the ending moment of Sukla ashtami, 14 July 1910, at Madras, is (52 ghat. + 1 ghat. 27palas =) 53 ghat. 27 palas. The "College" Panchāngam gives 52 ghat. 33 palas.
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167. We will now show the reader how to calculate successive tithis from the Eye-table and Table XII combined. The duration of 1 tithi, according to the Eye-table, is 59 ghatikas 4 palas, that is, 1 day less 1 ghaṭika plus 4 palas. Bearing this in mind, we shall proceed to calculate the successive tithis after new moon, 7 July 1910; the mean time of which is 10 ghat. 57 palas, or nearly 11 ghaṭikas.
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Details for determining the ending moments (in Madras Time) of 30 tithis, Lunar Ashadha, A.D. 1910.
c26.27.1
| A.D. 1910 Tithi | July | Mean ending moment of tithi | ʘ's Anom. | ʘ's Equa- tion. | Add or sub- tract ʘ's Equation. | Sum of ʘ's & ☾'s Equations. | Actual end of tithi, Lanka time. | Correction for Madras time. | Ending moment of tithi, Madras time. | Ending moment, "College" Panchangam. | Ending moment Kanjanur Panchangam | Difference between Surya Siddh- anta and naut. almanac. | Full name of tithi. |
| | | gh. p. | d. gh. p. | gh. p. | d. gh. p. | gh. p. | July gh. p. | gh. p. | July gh. p. | July gh. p. | July gh. p. | gh. p. | |
| Amavasya | 7 | 10, 57 | 84, 58 | - 1, 0 | 4 28, 23 | - 21, 2 | 6 49, 56 | + 1, 31 | 6 51, 26 | 6 52, 33 | 6 53, 11 | - 1, 7 | |
| Sukla paksha. | | | | | | | | | | | | | |
| 1 | 8 | 9, 5 | 86, 56 | - 1, 12 | 5 27, 17 | - 21, 2 | 7 46, 13 | + 1, 30 | 7 47, 44 | 7 47, 7 | 7 49, 41 | + 0, 37 | Pratipada. |
| 2 | 9 | 8, 9 | 87, 55 | - 1, 23 | 6 26, 9 | - 23, 35 | 8 43, 30 | + 1, 30 | 8 45, 0 | 8 42, 52 | 8 47, 16 | + 2, 8 | Dvitiya. |
| 3 | 10 | 7, 13 | 88, 54 | - 1, 34 | 7 25, 2 | - 25, 35 | 9 41, 59 | + 1, 29 | 9 43, 28 | 9 40, 10 | 9 45, 58 | + 3, 18 | Tritiya. |
| 4 | 11 | 6, 17 | 89, 53 | - 1, 45 | 8 24, 55 | - 26, 10 | 10 41, 40 | + 1, 29 | 10 43, 9 | 10 39, 13 | 10 45, 55 | + 3, 56 | Chaturthi. |
| 5 | 12 | 5, 21 | 90, 52 | - 1, 56 | 9 24, 48 | - 23, 40 | 11 42, 37 | + 1, 28 | 11 44, 5 | 11 47, 7 | 11 47, 7 | + 3, 55 | Panchami. |
c26.27.2
| 6 | 13 | 4, 25 | 91, 51 | - 2, 7 | 10 21, 41 | - 18, 36 | 12 44, 49 | + 1, 27 | 12 46, 16 | 12 43, 1 | 12 46, 36 | + 3, 15 | Shashti. |
| 7 | 14 | 3, 29 | 92, 50 | - 2, 17 | 11 20, 34 | - 16, 18 | 13 48, 7 | + 1, 27 | 13 49, 34 | 13 47, 25 | 13 52, 56 | + 2, 9 | Saptami. |
| 8 | 15 | 2, 33 | 93, 49 | - 2, 28 | 12 19, 28 | - 11, 9 | 14 52, 20 | + 1, 26 | 14 53, 46 | 14 52, 59 | 14 57, 23 | + 0, 47 | Ashtami. |
| 9 | 16 | 1, 37 | 94, 48 | - 2, 39 | 13 18, 21 | - 5, 26 | 15 57, 7 | + 1, 26 | 15 58, 33 | 15 59, 12 | 15 60, 0 | - 0, 39 | Navami. |
| 10 | 17 | 14 19, 53 | 14 17, 14 | + 3, 11 | + 0, 32 | 16 60, 0 | 16 60, 0 | 16 60, 0 | 16 60, 0 | 16 60, 0 | 16 60, 0 | 2, 16 | Dasami |
| 11 | 18 | 15 18, 57 | 15 16, 7 | + 9, 4 | + 6, 14 | 17 2, 9 | 17 2, 9 | 17 6, 55 | 17 5, 22 | 17 7, 11 | - 1, 48 | Dasami. |
| 12 | 19 | 16 18, 1 | 16 15, 1 | + 14, 22 | + 11, 22 | 18 6, 55 | 18 6, 55 | 18 8, 19 | 18 10, 56 | 18 11, 46 | - 2, 37 | Ekadasi. |
| 13 | 20 | 17 17, 5 | 17 13, 54 | + 18, 45 | + 15, 34 | 19 11, 7 | 19 11, 7 | 19 12, 31 | 19 15, 24 | 19 15, 44 | - 2, 53 | Dvadasi. |
| 14 | 21 | 18 16, 9 | 18 12, 48 | + 22, 1 | + 18, 40 | 20 14, 23 | 20 14, 23 | 20 15, 46 | 20 18, 35 | 20 18, 44 | - 2, 49 | Trayodasi. |
| 15 | 22 | 19 15, 13 | 19 11, 41 | + 24, 4 | + 20, 32 | 21 16, 33 | 21 16, 33 | 21 17, 56 | 21 20, 10 | 21 20, 34 | - 2, 14 | Chaturdasi. |
| | | | | | | | | | | | | | |
| Krishna | | | | | | | | | | | | | |
| Paksha | | | | | | | | | | | | | |
c26.27.3
| 1 | 22 | 56, 1 | 100, 42 | - 3, 42 | 20 10, 35 | + 21, 6 | 22 17, 29 | + 1, 22 | 22 18, 51 | 22 20, 24 | 22 21, 31 | - 1, 33 | Paurṇami. |
| 2 | 23 | 55, 10, 41 | 101, 42 | - 3, 52 | 21 9, 29 | + 24, 21 | 23 17, 7 | + 1, 22 | 23 18, 29 | 23 17, 10 | 23 20, 34 | + 1, 19 | Pratipada. |
| 3 | 24 | 54, 9 | 102, 40 | - 4, 2 | 22 8, 23 | + 22, 43 | 24 16, 34 | + 1, 21 | 24 16, 55 | 24 16, 53 | 24 18, 43 | + 0, 2 | Dvitiya. |
| 4 | 25 | 53, 13 | 103, 39 | - 4, 12 | 23 7, 17 | + 20, 4 | 25 15, 50 | + 1, 21 | 25 14, 11 | 25 18, 41 | 25 15, 46 | + 0, 30 | Tritiya. |
| 5 | 26 | 52, 17 | 104, 38 | - 4, 22 | 24 6, 11 | + 16, 34 | 26 9, 5 | + 1, 20 | 26 10, 25 | 26 9, 44 | 26 11, 49 | + 0, 41 | Chaturthi. |
| 6 | 27 | 51, 21 | 105, 37 | - 4, 32 | 25 5, 5 | + 12, 12 | 27 4, 29 | + 1, 20 | 27 5, 49 | 27 5, 18 | 27 7, 11 | + 0, 31 | Panchami. |
| 7 | 28 | 50, 25 | 106, 36 | - 4, 42 | 26 3, 59 | + 12, 23 | 28 59, 12 | + 1, 19 | 28 0, 31 | 28 0, 29 | 28 1, 47 | + 0, 2 | Shashti. |
| 8 | 29 | 49, 29 | 107, 35 | - 4, 52 | 27 2, 53 | + 7, 40 | 29 53, 23 | + 1, 19 | 29 54, 42 | 29 50, 5 | 29 49, 48 | - 1, 29 | Saptami. |
| 9 | 30 | 48, 33 | 108, 34 | - 5, 2 | 28 0, 30 | + 2, 40 | 30 47, 17 | + 1, 19 | 30 48, 36 | 30 44, 37 | 30 43, 44 | - 2, 17 | Ashtami. |
| 10 | 31 | 47, 37 | 109, 33 | - 5, 11 | 1 27, 25 | - 7, 31 | 31 34, 55 | + 1, 18 | 31 36, 13 | 31 39, 6 | 31 37, 49 | - 2, 53 | Navami. |
c26.27.4
| 11 | Aug. 1 | 46, 41 | 110, 32 | - 5, 21 | 2 26, 19 | - 12, 14 | Aug. 1 29, 6 | + 1, 18 | 1 30, 34 | 1 33, 34 | 1 32, 10 | - 3, 10 | Dasami. |
| 12 | 2 | 45, 45 | 111, 31 | - 5, 30 | 3 25, 14 | - 16, 27 | 2 22, 48 | + 1, 17 | 2 25, 5 | 2 28, 11 | 2 27, 3 | - 3, 6 | Ekadasi. |
| 13 | 3 | 44, 49 | 112, 30 | - 5, 39 | 4 24, 9 | - 19, 58 | 3 19, 12 | + 1, 17 | 3 20, 29 | 3 23, 14 | 3 22, 38 | - 2, 45 | Dvadasi. |
| 14 | 4 | 43, 53 | 113, 29 | - 5, 49 | 5 23, 3 | - 22, 39 | 4 16, 25 | + 1, 17 | 4 16, 42 | 4 18, 57 | 4 19, 4 | - 2, 15 | Trayodasi. |
| 15 | 5 | 42, 57 | 114, 28 | - 5, 58 | 6 22, 2 | - 24, 19 | 5 12, 40 | + 1, 16 | 5 13, 56 | 5 15, 24 | 5 16, 33 | - 1, 28 | Chaturdasi. |
| Amavasya | | | | | | | | | | | | | Amavasya. |
c26.28
168. The reader is presented in the above table with a full decursus of tithis for a whole lunar month, calculated in accordance with the Sûrya Siddhânta; and he is also furnished in the same table with the ending moments of the same tithis, as arrived at by two standard Panchângams current in Southern India, namely, (1) the Kanjanûr Panchângam of Annâvaiyangâr, commonly called "No. 28 Panchângam", and (2) the Nungambaukum Panchângam by Mr. Râghavachâri and his son, commonly called "The College Panchângam." The former of these panchângams is typical of the class known as "Vâkya Panchângams".
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169. A vâkyam (meaning in Tamil a sentence) is simply a phrase or series of phrases, employed in accordance with a very ancient "Transnumeration" table, called "Kadapayadi", closely analogous to the present writer's Transnumeration Table, published at page 37 of his "Secret of Memory". It is generally believed that the vâkya process is based on the Ârya Siddhânta, but this is by no means well established.
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Only solar dates in Southern India seem to follow the Ârya Siddhânta, but for lunar tithis and nakshatras, that Siddhânta, as proved by the above, as well as by the next, table, seems to have been given up long ago, even in Southern India, in favour of the more accurate Sûrya Siddhânta. If the Kanjanûr Panchângam had followed the Ârya Siddhânta, all mean tithis, including Amâvâsyâs, would have occurred according to that Panchângam 3 ghaṭikas 42 palas later than by the Sûrya Siddhânta, whereas the above table, and more especially the next table, shows a much smaller difference as a rule between that panchângam and the results arrived at on strict Sûrya Siddhânta principles.
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170. The Nungumbaukum or "College" Panchângam belongs to the class of what are called in Southern India Drig-ganita panchângams, i.e., those in which computation is checked by observation, the observation being understood to be that carried on at a modern standard observatory, like those at Greenwich, Paris and New York.
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171. Practically, all the Drig-ganita panchângams are based on the Greenwich Nautical Almanac, which is published some three years in advance for each year. It is not to be supposed that the moon's and sun's places given in the Nautical Almanac are those observed then and there, for they are also calculated on the best available data, which include several elements of the lunar theory, neglected in the Siddhântas.
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N.B.—For the manner in which Drig-ganita tithis are computed, see Section 121 Supra.
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172. The reader will observe that in our specimen table the difference between Sûrya Siddhânta tithis and what we may call Nautical Almanac tithis, does not exceed 2 ghaṭikas at new moon or 4 ghaṭikas at other times. (The extreme difference between the two systems may amount at times to as much as 17 ghaṭikas, or 7 hours.) This no doubt is a disadvantage of the Sûrya Siddhânta system for strictly astronomical purposes (i.e., on the assumption that Hindu Astronomy must agree with the most accurate results of European Astronomy); but if it is remembered that a Nautical Almanac panchângam cannot be constructed unless one has in hand the Nautical Almanac of the particular year, whereas a Siddhânta panchângam can be constructed in a few hours' time for any year, past, present or future, without any other materials than those furnished by Tables VI, VII, VIII and IX of this work, it will be seen that a panchângam on the purely Indian system is a great convenience.
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Moreover, the Indian panchângam is not a thing of to-day or yesterday, but has been constructed on the lines now followed for at least 1500 years, and no person, unacquainted with the system, can hope to understand the thousands of inscriptions scattered all over India, on the proper reading of which the reconstruction of scientific Indian history largely depends.
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173. *Adhika* or *Trisparsa Tithis* and *Kshaya Tithis*: A tithi which begins on one day, is current for the whole of the next day and ends in the morning of the third day is called an *Adhika* or *Trisparsa* tithi. *Trisparsa* means "touching three days." An example is furnished in the above table by *sukla navami* of the Kanjarûr Panchângam, and *sukla dasami* of our own and the College Panchângam. The fact that a tithi is current for the whole of a day is indicated in the panchângams by entering '60 ghat' as its ending moment, *i.e.*, it does not come to an end all that day. On the other hand, a tithi which begins and ends between one sunrise and another is *kshaya* or in defect and is suppressed. Such a tithi is *krishna saptami* by all three Panchângams in the above table.
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*N.B.*—If the reader wishes to apply to the Siddhânta results, the corrections necessary to bring them up to the Nautical Almanac standard, he is enabled to do so in Chapter XVII of Part I of this work, Sections 116 to 122.