Page:Sewell Indian chronography.pdf/50

These then are the required values for sunrise of the first civil day of the luni-solar year beginning in 155 B.C.; and we work by them exactly as if those values had been entered in cols. 23 to 25 of Table I.

84. I have tested the values by working the same moment with the use of Professor Jacobi's Indian Antiquary Tables, his Table 6 being extended backwards to a distance of 20 centuries from the 19th century, and using in his Table 5 the year A.D. 1846, which is the proper one to take for a year 55 or 155 B.C. The result is precisely the same, namely, for mean sunrise on March 2nd, 155 B.C., ${\displaystyle a=4648}$, ${\displaystyle b=662}$, ${\displaystyle c=249}$. (See Tables XXXIX., XL., below.)

85. But the warning cannot too often be repeated—these figures only show the results of calculation by the First Ārya Siddhānta, which was not composed for over six centuries after the date in question. What was the exact day and hour of the occurrence of new moon according to any other authority in use at the time it is not as yet possible to say. It is however not improbable that the result, in the matter of the state of the moon on the day in question, is fairly accurate, for reasons that need not now be gone into.

85A. Since it is often exceedingly useful, and even necessary, to check the accuracy of calculations made for the of mean sunrise of any day by the method suggested in the text (above, § 84), and in general for all luni-solar dates, I have framed Tables XXXIX. and XL. in order to enable this to be done. They are based on Tables 6 and 5, respectively, of Professor Jacobi's treatise in the Indian Antiquary, Vol. XVII., on the "Methods and Tables for verifying Hindū dates."

85B. Table XXXIX. consists of the Professor's Table 6 taken without alteration; except that his last column is omitted, columns are added showing, throughout, the years for which the entries are to be used, and the calculations are carried back for 23 centuries, or eight centuries earlier than the entries in Table 6. This enables the Table to be used for any year from 500 B.C. downwards. At the end, seeing that the New Style was introduced into Europe at different times from A.D. 1582, I have given the figures for both styles from A.D. 1500 to 2000.

That no mistake may be made in using this Table from the years A.D. 1582 to 1752 I repeat here the note in the Indian Calendar, p. 103. The New Style was introduced into all the Roman Catholic countries in Europe from October 5th, 1582, the year 1600 remaining a leap-year; while it was ordained that 1700, 1800 and 1900 should be common and not leap-years. This was not introduced into England till September 3rd, A.D. 1752.

85C. This Table has always to be used in conjunction with the Professor's Table 5, which gives the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for odd years of a century, and with Table 7, which gives the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for the days of any month in the year. The three Tables, worked together, enable us to find the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for mean sunrise of any day of any year. But the entries in his Table 6 were limited to a period of 15 centuries earlier than the 19th century A.D., and, now that I have carried this back to 23 centuries earlier than the 19th century A.D., it is necessary to adapt Table 5 so as to suit the odd years of any century B.C. as well as those of the period A.D. This is now done in my Table XL. And, by the use of my Table XXXIX., Jacobi's Table 5 (taking, as given in my Table XL., for the odd years of a century B.C., the figures for the years quoted by him of the 19th century ) and Jacobi's Table 7, we can now calculate the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, and find thereby the moon's place, both true and mean, at mean sunrise of any day from A.D. 1900 back to January 1st, 500 B.C. The Tables apply, for luni-solar calculation, to all the Siddhāntas.

85D. Thus in § 83 I showed, according to one method of calculation, that for mean sunrise on March 2nd, 155 B.C., the values for ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ were respectively 4648, 662 and 249; and in § 84 I stated