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 c29: Chapter XXIII. Theory of Anomalies and Equations of the Centre. Construction of Table IX
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 c29
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Chapter XXIII. Theory of Anomalies and Equations of the Centre. Construction of Table IX
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THEORY OF ANOMALIES AND EQUATIONS OF THE CENTRE.
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187. The uses of Table IX, to which we come after discussing Tables XI and X, will be sufficiently obvious from examples already worked out. We therefore give in this place in popular language a theory of anomalies and their equations.
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188. From the fact that the orbit of the moon as well as that of the earth is elliptical, not circular, it follows that the motions of these bodies cannot be uniform from day to day or from hour to hour. This irregularity is called the eccentricity of the orbit and the correction to be applied on this account is called the equation of the centre.
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189. The following extract from Prof. Jacobi's Table in Vol. I of *Epigraphia Indica* will serve to introduce the reader to the general theory of Solar and Lunar anomalies, and it will also show how the material furnished by the Siddhântas has been worked into Table IX of the present work :—
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Surya Siddhanta.
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| ☽'s Eqn. + ; ☉'s Eqn. – | | ☽'s Eqn. – ; ☉'s Eqn. + | | Moon's Equation of the centre. | | Sun's Eqn. of the centre. | |
| Deg. | Min. | Deg. | Min. | Deg. | Min. | Deg. | Min. | Deg. | Min. | Sec. | Deg. | Min. | Sec. |
| 0 | 0 | 180 | 0 | 180 | 0 | 360 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 0 | 150 | 0 | 210 | 0 | 330 | 0 | 2 | 32 | 0 | 1 | 6 | 3 |
| 60 | 0 | 120 | 0 | 240 | 0 | 300 | 0 | 4 | 22 | 30 | 1 | 53 | 26 |
| 90 | 0 | 90 | 0 | 270 | 0 | 270 | 0 | 5 | 2 | 46 | 2 | 10 | 31 |
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190. Let us try to interpret in detail the meaning of this table. We are supposed to measure the moon's rate of progress, beginning from perigee, the point when she is nearest the earth, and at every step we must distinguish the moon's mean position, *i.e.*, the position which she would have attained at a uniform rate of motion equal to the mean, and the actual position which she attains on account of the eccentricity of her orbit.
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191. The mean and actual positions are the same at 0° or 360°, *i.e.*, at perigee and at 180°, *i.e.*, at apogee.
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When the moon's mean position is 30° from perigee, her actual position has advanced by 2 degrees 32 minutes.
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When her mean position ought to be 60°, we find her actually at 64° 22' 30" from perigee.
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When her mean position ought to be 90°, that is half way between perigee and apogee, she is actually 95° 2' 46" from perigee.
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From this point she begins to move more slowly, though her actual position is still in advance of the mean.
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At 120° from perigee, she is 4° 22' in advance of the mean position, that is, exactly as she was at mean 60°.
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At 150° from perigee she is only 2° 32' in advance of the mean position.
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From 180° onwards she begins to slow down, and when she ought to be 210° from perigee, or 30° from apogee, we find she has reached only 210° minus 2° 32' or 207° 28'. When she ought to be 270° from perigee, she is only 270° minus 5° 2' 46" or 264° 57' 14".
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From 270° onwards she begins to move quicker, though she is still behind her mean position. At 300° she is behind by 4° 22' 30" and at 330° she is behind her mean position by only 2° 32' and at 360° or at perigee she is even with her mean position.
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192. In like manner we might trace the Sun's mean and actual positions from perigee through apogee back to perigee, using the figures in the last column of the above table, from which we see that the maximum equation of the centre for the sun is 2° 10' 31".
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193. In our tables (except under Planets' Tables XVII, XVIII) we do not refer to the sun's or moon's position by degrees, but by days, which is more readily intelligible and handier for purposes of calculation.
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194. Our Table IX with its numerous divisions (a) to (l) is simply the result of a careful expansion of the smaller tables from which the figures in Section 196 have been extracted, and we shall see presently how far our Table IX agrees with the original.
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We saw that when the moon is actually 32° 32' from perigee, she is 2° 32' in advance of her mean position.
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We turn the first of these figures into days with the help of Subsidiary Table VII (a) and for turning 2° 32' into days we use Subsidiary Table VI (a).
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By Sub. Table VII (a) 30° = 2·2962 days. 2° = ·1531 day. 32' = ·0408 day. 32° 32' = 2·4901 days.
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By Sub. Table VI (a) 2° = ·1641 day. 32' = ·0437 day. 2° 32' = ·2078 day. ... (1)
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195. We now turn to Table IX (a), and look up the equation for an anomaly of 2·4901 days. We there find,
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| ☾'s Anomaly. | Equation. |
| 2·479 days | ·207 day. |
| 2·493 days | ·208 day. |
| ·014 day | ·001 day. |
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(·490 minus ·479 =) ·011 day ; (·011)/(·014) × ·001 = ·008 day.
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∴ The equation for 2·490 days = ·2078 day . . . . . (2)
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Our result (2) according to Table IX (a) is in exact agreement with result (1) deduced from the Siddhānta Table, which shows the anomaly and the equation of the centre in degrees minutes and seconds.
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196. The reader will naturally ask why we turned the anomaly 32° 32' into days by means of Table VII (a) and the equation 2° 32' by means of Table VI (a). We proceed to explain.
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197. Let us suppose the sun and the moon to move in the same plane and in circular orbits, describing equal spaces in equal times. Let AB, BC, CD be the mean spaces described by the sun in three successive tithis, and let AM, MN, NP be the corresponding mean spaces described by the moon in the same tithis.
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198. If there were no irregularity or eccentricity in the sun's and moon's motions, the first tithi would be the time taken by the moon to gain 12°, that is, the space BM over the sun ; similarly the second and third tithis would be the periods in which the moon gains (MN minus BC) and (NP minus CD) over the sun.
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199. But owing to the eccentricity of their orbits we will suppose the sun to be at B' (actual position) when he ought to be at B (mean position) and the moon to be at M' (actual position) when she ought to be at M (mean position).
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200. Then in the period of a mean tithi (·9843 day) the moon gains over the sun the space B'M' but our tithi is the period during which the moon actually gains over the sun 12°, i.e., we must cut off from B'M' a space equal to BM and determine the time during which that space is gained by the moon. Our problem would be solved if we knew the time during which B'M' minus BM was gained by the moon.
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201. Now B'M' minus BM = MM' minus BB'. The time during which MM' - BB' is gained, is evidently the time during which MM' is gained minus the time during which BB' is gained. The times during which MM' and BB' are respectively gained are obtained by turning MM' and BB' into days according to Sub. Table VI (a). This is the reason why in the Tables for the moon's and also the sun's equation of the centre we turn the equation into days invariably by Table VI (a). For Nakshatra and Yoga equations other scales which it was unnecessary to give in detail, were used for converting degrees into days.
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202. As the equation for the sun as well as the equation for the moon is sometimes positive and sometimes negative, and for tithis we have to take the difference between the two equations, some confusion would result from our having to change signs so often. Therefore the sun's equations are tabulated, as in the extract given above, with the signs reversed. That is why the sun's eccentricity, starting from perigee, is shown in the table in Sec. 189 as negative whereas it is really positive.
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203. The reader will also observe that the moon's equation in the above table is, as it should be, positive between perigee and apogee, whereas in Table IX (a) it is negative for the same period. The reason is that when the moon does in a given time more than the mean space, this is equivalent to a given space being done in less than mean time. This is why in Table IX, which derives equations of time from equations of space, the signs of space-equations are reversed. In the case of the sun's equations, a double reversion of signs takes place, first a reversion in order to make the operation of combining the sun's and moon's equations always an addition, and secondly a reversion in order to derive equations of time from equations of space. As a final result, the sun's equation of time is shown in Table IX (c) as positive after perigee and negative after apogee. A third reversion takes place in calculating sunrise, where we have to derive equations of space from equations of time. (Vide Sec. 255 infra.)
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204. In calculating tithis, all we have to do, is to sum up the equations of the sun and the moon according to Table IX, whether they are positive or negative. Before taking the moon's equation, however, we in practice add to or deduct from the moon's anomaly the sun's equation and we determine the equations for the net moon's anomaly. The reason is that when the effect of the Sun's anomaly is to diminish or increase the mean time that would be necessary for a tithi, it is necessary to take the moon's anomalies for the altered mean time.
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205. Conversely, in calculating Yogas, where we have to take sum of the proper anomalistic equations of the sun and the moon, our Yoga equations in Table IX (i) [p. 20 (a),] have had to be suitably altered as to their signs.
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206. For Nakshatras, we have to convert the moon's eccentricity, in other words her equation of the centre, into days at the rate of the moon's sidereal motion, i.e., 360° for 27.32166 days and this has accordingly been done in Table IX (j), (k), and (l).
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207. The sun's anomaly in Tables IX (c), (f) and (h) is expressed, not in days of the anomalistic year, but as days of the solar year, and this is done for convenience of use, as explained in Part III, Sec. 253.
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208. For instance, in the above table, the sun's equation for an anomaly of 270° is entered as 2° 10' 31."
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Now by Table VII (b) 270° = 202·92 + 71·02 = 273·94 days. And by Table VI (a) 2° = ·1641 day. 10' = ·0137 day. 31" = ·0007 day.
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2° 10' 31" = ·1785 day.
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Since the solar year always begins with a mean anomaly of 102·0802 days, an anomaly of 273·94 days of the anomalistic year really belongs to 273·94 *minus* 102·08, i.e., the 171·86th day of the solar year.
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Accordingly we find in Table IX (c) the equation ·1785 day set down against the 171·86th day of the solar year.
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209. The length of the modern anomalistic year is 365·2596 days while that of the Hindu solar year is 365·2587 days. There is no practical error in adopting, as the Indian Siddhântis have done, an identical period for the anomalistic and solar (sidereal) years. The modern anomalistic lunar month is, like the Hindu anomalistic month, 27·5546 days. In converting degrees of anomaly into days we, therefore, put—
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Sun's anomaly : [Sub. Table VII (b)] 360° = 365·258756484 days. Moon's anomaly : [Sub. Table VII (a)] 360° = 27·45459999 days.
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210. The above theory of anomalies and equations is subject to an important variation in practice, which we alluded to briefly in Sec. 208, but which it is now necessary to dwell on specially. Supposing a mean tithi is accomplished at A days of the solar year, when the Moon's anomaly is B days, then, if the ☉'s equation for A days is —a, this means that the tithi would be accomplished (supposing for the moment that the ☾'s anomaly had no influence) in A—a days of the solar year, at a moment when the moon's anomaly would be B—a. Now we proceed to determine the influence of this moon's anomaly B—a and find the moon's equation to be, let us suppose, —b. We then put down, as the actual ending moment of the tithi, A—a—b days of the solar year.
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211. Strictly speaking, we ought to take (1) the sun's equation for A+b + a days of the solar year, where b is the equation (positive or negative) of the moon for anomaly B, and a is the ☉'s equation, positive or negative, for A days of the solar year, (2) the moon's equation for anomaly B+a days, and then add the equations so found.
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212. But in practice this refinement is not necessary for the sun's equation since the maximum value of b + a is (·4138 + ·1784=) ·5922 day, and the maximum variation of the sun's equation for this period is about ·0018 day or 6 palas only. We may note, however, that the test example, worked by Prof. Jacobi for Âshâḍha Śukla 12, K.Y. 3585, and by ourselves in sections 228 and 259 to 262 infra, a difference of 4 palas does occur between his method and Mr. Dikshit's, and the learned Professor rightly surmises in a footnote (Ep. Ind. Vol. I, p. 430) that the difference must be due to an abridgment in the Hindu method. We now see what the abridgment consists in.