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of the moon from the sun) for that moment the equations, also for that moment, of the mean anomaly of the moon, ${\displaystyle b}$, and of the sun, ${\displaystyle c}$. This yields us the "true" position required.

79. To find the true tithi, therefore, for any moment we take down the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ (cols. 19–21) at mean sunrise on the first civil day of the luni-solar year as given in Table I.; and we add to these the values of ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for the interval of days, hours and minutes to the given moment (Tables IV. and V.). We add to the ${\displaystyle a}$ of the quotient the amounts of equation ${\displaystyle b}$ and equation ${\displaystyle c}$ (Tables VI. and VII.), and thus arrive at ${\displaystyle t}$, or the tithi-index which (Table VIII.) gives us the apparent position of the moon at the given moment. Taking the figures in col. 3 of Table VIII. for the beginning and ending of the thus-found current tithi, and applying to them the value of the ${\displaystyle t}$ we have obtained, we have the interval in terms of ${\displaystyle t}$ between the given moment and the beginning and ending of the tithi. We turn this into mean time by Table X. This results in a first approximation. If we desire greater exactness we add or subtract, as the case may be (using Tables IV., V.), the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for the interval of time last found to or from the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ already found for the given moment, and by again applying to the resulting ${\displaystyle a}$ the equations of the resulting ${\displaystyle b}$ and ${\displaystyle c}$ we find the final ${\displaystyle t}$, which gives us the true tithi-index of the given moment. After one or two such operations we obtain in mean time (using Table X.) a result absolutely accurate so far as the system permits.

How to find the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for the first civil day of a year earlier than one embraced by Table I. of the "Indian Calendar."

80. It is clear from what has been said that if we have given us, or capable of being found, the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ of the starting point, viz., at mean sunrise on the first civil day of the lunar year, all the rest is easy.

Now Table I. (cols. 19–25) supplies this information for every year from A.D. 300 to 1900, but it does not contain the information for any earlier year, and we have to learn how this can be found.

81. It must first be observed, incidentally, that for many, especially the earlier, inscription-dates we have no certain knowledge of the authority by which they were calculated, and cannot therefore depend for practical work on any Tables yet published. The bulk of the present volume is concerned with the Ārya and Sūrya Siddhāntas, and the former did not come into use till after A.D. 500. Prior to that some other authority, possibly the Original Sūrya Siddhānta or another, was used for the preparation of almanacs. It may be possible hereafter to prepare Tables for these authorities, but at present all that we can do is to use the First Ārya Siddhānta as our authority, and ascertain the moon's position as it would have been determined had that authority been the basis of calculation.

82. The following process will enable the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ of mean sunrise on the first civil day of a luni-solar year earlier than A.D. 300 to be found, according to the Ārya Siddhānta, from the Indian Calendar Tables and those in the present paper. When these values for ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ have been ascertained the rest of the work will proceed according to regular rule, the figures standing as if they belonged to cols. 19 to 25 of Table I.

83. We assume that the luni-solar year began astronomically with the new-moon next after the moment of the sidereal Mīna saṁkrānti, and that the first civil day of that year is the day on which the first tithi of the bright fortnight following that new-moon is current at sunrise. Having ascertained the facts on that theory it is easy to arrive at the correct result should the month have to be altered owing to the pūrṇimānta system of fortnights having been in use, or should the number of the year have to be altered because the solar year was taken to begin with any solar saṁkrānti other than that of Mēsha.

We must therefore first find the moment of Mēsha saṁkrānti for the given year. Let us take the year 155–4 B.C., K.Y. 2947 expired, as our example. (For rule of work, see §§ 48-51 above.)