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Patell's "Chronology", and in the late Gen. Sir A. Cunningham's "Indian Eras,"^[1] But in none of these three works is a single word said as to how, or following what authority, the calculations were made, so that we have no guide to aid us in checking the correctness of their results.

79. An added lunar month being one in which no saṅkrânti of the sun occurs, it is evident that a saṅkrânti must fall shortly before the beginning, and another one shortly after the end, of such a month, or in other words, a solar month must begin shortly before and must end shortly after the added lunar month. It is further evident that, since such is the case, calculation made by some other Siddhânta may yield a different result, even though the difference in the astronomical data which form the basis of calculation is but slight. Hence we have deemed it essential, not only to make our own calculations afresh throughout, but to publish the actual resulting figures which fix the months to be added and suppressed, so that the reader may judge in each case how far it is likely that the use of a different authority would cause a difference in the months affected. Our columns fix the moment of the saṅkrânti before and the saṅkrânti after the added month, as well as the saṅkrânti after the beginning, and the saṅkrânti before the end, of the suppressed month; or in other words, determine the limits of the adhika and kshaya masas. The accuracy of our calculation can be easily tested by the plan shewn in Art. 90 below. (See also Art. 88 below.) The moments of time are expressed in two ways, viz., in lunation-parts and tithis, the former following Prof. Jacobi's system as given in Ind. Ant., Vol. XVII.

80. Lunation-parts or, as we elsewhere call them, "tithi-indices" (or "${\displaystyle t}$") are extensively used throughout this work and require full explanation. Shortly stated a lunation-part is ⁠1/10000⁠th of an apparent synodic revolution of the moon (see Note 2, Art. 12 above). It will be well to put this more clearly. When the difference between the longitude of the sun and moon, or in other words, the eastward distance between them, is nil, the sun and moon are said to be in conjunction; and at that moment of time occurs (the end of) amâvâsyâ, or new moon. (Arts. 7.29 above.) Since the moon travels faster than the sun, the difference between their longitudes, or their distance from one another, daily increases during one half and decreases during the other half of the month till another conjunction takes place. The time between two conjunctions is a synodic lunar month or a lunation, during which the moon goes through all its phases. The lunation may thus be taken to represent not only time but space. We could of course have expressed parts of a lunation by time-measure, such as by hours and minutes, or ghaṭikâs and palas, or by space-measure, such as degrees, minutes, or seconds, but we prefer to express it in lunation-parts, because then the same number does for either time or space (see Art. 89 below). A lunation consists of 30 tithis. ⁠1/30⁠th of a lunation consequently represents the time-duration of a tithi or the space-measurement of 12 degrees. Our lunation is divided into 10,000 parts, and about 333 lunation-parts (⁠1/10000⁠ths) go to one tithi, 667 to two tithis, 1000 to three and so on. Lunation-parts are therefore styled "tithi-indices", and by abbreviation simply "${\displaystyle t}$". Further, a lunation or its parts may be taken as apparent or mean. Our tithi-, nakshatra-, and yoga-indices are apparent and not mean, except in the case of mean added months, where the index, like the whole lunation, is mean.