true angular distance of the moon from the sun at any moment, or the distance which proves what true tithi was current at that moment, we have to find, first, the angular distance of the moon from the sun in mean longitude (this is the "a" of our Tables), then the moon's mean anomaly (our "b"), and the sun's mean anomaly (our "c"), and then to apply to the a so found the equations of the centre of the moon and the sun, viz., our "equation b" and "equation c." We then have the true angular distance of the moon from the sun, namely our t, or tithi-index, and can decide correctly by the help of Table VIII. (Ind. Cal.) which tithi was current at the given moment.
20. The following is the method adopted for stating the quantities expressed by a, b, c, respectively: a is stated in 10,000ths of a circle, so that the unit when measured in time varies constantly with the length of the lunation, the average time for each unit being about 4.25 minutes. b and c are stated in 1000ths of the circle. Now, in the case of both the moon and the sun the equations of the centre are necessarily sometimes minus and sometimes plus. Thus, when the moon is on her perigee point the mean anomaly is 0. She is then moving rapidly, and during the period covered by her in travelling to a distance of 90° from the perigee point the difference between her mean and true longitudes constantly increases, the true gaining on the mean; or in other words, equation b, applied to bring the mean up to the true, is a quantity plus, which becomes constantly larger. From 90° to 180° equation b decreases, till at the apogee point it is again 0, the mean and true anomalies being then at one. From 180° to 270° the mean gains on the true, the moon's true motion being slow, so that equation b is here a constantly increasing minus quantity. From 270° back to perigee point the equation is still minus, but in ever lessening degree, till at perigee it is again 0. And the same with the sun. The greatest possible equation b (moon) is, in terms of a, 140.2, sometimes minus and sometimes plus; and the greatest possible equation c (sun) is 60.4. In order that these equations may be always additive when applied to a, the sum of 140.2 is, in our Tables VI., VII., added invariably to equation b, and 60.4 to equation c; the total 200.6 being in compensation deducted from every value of a. Thus in Table VI. when the true value of equation b is 0 we have tabulated 140; when its true value is 140.2 we have tabulated 280, and so on; and similarly with equation c. Thus, opposite argument 0 is tabulated 60, for 60.4, instead of 0; and opposite argument 750 is tabulated 121 for 120.8 instead of 60.4. Every calculation is then additive. When, therefore, we have, working by the mean system, found the value of a in the work of testing the date of an inscription we must add 200.6, or 201, to a in order to arrive at the real mean longitudinal difference between the moon and the sun. The value of a when we calculate by mean months and the value of t when we calculate by true months, or the tithi-index resulting from such calculation in either case, is converted into tithis and decimals by use of Table VIII.B, cols. 1, 2, or by multiplying a or t by 3 and marking off three places of decimals; and is converted with fair accuracy into degrees and minutes by col. 3 of that Table. But employment of this last process is not recommended as, if degrees and minutes are wanted, it would be safer to make the calculation by Professor Jacobi's "Special Tables." (Epig. Ind., Vol. I.)
21. That one point may be thoroughly understood I think it well to repeat what has already been stated. The addition of 200.6 or 201 has only to be added to the resulting a in calculations for mean intercalations or suppressions of lunar months, or for mean lunar months and tithis, and not to the resulting in calculations for true intercalations and suppressions, or for true lunar months and tithis; since in the latter case the addition of equations b and c to the value of a cancels the arrangement, made for convenience, whereby the maximum value of those equations was deducted from a and added to their values in the Tables.
a and t being stated in 10,000ths of a circle, a result which shows that a + 201 (in calculations by mean longitude) or t (in calculations by true longitude) = (say) 1 or 9823, means that the moon's distance from the sun is, in terms of a or t, 0.0001 or 0.9823. (See Ind. Cal., §§ 102, 107, 108, pp. 56, 60, 61.)
