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 c10: Chapter V. The Lunar Months in Relation to Solar Months
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 c10
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Contents
- c10.s1 Tithis — 8 parts
Chapter V. The Lunar Months in Relation to Solar Months
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18. The lunar months are the doors to the Indian calendar, but the solar months are the hinges on which the doors move. Every lunar month takes its name in Bengal from the solar month in which it occurs, and in the Tamil country from the next solar month after that in which it occurs.
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19. Thus the lunar Vaiśākha must begin sometime in the solar Vaiśākha (Bengal) or sometime in the solar Chittirai (Tamil). The lunar month being only 29.53 days in length, there may be two new moons between the beginning and end of a solar month, and in this case both receive the same name, the first being called *Adhika* and the second *Nija* or true. The second is called 'true' because it immediately precedes a saṅkrānti. *Adhika* months occur ordinarily once in three years, as may be seen from Tables X and XII.
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20. More rarely, that is about once or twice a century, a lunar month may begin and end without a solar month beginning between, and then some lunar month must be suppressed as there is no hinge on which this particular door can turn. The suppressed lunar month is said to be *kshaya* or in defect.
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21. We will now take three examples illustrating the different kinds of lunar years: (1) a year consisting of twelve lunar months, corresponding to as many solar months; (2) a year consisting of thirteen lunar months, which will include an *adhika*, i.e., an extra or intercalary lunar month; (3) a year of thirteen lunar months, exhibiting two intercalary or *adhika* lunar months and one suppressed or *kshaya* lunar month.
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∗ Beginners find great difficulty in understanding *adhika* and *kshaya* months, because the idea is so utterly unlike anything in any other calendar; but the examples given on the next and following pages will enable the reader to cross safely over this *pons asinorum* of the Indian calendar.
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SCHEME OF MONTHS IN K.Y. 5011, A.D. 1910-11.
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| Lunar Months | A.D. Date and Fraction of Day of mean New Moon. | Day and Fraction of Day of Solar Year when mean New Moon occurs, also Sun's Anomaly for New Moon. | Day and Fraction of day of Solar Year when Sankranti occurs: names of Solar Months commencing at each sankranti. S.M. = SOLAR MONTH | Moon's Anomaly at moment of each mean New Moon. (To the C's An. at first new moon of Solar Year add 1.976 days for each lunar month) |
| | | | **K.Y. 5011.** | |
| | **A.D. 1910.** Indian Solar Year commences Ap. 13·2080. (Tab. X, p. 126.) | | Mesha sankranti Vaisakha S.M. ; Chittirai and Medam S.M. ; solar year begins. | |
| **1. Vaisakha** | May 9·1214. | 25·9134 (Tab. X.) | 30·9353. Vrishabha sank. Jyeshta S.M.; Vaikasi or Edavam S.M. | 0·534 (Table X.) |
| **2. Jyeshta** | June 7·6520. | 55·4440 | 62·3555. Mithuna sank. Ashada S.M.; Ani S.M. | 2·510 |
| **3. Ashada** | July 7·1826. | 84·9746 | 94·0003. Karkata sank. Sravana S.M.; Adi S.M. | 4·486 |
| **4. Sravana** | Aug. 5·7132. | 114·5052 | 125·4755. Simha sank. Bhadrapada S.M.; Avani S.M. | 6·462 |
| **5. Bhadrapada.** | Sep. 4·2437. | 144·0357 | 156·4942. Kanya sank. Asvina S.M.; Purattasi S.M. | 8·438 |
| **6. Asvina** | Oct. 3·7743. | 173·5663 | 186·9355. Tula sank. Kartika S.M. (Beng.); Aippasi S.M. | 10·414 |
| **7. Karttika** | Nov. 2·3049. | 203·0969 | 216·8289. Vrischika sank. Margasira S.M; Kartigai S.M. | 12·390 |
| **8. Margasira** | Dec. 1·8355. | 232·6275 | 246·3192. Dhanus sank. Pausha S.M.; Margali S.M. | 14·366 |
| **9. Pausha** | Dec. 31·3661. | 262·1581 | 275·6369. Makara sank. Magha S.M.; Tai S.M. | 16·342 |
| | **A.D. 1911.** | | | |
| **10. Magha** | Jan. 29·8967. | 291·6887 | 305·0850. Kumbha sank. Phalguna S.M.; Masi S.M. | 18·318 |
| **11. Phalguna** | Feb. 28·4273. | 321·2193 | 334·9053. Mina sank. Chaitra S.M.; Panguni S.M. | 20·294 |
| **12. Chaitra** | Mar. 29·9579. | 350·7499 | 365·2587. Mesha sank. Vaisakha S.M.; Chittirai S.M. | 22·270 |
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N.B.—The main point to which the reader's attention should be directed is that the lunar month following a new moon has its name determined by its occurring before a particular sankranti. Thus, a lunar month commencing at any time between 0 day of the solar year and the 30·9353rd day is called Vaisakha; similarly, a lunar month commencing at any time between 156·4942 days and 186·9355 days of the solar year is called Asvina and so forth. The lunar month commencing before the Mesha Sankranti is called Chaitra.
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22. For the first we shall select the current Indian year K.Y. 5011, A.D. 1910-11; for the second we shall select last year, K.Y. 5010, A.D. 1909-10; and for the third, because there has been no suppressed month since A.D. 1822, and there will be none again till A.D. 1963, we will select the very first year of our chronology, K.Y. 3101, B.C. 1.
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23. The solar year K.Y. 5011, A.D. 1910, which is a normal year, opens, as every year does, with the *Mesha Sankrânti* or entrance of the sun into the Indian *first point of Aries*, from which our celestial longitudes are reckoned. We have already seen how this moment is determined, namely, by the successive addition of 365·25875 days since the first moment of the year 0 *Kaliyuga*, less the *sodhya* of 2·1707 days. The *Mesha Sankrânti* determines all kinds of solar years in use in India, and it also determines directly the lunar year, since the first lunar month *Chaitra* is defined to be that whose commencement precedes the *Mesha Sankrânti* and the first day of the lunar year is that on which *Chaitra Sukla Pratipadâ* or the first tithi of the bright fortnight of Chaitra was current at sunrise. In our scheme of months in Table X and elsewhere the reader will observe that the lunar month *Chaitra* stands last, but this is only for purposes of computation, and, after all, the lunar month *Chaitra* begins in a previous solar year, and so it stands last among the lunar months of that year.
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The moment of the *Mesha Sankrânti* marks the commencement of the solar month *Vaisâkhâ* in Bengal, of the solar month *Chittirai* in the Tamil country, and of the solar month *Medam* in Malabar, Travancore, and Cochin.
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Each month begins and ends with a sankrânti, and the second and other sankrântis are named in the order of the signs of the zodiac, *Vrishabha*, *Mithuna*, etc.
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24. There is no difficulty about the lunar months in A.D. 1910-11, since the new moons and sankrântis occur *alternately*, each door having its own hinge to turn on. If you like to calculate the actual moment of occurrence of each new moon, it will be a useful exercise for you to do so, and you can use the anomalies noted in columns 2 and 4.
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SCHEME OF MONTHS IN THE YEAR A.D 1909-10, K.Y. 5010.
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| Lunar Months | A.D. Date and Fraction of Day of mean New Moon. | Day and Fraction of Day of Solar Year when mean New Moon occurs, also Sun's Anomaly for New Moon. | Day and Fraction of Day of Solar Year when sankrânti occurs ; names of Solar Months commencing at each sankranti. S.M.=SOLAR MONTH. | Moon's Anomaly at moment of each mean New Moon. [See this col. on p.(9).] | Sum of Sun's and Moon's equations (by Table IX a to e). |
| | | | **A.D. 1909.** Indian Solar Year commences. Ap. 12·2492 (Tab. X). | **K.Y. 5010.** Mesha sankranti Vaisakha S.M.; Chittirai or Medam S.M. Solar Year begins. | | |
| 1. Vaisakha | Ap. 20·2288 | 7·2745 (Tab. X). | 30·9353. V r i s h a b h a sank. Jyeshta S.M.; Vaikasi or Edavam S.M. | 2·401 (Tab.X) | |
| 2. Jyeshta | May 19·7543. | 36·8051. | 62·3555. Mithuna sank. Ashada S. M.; Ani S.M. | 4·377. | |
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SCHEME OF MONTHS IN THE YEAR A.D 1909-10, K.Y. 5010.
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| Lunar Months | A.D. Date and Fraction of Day of mean New Moon. | Day and Fraction of Solar Year when New Moon occurs, also Sun's Anomaly for New Moon. | Day and Fraction of Solar Year when sankranti occurs, names of Solar Months commencing at each S.M.=SOLAR MONTH. | Moon's Anomaly at mean New Moon. | Sum of Sun's and Moon's equation (by Table IX a to e). |
| | **A.D. 1909.** | | **K.Y. 5010.** | | |
| 3. Ashada ... | June 18·2849 | 66·3357 | 94·0003. Karkata sankranti Sravana S.M.; Adi S.M. | 6·353 | |
| 4. Sravana 5. | July 17·8155 Aug. 16·3461 | 95·8663 125·3970 — 4470 | 125·4755. Simha sank. Bhadrapada S.M.; Avani S.M. | 8·329 10·305 — 1253(⊙'s Eq.) 10·180 | — 1253 — 3217 — 4470 |
| | | *124·9500 | | | |
| 6. Bhadrapada. | Sep. 14·8767 | 154·9275 | 156·4942. Kanya sank. Asvina S.M.; Purattasi S.M. | 12·281 | |
| 7. Asvina ... | Oct. 14·4073 | 184·4581 | 186·9355. Tula sank. Karttika S.M. (Beng.); Aippasi S.M. (Tam.). | 14·257 | |
| 8. Karttika ... | Nov. 12·9379 | 213·9886 | 216·8289. Vrischaka sank. Margasira S.M.; Kartigai S.M. (Tam.). | 16·233 | |
| 9. Margasira ... | Dec. 12·4685 | 243·5192 | 246·3192. Dhanus sank. Pausha S.M.; Margali S.M. | 18·209 | |
| | **A.D. 1910.** | | | | |
| 10. Pausha ... | Jan. 10·9990 | 273·0498 | 275·6369. Makara sank. Magha S.M.; Tai. S.M. | 20·185 | |
| 11. Magha ... | Feb. 9·5296 | 302·5804 | 305·0850. Kumbha sank. Phalguna S.M.; Masi S.M. | 22·161 | |
| 12. Phalguna ... | Mar. 11·0602 | 332·1110 | 334·9053. Mina sank. Chaitra S.M.; Panguni S.M. | 24·137 | |
| 13. Chaitra ... | Ap. 9·5908 | 361·6416 | 365·2587. Mesha sank. Vaisakha S.M.; Chittirai S.M. | 26·113 | |
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25. In this year, as in 1910-11, the lunar months, up to Śrâvana, alternate with the solar months; but between the Sankrântis due at 94·30003 d. and 125·4755 d. of the solar year, we have two mean new moons at 95·8663 d. and 125·3970 d., respectively. We may satisfy ourselves by computation of the anomalies, as shown above, that the second new moon really precedes the sankrânti at 125·4755 d. In such a case both the new moons receive the same name (here Śrâvana), and the first is called Adhika Śrâvana, the second Nija Śrâvana. After this, the numerical order of the lunar months is disturbed, for the 6th new moon is called Bhâdrapada, whereas in 1910-11, the 5th is Bhâdrapada, and the remaining new moons, including Bhâdrapada, alternate with the remaining sankrântis, the total number of new moons for the year being 13 instead of 12.
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We now pass to the third example.
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∗ Actual moment of occurrence of new moon after allowing for solar and lunar omaly.
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SCHEME OF MONTHS FOR 1 B.C., K.Y. 3101.
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| Lunar Months | Date of mean New Moon by Christian era and fraction of day. | Day of Solar Year when mean New Moon occurs; also Sun's anomaly. | Day of Solar Year when sankranti occurs, also dates of commencement and end of Solar Months. S.M.=Solar Month. | Moon's anomaly at moment of New Moon. | Sun's and Moon's equations (by Table IX a to e). |
| | | | 1. B.C. Indian Solar Year commences Mar. 13.9831 (Table X, page 24) | K.Y. 3101. Mesha sankranti Vaisakha S.M.; Chittirai or Madam S.M. | | |
| 1. Vaisakha | ... Mar. 23.98 | 9.99810 | 30.93523. Vrishabha sank. Jyeshta S.M.; Vaikasi or Edavam S.M. | 22.736 | |
| 2. Jyeshta | ... Apr. 22.51 | 39.52869 | 62.3555. Mithuna sank. Ashada S.M.; Ani S.M. | 24.712 | |
| 3. Ashada | ... May 22.04 | 69.05928 | 94.0003. Karkata sank. Sravana S.M.; Adi S.M. | 26.688 | |
| 4. Sravana | ... June 20.57 | 98.58986 | 125.4755. Simha s a n k. Bhadrapada S.M.; A v a n i S.M. | 1.109 | |
| 5. Bhadrapada | July 20.10 | 128.12045 | 156.4942. Kanya sank. A s v i n a S.M.; Purattasi S.M. | 3.085 | |
| 6. } Asvina 7. | { Aug. 18.63 Sep. 17.16 | 157.65104 187.18163 - .5844 | 186.9355. Tula sank. K a r t t i k a S.M.; Aippasi S.M. | 5.061 7.037 - 1724 | - 1724 - 4120 } |
| 8. Karttika | ... Oct. 16.69 | *186.5972 216.71221 - .5143 | 216.8289. Vrishchika sank. Margasira S. M.; Kartigai (Tam.) S.M. | 6.865 9.013 - 1288 | - 5844 - 1288 - 3855 } |
| 9. Margasira | ... Nov. 15.23 | *216.1979 246.24280 - .3212 | 246.3192. Dhanus sank. Pausha S.M.; Margali S.M. | 8.884 10.089 - 0522 | - 5143 - 0522 - 2690 } |
| Pausha | ... (Kshaya) | *245.9216 | 275.6369. Makara sank. Magha S.M.; Tai S.M. | 10.937 | - 3212 |
| 10. Magha | ... Dec. 14.176 | 275.77339 - .0414 | 305.0850. Kumbha s a n k. P h a l - guna S.M.; M a s i S.M. | 2.965 + 0391 13.004 | + 0391 } - 0805 - 0414 |
| | | 1 A.D. *275.7319 | | | |
| 11. Phalguna | ... Jan. 13.29 | 305.30398 + .2501 | 334.9053. Mina sank. Chaitra S.M.; Panguni S.M. | 14.941 + 1693 15.060 | + 1193 } + 1308 + 2501 |
| 12. } Chaitra 13. | { Feb. 11.82 Mar. 13.35 | 305.5540 334.83457 *+ .4710 335.8055 364.36515 | 365.2587. Mesha sank. Vaisakha S.M.; Chittirai or Madam S.M. | 16.917 + 1685 17.085 18.893 | + 1685 } + 3025 + 4710 |
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* Day and fraction of day, marking exact moment of actual New Moon, after allowing for anomaly.
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26 In this year there is nothing specially worthy of note till we reach Asvina: but for that month there is an Adhika Asvina just as we had an Adhika Srávana in A.D. 1909. There is a peculiarity about Nija Asvina in the year B.C. 1, which we must note carefully. The mean Nija Asvina at 187·1816 days is not before the sankrânti at 186·9355 days, but the actual new moon, after calculation of anomalies, is found to be before 187·1816 days and is therefore Nija Asvina, not Kârttika. Generally speaking, when a nija or adhika month is within ·6 day on either side of a sankrânti, calculation, or at least a consideration, of the anomalies is necessary before we can determine its true character.
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27 Passing Âsvina, in the year 1 B.C., we find that each of the subsequent mean new moons is within ·6 day of a sankrânti, and we must therefore calculate the anomalies in order to be quite sure whether each new moon is on this or that side of a sankrânti. Having made these calculations, we find that between two sankrântis at 246·3192 d. and 275·6369 d. there is no new moon, although there is a new moon just under six ghatikas (·095 d.) after the second sankrânti. Had this new moon preceded the sankrânti ever so little, instead of following it, or had our method been inaccurate at this stage to the extent of 2 hours, the consequences we are about to state would not have followed. As it is, there is no new moon between the two sankrântis at 246·3192 d. and 275·6369 d., and therefore the lunar month which has no hinge to turn on is shut; this fact is expressed by saying that Pausha lunar month is kshaya or suppressed.
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28 How do we know that the lunar month to be suppressed is Pausha and not any other? Because that is the first new moon which we are unable to place before a sankrânti. We find that Mâgha and Phâlguna are each followed by a sankrânti, though a long way off, and between the sankrântis at 334·9053 d. and 365·2587 d. there are again two new moons, that is to say Chaitras, the first of these being adhika and the second nija. Here also we note that appearances are deceptive, for the mean new moon at 334·8346 is apparently a Phâlguna, but the calculation of the anomalies shows it to be an adhika Chaitra.
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29 Generally, (1) a kshaya month is preceded and followed, though not immediately, by an adhika month; (2) there are, as a rule, only one or two kshaya months in a century; (3) the kshaya months must be calculated from the mean moment by means of anomalies before we can prove them to be kshaya; and (4) only one of three months, Mârgasira, Pausha, and Mâgha, can be kshaya, because these are the three lunar months which turn on solar months of 29 days each.
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The following table enables us to know an adhika month or a kshaya month by mere inspection of the date of occurrence of the first new moon in a solar year :—
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| Names of Lunar Months. | Ending moments of Solar Months. | Periods of Lunar Months. | Limits of Adhika and Kshaya Months. |
| | | | The lunar month noted in column I will be an adhika month if first New Moon in Solar Year occurs before 1·40469 d. |
| Vaisakha | ... | 30·93528 d. | 29·53059 d. | " 3·29438 d. |
| Jyeshta | ... | 62·35555 d. | 59·06117 d. | |
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| Names of Lunar Months. | Ending Days of Solar Months. | Periods of Lunar Months. | Limits of Adhika and Kshaya Months. |
| **Ashada** | ... | 94·00028 d. | 88·59176 d. | The lunar month noted in column I will be *adhika* if first New Moon in Solar Year occurs before 5·40851 d. |
| **Sravana** | ... | 125·47555 d. | 118·12235 d. | " 7·35320 d. |
| **Bhadrapada** | ... | 156·49417 d. | 147·65293 d. | " 8·84122 d. |
| **Asvina** | ... | 186·93555 d. | 177·18353 d. | " 9·75201 d. |
| **Karttika** | ... | 216·82888 d. | 206·71411 d. | " 10·11475 d. |
| **Margasira** | ... | 246·31916 d. | 236·24470 d. | The lunar month noted in column I will be *kshaya* if first New moon in Solar year occurs after 10·07446 d. and before 10·11475 d. |
| **Pausha** | ... | 275·63694 d. | 265·77529 d. | { after 9·86164 d. and before 10·07447 d. |
| **Magha** | ... | 305·08499 d. | 295·30588 d. | { after 9·77910 d. and before 9·86164 d. |
| **Phalguna** | ... | 334·90527 d. | 324·83647 d. | The lunar month noted in column I will be *adhika* if first new moon in solar year occurs { after 9·77912 d. and before 10·06880 d. |
| **Chaitra** | ... | 365·25875 d. | 354·86705 d. | { after 10·06880 d. and before 10·89170 d. |
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*N.B.—All the figures in column 4 are obtained by subtracting the corresponding figures in column 3 from those in column 2.*
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30. We see that if the first new moon in a solar year occurs before 1·40469 days (the difference between 30·93528 and 29·53059) of the solar year, there will be a second new moon before the end of the first solar month. Reasoning in the same manner, we see that if the first new moon in a solar year occurs after 1·40469 days, but before 3·29438 days of the solar year, there will be two new moons between the commencement of the first and second solar months, that is, there will be an *Adhika* as well as a *Nija Jyeshtha*. Similar reasoning will enable us to connect the possibility of occurrence of the several *adhika* and *kshaya* months with the occurrence of the first new moon before or between the days mentioned in the fourth column. It will be a useful exercise for the reader to try and reason out for himself each line of this table. For determining mean intercalations and mean suppression, the table can be used as it stands; and taken with suitable anomalies, it is a safe and reliable guide for ascertaining true *adhika* and true *kshaya* months. We believe that this is the first time that so simple a method has been announced for recognizing *adhika* and *kshaya* months.
Tithis
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31. Tithis are in use over the whole of India for religious purposes and over the greater part of it for civil purposes also. To understand tithis thoroughly is to have mastered the system of Hindu chronology. Hence "Tithis" take the lead in the second title of this work.
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32. A lunation or synodical month is divided into thirty tithis or lunar days of equal mean length. The names of tithis are familiar to all Hindus, but they are given for convenience of reference on the same page as Table II. The first fifteen tithis, corresponding to the bright half of the month, are called *Sukla paksha*; and the second fifteen are called *Krishna paksha* or *Bahula paksha*. The last or 30th tithi is new moon or *Amāvāsyā*, and it is called sometimes by the name of the month of which it marks the end, and sometimes by the name of the following month. Thus the moment of Amâvasyâ, which marks the beginning of Vaisâkha and which in this work is invariably called the "Vaisâkha new moon" is the same as the ending moment of the 30th tithi of Chaitra. In an inscription* "Vaisâkha Amâvasyâ" often means the Amâvasyâ at the end of Vaisâkha, but for convenience of computation, the first new moon in the solar year is in this work called the Vaisâkha new moon, while the 30th tithi of Vaisâkha is called the Jyeshtha new moon, and so on with the rest of the lunar months. (See remarks by Dr. Thibout, Indian Antiq., Ap. 1895, p. 88.)
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33. To find the mean ending moment of a particular tithi, all you have to do is to add the corresponding duration in days, according to either Table II or the Eye-table, to the moment of the previous new moon.
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Thus, to find the mean ending moment of the 18th tithi of Mâgha lunar month, called krishna or badi (contraction for bahula divasa) Tritiya in K.Y. 5010, A.D. 1909-10, we proceed as follows:— Mâgha new moon, A.D. 1910 (Table X) ... February 9·5296 d. Duration of 18 tithis (Table II or Eye-table) ... 17·7183 d. Mâgha 3 badi, i.e., end of 18th tithi ... February 27·2479 d.
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The mean tithi in question ended at ·2479 of the 27th day of February, A.D. 1910.
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34. For the *actual* ending moment of the tithi, we add the equations of sun's and moon's anomalies.
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| | Anomaly at new moon. | | |
| | ☉ | ☾ | A.D. 1910, February 27·2479 d. |
| | 302·58 d. | 22·161 d. | ☉Eq. for 320·30 d = +·1491 d. |
| Add for 18 tithis | 17·72 d. | 17·718 d. | ☾Eq. for 12·473 d = —·1332 d. |
| | 320·30 d. | 39·879 d. | |
| | Deduct | 27·555 (1 anom. month.) | +·0159 d. +·0159 d. |
| | | 12·324 d. | 27·2638 d. |
| | | +·1491 (☉'s equation.) | |
| | | 12·473 | |
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27·2638 d. or, in ghatikas and palas, 15 ghatikas 50 palas after mean sunrise on 27th February 1910. This is the absolute ending moment of Magha badi 3, in K.Y. 5010.