Page:Sewell Indian chronography.pdf/108

The beginning of the 15th tithi. At mean sunrise on March 25th the tithi-index (${\displaystyle t}$) was found to be 4691. The beginning of the tithi was (Table VIII., 15th śukla tithi, col. 3) 4667, and the ending 5000. 4691 − 4667 = 24 = (Table X.) 1 h. 42 m. in mean time. The 15th śukla tithi, therefore, began approximately 1 h. 42 m. before mean sunrise on March 25th. But let us suppose that we want perfect accuracy. We calculate for the value of ${\displaystyle t}$ at 1 h. 42 m. before sunrise by deducting from the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for sunrise the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for that period by Table V.

	a.	b.	c.
Values (found) for sunrise	4484	423	282
Values for 1 hour (Table V.)	−0014	2	0
Values for 42 minutes (Table V.)	−0010	1	0
	4460	420	282
Equation ${\displaystyle b}$ (Table VI.)	208
Equation ${\displaystyle c}$ (Table VII.)	1
${\displaystyle t=}$	4669

4669 − 4667 (the beginning-point of the 15th śuk. tithi ) = 2 = (Table X.) 9 m., by which amount earlier than 1 h. 42 m. the tithi really began. This gives us 1 h. 51 m. before sunrise, in mean time, as the exact beginning time of the tithi.

The ending of the 15th śukla tithi. 5000 − 4691 = 309 = (Table X.) 21 h. 16 m. + 38 m. = 21 h. 54 m. The tithi expired approximately at 21 h. 54 m. after mean sunrise on March 25th. For exact calculation we work as before:—

	a.	b.	c.
Values (found) for sunrise	4484	423	282
Values for 21 h.(Table V.)	+0296	32	2
Values for 54 m.(Table V.)	+0013	1	0
	4793	456	284
Equation ${\displaystyle b}$ (Table VI.)	179
Equation ${\displaystyle c}$ (Table VII.)	1
${\displaystyle t=}$	4973

5000 − 4973 = 27 = (Table X.) 1 h. 55 m. We have to arrive at exactly 5000, and have discovered that the time, 21 h. 54 m after sunrise, was not late enough by 1 h. 55 m. We now therefore try for 23 h. 49 m.:—

	a.	b.	c.
Values (found) for sunrise	4484	423	282
Values for 23 h.(Table V.)	+0325	35	3
Values for 49 m.(Table V.)	+0012	1	0
	4821	459	285
Equation ${\displaystyle b}$ (Table VI.)	176
Equation ${\displaystyle c}$ (Table VII.)	2
${\displaystyle t=}$	4999

This is close enough to 5000, and we may take the mean-time ending of the tithi as 23 h. 49 m. after sunrise; or we may add 1, or 44 m., to the mean-time, making the ending 23 h. 53 m. We cannot be absolutely certain, by this method of calculation, which result is the correct one; for all the five terms which have been added together under the heading ${\displaystyle a}$ to produce the final result ${\displaystyle t}$ consist of whole figures with the fractions omitted, and in the end it can only be said that the result is exact within one