Page:Sewell Dikshit The Indian Calendar (1896) proc.djvu/66

Our tithi-index, or "${\displaystyle t}$", therefore shows in the case of true added months as well as elsewhere, the space-difference between the apparent, and in the case of mean intercalations between the mean, longitudes of the sun and moon, or the time required for the motions of the sun and moon to create that difference, expressed in 10,000ths of a unit, which is a circle in the case of space, and a lunation or synodic revolution of the moon in the case of time. Briefly the tithi-index "${\displaystyle t}$" shews the position of the moon in her orbit with respect to the sun, or the time necessary for her to gain that position., e.g., "0" is new moon, "5000" full moon, "10,000" or "0" new moon; "50" shews that the moon has recently (i.e., by ⁠50/10000⁠ths, or 3 hours 33 minutes—Table X.. col. 3) passed the point or moment of conjunction (new moon); 9950 shews that she is approaching new-moon phase, which will occur in another 3 hours and 33 minutes.

81. A lunation being equal to 30 tithis, the tithi-index, which expresses the 10,000th part of a lunation, can easily be converted into tithi-notation, for the index multiplied by 30 (practically by 3), gives, with the decimal figures marked off, the required figure in tithis and decimals. Thus if the tithi-index is 9950, which is really 0.9950, it is equal to (0.9950 × 30 =) 29.850 tithis, and the meaning is that ⁠9950/10000⁠ths of the lunation, or 29.850 tithis have expired. Conversely a figure given in tithis and decimals divided by 30 expresses the same in 10,000ths parts of a lunation.

82. The tithi-index or tithi is often required to be converted into a measure of solar time, such as hours or ghaṭikâs. Now the length of an apparent lunation, or of an apparent tithi, perpetually varies, indeed it is varying at every moment, and consequently it is practically impossible to ascertain it except by elaborate and special calculations; but the length of a mean lunation, or of a mean tithi, remains permanently unchanged. Ignoring, therefore, the difference between apparent and mean lunations, the tithi-index or tithi can be readily converted into time by our Table X.. which shews the time-value of the mean lunation-part (⁠1/10000⁠th of the mean lunation), and of the mean tithi-part (⁠1/1000⁠th of the mean tithi). Thus, if ${\displaystyle t=50}$, Table X. gives the duration as 3 hours 33 minutes; and if the tithi-part^[1] is given as 0.150 we have by Table X. (2 h. 22 m. + 1 h. 11 min. = ) 3 h. 33 m.

It must be understood of course that the time thus given is not very accurate, because the tithi-index (${\displaystyle t}$) is an apparent index, while the values in Table X. are for the mean index. The same remark applies to the nakshatra (${\displaystyle n}$) or yoga (${\displaystyle y}$) indices, and if accuracy is desired the process of calculation must be somewhat lengthened. This is fully explained in example 1 in Art. 148 below. In the case of mean added months the value of (${\displaystyle t}$) the tithi-index is at once absolutely accurate.

83. The saṅkrântis preceding and succeeding an added month, as given in our Table I., of course take place respectively in the lunar month preceding and succeeding that added month.

84. To make the general remarks in Arts. 80, 81, 82 quite clear for the intercalation of months we will take an actual example. Thus, for the Kali year 3403 the entries in cols. 9 and 11 are 9950 and 287, against the true added month Âśvina in col. 8. This shews us that the saṅkrânti preceding the true added, or Adhika, Âśvina took place when 9950 lunation-parts of the natural month Bhâdrapada (preceding Adhika Âśvina) had elapsed, or when (10,000 − 9950 =) 50 parts had to elapse before the end of Bhâdrapada, or again when 50 parts had to elapse