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The Indian Calendar, with Tables for the Conversion of Hindu and Muhammadan into A.D. Dates, and vice versa c8.s2: Method C. Exact Calculation of Dates

The Indian Calendar, with Tables for the Conversion of Hindu and Muhammadan into A.D. Dates, and vice versâ c8.s2

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Contents

Method C. Exact Calculation of Dates

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(A.) *Conversion of Hindu luni-solar dates into dates A.D.*

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139. To calculate the week-day, the equivalent date A.D., and the moment of beginning or ending of a tithi. Given a Hindu year, month, and tithi.—Turn the given year into a Chaitrâdi Kali, Śaka, or Vikrama year, and the given month into an amânta month (if they are not already so) and find the corresponding year A.D., by the aid of columns 1 to 5¹ of Table I., and Table II., Parts i., ii., iii. Referring to Table I., carry the eye along the line of the Chaitrâdi year so found, and write down² in a horizontal line the following five quantities corresponding to the day of commencement (Chaitra śukla pratipadâ) of that Chaitrâdi-year, viz., (*d*) the date-indicator given in brackets after the day and month A.D. (Table I., col. 19), (*w*) the week-day number (col. 20), and (*a*), (*b*), (*c*) (cols. 23, 24, 25). Find the number of tithis which have intervened between the initial day of the year (Chaitra śukla pratipadâ), and the given tithi, by adding together the number of tithis (collective duration) up to the end of the month previous to the given one (col. 3, Table III.), and the number of elapsed tithis of the given month (that is the serial number of the given tithi reduced by one), taking into account the extra 15 days of the śukla paksha if the tithi belongs to the kṛishṇa paksha, and also the intervening intercalary month,³ if any, given in col. 8 (or 8*a*) of Table I. This would give the result in tithis. But days, not tithis, are required. To reduce the tithis to days, reduce the sum of the tithis by its 60th part,⁴ taking fractions larger than a half as one, and neglecting half or less. The result is the (*d*), the approximate number of days which have intervened since the initial day of the Hindu year. Write this number under head (*d*), and write under their respective heads, the (*w*), (*a*), (*b*), (*c*) for that number of days from Table IV. Add together the two lines of five quantities, but in the case of (*w*) divide the result by 7 and write only the remainder, in the case of (*a*) write only the remainder under 10000, and in the case of (*b*) and (*c*) only the remainder under 1000.⁵ Find separately the equations to arguments (*b*) and (*c*) in Tables VI. and VII. respectively, and add them to the total under (*a*). The sum (*t*) is the tithi-index, which, by cols. 2 and 3 of Table VIII., will indicate the tithi current at mean sunrise on the week-day found under (*w*). If the number of the tithi so indicated is not the same as that of the given one, but is greater or less by one (or by two in rare cases), subtract one (or two) from, or add one (or two) to, both (d) and (w);¹ subtract from, or add to, the (a) (b) (c) already found, their value for one (or two) days (Table IV.); add to (a) the equations for (b) and (c) (Tables VI. and VII.) and the sum (t) will then indicate the tithi. If this is the same as given (if not, proceed again as before till it corresponds), the (w) is its week-day, and the date shewn in the top line and side columns of Table IX. corresponding with the ascertained (d) is its equivalent date A.D. The year A.D. is found on the line of the given Chaitrâdi year in col. 5, Table I. Double figures are given in that column; if (d) is not greater than 365 in a common year, or 366 in a leap-year, the first, otherwise the second, of the double figures shows the proper A.D. year.

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¹ The initial days in cols. 13 and 19, Table I., belong to the first of the double years, A.D. given in col. 5.

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² It will be well for a beginner to take an example at once, and work it out according to the rule. After a little practice the calculations can be made rapidly.

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³ When the intercalary month is Chaitra, count that also. See Art. 99 above.

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⁴ This number is taken for easy calculation. Properly speaking, to convert tithis into days the 64th part should be subtracted. The difference does not introduce any material error.

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⁵ Generally with regard to (*w*), (*a*), (*b*), (*c*) in working addition sums, take only the remainder respectively over 7, 10000, 1000 and 1000; and in subtracting, if the sum to be subtracted be greater, add respectively 7, 10000, 1000 and 1000 to the figure above.

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¹ Thus far the process will give the correct result if there be no probability by the rule given below of the expunction (kshaya) or repetition (vriddhi) of a tithi shortly preceding or following; and the (d) and (w) arrived at by this stage will indicate by use of Table IX. the A.D. equivalent, and the week-day of the given tithi.

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140. For all practical purposes and for some ordinary religious purposes a tithi is connected with that week-day at whose sunrise it is current. For some religious purposes, however, and sometimes even for practical purposes also, a tithi which is current at any particular moment of a week-day is connected with that week-day. (See Art. 31 above.)

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141. In the case of an expunged tithi, the day on which it begins and ends is its week-day and equivalent. In the case of a repeated tithi, both the civil days at whose sunrise it is current,² are its week-days and equivalents.

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² For the definitions of expunged and repeated tithis see Art. 32 above.

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142. A clue for finding when a tithi is probably repeated or expunged. When the tithi-index corresponding to a sunrise is greater or less, within 40, than the ending index of a tithi, and when the equation for (b) (Table VI.) is decreasing, a repetition of the same or another tithi takes place shortly after or before that sunrise; and when the equation for (b) is increasing an expunction of a tithi (different from the one in question) takes place shortly before or after it.

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143. The identification of the date A.D. with the week-day arrived at by the above method, may be verified by Table XIII. The verification, however, is not in itself proof of the correctness of our results.

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144. To find the moment of the ending of a tithi. Find the difference between the (t) on the given day at sunrise and the (t) of the tithi-index which shews the ending point of that tithi (Table VIII.). With this difference as argument find the corresponding time either in ghaṭikâs and palas, or hours and minutes, according to choice, from Table X. The given tithi ends after the given sunrise by the interval of time so found. But this interval is not always absolutely accurate. (See Art. 82). If accuracy is desired add the (a) (b) (c) for this interval of time (Table V.) to the (a) (b) (c) already obtained for sunrise. Add as before to (a) the equations of (b) and (c) from Tables VI. and VII., and find the difference between the (t) thus arrived at and the (t) of the ending point of the tithi (Table VIII.). The time corresponding to that difference, found from Table X., will show the ending of the tithi before or after the first found time. If still greater accuracy is desired, proceed until (t) amounts exactly to the (t) of the ending point (Table VIII.) For ordinary purposes, however, the first found time, or at least that arrived at after one more process, is sufficiently accurate.

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145. The moment of the beginning of a tithi is the same as the moment of ending of the tithi next preceding it; and this can be found either by calculating backwards from the (t) of the same tithi, or independently from the (t) of the preceding tithi.

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146. The moment of beginning or ending of tithis thus found is in mean time, and is applicable to all places on the meridian of Ujjain, which is the same as that of Laṅkâ. If the exact mean time for other places is required, apply the correction given in Table XI., according to the rule given under that Table. If after this correction the ending time of a tithi is found to fall on the previous or following day the (d) and (w) should be altered accordingly.

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Mean time is used throughout the parts of the Tables used for these rules, and it may sometimes differ from the true, used, at least in theory, in Hindu pañchâṅgs or almanacks.

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The ending time of a tithi arrived at by these Tables may also somewhat differ from the ending time as arrived at from authorities other than the Sûrya Siddhânta which is used by us. The results, however, arrived at by the present Tables, may be safely relied on for all ordinary purposes.¹

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¹ See Arts. 36 and 37 in which all the points noted in this article are fully treated of.

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147. *N.B. i.* Up to 1100 A.D. both mean and true intercalary months are given in Table I. (see Art. 47 above). When it is not certain whether the given year is an expired or current year, whether it is a Chaitrâdi year or one of another kind, whether the given month is amânta or pûrṇimânta, and whether the intercalary month, if any, was taken true or mean, the only course is to try all possible years and months.

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*N.B. ii.* The results are all Old Style dates up to, and New Style dates from, 1753 A.D The New Style was introduced with effect from after 2nd September, 1752. Since only the initial dates of 1752 and 1753 are given, remember to apply the correction (+ 11 days) to any date between 2nd September, 1752, and 9th April, 1753, in calculating by the Hindu solar year, and between 2nd September, 1752, and 4th April, 1753, in calculating by the Hindu luni-solar year, so as to bring out the result in New Style dates A.D. The day of the week requires no alteration.

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*N.B. iii.* If the date A.D. found above falls after February 28th in a leap-year, it must be reduced by 1.

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*N.B. iv.* The Hindus generally use expired (gata) years, while current years are given throughout the Tables. For example, for Śaka year 1702 "expired" 1703 current is given.

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148. EXAMPLE I. Required the week-day and the A.D. year, month, and day corresponding to Jyeshṭha śukla pañchamî (5th), year Śârvari, Śaka year 1702 expired (1703 current), and the ending and beginning time of that tithi.

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The given year is Chaitrâdi (see N.B. ii., Table II., Part iii.). It does not matter whether the month is amânta or pûrṇimânta, because the fortnight belongs to Jyeshṭha by both systems (see Table II., Part i.). Looking to Table I. along the given current Śaka year 1703, we find that its initial day falls in A.D. 1780 (see note 1 to Art. 139), a leap-year, on the 5th April, Wednesday; and that d (col. 19), w (col. 20), a (col. 23), b (col. 24) and c (col. 25) are 96, 4, 1, 657 and 267 respectively. We write them in a horizontal line (see the working of the example below). From Table I., col. 8, we find that there is no added month in the year. The number therefore of tithis between Chaitra ś. 1 and Jyeshṭha ś. 5 was 64, viz., 60 up to the end of Vaiśâkha (see Table III., col. 3), the month preceding the given one, and 4 in Jyeshṭha. The sixtieth part of 64 (neglecting the fraction 4/60 because it is not more than half) is 1. Reduce 64 by one and we have 63 as the approximate number of days between Chaitra ś. 1 and Jyeshṭha ś. 5. We write this number under (d). Turning to Table IV. with the argument 63 we find under (w) (a) (b) (c) the numbers 0, 1334, 286, 172, respectively, and we write them under their respective heads, and add together the two quantities under each head. With the argument (b) (943) we turn to Table VI. for the equation. We do not find exactly the number 943 given, but we have 940 and 950 and must see the difference between the corresponding equation-figures and fix the appropriate figure for 943. The auxiliary table given will fix this, but in practice it can be easily calculated in the head. (The full numbers are not given so as to avoid cumbrousness in the tables.) Thus the equation for (b) (943) is found to be 90, and from Table VII. the equation for (c) is found to be 38. Adding 90 and 38 to (a) (1335) we get 1463, which is the required tithi-index (t). Turning with this to Table VIII., col. 3, we find by col. 2 that the tithi current was śukla 5, i.e., the given date. Then (w) 4, Wednesday, was its week-day; and the tithi was current at mean sunrise on the meridian of Ujjain on that week-day. Turning with (d) 159 to Table IX., we find that the equivalent date A.D. was 8th June; but as this was after 28th February in a leap-year, we fix 7th June, A.D. 1780, (see N.B. iii., Art. 147) as the equivalent of the given tithi. As (t) is not within 40 of 1667, the (t) of the 5th tithi (Table VIII.), there is no probability of an expunction or repetition shortly preceding or following (Art. 142). The answer therefore is Wednesday, June 7th, A.D. 1780.

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To find the ending time of the tithi. (t) at sunrise is 1463; and Table VIII., col. 3, shews that the tithi will end when (t) amounts to 1667. (1667—1463 =) 204 = (Table X.) 14 hours, 27 minutes, and this process shews us that the tithi will end 14 hours, 27 minutes, after sunrise on Wednesday, June 7th. This time is, however, approximate. To find the time more accurately we add the increase in (a) (b) (c) for 14 h. 27 m. (Table V.) to the already calculated (a) (b) (c) at sunrise; and adding to (a) as before the equations of (b) and (c) (Tables VI. and VII.) we find that the resulting (t) amounts to 1686. 1686—1667 = 19 = 1 hour and 21 minutes (Table X.). But this is a period beyond the end of the tithi, and the amount must be deducted from the 14 h. 27 m. first found to get the true end. The true end then is 13 h. 6 m. after sunrise on June 7th. This time is accurate for ordinary purposes, but for still further accuracy we proceed again as before. We may either add the increase in (a) (b) (c) for 13 h. 6 m. to the value of (a) (b) (c) at sunrise, or subtract the increase of (a) (b) (c) for 1 h. 21 m. from their value at 14 h. 27 m. By either process we obtain (t) = 1665. Proceed again. 1667—1665 = 2 = (Table X.) 9 minutes after 13 h. 6 m. or 13 h. 15 m. Work through again for 13 h. 15 m. and we obtain (t) = 1668. Proceed again. 1668—1667 = 1 = (Table X.) 4 minutes before 13 h. 15 m. or 13 h. 11 m. Work for 13 h. 11 m., and we at last have 1667, the known ending point. It is thus proved that 13 h. 11 m. after sunrise is the absolutely accurate mean ending time of the tithi in question by the Sûrya-Siddhânta.

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To find the beginning time of the given tithi. We may find this independently by calculating as before the (t) at sunrise for the preceding tithi, (in this case śukla 4th) and thence finding its ending time. But in the example given we calculate it from the (t) of the given tithi. The tithi begins when (t) amounts to 1333 (Table VIII.), or (1463—1333) 130 before sunrise on June 7th. 130 is (Table X.) 9 h. 13 m. Proceed as before, but deduct the (a) (b) (c) instead of adding, and (see working below) we eventually find that (t) amounts exactly to 1333 and therefore the tithi begins at 8 h. 26 m. before sunrise on June 7th, that is 15 h. 34 m. after sunrise on Tuesday the 6th. The beginning and ending times are by Ujjain or Laṅkâ mean time. If we want the time, for instance, for Benares the difference in longitude in time, 29 minutes, should be added to the above result (See Table XI.). This, however, does not affect the day.

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It is often very necessary to know the moments of beginning and ending of a tithi. Thus our result brings out Wednesday, June 7th, but since the 5th tithi began 15 h. 34 m. after sunrise on Tuesday, i.e., about 9 h. 34 m. p.m., it might well happen that an inscription might record a ceremony that took place at 10 p.m., and therefore fix the day as Tuesday the 5th tithi, which, unless the facts were known, would appear incorrect.

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From Table XII. we find that 7th June, A.D. 1780, was a Wednesday, and this helps to fix that day as current.

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We now give the working of EXAMPLE I.

Working of Example I

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(a) The day corresponding to Jyeṣṭha śukla 5th.

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| Śaka 1703 current, Chaitra śukla 1st, (Table I., cols. 19, 20, 23, 24, 25) | | 96 | 4 | 1 | 657 | 267 |
| Approximate number of days from Chaitra śukla 1st to Jyeṣṭha śuk. 5th, (64 tithis reduced by a 60th part, neglecting fractions, = 63) with its (w) (a) (b) (c) (Table IV.) | | | | 63 | 0 | 1334 | 286 | 172 |
| | | | | | 159 | 4 | 1335 | 943 | 439 |
| Equation for (b) (943) (Table VI.) | | | | | | | | 90 | |
| Do. (c) (439) (Table VII.) | | | | | | | | | 38 | |
| | | | | | | | | | 1463 = *t.* |

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(t) gives śukla 5th (Table VIII., cols. 2, 3) (the same as the given tithi). (d)—1, (*N. B. iii.*, Art. 147), or the number of days elapsed from January 1st, = 158 158 = June 7th (Table IX.). A.D. 1780 is the corresponding year, and 4 (w) Wednesday is the week-day of the given tithi. Answer.—Wednesday, June 7th, 1780 A.D.

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(b) The ending of the tithi Jyeṣṭha śuk. 5. (Table VIII.) 1667—1463 = 204 = (14 h. 10 m. + 0 h. 17 m.) = 14 h. 27 m. (Table X.). Therefore the tithi ends at 14 h. 27 m. after mean sunrise on Wednesday. For more accurate time we proceed as follows:

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| | *a.* | *b.* | *c.* |
| At sunrise on Wednesday (see above) | 1335 | 943 | 439 |
| For 14 hours (Table V.) | 198 | 21 | 2 |
| For 27 minutes, (Do.) | 6 | 1 | 0 |
| | 1539 | 965 | 441 |
| Equation for (b) (965) (Table VI.) | 109 | | |
| Do. (c) (441) (Do. VII.) | 38 | | |
| | 1686 = *t.* | | |

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1686—1667 (Table VIII.) = 19 = 1 h. 21 m.; and 1 h. 21 m. deducted from 14 h. 27 m. gives 13 h. 6 m. after sunrise on Wednesday as the moment when the tithi ended. This is sufficient for all practical purposes. For absolute accuracy we proceed again.

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| | *a.* | *b.* | *c.* |
| For sunrise (as before) | 1335 | 943 | 439 |
| For 13 hours (Table V.) | 183 | 20 | 1 |
| For 6 minutes (Do.) | 1 | 0 | 0 |
| | 1519 | 963 | 440 |
| Equation for (b) (963) (Table VI.) | 108 | | |
| Do. (c) (440) (Do. VII.) | 38 | | |
| | 1665 = *t.* | | |

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| | *a.* | *b.* | *c.* |
| 1667-1665=2=9 m. after 13 h. 6 m.=13 h. 15 h. | | | |
| Again for sunrise (*as before*) . . . . . . . | 1335 | 943 | 439 |
| For 13 hours (Table V.) . . . . . . . | 183 | 20 | 1 |
| For 15 minutes (Do.) . . . . . . . | 4 | 0 | 0 |
| | ---: | ---: | ---: |
| | 1522 | 963 | 440 |
| Equation for (*b*) (963) . . . . . . . . . . | 108 | | |
| Equation forDo.r(*c*) (440) . . . . . . . . . | 38 | | |
| | ---: | | |
| | 1668=t. | | |
| 1668-1667=1=4 m. before 13 h. 15 m.=13 h. 11 m. | | | |
| Again for sunrise (*as before*) . . . . . . . | 1335 | 943 | 439 |
| For 13 hours (Table V.) . . . . . . . | 183 | 20 | 1 |
| For 11 minutes (Do.) . . . . . . . | 3 | 0 | 0 |
| | ---: | ---: | ---: |
| | 1521 | 963 | 440 |
| Equation for (*b*) (963) . . . . . . . . . . | 108 | | |
| Equation forDo.r(*c*) (440) . . . . . . . . . | 38 | | |
| | ---: | | |
| Actual end of the tithi . . . . . . . . . | 1667=t. | | |

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Thus 13 h. 11 m. after sunrise is the absolutely accurate ending time of the tithi.

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(*c*) *The beginning of the tithi, Jyeshṭha śuk. 5.* Now for the beginning. 1463 (the original t. as found)-1333 (beginning of the tithi, (Table VIII.))=130=(Table X.)(7 h. 5 m.+2 h.8 m.)=9 h. 13 m.; and we have this as the point of time before sunrise on Wednesday when the tithi begins.

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| | *a.* | *b.* | *c.* |
| For sunrise (*as before*) . . . . . . . . . . | 1335 | 943 | 439 |

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| | *a.* | *b.* | *c.* | | | |
| For 9 h. (Table V.) . . . . . | 127 | 14 | 1 | | | |
| For 13 m. (Do.) . . . . . | 3 | 0 | 0 | | | |
| | ---: | ---: | ---: | | | |
| Deduct . . . . . . . | 130 | 14 | 1 . . . | 130 | 14 | 1 |
| | | | | ---: | ---: | ---: |
| | | | | 1205 | 929 | 438 |
| Equation for *b.* (929) . . . . . . . . . . | | | | 79 | | |
| Equation forDo.rc. (438) . . . . . . . . . | | | | 37 | | |
| | | | | ---: | | |
| | | | | 1321=t. | | |

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(The beginning of the tithi) 1333-1321=12=Table X.) 51 m. after the above time (9 h 13 m.), and this gives 8 h. 22 m. before sunrise. We proceed again.

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| | *a.* | *b.* | *c.* |
| For 9 h. 13 m. before sunrise (*found above*) . . . . | 1205 | 929 | 438 |
| Plus for 51 minutes (Table V.) . . . . . | 12 | 1 | 0 |
| | ---: | ---: | ---: |
| | 1217 | 930 | 438 |
| Equation for *b.* (930) . . . . . . . . . | 80 | | |
| Equation forDo.rc. (438) . . . . . . . . . | 37 | | |
| | ---: | | |
| | 1334=t. | | |

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1334—1333 = 1 = 4 m. before the above time (viz., 8 h. 22 m.) i.e., 8 h. 26 m. before sunrise. Proceed again.

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| | *a.* | *b.* | *c.* |
| For 8 h. 22 m. before sunrise (*found above*) . . . . . | 1217 | 930 | 438 |
| Deduct for 4 m. (Table V.) . . . . . . . . . | 1 | 0 | 0 |
| | 1216 | 930 | 438 |
| Equation for *b.* (930) . . . . . . . . . . . | | 80 | |
| Do. *c.* (438) . . . . . | | | 37 |
| | | | 1333 = *t.* |

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The result is precisely the same as the beginning point of the tithi (Table VIII.), and we know that the tithi actually began 8 hours 26 minutes before sunrise on Wednesday, or at 15 h. 34 m. after sunrise on Tuesday, 6th June.

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EXAMPLE II. Required the week-day and equivalent A.D. of Jyeshṭha śuk. dasamī (10th) of the southern Vikrama year 1836 expired, 1837 current. The given year is not Chaitrādi. Referring to Table II., Parts ii., and iii., we find, by comparing the non-Chaitrādi Vikrama year with the Śaka, that the corresponding Śaka year is 1703 current, that is the same as in the first example. We know that the months are amānta.

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| State the figures for the initial day (Table I., cols. 19, 20, 23, 24, 25) | 96 | 4 | 1 | 657 | 267 |
| The number of intervened tithis down to end of Vaiśākha, 60, (Table III.) + the number of the given date minus 1, is 69; reduced by a 60th part = 68, and by Table IV. we have . . . . . . | | | 68 | 5 | 3027 | 468 | 186 |
| | | | 164 | 2 | 3028 | 125 | 453 |
| Equation for (*b*) 125 (Table VI.) . . . . . . . . | | | | | 239 | |
| Do. (*c*) 453 (Table VII.) . . . . . . . . . . | | | | | 42 | |
| | | | | | 3309 = *t.* |

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(*d*) (164)—1 (*N. B. iii.*, Art. 147) = 163.

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The result, 3309, fixes the day as śukla 10th (Table VIII., cols. 2, 3), the same as given.

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Answer.—(By Table IX.) 163 = June 12th, 2 = Monday. The year is A.D. 1780 (Table II., Part ii.). The tithi will end at (3333—3309 = 24, or by Table X.) 1 h. 42 m. after sunrise, since 3309 represents the state of that tithi at sunrise, and it then had 24 lunation-parts to run. Note that this (*t*) (3309) is less by 24 than 3333, the ending point of the 10th tithi; that 24 is less than 40; and that the equation for (*b*) is increasing. This shows that an expunction of a tithi will shortly occur (*Art. 142.*)

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EXAMPLE III. Required the week-day and equivalent A.D. of Jyeshṭha śukla ekādaśī (11th) of the same Śaka year as example 2, *i.e.*, Ś. 1703 current.

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| See (Table I.) example 2 | 96 | 4 | 1 | 657 | 267 |
| Intervened days (to end of Vaiśâkha 59, + 11 given days—1) = 69. | | | | | |
| By Table IV. | 69 | 6 | 3366 | 504 | 189 |
| | 165 | 3 | 3367 | 161 | 456 |
| Equation for (*b*) (161) (Table VI.) | | | | 258 | |
| Do. (*c*) (456) (Table VII.) | | | | 43 | |
| | | | | 3668 | = *t.* |

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This figure (*t* = 3668) by Table VIII., cols. 2, 3, indicates śukla 12th. (*d*—1 (N.B. *iii.*, Art. 147) = 164 and Table IX. gives this as June 13th. The (*w*) is 3 = Tuesday. The year (Table II., Part iii.) is 1780 A.D.

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The figure of (*t*), 3668, shows that the 12th tithi and not the required tithi (11th) was current at sunrise on Tuesday; but we found in example 2 that the 10th tithi was current at sunrise on Monday, June 12th, and we therefore learn that the 11th tithi was expunged. It commenced 1 h. 42 min. after sunrise on Monday and ended 4 minutes before sunrise on Tuesday, 13th June.¹ The corresponding day answering to śukla 10th is therefore Monday, June 12th, and that answering to śukla 12 is Tuesday the 13th June.

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¹ This is shewn by (*t*) = 3668 at sunrise, the end being indicated by 3667. Difference 1 lunation-unit, or 4 minutes.

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EXAMPLE IV. Required the week-day and equivalent A.D. of the pûrṇimânta Âshâḍha kṛishṇa dvitîyâ (2) of the Northern Vikrama year 1837 expired, 1838 current. The northern Vikrama is a Chaitrâdi year, and so the year is the same as in the previous example, viz., A.D. 1780—1 (Table II., Part iii.). The corresponding amânta month is Jyeshṭha (Table II., Part i.). Work therefore for Jyeshṭha kṛishṇa 2nd in A.D. 1780—1 (Table I.).

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| See example 1 (Table I.) | 96 | 4 | 1 | 657 | 267 |
| 60 (coll. dur. to end Vaiś.) + 15 (for kṛishṇa fortnight) + 1 (given date minus 1) = 76 tithis = 75 days (as before); Table IV. gives | 75 | 5 | 5397 | 722 | 205 |
| | 171 | 2 | 5398 | 379 | 472 |
| Equation for (*b*) (379) | | | | 237 | |
| Do. (*c*) (472) | | | | 50 | |
| | | | | 5685 | = *t.* |

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(*d*—1 (N.B. *iii.*, Art. 147) = 170 = (Table IX.) 19th June. (2) = Monday. The year is 1780 A.D.

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So far we have Monday, 19th June, A.D. 1780. But the figure 5685 for (*t*) shows that kṛi. 3rd and not the 2nd was current at sunrise on Monday the 19th June. It commenced (5685—5667 = 18 =) 1 h. 17 m. before sunrise on Monday. (*t*) being greater, but within 40, than the ending point of kṛi. 2nd, and the equation for (*b*) decreasing, it appears that a repetition of a tithi will shortly follow (but not precede). And thus we know that Sunday the 18th June is the equivalent of kṛi. 2nd.

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EXAMPLE V. Required the week-day and equivalent A.D. of the amânta Jyeshṭha kṛi. 3rd of the Śaka year 1703 current, the same as in the last 4 examples.

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| (See example 1) . . . . . | 96 | 4 | 1 | 657 | 267 |
| 60 (coll. dur. to end Vaiś.) + 15 + 2 = 77 tithis = 76 days. (Table IV.) | 76 | 6 | 5736 | 758 | 208 |
| | 172 | 3 | 5737 | 415 | 475 |
| Equation for (*b*) (415) . . . . . | | | | 211 | |
| Do. (*c*) (475) . . . . . | | | | 51 | |
| | | | | 5999 | |

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This indicates kṛṣṇa 3rd, the same tithi as given. (*d*) - 1 = 171 = 20th June, 1780 A.D.

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From these last two examples we learn that kṛṣṇa 3rd stands at sunrise on Tuesday 20th as well as Monday 19th. It is therefore a repeated or *vṛddhi* tithi, and both days 19th and 20th correspond to it. It ends on Tuesday (6000 - 5999 = 1 =) 4 minutes after sunrise.

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EXAMPLE VI. Required the week-day and A.D. equivalent of Kārttika śukla 5th of the Northern Vikrama year 1833 expired (1834 current). (See example 2, page 70.)

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The given year is Chaitrādi. It matters not whether the month is amānta or pūrṇimānta because the given tithi is in the śukla fortnight. The initial day of the given year falls on (Table I., col. 19) 20th March (80), (col. 20) 4 Wednesday; and looking in Table I. along the line of the given year, we find in col. 8 that the month Bhādrapada was intercalated or added (*adhika*) in it. So the number of months which intervened between the beginning of the year and the given tithi was 8, one more than in ordinary year.

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| (Table I., cols. 19, 20, 23, 24, 25) . . . . . . . . . . | 80 | 4 | 9841 | 54 | 223 |
| (Coll. dur.) 240 + 4 = 244 = 240 days (Table IV.,) . . . . | 240 | 2 | 1272 | 710 | 657 |
| | 320 | 6 | 1113 | 764 | 880 |
| Equation for (*b*) (764) . . . . . | | | | 0 | |
| Do. (*c*) (880) . . . . . | | | | 102 | |
| | | | | 1215 | = *t.* |

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This indicates, not kṛ. 5 as given, but kṛ. 4 (Table VIII.)

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Adding 1 to (*d*) and (*w*) (see Rule above, Art. 139) . . . . . . . 321 0

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*a* - 1 (N.B. iii., Art. 147) 320 = (Table IX.) Nov. 16th, A.D. 1776. 0 = Saturday.

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(*t*) being not within 40 of the ending point of the tithi there is no probability of a repetition or expunction shortly preceding or following, and therefore Saturday the 16th November, 1776 A.D., is the equivalent of the given tithi.

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EXAMPLE VII. Required the week-day and A.D. equivalent of amānta Māgha kṛṣṇa 1st of Kali 4923 expired, 4924 current. (See example 4, page 71.)

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The given year is Chaitrādi. Looking in Table I. along the line of the given year, we see that its initial day falls on 24th March (83), 1822 A.D., 1 Sunday, and that (col. 8) the month (7) Āśvina was intercalated and (10) Pausha expunged. So that, in counting, the number of intervened months is the same, viz., 10, as in an ordinary year, Māgha coming after Pausha.

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| (Table I., cols. 19, 20, 23, 24, 23) . . . . . . . . . . | 83 | 1 | 212 | 899 | 229 |
| (Coll. dur.) 300 + 15 (śukla paksha) + (1-1=) 0 = 315 tithis = 310 | | | | | |
| days. By (Table IV.) . . . . . | 310 | 2 | 4976 | 250 | 849 |
| | 393 | 3 | 5188 | 149 | 78 |
| Equation for (*b*) (149) (Table VI.) . . . . . . . . . . | | | 252 | | |
| Do. (*c*) (78) (Table VII.) . . . . . | | | 32 | | |
| | | | 5472 = *t*. | | |

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The figure 5472 indicates (Table VIII.) kṛi. 2nd, *i.e.*, not the same as given (1st), but the tithi following. We therefore subtract 1 from (*d*) and (*w*) (Art. 139) making them 392 and 2.

c8.s2.t1.47

Since (*t*) is not within 40 of the ending point of the tithi, there is no probability of a *kshaya* or *vṛiddhi* shortly following or preceding. (*w*) 2 = Monday. 392 = (Table IX.) 27th January. And therefore 27th January, A.D. 1823, Monday, is the equivalent of the given tithi.

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EXAMPLE VIII. Required the week-day and the A.D. equivalent of śukla 13th of the Tuḷu month Puntelu, Kali year 4853 expired, 4854 current, “Aṅgiras samvatsara” in the luni-solar or southern 60-year cycle. (See example 5, page 72.)

c8.s2.t1.49

The initial day (Table I.) is Old Style 5th March (65), A.D. 1752, a leap-year, (5) Thursday; and Āshāḍha was intercalated. The Tuḷu month Puntelu corresponds to the Sanskrit Pausha (Table II., Part ii.), ordinarily the 10th, but now the 11th, month on account of the intercalated Āshāḍha.

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| | *d.* | *w.* | *a.* | *b.* | *c.* |
| (Table I., cols. 19, 20, 23, 24, 25) . . . . . . . . . . | 65 | 5 | 39 | 777 | 213 |
| (Coll. dur.) 300 + 12 (given tithi minus 1) = 312 tithis = 307 days | | | | | |
| (Table IV.) . . . . . | 307 | 6 | 3960 | 142 | 840 |
| | 372 | 4 | 3999 | 919 | 53 |
| Equation for (*b*) (919) . . . . . | | | 71 | | |
| Do. (*c*) (53) . . . . . | | | 40 | | |
| | | | 4110 = *t*. | | |

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The result, 4110, indicates śukla 13th, *i.e.*, the same tithi as that given. (*d*)-1 (N.B. *iii.*, Art. 147) = 371 = (by Table IX.) January 6th, A.D. 1753.

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We must add 11 days to this to make it a New Style date, because it falls after September 2nd, 1752, and before 4th April, 1753, the week-day remaining unaltered (see N.B. *ii.*, Art. 147), and 17th January, 1753 A.D., is therefore the equivalent of the given date.

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(B.) *Conversion of Hindu solar dates into dates A.D.*

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149. To calculate the week-day and the equivalent date A.D. Turn the given year into a Meshādi Kali, Śaka, or Vikrama year, and the name of the given month into a sign-name, if they are not already given as such, and find the corresponding year A.D. by the aid of columns 1 to 5, Table I., and Table II., Parts ii., and iii. Looking in Table I. along the line of the Meshādi year so obtained, write down in a horizontal line the following three quantities corresponding to the commencement of that (Meshâdi) year, viz., (*d*) the date-indicator given in brackets after the day and month A.D. in col. 13, (*w*) the week-day number (col. 14), and the time—either in ghaṭikâs and palas, or in hours and minutes as desired—of the Mesha saṅkrânti according to the *Ârya-Siddhânta* (cols. 15, or 17). For a Bengali date falling between A.D. 1100 and 1900, take the time by the *Sûrya-Siddhânta* from cols. 15*a* or 17*a*. When the result is wanted for a place not on the meridian of Ujjain, apply to the Mesha saṅkrânti time the correction given in Table XI. Under these items write from Table III., cols. 6, 7, 8, or 9 as the case may be, the collective duration of time from the beginning of the year up to the end of the month preceding the given one—days under (*d*), week-day under (*w*), and hours and minutes or ghaṭikâs and palas under *h. m.*, or *gh. p.* respectively. Add together the three quantities. If the sum of hours exceeds 24, or if the sum of ghaṭikâs exceeds 60, write down the remainder only, and add one each to (*w*) and (*d*). If the sum of (*w*) exceeds 7, cast out sevens from it. The result is the time of the astronomical beginning of the current (given) month. Determine its civil beginning by the rules given in Art. 28 above.

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When the month begins civilly on the same day as, on the day following, or on the third day after, the saṅkrânti day, subtract 1 from, or add 0, or 1, to both (*d*) and (*w*), and then to each of them add the number of the given day, casting out sevens from it in the case of (*w*). (*w*) is then the required week-day, and (*d*) will show, by Table IX., the A.D. equivalent of the given day.

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*N.B. i.* When it is not certain whether the given year is Meshâdi or of another kind, or what rule for the civil beginning of the month applies, all possible ways must be tried.

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*N.B. ii.* See *N.B. ii.*, *iii.*, *iv.*, Art. 147, under the rules for the conversion of luni-solar dates.

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Example IX. Required the week-day and the date A.D. corresponding to (Tamil) 18th Puraṭṭâśi of Rudhirodgârin, Kali year 4904 expired, (4905 current). (See example 7, p. 73.)

c8.s2.t1.59

The given year, taken as a solar year, is Meshâdi. The month Puraṭṭâḍi, or Puraṭṭâśi, corresponds to Kanyâ (Table II., Part ii.), and the year is a Tamil (Southern) one, to which the *Ârya Siddhânta* is applicable (*see Art. 21*). Looking in Table I. along the line of the given year, we find that it commenced on 11th April (col. 13), A.D. 1803, and we write as follows:—

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| | *d.* | *w.* | *h.* | *m.* |
| (Table I., cols. 13, 14, 17) . . . . . | 101 | 2 | 10 | 7 |
| (Table III., col. 7) collective duration up to the end of Siṁha . . . . . . . . | 156 | 2 | 10 | 28 |
| | --- | --- | --- | --- |
| | 257 | 4 | 20 | 35 |

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This shows that the Kanyâ saṅkrânti took place on a (4) Wednesday, at 20 h. 35 m. after sunrise, or 2.35 a.m. on the European Thursday. (Always remember that the Hindu week-day begins at sunrise.) The month Kanyâ, therefore, begins civilly on Thursday.¹ (*Rule 2(a), Art. 28.*) We add, therefore 0 to (*d*) and (*w*) 0 0

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¹ It would have so begun if the saṅkrânti occurred at 7 p.m. on the Wednesday, or at any time after sunset (6 p.m.)

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Add 18, the serial number of the given day, to (*d*) and, casting out sevens from the same figure, 18, add 4 to (*w*) . . . . . . . . . . 18 4

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--- --- Then (*w*) = 1, *i.e.*, Sunday, and 275 = (Table IX.) 2nd October. 275 1 *Answer.*—Sunday, 2nd October, 1803 A.D.

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Example X. Required the week-day and A.D. date corresponding to the 20th day of the Bengali (solar) month Phâlguna of Śaka 1776 expired, 1777 current, at Calcutta.

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The year is Meshâdi and from Bengal, to which the *Sûrya Siddhânta* applies (*see Art. 21*). The Bengâli month Phâlguna corresponds to Kumbha (Table II., Part ii.). The year commenced on 11th April, 1854, A.D. (Table I.).

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| | *d.* | *w.* | *h.* | *m.* |
| (Table I., cols. 13, 14, 17*a*) . . . . . . . . . | 101 | 3 | 17 | 13 |
| Difference of longitude for Calcutta (Table XI.) . . . . . | | | + 50 | |
| Collective duration up to the end of Makara (Table III., col. 9.) | 305 | 4 | 2 | 2 |
| | — | — | — | — |
| | 406 | 0 | 20 | 5 |

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This result represents the moment of the astronomical beginning of Kumbha, which is after midnight on Saturday, for 20 h. 5 m. after sunrise is 2.5 a.m. on the European Sunday morning. The month, therefore, begins civilly on Monday (*Art.* 28, *Rule 1 above*).

c8.s2.t1.69

| Add, therefore, 1 to (*d*) and (*w*) . . . . . . . . . . | 1 | 1 |
| Add 20 (given day) to (*d*), and, casting out sevens from 20, add 6 to (*w*) . . . . . | 20 | 6 |
| | — | — |
| 0 = Saturday, 427 = 3rd March (Table IX.) . . . . | 427 | 0 |

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*Answer.*—Saturday, 3rd March, A.D. 1855.

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EXAMPLE XI. Required the week-day and A.D. date corresponding to the Tinnevelly Âṇḍu 1024, 20th day of Âvaṇi. (See example 8, p. 73.)

c8.s2.t1.72

The year is South Indian. It is not Meshâdi, but Siṁhâdi. Its corresponding Śaka year is 1771 current; and the sign-name of the month corresponding to Âvaṇi is Siṁha (Table I., and Table II., Parts ii., and iii.) The Śaka year 1771 commenced on 11th April (102), A.D. 1848 (a leap-year), on (3) Tuesday. Work by the *Ârya-Siddhânta* (*Art. 21*).

c8.s2.t1.73

| | *d.* | *w.* | *h.* | *m.* |
| (Table I., cols. 13, 14, 17) . . . . . . . . . . | 102 | 3 | 1 | 30 |
| Collective duration up to the end of Karka . . . . . . . | 125 | 6 | 9 | 38 |
| | — | — | — | — |
| | 227 | 2 | 11 | 8 |

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The month begins civilly on the same day by one of the South Indian systems (Art. 28, Rule 2, *a*); therefore subtract 1 from both (*d*) and (*w*).

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| | 1 | 1 |
| | — | — |
| | 226 | 1 |
| Add 20, the serial number of the given day, to (*d*) and (less sevens) to (*w*) . . . . . | 20 | 6 |
| | — | — |
| | 246 | 0 |
| Deduct 1 for 29th February (*N.B. ii.*, Art. 149 and *N.B. iii.*, Art. 147) | 1 | |
| | — | |
| | 245 | |

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o = Saturday. 245 = (Table IX.) Sept. 2nd. Answer.—Saturday, September 2nd, 1848 A.D.

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EXAMPLE XII. Required the week-day and A.D. date corresponding to the South Malayâḷam Aṇḍu 1024, 19th Chiṅgam. (The calculations in Example xi. shew that the South-Malayâḷam month Chiṅgam began civilly one day later (Art. 28, Rule 2b). Therefore the Tamil 20th Avaṇi was the 19th South-Malayâḷam.)

c8.s2.t1.78

Referring to Table II., Part ii., we see that the date is the same as in the last example.

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EXAMPLE XIII. Required the week-day and A.D. date corresponding to the North Mala-yâḷam Aṇḍu 1023, 20th Chiṅgam.

c8.s2.t1.80

Referring to Table II., Part ii., we see that the date is the same as in the last two examples.

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(C.) Conversion into dates A.D. of tithis which are coupled with solar months.

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150. Many inscriptions have been discovered containing dates, in expressing which a tithi has been coupled, not with a lunar, but with a solar month. We therefore find it necessary to give rules for the conversion of such dates.

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Parts of two lunar months corresponding to each solar month are noted in Table II., Part ii., col. 14. Determine by Art. 119, or in doubtful cases by direct calculation made under Arts. 149 and 151, to which of these two months the given tithi of the given fortnight belongs, and then proceed according to the rules given in Art. 139.

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It sometimes happens that the same solar month contains the given tithi of both the lunar months noted in Table II., Part ii., col. 14, one occurring at the beginning of it and the other at the end. Thus, suppose that in a certain year the solar month Mesha commenced on the luni-solar tithi Chaitra śukla ashṭami (8th) and ended on Vaiśâkha śukla daśamî (10th). In this case the tithi śukla navamî (9th) of both the lunar months Chaitra and Vaiśâkha fell in the same solar month Mesha. In such a case the exact corresponding lunar month cannot be determined unless the vâra (week-day), nakshatra, or yoga is given, as well as the tithi. If it is given, examine the date for both months, and after ascertaining when the given details agree with the given tithi, determine the date accordingly.

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EXAMPLE XIV. Required the A.D. year, month, and day corresponding to a date given as follows;—"Śaka 1187, on the day of the nakshatra Rohiṇî, which fell on Saturday the thirteenth tithi of the second fortnight in the month of Mithuna."¹

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¹ This date is from an actual inscription in Southern India. (See Ind. Ant., XXII., p. 219).

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It is not stated whether the Śaka year is expired or current. We will therefore try it first as expired. The current year therefore is 1188. Turning to Table I. we find that its initial day, Chaitra śukla 1st, falls on 20th March (79), Friday (6), 1265. From Table II., Part ii., col. 14, we find that parts of the lunar months Jyeshṭha and Âshâḍha correspond to the solar month Mithuna. The Mesha saṅkrânti in that year falls on (Table I., col. 13) 25th March, Wednesday, that is on or about Chaitra śukla shashṭhî (6th), and therefore the Mithuna saṅkrânti falls on (about) Jyeshṭha śukla daśamî (10th) and the Karka saṅkrânti on (about) Âshâḍha śukla dvâdaśî (12th) (see Art. 119). Thus we see that the thirteenth tithi of the second fortnight in the solar month of Mithuna of the given date must belong to amânta Jyêshṭha.

c8.s2.t1.88

| | *d.* | *w.* | *a.* | *b.* | *c.* |
| S. 1188, Chaitra ś. 1st (Table I., cols. 19, 20, 23, 24, 25) | 79 | 6 | 287 | 879 | 265 |
| Approximate number of days from Ch. ś. 1st to Jyesh. kṛi. 13th (87 tithis reduced by 60th part = 86) with its (*w*) (*a*) (*b*) (*c*) (Table IV.) | 86 | 2 | 9122 | 121 | 235 |
| | 165 | 1 | 9409 | 0 | 500 |
| Equation for (*b*) (*o*) (Table VI.) | | | 140 | | |
| Do. (*c*) (500) Table VII.) | | | 60 | | |
| | | | 9609 | = *t.* | |

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The resulting number 9609 fixes the tithi as kṛishṇa 14th (Table VIII., cols. 2, 3), *i.e.*, the tithi immediately following the given tithi. There is no probability of a *kshaya* or *vṛiddhi* shortly before or after this (Art 142). Deduct, therefore, 1 from (*d*) and (*w*)

c8.s2.t1.90

164 = (Table IX.) 13th June; o = Saturday. Answer.—13th June, 1265 A.D., Saturday, (as required). ¹

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¹ This problem is easier than its converse, the number of intervening days here being certain.

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(D.) *Conversion of dates A.D.* ² *into Hindu luni-solar dates.*

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² If the Rule I(*a*) in Art. 104 (Table II., Part iii.) be applied, this latter part of the rule necessarily follows.

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151. Given a year, month, and date A.D., write down in a horizontal line (*w*) the weekday number, and (*a*), (*b*), (*c*) (Table I., cols. 20, 23, 24, 25) of the initial day (Chaitra ś. 1) of the Hindu Chaitrâdi (*Śaka*) year corresponding to the given year; remembering that if the given date A.D. is earlier than such initial day, the (*w*) (*a*) (*b*) (*c*) of the previous Hindu year ³ must be taken. Subtract the date-indicator of the initial date (Table I., col. 19) from the date number of the given date (Table IX.), remembering that, if the initial day of the previous Hindu year has been taken, the number to be taken from Table IX. is that on the right-hand side, and not that on the left (see also *N.B. ii.* below). The remainder is the number of days which have intervened between the beginning of the Hindu year and the required date. Write down, under their respective heads, the (*w*) (*a*) (*b*) (*c*) of the number of intervening days from Table IV., and add them together as before (see rules for conversion of luni-solar dates into dates A.D.). Add to (*a*) the equation for (*b*) and (*c*) (Tables VI., VII.) and the sum (*t*) will indicate the tithi (Table VIII.) at sunrise of the given day; (*w*) is its week-day. To the number of intervening days add its sixtieth ⁴ part. See the number of tithis next lower than this total ⁵ (Table III., col. 3) and the lunar month along the same line (col. 2). Then this month is the month preceding the required month, and the following month is the required month.

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³ A 59th part, or more properly 63rd, should be added, but by adding a 60th, which is more convenient, there will be no difference in the ultimate result. Neglect the fraction half or less, and take more than half as equivalent to one.

c8.s2.t1.96

⁴ This total is the approximate number of tithis which have intervened. When it is the same as, or very near to, the number of tithis forming the collective duration up to the end of a month (as given in col. 3, Table III.), there will be some doubt about the required month; but this difficulty will be easily solved by comparing together the resulting tithi and the number of tithis which have intervened.

c8.s2.t1.97

When there is an added month in the year, as shown along the line in col. 8 or 8*a* of Table I., if it comes prior to the resulting month, the month next preceding the resulting month

c8.s2.t1.98

It is found by actual calculation under Art. 156 that the given nakshatra falls on the same date, and therefore we know that the above result is correct.

c8.s2.t1.99

is the required month; if the added month is the same as the resulting month, the date belongs to that added month itself; and if the resulting month comes earlier than the added month, the result is not affected.

c8.s2.t1.100

When there is a suppressed month in the year, if it is the same as, or prior to, the resulting month, the month next following the resulting month is the required month. If it is subsequent to the resulting month the result is not affected. If the resulting month falls after both an added and suppressed month the result is unaffected.

c8.s2.t1.101

From the date in a Chaitrâdi year thus found, any other Hindu year corresponding to it can be found, if required, by reference to Table II., Parts ii., and iii.

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The tithi thus found is the tithi corresponding to the given date A.D.; but sometimes a tithi which is current at any moment of an A.D. date may be said to be its corresponding tithi.

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*N.B. i.* See *N.B. ii.*, Art. 147; but for "+ 11" read "- 11".

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*N.B. ii.* If the given A.D. date falls in a leap-year after 29th February, or if its date-number is more than 365 (taken from the right-hand side of Table IX.) and the year next preceding it was a leap-year, add 1 to the date-number before subtracting the date-indicator from it.

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*Example XV.* Required the tithi and month in the Śaka year corresponding to 7th June, 1780 A.D.

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The Śaka year corresponding to the given date is 1703 current. Its initial day falls on (4) Wednesday, 5th April, the date-indicator being 96.

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| | *w.* | *a.* | *b.* | *c.* |
| (Table I., cols. 20, 23, 24, 25) . . . . . . | 4 | 1 | 657 | 267 |
| 7th June = . . . . . . 158 (Table IX.) | | | | |
| Add . . . . . . . + 1 for leap-year (*N.B. ii.*) | | | | |
| | 159 | | | |
| Deduct . . . . . . 96 the (*d*) of the initial date | | | | |
| | — | | | |
| Days that have intervened 63. By Table IV. 63 = . . . | 0 | 1334 | 286 | 172 |
| | 4 | 1335 | 943 | 439 |
| Equation for (*b*) (943) (Table VI.) . . . . . . | | | 90 | |
| Do. (*e*) (439) (Table VII.) . . . . . . | | | 38 | |
| | — | | — | |
| | 4 | 1463 | = *t.* | |

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Śukla 5th (Table VIII.) is the required tithi, and (4) Wednesday is the week-day. Now 63 + 63/60 = 64 3/60. The next lowest number in col. 3, Table III., is 60, which shows Vaiśâkha to be the preceding month. Jyeṣṭha is therefore the required month.

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*Answer.*—Śaka 1703 current, Jyeṣṭha śukla 5th, Wednesday.

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If the exact beginning or ending time of the tithi is required, proceed as in example I above (*Art. 148*.)

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We have seen in example I above (*Art. 148*) that this Jyeṣṭha 5th ended, and śukla 6th commenced, at 13 h. 11 m. after sunrise on the given date; and after that hour śukla 6th corresponded with the given date. Śukla 6th therefore may be sometimes said to correspond to the given date as well as śukla 5th.

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*Example XVI.*—Required the tithi and month in the southern Vikrama year corresponding to 12th September, 1776 A.D.

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The Śaka year corresponding to the given date is 1699 current. Its initial date falls on 20th March (80), 4 Wednesday, A.D. 1776. Bhâdrapada was intercalated in that year.

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| | *w.* | *a.* | *b.* | *c.* |
| (Table I., cols. 20, 23, 24, 25) . . . . . | 4 | 9841 | 54 | 223 |
| 12 September = . . . 255 (Table IX.) | | | | |
| Add . . . . . . 1 for leap-year (*N.B. ii.*) | | | | |
| 256 | | | | |
| Deduct . . . . . 80 the (*d*) of the initial day. | | | | |
| Days that have intervened 176 = (Table IV.) . . . . . . . . . | 1 | 9599 | 387 | 482 |
| | --- | ---- | --- | --- |
| | 5 | 9440 | 441 | 705 |
| Equation for (*b*) (441) (Table VI.) . . . . . . . . . . | | 191 | | |
| \ \ \ \ \ Do. \ \ \ \ (*c*) (705) (Table VII.) . . . . . . . . . . | | 118 | | |
| | --- | ---- | --- | --- |
| | 5 | 9749 | = *t*. | |

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This indicates (Table VIII.) kṛishṇa 30th (amâvâsyâ, or new moon day), Thursday. The intervening tithis are 176 + 176/60 = 179. The number next below this in col. 3, Table III., is 150, and shows that Śrâvaṇa preceded the required month. But Bhâdrapada was intercalated this year and it immediately followed Śrâvaṇa. Therefore the resulting tithi belongs to the intercalated or adhika Bhâdrapada.

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*Answer.*—Adhika Bhâdrapada kṛi: 30th of Śaka 1699 current, that is adhika Bhâdrapada kṛi. 30th of the Southern Vikrama Kârttikâdi year 1833 current, 1832 expired. (Table II., Part ii.).

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EXAMPLE XVII. Required the Telugu and Tuḷu equivalents of December 1st, 1822 A.D. The corresponding Telugu or Tuḷu Chaitrâdi Śaka year is 1745 current. Âśvina was intercalary and Pausha was expunged (col. 8, Table I.). Its initial date falls on 24 March (83), A.D. 1822, (1) Sunday.

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| | *w.* | *a.* | *b.* | *c.* |
| Table I., cols. 20, 23, 24, 25) . . . . . | 1 | 212 | 899 | 229 |
| 1st December = . . 335 (Table IX.) | | | | |
| Deduct . . . . . 83 (The *d.* of the initial day) | | | | |
| Days that have intervened 252 = (Table IV.) . . . . . . . . . | 0 | 5335 | 145 | 690 |
| | --- | ---- | --- | --- |
| | 1 | 5547 | 44 | 919 |
| Equation for (*b*) (44) (Table IV.) . . . . . . . . . . | | 180 | | |
| \ \ \ \ \ Do. \ \ \ \ (*c*) (919) (Do. VII.) . . . . . . . . . | | 90 | | |
| | --- | ---- | --- | --- |
| The results give us kṛishṇa 3, Sunday (1), (Table VIII.) . . . | 1 | 5817 | = *t*. | |

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252 + 252/60 = 256. The number next below 256 in col. 3, Table III., is 240, and shews that Kârttika preceded the required month, and the required month would therefore be Mârgaśīrsha. But Āśvina, which is prior to Mārgaśīrsha, was intercalated. Kārttika therefore is the required month. Pausha was expunged, but being later than Kārttika the result is not affected.

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Answer.—Sunday, Kārttika (Telugu), or Jārde (Tuḷu) (Table II., Part ii.), kṛ. 3rd of the year Chitrabhānu, Śaka 1745 (1744 expired), Kali year 4923 expired.

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EXAMPLE XVIII. Required the tithi and pūrṇimānta month in the Śaka year corresponding to 18th January, 1541 A.D.

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The given date is prior to Chaitra śukla 1 in the given year. We take therefore the initial day in the previous year, A.D. 1540, which falls on Tuesday the 9th March (69). The corresponding Śaka year is 1463 current.

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| | *w.* | *a.* | *b.* | *c.* |
| (Table I., cols. 20, 23, 24, 25) . . . . . . . . | 3 | 108 | 756 | 229 |
| 18th January = . . 383 (Table IX.) | | | | |
| Add for leap-year . . . 1 (N.B. ii., latter part.) | | | | |
| | | | 384 | | |
| Deduct . . . . . 69 (The *d.* of the initial day.) | | | | |
| No. of intervening days. . 315 = (by Table IV.) . . . . . | 0 | 6669 | 432 | 862 |
| | | 3 | 6777 | 188 | 91 |
| Equation for (*b*) (188) (Table VI.) . . . . . . . | | | 269 | |
| Do. (*c*) (91) (Do. VII.) . . . . . . . . | | | 28 | |
| | | 3 | 7074 | = *t.* | |

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The result gives us kṛishṇa 7th, Tuesday (3) (Table VIII.).

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