Page:Sewell Dikshit The Indian Calendar (1896) proc.djvu/96

full numbers are not given so as to avoid cumbrousness in the tables.) Thus the equation for (${\displaystyle b}$) (943) is found to be 90, and from Table VII. the equation for (${\displaystyle c}$) is found to be 38. Adding 90 and 38 to (${\displaystyle a}$) (1335) we get 1463, which is the required tithi-index (${\displaystyle t}$). Turning with this to Table VIII., col. 3, we find by col. 2 that the tithi current was śukla 5, i.e., the given date. Then (${\displaystyle w}$) 4, Wednesday, was its week-day; and the tithi was current at mean sunrise on the meridian of Ujjain on that week-day. Turning with (${\displaystyle d}$) 159 to Table IX., we find that the equivalent date A.D. was 8th June; but as this was after 28th February in a leap-year, we fix 7th June, A.D. 1780, (see N.B. iii.. Art. 147) as the equivalent of the given tithi. As (${\displaystyle t}$) is not within 40 of 1667, the (${\displaystyle t}$) of the 5th tithi (Table VIII.), there is no probability of an expunction or repetition shortly preceding or following (Art. 142). The answer therefore is Wednesday, June 7th, A.D. 1780.

To find the ending time of the tithi. (${\displaystyle t}$) at sunrise is 1463; and Table VIII., col. 3, shews that the tithi will end when (${\displaystyle t}$) amounts to 1667. (1667 − 1463 =) 204 = (Table X.) 14 hours, 27 minutes, and this process shews us that the tithi will end 14 hours, 27 minutes, after sunrise on Wednesday, June 7th. This time is, however, approximate. To find the time more accurately we add the increase in (${\displaystyle a}$) (${\displaystyle b}$) (${\displaystyle c}$) for 14 h. 27 m. (Table V.) to the already calculated (${\displaystyle a}$) (${\displaystyle b}$) (${\displaystyle c}$) at sunrise; and adding to (${\displaystyle a}$) as before the equations of (${\displaystyle b}$) and (${\displaystyle c}$) (Tables VI. and VII.) we find that the resulting (${\displaystyle t}$) amounts to 1686. 1686 − 1667 = 19 = 1 hour and 21 minutes (Table X.). But this is a period beyond the end of the tithi, and the amount must be deducted from the 14 h. 27 m. first found to get the true end. The true end then is 13 h. 6 m. after sunrise on June 7th. This time is accurate for ordinary purposes, but for still further accuracy we proceed again as before. We may either add the increase in (${\displaystyle a}$) (${\displaystyle b}$) (${\displaystyle c}$) for 13 h. 6 m. to the value of (${\displaystyle a}$) (${\displaystyle b}$) (${\displaystyle c}$) at sunrise, or subtract the increase of (${\displaystyle a}$) (${\displaystyle b}$) (${\displaystyle c}$) for 1 h. 21 m. from their value at 14 h. 27 m. By either process we obtain (${\displaystyle t}$) = 1665. Proceed again. 1667 − 1665 = 2 = (Table X.) 9 minutes after 13 h. 6 m. or 13 h. 15 m. Work through again for 13 h. 15 m. and we obtain (${\displaystyle t}$) = 1668. Proceed again. 1668 − 1667 = 1 = (Table X.) 4 minutes before 13 h. 15 m. or 13 h. 11 m. Work for 13 h. 11 m., and we at last have 1667, the known ending point. It is thus proved that 13 h. 11 m. after sunrise is the absolutely accurate mean ending time of the tithi in question by the Sûrya-Siddhânta.

To find the beginning time of the given tithi. We may find this independently by calculating as before the (${\displaystyle t}$) at sunrise for the preceding tithi, (in this case śukla 4th) and thence finding its ending time. But in the example given we calculate it from the (${\displaystyle t}$) of the given tithi. The tithi begins when (${\displaystyle t}$) amounts to 1333 (Table VIII.). or (1463 − 1333) 130 before sunrise on June 7th. 130 is (Table X.) 9 h. 13 m. Proceed as before, but deduct the (${\displaystyle a}$) (${\displaystyle b}$) (${\displaystyle c}$) instead of adding, and (see working below) we eventually find that (${\displaystyle t}$) amounts exactly to 1333 and therefore the tithi begins at 8 h. 26 m. before sunrise on June 7th, that is 15 h. 34 m. after sunrise on Tuesday the 6th. The beginning and ending times are by Ujjain or Laṅkâ mean time. If we want the time, for instance, for Benares the difference in longitude in time, 29 minutes, should be added to the above result (See Table XI.). This, however, does not affect the day.

It is often very necessary to know the moments of beginning and ending of a tithi. Thus our result brings out Wednesday, June 7th, but since the 5th tithi began 15 h. 34 m. after sunrise on Tuesday, i.e., about 9 h. 34 m. p.m.. it might well happen that an inscription might record a ceremony that took place at 10 p.m., and therefore fix the day as Tuesday the 5th tithi, which, unless the facts were known, would appear incorrect.

From Table XII. we find that 7th June, A.D. 1780, was a Wednesday, and this helps to fix that day as current.

We now give the working of Example 1.