Page:Sewell Dikshit The Indian Calendar (1896) proc.djvu/76

(${\displaystyle c}$) for any required number of civil days, hours, and minutes, according to the Sûrya Siddhânta. It will be seen that the figures given in these Tables are calculated by the value for one day given in Art. 102.

Table IV. is Prof. Jacobi's Indian Antiquary (Vol. XVII.) Table 7, slightly modified to suit our purposes; the days being run on instead of being divided into months, and the figures being given for the end of each period of 24 hours, instead of at its commencement. Table V. is Prof. Jacobi's Table 8.

107. Tables VI. and VII. These are Prof. Jacobi's Tables 9 and 10 re-arranged. It will be well that their meaning and use should be understood before the reader undertakes computations according to our method "C". It will be observed that the centre column of each columntriplet gives a figure constituting the equation for each figure of the argument from 0 to 1000, the centre figure corresponding to either of the figures to right or left. These last are given only in periods of 10 for convenience, an auxiliary Table being added to enable the proper equation to be determined for all arguments. Table VI. gives the lunar equation of the centre, Table VII. the solar equation of the centre. (Art. 75 note 3 above). The argument-figures are expressed in 1000ths of the circle, while the equation-figures are expressed in 10,000ths to correspond with the figures of our "${\displaystyle a}$," to which they have to be added. Our (${\displaystyle b}$) and (${\displaystyle c}$) give the mean anomaly of the moon and sun for any moment, (${\displaystyle a}$) being the mean longitudinal distance of the moon from the sun. To convert this last (${\displaystyle a}$) into true longitudinal distance the equation of the centre for both moon and sun must be discovered and applied to (${\displaystyle a}$) and these Tables give the requisite quantities. The case may perhaps be better understood if more simply explained. The moon and earth are constantly in motion in their orbits, and for calculation of a tithi we have to ascertain their relative positions with regard to the sun. Now supposing a railway train runs from one station to another twenty miles off in an hour. The average rate of running will be twenty miles an hour, but the actual speed will vary, being slower at starting and stopping than in the middle. Thus at the end of the first quarter of an hour it will not be quite five miles from the start, but some little distance short of this, say ${\displaystyle m}$ yards. This distance is made up as full speed is acquired, and after three-quarters of an hour the train will be rather more than 15 miles from the start, since the speed will be slackened in approaching the station,—say ${\displaystyle n}$ yards more than the 15 miles. These distances of ${\displaystyle m}$ yards and ${\displaystyle n}$ yards, the one in defect and the other in excess, correspond to the "Equation of the Centre" in planetary motion. The planetary motions are not uniform and a planet is thus sometimes behind, sometimes in front of, its mean or average place. To get the true longitude we must apply to the mean longitude the equation of the centre. And this last for both sun (or earth) and moon is what we give in these two Tables. All the requisite data for calculating the mean anomalies of the sun and moon, and the equations of the centre for each planet, are given in the Indian Siddhântas and Karaṇas, the details being obtained from actual observation; and since our Tables generally are worked according to the Sûrya Siddhânta, we have given in Tables VI. and VII. the equations of the centre by that authority.

Thus, the Tables enable us to ascertain (${\displaystyle a}$) the mean distance of moon from sun at any moment, (${\displaystyle b}$) the correction for the moon's true (or apparent) place with reference to the earth, and (${\displaystyle c}$) the correction for the earth's true (or apparent) place with reference to the sun; and with these corrections applied to the (${\displaystyle a}$) we have the true(or apparent) distance of the moon from the sun, which marks the occurrence of the true (or apparent) tithi; and this result is our tithi-index, or (${\displaystyle t}$). From this tithi-index (${\displaystyle t}$) the tithi current at any given moment is found from Table VIII.. and the time equivalent is found by Table X. Full explanation for actual work is given in Part IV. below (Arts. 139—160).