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it even easier by providing a Table by which ${\displaystyle s}$ can be found at once from the ${\displaystyle c}$ of the given moment.

This was first published in the Indian Antiquary (1896, pp. 287–88), republished in my Eclipses of the Moon in India (pp. 10, 11), and by his kind permission I again republish it as Table XXIII. below for general use by those who prefer it. Properly used it yields the same result as is obtained from the method stated above. Thus in the above example Table XXIII. gives, for argument ${\displaystyle c}$ 430, 1473. Add 86, for difference of ${\displaystyle s}$ with last figure 9, by the auxiliary Table. Total ${\displaystyle s=1559}$ as above. With the ${\displaystyle n}$ so found we turn to Table VIII. where cols. 7 to 10 give the ending-points of the several nakshatras by the three systems. Our ${\displaystyle n}$ gives us the state of the nakshatra-index at mean sunrise on the day in question, and we take the difference between its value and the value for the beginning and ending of the nakshatra (Table VIII.), and so find the resulting argument, which we turn into mean time by Table X. (See the concluding paras. of § 158, p. 97, Ind. Cal.) This gives us, in mean time, the beginning and ending times of the nakshatra.

120. Both these systems entail an unavoidable possibility of error consequent on the multiplication of ${\displaystyle c}$ by 10 an error which may amount to as much as 42 minutes in some cases, but never more than that. This multiplication is necessary in order that the value of ${\displaystyle c}$, which is expressed in thousandths of the circle, may conform to the value of ${\displaystyle t}$, which is expressed in ten thousandths. By Prof. Jacobi's Special Tables, where the work is carried out in degrees, minutes and seconds, this error is reduced to 15 minutes, his ${\displaystyle n}$ being obtained by multiplying his tithi (with two places of decimals) by 12 in order to obtain degrees and decimals.

121. I have taken great pains to ascertain the system of nakshatra calculation adopted by the late Prof. Kielhorn, whose figures for the nakshatra I found constantly to differ from mine and from Prof. Jacobi's by a few minutes. He appears to have followed our Indian Calendar method, but with a difference. Having observed that because of our multiplication of ${\displaystyle c}$ by 10 the difference of one unit in ${\displaystyle c}$ causes a leap in ${\displaystyle c\times 10}$ of 10, or 42 minutes, for each unit so multiplied—which leap is modified by the changing values of equation ${\displaystyle c}$, which is the quantity actually added—he stated the ending-point of the current nakshatra not in, as we do, the mean-time equivalent (Table X.) of the ending-point (Table VIII.) minus our final value of ${\displaystyle n}$, but in the mean-time equivalent of the nearest figure to this, with the last unit ignored. He habitually, that is, gave the mean-time equivalents of 10, 20, 30, &c., and their multiples, as the ending-points of the nakshatra, and never stated the mean-time equivalents of 11, 12, 13, &c. Supposing, for instance, that we have a series of ending-times of nakshatras in different inscription-dates resulting from (Table X.) arguments 1, 2, 3, &c., up to 20. According to Table X. the ending-times would be respectively 4 m., 8 m., 12 m., &c., up to 1 h. 11 m., 1 h. 15 m., 1 h. 19 m. Now the time-equivalents for arguments 10 and 20 are 39 m. and 1 h. 19 m., and Prof. Kielhorn's plan was, for all ending-times between 0 m. and 1 h. 19 m., to state the result as either 0 m. or 39 m. or 1 h. 19 m.; and so for all values, advancing by the equivalents of tens instead of units in the argument.

122. Some examples will best illustrate Prof. Kielhorn's method.

(Ep. Ind. VIII., 278) Pāṇḍya date 51. By the Ind. Cal. system the nakshatra ended at 14 h. 14 m., the argument (${\displaystyle n}$) being (Table X.) 217. Kielhorn stated the ending-time at 14 h. 27 m., as if with argument 220; this by the equal-space system. According to Garga the argument was 32 and the ending 2 h. 6 m. Kielhorn quoted this ending as 2 h. 38 m. as if with argument 40.

(Ep. Ind. IX., 211–12) Chōḷa date 147. With the Ind. Cal. system the ending (Garga) was 2 h. 14 m., with argument 34. Kielhorn gave it, as if with argument 40, at 2 h. 38 m.

(Ep. Ind. IV., 262.) Chōḷa date 20. By the Ind. Cal. system (equal-space) the ending was, with argument 45, 2 h. 57 m. Kielhorn gave it, as if with argument 40, at 2 h. 38 m.

It will be noticed here that in the first two instances the actual argument was, in one 32 and in the next 34; and it would be imagined that the Professor would have stated the time-equivalent for 30, and not for 40, as the ending-time of the nakshatra, but he selected 40. The time-equivalent for