The Lining Classic of the Five Stars: The Five Stars Method

歲星：率，五十三萬四千四百八十二。奇，四十五。伏分，二萬四千三十一。奇，二十二半。熒惑：率，一百四萬五千八十。奇，六十。伏分，九萬七千九十。奇，六十。填星：率，五十萬六千六百二十三。奇，二十九。伏分，二萬四千八百四十一。奇，六十四半。太白：率，七十八萬二千四百四十九。奇，九。伏分，五萬六千二百二十四。奇，五十四半。辰星：率，一十五萬五千二百七十八。奇，六十六。伏分，二萬六百九十九。奇，三十三。

The following lists two fundamental constants used in the calculation of planetary motion. "Rate" is the total fraction of one conjunction cycle for each planet, "concealed fraction" is the fraction during which the planet is hidden and not visible, and "remainder" is a finer subdivision of the fraction. For the planet Jupiter: rate 534482, remainder 45; concealed fraction 24031, remainder 22 and a half. For the planet Mars: rate 1,045,080, remainder 60; concealed fraction 97,090, remainder 60. For the planet Saturn: rate 506,623, remainder 29; concealed fraction 24,841, remainder 64 and a half. For the planet Venus: rate 782,449, remainder 9; concealed fraction 56,224, remainder 54 and a half. For the planet Mercury: rate 155,278, remainder 66; concealed fraction 20,699, remainder 33. Both the calculation of the time of appearance and the fixed appearance will use these two sets of numbers for division and subtraction.

木終日：三百九十八。餘，一千一百六十二。奇，四十五。火終日：七百七十八。餘，一千二百二十。奇，六十。土終日：三百七十八。餘，一百三。奇，二十九。金終日：五百八十三。餘，一千二百二十九。奇，九。夕見：伏二百五十六日。晨見：伏三百二十七日。餘、奇、同終分奇。水終日：百一十五。餘，一千一百七十八日。奇，六十六。夕見：伏五十二日。晨見：伏六十三日。餘、奇、同終分奇。

The term "率" is converted into days, referred to as "终日," which represents the total number of days in a conjunction cycle, along with the remainder and the leftover fraction. The Jupiter's 终日 is 398 days, remainder 1162, leftover 45. Mars' 终日 is 778 days, remainder 1220, leftover 60. Saturn's 终日 is 378 days, remainder 103, leftover 29. Venus' 终日 is 583 days, remainder 1229, leftover 9; the period when it appears in the west in the evening including its occultation is 256 days, and the period when it appears in the east in the morning including its occultation is 327 days, with the remainder and leftover fraction the same as those of the 终日. Mercury's 终日 is 115 days, remainder 1178, leftover 66; the period when it appears in the west in the evening including its occultation is 52 days, and the period when it appears in the east in the morning including its occultation is 63 days, with the remainder and leftover fraction the same as those of the 终日. The planets Mercury and Venus are attached to the Sun, so their appearances are recorded in two segments, morning and evening. The planets Jupiter, Mars, and Saturn do not require this division.

推五星平見術，各以伏分減總實，餘以其星率去之。餘反減率餘總法，約為日。不盡，為餘奇，即所求年天正恆朔夜半後晨夕見，見日算及餘奇。（天正定朔進退日者，進減加一日為定朔半夜後星平見日及餘奇，其金水二星，知得夕平見，其滿晨見伏日及餘奇者去之，餘為晨平見之。）命起天正定月大小，去之，不滿月，命日，算外，即各所求。

The method for calculating the 'initial appearance' (without correction) of the five planets is as follows: for each planet, subtract its hidden period from the total actual value. The remainder is then divided by the planet's rate until it is no longer sufficient for one rate. Then, the remaining value is subtracted from the rate, and the remainder is simplified into days using the 'total method', with any remainder and odd numbers noted. This gives the number of days and the remainder and odd number for the planet's morning or evening appearance after the winter solstice eleventh month new moon at midnight of that year. (Note from the original text: if the fixed new moon is earlier or later than the average new moon, subtract or add one day respectively to obtain the number of days and remainder for the planet's initial appearance after the fixed new moon at midnight. For Mercury and Venus, the initial calculation gives the evening appearance. If the result exceeds the combined hidden period and morning appearance days, subtract this amount to obtain the number of days and remainder for the morning appearance.) Finally, starting from the winter solstice month, subtract the days month by month according to the size of each month until the remaining days are less than a month. The remaining days are counted, and the starting day is not counted. This gives the specific date of the planet's initial appearance.

推五星定見術，各置其星平見所入氣，計日損益，滿半總為日；不滿為分，以損益所加減。訖，餘以加減平見日及分，為半其日消息定數；息減消加之，即其定見分。

The method to calculate the corrected initial appearance date ('dingjian') of the five planets involves taking the solar term that the planet enters on the day of its mean appearance. According to the following rules for each solar term, calculate the daily amount to be added or subtracted. Accumulate these daily changes until reaching half the denominator, which is then converted into one day, and any remainder is kept as fractions. Apply the additions and subtractions according to the rules, and after completing the calculation, add or subtract the remaining amount to the mean appearance day and fraction to obtain the 'xixiang dingshu' (corrected number according to the solar term). If it is an 'xi' (diminishing), subtract; if it is an 'xiao' (increasing), add. After completing the addition and subtraction, the result gives the corrected day and fraction for the planet's appearance.

歲星：（初見，去日起十四度）見入冬至，（旱小寒）減六日。（已後日損十七分）春分初日，平。（已後日加八十五分）立夏畢小滿，加六日。（已後日損八十九分）夏至畢立秋，加四日。（已後日損百七十八分）自白露初日，平。（已後日減五十三分）小雪畢大雪，減六日。

Critical: The following is an English translation of the original text. Do not rewrite or paraphrase in Chinese. Every sentence must be in English. Rules for Correcting the Fixed Appearance of the Year Star (Jupiter): When Jupiter first appears, it is 14 degrees away from the Sun. The fixed appearance falls on the Winter Solstice (note: up to the Minor Cold). Subtract 6 days (note: from then on, each day reduces by 17 parts). On the day before the Vernal Equinox, the correction is set to zero, and the fixed appearance is used without addition or subtraction (note: from then on, each day adds 85 parts). From the Beginning of Summer to the Grain in Ear, add 6 days (note: from then on, each day reduces by 89 parts). From the Summer Solstice to the Beginning of Autumn, add 4 days (note: from then on, each day reduces by 178 parts). Starting from the Beginning of White Dew, it returns to the fixed appearance (note: from then on, each day reduces by 53 parts). From the Minor Snow to the Major Snow, subtract 6 days.the so-called "依平" means that the fixed appearance is the same as the mean appearance, and no correction is needed.the so-called "加减若干日" means that the actual appearance is delayed or advanced by that many days compared to the mean appearance.

熒惑：（初見，去日十七度。）見入冬至初日，減二十七日。（已後日損六十三分）大寒初日，平。（自後日加四十二分）雨水畢穀雨，加二十七日。（自入立夏已後，日損一百九十八分。）立秋，平。（自入處暑已後，減百九十八分。）小雪畢大雪，減二十七日。

The fixed correction rules for the planet Mars (Huo Xing). When Mars first appears, it is 17 degrees away from the Sun. If it appears on the day before the winter solstice, subtract 27 days. (Note: Each subsequent day reduces by 63 parts.) By the day before the great cold (Dahanshan), the correction returns to zero, following the normal calculation. (Note: Each subsequent day adds 42 parts.) From the start of rainwater to the start of grain rain, add 27 days. (Note: After the start of summer, each day reduces by 198 parts.) By the start of autumn, it follows the normal calculation. (Note: After the start of the heat (Chushu), each day reduces by 198 parts.) From the start of light snow to the start of heavy snow, subtract 27 days. The correction for Mars reaches its maximum, up to 27 days, because it is close to the earth and its apparent motion changes rapidly, unlike Jupiter, which differs by only six days.

填星：（初見，去日十七度。）見入冬至，（初日）減四日。（自後日益八十九分）大寒畢春分，減八日。（自入清明已後，日損五十九分。）小暑初，依平。（自後日加八十九分）白露初日，加八日。（自後日損七十分）秋分，加四日。（自後寒露已後，日損五十九分。）小雪初日，依平。（自後日減八十七份）

The regulations for correcting the fixed observation of the planet Tianshi (Fill Star). (Original note: When the planet Tianshi first appears, it is seventeen degrees away from the Sun.) When the planet appears on the day of Winter Solstice, (Original note: meaning the previous day.) subtract four days. (Original note: After this, each day increases by 89 parts.) From the beginning of Greater Cold to the beginning of Spring Equinox, subtract eight days. (Original note: After entering Qingming, each day decreases by 59 parts.) At the beginning of Minor Heat, follow the standard without adding or subtracting. (Original note: After this, each day increases by 89 parts.) On the first day of White Dew, add eight days. (Original note: After this, each day decreases by 70 parts.) At the beginning of Autumn Equinox, add four days. (Original note: After the beginning of Cold Dew, each day decreases by 59 parts.) On the first day of Minor Snow, follow the standard. (Original note: After this, each day decreases by 87 parts. The final character '份' should be '分'.) The planet Tianshi moves slowly, covering only one zodiac mansion in a year, so the correction amount is between the values for wood and fire.

太白：初見，（加八日，自後日損，去日十一度。）夕見，入冬至初日，依平。（自後日減一十分）驚蟄畢春分，減九日。（自入清明已後，日損一十分。）芒種，依平。（自入夏至後，日加一百分。）處暑畢秋分，減九日。（自入寒露已後，日損一百分。）大雪，依平。（自入大寒已後，日加六十七分。）立春畢立夏，加三日。（自入小滿已後，日損六十七分。）

The fixed regulations for the correction of the position of Tai Bai are first explained regarding its appearance in the west in the evening. (Original book note: The initial appearance adds eight days, and thereafter each day decreases by a certain amount; the initial appearance is eleven degrees away from the Sun.) When the evening appearance coincides with the first day of the Winter Solstice, it follows the standard without addition or subtraction. (Original book note: Thereafter, each day decreases by ten parts.) From the Awakening of Insects to the Beginning of Spring, it decreases by nine days. (Original book note: After entering the Clear and Bright period, each day decreases by ten parts.) By the time it reaches the Grain in Ear, it follows the standard. (Original book note: After entering the Summer Solstice, each day increases by one hundred parts.) From the Beginning of Autumn to the Beginning of Autumn, it decreases by nine days. (Original book note: After entering the Cold Dew period, each day decreases by one hundred parts.) By the time it reaches the Great Snow, it follows the standard. (Original book note: After entering the Greater Cold period, each day increases by sixty-seven parts.) From the Beginning of Spring to the Beginning of Summer, it increases by three days. (Original book note: After entering the Grain Full period, each day decreases by sixty-seven parts.)

晨見，入冬至，依平。（自入大寒已後，日加六十七分。）大雪，依平。晨見，入冬至，依平。（自入大寒已後，日加六十七分。）立春畢立夏，加三日。（自入小滿已後，日減六十七分。）夏至，依平。（自入小暑已後，日減六十七分。）立秋畢立冬，減三日。（自入小雪已後，日損六十七分。）

Now continue with the part about the bright star appearing in the morning in the east. If the morning appearance falls on the day of Winter Solstice, it is considered as the standard. (Original note: After entering the start of Greater Cold, add 67 parts each day.) By the time of Heavy Snow, it is considered as the standard. The following passage about the morning appearance is repeated again: If the morning appearance falls on Winter Solstice, it is considered as the standard. (Original note: After entering the start of Greater Cold, add 67 parts each day.) From the start of Start of Spring to the start of Start of Summer, add three days. (Original note: After entering the start of Grain Bearer, subtract 67 parts each day.) By the time of Summer Solstice, it is considered as the standard. (Original note: After entering the start of Minor Heat, subtract 67 parts each day.) From the start of Start of Autumn to the start of Start of Winter, subtract three days. (Original note: After entering the start of Minor Snow, subtract 67 parts each day.) The sentence "Morning appearance entering Winter Solstice follows the standard, from entering Greater Cold onwards, add 67 parts each day, Heavy Snow follows the standard" is a repetition in the original text, likely due to a copying error. It should be considered as a duplicate, but it is still recorded as is without deletion.

辰星：（初見，去日十七度。）夕見，冬至畢清明，依平。穀雨畢芒種，減二日。夏至畢大暑，依平。立秋畢霜降，應見不見，其在立秋及霜降二氣之內，夕去日十八度外，三十六度內，有木火土金星已上者夕見，立冬畢大雪，依平。

The regulations for correcting the fixed observations of the planet Chén Xīng (Water Star) begin with the section on its appearance in the west in the evening. (Original note: When Water Star first appears, it is seventeen degrees away from the Sun.) The evening appearance is considered normal between the Winter Solstice and the Beginning of Spring, with no addition or subtraction. Between the Grain Rain and the Beginning of Summer, two days are subtracted. Between the Summer Solstice and the Great Heat, it is considered normal. Between the Beginning of Autumn and the Cold Dew, it should be visible in principle but is actually not seen; if it is during the Beginning of Autumn or the Cold Dew, and the Water Star is eighteen degrees away from the Sun but not more than thirty-six degrees, and there is at least one of the stars of Wood, Fire, Earth, or Metal visible in the sky, then the Water Star can be seen in the evening. Between the Beginning of Winter and the Snowfall, it is considered normal. Since the Water Star is closest to the Sun, it is the most difficult to see, so the regulations specifically include an item called "should be seen but is not seen," and its visibility is determined by the presence of other stars.

晨見，入冬至，減四日。小寒畢大寒，依平。立春畢啟蟄，減三日。（其在啟蟄氣內，去日度如前，晨無木火土金一星已上者，不見。）雨水畢立夏，應見不見。（其在立夏氣內，去日度如前，有木火金一星已上者，亦見。）小滿畢寒露，依平。霜降畢立冬，加一日。小雪畢大雪，依平。

Continuing with the section about the morning appearance of the planet Chén Xīng (Dawn Star). If the morning appearance occurs during the period of the Winter Solstice, it is reduced by four days. From the Lesser Cold to the Greater Cold, it follows the normal calculation. From the Beginning of Spring to the Awakening of Insects (i.e., Start of Thunder), it is reduced by three days. (Original note: If it occurs within the period of Awakening of Insects, and the distance from the Sun is the same as previously mentioned, but there is no appearance of one or more of the stars of Wood, Fire, Earth, or Metal in the morning sky, it will not be visible.) From the Rain Water to the Beginning of Summer, it should be visible but is not. (Original note: If it occurs within the period of the Beginning of Summer, and the distance from the Sun is the same as before, but there is an appearance of one or more of the stars of Wood, Fire, or Metal in the morning sky, it will be visible.) From the Grain Full to the Cold Dew, it follows the normal calculation. From the Frost Descent to the Beginning of Winter, it is increased by one day. From the Minor Snow to the Major Snow, it follows the normal calculation.

推星見所在度術，置星定見日夜半日所在宿度，算及分，以定見餘，加其夜半度分，乃以其星初見去日度數，晨減，夕加之，即星初見辰所在宿度。

The method to determine the degree of the five planets when they first appear: first, set out the degree of the Star of brightness and prestige at midnight on the day the planet is first seen, including the minutes. Then take the remaining minutes from the time of first appearance and add them to this midnight degree and minutes. Next, according to the degrees between the planet and the Sun at first appearance (the fourteen degrees, seventeen degrees, eleven degrees, etc., mentioned in the previous notes), subtract this degree from the Sun's degree if it is a morning appearance, or add it if it is an evening appearance. The resulting degree is the degree of the planet at the moment of its first appearance. Subtracting for morning appearances and adding for evening appearances is because a morning appearing planet is west of the Sun and rises before the Sun, while an evening appearing planet is east of the Sun and sets after the Sun.

轉求次日夜半星行所至，各以一日所行度及分，順行逆減之，有小分以日率為母，小分滿母，去從行分一，行分滿半，總去，從度一，其行有益疾益遲者，副置一日行分，各以其差疾益遲，損乃加之。留者因前，逆則依減，得每日所至。（其五星後順流退所終日度，各依伏求其去日遠近消息日度之率，以定伏日所在，若注歷，其日月及金水等，皆棄其分。）

To determine where the star will be at the midpoint of the next day, calculate its daily movement in degrees and minutes: for a star moving forward, add the daily movement; for a star moving backward, subtract it. The small minutes are divided by the "daily rate" as the denominator. When the small minutes reach one "mother," they are converted into one movement minute. When the movement minutes reach half the total, they are converted into one degree. If a star's movement has a gradual increase or decrease in speed, set aside a separate daily movement for each day, adjusting according to the daily difference in speed. For a gradually increasing speed, add the difference; for a gradually decreasing speed, subtract it, then combine with the main rate. On days when the star is stationary, use the degree from the previous day without change; on days of retrograde motion, subtract the corresponding amount. By proceeding day by day in this way, you can determine the position of the star each day. (Original note: When the five planets complete their forward or retrograde motion, the number of days and degrees at the end are determined according to the method of rising and setting, taking into account the rate of distance from the Sun to determine the days and positions of emergence and disappearance. If included in the calendar, the fractional parts of the Sun, Moon, and the planets Mercury and Venus are all omitted.)

推平行度分，置定度，率以半總乘，有分者從之，以日率除，所得為一日行分。求差行初日行度分術，置定日，率減一，以所差分乘；半之，為差率，益疾，以差率減；益遲，以差率加，平行分，即初日行分。

The method for calculating the daily progression of equal motion: first determine the total degrees for this period, then multiply by the denominator "half total." If there are additional minutes under the degrees, add them as well. Then divide by the daily rate for this period. The result is the daily progression in fractions, which is the method for equal motion. The method for calculating the initial daily progression for varying motion (accelerating or decelerating each day): first determine the fixed daily rate for this period, subtract one, then multiply by the daily difference in fractions. Take half of this result, which is called the "difference rate." If the motion is increasing, subtract the difference rate from the equal motion fraction; if the motion is decreasing, add the difference rate to the equal motion fraction. The result is the initial daily progression. Subsequently, each day increases or decreases by the difference amount, so that by the last day, the total matches the total degrees.

歲星：初順，差行（先疾日益遲一分）一百一十四日。行一十八度。（五百九十四分）前留二十六日。旋退，差行（先遲益疾二分）四十二日。退六度十二分，又退（先疾益遲二分）四十二日，退六度。（十二分）後留二十五日。後順，差行（先遲日益疾一分）一百一十四日。行一十八度。（五百九十四分，日盡而夕伏。）

The movement of the Star of Years is divided into segments during its orbital period. After its first appearance, it moves forward at a decreasing rate (initially fast, then slowing by 1 part each day), taking 114 days to cover 18 degrees (with an additional 594 parts). Then it remains stationary for 26 days. Next, it moves backward at a decreasing rate (initially slow, then speeding up by 2 parts each day), covering 6 degrees and 12 parts in 42 days. Then it continues to move backward at a decreasing rate (initially fast, then slowing by 2 parts each day), taking another 42 days to cover 6 degrees (with an additional 12 parts). Then it remains stationary for 25 days. Finally, it moves forward again at a decreasing rate (initially slow, then speeding up by 1 part each day), taking 114 days to cover 18 degrees (with an additional 594 parts; after the days are completed, it disappears at sunset).

熒惑：（前日加加減度率，加疾率行限，減率。）前遲日率，度率，退，（率度率）後留，（日率）後疾。（日率行限）冬至（二百四十三依平），百六十五，六十，二十五，六十三，二十二，十三（二百一十平行），一百三，十二。小寒（二百二十二依平平行），一百五十五，同前，同前，同前，二十六，十五（一百九十五正行），一百四十一，十七。

The planet Mars (荧惑) has the most significant changes in its movement. Therefore, data must be established for each of the twenty-four solar terms (二十四气), with each entry representing a separate set of figures. The columns at the top of the table are: the daily rate and degree rate of the initial rapid motion (with notes on when to add or subtract and when the motion is limited), the daily rate and degree rate of the initial slow motion, the rate and degree rate of retrograde motion, the daily rate of the stationary phase after retrograde motion, and the daily rate and motion limit of the subsequent rapid motion. The following entries are listed for each solar term. If it is marked as "同前" (same as before), it means the data is the same as the previous term and is not repeated. For the term 冬至 (Winter Solstice), the data is: initial rapid motion 243, following the average without addition or subtraction; the following numbers are 165, 60, 25, 63, 22, 13 (small note 210, marked as parallel motion), 103, 12. For the term 小寒 (Minor Cold), the data is: 222, following the average as parallel motion; the following number is 155, the remaining three items are the same as before, 26, 15 (small note 195, marked as direct motion), 141, 17. The original text of this table has many errors, omissions, and misarrangements, making it difficult to match each number with its corresponding column. The current recording preserves the original appearance, only explaining the structure.

大寒（二百二十五依平行，一百四十七加四），五十五，二十，同前，二十二，二十五（一百八十八日平行一百二九十四）。立春（二百一十八減二一百四十同前），六十，二十五，同前，十八，同前（一百六十八平行），九十。雨水（二百三減七差行二百一十五平），同前，八百六十七，同前，同前。

Great Cold: 225, following the level line, add four to 147; the following numbers are 55 and 20, same as before, 22 and 25 (note 188: on the 188th day, follow the level line, total 1294, this number is incorrect). Start of Spring: 218, subtract two, 140, same as before; the following numbers are 60 and 25, same as before, 18, same as before (note 168: follow the level line). Rain Water: 203, subtract seven, follow the difference line, 215 follow the level line; the following numbers are same as before, 867, the remaining two numbers same as before. Among the three periods, the rate of the first rapid days decreases from 225 to 203, indicating that Mars moves from winter to spring, and the number of days decreases gradually with each period. The number 'eight hundred and sixty-seven' does not match the scale of the entire table, so it is likely incorrect.

驚蟄（二百一十五差行，一百三十二同前），同前，同前，同前，十九（一百六十五平行），同前，八十三。春分（一百九十五差行一百一十七平），同前，同前，同前（四百六十三），同前（十七），十八（百六十七差行），八十八。清明（一百八十八日前差行一百一十加二），同前，同前，同前，同前，二十三（一百七十二差行），九十三。

The entry for惊蛰 is 215, a difference row, 132 the same as before; the following three items are the same as before, 19 (small note 165 says parallel), same as before, 83. The entry for春分 is 195, a difference row, 117 according to parallel; the following three items are the same as before (small note 463), same as before (small note 17), 18 (small note 167 says difference row), 88. The entry for清明 is 188 days, previously a difference row, 110 plus two; the following four items are the same as before, 23 (small note 172 says difference row), 93. In the previous three periods, the rate of rapid days decreased from 215 to 188, while the last column increased from 83 to 93, one decreasing and one increasing, which exactly reflects the transition of the planet Mars from its slow movement after the winter solstice to its rapid movement in summer.

穀雨（一百八十四百一十八同前差行），同前，二十四，同前，同前，同前（一百七十六一百七十七，九十八，九十八）。立夏（同前同前差行同前加四十），同前，二十三，同前，同前，同前（一百八十九，度行六十六日），一百一十一。小滿（九日同前差行，同前同前），同前，同前，同前，同前，同前，（一百四同前），一百，二十六。

Grain Rain Yixing: 184, 118, same as before, subtract to get the row; the following are same as before, 24, the other three items same as before (note 176, 177, 98, 98). Beginning of Summer Yixing: all items same as before, subtract to get the row, same as before plus forty; the following are same as before, 23, three items same as before (note 189, half degree and moving sixty-six days). Minor Fullness Yixing: nine days, same as before, subtract to get the row, same as before same as before; the following items are all same as before (note 104 same as before), 100, 26. Among these three periods, "same as before" appears most frequently, indicating that the planet Mars during the late spring and early summer period has relatively minor changes in daily rate and degree rate for each segment, and the table only makes slight additions or subtractions in individual columns.

芒種（一百七十三平差，九十五同前），二十四，同前，同前，同前，同前，（二百一十九十四日同前，二百四十一，一百五十五）。夏至（一百七十一六日同前平差行，九十三同前），同前，二十五，同前，同前，同前，（十日，同前同前），同前。小暑（一百七十四平差行，九十六平），同前，同前，六十，十四，同前，（五日二百五十三半度行五十日），一百七十五。

Mangzhong Yixing: 173, parallel and difference lines combined, 95 same as before; the following 24, four items same as before (small note 219, fourteenth day same as before, 241, 155). Summer Solstice Yixing: 171, sixth day, same as before, parallel and difference lines combined, 93 same as before; the following same as before, 25, three items same as before (small note ten days, same as before same as before). Xiaozhu Yixing: 174, parallel and difference lines, 96 according to parallel; the following two items same as before, 60, 14, same as before (small note five days, 253, moving fifty days at half degree rate). The rate of rapid movement before and after Summer Solstice drops to 171, which is the smallest in the whole table. This is exactly when the planet Mars moves forward the fewest days in a year and moves at the fastest rate.

大暑（一百七十九平差行，一百加四），同前，同前，五十六，十二，同前，（一百六十三同前），一百八十五。立秋（一百八十四平平行，一百六平），同前，同前，五十七，十一，同前，（同前行半度二十日），同前。處暑（九十九半度行，加五十六日，一百一十一平），同前，同前，同前，同前，（同前平行），同前。

Great Heat (Dàshǔ) Yixing: 179, parallel difference combined, 100 plus four; the following two items are the same as before, 56, 12, same as before (note 163 same as before), 185. Beginning of Autumn (Lìqiū) Yixing: 184, parallel; 106 according to parallel; the following two items are the same as before, 57, 11, same as before (note same as before, moving at half degree rate for twenty days), same as before. Beginning of Autumn (Chǔshǔ) Yixing: 99, moving at half degree rate, adding fifty-six days, 111 according to parallel; the following four items are the same as before (note same as before, as parallel), same as before. From the summer solstice onwards, the rate of rapid movement starts from 171 and rises to 184, the last column numbers rise from 175 to 185, indicating the beginning of Mars moving from rapid to slow.

白露（二百一十四加十半度行四十四日，一百一十六平），同前，同前，（同前十二日六十三），十七，同前，（同前平行），同前。秋分（二百三十三半度行，同前同前），百五十四半，同前，（同前二十），同前，（五十一日，二百五十五平行），一百七十七。寒露（二百四十七同前平行，一百六十五同前平行），七十五，三十，（同前九日六十六，同前二十），四，（二百五十五平行），一百六十七。

White Dew Yixing: 214, add ten, move at half degree rate for forty-four days, 116 according to level; the following two items are the same as before (small note same as before, twelve days 63), 17 same as before (small note same as before, made level), same as before. Autumn Equinox Yixing: 233, move at half degree rate, two items same as before; the following 154 half, same as before (small note same as before 20), same as before (small note fifty-one days, 255 made level), 177. Cold Dew Yixing: 247, same as before, made level, 165 same as before made level; the following 75, 30 (small note same as before nine days 66, same as before 20), 4 (small note 255 made level), 167. The rate of rapid days starts from 214 at White Dew and gradually increases to 247 at Cold Dew, the number of days Mars moves in the south after autumn is clearly longer.

霜降（二百五十六八百二十五十九平差行，二百五十六，二百一百二十九，一百七十八一百八十一加四），六十，二十五，（同前同前六日一百三七十，二百二十五平行），一百五十七。立冬（十三日同前平平行，同前同前，同前），十七，（同前十二日六十七，同前二十一，二百二十五平行），一百四十七。

The entry for Frost Descent includes the numbers 256, 825, 19, with parallel differences combined as 256, 200, 129, 178, 181, adding four; below that 60, 25 (small note same as before, six days, 103, 70, 225 as parallel), 157. The entry for Start of Winter is thirteen days, same as before, parallel, all items same as before; below that 17 (small note same as before twelve days 67, same as before 21, 225 as parallel), 147. The numbers in the Frost Descent entry are most complicated, with numbers like 'eight hundred twenty-five nineteen' and 'one hundred thirty-seven' not connecting with the numbers of the preceding and succeeding solar terms, clearly indicating errors in the original text, making it impossible to correct. The numbers are now transcribed as they originally appeared.

小雪（二百五十八平平行，同前同前），同前，同前，（同前八日六十三，同前十七），同前，（二百一十五平行），一百三十七。大雪（二百五十平平行，一百七十二平），同前，二十，同前，同前，同前，（二百五平行），一百二十七。

Small Snow (Xiaoxue) Yixing: 258, parallel, both items same as before; the following two items same as before (small note same as before eight days 63, same as before 17), same as before (small note 215 says parallel), 137. Great Snow (Daxue) Yixing: 250, parallel, 172 follows the level; the following same as before, 20, three items same as before (small note 205 says parallel), 127. By this point, the twenty-four solar terms have completed one cycle: the rate of rapid days starts from 243 at the winter solstice and decreases to 171 at the summer solstice, then rises to 258 at Small Snow and decreases again. The numbers in the last column rise from 12 to 185 and then drop back to 127. Throughout the year, the numbers rise and fall, returning to the starting point of the winter solstice, matching the actual movement of Mars as it varies in distance from the sun.

前留，十三（前疾減日率者，以其數分益，比留及後遲日益。前疾加日率者，以其數分減，比留及後遲日益。）　後順，遲差行（先遲日益疾二分）六十日。行二十五度。（前疾加度者，此遲依數減之，為定疾無加度者，此遲入秋分至立冬，減三度，入立冬減五度，後留，定日縮十三日者，以所縮日數，加此遲日率之。）

The "before staying" period of the planet Mars (荧惑) lasts thirteen days. (Original note: If during the initial rapid movement period the daily rate is reduced, the reduced amount is added to the later period of staying and slow movement. If during the initial rapid movement period the daily rate is increased, the increased amount is subtracted back, and the same calculation is applied to the later period of staying and slow movement.) The "after following" period is a slow movement phase, during which it moves 2 parts faster each day, lasting 60 days and covering 25 degrees. (Original note: If during the initial rapid movement period an excessive amount is added, this slow movement period is reduced by the same amount to get the final value. If no excess was added during the initial rapid movement period, then during this slow movement period from the autumn equinox to the beginning of winter, three degrees are subtracted, and five degrees are subtracted from the beginning of winter. In the final staying period, if the fixed number of days is reduced by thirteen days, the reduced days are added back to the daily rate of this slow movement period.)

推熒惑平率及變率術，各以其所入氣數，與後氣數互減，餘計日以加減本率，皆為平率。（其後疾變率，與前遲盈縮，六十日，行二十五度，退行六十三日十五度者，以所盈縮日度，盈減，縮加之，唯退行度，即盈加縮減其差行者，前疾先疾，後疾先遲，各日益遲疾，分餘，皆平行，後疾分，日盡而夕伏。）

The method for calculating the mean rate and variation rate of the planet Mars: take the numbers from the table for the period during which the planet enters and subtract them from the numbers of the next period. The remainder is divided by the number of days to distribute daily, which is then added to or subtracted from the rate of the original period. The result is the 'mean rate'. (Original note: regarding the variation rate of the later rapid motion, it relates to the excess and deficiency of the previous slow motion. The previous slow motion originally moves 25 degrees in 60 days, while the retrograde motion originally retreats 15 degrees in 63 days. For any case of excess or deficiency, the excess days and degrees are subtracted and the deficient ones are added. However, for the retrograde motion, the degrees are reversed, so the excess is added and the deficiency is subtracted. For the motion of difference, the first rapid motion starts fast and gradually slows down, while the second rapid motion starts slow and gradually speeds up. Each is calculated by daily increments or decrements. Once the daily divisions are exhausted, it returns to a uniform motion. When the daily divisions of the second rapid motion are completed, Mars becomes invisible in the evening.)

填星：初順，差行（先疾，日益遲半分。）八十三日，行七度。（二百九十分）前留三十六日。旋退，差行（先遲，日益疾退半分。）五十一日。退三度。（三十分）又退（先疾，日益遲退半分。）五十一日。退三度。（三十分）後留三十六日。後順，差行（先遲，日益疾半分。）八十三日。行七度。（二百九十分日盡而夕伏）

The movement of the planet Jupiter over one conjunction cycle is divided into segments. Initially, after appearing, it moves forward at a decreasing rate (originally noted: it starts fast, then slows by half a degree each day), taking 83 days to cover 7 degrees (originally noted: another record of 290 parts). Then it remains stationary for 36 days. Next, it moves backward at a decreasing rate (originally noted: it starts slow, then speeds up by half a degree each day), taking 51 days to retreat 3 degrees (originally noted: another record of 30 parts). Then it moves backward again (originally noted: it starts fast, then slows by half a degree each day), taking another 51 days to retreat another 3 degrees (originally noted: another record of 30 parts). Then it remains stationary for 36 days. Finally, it moves forward again at a decreasing rate (originally noted: it starts slow, then speeds up by half a degree each day), taking 83 days to cover 7 degrees (originally noted: another record of 290 parts; after these days, it sets at dusk). Saturn moves the slowest, covering less than 14 degrees in one cycle, so its stationary periods are longer than those of Jupiter.

太白：夕見，疾順（入冬至畢立夏，入立秋畢大雪。）一百七十二日。行二百六度。（自入小滿後十日益一度，為定度，初入白露畢春分，差行先疾，自益遲二分，自餘平行。）夏至畢小暑，一百七十二日，行二百九度。（自入大暑已後，五日損一度半，氣盡。）啟蟄畢芒種，七日行七度。（自入夏至後五日，益一，畢於小暑。）寒露初日，二十三日行二十三度。（自後六日損一，畢於小暑。）

The Star of Brightness and Prestige appears in the western sky in the evening. Initially, it moves swiftly in a direct path during the periods from winter solstice to spring beginning and from autumn beginning to great snow, lasting 172 days and covering 206 degrees. After entering the period of small fullness, it progresses at a rate of one degree every ten days. From the beginning of white dew to spring beginning, it moves with a varying speed, starting fast and then decreasing by two parts per day. At other times, it moves at a constant rate. From summer solstice to small heat, it lasts 172 days and covers 209 degrees. After entering great heat, it decreases by one and a half degrees every five days until it reaches its limit. From the beginning of the awakening of insects to the beginning of wheat growth, it moves seven degrees over seven days. After summer solstice, it increases by one degree every five days until small heat. Starting from the first day of cold dew, it covers twenty-three degrees over twenty-three days. After that, it decreases by one degree every six days until small heat. The term "evening appearance" refers to when the Star of Brightness appears in the western sky after sunset, east of the Sun. At this time, it moves swiftly in a direct path, covering nearly more than one degree per day.

順遲，差行（先疾日益遲八分，前疾加度，過二百六度者，唯數損此度。）四十二日行十十度。夕留七日，夕退，西行十日，退五度。（日盡而夕伏）

Then it changes to a slow and steady motion, moving at a decreasing rate (the original commentary notes that it starts off fast and then slows down by 8 parts each day; if the initial rapid phase exceeds 206 degrees, the excess is subtracted back within that segment). It takes 42 days to cover a distance of ten degrees (the original text says 'ten ten degrees,' with an extra 'ten' character). Then it 'stays in the west' for seven days, remaining stationary; afterwards, it begins to 'retrograde westward,' moving west for ten days and receding five degrees. (The original commentary notes that once the days are completed, it disappears in the evening.) The transition of Venus from rapid motion to slow, then to stationary, and then to retrograde movement signifies its gradual approach to the Sun, about to conjunction with the Sun and then moving westward relative to the Sun. After retrograding five degrees, it disappears into the sunlight.

晨見，（初退）西行十日。退五度。（日退半度）晨留七。順遲，差行（冬至畢立夏大雪畢氣盡）四十二日。行三十度。（先遲，日益疾八分，自入小滿已後，率十日損一度，畢芒種。）夏至畢寒露，四十二日行二十七度。（差行依前，自入霜降已後，氣數益一度，畢於小雪。）

The planet Jupiter appears in the east in the morning and moves along its path. (Original note: It initially moves retrograde.) It moves westward for ten days, retreating five degrees. (Original note: It retreats half a degree each day.) Then it remains in the morning sky for seven days. After that, it changes to a slow prograde motion, moving in a retrograde manner (Original note: This applies during the periods from winter solstice to spring beginning and from great snow to the end of the season), lasting forty-two days and covering thirty degrees. (Original note: It starts slow and then increases by eight parts each day; after the beginning of summer, it decreases by one degree every ten days, until the beginning of summer solstice.) From summer solstice to cold dew, it lasts forty-two days and covers twenty-seven degrees. (Original note: The retrograde motion follows the same method as before; after the frost descent, it increases by one degree for each season, until the beginning of snow.) Jupiter's retrograde motion only covers five degrees, which is the shortest retrograde among the five planets, because it is an inner planet, and its retrograde motion occurs only around the time of inferior conjunction.

平行。（冬至畢氣盡，立夏畢氣盡。）十三日，行十三度。（自行一度，自入小寒已後六日，益日及度各一，畢於啟蟄，入小滿，後七日損日及度各一，畢立秋。）雨水初日，二十三日行二十三度。（各後六日損度各一，畢於穀雨。）處暑畢寒露，無此平行。（自霜降後五日，益日及度各一，畢大雪。）度行一百七十二日行二百六。（前遲行，損不滿三十度者，此疾依數益之，處暑寒露差行先遲，日益疾一分，盡餘平行，日盡而晨伏。）

Following this is a period of parallel movement. (Original book note: Applicable from Winter Solstice to the end of the season, and from Start of Summer to the end of the season.) Over thirteen days, it moves thirteen degrees. (Original book note: Each day moves one degree; after entering Minor Cold, each six days increases the days and degrees by one, up to the end of Awakening of Insects; after entering Grain Buds, each seven days decreases the days and degrees by one, up to the end of Start of Autumn.) Starting from the first day of Rain Water, over twenty-three days, it moves twenty-three degrees. (Original book note: After this, each six days decreases the degrees by one, up to the end of Grain Rain.) From Limit of Heat to Cold Dew, there is no period of parallel movement. (Original book note: After Frost Descent, each five days increases the days and degrees by one, up to the end of Greater Snow.) Finally comes the rapid movement, which covers 206 degrees over 172 days. (Original book note: If the preceding slow and gradual period decreases by less than thirty degrees, this rapid period will make up the difference; during Limit of Heat and Cold Dew, the movement is adjusted, starting slow and then increasing by one part each day, until the parts are exhausted and it returns to parallel movement; when the days are completed, it disappears before dawn.)

辰星：夕見，順行，十二日行二十一度六分。（日行一度，五百三分。）大暑畢處暑，十二日行一十七度一十分。（日行一度，二百八十分。）平行七日，行七度。（自入暑後二日，損日及度各一，入立秋無此平行。）順遲，行六日，行七度。（四月行二百二十四分，疾行分一十七度者，無此遲行。）夕留五日。（日盡而夕伏）

The planet Chén Xīng (Water Star) appears in the western sky at dusk. Initially it moves forward, covering 21 degrees and 6 parts in twelve days. (Original note: It moves 1 degree and 503 parts each day.) From the beginning of Dà Shǔ (Great Heat) to Chǔ Shù (Limit of Heat), it covers 17 degrees and 10 parts in twelve days. (Original note: It moves 1 degree and 280 parts each day.) Then it moves evenly for seven days, covering seven degrees. (Original note: After Dà Shǔ, it decreases by one day and one degree every two days; after Lì Qiū (Beginning of Autumn), this stage of even motion ceases.) It then moves forward slowly for six days, covering seven degrees. (Original note: It moves 224 parts each day; if the previous rapid motion only covered 17 degrees, this stage of slow motion does not occur.) Finally, it remains visible for five days in the evening. (Original note: After these days, it disappears at dusk.) Since the Water Star is closest to the Sun, its visibility period is only about twenty days, hence the short duration of each stage.

晨見，留五日，行二度。（四日行二百四分，自入大寒畢行於啟蟄，無遲行。）順遲行六日，平行七日，行七度。（日行一度，入大寒已後二日，損日及度各一，入立春無平行。）順，疾行十二日，行二十一度。（六日一行度，五百三分。）前無遲行者，十二日，行一十七度一十分。（日行一度，二百八十分，各日盡而夕伏。）

The planet Chén Xīng (Morning Star) appears in the east in the early morning and moves along a certain path. It first remains for five days and moves two degrees. (Original note: It moves 204 parts over four days; during the period from the Great Cold to the Awakening of Insects, there is no slow movement.) Then it moves slowly for six days, followed by a period of uniform movement for seven days, covering seven degrees. (Original note: It moves one degree per day; after the Great Cold, it decreases one day and one degree every two days; after the Beginning of Spring, this uniform movement does not occur.) Then it changes to a rapid movement in the forward direction, covering 21 degrees in twelve days. (Original note: It moves one degree and 503 parts per day.) If there is no prior slow movement, it covers 17 degrees and 10 parts in twelve days. (Original note: It moves one degree and 280 parts per day; after completing the days, it disappears and is not visible.) The last sentence reads 'xī fú' (setting hidden), which does not match the morning appearance, so it should be 'chén fú' (morning hidden), indicating an error in the original text.

臣等謹案，《九執曆》法，梵天所造，五通仙人承習傳授，肇自上古，百博義二月春分朔，於時曜躔婁宿，道歷景止，日中氣和，庶物漸榮，一切漸長，動植歡喜，神祗交泰，棹茲令節，命為曆元。

The ministers Qutan Xida and others respectfully state: The "Nine Stars Calendar" is said to have been created by the Brahma, passed down through generations by sages who possess the five supernatural powers. Its origin is extremely ancient. It begins with the spring equinox of the second month of "Bai Boyi", at which time the sun is in the Bond mansion, and the shadow of the sun is exactly at the position determined by the calendar, with day and night equal, cold and heat in harmony, and all things gradually flourishing, everything growing and expanding, animals and plants all rejoicing, and heavenly spirits and earthly deities harmoniously communicating. Because of the auspiciousness of this time, it was established as the beginning of the calendar cycle, the starting point for all calculations. The term "Nine Stars" refers to the sun, moon, five planets, and the two additional celestial bodies, Rohini and Kethu, making a total of nine celestial bodies. This calendar is named after these nine celestial bodies which govern time.

竊稽開設法數，建立章率，述而不作，信而好古，竊簡易之智陳，得希夷之妙術，河帶山礪，久而逾新，藏往知來，挹而靡竭。嘗試言之，蓋以其國人多好道，苟非其氣，雖曰子弟，終不傳也。臣等謹憑天旨，專精鑽仰，凡在隱秘，咸得解通，今削除繁冗，開明法要，修仍舊貫，緝綴新經，備列算術，目標如左，自作口訣，亦題目，附本章。

The ministers have examined the various methods and established rates of this calendar. They merely follow the saying from the "Analects" of Confucius, "to narrate rather than to create, to be faithful and fond of antiquity," transmitting them as they are without any reckless alteration. Among these, they have selected the simple and clear principles, and obtained the subtle and profound techniques. Even if the Yellow River were as thin as a sash and Mount Tai were ground into a grinding stone, this method would become more evident with the passage of time. It can both understand the past and predict the future, and its resources are inexhaustible. Let us now explain why: the people of India often love the Dao; if they are not of the same spirit, even their own children would not be willing to pass on the knowledge. The ministers have carefully followed the imperial decree, devoted themselves to research and inquiry, and have thoroughly understood all the hidden aspects. Now, we have removed the redundant parts, clarified the essential points, and still arranged them in an orderly manner, compiling them into this new scripture. All the mathematical calculations are listed afterwards, with the following headings. Each heading has its own mnemonic verse, and the titles are attached to this chapter.

算字法：樣（一字、二字、三字、四字、五字、六字、七字、八字、九字）點，右天竺算法，用上件九個字，乘除其字，皆一舉禮而成。凡數至十，進入前位，每空位處，恆安一點，有間咸記，無由輒錯，運算便眼，趁須先及歷度。

The method of using numerals is as follows: the pattern is as shown above (original book note: one character, two characters, three characters, four characters, five characters, six characters, seven characters, eight characters, nine characters), and there is also a dot. This is the Indian method of calculation, which uses only these nine numerals; multiplication and division can all be written with a single stroke, without needing to use Chinese counting rods as in China. Whenever a number reaches ten, it carries over to the previous position; whenever there is an empty position, a dot is placed there as a marker, making each position clearly distinct, so there is no confusion. This method of calculation is clear and straightforward, and is suitable for use in calculating the calendar and celestial phenomena. What is recorded here is exactly the Indian numerals, the decimal place value system, and the use of a dot to represent a zero position, which is the earliest actual record found in Chinese classics.

右天竺度法，三百六十。權符管律，更無奇剩。（中國勝五度四分度一，今斗（闕）家術源天竺，則棄沒日，不入曆度。中國則收沒日，推日曆度。由是度數不合，彼此有異。又凡稱沒者，虛數之謂也。所以二十四氣，遇沒十六日移節，在漏刻，遇沒，十日移然。天地所產，人最靈焉，骸骨之數，有法象乎，玩同管律，理亦詳矣。）

The above is the method of measurement used in India: the entire celestial circle is divided into 360 degrees, exactly matching the number of the bamboo pipes used in the system of pitch-pipes, with no remainder. (Original note: The Chinese system of celestial degrees is five and a quarter degrees more than the Indian system. Today, the Indian system omits the 'no-day' (mei ri) and does not include it in the calculation of the calendar; the Chinese system includes the 'no-day' and uses it to calculate the daily progression of the calendar. Therefore, the two systems do not align, and there are differences between them. Any mention of 'no-day' is an artificial number. Among the twenty-four solar terms, when encountering a 'no-day', the season changes every sixteen days; in the water clock system, when encountering a 'no-day', the time changes every ten days. The things generated by heaven and earth, with humans being the most refined and delicate, the numerical structure of the human body's bones seems to have its own pattern, matching the number of the bamboo pipes. The principles behind this are also well explained.)

推積日及小餘章：（閏及甲子算、七曜直等，在術中。）上古積年，數太繁廣，每因章首，遂便刪除，務從簡易，用舍隨時。今起明慶二年丁巳歲二月一日，以為歷首，至開元二年甲寅歲，置積年五十七算。（甲子五十算）術曰：置積年。（假令推開元二年甲寅歲，事置五十七算為積年；若推向前一年癸丑歲，事即減一算；若推向其年三月五日事，既歷後一年，乙卯歲事，即加一算，他皆仿此。）

This chapter deals with the calculation of accumulated days and the remainder of the day. (Original note: The calculation of leap months, the determination of the Jiazi stem-branch, and the method of determining the seven luminaries for each day are all included in this method.) The accumulated years since ancient times are too large in number, so whenever a new chapter begins, they are simplified to make the calculation easier, with the selection and omission varying according to the time. Now, taking the first day of the second month in the year Dingsi (丁巳) of the Mingqing era as the starting point of the calendar, the accumulated years up to the year Jiayin (甲寅) of the Kaiyuan era are fixed at 57. (Original note: The number of Jiazi stem-branch cycles is 50.) The method states: first, write down the accumulated years. (Original note: For example, if calculating the year Jiayin (甲寅) of the Kaiyuan era, write 57 as the accumulated years; if calculating the previous year Guichou (癸丑), subtract one; if calculating an event on the fifth day of the third month of that year, since it has already passed into the next year Yimao (乙卯), add one. Other examples follow the same pattern.)

以十二乘之，加自入年已來所積月。（假令推其年三月五日事，即歷起二月一日為首，於二乘訖數上，更加一算，即是加入年所經一個月了。）加訖，重張位下，以七乘之；恆加一百三十二，以二百二十八除之，得閏月。（不盡，為閏餘，既未滿閏，棄之。）以閏月加上位，為積月，以三十乘之，加自入月已來所經日；（假令推三月五日事，即於三十乘訖數上，更五算，即是加入月所經五日了。）

Multiply the number of years by 12, then add the number of months that have already passed in the current year. (Original note: For example, if calculating an event on the fifth day of the third month, and the calendar starts on the first day of the second month, add one after multiplying by 12, which accounts for the one month already passed in the current year.) After adding, arrange another set of the same number below, multiply it by 7, then add 132 fixedly, and divide by 228. The quotient obtained is the number of leap months. (Original note: If the division is not exact, the remainder is considered the leap remainder; since it is not yet a full leap month, it is discarded.) Add the number of leap months to the accumulated months above to get the total accumulated months; then multiply by 30 and add the number of days that have already passed in the current month. (Original note: For example, if calculating an event on the fifth day of the third month, add five to the number after multiplying by 30, which accounts for the five days already passed in the current month.)

重張位下位，十一乘之，恆加差四百二十九，一百六十九以七百三除之，得自入曆已來所經小月。（其小月，梵雲欠夜。）不盡為小餘。（其小，梵雲小月餘。）以小月減上位，為積月。其小餘及積日，各列為位，又置積日，以六十除，棄之餘。從庚申算上命之，得甲子之次，又置積日，以七除，棄之餘，從熒惑月，命得之七曜直日次。（一算為熒惑，二算為辰星，三算為歲星，四算為太白，五算為填星，算定為日。）其七曜直用事法，別具本占。

Now, arrange another set below. Multiply by 11, add the fixed difference of 429 (the text below also mentions 'one hundred sixty-nine', which is likely a correction or a later addition), then divide by 703. The result is the number of small months that have passed since the beginning of the calendar. (Original note: A small month is called 'kianya' in Sanskrit.) If the division is not exact, the remainder is called the small remainder. (Original note: This small remainder is called 'small month remainder' in Sanskrit.) Subtract the number of small months from the number above to get the accumulated months. The small remainder and the accumulated days are each arranged as a single digit. Then take the accumulated days, divide by 60, discard the whole number and keep only the remainder, counting from Gengshen to determine the sequence of the Jiazi stem and branch for that day. Take the accumulated days again, divide by 7, discard the whole number and keep only the remainder, counting from Yinghuo to determine the sequence of the seven planetary days. (Original note: One is Yinghuo, the fire planet; two is Chenxing, the water planet; three is Suixing, the wood planet; four is Taibai, the metal planet; five is Tianxing, the earth planet. After calculation, this determines the planetary day for that day.) As for the divination method based on the planetary day, see the relevant divination texts in this book.

推中日章：凡在梵曆，大例分積滿六十，成一度；其度積滿三十，成一相；其相積滿十二，乘棄之，他皆仿此。（其相，梵音呼為星施，是聚義也。承前或譯為次，或譯為辰，今從相也；其度，梵音呼為薄伽，承前譯為大分，今從度也；其分，梵音呼為立多，承前譯為小分，今從分也。）

In this chapter on the calculation of days and months, it is generally stated in the Indian calendar system that when the accumulation reaches 60, it advances to one degree; when the degrees accumulate to 30, it advances to one phase; when the phases accumulate to 12, it completes a full cycle and starts again. This pattern is followed in the subsequent calculations. The term 'phase' is derived from the Sanskrit 'starasi', meaning aggregation; previously it was translated as 'next' or 'hour', but now it is translated as 'phase'. 'Degree' comes from the Sanskrit 'bagga', previously translated as 'great division', but now as 'degree'. 'Division' comes from the Sanskrit 'litta', previously translated as 'small division', but now as 'division'. The twelve phases refer to the twelve signs of the Yellow Path; each phase consists of thirty degrees, making a total of three hundred and sixty degrees, which differs from the Chinese system of twelve times with a full circle of three hundred and sixty-five and one fourth degrees.

術曰：置積日重張位，下位以十二乘，以九百除之，得沒度。（其沒度，中國在曆法為沒日者是也。）不盡十五，除之，得沒分。恆加差三十分，（其分薄六十成一度）以沒度減上積日，又每退積日一，置為六十分，以沒分減之；減餘列為中日分位，其減訖積日，以三百六十除之，得自入曆已來所經年；棄之（假令置積年五十七算，還只除得五十七。）餘，以三十除之，得相，不盡為度。其相及度，與前所列中日分並之，置為日中位。（置位皆三重，從戴而列之，其下位列分，其中位列度，其上位列相，他皆仿此。）

The text says: Arrange the accumulated days, and then arrange another set below it. For the lower set, multiply by 12 and divide by 900; the result is called "mei du" (no degree). If the division is not exact, divide again by 15 to get "mei fen", and add a fixed 30 minutes. (Note: Here, "fen" is a unit of measure, and 60 minutes make one degree.) Use the "mei du" to subtract from the accumulated days above. If it is not enough, subtract one day from the accumulated days, which is equivalent to 60 minutes, and then subtract the "mei fen". The remainder is placed in the "zhong ri fen" position. After subtraction, take the remaining accumulated days and divide by 360; the result is the number of years since the start of the calendar, which is discarded. (Note: For example, if the accumulated years are 57, the result is still 57.) Take the remainder and divide by 30 to get "xiang"; the remainder is the "du". Combine the obtained "xiang" and "du" with the previous "zhong ri fen" and place them in the "zhong ri" position. (Note: All positions are arranged in three layers, from bottom to top: the lower layer is "fen", the middle layer is "du", and the upper layer is "xiang", and this pattern is followed in all other positions.)

推中月章：術曰：置小餘重張位，下位二十五餘之，得者加上位；加訖，以六十除之，得度；不盡為分，其度分列為位；又置自入月已來所經日，（假令前推積日，加自入月五算，推此亦須準前數置止算。）以十二除之，以三十除之，得相，不盡為度。以其相及度，與前所列度及分並之，又與中日並之，置為中月位。

In this chapter on determining the middle month, the method says: first arrange the small remainder, and then place another copy below it. Below, divide by 25, and add the result to the upper copy. After adding, divide again by 60 to get degrees, and if there is a remainder, it becomes minutes. Place these degrees and minutes together. Then, arrange the number of days since the beginning of the month (note: for example, if previously added five days for entering the month, the same number should be placed here). Divide this by 12, then divide again by 30 to get phases, and if there is a remainder, it becomes degrees. Combine the obtained phases and degrees with the previously arranged degrees and minutes, and then combine them with the middle day number to form the 'middle month' position. The middle day represents the mean position of the sun, and the middle month represents the mean position of the moon, both without any correction for their irregularities.

推高月章：術曰：置積日，以九除之，得度；餘以六十乘之，依前除之，謂亦九除也，得分；其度以三百六十除，棄之餘以三十除之，得相，不盡為度，其相及度兼分列為位。又置積日，以六十除之，得分，（其分滿六十，成一度。）以其分並前所列分位，恆加差十八相二十六度四十一分，一相十三度四十五分，置為高月位。

To calculate the position of the high moon, first arrange the accumulated days, divide them by 9 to get the degrees; take the remainder and multiply it by 60, then divide by 9 again as before to get the minutes. Divide the obtained degrees by 360, discard the whole number, and divide the remainder by 30 to get the phases; if there is a remainder, it is the degrees. Combine the phases, degrees, and minutes into one position. Then arrange the accumulated days again, divide by 60 to get the minutes, and add them to the previously obtained minutes. Finally, add the fixed difference of 18 phases 26 degrees 41 minutes and 1 phase 13 degrees 45 minutes to form the position of the high moon. The high moon refers to the position of the moon's apogee. It is compared with the mean moon to determine the moon's varying speed.

推月藏章：（承前或譯為月損益率）術曰：置中月，以高月減之，（如不是減於月中相位，上更加十二相藏之。）減訖，置為月藏位。推日藏章：（承前或譯為日損益率）術曰：置中日，減二相二十度，（如不足減，於中月相位上更加十二相減之也，他皆仿此。）減訖，置為日藏位。

This chapter explains how to calculate the "month concealment." The original notes mention that this was previously translated as "monthly rate of loss and gain." The method states: first set up the mean moon, then subtract the high moon from it. If the subtraction is not possible, add twelve phases to the mean moon's phase before subtracting. The result of this subtraction is placed as the "month concealment" position. This chapter explains how to calculate the "day concealment." The original notes mention that this was previously translated as "daily rate of loss and gain." The method states: first set up the mean sun, then subtract two phases and twenty degrees. If the subtraction is not possible, add twelve phases to the mean sun's phase before subtracting. The result of this subtraction is placed as the "day concealment" position. "Concealment" refers to the angular distance from the apogee. The day concealment and month concealment are the arguments used in the subsequent calculation of the corrections for the sun's and moon's apparent slow and fast motion.

推定日章：日段六：第一段，（三十五）第二段，（三十二）第三段，（二十七）第四段，（二十二）第五段，（十三）第六段，（五）右一段，每管十五度，兩段管一相，凡在六段，用管三相。

In this chapter on determining the fixed day, the correction numbers for the Sun are divided into six sections, each with the following values: the first section 35, the second 32, the third 27, the fourth 22, the fifth 13, and the sixth 5. Each section covers fifteen degrees, so two sections cover thirty degrees, and six sections cover ninety degrees, which corresponds to a quarter of the orbit from the apocenter to the pericenter. These six numbers decrease gradually in each section, indicating that the increment of the Sun's variation correction becomes smaller as it approaches the turning point. This is a correction table with a linear decreasing difference.

術曰：置日藏，若相及度位俱定，唯有分者；置分，以第一段三十五乘之，以九百除之，得分。（凡此分滿六十成一度）恆視日藏位，（相定及一二三四五相者，命曰羖首；六七八九十及十一相者，命曰秤首。又，凡在梵曆，相定是一相法一相，是二相法二相，是法，他皆仿此。）得羖首，即以此度分損中日位；得秤首，即以此分益中日位。（以度損益度以損益分）如是損益訖，置為定日位。

The text says: When the solar position is set, if both the phase and degree positions are fixed and only the minute remains, take this minute, multiply it by 35 from the first section, then divide by 900. The result is the minute. (Original note: If the minute reaches 60, it becomes one degree.) Continuously observe the phase where the solar position lies: (Original note: Phases zero and one, two, three, four, five are called "Goat Head"; phases six, seven, eight, nine, ten, and eleven are called "Scale Head." Also, in the Indian calendar, if the phase is determined as one phase, it is treated according to the method of one phase; if it is determined as two phases, it is treated according to the method of two phases, and so on.) If it is Goat Head, subtract the obtained degrees and minutes from the mean solar position; if it is Scale Head, add the obtained degrees and minutes to the mean solar position. (Original note: Subtract degrees from degrees and add degrees, subtract minutes from minutes and add minutes.) After completing the addition or subtraction, the result is placed in the "fixed solar position" position, which is the actual position of the Sun after correction according to the solar path. Goat Head refers to the beginning of Aries, and Scale Head refers to the beginning of Libra, which are two opposite points on the Yellow Path.

推定月章：（承前為月或）月段六：第一段，（七十七）第二段，（七十一）第三段，（六十一）第四段，（四十七）第五段，（三十）第六段，（十）右一段，每管十五度，兩段管一相，凡在六段，用管三相。術曰：置月藏，若相及度位俱定，唯有分者；置其分，以第一段七十七乘之，以九百除之，得分。（凡此分滿六十成一度）恆視月藏位，（相定段一二三四五相者，命曰羖首；六七八九十及十一相者，命曰秤首。）得羖首，即以此度分損中月位；得秤首，即以此度分益中月位；如是損益訖，置為定月位。

This chapter explains the method of determining the fixed moon. The correction for the moon is divided into six sections: the first section is 77, the second is 71, the third is 65, the fourth is 47, the fifth is 30, and the sixth is 10. Each section covers fifteen degrees, and two sections cover one phase, with all six sections covering three phases in total. The text says: when the moon's position is determined, if both the phase and degree are known and only the minute remains, take this minute, multiply it by the first section's number 77, then divide by 900. The result is the minute. (Note: here, when the minute reaches 60, it becomes one degree.) Always observe the phase in which the moon's position lies: (Note: phases zero and one, two, three, four, five are called "Goat Head," and phases six, seven, eight, nine, ten, and eleven are called "Scale Head.") For all Goat Heads, subtract the obtained degrees and minutes from the mean moon. For all Scale Heads, add the obtained degrees and minutes to the mean moon. After completing the addition or subtraction, the result is placed as the "fixed moon" position. The six sections of the moon's numbers are more than double those of the sun, because the difference in the moon's slow and fast movement is much greater than the difference in the sun's expansion and contraction.

敘三相已下藏例：（日與月並同此法）置藏位，（若相定位，其度不滿十五兼有入者，而置其度，以六十乘之內分，在梵曆，是名通作分也，亦以第一段乘之，以九百除之，得分；其分命用，並亦準前。）置藏位，（若有十五度已上者，直將除棄十五度訖，十乘度內分也，他皆仿此。以次第二段乘之，準前除也，他皆仿此。以此第二段乘之，準前除之。凡言準前者，用舊術也；今亦用九百除之，他皆仿此。得分，其分加上位，不滿六十成一度，其度及分命用，並已準前。）

Here are examples of how to calculate the hidden numbers when the number of phases is below three. (Original book note: both the hidden sun and hidden moon use the same method.) First, set out the hidden position: (Original book note: if the position is determined but the degree is less than fifteen and has additional minutes, take this degree, multiply it by 60 and convert it into minutes — in the Indian calendar this is called 'general minutes' — also multiply by the number of the first segment, divide by 900 to get the minutes; the way these minutes are named and used is the same as before.) Then set out the hidden position again: (Original book note: if the degree is more than fifteen, subtract fifteen first, then proceed as before by multiplying the degree and converting it into minutes, and so on; then use the number of the second segment for multiplication, still following the previous method of division. Whenever it says 'according to the previous method,' it means continuing with the old technique; now, all calculations are uniformly divided by 900, and the same applies below. The result is minutes, which are added to the previous position. If the total is less than 60, it becomes one degree; the way the degree and minutes are named and used is the same as before.)

置藏位，（若有一相十五度已下者，直除去一相訖，即併到第一段，第二段為上位餘，通分，內子以次段乘之，自餘命用，並亦準前。）置藏位，（若有一相十五度已上者，直除訖，一相兼十五訖，即並第一段，訖至第三段，為之位，旬餘命用，並亦準前。）置藏位，（若有二相十五度已下者，除訖，二相訖，而開列第一段，迄至第四段，為上位，自餘命用，並亦準前。）置藏位，（若有二相十五度已下者，有除訖二相兼十五度訖，即列第一段迄至第五段為上位，自餘命用，並亦準前。）置藏位，（若有三相更無餘度分者，直棄三相訖，即並列第一段迄至第六段為上位，自餘命用，並亦準前。）

Now arrange the hidden positions: (Original book note: If it is one phase and less than fifteen degrees, first subtract one phase, combine the numbers of the first and second segments as the upper position, the remaining folded degrees are then multiplied by the number of the next segment, and the rest is handled as before.) Now arrange the hidden positions: (Original book note: If it is one phase and more than fifteen degrees, subtract one phase together with fifteen degrees, combine the numbers from the first to the third segment as the upper position, and the rest is handled as before.) Now arrange the hidden positions: (Original book note: If it is two phases and less than fifteen degrees, subtract two phases, combine the numbers from the first to the fourth segment as the upper position, and the rest is handled as before.) Now arrange the hidden positions: (Original book note: If it is two phases and more than fifteen degrees, subtract two phases together with fifteen degrees, combine the numbers from the first to the fifth segment as the upper position, and the rest is handled as before. The first two characters of this entry should still be '已下', which repeats the previous entry, so it should be a mistake for '已上'.) Now arrange the hidden positions: (Original book note: If it is exactly three phases with no remaining degrees, directly remove three phases, combine all the numbers from the first to the sixth segment as the upper position, and the rest is handled as before.) This entire method actually involves adding the six segments' corrected numbers sequentially according to the size of the argument, taking the accumulated value, which is the same as the cumulative difference table used in later calendar systems.

敘三相已上藏例：（日與月並用此法，凡在梵曆，（闕）皆仿此。）置藏位，（若有三四五相者，別置六相，以減之，減餘相度分至於排段命用，並亦準前。此承前（闕）云：傍五六相，以本減傍，去上張下，命用者是也。）置藏位，（如有六七八相者，直棄六相，餘相度分至於排段，命用，並亦準前。）置藏位，（如有九十及十一相者，別置十二相減之，減餘相度分至於排段，命用，並亦準前。）

Here is an example of how to handle the concealed numbers when they are above three phases. (Original note: Both the day and month concealed numbers use this method. In the Indian calendar, there is a missing text here, and the rest follows this pattern.) Arrange the concealed position: (Original note: If it is three, four, or five phases, then arrange six phases and subtract them. The remaining phases, degrees, and fractions are used according to the previous segment arrangement method. This is what was previously mentioned: listing five or six phases, subtracting the listed number from the original number, removing the upper position and arranging the lower position, then using it.) Arrange the concealed position: (Original note: If it is six, seven, or eight phases, then directly remove six phases, and the remaining phases, degrees, and fractions are used according to the previous segment arrangement.) Arrange the concealed position: (Original note: If it is nine, ten, or eleven phases, then arrange twelve phases and subtract them. The remaining phases, degrees, and fractions are used according to the previous segment arrangement.) In this way, no matter where the reference number falls within the twelve phases, it can be reduced back to the range of zero to three phases for table lookup, because the correction number within one cycle is symmetric and reciprocal.

推晝刻及夜刻章：（梵曆晝夜刻，共有六十刻，凡一刻即六十分。）刻段三：第一段，（一百六十）第二段，（一百三十二）第三段，（五十四）右一段，每管一相，凡在三段，用管三相。（至於排段別位，受及乘除，敘例命用，亦同前定日法。）術曰：置定日，若相空，即置其度，通作分；以第一段一百六十乘之，以一千八百除之，得分。（其分滿六十成一刻）其分一，六十除之，得刻，不盡為分，恆加三十刻，置為夜刻分位。

This chapter discusses the calculation of daytime and nighttime divisions. (Original note: The Indian calendar divides a day and night into sixty divisions, with each division further divided into sixty parts, differing from China's system of one hundred divisions.) The correction numbers for the divisions are divided into three sections: the first section is 160, the second is 132, and the third is 54. Each section corresponds to one phase, and the three sections together cover three phases. (Original note: As for arranging the sections, placing them in order, how to take numbers and perform multiplication and division, and the rules for using them, they are the same as in the previous method for determining the day.) The text says: set out the fixed day, if the phase is zero, take its degree and convert it into parts, multiply it by the first section's 160, then divide by 1800, to get the result. (Original note: Here, when the parts reach 60, they form one division.) Divide the resulting parts by 60 to get the divisions, and any remainder is expressed as parts; then add thirty divisions fixedly, and place it as the 'night division parts'. Thirty divisions are exactly half of sixty divisions, which is the length of the night at the time when day and night are equal.

又恆別置六十刻，以所置刻及減之，減餘刻及分，置為短刻分位。（凡春分後，晝漸長，夜漸短，其長刻晝也，短刻夜也，春分羖首也；秋分後，夜漸長，晝漸短，其長刻夜也，其短刻晝也，秋分秤首也。）其長刻及其短刻及分，合置為全晝全夜刻位。其全晝全夜刻及分，並各半之置為半晝半夜位。（置定日，若有一相，直棄一相，即列第一段，一百六十為上位，餘通作分，以第二段一百二十乘，以一千八百除，自餘命用，並亦準前。置定日，若有二相，亦直棄二相，並列第一段，第二段二百九十為上位，餘通作分，以第三段五十四乘之，以一千八百除之，自餘命用，並亦準前。）

Another fixed arrangement of sixty divisions is placed separately. The number of divisions obtained earlier is subtracted from it, and the remaining divisions and fractions are arranged as the "short divisions and fractions" position. (Original note: After the spring equinox, daylight gradually increases while night decreases; thus, the longer part is day and the shorter part is night. The spring equinox belongs to the beginning of the goat. After the autumn equinox, night gradually increases while daylight decreases; thus, the longer part is night and the shorter part is day. The autumn equinox belongs to the beginning of the scale.) The longer part and the shorter part, along with their fractions, are combined and arranged as the "full day and night divisions" position. Then, take half of the divisions and fractions of the full day and full night and arrange them as the "half day and half night" position. (Original note: When arranging the fixed day, if there is one phase, simply remove one phase, and set the first segment as 160 as the upper position. The remaining is converted into fractions, multiplied by the number of the second segment, divided by 1800, and the rest is handled as before. When arranging the fixed day, if there are two phases, simply remove two phases, combine the first and second segments to get 290 as the upper position. The remaining is converted into fractions, multiplied by the number of the third segment, which is 54, divided by 1800, and the rest is handled as before. However, the number of the second segment here is stated as "one hundred twenty," which does not match the previous "one hundred thirty-two," indicating an error or omission.)

推月域章：（承前或譯為明量，確據梵音，呼為勃夜，其義雲月食限也。謂每經一晝一夜，月行吞得度數之量也。譯為域者，亦得劑域之限也。此月域內兼日行，分合在其中。）術曰：置今日定月，以昨日定月減之，餘通作分，凡置為月域位。（又法置七百九十為本位，又取通，乘月段以九乘之訖，直棄一位，餘者恆視月藏三四五六七八相者，命曰蟹首；九十十一兼相位，定及一二相，命曰龜首。）蟹首益本位，龜首損本位，即是月域。

In this chapter, the calculation of the lunar domain is discussed. The original note explains that previously it was translated as "ming liang" (brightness and measure), but according to the Sanskrit pronunciation, it should be read as "Boye," meaning the lunar limit, that is, the amount of degrees the moon travels through each day and night. The term "domain" is also used to indicate a limit or boundary. Within this lunar domain, the movement of the sun is also included, so both the sun and the moon's portions are combined. The method says: set out the fixed moon of today, subtract the fixed moon of yesterday, convert the remainder into parts, and this becomes the "lunar domain" position. Another method: set 790 as the base, then take the total number of parts multiplied by the lunar segment, multiply by 9, and discard one digit. The remainder is then checked against the lunar concealment. If it falls in the third, fourth, fifth, sixth, seventh, or eighth phases, it is called "crab's head." If it falls in the ninth, tenth, eleventh phases, or the zeroth, first, or second phases, it is called "turtle's head." For those with "crab's head," add it to the base; for those with "turtle's head," subtract it from the base. The result is the lunar domain, which represents the actual degree the moon travels through in one day and night.

推日域章：（承前譯為日法，明量其義，日以減卻日行分，故標日為前也。）日行分法，（相位定及一相、二相、三相，行分五十七；四相，行分五十八；五相，行分五十九；六相，行分六十；七相、八相、九相，行分六十一；十相，行分六十；十一相，行分五十九。）術曰：恆視定月相位，以前行分，於月域數內（假令相位空，即於月域數內減卻行分五十七，他皆仿此。）減訖，置為日域位。

This chapter explains the method of calculating the "day domain." (Original note: Previously translated as "day method," taking the meaning of measuring brightness; since it involves subtracting the movement of the sun from the moon domain, the character "day" is placed before.) The method for calculating the daily movement of the sun is as follows: (Original note: For phases zero, one, two, and three, the movement is 57; for phase four, it is 58; for phase five, it is 59; for phase six, it is 60; for phases seven, eight, and nine, it is 61; for phase ten, it is 60; for phase eleven, it is 59.) The text states: observe the fixed position of the moon's phase at any time, take the corresponding movement from the table above, and subtract it from the moon domain number. (Original note: For example, if the phase is zero, subtract the movement of 57 from the moon domain number, and so on for the rest.) After subtraction, the result is placed in the "day domain" position. The day domain actually represents the degrees the moon travels relative to the sun in one day and night. It is the fundamental speed used in calculating the time of conjunction and solar eclipses.

推宿刻章：（宿法，於此術中，凡是宿，平等為八百分，天竺每以月臨宿，占其日一，即休咎仍取其宿用事，又唯用二十七宿，命婁為始，失牛，終奎。其牛宿，恆著吉祥之時，不拘諸宿之例，別有占算法。）術曰：置定月，通作分，（謂三十乘內度，六十乘度內分，他皆仿此。）以八百除之，得已通宿次，餘者是用宿。（假令除得為婁二百，胃三百，即是已過宿次，餘者是所臨畢宿用事也，他皆仿此。）以六十乘之，以月域除之，得宿刻。又乘，又除，（謂亦以六十乘，亦以月域除，他皆仿此。）得分，置其刻及分，為宿刻位。

This chapter explains the method of calculating the "host and moment" (宿刻). The method used here divides each of the twenty-eight mansions into eight hundred equal parts. The Indians often use the mansion where the moon is located to determine the auspiciousness of the day, and they use that mansion for the current activity. They only use twenty-seven mansions, starting with the Bond mansion (娄宿) and ending with the Legs mansion (奎宿), excluding the Ox mansion (牛宿). The Ox mansion usually governs a time of good fortune and is not subject to the general rules of the other mansions, but has its own method of calculation. The text says: first, determine the fixed moon, then convert the degrees into minutes. Multiply 30 by the degrees and 60 by the minutes, and proceed in this way. Divide by 800; the quotient is the number of mansions already passed, and the remainder is the mansion currently occupied. For example, if dividing results in 200 for the Bond mansion and 300 for the Stomach mansion, then those are the mansions already passed, and the remainder indicates the current mansion, which is the Net mansion (毕宿), and so on. Multiply the remainder by 60, then divide by the moon's domain to get the "host moment" (宿刻); multiply again and divide again to get the minutes. When these are arranged, they form the "host moment" position.

推宿斷章：術曰：置半（闕）刻及分，兼全晝刻及分，以宿刻及分減之；先減夜刻，（謂從夜半子時，向亥申至於戌酉而減之。）如夜刻盡，餘以減晝刻，（亦謂從酉向申未等而有減之也。）如減夜不盡，即直只減夜，不減晝也。知夜晝俱盡，入以減往夜刻。（謂從卯向寅丑等而減之也）如減往夜全刻亦盡，餘以減往晝刻。（謂從酉向申等而減之也）凡減晝夜刻，至所止處，是正著兩宿界中央刻時，（謂已遇宿位未所臨宿之初也）以此時名宿斷時，置其刻及分，為宿斷位。

This chapter explains the method of determining the "host break time" (宿断时). The technique says: first, set out the time and minutes of the middle of the night, then combine it with the time and minutes of the entire day. Use the previously calculated time and minutes of the host to subtract from it. First subtract the night time (note: this is subtracted by counting backward from the middle of the night, starting from the hour of Hai and moving toward You and Xu). If the night time is fully subtracted, then use the remaining time to subtract from the day time (note: this is subtracted by counting backward from You, moving toward Shen and Wei). If the night time is not fully subtracted, only subtract the night time, without touching the day time. If both day and night times are fully subtracted, then use the remaining time to subtract from the previous night's time (note: this is subtracted by counting backward from Mao, moving toward Yin and Chou). If the entire time of the previous night is also fully subtracted, then use the remaining time to subtract from the previous day's day time (note: this is subtracted by counting backward from You, moving toward Shen). Whenever you subtract the time of day and night in this way, the point at which the subtraction stops is the midpoint of the boundary between two hosts (note: this is the beginning of the division between the host that has already passed and the one that is about to arrive). This moment is called the "host break time" (宿断时). Setting out the time and minutes of this moment is called the "host break" (宿断).

推節刻章：（或譯為著蝕時，或譯為日節，中國名為加時，梵雲即初，詳意義如竹以節隔其間。今日一晝一夜（闕）其昨日一晝一夜相分，每刻之處，亦如竹節，由是名焉。）術曰：置定月，以定日減之，（如不足減，於定月相位上更加十二相減之。）減餘通作分，以七百二十除棄之，（其棄者，是加自入月已來日，若少於本數，名未來節數，若多於本數，名過去節數。）餘者名為節除，以六十乘，以日域除之，得節刻，不盡，又乘，又除，（凡言又乘又除，皆是依前數乘之，依前數除之。今此以六十乘，以日域除，他皆仿此。）得分，置其刻及分，為節刻位。

This chapter discusses the calculation of the "section and moment" (节刻). The original text notes that this term has been translated as "著蚀时" by some, "日节" by others, and in Chinese it is called "加时," while in Sanskrit it is transliterated as "即初." Upon closer examination, the term is like the joints of bamboo that divide the bamboo into segments: the current day and night and the previous day and night are separated by these joints, and each moment is like a joint, hence the name. The method states: set the fixed month, subtract the fixed day from it (original note: if it is not enough to subtract, add twelve phases to the fixed month before subtracting). The remainder is converted into parts by multiplying by 720 and discarding the integer part (original note: the discarded part represents the number of days since the beginning of the month; if it is less than the original number, it is called "future phase number," if it is more, it is called "past phase number"). The remainder is called "section remainder." Multiply this by 60 and divide by the day domain to get the "section and moment." If there is a remainder, multiply it again by 60 and divide again by the day domain (original note: whenever it says "multiply and divide again," it means to multiply by the previous number and divide by the previous number; here it means to multiply by 60 and divide by the day domain, and so on for the following steps). The resulting minutes and parts are arranged to form the "section and moment" position.

推節斷章：（謂正著蝕時也，亦是往日今日每兩界中央分判檢劑節斷之處也。）術曰：置半夜刻及分，兼全晝刻及分，以節刻及分，一如取宿斷法，減之，至所止刻，為節斷刻時。（謂正著蝕時也）置其刻及分，為節斷位。

This chapter discusses the concept of "sectional division" (节断). The original commentary explains that "sectional division" refers to the exact moment when an eclipse occurs, which is also the boundary between yesterday and today. The text states: arrange the hours and minutes of midnight, and combine them with the hours and minutes of the entire day. Then use the previously calculated "sectional hour" and "sectional minute" to subtract from them, following exactly the method used for "previous division" (宿断). When the subtraction stops at a specific moment, that is the "sectional division time" (节断刻时). The original commentary further explains that this refers to the exact moment of the eclipse. When the hours and minutes of this moment are arranged, it is called the "sectional division" (节断). "Previous division" determines the boundary between two constellations where the moon passes, while "sectional division" determines the exact moment when the sun and moon meet or oppose each other. Both methods are based on the same principles but seek different results.

推均分章：（承前或譯為月度分法，在梵曆，此術九妙，朔下日月相及度分算三位，並均；望即度及分二位，均；弦即準只分一位，均；推得朔望均分路日月交蝕。）根法，（置定月以日定，減之，減餘有六相者，棄有有相，餘通作分，名為過去根法，如減餘，通五相者，別置六相減之，減餘作分，名為未來根法。）術曰：置根法以六十乘之，以日減除之；如是過去，以除得數損之日分；如是未來，以除得數益定日分。又以除得數加根法，以六十除之，得度不盡，為分；如是過去，損定月度分；如是未來，益定月度分。日月度分均平齊等，即並列之，置為均分位。

This chapter discusses the method of equal distribution. (Original note: Previously, this was translated as "Monthly Division Method." In the Indian calendar system, this technique is very precise: at the new moon, the positions, degrees, and minutes of the sun and moon must all align; at the full moon, the degrees and minutes must align; and at the first and last quarters, only the minutes need to align. By determining where the new and full moons align, one can predict solar and lunar eclipses.) First, the "root method" is explained: (Original note: Set a fixed month, subtract it from the fixed day. If the remainder exceeds six phases, subtract six phases and convert the remainder into minutes, which is called the "past root method." If the total is within five phases, subtract six phases from it and convert the remainder into minutes, which is called the "future root method.") The method states: Set the root method, multiply it by 60, then divide by the daily movement in minutes. If it is in the past, subtract the result from the fixed day's minutes; if it is in the future, add the result to the fixed day's minutes. Then add the result to the root method, divide by 60 to get degrees, and the remainder is the minutes. If it is in the past, subtract the result from the fixed month's degrees and minutes; if it is in the future, add the result to the fixed month's degrees and minutes. When the degrees and minutes of the sun and moon are equal and aligned, they are listed together as the "equal distribution" position.