The Burmese & Arakanese Calendars/Chapter 4

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The Burmese & Arakanese Calendars
by Alfred Macdonald Bulteel Irwin

Chapter 4: Methods of Calculation

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4637055The Burmese & Arakanese Calendars — Chapter 4: Methods of CalculationAlfred Macdonald Bulteel Irwin

CHAPTER IV.

Methods of Calculation.

52. The rules of both Makaranta and Thandeikta commence with the calculation of hnit-bo and yet-bo. Hnit-bo are the total lunations, with fraction, and total days, with fraction, expired at the moment when a given solar year ends, and the day of the week on which solar new year's day falls. Yet-bo are similar figures for midnight of any given day in any given year.

53. In this chapter the following symbols are used:—

R	represents Ratha or Hnit kywin.
H	represents„ Haragon or Thawana.
Ky	represents„ Kyammat.
D	represents„ Didi.
Y	represents„ Yetlun.
Ka	represents„ Kaya.
W	represents„ Awaman.
S. M.	represents„ Solar months.
L. M.	represents„ Lunar months.
Ad	represents„ Adimath.
AT	represents„ Adimath thetha.
T	represents„ Thokdadein.
m	represents„ the expired months of the current year, counting from and including Tagu.
n	represents„ the given day of the month.

HNIT BO.

54. Thandeikta calculations start from 1100 B. E. Thagayit − 1100 = Ratha. Makaranta calculations stare from 0 B. E., Poppasaw's epoch. But there is a second Makaranta method, starting from 798 B. E., the epoch of Mohnyin, King of Ava. Thagayit − 798 = Ratha by this method.

55. By Thandeikta, the total days expired at the end of the year R are

${\frac {292207R+{\frac {R}{193}}+17742}{800}}$

292000 is the kymmat of 365 days, 207 is the kyammat of 15 nayr 31 bizana and 30 kaya. 1/193 is the kyammat of 1 kaya and 24 anukaya (.56 second) which forms the difference between the Thandeikta and Makaranta years. By the formula given above the difference is disregarded from 1100 to 1292 B. E. In 1293 the end of the solar year is postponed by one kyammat unit (108 seconds or 270 kaya). The constant 17742 kyammat equals 22 yet 10 nayi and 39 bizana, the time which elapsed from 1099 Tabaung Lagwè midnight to the moment when the mean sun entered Meiktha, when the 1100th year expired.

56. By Makaranta the fraction 1/193 is disregarded, and omitted altogether. R = Thagayit, and the total days expired are

${\frac {292207R+373}{800}}$

The constant 373 equals 27 nayi 58 bizana and 30 kaya, the time which elapsed from Kali Yug 3738 Tabaung Lagwè midnight to the moment when the mean sun entered Meiktha and the year 0 of Poppasaw's era (Kali Yug 3739) commenced. By the second Makaranta method, in which Ratha = Thagayit − 798, the constant is 8759.

57. The Haragon or Thawana is the quotient of

${\frac {292207R+{\frac {R}{193}}+17742}{800}}$

plus 1, because the haragon includes new year's day. The remainder is the portion of new year's day which belongs to the old year. 800 minus the remainder is the Ata Ne Kyammat.

58. The day of the week, by Thandeikta, is the remainder of ${\frac {H-2}{7}}$ because Ata Ne in 1100 B. E. was Saturday, and its haragon was 23. 21 divided by 7 gives remainder 0, which represents Saturday. By either of the Makaranta methods the remainder of ${\frac {H}{7}}$ is the day of the week. In 0 B. E. Ata Ne was Sunday, and its haragon was 1. In 798 B. E. Ata Ne was Wednesday, and its haragon was 11; remainder 4.

59. The kaya, or excess of didi over days, is by Thandeikta the quotient of

${\frac {11H+{\frac {R}{25}}+176}{692}}$

and the remainder is the awaman. The ratio 11 : 692 is the difference between unity and 703 : 692, which is an approximation to 400,750,020 : 394,479,457, the Surya Siddhanta ratio of the length of a day to the length of a didi. (Para. 34). The fraction ${\frac {R}{25}}$ represents the 0.000004731 day, by which a Thandeikta mean lunation exceeds a Makaranta mean lunation. (See para. 39). The constant 176/692 day = 15 nayi, 15 bizana and 45 kaya, the time which elapsed from the moment of mean new moon to midnight of Tabaung Lagwè 1099 B. E. By Makaranta the kaya is the quotient of

${\frac {11H+650}{692}}$

the constant 650/692 day being = 55 nayi, 28 bizana and 35.5 kaya, the time which elapsed from the moment of mean new moon to midnight of Tabaung Lagwè 3738 Kali Yug. By the second Makaranta method the constant is 48/692.

60. Haragon + kaya = total didi. Didi divided by 30 gives quotient Sandra Matha, and remainder yet lun or epact at midnight of solar new year's day expressed in whole didi, the fraction of the epact being ${\frac {W}{692}}$ .

61. Thandeikta rules for hnit-bo end here. Makaranta adds the following:— ${\frac {7L.M.}{235}}$ gives quotient adimath and remainder adimath thetha.

$L.M.-Ad=S.M.$

Divide solar months by 12. The quotient is the year or Ratha with which the calculations commenced. If there is no remainder the Ata Ne or Thingyan Tet falls in Tagu. If the remainder is 1 it falls in Kason. This remainder is the La lun.

62. The reason why this rule is not in Thandeikta is evident. The ratio 7/235 is the ratio of the Metonic cycle, which Thandeikta does not follow. It is the error of this ratio with reference to the Burmese solar year that causes Ata Ne to fall in Kason instead of Tagu. If the adimath were always adjusted in harmony with the solar year there would be no la lun.

63. If Thandeikta hnit-bo were calculated from Poppasaw's epoch, without cutting off 1100 years, the constant for the haragon would be 442 instead of 373. The added 69 kyammat consist of two items, 25 and 44. 25 represents the accumulated error of 1 kaya and 24 anukaya per year during 4839 years, from the beginning of the Kali Yug to 1100 B. E. We have seen (para. 55) that the error amounts to one kyammat in 193 years. 193 × 25 = 4825. The 44 kyammat represent the difference between Amarapura time and Lanka time which is used in Hindu calculations.

64. The following table of Thandeikta hnit-bo is copied from the Thandeikta Calendar for forty years, published by Saya Wizaya of Mandalay.

THANDEIKTA HNIT BO FOR THAGAYIT 1230 to 1269.

Thagayit.	Kali Yug.	Hnit Kywin.	Kyammat.	Thawana.	Awaman.	Kaya.	Yet lun.	Sandra masa.	Ata.
1230	4969	130	148	47506	277	755	21	1608	2
1231	4970	131	741	47872	151	761	3	1621	4
1232	4971	132	534	48237	14	767	14	1633	5
1233	4972	133	327	48602	569	772	24	1645	6
1234	4973	134	120	48967	432	778	5	1658	0
1235	4974	135	713	49333	306	784	17	1670	2
1236	4975	136	506	49698	169	790	28	1682	3
1237	4976	137	299	50063	32	796	9	1695	4
1238	4977	138	92	50428	587	801	19	1707	5
1239	4978	139	685	50794	461	807	1	1720	0
1240	4979	140	478	51159	324	813	12	1732	1
1241	4980	141	271	51524	187	819	23	1744	2
1242	4981	142	64	51889	50	825	4	1757	3
1243	4982	143	657	52255	616	830	15	1769	5
1244	4983	144	450	52620	479	836	26	1781	6
1245	4984	145	243	52985	342	842	7	1794	0
1246	4985	146	36	53350	205	848	18	1806	1
1247	4986	147	629	53716	79	854	0	1819	3
1248	4987	148	422	54081	634	859	10	1831	4
1249	4988	149	215	54446	497	865	21	1843	5
1250	4989	150	8	54811	360	871	2	1856	6
1251	4990	151	601	55177	233	877	14	1868	1
1252	4991	152	394	55542	96	883	25	1880	2
1253	4992	153	187	55907	651	888	5	1893	3
1254	4993	154	780	56273	525	894	17	1905	5
1255	4994	155	573	56638	388	900	28	1917	6
1256	4995	156	366	57003	251	906	9	1930	0
1257	4996	157	159	57368	114	912	20	1942	1
1258	4997	158	752	57734	680	917	1	1955	3
1259	4998	159	545	58099	543	923	12	1967	4
1260	4999	160	338	58464	406	929	23	1979	5
1261	5000	161	131	58829	269	935	4	1992	6
1262	5001	162	724	59195	143	941	16	2004	1
1263	5002	163	517	59560	6	947	27	2016	2
1264	5003	164	310	59925	561	952	7	2029	3
1265	5004	165	103	60290	424	958	18	2041	4
1266	5005	166	696	60656	298	964	0	2054	6
1267	5006	167	489	61021	161	970	11	2066	0
1268	5007	168	282	61386	24	976	22	2078	1
1269	5008	169	75	61751	579	981	2	2091	2

YET BO.

65. The total solar months expired $=12R+m$ . The Adimath, by Thandeikta, is the quotient of

${\frac {28S.M.+{\frac {R}{475}}+690}{911}}$

and the remainder is Adimath thetha, The ratio 28 : 911 is the difference between unity and 939 : 911, which is an approximation to 13,358,334 : 12,960,000, the Surya Siddhanta ratio of the length of a mean solar month to the length of a mean lunar month (para. 34). ${\frac {R}{475}}$ is a correction to obtain a closer approximation. The constant 690 is the adimath thetha at the end of the 1100th year of Poppasaw's era.

66. Makaranta not only omits the correction ${\frac {R}{475}}$ , but takes a slightly rougher approximation of the Surya Siddhanta ratio, viz., 940 : 912, because this ratio can be reduced to lower terms, namely 235 : 228. This is the ratio of the Metonic cycle; nineteen years contain 235 lunar months, 228 solar months.

67. Total lunar months (L.M.) $=S.M.+Ad$ . $D=30{L.M.}+n-1$ , that is to midnight preceding the day $n$ . Kaya, by Thandeikta, is the quotient of

${\frac {11D-{\frac {R}{25}}+176}{703}}$

and the remainder is awaman. $H=D-Ka$ . The day of the week is the remainder of ${\frac {H-2}{7}}$ .

68. The kyammat of the whole period is $800H-{\frac {R}{193}}-17742$ . Divide this by 292207. The quotient is Ratha, and the remainder is kyammat-pon, or the kyammat of the fraction of a year elapsed during the current solar year. Divide kyammat-pon by 800. The quotient is Thokdadein, and the remainder is Ata kyammat.

69. Makaranta, by using the Metonic cycle, takes a different method of arriving at the total didi, and at the same time ascertains the intercalary months and days. The watat years are first found by the Metonic cycle, and then the yet-bo for midnight preceding the Labyi of Second Wazo of each watat year are calculated, as follows.

70. Divide the year by 19. The quotient is the expired cycles. If the remainder is 2, 5, 7, 10, 13, 15 or 18 there is an intercalary month. These alone are the years with which we are concerned at present.

71. The expired cycles multiplied by 7050 (235 × 30) = the total didi of the completed cycles, ending on the Lagwè of Tabaung. To find the didi of the remaining years and fraction, first multiply these years by 12. Add 4 for the months of Tagu, Kason, Nayon and First Wazo, and one month for each watat year expired during the cycle, thus:—

In the first watat year of each cycle add		4
In the„ second	at yea„ r of each„ cycle add„	5
In the„ third	at yea„ r of each„ cycle add„	6
In the„ fourth	at yea„ r of each„ cycle add„	7
In the„ fifth	at yea„ r of each„ cycle add„	8
In the„ sixth	at yea„ r of each„ cycle add„	9
In the„ seventh	at yea„ r of each„ cycle add„	10

Multiply the total months by 30 and add 14 didi of Second Wazo. Add this total to the total didi of the completed cycles. The result is the total didi of the whole period from the beginning of the era to the midnight preceding the Labyi of Second Wazo of the given year.

72. To reduce these didi to days, Kaya is the quotient of ${\frac {11D+650}{703}}$ and the remainder is the awaman.

$H=D-Ka.$

Divide H by 7. The remainder indicates the day of the week on which the Labyi of Second Wazo falls.

73. The intercalary day is determined by the changes in the awaman from watat year to watat year. These changes can easily be found without calculating the haragon in full for each watat year. In the arithmetical operation expressed by ${\frac {11D+650}{703}}$ it is obvious that the change in the remainder depends solely on the increment of total didi. When the interval from watat to watat is two years, the increase of total didi is 25 × 30 = 750. Multiply this by 11 and divide by 703; the remainder is 517. Therefore in every case of two years' interval the awaman is found by simply adding 517 to the last preceding awaman and then subtracting 703 if the total is 703 or greater. In like manner in every case of three years' interval the awaman is found by adding 259 and subtracting 703 if the total is 703 or greater.

74. A still easier method of calculating the awaman for a long period is this: the awaman for any watat year is obtained from the awaman for the corresponding year in the last preceding cycle by adding 220, or subtracting 483 if the preceding one is 483 or greater.

75. The kaya found from the equation in paragraph 72 is subtracted from didi. Hence, when the addition of 517 or 259 does not raise the awaman to 703, the increase of the haragon is greater by 1 than when the awaman becomes 703 or more, and has to be reduced by subtracting 703. A little calculation will show that the increase of the haragon in 25 months is 738 when 703 is subtracted and 739 when 703 is not subtracted. The corresponding figures for 37 months are 1092 and 1093.

76. Hence the rule. When the awaman of Second Wazo Labyi is less than in the last preceding watat year Nayon has 29 days. When the awaman is greater than in the last preceding watat year Nayon has 30 days.

77. The day of the week on which the Labyi of Second Wazo falls in any watat year may be deduced from the last preceding watat year by dividing the increase of the haragon by 7. The result may be expressed thus:—

Interval, years.	Days in Nayon.	Increase of haragon.	Increase of week day.
2	29	738	3
2	30	739	4
3	29	1092	0
3	30	1093	1

78. From the 1st of Tagu to the 15th of Second Wazo is in a wa-ngè-tat year 132 days, in a wa-gyi-tat year 133 days. Dividing by 7, we find that in a wa-ngè-tat year the 1st of Tagu falls one day later in the week than the Labyi of Second Wazo; in a wa-gyi-tat year they fall on the same day of the week. The Table in paragraph 77, therefore, gives the sequence of luni-solar New Year's Days from watat year to watat year, with this difference, in the case of New Year's Day the column "Days in Nayon" refers to the former watat year; in the case of Second Wazo it refers to the latter watat year.

THANDEIKTA WATAT

79. Thandeikta does not give any clear and invariable rule for determining which years shall be watat, and the reason probably is that the Surya Siddhanta does not contemplate the Burma practice of placing the intercalary month always near the summer solstice. The Burmese sayas who framed the Thandeikta rules were thus thrown on their own wits for guidance, and the result is that several different tests are applied.

80. Dividing the number of days in one-fourth of a Maha Yug (394,479,457) by the number of adimath in the same period (398,334), we find that the average time from one intercalary month to the next should be 990 yet 19 nayi, 24 bizana, 1 kaya and 16.269 anukaya. Consequently it is laid down in Thandeikta that the period from one intercalary month to the next is 990 yet and 19 nayi.

81. To apply this principle, one method is to reduce the adimath thetha to days; another is to reduce the yet-lun to days. In each case the resulting days being subtracted from 990 days and 19 nayi, the difference is the number of days to run from solar new year's day before the adimath thetha amounts to a full month) or the yet-lun amounts to a full month, as the case may be. It is not expressly stated that an intercalary month should be inserted if the full number of days expires before the Labyi of Wazo, but it may be inferred that that is what is meant.

82. The rule for reducing adimath thetha to days is to multiply the adimath thetha by 100 and divide by 92. The rule for reducing yet-lun to days is

$H=33Y+{\frac {W}{21}}-{\frac {44+Ky}{24}}$ .

83. A third rule is that every yet-lun pyo year should be a watat year. Yet-lun pyo means that the didi-epact, which has been increasing every year by about 11, amounts to 30 or more, when 30 is deducted from it, one lunar month is added, and the total lunar months exceed those reckoned to the end of the previous year by 13 instead of 12.

84. A fourth rule is that the Labyi on the day following which Lent begins must fall on a day when the moon is within the nekkat Athanli, that is, between longitude 266° 40′ and 270°. Athanli is a Pāli name for the month of Wazo. It is not one of the 27 lunar nekkats.

85. None of these rules seems to have been consistently followed since 1215 B. E. The third is contradictory to the fourth, for when the yet-lun exceeds 19 the full moon of the third succeeding month never reaches Athanli. This point is further discussed in para. 112. The actual practice since 1215 has been that watat has always occurred either in yet-lun pyo year, or in the year preceding yet-lun pyo when the yet-lun amounted to 27, 28 or 29. It is to be observed that under this practice although the rule of Athanli is fulfilled in watat years, yet there are many common years in which the moon's longitude on the Labyi of Wazo falls short of Athanli, namely every year in which the yet-lun is 20, 21, 22, 23, 24, 25 or 26.

THANDEIKTA YET-NGIN.

86. For determining the places of intercalary days there are three rules given in Thandeikta. One is that every year in which the kaya increases by 5 (not 6) or in other words every awaman pyo year, should have an intercalary day. As thus stated the rule is impossible, for awaman pyo years are frequently not watat years, and yet-ngin never occur except in watat years. If the awaman of watat years alone be considered, the rule is practically the Makaranta rule stated in paragraph 76, and this has certainly not been followed in Burma proper since 1100 B. E.

87. Another rule is based on the average time which should elapse between one yet-ngin and the next. Taking the figures for a quarter Maha Yug in paragraph 34, the total days in 13,358,334 Burmese months, if there were no intercalary days, would be $\textstyle 30(13,358,334)-{\frac {1}{2}}(12,960,000)$ , or 394,270,020 days. Subtracting this number from the total days, viz: 394,479,457, there remain 209,437 intercalary days. Dividing the total days by the intercalary days we obtain the quotient 1883.5, the average period from one intercalary day to the next. The rule then is that the quotient of

${\frac {(H+720)(1+{\frac {1}{3951}})}{1884}}$

is the number of intercalary days included in the given $H$ , and the remainder is the number of days since the last intercalary day fell due. The wagyitat since 1250 B. E. agree with this rule. In earlier years they do not.

88. The third rule is to try whether the beginning of the 15th didi of Second Wazo brings the moon within Athanli. If the intercalary month does not suffice to bring the full moon within Athanli, an intercalary day may do so, because the moon moves every day through about 13° of longitude, and the difference of the moon's longitude between one full moon and the next is about 29°.

89. To find the moon's longitude, first find the Thokdadein, i.e. the number of days expired from midnight of solar new year's day to midnight of 14th Waxing Second Wazo. From the Thokdadein, the sun's longitude and the difference of longitude between the sun and the moon are separately calculated. The sum of these two is the moon's longitude.

90. A complete circle of 360 degrees consists of 21600 minutes. Therefore, if $Ky$ be Ata Kyammat, the sun's longitude, expressed in minutes, at the midnight indicated by thokdadein $T$ , bears the same ratio to 21600 as $800T+Ky$ bears to 292207.

∴ ${\text{Sun's long.}}\times 292207=21600(800T+Ky)$ .

Now 216 happens to be a very close approximation to a factor of 292207. The result of dividing the latter by the former is 1352811/108. The last equation, therefore may be written

${\text{Sun's long.}}\times 13528{\frac {11}{108}}=1000(800T+Ky)$ .

Thandeikta disposes of the fraction 11/108 by subtracting $6T$ from the other side of the equation, and the Sun's longitude, expressed in minutes, is taken to be

${\frac {1000(800T+Ky)-6T}{13528}}$ .

Reduce the result to degrees and minutes by dividing by 60.

91. The difference of longitude between sun and moon is found by didi. A didi is the time in which the mean moon increases her longitudinal distance from the mean sun by 12 degrees. The didi elapsed from mean new moon next before Ata Ne to the midnight indicated by the given Thokdadein equal the sum of yet-lun and its fraction plus Thokdadein reduced to didi. That is to say

$D=Y+{\frac {W}{692}}+T(1+{\frac {11}{692}})=Y+T+{\frac {11T+W}{692}}$ .

92. Having found the sum of didi, divide by 30, and reject the quotient, as it represents complete lunations, and at every new moon the difference of longitude between sun and moon is zero. The remainder multiplied by 12 is degrees of longitude.

93. The remainder of ${\frac {11T+W}{692}}$ is the awaman of the day. If it be denoted by $Wd$ , then the increase of difference of longitude during the fraction of a didi is, in minutes,

$Wd+{\frac {7Wd}{173}}$ .

Reduce it to degrees and minutes by dividing by 60.

94. Add together the sun's longitude and the two parts of the difference between sun and moon. Subtract from the sum 52 minutes. The result is the moon's longitude.

95. Add the week-day figure of Ata Ne to the Thokdadein of Second Wazo Labyi. Divide the sum by 7. The remainder indicates the day of the week of Second Wazo Labyi. The sequence of this day from watat to watat ought to agree with the table in paragraph 77.

96. If the moon's longitude as calculated does not lie within Athanli, a day may be added or subtracted, provided it does not set the week-day wrong. That is to say, if the increase of week-day indicates a wangètat one day may be added; if it indicates a wagyitat one day may be subtracted. Thus, in 1234 the increase of week-day since 1231 as obtained from the Thokdadein was 0, indicating a wangètat. The moon's longitude as calculated was 254° 32′, falling short of Athanli. One day was added to the Thokdadein, making the year a wagyitat, with week-day increase 1. The same occurred in 1245, when the calculated moon's longitude was 261° 10′. In 1261 the calculated longitude was 276° 24′, and increase of week-day 1, indicating a wagyitat. One day was deducted, making a wangètat. The object of this is not apparent, as the moon's longitude often exceeds Athanli. These are the only occasions on which a correction has been applied to the calculated Thokdadein for Second Wazo Labyi since 1215 B. E.

97. Table VI shows the week-day and moon's longitude of Second Wazo Labyi, calculated as above described, for all the watat years from B. E. 1217 to 1361, and the resulting wagyi and wangè tat. For past years the three corrections mentioned in the last paragraph have been made. For future years four corrections are made.

1291	moon's longitude	263° 20′
1307	moon's„ longitude„	265° 29′
1337	moon's„ longitude„	265° 35′
1348	moon's„ longitude„	261° 50′

98. In each of these four years the moon's calculated longitude falls short of Athanli, and the addition of one day would bring it within Athanli. Without correction each of these years would be wangètat. The Thokdadein is therefore increased

from	96 Saturday	to	97 Sunday
	99 Monday	to	100 Tuesday
	97 Tuesday	to	98 Wednesday
and	95 Sunday	to	96 Monday respectively,

and all four years will be wagyitat.

99. In consequence of these alterations the years 1293, 1310, 1339 and 1350 are ipso facto altered from wagyitat to wangètat, the Thokdadein of Second Wazo Labyi in each case remaining unaltered.

100. The conclusion I arrive at is that in calendars computed under Thandeikta rules watat will continue to be placed in yet-lun-pyo years, but in the year before yet-lun-pyo when yet-lun is 27, 28 or 29; and that yet-ngin will be determined by computation of the moon's longitude by means of Thokdadein, four corrections being made within the next 90 years, namely those in the years 1291, 1307, 1337 and 1348, shown in Table VI.^[1]

↑ U Kyaw Yan interprets the rule mentioned in para. 81 as meaning that if adimath pyoes while the sun is in Meiktha, Pyeiktha, Medon or Karakat, there is a watat in the current year. This is equivalent to saying that watat occurs in any year in which the epact exceeds yet lun 26 awaman 130. I venture to think that such a rule as this has not been followed hitherto. In 1244 the epact was yet lun 26 awaman 450, but there was no watat.

[1] U Kyaw Yan interprets the rule mentioned in para. 81 as meaning that if adimath pyoes while the sun is in Meiktha, Pyeiktha, Medon or Karakat, there is a watat in the current year. This is equivalent to saying that watat occurs in any year in which the epact exceeds yet lun 26 awaman 130. I venture to think that such a rule as this has not been followed hitherto. In 1244 the epact was yet lun 26 awaman 450, but there was no watat.

[1]