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 c34: Chapter XXVII. Tables for Converting Space into Time
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 c34
- Language
- English
Contents
- c34.s1 Surya Siddhanta — 16 parts
Chapter XXVII. Tables for Converting Space into Time
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244. The method of calculating and verifying Indian dates, presented in this work, is intended to be of service to the general reader as well as to the scientific expert, the epigraphist, the archæologist, and the historian. The method is perfectly simple and at the same time absolutely correct according to the Siddhântas, so that it has become possible for the first time to dispense with all manner of approximations and rough and ready methods which, however valuable in the hands of an expert, are apt to mislead and confuse, more often than they assist, the general reader. The principal device by which this combination of extreme simplicity with accuracy and absolute fidelity to the original authorities, has been accomplished, is the reduction of all quantities required for calculation to whole days and fractions of a day.
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245. The civil day (with its multiple, the week,) is the one measure of time that is common to European and Indian reckoning, everything else (year, month, ghaṭikas, palas hours, minutes, seconds,) being different in the two systems. Accordingly, the civil day and decimals of a day have been adopted throughout this work for expressing all manner of Indian dates as well as for working out ending moments of tithis, etc., and also for verifying the correspondence of English and Indian dates. Any decimal of a day can be converted readily into Indian ghaṭikas and palas (Tamil, naligais and vinâdis) or English hours, minutes, and seconds by means of Tables XIX and XX. To assist the reader in very exact computation, the fractions expressing minutes have been carried far enough to show the recurring places. If, for instance, we wish to know how many hours, minutes, and seconds are equivalent to ·40490 of a day, we turn to Table XX and find that ·40486 of a day is equal to 9 hours 43 minutes. The remainder of the decimal fraction is ·00004 which, the same table informs us, is between 3 and 5 seconds. So the answer is, 9 hours 43 minutes 4 seconds. The same decimal fraction is equivalent in Indian time (as we may see from Table XIX) to 24 ghaṭikas 18 palas. Ghaṭikas and hours, as fractions of a day, are also shown in the Eye-table at the end of the book.
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246. The reader will notice that in the first three parts of the present work as well as in the connected tables (except the Subsidiary Tables VI-a, VII-a, and VII-b, which are intended mainly for purposes of comparison between the present tables and those of previous writers on the same subject,) measures of time alone are used, and that measures of space, i.e. (degrees of celestial longitude, degrees of mean anomaly of the sun and moon, etc.) have been altogether excluded. This is the principle known as Largeteau's method, which was first applied to Indian astronomical computation by Professor Jacobi in 1888. Messrs. Sewell and Dikshit have applied the same principle in their "*Indian Calendar*" (1896).
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247. The present method is founded on Largeteau's principle, but differs essentially from it as well as all previous applications of it in one important respect. Instead of using Largeteau's method to discover how much space has been accomplished at a particular moment of time, the present writer has used the method of day-spaces or space days to discover the moment of time at which a particular extent of space has been accomplished. Thus, instead of determining the expired portion of a tithi, corresponding to a given moment of time (i.e., generally, to mean sunrise on a particular day,) as is done by Messrs. Jacobi and Sewell, and then calculating the unexpired portion of the tithi by means of successive approximations, the present writer investigates, directly and once for all, the ending moment of a tithi, the very thing required by Indian usage.
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248. By setting this object steadily in view, the author has been enabled to reduce to two or three very simple and easy steps, Messrs. Sewell and Dikshit's method, which covers a page and a half (pp. 81, 82) of their "*Indian Calendar*"; likewise he has considerably abridged Professor Jacobi's process, which consists, in the first place, of an approximation, on Largeteau's method, and thirteen or fourteen subsequent steps (a) to (m), as expounded in Volumes I and II of "*Epigraphia Indica*". As regards Mr. Dikshit's own very accurate but very tedious method, covering several pages of the introduction to Dr. Fleet's "*Gupta Inscriptions*", the present writer has been successful in arriving at absolutely the same result as Mr. Dikshit in two lines of working. (See Sections 223, 259.)
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249. The principles upon which space was converted into time for the purposes of the present work are set forth in the following paragraphs.
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The principal measure of space, the distance of the moon from the sun, was converted into days in the ratio of 29·530587946 days to 360° in the case of the Sûrya Siddhânta and 29·5305925 days to 360° in the case of the Ârya Siddhânta. Subsidiary Table VI-A is the conversion Table for Sûrya Siddhânta.
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The increase of the moon's age, according to the Sûrya Siddhânta, for each solar year is according to the above rate of conversion, 10·891701134 days. Instead of reckoning the increase of the moon's age, however, the present method reckons directly the retardation in the date of appearance of the first new moon in each solar year, for which purpose it is, of course, necessary to deduct 10·891701134 days from 29·530587946 days: result, 18·638886812 days. This, then, is the number of days by which the appearance of the first new moon is retarded each year, and the first thing to do every year is to calculate the interval of retardation for that year. The interval (if we take the retardation for one year) will, *ipso facto*, be the date of appearance of the first or Vaisâkha new moon in solar year 1 of Kaliyuga (expired). From this date all other mean new moons for that or any subsequent solar year may be found by the successive addition of multiples of 29·53059 days; and the mean ending moment of every *tithi* is given by the addition of the *tithi* equivalent in days (according to the Eye-Table) to the date of mean new moon. Precisely the same method was followed for the Ârya Siddhânta, Lalla's corrections being introduced at the appropriate date.
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250. The mean anomaly, in the case of the moon as well as that of the sun, was reckoned from perigee, as in Professor Jacobi's article in the *Indian Antiquary* (1888), and not from apogee, as in his articles in Volumes I and II of *Epigraphia Indica*. For the purpose of Table IX, the moon's mean anomaly was converted into days in the ratio of 27·554599899 days to 360° in the case of the Sûrya Siddhânta, and of 27·554566986 days to 360° in the case of the Ârya Siddhânta. The increase of the moon's mean anomaly for a single solar year is thus:—
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Sûrya Siddhânta, 7·048957797 days.
Ârya Siddhânta, 7·049310381 days.
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The anomaly of the moon at the first moment of Kaliyuga was taken as 90° from perigee, that being the figure according to all the authorities. Subsidiary Table VII (*a*) is the conversion table for moon's anomaly according to the Sûrya Siddhânta. From the year A.D. 1600, the corrected period of the anomalistic month (27·55459797 days) has been adopted for Sûrya Siddhânta calculations.
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251. The moon's mean anomaly, as entered in Table IX, corresponds to the *tithi* or space accomplished, while the equation is the addition to or deduction from the *tithi*, to be made in order to arrive at the time or ending moment of the *tithi*. Consequently, the equation in degrees was in every case added to or deducted from the mean anomaly, and the result, converted into days at the rate of 27·5546 days to 360°, is entered as the anomaly in Table IX.
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252. For the purpose of Table IX-*c*, *f*, and *h*, the sun's mean anomaly, reckoned from perigee, was converted into days at the rate of 365·25875 days to 360° [Subsidiary Tab. VII (*b*)] and the sun's equation of the centre was converted at the same rate as the distance of moon from sun (29·53059 days to 360°), because the principal use of the sun's equation of the centre is to correct the mean into the true distance of the moon from the sun. The moon's equation of the centre in Table IX-*a*, *b*, *d*, *e*, *g*, was converted at the same rate, and for the same reason, from degrees into decimals of a day.
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*N.B.*—In the Nakshatra and Yoga Equation Tables, IX-*i*, *j*, *k*, *l*, equations in degrees were converted into days at the rate of 27·32167 days for 360° in the case of Nakshatras, and of 25·4202 days for 360° in the case of Yogas.
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²⁵³ The actual sun's anomaly, as entered in Table IX, is the result of a series of transformations which had to be carefully executed for each of the 680 stages of the anomaly. According to the Sûrya Siddhânta, the mean anomaly of the sun at the commencement of each solar year, neglecting the slow motion of the perigee which is allowed for in that Siddhânta but not in the others, is 102° 45' or, in solar days, 104·25093 days. This amount has therefore to be added to the number of days expired in the solar year at a given moment in order that the sun's anomaly for that moment may be correctly expressed in days. From the sum, 2·1707 days have to be deducted for Sodhya. Thus the equation entered in Table IX (*c*) against the 182nd day of the solar year is the equation which really belongs to the following mean anomaly: *viz.*, 182 days plus 104·25093 days *minus* 2·1707 days, that
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is, an anomaly of 284·0802 days : the latter figure, which is not required in practice, has wholly disappeared from Table IX (c), and the equation is simply entered against the day of the solar year for which, in practice, it is likely to be required.
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254. For *nakshatras*, the following formula was used : (moon's longitude minus sun's longitude) plus sun's longitude = moon's longitude for *nakshatras*. As moon's minus sun's longitude is expressed according to the ratio 360° = 29·53059 days, the sun's longitude is expressed in the same ratio. Solar days are thus expressed in terms of the synodical month and the result of the reduction, after allowing for *śodhya*, is exhibited for whole days in the last column of Table VIII, "Sun's longitude for Nakshatras". Fractions of solar days can be converted into lunation-longitude by means of the third multiplication table in Table V. The first two multiplication tables in Table V were originally intended to be of use in determining the ending moment of a yoga or nakshatra, but in practice, either Table XI or Tables XVII-A and XVII-C will be found much easier and handier than Table V for the investigation of all kinds of problems connected with nakshatras and yogas. Table XI is referred to in Secs. 263 to 265 *infra* and Tables XVII-A and XVII-C in Sec. 266 *infra*.
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255. The moment of sunrise for any latitude and longitude in India can be ascertained by means of Table XIII, based on the rules and table of *asus* given by Professor Jacobi in Volume I of the *Epigraphia Indica*. Professor Jacobi has himself given detailed tables for sunrise in Volume II of the same publication, but the results achieved by means of those tables can, it is believed, be more easily arrived at by the present Table XIII. For the purpose of determining the equation of time for each day of the solar year, the sun's equation of the centre, according to Table IX (c), was used, with the sign changed : likewise the *asus* given in Professor Jacobi's table and reproduced in Sec. 76 *supra*, had to be suitably modified.
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Thus (according to Professor Jacobi's table) in the 10th degree of Northern latitude 30 degrees of Sign 1 of the Zodiac take 1544 *asus* or 1544 × 4 seconds of time to rise; or in lunation-longitude, (30/360) × 29·53059, or 2·46088 units of space take 1544 × 4 seconds of time to rise ; ∴ 1 unit of space takes (1544 × 4) / 2·46088 = 1544 × 1·62 = 2510 seconds.
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This 2510, then, is the factor by which each day's equation of the centre according to Table IX (c) should be multipled (so long as the sun is in the first Sign) in order to give that day's equation of time.
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And generally, all the *asus* in Professor Jacobi's table were multiplied by 4 / 2·46088 = 1·62, and the factors thus obtained were multiplied again by each day's equation according to Table IX (c) in the present work : the result was each day's equation of time in seconds as entered in Table XIII of this work. The same result could of course have been arrived at directly from Table XVII-A, "Sun's Equation in degrees for each day of Solar Year".
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²⁵⁶ For using the tables in *Epigraphia Indica*, Vol. II, the sun's longitude (first sidereal and then tropical), corresponding to a given day of the solar year, has to be first determined from special tables in Vol. I, *Epigraphia Indica*; then the *vināḍis* given under each degree of latitude and each Sign of the Zodiac in Vol. II, *ibid.*, have to be multiplied
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by the equation of the centre, which in turn has to be calculated from the sun's mean anomaly, applied with proportional parts to Special Table XXIV-B in Vol. I, *Epigraphia Indica*: only then can the moment of sunrise for a given day in the solar year be ascertained.
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**257.** In the present Table XIII on the other hand, the moment of sunrise for any day of any solar year is obtained by simply adding a figure in the column "Tropical Longitude" to the figure opposite the given day of the solar year in the column "Equation of Time"; and the result is the correction, in seconds of time, to be applied to mean sunrise at Lankâ (6 a.m.) to determine the local sunrise for the given latitude.
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**258.** Column 3 under each degree of latitude in Table XIII is a further step towards simplification, for it gives, for the 80 years ending 1920, the total correction to be applied to mean Lankâ time in order to arrive at the local time, for each of 30 important places, including Calcutta, Madras and Bombay. In this column the difference in time corresponding to the longitude of each place, measured from Ujjain, is added to the correction for sunrise for each day of the solar year and for each degree of latitude.
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*Example.*—Ashâdha Sukla 12, Kaliyuga 3585 (expired) A.D. 484.
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**259.** A single test problem, the same as that selected by Mr. Dikshit in his Introduction to Dr. Fleet's *Gupta Inscriptions* as well as by Professor Jacobi in Vol. I of *Epigraphia Indica*, will suffice to demonstrate the absolute reliability and extreme simplicity of the above processes as carried out in the present work. The problem is to determine the ending moment of Ashâdha sukla dvâdasî in Kaliyuga 3585 (expired), A.D. 484. The ending moment has to be determined (a) for Lankâ and (b) for the latitude and longitude of Eran (Lat. 24°, Long. 78° 15'). Lastly, the problem has to be worked out, first according to the *Sûrya Siddhânta*, and then according to the *First Arya Siddhânta*.
Surya Siddhanta
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| References to Tables. | Days of Solar Year. | Moon's mean Anomaly in days. |
| (Table X): Kaliyuga 3585 ; A.D. 484 ; * March 18·2212. | | |
| First New Moon in Solar year ... | 24·3900 | 4·494 |
| (Table VIII) : Ashâdha śukla 12 ... | 70·8734 | 15·764 |
| | 95·2634 | 20·258 |
| | | (☉'s Eqn.) — ·0455 |
| Table IX (c) : ☉'s Eqn. for 95·26 days of Solar Year = | — ·0455 | |
| Table IX (b) : ☽'s Eqn. for anom. of 20·212 days = | + ·4138 | 20·212 |
| | + ·3683 + ·3683 | |
| * English month, day, and fraction of day marking commencement of solar year ... | 95·6317 *March 18·2212 | |
| | 113·8529 | |
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Our result is : the tithi ended at ·8529 of a day, *i.e.*, at 51 ghaṭikas 11 palas on the 113th day of the English Calendar, counting from 1st March.
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The reader will be pleased to note that this absolutely correct result for the ending moment of a tithi is obtained by simply adding up six or seven figures from Tables VIII, IX, and X, and that absolutely no other process is required for any tithi in any year.
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260. Now, by Table VIII, the 113th day of the English Calendar, counting from 1st March, is 21st June;
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by Table XIX, ·8529 of a day = 51 ghaṭikas 11 palas; by Table IV we arrive at the week-day as follows:— Co-efficient of 400 A.D. ... 2 Co-efficient of odd year 84 ... 0 Co-efficient of June ... 3 Day of the month ... 21 --- 26
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Since 26, divided by 7, leaves remainder 5 = THURSDAY, the final answer is THURSDAY, 21st JUNE, A.D. 484, 51 GHAṬIKAS 11 PALAS after mean sunrise at Laṅkā.
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NOTE.—This is the absolutely correct ending moment, according to Mr. Dikshit; but Professor Jacobi arrives at a result which is 4 palas short. To arrive at the latter result, ☉'s + ☉'s Eqn. must be added as in foot-note to p. (83) to ☉'s Anom. before ascertaining ☉'s Eqn., just as ☉'s Eqn. is in the actual working *supra*, added to ☉'s anom. before ascertaining ☉'s Eqn. This extra step, however, is a nicety seldom required in practice, since the error on this account can never exceed 6 palas. See Sec. 212 *supra*.
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TO DETERMINE THE ENDING MOMENT OF THE ABOVE TITHI IN TRUE LOCAL TIME AT ERAN.
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261. We first of all find the sun's sidereal longitude for 95·85 days, for which purpose we deduct the sodhya, 2·17 days. Remainder, 94 days nearly.
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We turn to Table XIII and bring down the entry under Latitude 24° corresponding to the 94th day of the Solar Year, for ☉'s trop.* long. and the entry corresponding to the 95th day for the equation of time.
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(1) Equation of time (95th day): + 148 seconds of time.
(2) ☉'s tropical longitude (94th day): + 2647 seconds of time.
(3) To these figures from Table XIII we add the time-difference for the longitude of Eran (+ 2·53 degrees Ujjain longitude), namely, + 2·53 × 240 or + 608 seconds of time. Total: + 148 + 2647 + 608 = + 3403 seconds of time.
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Now 3403 seconds of time, divided by 60, are 56 minutes 43 seconds or ·0393 of a day. Adding this to the mean Laṅkā time already arrived at, *viz.*, ·8529, we obtain, as TRUE LOCAL TIME at ERAN for Āṣāḍha Sukla 12, 3585 Kaliyuga, ·8922 of a day or 53 GHAṬIKAS 32 PALAS which is exactly the same as Mr. Dikshit's result.
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ACCORDING TO 1st ARYA SIDDHANTA.
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262. The simplest way of arriving at the ending moment of the above tithi according to the first Ārya Siddhānta—and this was apparently the method adopted by Mr. Dikshit—is to deduct from the ending moment, already arrived at, the difference between the Sodhyas of the Sūrya and Ārya Siddhāntas (2·1707 days *minus* 2·1476 days = ·0231 day), this being the only difference between the Sûrya and Ârya Calendars at the epoch 3585 K.Y. We thus obtain 8530 day less 0231 day = 8299 day = 49 GHATIKAS 48 PALAS (Table XIX). This is exactly Mr. Dikshit's result for the mean Lankâ time of the ending moment of the tithi according to the 1st Ârya Siddhânta. We may make the same deduction for true local time at Eran according to the same Siddhânta, and obtain Mr. Dikshit's result for this also.
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∗ NOTE.—In the solar year, 3585 K.Y. we make no correction for the difference between ☉'s sidereal and tropical longitude because in 3600 K.Y., *i.e.*, only 15 years later, the ☉'s sidereal coincided with the ☉'s tropical longitude.
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NOTE.—There was also a difference of 001 of a day, i.e., 1½ minutes or 4 palas between the commencement of the solar year, K.Y. 3585, according to the Surya and Arya Siddhantas (vide Tables VI and VII in the present work); but apparently Mr. Dikshit neglected this trivial difference.