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The Indian Calendar, with Tables for the Conversion of Hindu and Muhammadan into A.D. Dates, and vice versa c8: Part IV. Use of the Tables

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

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English

Contents

Part IV. Use of the Tables

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118. The Tables now published may be used for several purposes, of which some are enumerated below.

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(1) For finding the year and month of the Christian or any Indian era corresponding to a given year and month in any of the eras under consideration.

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(2) For finding the samvatsara of the sixty-year cycle of Jupiter, whether in the southern (luni-solar) or northern (mean-sign) scheme, and of the twelve-year cycle of Jupiter, corresponding to the beginning of a solar (Meshâdi) year, or for any day of such a year.

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(3) For finding the added or suppressed months, if any, in any year.

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But the chief and most important use of them are;

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(4) The conversion of any Indian date—luni-solar (tithi) or solar—into the corresponding date A.D. and vice versâ, from A.D. 300 to 1900, and finding the week-day of any such date;

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(5) Finding the karaṇa, nakshatra, and yoga for any moment of any Indian or European date, and thereby verifying any given Indian date;

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(6) Turning a Hindu solar date into a luni-solar date, and vice versâ.

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(7) Conversion of a Muhammadan Hijra date into the corresponding date A.D., and vice versâ. This is fully explained in Part V. below.

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119. (1) For the *first purpose* Table I., cols. 1 to 5, or Table II., must be used, with the explanation given in Part III. above. For eras not noted in these two Tables see the description of them given in Art. 71. In the case of obscure eras whose exact nature is not yet well known, the results will only be approximate.

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(N.B.—It will be observed that in Table II., Part ii., portions of two solar months or of four¹ Christian months are made to correspond to a lunar month and vice versâ, and therefore that if this Table only be used the results may not be exact).

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¹ Of course only two in a single case, but four during the entire period of 1600 years covered by our Tables.

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The following note, though not yielding very accurate results, will be found useful for finding the corresponding parts of lunar and solar months. The tithi corresponding to the Mesha-saṅkrānti can be approximately² found by comparing its English date (Table I., col. 13) with that of the luni-solar Chaitra śukla 1st (Table I., col. 19); generally the saṅkrāntis from Vṛishabha to Tulâ fall in successive lunar months, either one or two tithis later than the given one. Tulâ falls about 10 tithis later in the month than Mesha; and the saṅkrāntis from Vṛischika to Mîna generally fall on the same tithi as that of Tulâ. Thus, if the Mesha saṅkrānti falls on śukla pañchamî (5th) the Vṛishabha saṅkrānti will fall on śukla shashṭhî (6th) or saptamî (7th), the Mithuna saṅkrānti on śukla ashṭamî (8th) or navamî (9th), and so on.

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² The exact tithi can be calculated by Arts. 149 and 151.

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120. (2) For the *samvatsara* of the southern sixty-year cycle see col. 6 of Table I., or calculate it by the rule given in Art. 62. For that of the sixty-year cycle of Jupiter of the mean sign system, according to Sûrya Siddhânta calculations, current at the beginning of the solar year, *i.e.*, at the true (or apparent) Mesha saṅkrānti, see col. 7 of Table I.; and for that current on any day in the year according to either the Sûrya or Ârya Siddhântas, use the rules in Art. 59. To find the samvatsara of the twelve-year cycle of the mean-sign system corresponding to that of the Jupiter sixty-year cycle see Table XII.

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121. (2) *To find the added or suppressed month* according to the Sûrya Siddhânta by the true (apparent) system see col. 8 of Table I. throughout; and for an added month of the mean system according to either the Original or Present Sûrya Siddhântas, or by the Ârya Siddhânta, see col. 8a of Table I. for any year from A. D. 300 to 1100.

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122. (4) *For conversion of an Indian date into a date A. D. and vice versâ, and to find the week day of any given date*, we give below three methods, with rules and examples for work.

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123. The first method A (Arts. 135, 136), the invention of Mr. T. Lakshmiah Naidu of Madras, is a method for obtaining approximate results without any calculation by the careful use of mere eye-tables, viz., Tables XIV. and XV. These, with the proper use of Table I., are alone necessary. But it must never be forgotten that this result may differ by one, or at the utmost two, days from the true one, and that it is not safe to trust to them unless the era and bases of calculation of the given date are clearly known. (See Art. 126 below.)

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124. By our second method B (Arts. 137, 138), which follows the system established by Mr. W. S. Kṛiṣhṇasvâmi Naidu of Madras, author of *South Indian Chronological Tables* (Madras 1889), and which is intended to enable an approximation to be made by a very simple calculation, a generally accurate correspondence of dates can be obtained by the use of Tables I., III., and IX. The calculation is so easy that it can be done in the head after a little practice. It is liable to precisely the same inaccuracies as method A, neither more nor less.

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125. Tables II. and III. will also be sometimes required for both these methods.

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126. The result obtained by either of these methods will thus be correct to within one or two days, and as often as not will be found to be quite correct; but there must always be an element of uncertainty connected with their use. If, however, the era and original bases of calculation of the given date are certainly known, the result arrived at from the use of these eye-Tables may be corrected by the week-day if that has been stated; since the day of the month and year will not be wrong by more than a day, or two at the most, and the day of the week will determine the corresponding civil day. Suppose, for instance, that the given Hindu date is Wednesday, Vaiśâkha śukla 5th, and it is found by method A or method B that the corresponding day according to European reckoning fell on a Thursday, it may be assumed, presuming that all other calculations for the year and month have been correctly made, that the civil date A.D. corresponding to the Wednesday is the real equivalentof Vaiśâkha śukla 5th. But these rough methods should never be trusted to in important cases. For a specimen of a date where the bases of calculation are not known see example xxv., Art. 160 below.

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127. When Tables XIV. and XV. are once understood (and they are perfectly simple) it will probably be found advisable to use method A in preference to method B.

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128. As already stated, our method "C" enables the conversion of dates to be made with precise accuracy; the exact moments of the beginning and ending of every tithi can be ascertained; and the corresponding date is obtained, simultaneously with the week-day, in the required reckoning.

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129. The week-day for any European date can be found independently by Table XIII., which was supplied by Dr. Burgess.

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131¹ (5) *To find the karaṇa, nakshatra, or yoga current on any Indian or European date; and to verify any Indian date.*

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¹ Art. 130 has been omitted.

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Method C includes calculations for the karaṇa, nakshatra and yoga current at any given moment of any given day, as well as the instants of their beginnings and endings; but for this purpose, if the given date is other than a tithi or a European date, it must be first turned into one or the other according to our rules (Art. 139 to 152.)

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132. It is impossible, of course, to verify any tithi or solar date unless the week-day, nakshatra, karaṇa, or yoga, or more than one of these, is also given; but when this requirement is satisfied our method C will afford proof as to the correctness of the date. To verify a solar date it must first be turned into a tithi or European date. (Art. 134 or 149.)

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133. For an explanation of the method of calculating tithis and half-tithis (karaṇas) see Art. 107 above. Our method of calculation for nakshatras and yogas requires a little more explanation. The moon's nakshatra (Arts. 8, 38) is found from her apparent longitude. By our method C we shew how to find t (= the difference of the apparent longitudes of sun and moon), and equation¹ c (= the solar equation of the centre) for any given moment. To obtain (t) the sun's apparent longitude is subtracted from that of the moon, so that if we add the sun's apparent longitude to (t) we shall have the moon's apparent longitude. Our (c) (Table I., last column) is the sun's mean anomaly, being the mean sun's distance from his perigee. If we add the longitude of the sun's perigee to (c), we have the sun's mean longitude, and if we apply to this the solar equation of the centre (+ or -) we have the sun's apparent longitude.² According to the Sûrya-Siddhânta the sun's perigee has only a very slight motion, amounting to 3' 5".8 in 1600 years. Its longitude for A.D. 1100, the middle of the period covered by our Tables, was 257° 15' 55".7 or .7146,3 of a circle, and therefore this may be taken as a constant for all the years covered by our Tables.

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¹ Equation c is the equation in Table VII.

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² Reference to the diagram in Art. 108 will make all this plain, if PSE be taken as the sun's mean anomaly, and ESE' the equation of the centre, PSE' + longitude of the sun's perigee being the sun's true or apparent longitude.

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Now, true or apparant sun = mean sun + equation of centre. But we have not tabulated in Table VII., col. 2, the exact equation of the centre; we have tabulated a quantity (say x) the value of which is expressed thus;— x = 60,4—equation of centre (see Art. 108). So that equation of centre = 60,4—x. Hence, apparent sun = mean sun + 60,4—x. But mean sun = c + perigee, (which is 7146,3 in tithi-indices.) = c + 7146,3. Hence apparent sun (which we call s) = c + 7146,3 + 60,4—x. = c + 7206,7—x; or, say, = c + 7207—x where x is, as stated, the quantity tabulated in col. 2, Table VII.

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(c) is expressed in 1000ths, while 7207 and the solar equation in Table VII. are given in 10000ths of the circle, and therefore we must multiply (c) by 10. t + s = apparent moon = n (the index of a nakshatra.) This explains the rule given below for work (Art. 156).

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For a yoga, the addition of the apparent longitude of the sun (s) and moon (n) is required. s + n = y (the index of a yoga.) And so the rule in Art. 159.

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134. (6) To turn a solar date into its corresponding luni-solar date and vice versâ. First turn the given date into its European equivalent by either of our three methods and then turn it into the required one. The problem can be worked direct by anyone who has thoroughly grasped the principle of these methods.

Method A. Approximate Computation of Dates by Use of the Eye-Table

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This is the method invented by Mr. T. Lakshmiah Naidu, nephew of the late W. S. Krishnasvâmi Naidu of Madras, author of "South Indian Chronological Tables."

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Results found by this method may be inaccurate by as much as two days, but not more. If the era and bases of calculation of the given Hindu date are clearly known, and if the given date mentions a week-day, the day found by the Tables may be altered to suit it. Thus, if the Table yield result Jan. 10th, Thursday, but the inscription mentions the week-day as "Tuesday", then Tuesday, January 8th, may be assumed to be the correct date A.D. corresponding to the given Hindu date, if the principle on which the Hindu date was fixed is known. If not, this method must not be trusted to.

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135. (A.) Conversion of a Hindu solar date into the corresponding date A. D. Work by the following rules, always bearing in mind that when using the Kaliyuga or Śaka year Hindus usually give the number of the expired year, and not that astronomically current, (e.g., Kaliyuga 4904 means in full phrase "after 4904 years of the Kaliyuga had elapsed")—but when using the name of the cyclic year they give that of the one then current. All the years given in Table I. are current years. The Table to work by is Table XIV.

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Rule I. From Table I., cols. 1 to 7, and Table II., as the case may be, find the year (current) and its initial date, and week-day (cols. 13, 14, Table I.). But if the given Hindu date belongs to any of the months printed in italics at the head of Table XIV., take the next following initial date and week day in cols. 13, 14 of Table I. The months printed in the heading in capitals are the initial months of the years according to the different reckonings.

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Rule II. For either of the modes of reckoning given at the left of the head-columns of months, find the given month, and under it the given date.

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Rule III. From the given date so found, run the eye to the left and find the week-day in the same line under the week-day number found by Rule I. This is the required week-day.

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Rule IV. Note number in brackets in the same line on extreme left.

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Rule V. In the columns to left of the body of the Table choose that headed by the bracket-number so found, and run the eye down till the initial date found by Rule I. is obtained.

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Rule VI. From the month and date in the upper columns (found by Rule II.) run the eye down to the point of junction (vertical and horizontal lines) of this with the initial date found by Rule V. This is the required date A. D.

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Rule VII. If the date A. D. falls on or after 1st January in columns to the right, it belongs to the next following year. If such next following year is a leap-year (marked by an asterisk in Table I.) and the date falls after February 28th in the above columns, reduce the date by one day.

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N.B.—The dates A.D. obtained from this Table for solar years are Old Style dates up to 8th April, 1753, inclusive.

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EXAMPLE. Find date A.D. corresponding to 20th Paṅguni of the Tamil year Rudhirodgâri, Kali 4904 expired.

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By Rule I. Kali 4905 current, 2 (Monday), 11th April, 1803. " " II. Tamil Panguni 20. " " III. (under " 2") Friday. " " IV. Bracket-number (5). " " V. [Under (5)]. Run down to April 11th. " " VI. (Point of junctions) March 31st. " " VII. March 30th. (1804 is a leap year.) Answer.—Friday, March 30th, 1804 N.S. (See example 11, p. 74.)

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(B.) Conversion of a date A.D. into the corresponding Hindu solar date. (See Rule V., method B, Art. 137, p. 70.) Use Table XIV.

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Rule I. From Tables I., cols. 1 to 7 and 13, 14, and Table II., as the case may be, find the Hindu year, and its initial date and week-day, opposite the given year A.D. If the given date falls before such initial date, take the next previous Hindu year and its initial date and week-day A.D.

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Rule II. From the columns to the left of the body of Table XIV. find that initial date found by Rule I. which is in a line, when carrying the eye horizontally to the right, with the given A.D. date, and note point of junction.

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Rule III. Note the bracket-figure at head of the column on left so selected.

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Rule IV. From the point of junction (Rule II.) run the eye vertically up to the Hindu date-columns above, and select that date which is in the same horizontal line as the bracket-figure on the extreme left corresponding with that found by Rule III. This is the required date.

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Rule V. If the given date falls in the columns to the right after the 28th February in a leap-year (marked with an asterisk in Table I.), add 1 to the resulting date.

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Rule VI. From the date found by Rule IV. or V., as the case may be, carry the eye horizontally to the week-day columns at the top on the left, and select the day which lies under the week-day number found from Table I. (Rule I.). This is the required week-day.

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Rule VII. If the Hindu date arrived at falls under any of the months printed in italics in the Hindu month-columns at head of Table, the required year is the one next previous to that given in Table I. (Rule I.).

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EXAMPLE. Find the Tamil solar date corresponding to March 30th, 1804 (N.S.).

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(By Rule I.) Rudhirodgâri, Kali 4905 current. 2 (Monday) April 11th. (March 30th precedes April 11th.)

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(By Rules II., III.) The point of junction of March 30th (body of Table), and April 11th, (columns on left) is under "(4)." Other entries of April 11th do not correspond with any entry of March 30th).

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(By Rule IV.) The date at the junction of the vertical column containing this "March 30th" with "(4)" horizontal is 19th Paṅguni.

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(By Rule V.) (1804 is a leap-year) 20th Paṅguni.

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(By Rule VI.) Under "2" (Rule I.), Friday.

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Answer.—Friday, 20th Paṅguni, of Rudhirodgâri, Kali 4905 current. (See example 15, p. 76.

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136. (A.) Conversion of a Hindu luni-solar date into the corresponding date A.D. Work by the following rules, using Tables XV.A., and XV.B.

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Rule I. From Table I. find the current year and its initial day and week-day in A.D. reckoning, remembering that if the given Hindu date falls in one of the months printed in italics at the head of Table XV. the calculation must be made for the next following A.D. year. (The months printed in capitals are the initial months of the years according to the different reckonings enumerated in the column to the left.)

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Rule II. (a.) Find the given month, and under it the given date, in the columns at the head of Table XV., in the same line with the appropriate mode of reckoning given in the column to the left. The dates printed in black type are kṛishṇa, or dark fortnight, dates.

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(b.) In intercalary years (cols. 8 to 12, 8a to 12a of Table I.), if the given month is itself an adhika mâsa (intercalary month), read it, for purpose of this Table, as if it were not so; but if the given month is styled nija, or if it falls after a repeated month, but before an expunged one (if any), work in this Table for the month next following the given one, as if that and not the given month had been given. If the given month is preceded by both an intercalated and a suppressed month, work as if the year were an ordinary one.

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Rule III. From the date found by Rule II. carry the eye to the left, and find the week-day in the same horizontal line, but directly under the initial week-day found by Rule I.

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Rule IV. Note the number in brackets on the extreme left opposite the week-day last found.

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Rule V. In the columns to the left of the body of the Table choose that headed by the bracket-number so found, and run the eye down till the initial date found by Rule I. is obtained.

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Rule VI. From the Hindu date found by Rule II. run the eye down to the point of junction, (vertical and horizontal lines) of this date with the date found by Rule V. The result is the required date A.D.

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Rule VII. (a.) If the date A.D. falls on or after January 1st in the columns to the right, it belongs to the next following year A.D.

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(b.) If it is after February 28th in a leap-year (marked by an asterisk in col. 5, Table I.) reduce the date by one day, except in a leap-year in which the initial date (found in Table I.) itself falls after February 28th.

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(c.) The dates obtained up to April 3rd, A.D. 1753, are Old Style dates.

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EXAMPLE. To find the date A. D. corresponding to amânta Kârttika kṛiṣhṇa 2nd of Kali 4923 expired, Śaka 1744 expired, Kârttikâdi Vikrama 1878 expired, Chaitrâdi Vikrama 1879 expired (1880 current), "Vijaya" in the Bṛihaspati cycle, "Chitrabhânu" in the luni-solar 60-year cycle.

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(By Rule I.) (Kali 4924 current), 1 Sunday, March 24th, 1822.

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(By Rule II.) (Kârttika, the 8th month, falls after the repeated month, 7 Âśvina, and before the suppressed month, 10 Pausha), Mârgaśîrsha kṛiṣhṇa 2nd.

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(By Rule III.) (Under "1"), 1 Sunday.

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(By Rule IV.) Bracket-number (1).

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(By Rule V.) Under (1) run down to March 24th (Rule I.)

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(By Rule VI.) (Point of junction) December 1st.

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Answer.—Sunday, December 1st, 1822.

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(B.) Conversion of a date A. D. into the corresponding luni-solar Hindu date. (See Rule V. method B, p. 67 below). Use Tables XV.A., XV.B.

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Rule I. From Table I. find the Hindu year, and its initial date and week-day, using also Table II., Parts ii., iii. If the given date falls before such initial date take the next previous Hindu year, and its initial date and week-day.

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Rule II. In the columns to the left of the body of Table XV. note the initial date found by Rule I., which is in the same horizontal line with the given date in the body of the Table.

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Rule III. Carrying the eye upwards, note the bracket-figure at the head of the initial date-column so noted.

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Rule IV. From the given date found in the body of the Table (Rule II.) run the eye upwards to the Hindu date-columns above, and select the date which is in the same horizontal line as the bracket-figure in the extreme left found by Rule III. This is the required Hindu date.

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Rule V. Note in Table I. if the year is an intercalary one (cols. 8 to 12, and 8a to 12a). If it is so, note if the Hindu month found by Rule IV. (a) precedes the first intercalary month, (b) follows one intercalated and one suppressed month, (c) follows an intercalated, but precedes a suppressed month, (d) follows two intercalated months and one suppressed month. In cases (a) and (b) work as though the year were a common year, i.e., make no alteration in the date found by Rule IV. In cases (c) and (d) if the found month immediately follows the intercalated month, the name of the required Hindu month is to be the name of the intercalated month with the prefix "nija," and not the name of the month actually found; and if the found month does not immediately follow the intercalated month, then the required Hindu month is the month immediately preceding the found month. If the found month is itself intercalary, it retains its name, but with the prefix "adhika." If the found month is itself suppressed, the required month is the month immediately preceding the found month.

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Rule VI. If the given date A.D. falls after February 29th in the columns to the right, in a leap-year (marked with an asterisk in Table I.), add 1 to the resulting Hindu date.

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Rule VII. From the date found by Rule IV. carry the eye horizontally to the week-day columns on the left, and select the day which lies under the initial week-day number found by Rule I. This is the required week-day.

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Rule VIII. If the Hindu date arrived at falls under any of the months printed in italics in the Hindu month-columns at head of the table, the required year is the one next previous to that given by Table I. (Rule I. above.)

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EXAMPLE. Find the Telugu luni-solar date corresponding to Sunday, December 1st, 1822. (By Rule I.) A.D. 1822—23, Sunday, March 24th, Kali 4923 expired, Śaka 1744 expired, Chitrabânu samvatsara in the luni-solar 60-year or southern cycle reckoning, Vijaya in the northern cycle. (By Rules II., III.) (Bracket-figure) 1. (By Rule IV.) Mârgaśîrsha kṛiṣhṇa 2nd. (By Rule Vc.) (Âśvina being intercalated and Pausha suppressed in that year), Kârttika kṛiṣhṇa 2nd. (By Rule VI.) The year was not a leap-year. (By Rule VII.) Sunday. (By Rule VIII.) Does not apply. Answer.—Sunday, Kârttika kṛiṣhṇa 2nd, Kali 4923 expired, Śaka 1744 expired. (This can be applied to all Chaitrâdi years.) (See example 12 below, p. 75.)

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Method B.

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APPROXIMATE COMPUTATION OF DATES BY A SIMPLE PROCESS.

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This is the system introduced by Mr. W. S. Krishṇasvâmi Naidu of Madras into his "South-Indian Chronological Tables."

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137. (A.) Conversion of Hindu dates into dates A.D. (See Art. 135 above, para. 1.) Rule I. Given a Hindu year, month and date. Convert it if necessary by cols. 1 to 5 of Table I., and by Table II., into a Chaitrâdi Kali or Śaka year, and the month into an amânta month. (See Art. 104.) Write down in a horizontal line (d) the date-indicator given in brackets in col. 13 or 19 of Table I., following the names of the initial civil day and month of the year in question as so converted, and (w) the week-day number (col. 14 or 20) corresponding to the initial date A.D. given in cols. 13 or 19. To both (d) and (w) add, from Table III., the collective duration of days from the beginning of the year as given in cols. 3a or 10 as the case may be, up to the end of the month preceding the given month, and also add the number of given Hindu days in the given month minus 1. If the given date is luni-solar and belongs to the kṛiṣhṇa paksha, add 15 to the collective duration and proceed as before.

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Rule II. From the sum of the first addition find in Table IX. (top and side columns)

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the required English date, remembering that when this is over 365 in a common year or 366 in a leap-year the date A.D. falls in the ensuing A.D. year.

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Rule III. From the sum of the second addition cut out sevens. The remainder shews the required day of the week.

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Rule IV. If the Hindu date is in a luni-solar year where, according to cols. 8 to 12, there was an added (adhika) or suppressed (kshaya) month, and falls after such month, the addition or suppression or both must be allowed for in calculating the collective duration of days; i.e., add 30 days for an added month, and deduct 30 for a suppressed month.

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Rule V. The results are Old Style dates up to, and New Style dates from, 1752 A.D. The New style in England was introduced with effect from after 2nd September, 1752. Since the initial dates of 1752, 1753 only 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, or 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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Rule VI. If the date A.D. found as above falls after February 29th in a leap-year, it must be reduced by one day.

(a) Luni-Solar Dates

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EXAMPLE 1. Required the A.D. equivalent of (luni-solar) Vaiśâkha śukla shashṭhî (6th), year Śârvari, Śaka 1702 expired, (1703 current).

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The A.D. year is 1780 (a leap-year). The initial date (d) = 5th April (96), and (w) = 4 Wednesday, (Table I., cols. 5, 19, 20).

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| | *d.* | *w.* |
| State this accordingly | 96 | 4 |
| Collective duration (Table III., col. 3a) | 30 | 30 |
| Given date (6)—1 | 5 | 5 |
| | --- | --- |
| | 131 | |
| | 1 (Rule VI.) | |
| | --- | --- |
| | 130 | 39÷7 = Rem. 4 |

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The result gives 130 (Table IX.) = May 10th, and 4 = Wednesday. The required date is therefore Wednesday, May 10th, A.D. 1780.

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EXAMPLE 2. Required the A.D. equivalent of (luni-solar) Kârttika śukla pañchamî (5th) Śaka 1698 expired (1699 current).

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The A.D. year is 1776, and the initial date is (d) = 20th March (80), (w) = Wednesday (4). This is a leap-year, and the Table shews us that the month (6) Bhâdrapada was intercalated. So there is both an adhika Bhâdrapada and a nija Bhâdrapada in this year, which compels us to treat the given month Kârttika as if it were the succeeding month Mârgaśîrsha in order to get at the proper figure for the collective duration.

c8.s1.t1.7

| | *d.* | *w.* |
| The given figures are . . . | 80 | 4 |
| Collective duration (Table III.) | 236 | 236 |
| for Mârgaśîrsha . . . | | |
| Given date (5)—1 . . . | 4 | 4 |
| | 320 | |
| | —1 | (Rule VI.) |
| | 319 | 244 ÷ 7 = Rem. 6. |

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319 = (Table IX.) November 15th. 6 = Friday *Answer.*—Friday, November 15th, A.D. 1776.

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EXAMPLE 3. Required the A.D. equivalent of Kârttika kṛishṇa pañchamî (5th) of the same luni-solar year.

c8.s1.t1.10

| | *d.* | *w.* |
| As before . . . . . . . | 80 | 4 |
| Collective duration (Table III., col. 3a.) | 236 | 236 |
| Given date (5 + 15)—1 . . . | 19 | 19 |
| | 335 | |
| | —1 | (Rule VI.) |
| | 334 | 259 ÷ 7, Rem. 0. |

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334 = (Table IX.) November 30th. 0 = Saturday. *Answer.*—Saturday, November 30th, A.D. 1776.

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EXAMPLE 4. Required the A.D. equivalent of Mâgha kṛishṇa pâdyami (1st) of K.Y. 4923 expired (4924 current). This corresponds (Table I., col. 5) to A.D. 1822, the Chitrabhânu samvatsara, and col. 8 shews us that the month Âśvina was intercalated (*adhika*), and the month Pausha suppressed (*kshaya*). We have therefore to add 30 days for the adhika month and subtract 30 days for the kshaya month, since Mâgha comes after Pausha. Hence the relative place of the month Mâgha remains unaltered,

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Table I. gives 24th March (83), (1) Sunday, as the initial day.

c8.s1.t1.14

| | *d.* | *w.* |
| Initial date . . . . . . | 83 | 1 |
| Collective duration (Table III., col. 3a) | 295 | 295 |
| Given date (1 + 15)—1 . . . | 15 | (Rule I.) 15 |
| | 393 | 311 ÷ 7, Rem. 3. |

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3 = Tuesday. 393 = January 28th of the following A.D. year (Table IX.). *Answer.*—Tuesday, January 28th, A.D. 1823.

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This is correct by the Tables, but as there happened to be an expunged tithi in Mâgha śukla, the first fortnight of Mâgha, the result is wrong by one day. The corresponding day was really Monday, January 27th, and to this we should have been guided if the given date had included the mention of Monday as the week-day. That is, we should have fixed Monday, January 27th, as the required day A.D. because our result gave Tuesday, January 28th, and we knew that the date fell on a Monday, EXAMPLE 5. Required the A.D. equivalent of Pauṣa śukla trayodaśī (13th) K.Y. 4853 expired, Aṅgiras saṃvatsara in luni-solar or southern reckoning. This is K. Y. 4854 current.

c8.s1.t1.17

The year (Table I., col. 5) is A.D. 1752, a leap-year. The initial date (cols. 19, 20) is 5th March (65), (5) Thursday. The month Āṣāḍha was intercalated. Therefore the given month (Pauṣa) must be treated, for collective duration, as if it were the succeeding month Māgha.

c8.s1.t1.18

| | *d.* | *w.* |
| Initial date . . . . . . . . . . | 65 | 5 |
| Collective duration (Table III., col. 3a) | 295 | 295 |
| Given date (13)—1 . . . . . . . | 12 | 12 |
| | 372 | |
| | —1 | (Rule VI) |
| | 371 | 312 ÷ 7, Rem. 4. |

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We must add eleven days to the amount 371 to make it a New Style date, because it falls after September 2nd, 1752, and before 4th April, 1753, (after which all dates will be in New Style by the Tables). 371 + 11 = 382 = January 17th (Table IX.). 4 = Wednesday.

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*Answer.*—Wednesday, January 17th, A.D. 1753.

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EXAMPLE 6. Required the A.D. equivalent of Vikrama saṃvatsara 1879 Āṣāḍha kṛṣṇa dvitīyā (2nd). If this is a southern Vikrama year, as used in Gujarāt, Western India, and countries south of the Narmadā, the year is Kārttikādi and amānta, *i.e.*, the sequence of fortnights makes the month begin with śukla 1st. The first process is to convert the date by Table II., Part iii., col. 3, Table II., Part ii., and Table I., into a Chaitrādi year and month. Thus—Āṣāḍha is the ninth month of the year and corresponds to Āṣāḍha of the following Chaitrādi Kali year, so that the given month Āṣāḍha of Vikrama 1879 corresponds to Āṣāḍha of Kali 4924. Work as before, using Table I. for Kali 4924. Initial date, 24th March (83), (1) Sunday.

c8.s1.t1.22

| | *d.* | *w.* |
| Initial date . . . . . . . . . . | 83 | 1 |
| Collective duration (Table III., col. 3a) | 89 | 89 |
| Given date (2 + 15)—1 . . . . . | 16 | 16 |
| | 188 | 106÷7 Rem. 1 |
| | 188 (Table IX.) = July 7th. 1 = Sunday. |

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*Answer.*—Sunday, July 7th, A.D. 1822.¹

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¹ This is actually wrong by one day, owing to the approximate collective duration of days (Table III., 3*a*) being taken as 89. It might equally well be taken as 88. If it is desired to convert tithis into days (p. 75, note 2) a 64th part should be subtracted. The collective duration of the last day of Jyeṣṭha in tithis is 90. 90 ÷ 64 = 1.40. 90 — 1.40 = 88.60. If taken as 88 the answer would be Saturday, July 6th, which is actually correct. This serves to shew how errors may arise in days when calculation is only made approximately.

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If the year given was a northern Vikrama year, as used in Mālwa, Benares, Ujjain, and countries north of the Narmadā, the Vikrama year is Chaitrādi and corresponds to the Kali 4923, except that, being pūrṇimānta, the sequence of fortnights differs (see Table II., Part i.). In such a case Āṣāḍha kṛṣṇa of the Vikrama year corresponds to Jyeṣṭha kṛṣṇa in amānta months, and we must work for Kali 4923 Jyeṣṭha kṛṣṇa 2nd. By Table I. the initial date is April 3rd (93), (3) Tuesday. The A.D. year is 1821—22.

c8.s1.t1.26

| | *d.* | *w.* |
| | 93 | 3 |
| Collective duration (Table III., col. 3*a*) | 59 | 59 |
| Given date (2 + 15) - 1 | 16 | 16 |
| | — | — |
| | 168 | 78 ÷ 7, Rem. 1. |

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168 = June 17th. 1 = Sunday. *Answer.*—Sunday, June 17th, A.D. 1821.

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(*b*) Solar Dates.

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EXAMPLE 7. Required the date A.D. corresponding to the Tamil (solar) 18th Puraṭṭāśi of Rudhirodgârin = K.Y. 4904 expired, or 4905 current. Table I., cols. 13 and 14, give (*d*) = April 11th (101), (*w*) = (2) Monday, and the year A.D. 1803.

c8.s1.t1.30

| | *d.* | *w.* |
| Initial date | 101 | 2 |
| Collective duration (Table III., col. 10) | 156 | 156 |
| Given date (18) - 1 | 17 | 17 |
| | — | — |
| | 274 | 175 ÷ 7, Rem. 0. |

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274 (Table IX.) gives October 1st. 0 = Saturday. *Answer.*—Saturday, October 1st, A.D. 1803.

c8.s1.t1.32

EXAMPLE 8. Required the equivalent A.D. of the Tinnevelly Aṇḍu 1024, 20th Āvaṇi. The reckoning is the same as the Tamil as regards months, but the year begins with Āvaṇi. Aṇḍu 1024 = K.Y. 4950. It is a solar year beginning (see Table I.) 11th April (102), (3) Tuesday, A.D. 1848 (a leap-year).

c8.s1.t1.33

| | *d.* | *w.* |
| Initial date | 102 | 3 |
| Tables II., Part ii., cols. 10 & 7, and III., col. 10. | 125 | 125 |
| Given date (20) - 1 | 19 | 19 |
| | — | — |
| | 246 | |
| | - 1 | (Rule VI.) |
| | — | — |
| | 245 | 147 ÷ 7, Rem. 0. |

c8.s1.t1.34

0 = Saturday; 245 = (Table IX.) September 2nd. *Answer.*—Saturday, September 2nd, A.D. 1848.

c8.s1.t1.35

EXAMPLE 9. Required the equivalent date A.D. of the South Malayāḷam Aṇḍu 1024, 20th Chiṅgam. The corresponding Tamil month and date (Table II., part ii., cols. 9 and 11) is 20th Āvaṇi K.Y. 4950, and the answer is the same as in the last example.

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EXAMPLE 10. Required the equivalent date A.D. of the North Malayāḷam (Kollam) Aṇḍu 1023, 20th Chiṅgam. This (Chiṅgam) is the 12th month of the Kollam Aṇḍu year which begins with Kanni. It corresponds with the Tamil 20th Āvaṇi K.Y. 4950 (Table II., Part ii., cols. 9, 12, and Table II., Part iii.), and the answer is similar to that in the two previous examples. [The difference in the years will of course be noted. The same Tamil date corresponds to South Malayâḷam Aṇḍu 1024, 20th Chiṅgam, and to the same day of the month in the North Malayâḷam (Kollam) Aṇḍu 1023, the reason being that in the former reckoning the year begins with Chiṅgam, and in the latter with Kanni.]

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EXAMPLE II. Required the A.D. equivalent of the Tamil date, 20th Paṅguni of Rudhirodgârin, K.Y. 4905 current (or 4904 expired.)

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Table I. gives (d) 11th April (101), 1803 A. D. as the initial date of the solar year, and its week-day (w) is (2) Monday.

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| | *d.* | *w.* |
| Initial date . . . . . . . . . . . | 101 | 2 |
| Collective duration (Table III., col. 10) | 335 | 335 |
| Given date, (20)—I . . . . . . . | 19 | 19 |
| | 455 | |
| | —1 | (Rule VI.) |
| | 454 | 356 ÷ 7, Rem. 6. |

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6 = Friday; 454 (Table IX.) = March 30th in the following A.D. year, 1804. *Answer.*—Friday, March 30th, 1804. (See example I, above.)

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138. (B.) Conversion of dates A.D. into Hindu dates. (See Art. 135 above, par. 1.)

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Rule I. Given a year, month, and date A.D. Write down in a horizontal line (d) the date-indicator of the initial date [in brackets (Table I., cols. 13 or 19, as the case may be)] of the corresponding Hindu year required, and (w) the week-day number of that initial date (col. 14 or 20), remembering that, if the given date A.D. is earlier than such initial date, the (d) and (w) of the previous Hindu year must be taken. Subtract the date-indicator from the date number of the given A.D. date in Table IX., remembering that, if the previous Hindu year has been taken down, the number to be taken from Table IX. is that on the right-hand side of the Table and not that on the left. From the result subtract (Table III., col. 3a or 10) the collective-duration-figure which is nearest to, but lower than, that amount, and add 1 to the total so obtained; and to the (w) add the figure resulting from the second process under (d), and divide by 7. The result gives the required week-day. The resulting (d) gives the day of the Hindu month following that whose collective duration was subtracted.

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Rule II. Observe (Table I., cols. 8 or 8a) if there has been an addition or suppression of a month prior to the month found by Rule I. and proceed accordingly.

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An easy rule for dealing with the added and suppressed month is the following. When the intercalated month (Table I., col. 8 or 8a) precedes the month immediately preceding the one found, such immediately preceding month is the required month; when the intercalated month immediately precedes the one found, such immediately preceding month with the prefix "nija," natural, is the required month; when the intercalated month is the same as that found, such month with the prefix "adhika" is the required month. When a suppressed month precedes the month found, the required month is the same as that found, because there is never a suppression of a month without the intercalation of a previous month, which nullifies the suppression so far as regards the collective duration of preceding days. But if the given month falls after two intercalations and one suppression, act as above for one intercalation only.

c8.s1.t1.45

Rule III. See Art. 137 (A) Rule V. (p. 70), but subtract the eleven days instead of adding.

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Rule IV. If the given A.D. date falls in a leap-year after 29th February, or if its date-number (right-hand side of Table IX.) is more than 365, and the year next preceding it was a leap-year, add 1 to the date-number of the given European date found by Table IX., before subtracting the figure of the date-indicator

c8.s1.t1.47

Rule V. Where the required date is a Hindu luni-solar date the second total, if less than 15, indicates a śukla date. If more than 15, deduct 15, and the remainder will be a kṛiṣhṇa date. Kṛiṣhṇa 15 is generally termed kṛiṣhṇa 30; and often śukla 15 is called “pūrṇimā” (full-moon day), and kṛiṣhṇa 15 (or “30”) is called amāvāsyā (new-moon day).

c8.s1.t1.48

(a) Luni-Solar Dates.

c8.s1.t1.49

EXAMPLE 12. Required the Telugu or Tuḷu equivalent of December 1st, 1822. The luni-solar year began 24th March (83) on (1) Sunday (Table I., cols. 19 and 20.)

c8.s1.t1.50

| | *d.* | *w.* |
| (*d*) and (*w*) of initial date (Table I.) . . . . . 83 | 1 |
| (Table IX.) 1st December (335) . . . (335— 83 =) 252 | 252 |
| (Table III.) Collective duration to end of Kārttika | —236 | |
| Add 1 to remainder . . . . . . . . 16 + 1 = 17 | 253 ÷ 7, Rem. 1. |

c8.s1.t1.51

17 indicates a kṛiṣhṇa date. Deduct 15. Remainder 2. The right-hand remainder shews (1) Sunday.

c8.s1.t1.52

The result so far is Sunday Mārgaśīrsha kṛiṣhṇa 2nd. But see Table I., col. 8. Previous to this month Aśvina was intercalated. (The suppression of Pausha need not be considered because that month comes after Mārgaśīrsha.) Therefore the required month is not Mārgaśīrsha, but Kārttika; and the answer is Sunday Kārttika kṛiṣhṇa 2nd (Telugu), or Jarde (Tuḷu), of the year ChitRabhānu, K.Y. 4923 expired, Śaka 1744 expired. (See the example on p. 69.)

c8.s1.t1.53

(Note.) As in example 6 above, this date is actually wrong by one day, because it happened that in Kārttika śukla there was a tithi, the 12th, suppressed, and consequently the real day corresponding to the civil day was Sunday Kārttika kṛiṣhṇa 3rd. These differences cannot possibly be avoided in methods A and B, nor by any method unless the duration of every tithi of every year be separately calculated. (See example xvii., p. 92.)

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EXAMPLE 13. Required the Chaitrādi Northern Vikrama date corresponding to April 9th 1822. By Table I. A.D. 1822—23 = Chaitrādi Vikrama 1880 current. The reckoning is luni-solar. Initial day (*d*) March 24th (83), (*w*) 1 Sunday

c8.s1.t1.55

| | *d.* | *w.* |
| From Table I . . . . . 83 | 1 |
| (Table IX.) April 9th (99) . . . . . . 99—83 = 16 | 16 |
| Add . . . . . 1 | |
| | 17 | |
| For śukla dates . . . . . . . . . . —15 | |
| | 2 | 17 ÷ 7, Rem. 3. |

c8.s1.t1.56

This is Tuesday, amānta Chaitra kṛiṣhṇa 2nd.¹ But it should be converted into Vaiśākha kṛiṣhṇa 2nd, because the custom of beginning the month with the full-moon (Table II., Part i.).

c8.s1.t1.57

¹ The actual date was Tuesday, amānta Chaitra kṛiṣhṇa 3rd, the difference being caused by a tithi having been expunged in the śukla fortnight of the same month (see note to examples 6 and 12 above).

c8.s1.t1.58

Since the Chaitrâdi Vikrama year begins with Chaitra, the required Vikrama year is 1880 current, 1879 expired. But if the required date were in the Southern reckoning, the year would be 1878 expired, since 1879 in that reckoning does not begin till Kârttika.

c8.s1.t1.59

*(b) Solar Dates.*

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EXAMPLE 14. 1. Required the Tamil equivalent of May 30th, 1803 A.D. Table I. gives the initial date April 11th (101), and week-day number 2 Monday. *d.**w.* From Table I . . . . . 1012 (Table IX.) May 30th (150) . . . . . . . . 150—101 =4949 (Table III.) Collective duration to end of Śittirai (Mesha) .—31 —— 18 Add 1 . . . . . + 1 ———— 1951 ÷ 7, Rem. 2. The day is the 19th; the month is Vaiyâśi, the month following Śittirai; the week-day is (2) Monday. *Answer.*—Monday, 19th Vaiyâśi of the year Rudhirodgârin, K.Y. 4904 expired, Śaka 1725 expired.

c8.s1.t1.61

EXAMPLE 15. Required the Tamil equivalent of March 30th, 1804. The given date precedes the initial date in 1804 A.D. (Table I., col. 13) April 10th, so the preceding Hindu year must be taken. Its initial day is 11th April (101), and the initial week-day is (2) Monday. 1804 was a leap-year.

c8.s1.t1.62

*d.**w.* From Table I . . . . . 1012 (Table IX.) (March 30th) 454 + 1 for leap-year, 455—101 =354354 .(Table III., col. 10) Collective duration to end of / Mâśi = Kumbha (Table II., Part ii.) . . . . .—335 —— 19 Add 1 . . . . . + 1 ———— 20356 ÷ 7, Rem. 6.

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*Answer.*—Friday 20th Paṅguṇi of the year Rudhirodgârin K.Y. 4904 expired, Śaka 1725 expired. (See the example on p. 67.)

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EXAMPLE 16. Required the North Malayâḷam Âṇḍu equivalent of September 2nd, 1848. Work as by the Chaitrâdi year. The year is solar. 1848 is a leap-year.

c8.s1.t1.65

*d.**w.* From Table I . . . . . 1023 (Table IX.) September 2nd (245) + 1 for leap year . . . . . . . . . . 246—102 =144144 Coll. duration to end of Karka . . . . . . .—125 —— 19 Add 1 . . . . . + 1 ———— 20147 ÷ 7, Rem. 0

c8.s1.t1.66

Answer.—Saturday 20th Chingam. This is the 12th month of the North Malayâḷam Aṇḍu which begins with Kanni. The year therefore is 1023.

c8.s1.t1.67

If the date required had been in South Malayâḷam reckoning, the date would be the same, 20th Chingam, but as the South Malayâḷis begin the year with Chingam as the first month, the required South Malayâḷam year would be Aṇḍu 1024.

Method C. Exact Calculation of Dates

c8.s2.2

(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.

c8.s2.15

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.

c8.s2.16

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.

c8.s2.17

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.

c8.s2.18

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.

c8.s2.19

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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