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

又法：置節刻位，通作分，列為根法。術曰：置根法，以定日行分，（謂日域術中，相法之下，所標五十七等是也。）乘之，以三千六百除之，得分；其餘損益，定日分。（其損益法，損之而得均者，即便損之；益之而得均者，即便益之。）又置根法，以日域乘之，以三千六百除之，得分；其分又以六十除之，得度，不盡為分；以其度及分損益，定月度分。（日若益之，月亦益之；日若損之，月亦損之。）如是損益訖，置為均分法。（俱損俱益，是均分也；一損一益，非均分也。）

There is another method: set up the positions of the solar terms and the moments, convert them into fractions, and arrange them as the root method. The technique says: set up the root method, multiply it by the minutes of the fixed day, then divide by 3600 to get a result; use the remainder to add or subtract from the fixed day's minutes. (Original note: the rule for addition and subtraction is: if subtracting brings the sun and moon into alignment, subtract; if adding brings them into alignment, add.) Then set up the root method again, multiply it by the solar domain, divide by 3600 to get a result; take this result and divide by 60 to get degrees, with any remainder being minutes; use these degrees and minutes to add or subtract from the fixed moon's degrees and minutes. (Original note: if the sun is added, the moon is also added; if the sun is subtracted, the moon is also subtracted.) After completing the addition and subtraction, arrange it as the 'equal division method.' (Original note: only when the sun and moon are both subtracted or both added is it considered equal division; if one is added and the other is subtracted, it is not equal division.)

推阿修章：（承前或譯為風，或譯為蝕神，梵之日呼為羅睺。《釋典》所云：“羅睺，阿修王，即此臣靈也。”又《河圖》云：“暗虛值月，則月蝕；值星，則星亡；亦謂此怪靈也。”又諸曜則巡宿順行，其阿修則巡宿逆轉，掩蔽日月，以亦交蝕。）

In this chapter, we examine the concept of "Aśura." (Original note: This term has been translated by some as "Wind" and by others as "Ecliptic God." In Sanskrit, it is called "Rāhu." According to Buddhist texts, Rāhu is the king of the Aśuras, referring to this divine being. The "Yellow River Chart" also states that when darkness meets the moon, an eclipse occurs; when it meets a star, the star disappears, describing the same strange spirit. Furthermore, all celestial bodies move along the twenty-eight mansions in the same direction, but Aśura moves in the opposite direction, blocking the sun and moon, thus causing eclipses.) From a modern astronomical perspective, Aśura and Rāhu correspond to the intersection points of the ecliptic and the white path; these intersection points move backward along the ecliptic, and when the sun and moon approach these points, eclipses occur. The ancient people recorded these phenomena with divine names, but in reality, they represent an astronomical quantity that can be precisely calculated.

術曰：置積日，以六千七百九十四除之，得為已過遍數，棄之，餘以十二乘之，準前除之。（謂亦以六千九百九十四除之也）得相；餘以十三乘之，準前除之，得度；餘以六十乘之，準前除之，得分；列為前位。又別置五相二十四度四十分，以其前位減之，（若不足減，於五相位上，更加十二相，減之。）減訖，餘相度分，置為阿修位。

The text says: Set out the accumulated days, divide them by 6794. The quotient represents the number of cycles that have already been passed by the moon's apogee, which is discarded. The remainder is multiplied by 12, and then divided by 6794 again using the same method to obtain the phase. The remainder is then multiplied by 13 and divided by 6794 again to obtain the degree. The remainder is then multiplied by 60 and divided by 6794 again to obtain the minute. These phases, degrees, and minutes are arranged in the preceding position. Then set out another 5 phases, 24 degrees, and 40 minutes, and subtract the preceding position's number from it. If the preceding number is insufficient for subtraction, add twelve phases to the five phases before subtracting. The remaining phases, degrees, and minutes are arranged as the position of the apogee. The divisor 6794 days corresponds to the number of days required for the apogee to complete one cycle of retrograde motion, which matches the measured retrograde cycle of the apogee of more than eighteen years.

敘日月蝕法：凡算蝕者，先置均分及阿修位，從前蝕之後，斗至六個月白博義，（天竺每月二博義，從月初至十五日，為白博義，從十六日至月盡，為黑博義。其博義，譯雲翅也。）十五日，月當交蝕之限，從前蝕後，斗至六個月黑博義，（月盡日也）日當交蝕之限，月或個月白博義蝕，或五個月白博義蝕，或十四日蝕，或十六日蝕。日或七個月蝕，或五個月蝕，或十六日蝕，日或七個月蝕，或五個月蝕。

The following is a general explanation of the method for calculating solar and lunar eclipses. When calculating eclipses, first set up the positions of the mean and the Axiu. From the last eclipse, count six months of the White Bo Yi, and on the fifteenth day, it is the limit for a lunar eclipse to occur; from the last eclipse, count six months of the Black Bo Yi, which is the limit for a solar eclipse to occur. A lunar eclipse may occur after six months of the White Bo Yi, or after five months of the White Bo Yi, or on the fourteenth day, or on the sixteenth day; a solar eclipse may occur after seven months, or after five months, or on the sixteenth day. The text has a missing character "six" before "months" and the last sentence is repeated, both being errors in the original text.

又，日蝕初虧，皆在西方，月蝕初虧，皆在東方，蝕既者，雖亦帶隅，正方之數俱多也。（其正方，謂東西方也。）蝕鮮者，雖亦帶隅，正方之數小也。又蝕所從方，進而虧黑，還於其方，退而放明也。又蝕色初至如煙，於時亦如煙，又蝕不盡，缺處黑；如盡，外赤色，中赤黑色。

Again, solar eclipses begin to wane from the western side, while lunar eclipses begin to wane from the eastern side. Whenever an eclipse is deep, although it may also appear in the corners, the portion that corresponds to the main direction always occupies a larger share. (Original note: By "main direction," it refers to due east and due west.) Whenever an eclipse is shallow, although it may also appear in the corners, the portion that corresponds to the main direction is smaller. Also, an eclipse always comes from one direction, gradually darkens, and then exits from the same direction, restoring brightness. Moreover, when the eclipse first begins, its appearance is like smoke; when it ends, it also appears like smoke. If the eclipse is incomplete, the missing portion appears black; if the eclipse is complete, the outer ring shows red, while the center shows reddish-black. The descriptions of the colors and the starting directions of solar and lunar eclipses in this passage are based on empirical observations made by the naked eye, and they complement the previous chapters that rely purely on arithmetic calculations.

推間量府章：（日月有蝕，無蝕，及起虧方隅，並在此術中。）置均分，以阿修減之，（如不足減，加十二相於均分相位上減之。）記減，得羖首，為北行；（若得北行，其有日蝕，初起西北；其有月蝕，初起東南。）得秤首，為南行。（若得南行，其有日蝕，初起西南；其有月蝕，初起東北。）

This chapter discusses the method of determining the location where an eclipse begins. The method involves dividing the mean motion and subtracting the value of "Axiu." If the subtraction is not possible, add twelve phases to the mean motion before subtracting. The result, if it falls on the northern horn, indicates a northern movement. If it falls on the southern horn, it indicates a southern movement. The subtraction of Axiu from the mean motion actually calculates the angular distance between the sun and moon relative to the intersection point. The positive or negative value determines whether the moon passes north or south of the intersection point, thereby determining the direction from which the eclipse begins.

餘者，置為間量府。（凡有蝕法減阿修訖，餘者即是間量府也。如十二度已下，月即有蝕；十二度已上，無蝕。凡日蝕法減阿修訖，餘者即是間量府也。兼有日成間量訖，有十二度已上，日即有蝕；十二度已下，無蝕。）如其加十二相，減阿修者還，卻置十二相減訖蝕者，置為間量府；如其減阿修有六相已上者，置棄六相，餘者置為間量府；如其減阿修訖，有五相已上者，別置六相減之，減訖，餘者置為間量府。

After subtracting the value of "Axiu" from the remaining amount, it is arranged as the "Intercepting Measure." (Original note: When using the method for lunar eclipses, after subtracting Axiu, the remaining amount is the Intercepting Measure; if it is less than twelve degrees, the moon will be eclipsed, but if it is more than twelve degrees, it will not be eclipsed. When using the method for solar eclipses, after subtracting Axiu, the remaining amount is also the Intercepting Measure; then adding the Intercepting Measure calculated from the sun, if the total is more than twelve degrees, the sun will be eclipsed, but if it is less than twelve degrees, it will not be eclipsed.) If twelve phases were previously added before subtracting Axiu, then twelve phases should be subtracted again, and the result is the Intercepting Measure. If after subtracting Axiu there is more than six phases remaining, then six phases should be discarded, and the remaining amount is the Intercepting Measure. If after subtracting Axiu there is more than five phases remaining, then another six phases should be subtracted from it, and the remaining amount is the Intercepting Measure. All these rules are used to fold the angular distance back into the smallest range for reference in tables.

推月間量命段法：（凡一段，管三度四十五分，每八段管一相，總有二十四段，用管三相。其段下側注者，是積段，併成三數。）第一段，（二百二十五）第二段，（二百二十四，並四百四十九。）第一相；第三段，（二百二十二，六百七十一。）第四段，（一百一十九，並八百九十。）第五段，（二百一十五，一千一百五。）第六段，（二百一十，並一千三百一十五。）第七段，（二百五，並一千五百二十）第八段，（一百九十九，並一千七百一十九。）

The method of calculating the segments for each month involves dividing the total into segments, each covering 3 degrees and 45 minutes. Eight segments make up one phase, totaling 24 segments for three phases. The numbers noted beside each segment are cumulative totals, obtained by adding up the numbers for each segment. The first segment is 225. The second segment is 224, cumulative total 449, which completes the first phase. The third segment is 222, cumulative total 671. The fourth segment is 219, cumulative total 890 (the original text reads 'one hundred and nineteen', but based on the cumulative difference, it should be 'two hundred and nineteen', the original text is missing the character 'two'). The fifth segment is 215, cumulative total 1105. The sixth segment is 210, cumulative total 1315. The seventh segment is 205, cumulative total 1520. The eighth segment is 199, cumulative total 1719. The numbers for each segment gradually decrease from 225, while the cumulative totals steadily increase, forming a table of corrections based on angular distances.

第九段，（一百九十一，並一千九百一。）第十段，（一百八十三，並二千九十三。）第二相；第十一段，（一百七十四，並二千二百六十七。）第十二段，（一百六十四，並二千四百三十一。）第十三段，（一百五十四，並二千五百八十五。）第十四段，（一百四十三，並二千七百二十八。）第十五段，（一百三十一，並二千八百五十九。）第十六段，（一百一十九，並二千九百七十八。）第十七段，（一百六，並三千八十四。）第十八段，（九十三，並三千一百七十七。）第三相；第十九段，（七十九，並三千二百五十六。）第二十段，（六十五，並三千三百二十一。）第二十一段，（五十一，並三千三百七十二。）第二十二段，（三十七，並三千四百九。）第二十三段，（二十二，並三千四百三十一。）第二十四段。（七，並三千四百三十八。）

Paragraph 9, 191, cumulative total 1901 (based on paragraph 8, 1719 plus 191 should be 1910, the original text reads 'one thousand nine hundred and one', possibly missing a character). Paragraph 10, 183, cumulative total 2093. The above constitutes the second phase. Paragraph 11, 174, cumulative total 2267. Paragraph 12, 164, cumulative total 2431. Paragraph 13, 154, cumulative total 2585. Paragraph 14, 143, cumulative total 2728. Paragraph 15, 131, cumulative total 2859. Paragraph 16, 119, cumulative total 2978. Paragraph 17, 106, cumulative total 3084. Paragraph 18, 93, cumulative total 3177. The above constitutes the third phase. Paragraph 19, 79, cumulative total 3256. Paragraph 20, 65, cumulative total 3321. Paragraph 21, 51, cumulative total 3372. Paragraph 22, 37, cumulative total 3409. Paragraph 23, 22, cumulative total 3431. Paragraph 24, 7, cumulative total 3438. The numbers from paragraph 24 decrease from 225 to 7, with the cumulative total reaching 3438. This number 3438 corresponds to the fraction of the radius in Indian mathematics, and the entire table is essentially a sine table.

術曰：置間量府，通作分，以二百二十五除之，得者為段。以其段下並數列為上位，（假令除得一，其第一段下無並，即直列二百二十五為上位；如其除得二，即例側注並數四百四十九為上位；如其除得三，例側注並數六百七十一為上位。他皆仿此。）餘，以次段乘之。（假令除得三，例側注並數為上位訖，即以第四段二百一十九乘之，他皆仿此。）以二百二十五除之，得者並上位，置為間量命。（非月蝕用之）

The text says: Arrange the intervals and values, divide by 225, and the quotient is the segment number. Take the cumulative number listed under that segment as the upper value. (Original note: For example, if the division results in one, there is no cumulative number under the first segment, so directly take 225 as the upper value; if it results in two, take the cumulative number 449 as the upper value; if it results in three, take the cumulative number 671 as the upper value. The rest follows the same pattern.) The remainder is then multiplied by the number of the next segment. (Original note: For example, after taking the cumulative number as the upper value when the division results in three, multiply by the number of the fourth segment, 219, and so on.) Then divide again by 225, and combine the result with the upper value to form the "interval measurement of life." (Original note: This method is not used for lunar eclipses.) This entire calculation method is equivalent to taking values in segments from a sine table and interpolating the remainder proportionally, which is exactly the same as linear interpolation in later times.

推月間量法：術曰：置間量命，以四乘之，置為初位；又列置四萬三千四十一，以月域除之，得者（假令除得五十一，即以五十一除初位。）以除初位，得度，不盡，六十乘之，依前除之，得分，置為月間量位。（如推日蝕例算日間星法）

The method for calculating the lunar distance during the month. The technique says: arrange the lunar distance, multiply it by 4, and set it as the initial position. Then, separately list 43041, divide it by the lunar domain, take the quotient obtained (for example, if the division results in 51, use 51 to divide the initial position), use this quotient to divide the initial position to get the degree. If there is a remainder, multiply it by 60 and continue dividing according to the same method to get the minute, and set this as the "lunar distance" position. (Note from the original text: the method for calculating the solar distance during a solar eclipse is also done in the same way.) The lunar distance represents the apparent distance between the center of the moon and the center of the sun at the time of an eclipse. By comparing this lunar distance with the subsequent lunar measure and the Axiu measure, one can determine the extent of the eclipse.

推月量法：術曰：置月域，以二乘之，以四十九除之，得度，不盡，以六十乘之，依前除之，得分，置是月量位。推阿修量法：術曰：置月域，以五乘之，以四十八除之，得度，不盡，以六十乘之，依前除之，得分，置為阿修量位。推阿修及月全位半位法：置阿修量與月量，並之為全位；又半之，為半位；其全位、其半位，各列為位。

The method for calculating the lunar measure. The text says: set out the lunar domain, multiply by 2, divide by 49, to obtain degrees; if there is a remainder, multiply it by 60 and continue dividing as before to obtain minutes, and arrange it as the 'lunar measure' position. The method for calculating the eclipse measure. The text says: set out the lunar domain, multiply by 5, divide by 48, to obtain degrees; if there is a remainder, multiply it by 60 and continue dividing as before to obtain minutes, and arrange it as the 'eclipse measure' position. The method for calculating the eclipse measure and the full and half positions of the moon: add the eclipse measure and the lunar measure together to form the 'full position'; take half of that to form the 'half position'; the full and half positions each form one position. The lunar measure corresponds to the apparent diameter of the moon, and the eclipse measure corresponds to the apparent diameter of the shadow. Half of their sum is exactly the distance between the centers when they are just tangent.

推蝕經刻法：（謂初虧至復滿所經刻數也）術曰：置量，自相乘，（先以度自相乘，列為上位，又以分自相乘，以三上除之，加上位，凡三十分從度者，謂收半已上也。）又置半位，亦自相乘，（亦如收分法為之）置半位相乘訖數減之，減餘以開方除之，（其開方，梵音雲根法也。）得者，以六十乘之，又以日域除之，得刻；不盡，又乘又除，得分；其刻及分，二乘之，（謂位分也）置為虧滿刻法。

The method for calculating the number of time units between the beginning of an eclipse and its end involves the following steps. First, set out the measured quantities and square them once. The explanation states that first the degrees are squared and placed as the upper term, then the minutes are squared and divided by thirty, with the result added to the upper term. Whenever the total reaches thirty, it is carried over as one degree, meaning that half a degree or more is counted as a full degree. Then, set out the half term and square it in the same way. Subtract the squared half term from the previously squared term, and take the square root of the remainder. Multiply the result by sixty, then divide by the solar domain to obtain the time units. If there is a remainder, multiply it again and divide again to get the minutes. Add the time units and minutes together and double the result to form the "eclipse duration method." This process essentially involves calculating the difference of the squares of the distances between the centers, taking the square root to find the half chord, and converting it into time, which is consistent with the modern method of using the Pythagorean theorem to calculate the duration of an eclipse.

（又以其數加節斷刻上，節斷是，若初虧得此刻，通至復滿時。）推月規法：（此術中，備載日月虧缺多少，及蝕既深淺等事。）術曰：置月量半，準其數，或用綎，或用木，為規限，繞作光明壇。又置間量，準其數，或以綎，或以木，從光明壇正中心向蝕方引出至末際，置為位。

The original note says: Add this number to the time of the eclipse's interruption; the interruption is the time of maximum eclipse; if calculated from the beginning of the eclipse, this time can be used to predict the time of the full moon's restoration. The method for determining the lunar crescent. The original note says: This method details the extent of the moon's deficiency and the depth of the eclipse's maximum. The text explains: Set out half the moon's measure, using this number, and use a rope or a stick as the radius to draw a circle around it, called the "Bright Circle," representing the moon's disk. Then set out the intermeasure, using this number as well, and draw a line from the center of the Bright Circle in the direction of the initial eclipse until the end, fixing this position. The so-called lunar crescent is a method of drawing the intersection of the eclipse using graphical techniques, replacing complex calculations.

又起末際位，據為正中心，置阿修量半，準其數，或用綎，或用木，為規限，繞作黑暗壇，據黑暗壇掩著處，以定虧缺多少，蝕既深淺，一如其事。（若推日蝕掩規，置月量半，為光明壇，以日成間量府，得作間量者，為間量。以月量半，為黑暗壇，自餘算術，並同月規法。）

Taking the previously mentioned end as a new central point, set out half of the length of the Axiu quantity, and using either a rope or a stick as the radius, draw another circle around it, called the "Darkness Circle," representing the shadow that is covered. By observing the part of the Darkness Circle that covers the Light Circle, one can determine the extent of the eclipse, the depth of the total eclipse, and it matches exactly with what is actually seen. (Original note: If calculating the eclipse using the lunar method, use half the lunar measure as the Light Circle, and the interval measured from the solar interval as the interval measure; then use half the lunar measure as the Darkness Circle; the rest of the calculation methods are the same as those for the lunar method.) The overlapping part of the two circles is the eclipse fraction, which is an intuitive method of calculation through diagram.

推蝕甚法：（謂蝕後更停，經一刻，或二刻，或半刻，方始退蝕放明也。）術曰：置阿修量半，以月量半減之，餘又以月間量減之，（如其減間量盡，為蝕盡；如其減不盡，為蝕不盡。若盡即有蝕甚法，若不盡則無蝕甚法。）減餘，以六十乘之，以日域除之，得刻，不盡，又乘又除，得分，其刻及分，二乘之，（謂倍也）置為蝕甚刻位。

The method for determining the duration of maximum eclipse involves taking half of the Axiu measure, subtracting half of the lunar measure from it, and then using the remaining amount to subtract the intermeasure. If the intermeasure can be completely subtracted, it indicates a total eclipse; if not, it is not a total eclipse. Only when the intermeasure is fully subtracted does the step of maximum eclipse occur. After subtraction, the remaining amount is multiplied by 60 and divided by the solar domain to obtain the number of hours. If there is a remainder, multiply and divide again to get the minutes. The total hours and minutes are then doubled and placed in the position of "maximum eclipse time." This section calculates precisely the duration of a total eclipse, which is the time between the moment of greatest eclipse and the moment when the eclipse ends.

推蝕刻位：（謂左右用行數推步，蝕隅畔劑並圖如左。）術曰：置間量，以九十乘之，以半位除之，得度，置為蝕行法。蝕行法：度二十分州。度二十分州。

Determine the position of the eclipse. (Original note: This refers to using rows on both sides to calculate the steps, and the corner boundaries where the eclipse intersects, along with the diagram, are attached below.) The method states: Set out the measurement between, multiply by 90, divide by half the position, to obtain the degree, and arrange it as the 'method of eclipse rows'. Below is the diagram attached to the method of eclipse rows: The top and bottom of the diagram each mark 'twenty degrees and twenty minutes of state', where 'state' is likely a miswriting of the repetition mark for 'minute', indicating divisions such as twenty degrees and twenty minutes. This diagram has been damaged in transmission, leaving only scattered annotations of direction and degrees; the original form can no longer be restored.

右先為八方，訖東西正中加一晝，各中通，以成十方。諸方各置四十五度，其東西二分頭加一中晝，便是各半其方，即東西各四半方也。各置二十二度三十分，言觸從北虧者，是從中道北入也；言從南虧者，是從中道南入也者。度四十五，西，度四十五。南度四十五，度四十五北。度四十五，東，度四十五。

The diagram first draws out the eight cardinal directions, then adds a central line in the due east and due west, each line passing through the center, forming ten directions in total. Each direction occupies forty-five degrees. The central lines added at the east and west divide each of those directions into two halves, so the east and west each have four and a half directions. Each half direction occupies twenty-two and a half degrees. When it says starting from the north side, it refers to entering from north of the central line; when it says starting from the south side, it refers to entering from south of the central line. The diagram is labeled as: forty-five degrees on each side of the west; forty-five degrees on each side of the south and north; and forty-five degrees on each side of the east. The full circle of forty-five degrees multiplied by eight exactly equals three hundred and sixty degrees. This method divides the angular position of lunar and solar eclipses into eight cardinal directions and ten positions for recording.

度二十分州。度二十分州。若從東北隅入，月蝕即從東中道北行，以蝕行減方數盡，則蝕初之分。（南入法準此）若從西北隅入，日蝕即從西中道北行，以蝕行減方數盡，即蝕初之分。（南入法準此）

Below the diagram, add two lines labeled 「度二十分州」, which should match the markings above. For those entering from the Northeast corner, the lunar eclipse progresses from the eastern central line toward the north, using the eclipse progression number to subtract from the degrees of this direction; the place where the subtraction is completed is the position of the beginning of the eclipse. (Original note: Those coming from the south are also calculated in this way.) For those entering from the Northwest corner, the solar eclipse progresses from the western central line toward the north, similarly using the eclipse progression number to subtract from the degrees of this direction; the place where the subtraction is completed is the position of the beginning of the eclipse. (Original note: Those coming from the south are also calculated in this way.) Lunar eclipses begin to wane from the east, while solar eclipses begin to wane from the west, so their respective central lines are one in the east and one in the west.

推日量法：術曰：置日行分，（謂日減術先所標五十七等數）以六十乘之，以十一除之，得度，餘以六十乘之，依前除，得分，置為日量法。推日蝕法：（凡雲日蝕，太白從月，星伐阿修星；又並日月二為半位，其所用間量之，並以日間為之，日蝕術算，亦同月蝕也。）術曰：置節斷刻位，通作分，謂六十乘刻內分也，別置之，為刻分位。

The method for calculating the daily measure. The text says: Arrange the daily divisions (note: referring to the 57 divisions mentioned in the previous "Daily Domain Method"), multiply by 60, divide by 11 to get degrees; the remainder is multiplied by 60 again, and the same method is applied to get minutes. Arrange this as the "Daily Measure Method." The method for calculating solar eclipses. (Note: Generally, when discussing solar eclipses, it is due to the bright star following the moon and the star invading the Axiu star; also, the positions of the sun and moon are taken as their halves, and the interval measure used is always the solar interval measure. The calculation method for solar eclipses is the same as that for lunar eclipses except for the remaining parts.) The text says: Arrange the division of the solar term and the hour, convert them into minutes by multiplying the hours by 60 and adding them to the minutes, and place them separately as the "Minute Position."

推日上星駟法：術曰：置定日，以半夜刻及全晝刻並之，並訖，所行刻以減定日分行，減訖，置為日蝕出位。又別置三十度，以日出位度及分減分，（其減分法，退一度，破為六十分而減之。）減餘，通作分，置為上虛駟。

To determine the method of calculating the position of the stars above the sun. The technique states: set out the fixed day, add the number of divisions for the midnight to noon period to the total number of divisions for the entire day; after adding, subtract the number of divisions traveled from the fixed day's divisions. The result obtained after subtraction is placed as the "sun's emergence position," which is the location of the sun at sunrise. Then set out thirty degrees, subtract the degrees and minutes of the sun's emergence position from it (note: the method of subtracting minutes is: reduce one degree, convert it into sixty minutes, then subtract). The remaining amount is converted entirely into minutes and placed as the "upper empty quartile." "Quartile" here refers to a section on the Yellow Path; the upper empty quartile is the section remaining between the point of sunrise and the end of the current phase.

段法：第一段，（一百九十八）第二段，（二百三十二）第三段，（二百九十）第四段，（三百五十一）第五段，（二百六十）第六段，（三百五十八）右六段，從上向下，為羖首次；從下向上，為秤首；及置上虛駟，恆視日出相，得羖首、秤首次第；（假令日出相定，即得羖首也，謂即須用羖首第一段乘也，他皆仿此。）

The method of selecting segments: the first segment is 198, the second is 232, the third is 290, the fourth is 351, the fifth is 260, and the sixth is 358. These six segments, counted from top to bottom, represent the order of the goat's head, while counted from bottom to top, they represent the order of the scale's head. After arranging the upper empty four, observe the phase at sunrise to determine whether it belongs to the order of the goat's head or the scale's head. (Original note: for example, if the phase at sunrise is fixed as zero, it belongs to the goat's head, and the number of the first segment of the goat's head should be used for calculation; the following should be done in the same way.) The number of the fifth segment is smaller than that of the fourth, which contradicts the trend of increasing numbers segment by segment, suggesting that the original text may have an error.

其段乘之，以一千八百除之，所得者，謂所得數也。以減刻分位，成減為一相，即以一相加日出相位。（日出位中度及分並棄之）即以次段減段，令用羖首第段棄上虛駟訖，即第刻分位乘三十，段二百三十二減之，又以一相準前加日出相位，又以其次段減刻分位，成減，又以一相加日出相位，視刻分位數，堪更減之，他皆仿此。至不成減止，餘刻分位不成減，雲餘也。

Use the number from the designated section to multiply, then divide by 1800; the result is the number to be applied. Subtract this from the degree and minute position of the hour division. When subtracted once, it counts as one phase, and then add this one phase to the rising sun's position. (Original note: The degrees and minutes originally present at the rising sun position are all discarded.) Then proceed to subtract using the number from the next section: first subtract the entire previous section's number as determined by the goat's head, then multiply the hour division's degree and minute position by thirty, and subtract the number 232 from the second section; after subtraction, add another phase to the rising sun's position as before. Continue this process, using the next section's number to subtract from the hour division's degree and minute position, adding another phase each time, and keep checking how much of the hour division's degree and minute position remains. If there is enough to subtract, continue subtracting; proceed in this manner until there is not enough to subtract anymore. The remaining part of the hour division's degree and minute position that cannot be subtracted further is called the 'remainder'.

以三十乘，以所至段除，能止，從羖首加三相於星相位訖，即取決四段除之，他皆仿此。得此度不盡，以六十乘，依前除，得分，以所得度及分，並加日出位，加訖，即是節斷。恆減三相，減訖，（羖首為北行，秤首為南行。）日間如是量府三相已上，準減相例為如之，為其相定及三相已下，總通作分，謂三十乘內度，六十乘度也。一如前推月間量命法為之，置為月間量命，以一百四十六數除之，所得為度，餘以六十乘之，依前除之，所得為分，置為位。恆觀月間量府，若羖首減，謂隨方眼法。隨方眼法：（其隨方眼，中國用三十五分也。）

Multiply the remainder by 30, and divide it by the number of the segment that was subtracted, continuing until the division is exact. If, according to the order of the goat's head, the three phases have already been added to the star positions, then use the number of the fourth segment for division, and proceed in the same manner thereafter. The result is degrees; if the division is not exact, multiply the remainder by 60 and continue dividing as before to obtain the minutes. Add the resulting degrees and minutes to the position of sunrise; the total is the 'section break'. After this, subtract three phases regularly. If the original text's note says that those belonging to the goat's head are northward and those belonging to the scale's head are southward, then during the day, if the measure of the mansion is above three phases, follow the general rule of subtracting phases as before; if the phase is zero or below three phases, convert the entire amount into minutes by multiplying the phases by 30 to get degrees and multiplying the degrees by 60 to get minutes. After this, follow the method used previously for calculating the monthly measure, arranging it as 'monthly measure'. Divide by 146 to get degrees, and multiply the remainder by 60, then divide as before to get minutes, arranging them in one position. Continuously observe the monthly measure of the mansion; if it belongs to the goat's head, subtract it. This is called the method of following the square eye. The method of following the square eye: (Original text's note: This square eye, used in China, is thirty-five parts.) This section ends here; the following methods concerning solar eclipses and their patterns are missing from the original text.

若秤首以加隨方眼法之置以位，為中命，置中命又一如前命法為之，置為後命，月域乘之，以五萬一千五百六除之，所得為度，餘以六十乘之，依前除之，所得分。所得度及分，恆視間量府，（謂均分，減阿修訖，間量府也。）得羖首減之，（亦為均分，減阿修訖，間量府也。）得秤首加之，（亦謂如均分，間量府也。）減阿修訖，置為日間量；（如十一度已下，有蝕；十一度已上，無蝕。）

The "Nine Instruments Calendar" was translated from an Indian calendar by Qutan Xida in the sixth year of Kaiyuan. The terms used are mostly transliterations from Sanskrit, and the method of calculation differs from the calendars used in China. This passage continues the previous discussion on the method of predicting solar eclipses: if the calculated position falls in the "scale head" (the beginning of the Libra sign in the twelve zodiacal signs of the Yellow Path), then the "directional eye" (a geographical correction factor based on the observer's location) should be added according to the rules, and this value is called the "central mandate." Then, this central mandate should be recalculated again using the method mentioned earlier, and the result is called the "later mandate." Multiply the "moon domain" by the later mandate, then divide by 51506 (five ten-thousand one hundred and fifty-six). The integer obtained from the division is the degree; the remainder is multiplied by 60 and divided again using the same method, resulting in the minute. After calculating the degrees and minutes, one should constantly refer to the "inter-measurement chamber" (the angular distance between two celestial bodies. The original commentary explains: this refers to the inter-measurement chamber after equal division and after subtracting the "Asura" value). If the position is in the "ram head" (the beginning of the Aries sign), subtract the degree and minute from the inter-measurement chamber (the commentary also refers to the inter-measurement chamber after subtracting the Asura value). If the position is in the "scale head," add the degree and minute to the inter-measurement chamber (the commentary also refers to the inter-measurement chamber after equal division). After subtracting the Asura value, the result is set as the "solar inter-measurement," which is the measurement of the apparent diameter of the sun. The original commentary points out the criterion for determining an eclipse: if the result is less than eleven degrees, there will be an eclipse; if it is more than eleven degrees, there will be no eclipse.

又並日月二量為全位，復半之，為半位，置半位自相至，又置日間量，亦自相乘訖，即以半位數內減卻日成數，成減有蝕，不成減無蝕，餘並一如蝕中敘。（凡在歷大側，如其分不足減，退度一，置為六十分而減之；如其度不足，退相一，置為三十度而減之；如其相不足減，加十二相而減之。）

Add the "measure" of the sun and the "measure" of the moon, which are the apparent diameters of the sun and moon, to form the "full measure." Divide the full measure by two to get the "half measure," which is the sum of the radii of the sun and moon. Then square the half measure and also square the "measure" of the sun. Subtract the squared value of the sun's measure from the squared value of the half measure. If the subtraction is possible, there will be an eclipse; if not, there will be no eclipse. The rest of the calculations follow the same procedures as described in the section on calculating lunar eclipses in the "Nine Instruments Calendar." The original text notes the method of borrowing in calculations: when the fractions are insufficient for subtraction, borrow one degree from the degrees on the calendar table, converting it into sixty parts for subtraction; if the degrees are insufficient, borrow one sign (phase) from the signs, converting it into thirty degrees for subtraction; if the signs are insufficient, first add twelve signs (the twelve signs of the Yellow Path) before subtracting.

置上虛駟，恆日出相，依羖首、秤首次第，（假令日出相定，即得羖首也，謂即用羖首第二段乘也，他皆仿此。）以其段乘之，以一千八百除之，所得者，（謂所除得數也）以減刻分位，成減，為一相，即以一相加日出相位，（其日出位所有度及分並棄之）又即以次段減刻分位，（假令用羖首第一段乘上虛駟訖，即用以第二段二百三十二減之，他皆仿此。）成減，又以一相加日出相位，又以次段分減刻位，成減，又以一相加日出相位；

The number "Shang Xu Si" is placed and often used in conjunction with "Ri Chu Xiang" (the palace where the sun is located at sunrise). According to the order of "Gu Shou" and "Beng Shou", the segments of each palace are taken in sequence. The original text provides an example: if "Ri Chu Xiang" has already been determined and "Gu Shou" is selected, then the second segment of "Gu Shou" is used for calculation, and the other palaces are treated similarly. Multiply this segment number by "Shang Xu Si", then divide by 1800 (one thousand eight hundred). The quotient obtained from this division is used to subtract from "Ke Fen Wei" (the degree and minute position of the division). If the subtraction results in a positive number, it counts as one palace, and this palace is added to the position of "Ri Chu Xiang" (the original degrees and minutes of the position are entirely discarded). Then, use the next segment number to subtract from "Ke Fen Wei"; if the subtraction results in a positive number, another palace is added to the position of "Ri Chu Xiang". This process continues with the next segment number, and if the subtraction results in a positive number, another palace is added to the position of "Ri Chu Xiang".

每視刻分位數，堪更減段者，恆教此法，減而加之，至不成減止，餘以三千乘，以所至段除之，日得度，不盡，六十乘，依前除之，得分，以此度及分並加日出位，（其日出位度分先並棄之，令以此度加及分，置之。）加訖，即是節斷著也。

At each step, check the remaining number on the division scale: as long as there is still enough to subtract another segment, continue this method, subtracting a segment and adding one section each time, until no more subtraction is possible. The remaining remainder is multiplied by 3000, then divided by the number of the segment that was last subtracted to obtain the degrees. The part that does not divide evenly is multiplied by 60 and divided again using the same method to obtain the minutes. Add these calculated degrees and minutes to the position of sunrise (note: the original degrees and minutes at the sunrise position had already been discarded; now these newly calculated degrees and minutes are added to that position). The result obtained after this addition is called the "section break."

其節斷著恆減三相，減訖，（得羖首，為北行；得秤首，為南行。）置為日間量府。如其有三相已上，（謂日間量府有三相已上也）準減相例（其例在定日術後者是也）為之；如其三相已下，總通作分，（謂三十乘相內度，六十乘度內分也。）

The obtained section is usually reduced by three palaces. After reduction, if the result falls in the goat's head, it is heading north; if it falls in the scale's head, it is heading south. Then it is placed as the 'daytime measure palace,' which refers to the distance on the side of the sun. If the result is three palaces or more, it is handled according to the usual method of reducing palaces (this usual method is found after 'Determining the Day Method'). If the result is three palaces or less, the palaces, degrees, and minutes are all converted into minutes. (Note: thirty is multiplied by the number of palaces and added to the degrees, then sixty is multiplied by the degrees and added to the minutes.)

如推月間量命法為之，置為日間量命，以一百四十六除之，得度；餘以六十乘之，依前除之，得分；恆視日間量府，若得羖首，即以此度分數內減卻隨方；（其隨方眼，中國用三十五分。）若得秤首，即以此度分數內更並隨方眼，置為中命；置中命，又再更一如前命法為之，置為後命；

Following the method of "monthly measure" to determine the fate, the result is arranged as "daily measure." Divide this by 146 (one hundred forty-six) to obtain degrees; take the remainder and multiply it by 60, then divide again using the same method to obtain minutes. Always check which palace the daily measure falls into: if it falls into the goat's head, subtract the "directional correction" (which is thirty-five parts in the original text) from the degree and minute number; if it falls into the scale's head, add the directional correction to the degree and minute number. After making the necessary additions or subtractions, arrange it as the "central fate." Then recalculate the central fate using the same method as before and arrange it as the "final fate."

置後命，以月城乘之，以五萬一千五百六十六除之，得度，餘以六十乘，依前除之，得分；恆視間量府，（得均分，減阿修訖，間量府也。）得羖首，以此度分損之；（謂損其減阿修訖，間量府也。）得秤首，以此度分益之；（謂其減益阿修訖，間量府也。）如此損益訖，置為日間量位。

After extracting the life span, multiply it by the "month city" and then divide by 51566 to obtain the degree. Take the remainder, multiply it by 60, and divide again by the same method to obtain the minute. Continuously observe the "intermediate measure" (referring to the remaining intermediate measure after equal division and subtracting the "Axiu" portion): if it falls within the "goat's head" section, use the obtained degree and minute to reduce it (referring to the intermediate measure after subtracting the "Axiu" portion); if it falls within the "scale's head" section, use the obtained degree and minute to increase it (referring to the same intermediate measure after subtracting the "Axiu" portion). After completing this reduction and increase, the result is set as the "daily intermediate measure position."

其間量數，有十一度已下，日即占蝕；十一度已上，又並日月二量為全位，又半之，占無蝕；為半位，置半位自相乘，又置日間量亦自相乘，即以半位數，謂有相乘訖數也。內減卻日間量數，謂自相訖數也。成減，有蝕；不成減，無蝕。自餘術理咸悉，一如月蝕中術。

If the obtained intercalary measure is less than eleven degrees, the Sun is considered to have an eclipse; if it is more than eleven degrees, it is considered to have no eclipse. Additionally, the measures of the Sun and Moon should be added together to form the 'full measure,' then halved to get the 'half measure.' The half measure should be squared once, and the intercalary measure of the Sun should also be squared once. Then, subtract the squared number of the Sun's intercalary measure from the squared number of the half measure: if the subtraction is possible, there is an eclipse; if not, there is no eclipse. The rest of the calculation procedures are the same as those described in the section on predicting lunar eclipses.

古今曆積年及章率：古今曆上元已來，至今開元二年甲寅歲積。《黃帝曆》上元辛卯，至今二百七十六萬八百六十三算外。章歲十九，章閏七，章月二百三十五，蔀歲七十九，蔀月九百四十，蔀日二萬七千七百五十九，元法四千五百六十，紀法一千五百二十，蝕月一百三十五，蝕法二十三，蝕歲五百一十三，蝕數一千八十一。

This passage lists the accumulated years and the cycle rates of various historical and modern calendar systems: starting from the "upper beginning" (the starting year for calculation in each calendar system) to the current year, Kaiyuan Year 2, Jiayin, how many years have accumulated. The Huangdi calendar's upper beginning starts in the Xinmao year, and from that year to Kaiyuan Year 2, Jiayin, a total of 2,760,863 years have accumulated. "Calculated outside" means the number obtained does not include the current year. Its chapter year is nineteen years, with seven leap months arranged within nineteen years, chapter months are 235 months, the bù year is seventy-nine years, bù months are 940 months, bù days are 27,759 days, the original method is 4,560, the record method is 1,520, the eclipse month is 135, the eclipse method is 23, the eclipse year is 513, and the eclipse number is 1,811.

《顓頊曆》上元乙卯，至今二百七十六萬一千一十九算外。章歲十九，章閏七。《夏曆》上元乙丑，至今二百七十六萬五百八十九算外。章歲十九，章閏七。《殷曆》上元甲寅，至今二百七十六萬一千八十算外。章歲十九，章閏七。《周曆》上元丁巳，至今二百七十六萬一千一百三十算外。章歲十九，章閏七。《魯曆》上元庚子，至今二百七十六萬一千三百三十四算外。章歲十九，章閏七。

The "Zhuanyu Calendar" began its upper origin in the Yimao year, accumulating a total of 2,761,019 years up to the second year of Kaiyuan, excluding the starting year; its cycle of years is nineteen, with seven leap years in each cycle. The "Xia Calendar" began its upper origin in the Yichou year, accumulating a total of 2,760,589 years up to the same period, excluding the starting year; its cycle of years is nineteen, with seven leap years in each cycle. The "Yin Calendar" began its upper origin in the Jiayin year, accumulating a total of 2,761,080 years up to the same period, excluding the starting year; its cycle of years is nineteen, with seven leap years in each cycle. The "Zhou Calendar" began its upper origin in the Dingsi year, accumulating a total of 2,761,130 years up to the same period, excluding the starting year; its cycle of years is nineteen, with seven leap years in each cycle. The "Lu Calendar" began its upper origin in the Gengzi year, accumulating a total of 2,761,334 years up to the same period, excluding the starting year; its cycle of years is nineteen, with seven leap years in each cycle. These five calendars, together with the earlier "Huangdi Calendar," are known as the "Six Ancient Calendars" of old, all using the nineteen-year cycle with seven leap years.

前漢劉歆《三統曆》上元庚戌，至今一十四萬三千九百三十四算外。章歲十九，章閏七，日法八十一，統法一千五百三十九，朔望會一百三十五，元法四千六百一十七。冬至牛初，歲星一百四十四年起一次，歲數一百四十四，周數一百四十五。

The "Three Dynasties Calendar" (San Tong Li) was created by Liu Xin of the Western Han dynasty. Its cycle begins with the Geng Xu year and spans a total of 143,934 years up to the second year of the Kaiyuan era, excluding the starting year. The cycle of years is nineteen, with seven leap years in each cycle. The day is divided into eighty-one parts, hence this calendar is also known as the "Eighty-one Part Law Calendar." The overall cycle is 1,539 years, with 135 conjunctions of new and full moons, and the fundamental cycle is 4,617 years. It fixed the winter solstice at the beginning of the constellation Niutou, and also determined that the planet Jupiter (the star of wood) advances by one degree every 144 years, with a period of 144 years and a cycle of 145 years.

後漢《四分曆》上元庚辰，至今九千九百九十五算外。元法四千五百六十，紀法一千五百二十，紀月一萬八千八百，蔀法七十六，蔀月九百四十，章法十九，章月二百四十五，周天一千四百六十一，日法四，蔀日二萬七千七百五十九，沒數二十一，通法四百八十七，沒法七，（亦為章閏）日餘一百六十八，中法三十三，大周三十四萬三千三百三十五，月周一千一十六。

The Four Divisions Calendar used during the Eastern Han dynasty began its upper origin in the Gengchen year. From that time up to the second year of the Kaiyuan era, it totaled 9995 years, not counting the current year. The fundamental cycle was 4560, the cycle law was 1520, the monthly cycle was 18,800, the蔀 cycle was 76, the蔀 months were 940, the chapter cycle was 19, the chapter months were 245, and the full circle of the sky was 1461, which is one year of 365 and 1/4 days. When multiplied by four as the day cycle, this number is obtained. The name "Four Divisions" comes from this. The day cycle is four, the蔀 days are 27,759, the loss number is 21, the common method is 487, the method number is seven (note from the original text: this number is also the number of intercalary months in the chapter cycle), the day remainder is 168, the middle method is 33, the great three is 343,335, and the monthly cycle is 116.

吳劉洪《乾象曆》上元己丑，至今七千八百八十五算外。乾法一千一百七十八歲，會通七千一百七十一，紀法五百八十九，周天二十一萬五千一百三十，通法四萬三千二十六，通數三十一，日法一千四百五十七，歲中十二，餘數三千九十，章歲十九，沒法一百三，章閏七，會數四十七，會歲八百九十三，章月二百三十五，會率一千八百八十三，朔望合數九百四十一，會月一萬四千四十五，紀月七千二百八十五，元月一萬四千五百七十，月周七千八百七十四，小周二百五十四。

The calendar "Ganxiang Li" was created by Liu Hong of the Wu state during the Three Kingdoms period. It begins from the Jichou year and spans a total of 7885 years up to the second year of Kaiyuan, excluding the starting year. The Gan method is 1178 years, the combined cycle is 7171, the jiyi method is 589, the full circle of the sky is 215,130, the common method is 43,260, the common number is 31, the day method is 1,457, the number of middle terms in a year is 12, the remainder is 3,990, the chapter year is 19 years, the method is 13, the chapter leap is 7, the combined number is 47, the combined year is 893 years, the chapter month is 235, the combined rate is 1,883, the conjunction of new and full moons is 941, the combined month is 14,450, the jiyi month is 7,285, the original month is 14,570, the monthly cycle is 7,874, and the small cycle is 254. This calendar was the first to incorporate the slow and fast movement of the moon and the cycle of solar and lunar eclipses into calculations, making a significant contribution to the development of the calendar system in the central plains.

魏韓詡《黃初曆》上元壬午，至今三萬二千七十二算外。章歲一十九，章閏七，紀法二千八百八十三，斗分一千二百五，周天一百七十八萬三千五百，日法一萬二千七十九，經六千四百九，月法三十五萬六千七百。

The calendar "Huangchu Li" was created by Han Xu of the Cao Wei dynasty. Its upper beginning starts from the Renwu year. From the Renwu year to the second year of Kaiyuan, it totals 32,072 years, excluding the current year. The chapter age is nineteen years, with seven leap months, and the cycle law is 2,883. The division of the Big Dipper is 1,250, referring to the remaining fraction of the celestial sphere beyond full days, which astronomers use to determine the actual length of a year. The celestial sphere is 1,783,500, the day law is 12,790, the number of days is 6,490, and the lunar law is 356,700.

魏楊偉《景初曆》上元壬辰，至今四千五百二十三算上。元法一萬一千五十八，紀法一千八百四十三，紀月二萬二千七百九十五，章歲十九，章月二百三十五，章閏七，通數一十三萬四千六百三十，日法四千五百五十九，餘數九千六百七十，周天六十七萬三千一百五十紀日，歲中一十二，氣法一十二，沒分六萬七千三百一十五，沒法九百六十七，月周二萬四千六百三十八，通法四十七，會通七十九萬一百一十，朔望合數六萬七千三百一十五，入交限數七十二萬二千七百九十五，通周一十二萬五千六百二十一，周日日餘二千五百二十八，周虛二千三十一，斗分四百五十五。

The calendar "Jingchu Li" was created by Yang Wei of the Cao Wei dynasty. Its upper beginning (shangyuan) starts from the Renchen year. From that time to the second year of Kaiyuan, a total of 4523 years have passed. The term "suan shang" means that the number obtained includes the current year, which is opposite to "suan wai" mentioned in previous entries. The fundamental number (yuan fa) is 11,580, the cycle number (ji fa) is 1,843, the cycle for months (ji yue) is 22,795, the chapter for years (zhang sui) is 19 years, the chapter for months (zhang yue) is 235 months, the chapter for leap years (zhang run) is 7, the total number (tong shu) is 134,630, the day number (ri fa) is 4,559, the remainder (yu shu) is 9,670, the number of days in a cycle (zhou tian) is 673,150 days, the number of months in a year (sui zhong) is 12, the method for solar terms (qi fa) is 12, the division for the moon's cycle (mei fen) is 67,315, the method for the moon (mei fa) is 967, the number of days in a lunar cycle (yue zhou) is 24,638, the total method (tong fa) is 47, the conjunction number (hui tong) is 791,110, the number of conjunctions of new and full moons (shuo wang he shu) is 67,315, the limit number for the conjunction of the sun and moon (ru jiao xian shu) is 722,795 (the limit number used for calculating solar and lunar eclipses), the total cycle (tong zhou) is 125,621, the remainder of the day cycle (zhou ri ri yu) is 2,528, the empty cycle (zhou xu) is 231, and the division of the Big Dipper (dou fen) is 455.

晉劉智《正歷》上元甲子，至今九萬七千八百五十算外。章歲十九，章閏七，紀日一百四萬九百五十三，交會通六百一十萬九千一百七十四，紀歲二千八百五十，冬至斗二十一，紀月三萬五千二百五十，餘一萬八千七百三。《姜岌曆》上元甲子，至今八萬四千一百七十算外。章歲十九，章閏七，紀二千四百五十一，斗分六百五，日法六千六十三，經餘三千二百一十七，歷餘三千三百六十二，冬至在斗十七。

The calendar "Zhengli" created by Liu Zhi of the Western Jin dynasty began its upper origin in the Jiazi year. From the upper origin to the second year of Kaiyuan, it totaled 97,850 years, not counting the current year. The cycle of years is 19, with 7 leap years in the cycle, 1,495,300 days in the cycle of days, 6,109,174 cycles of conjunctions and eclipses, 2,850 years in the cycle of years, and the winter solstice was at 21 degrees in the constellation Dou. The calendar "Jiang Ji" also began its upper origin in the Jiazi year. From the upper origin to the second year of Kaiyuan, it totaled 84,170 years, not counting the current year. The cycle of years is 19, with 7 leap years in the cycle, 2,451 cycles of days, 650 parts of the constellation Dou, 6,630 as the day method, 3,217 as the celestial remainder, and 3,362 as the calendar remainder. It fixed the winter solstice at 17 degrees in the constellation Dou.

《梁趙曆》上元甲寅，至今六萬一千七百四十算上。元法四十三萬二千，紀法七萬二千，蔀法七千二百，章歲六百，章月七千四百二十一，（亦曰時法）章閏二百二十二，周天二百六十二萬九千七百五十九，（亦曰通數）餘數三萬七千七百五十九，斗分一千七百五十九，日法八萬九千五十二，（亦曰蔀日）月周九萬六千二百五十二，小周八千二十二，會數一百七十三，度餘二萬七千七百一十九，會虛六萬一千三百三十三，交會差一百四十七，度餘三千三百一十一，遲疾差六百餘四千五百三十，周日二十七日，餘四萬九千三百八十，周虛三萬九千六百七十二。

The upper beginning of the Liang Zhao calendar starts at the Jiayin year. From the Jiayin year to the second year of Kaiyuan, there is a total of 61,740 years, and "counting the current year" includes the year itself. The original method is 432,000, the cycle method is 72,000, the蔀 method is 720, the chapter age is 600 years, the chapter month is 7,421 (note: this item is also called the time method), the chapter leap year is 222, the full circle of the sky is 2,629,759 (note: also called the universal number), the remainder is 37,759, the division of the star is 1,759, the day method is 89,520 (note: also called the蔀 day), the monthly cycle is 96,252, the small cycle is 8,220, the conjunction number is 173, the degree remainder is 27,719, the conjunction void is 61,333, the conjunction difference is 147, the degree remainder is 3,311, the slow and fast difference is 600, the remainder is 4,530, the weekly day is 27 days, the remainder is 49,380, and the weekly void is 39,672.

宋何承天《元嘉曆》上元庚辰，至今五千九百七十四算外。章歲十九，章閏七，紀法六百八，日法九百五十二，歷餘四百一十七，經歷三百九十九，度法三百四，度分七十五，會月九百三十九，會數一百六十，雨水室二度。

The calendar "Yuanjia Li" was created by He Chengtian of the Liu Song dynasty. Its cycle begins in the Gengchen year and spans a total of 5974 years up to the second year of Kaiyuan, excluding the starting year. The cycle length is 19 years, with 7 leap years in each cycle. The system uses a base of 680, a day cycle of 952, a remainder of 417 days, a total of 399 years, a degree cycle of 340, 75 degree fractions, a lunar month cycle of 939, and a lunar month count of 160. This calendar determines that the solar term of Rain Water occurs when the Sun is at 2 degrees of Yingshi.

宋祖沖之《大明曆》上元甲子，至今五萬二千一百九十算外。元法五十九萬二千三百六十五，紀法三萬九千四百九十一，章歲三百九十一，章月四千八百三十六，小周五千二百二十七，章閏一百四十四，閏法十二，月周五十二萬七千九百二十七，日餘十六萬九千七百七十五，月法十一萬六千三百二十一，日法三千九百三十九，歲分一千四百四十二萬三千八百四，歲餘九千五百八十九，餘數二十萬七千四十四，沒分三百六十萬五千九百五十一，沒法五萬一千七百六十一，周天一千四百四十二萬四千六百六十四，度分一萬四百四十九，差八百六十，行分法二十三，小分法一千七百一十七，通周七十二萬六千八百一十，會周七十一萬七千七百七十七，通法二萬六千三百七十七。

The "Da Ming Li" (Great Brightness Calendar) created by Zu Chongzhi of the Liu Song dynasty began its calculation from the Jiazi year, and by the second year of Kaiyuan, it had accumulated a total of 52,190 years, not counting the current year. The fundamental cycle was 592,365, the cycle law was 39,491, the chapter years were 391, the chapter months were 4,836, the small five years were 5,227, and the chapter leap years were 144. Zu Chongzhi abolished the old system of adding a leap month every 19 years and instead arranged 144 leap months over 391 years, which was his most famous reform in the calendar; the leap month rule was 12, the monthly cycle was 527,927, the day remainder was 169,775, the monthly calculation was 116,321, the day calculation was 3,939, the annual division was 1,442,384, the annual remainder was 9,589, the remainder number was 207,440, the division of the remainder was 3,600,5951, the calculation of the remainder was 51,761, the circumference of the sky was 1,442,4664, the degree and minute measure was 14,490, the difference was 860 (which is the value of the precession of the equinoxes, and the introduction of the precession into calendar calculations began with this calendar), the movement division was 23, the small division calculation was 1,717, the overall cycle was 726,810, the conjunction cycle was 717,777, and the overall calculation was 26,377.

梁武《大同歷》上元甲子，至今一百二萬五千八百七十算外。章歲六百一十九，日法一千五百三十六，紀法三萬九千六百一十六。後魏張龍祥《正光曆》上元壬子，至今一十六萬七千九百四十二算外。章歲五百五，章閏一百八十六，章月六千二百四十六，蔀法六千六十，斗分一千四百七十七，紀法六萬六百，統法十二萬一千二百，元法三十六萬三千六百，日法七萬四千九百五十二，周天分二百二十一萬三千三百七十七，氣法二十四，經月大餘二十九，（小餘三萬九千七百六十九）會數一百七十三餘二萬三千二百八，含通一千二百九十八萬九千九百四，周日二十七，餘四萬一千五百六十二，通周二百六萬五千二百六十六，小周六千七百五十一，月周八萬一千一十二。

During the reign of Liang Wu Di, the calendar "Dadong Li" was used, starting from the Jiazi year as the beginning of the cycle. From the beginning of the cycle to the second year of Kaiyuan, it accumulated a total of 1,025,870 years, not counting the current year. The cycle length was 619 years, the day method was 15,360, and the record method was 39,616. The calendar "Zhengguang Li" was created by Zhang Longxiang of the Later Wei dynasty, starting from the Renzi year as the beginning of the cycle. From the beginning of the cycle to the second year of Kaiyuan, it accumulated a total of 167,942 years, not counting the current year. The cycle length was 555 years, the cycle leap years were 186, the cycle months were 6,246, the蔀 method was 6,600, the division of the斗 was 14,770, the record method was 66,000, the overall method was 121,200, the original method was 363,600, the day method was 74,952, the total heaven division was 2,213,377, the weather method was 24, the large remainder of the month was 29 (note in the original text: small remainder was 39,769), the conjunction number was 173 with a remainder of 23,280, the total comprehensive number was 12,989,940, the total day cycle was 27 with a remainder of 41,562, the overall cycle was 266,526, the small week was 6,751, and the monthly cycle was 81,120.

東魏《興和曆》上元甲子，至今二十九萬四千一百七十算外。章歲五百六十二，度法一萬六千八百八十，章閏二百七，斗分四千一百一十七，日法二十萬八千五百三十，餘十一萬六百四十七，周天六百一十五萬八千一十七，歷餘一十萬五千六百三十一，會數一百七十三，日餘六萬七千一百一十七，木終餘一萬五千六百八，火終餘一萬五千一百四十三，土終餘九百八十一，金終餘一萬四千五百二，水終餘一萬四千八百一十八，冬至斗。

The calendar "Xinghe Li" used by the Eastern Wei began its cycle from the Jiazi year. From that starting point to the second year of Kaiyuan, it totaled 294,170 years, not counting the current year. The cycle length is 562 years, the degree method is 16,880, the cycle leap year is 27, the division of the Dipper mansion is 4,117, the day method is 208,530, the remainder is 116,470, the circumference of the sky is 6,158,117, the calendar remainder is 105,631, the conjunction number is 173, and the day remainder is 67,117. The following are the remainders after the completion of each planet's cycle: the remainder for Jupiter is 15,680, for Mars is 15,143, for Saturn is 981, for Venus is 14,520, and for Mercury is 14,818. The calendar defines the winter solstice to occur in the Dipper mansion.

《劉孝孫曆》上元甲子，至今四十三萬五千二百三十算外。章歲六百一十九，元法一十六萬九百四十，紀法九千四十七，日法一千一百四十四，歲餘一千九百六十六，虛分六千四百七，差分五百九，度法二萬四千一百四十一，行分三十九，會月二千一十三，會率三百四十一，周法三萬四千三百二十，歷朔差分六萬七千四百四十二，冬至命度，起危前五度。

The "Liu Xiaosun Calendar" begins its upper origin at the Jiazi year. From the Jiazi year to the second year of Kaiyuan, it accumulates a total of 435,230 years, not counting the current year. The chapter age is 619 years, the fundamental law is 169,400, the calendar law is 9,447, the day law is 1,144, the surplus of years is 1,966, the empty fraction is 6,470, the difference fraction is 590, the degree law is 24,141, the movement fraction is 39, the conjunction month is 2,113, the conjunction rate is 341, the cycle law is 34,320, and the lunar month difference is 67,442. It calculates the degrees of the winter solstice starting from five degrees before the Rooftop Mansion.

齊宋景《天保曆》上元甲子，至今一十一萬六百九十算外。元法一百四十一萬九千六百，紀法二千三萬六千六百，蔀法三萬三千六百六十，（亦名日度法）章歲六百七十六，（亦名日度法）章閏二百四十九，（亦名閏法）章中八千一百一十二，章月八個三百六十一，日法二十九萬二十六百三十五，周天八百六十四萬一千六百八十五，（亦名通數，亦名蔀法，亦名沒分。）餘數一十二萬四千八十七，（亦名沒分）斗名五千七百八十七，歲中十二，氣法二十四，

The calendar called Tianbao Li, compiled by Song Jing of the Northern Qi dynasty, begins its upper origin in the Jiazi year. From the Jiazi year to the second year of Kaiyuan, it totals 110,690 years, not counting the current year. The fundamental number is 1,419,600, the cycle number is 23,6600, the蔀 number is 33,660 (note: this item is also called the day measure), the chapter year is 676 years (note: also called the day measure), the chapter leap years are 249 (note: also called the leap measure), the chapter middle is 8,112, and the chapter month is eight hundred thirty-six and one. The "eight" here is likely a mistake for "eight thousand," as the original manuscript shows, and it is still recorded as is without correction. The day measure is 292,635, though this number is also found to have an error in the original text. The full circle is 8,641,685 (note: also called the universal number, also called the蔀 number, also called the measure of disappearance), the remainder is 124,87 (note: also called the measure of disappearance), the name of the star is 5,787, the number of years is twelve, the measure of the solar term is twenty-four.

會數一百七十三，餘九萬一千五十八，會通五千七十一萬六千九百一十三，會虛二十七萬一千五百七十七，周日二十七，餘一十六萬二千二百六十一，通周八百六萬三千四百六，周虛一十三萬三百七十四，小周九千三十七，月周三十一萬六千二百九十五，望十四，餘二十二萬三千九百五十三半，交限數一百五十八，餘一十五萬九千七百三十九半，經月二十九，餘一十五萬五千二百七十二，虛分十三萬七千三百六十三。

The Heavenly Stems are: Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui. The Earthly Branches are: Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, Hai. The Trigrams are: Qian (Heaven), Dui (Lake), Li (Fire), Zhen (Thunder), Xun (Wind), Kan (Water), Gen (Mountain), Kun (Earth).

周甄鸞《天和曆》上元甲寅，至今八十七萬五千九百四十算外。章歲三百九十一，章閏一百四十四，蔀法三萬三千四百六十，日法二十九萬一百六十，朔餘一十九萬三千九百九十一，斗分五千七百三十一，會餘九萬三千五百一十六，歷餘一十六萬八百三十，冬至日在斗十五度。周馬顯《景寅元曆》上元景寅，至今四萬一千六百八十八算外。章歲四百四十八，章閏一百六十五，斗分三千一百六十七，蔀法一萬三千九百九十二，日法五萬三千五百六十三，（亦曰蔀會法）歷餘二萬九千六百九十三，會日一百七十三，會餘一萬六千六百一十九，冬至日在斗十二度。

The calendar "Tianhe Li" (Heavenly Harmony Calendar) was created by Zhen Luan of the Northern Zhou dynasty. Its cycle began in the year Jiayin, and by the second year of Kaiyuan, it had accumulated 875,940 years, not counting the current year. The cycle of years is 391, the cycle of intercalary months is 144 (following Zu Chongzhi's intercalation cycle), the method of the "pu" is 33,460, the method of days is 291,600, the remainder of the new moon is 193,991, the division of the Big Dipper is 5,731, the remainder of the conjunction is 93,516, and the remainder of the calendar is 168,300. The winter solstice was fixed at the fifteenth degree of Nandou (Southern Dipper). The calendar "Jingyin Yuanli" (Jingyin First Year Calendar) was created by Ma Xian of the Northern Zhou dynasty. Its cycle began in the year Jingyin (in the Tang dynasty, to avoid the name of Emperor Gaozu Li Yuan's father Li Bian, the character "bing" was changed to "jing", so Jingyin here refers to Bingyin). By the second year of Kaiyuan, it had accumulated 41,688 years, not counting the current year. The cycle of years is 448, the cycle of intercalary months is 165, the division of the Big Dipper is 3,167, the method of "pu" is 13,992, the method of days is 53,563 (note in the original text: also called "pu hui fa"), the remainder of the calendar is 29,693, the conjunction day is 173, and the remainder of the conjunction is 16,619. The winter solstice was fixed at the twelfth degree of Nandou.

隋《張賓曆》上元甲子，至今四百一十二萬九千一百三十算外。元法六百一十七萬七千六百，紀法一百二萬九千六百，蔀法十萬二千九百六十，（亦名度法）章歲四百二十九，章閏一百五十八，章月五千三百六，通月五百三十七萬二千二百九，日法十八萬一千一百二十，（亦曰周法）虛分八萬五千三百九十一，朔時法一萬五千六百一十，周天分三千七百六十萬五千四百六十三，（亦曰沒分）餘數五十三萬九千八百六十三，（亦曰沒法）歲中十二，斗分三萬五千六十三，氣法二十四，氣時法八千五百八十，

The calendar system created by Zhang Bin of the Sui dynasty begins its cycle from the Jiazi year. From the Jiazi year to the second year of Kaiyuan, it accumulates a total of 4,129,130 years, excluding the current year. The fundamental number is six million one hundred seventy-seven thousand six hundred, the cycle number is one hundred twenty-nine thousand six hundred, the蔀 number is ten thousand two hundred ninety-six (note: also called the degree number), the chapter year is four hundred twenty-nine years, the chapter leap year is one hundred fifty-eight, the chapter month is five thousand three hundred sixty, the total month is five million three hundred seventy-two thousand two hundred ninety, the day number is eighteen thousand one hundred twenty (note: also called the week number), the empty fraction is eight thousand five hundred thirty-one, the new moon time number is fifteen thousand six hundred ten, the total celestial circle number is thirty-seven million six hundred fifty-four thousand six hundred thirty-three (note: also called the disappearance number), the remainder is fifty-three thousand nine hundred sixty-three (note: also called the method number), the year has twelve months, the star division is thirty-five thousand six hundred thirty-three, the qi number is twenty-four, and the qi time number is eight thousand five hundred eighty. These constants of this calendar are extremely intricate. Below, they are listed in three categories: conjunctions, eclipses, and variations in speed and distance.

會月一千二百九十七，會率二百二十二，合數一百一十半，會分一十一億八千七百二十五萬八千一百八十九，會通六十九億六千七百七十五萬五千七十三，（亦曰交數）會日法四千二十萬四千三百二十，會日一百七十三，餘五萬六千一百四十三，小分一百一十，會虛一十二萬五千七百七十六，小分一百一十一，會日限一百五十八，會九萬八千八百三十八，小分二百二十半，朔望合日數十四，餘一十三萬九千二百二十四，小分一百一十半，

The numbers used in the calculation of conjunctions in the "Zhang Bin Li": the conjunction month is 1,297; the conjunction rate is 222; the combined number is 110 and a half (half refers to half, same below); the conjunction fraction is 118,725,8189; the conjunction passage is 69,677,5573 (original book note: also called the conjunction number); the conjunction day method is 42,4320; the conjunction day is 173, remainder 56,143, small fraction 110; the conjunction void is 125,776, small fraction 111; the conjunction day limit is 158; the conjunction nine is 98,838, small fraction 220 and a half; the number of days from new moon to full moon is 14, remainder 139,224, small fraction 110 and a half.

交法五億一千二百一十萬四千八百三，分法二千八百一十五，陰陽曆一十三，餘三萬二百六十三，小分二千三百二十八，歷合二十七，餘三萬八個六百七，小分一千八百四十一，朔差二，餘五萬七千九百二十二，小分九百七十四，望差一，餘二萬八千九百六十一，小分一千八百九十四半，蝕限十二，餘八萬一千三百三，小分四百三十三半，定差四萬四千五百四十八，通周五百一萬二千六百九十九，餘歲一萬三千八百九十七，限三千入百一十三，周日二十七，餘十萬八百五十九，（亦名小大法）周虛八萬一千六十一，通率七，差虛十三萬七千三百七十二，轉率二百四十，分率七百五十八，日周一百三十七萬六千四百，小周五千七百三十五，朔望合數十四，度餘一十三萬九千二百二十四半。

The numbers used in The Zhang Bin Calendar for calculating eclipses and the moon's slow and fast motion: the method of intersection is five hundred million one hundred twenty-one thousand four hundred eighty-three, and the division method is two thousand eight hundred fifteen; yin and yang calendar thirteen, remainder thirty-two thousand six hundred thirty, small fraction two thousand three hundred twenty-eight, which is the number of days the moon takes to move in and out of the Yellow Path north and south once; the conjunction cycle is twenty-seven, remainder thirty-eight thousand six hundred seventy (the term 'eight' is likely a mistake for 'eight thousand', as in the original text), small fraction one thousand eight hundred forty-one; the new moon difference is two, remainder fifty-seven thousand nine hundred twenty-two, small fraction nine hundred seventy-four; the full moon difference is one, remainder twenty-eight thousand nine hundred sixty-one, small fraction one thousand eight hundred ninety-four and a half; the eclipse limit is twelve, remainder eighty-one thousand three hundred thirty, small fraction four hundred thirty-three and a half, and only when the moon enters this limit can an eclipse be visible; the fixed difference is forty-four thousand five hundred forty-eight; the total cycle is five hundred eleven thousand two hundred sixty-nine, remainder ten thousand three hundred eighty-nine, limit three thousand eight hundred thirteen (the term 'enter' is likely a mistake for 'eight', as in the original text); the weekly cycle is twenty-seven, remainder one hundred thousand eight hundred fifty-nine (original note: also called the small and large method); the empty cycle is eighty-one thousand six hundred eleven, the general rate is seven, the difference in the empty cycle is thirteen thousand seven hundred seventy-two, the turning rate is two hundred forty, the division rate is seven hundred fifty-eight; the daily cycle is one hundred thirty-seven thousand six hundred forty, the small cycle is five thousand seven hundred thirty-five; the conjunction and opposition number is fourteen, degree remainder thirteen thousand nine hundred twenty-four and a half.

《張曹玄曆》上元甲子，至今一百四十二萬七千七百五十算外。章歲四百一十，章閏一百五十一，日法一千一百四十四，度法四萬二千六百四十，朔餘六百，斗分一萬八千八百六十六，歲差五百三，周法二千五百四十八，餘一千四百一十三，冬至命度虛，（初五，後七。）交會通一千六十四萬六千七百二十九，（比古會日）朔差九十萬七千五十七。（比古會數）劉焯《皇極曆》上元甲子，至今一百萬八千九百五十算外。章歲六百七十六，度法九萬三千二百八十八，日法一千二百四十二，餘六百五十九，歲餘二萬二千八百一十三，歲差一千四百二十三，會月二千七百二十九，會率四百六十五，周法二千二百六十三，餘一千二百五十五。

The "Zhang Caoyuan Li" begins its upper origin in the Jiazi year, and by the second year of Kaiyuan, it accumulates 1,427,750 years, excluding the current year. The chapter age is 410 years, the chapter leap years are 151, the day law is 1,144, the degree law is 42,640, the lunar remainder is 600, the degree of the zodiac is 18,866, the annual difference is 530, the week law is 2,548, and the remainder is 1,413; it states that the degree of the winter solstice falls in the mansion of Emptiness (note: originally it was five, later it became seven). The conjunctions total 1,646,729 (note: equivalent to the item of the conjunction day in the ancient calendar), and the lunar difference is 907,570 (note: equivalent to the item of the conjunction number in the ancient calendar). The "Huangji Li" created by Liu Zhuo of the Sui dynasty also begins its upper origin in the Jiazi year, and by the second year of Kaiyuan, it accumulates 1,008,950 years, excluding the current year. The chapter age is 676 years, the degree law is 93,288, the day law is 1,242, the remainder is 659, the annual surplus is 22,813, the annual difference is 1,423, the conjunction month is 2,729, the conjunction rate is 465, the week law is 2,263, and the remainder is 1,255.

傅仁均《戊寅歷》上元戊寅，至今一十六萬四千四百三十六算外。章歲六百七十六，（亦名行分法）章閏二百四十九，章月八千六百六十一，月法三十八萬四千七十五，日法一萬三千六，時法六千五百三，度法九千四百六十四，（亦名氣法）氣時法一千一百八十三，歲分三百四十五萬六千六百七十五，歲餘二千三百一十五，周天分三百四十五萬六千八百四十五，斗分二千四百八十五半，沒分七萬六千八百一十五，沒法一千一百三，歷日二十七，歷日餘一萬六千六十四，歷周七十九萬八千二百，曆法二萬八千九百六十八，餘數四萬九千六百三十五。

The calendar "Wuyin Li" was created by Fu Renjun in the early Tang dynasty. It began from the year Wuyin and spanned a total of 164,436 years up to the second year of Kaiyuan, not counting the current year. This was the first calendar issued after the founding of the Tang dynasty. The cycle of years is 676 years (also called "Xing Fen Fa"), the cycle of leap years is 249, the cycle of months is 8,661, the monthly method is 384,750, the daily method is 13,600, the hourly method is 6,530, the degree method is 9,464 (also called "Qi Fa"), and the method for calculating the solar terms is 1,183. The annual division is 3,456,675, the annual remainder is 2,315; the total circumference of the sky is 3,456,845, the division of the Big Dipper is 2,485.5; the division for the moon's disappearance is 76,815, the method for the moon is 1,130; the calendar days are 27, the remainder of the calendar days is 16,640, the calendar cycle is 798,200, the calendar method is 28,968, and the remainder is 49,635.

《神龍曆》上元乙巳，至今四十一萬四千三百七十算外。母法一百，旬周六十，辰法八刻三十三分二十半。期周三百六十五日，餘二十四，奇四十八。氣法十五日，餘二十一，奇八十五少半。月法二十九日，餘五十三，奇六。候法五日，餘七，奇二十八。遊分四，望法十四日，餘七十六，奇五十三。弦法七日，餘三十八，奇二十六半。閏差十日，餘八十七，奇七十六。沒數九十一，餘三十一，奇十二。沒法一，餘三十一，奇十二。日周法二十七日，餘五十五，奇四十五，小分五十九。日差法一百，餘九十七，奇六十，小分四十一。天周三百六十五，度餘二十五，奇七十一，小分七十一。

The Shennong Calendar begins its upper origin in the Yisi year. From the Yisi year to the second year of Kaiyuan, it accumulates 414,370 years, excluding the current year. This calendar uses 100 as the base number, and all remainders are calculated in percentages. A period of ten days (ten days) is sixty, meaning that the sixty heavenly stems and earthly branches form a cycle. The time of the hour is eight and a half hours and thirty-three parts, which is the number of divisions in one hour. A year is 365 days, 24 remainders, and 48 odd parts, which is the length of a solar year. A solar term is fifteen days, 21 remainders, and 85 odd parts, which is the number of days in a solar term. A synodic month is twenty-nine days, 53 remainders, and 6 odd parts, which is the length of a synodic month. A climate period is five days, 7 remainders, and 28 odd parts, which is the number of days in a climate period. The movement of the sun is four. The time from new moon to full moon is fourteen days, 76 remainders, and 53 odd parts, which is the number of days from new moon to full moon. The time from new moon to the first quarter moon is seven days, 38 remainders, and 26 and a half odd parts, which is the number of days from new moon to the first quarter moon. The leap month difference is ten days, 87 remainders, and 76 odd parts. The disappearance number is ninety-one days, 31 remainders, and 12 odd parts; the no-number is one day, 31 remainders, and 12 odd parts. The daily cycle is twenty-seven days, 55 remainders, 45 odd parts, and fifty-nine small parts; the daily difference is one hundred days, 97 remainders, 60 odd parts, and forty-one small parts. The celestial cycle is three hundred and sixty-five days, 25 remainders, 71 odd parts, and seventy-one small parts, which is the number of degrees in a full circle.

交周法二十七日，餘二十一，奇二十二，小分十六七分。交差法二日，餘二十一，奇八十三，小分八十三三分。交中十三日，餘六十，奇六十一，小分八三分半。陽前限十二日，餘四十四，奇六十九，小分十六七分。陽後限一日，餘十五，奇九十一，小分九十一六分半。陰前限二十六日，餘十五，奇三十，小分二十五分半。歲星合法三百九十三日，餘八十六，奇七十九，小分八十。熒惑合法七百七十九日，餘九十一，奇五十五，小分四十五。鎮星合法三千七十九日，餘八，奇四，小分八十。太白合法五百八十三日，餘九十一，奇七十七，小分七十。中合法二百九十日，餘九十五，奇八十八，小分八十五。辰星合法一百一十五日，餘八十七，奇九十五，小分七十。中合法五十七日，餘九十三，奇九十七，小分八十五。

The following are the various numbers used in the Shénlóng Calendar for calculating eclipses and the conjunctions of the five planets. The cycle of the lunar node is 27 days, 21 days, 22 parts, and 16 and 7 fractions, which is the length of one lunar month passing through a node. The difference in the node is 2 days, 21 days, 83 parts, and 83 and 3 fractions. The midpoint of the node is 13 days, 60 days, 61 parts, and 83 and a half fractions, which is half of the node cycle. The limits for the moon's visibility of an eclipse when it is ahead of the sun are 12 days, 44 days, 69 parts, and 16 and 7 fractions; when it is behind the sun, it is 1 day, 15 days, 91 parts, and 91 and 6 fractions. The limits for the moon's visibility of an eclipse when it is in the shadow of the sun are 26 days, 15 days, 30 parts, and 25 and a half fractions: these are the boundaries for the moon's movement inside and outside the ecliptic where eclipses can be seen. The cycle for the planet Jupiter is 393 days, 86 days, 79 parts, and 80 fractions; for Mars, it is 779 days, 91 days, 55 parts, and 45 fractions; for Saturn, it is 3,790 days, 8 days, 4 parts, and 80 fractions; for Venus, it is 583 days, 91 days, 77 parts, and 70 fractions; for Mercury, it is 115 days, 87 days, 95 parts, and 70 fractions. The term "cycle" refers to the number of days between one conjunction of a planet with the sun and the next conjunction.

李淳風見行《麟德曆》上元甲子，至今二十六萬九千九百三十算外。章歲三萬九千五百七十一，總法一千三百四十，章月三十八萬九千四百二十八，半總六百七十，章閏一萬四千五百七十六，辰率三百三十五，閏分一千二百一十四奇八，奇率一十二。歷周四十四萬三千七十七，旬周六十。月法二十九日，餘七百一十一。弦法七日，餘五百一十二，奇九。氣法十五日，餘二百九十二，奇十。曆法二十七日，餘七百四十三，奇。

Li Chunfeng compiled the "Linde Calendar," which is still in use today. Its cycle begins with the Jiazi year and spans a total of 269,930 years up to the second year of the Kaiyuan era, not counting the starting year. The chapter age is 39,571, the total method is 1,340, the chapter month is 389,428, half of the total method is 670, the chapter leap year is 14,576, the rate of the celestial stem is 335, the leap year fraction is 1,214 with a remainder of 8, and the remainder rate is 12. This calendar abandoned the traditional methods of chapters, parts, eras, and origins, using the total method as the universal rate, marking a major change in the field of calendar studies. The calendar cycle is 443,770, the ten-day cycle is 60. The lunar method is 29 days with a remainder of 711, which is the length of a synodic month. The method for the quarter moon is 7 days with a remainder of 512 and a remainder of 9, which is the number of days from new moon to quarter moon. The method for the solar term is 15 days with a remainder of 292 and a remainder of 10, which is the number of days for a solar term. The method for the lunar motion is 27 days with a remainder of 743 and a remainder of several, which is the number of days for the moon to complete one cycle of its slow and fast motion.