In depth translation and interpretation of the algorithm for planetary motion in Chapter 25 of the Brahma Revised System, including the calculation of the true ecliptic of the Sun and Moon, determination of the solar eclipse, and calculation of the planetary ecliptic. It specifically points out the unique lunar correction table (lunar departure table) in Indian astronomy and its association with Tibetan calendar documents, emphasizing its academic value. The translation adopts a literal translation method, with Sanskrit terms and explanatory sentences marked in parentheses, and additional sentences indicated in square brackets. The annotation section provides a detailed explanation of the scripture text, offering algorithm formulas, charts, and derivation processes. Each section also begins with a brief introduction to the content.

Section 18: Calculation of Slow Circle Correction Values for the Sun and Moon

Section 18 elaborates on the calculation method of the slow circle correction values for the sun and moon. The correction value of the sun is calculated by the difference between its ecliptic longitude and its apogee longitude, and the correction value of the moon is similar. The maximum correction value for the sun is 90 °, and for the moon it is about 300 ′, both reaching their maximum at an argument of 90 °. The polarity of the correction value will be explained in the next section.

Section 19 Calculation of Slow Circle Correction Values for Daily Motion of the Sun and Moon

Section 19 elaborates on the calculation method of the slow circle correction value for the daily motion of the sun and moon. The correction value of the sun is obtained by dividing the number of segments of the argument by 17, while the correction value of the moon needs to be subtracted by 1/8 and then multiplied by 2. The sign of the correction value depends on the house of the argument, with the 6th house starting from Cancer being positive and the 6th house starting from Capricorn being negative. In the slow circle model, the sun or moon has a correction value of 0 at the onset of Aries, a negative maximum value at the onset of Cancer, a correction value of 0 at perigee for celestial bodies, and a positive maximum value at the onset of Capricorn. The correction value varies with the number of arguments, with a positive value in the range of 90 ° -180 °, a continued positive value in the range of 180 ° -270 °, and a negative value in the range of 270 ° -360 °. In the calculation formula, B represents the string of the argument, and its calculation method is described in Section 16. The difference between the chord values of the daily movement of the sun and the correction value is equal, and the daily movement of the moon is the total movement minus the apogee movement. The correction value is twice the difference between the chord values.

Section 20: Correction of the Moon's Wrist and Correction of the Taiyin Day

Section 20 provides a detailed explanation of the calculation methods for correcting lunar wrist deviation and solar eclipse correction. Firstly, apply 1/27th of the correction value of the sun to the moon to adjust its arguments. The revised moon argument takes into account the sign of the sun correction value. Next, adjust the number of days in a synodic month by adjusting the correction values of the sun and moon for the lunar eclipse. The correction of the Taiyin day is based on the ecliptic longitude difference between the moon and the sun, as well as their true positions.

Section 21 Slow Circle Correction Difference Table for Arc Engraving Units

Section 21 elaborates on the calculation method of the slow circle correction difference table for the arc units of the moon and the sun. The correction value difference table for the moon is obtained through arguments, while the correction value for the sun is based on subtracting the apogee. The orbit of the moon's argument is divided into 28 equal parts, each with an angle of approximately 13 degrees. The chord and correction values of the argument are calculated in units of arc minutes and arc minutes. Table 2 shows the moon correction values and their differences, while Table 3 shows the sun correction values and their differences. These correction tables are rare in Indian calendar books, and their numerical results are concise and in line with astronomical calculation goals. The structure and numerical values of the moon correction table also appear in the Tibetan calendar "Shilun Calendar", indicating the similarity between the two and worthy of further research.

Section 22 Calculation of Taiyin Day

This article elaborates on the calculation method of a solar eclipse day, including subtracting the solar fraction, dividing by 720 to obtain the number of solar eclipse days, and how to calculate the passing and upcoming parts of a solar eclipse day based on the daily motion difference between the sun and the moon. By following these steps, the number of solar terms in the lunar calendar can be determined.

Section 23: Calculating the Constellation of a Planet

Section 23 provides a detailed explanation on how to calculate the position of a planet in the 27 constellations of India. Firstly, divide the planetary ecliptic fraction by 800 to obtain the number of constellations. Then, divide the part of the planet that has passed and will pass by by the daily motion to obtain the number of days, and multiply by 60 to obtain the number of minutes. The length of each constellation is determined by dividing the circumference of the ecliptic into 27 equal parts. The number of constellations passed by a planet is obtained by dividing its yellow longitude by the length of each constellation, and the remainder represents its current position within the constellation. Finally, divide the two parts by the daily exercise amount to obtain the corresponding number of days and minutes.

Section 24: Calculating the 'Union' of the Sun and Moon

Section 24 elaborates on the method of calculating the "conjunction" of the sun and moon, which involves dividing the "sum" score of the sun and moon by 800 to obtain the "conjunction" value, then multiplying the "sum" by 60 and dividing it by the "sum" of daily movements to obtain the number of moments. 'Union' is a concept in astrology, where the sum of the movements of the sun and moon equals the length of a constellation, known as a union. The calculation of the remainder follows the principle of equation (22), which represents the time that has passed since the current 'union' or the time required to reach the next 'union' in units of units of units. The article also mentioned Pinggeli's restoration work on the order and names of "He", as well as the naming of specific "He" in the "Calendar Ganlu".

Section 25: Calculating Half Day of Taiyin

Detailed explanation on how to calculate half lunar days, including the calculation methods for fixed and indefinite half lunar days. Introduced the concept of a semi lunar day, which is half of a lunar day and includes 60 semi lunar days in a synodic month. The article also mentioned 11 types of semi cloudy days, of which 4 are fixed and 7 are indefinite, appearing every 7 and a half cloudy days starting from Vava. Table 6 shows the numbering and arguments of semi cloudy days used to determine indefinite semi cloudy days.

Chapter 26: The Plain Yellow Meridian of Mars at the End of the Solar Year

Detailed explanation on how to calculate Mars' ecliptic longitude at the end of the solar year. Firstly, multiply the solar year by a specific value, and then add a constant to obtain the ecliptic longitude of Mars. The article cites relevant content from Chapter 1 of the Brahma Revised System, explaining how to convert the number of days per week of Mars into seconds and divide it by the number of solar years in one cycle to obtain Mars' ecliptic longitude. In addition, the epoch constant of Mars was mentioned, which refers to the position of Mars at the beginning of the epoch, and how to calculate the ecliptic longitude of Mars at a specific time by adding this constant.

Section 27: Calculating the Yellow Meridian of Mercury's Fast Circle High Point

Detailed explanation on how to calculate Mercury's fast circle high point ecliptic longitude. Firstly, a base value is obtained by multiplying the number of years by a specific degree, minute, and second, and then adding a specific degree, minute, and second to obtain the yellow longitude of Mercury's fast circle high point. The article mentions that Indian mathematical astronomy, in addition to the slow circle model, also uses the fast circle model to correct the retrograde phenomenon of five stars when correcting planetary motion. The fast circle model has a larger circle radius and faster speed, and its high point is equivalent to that of the sun for inner planets such as Mercury and Venus. The calculation method is based on the number of days per week of Mercury's rapid circle high point, derived through specific multipliers and appendages.

Sections 28-31: Calculate the ecliptic longitude of the high points of Jupiter, Venus, Saturn, and the intersection of the Moon

Detailed explanation on how to calculate the ecliptic longitude at the intersection of Jupiter, Venus, Saturn, and the Moon. Firstly, the position of Jupiter is determined by multiplying the historical years with a specific coefficient and adding a fixed value. Next, the high point position of Venus is calculated using a similar method, but with different coefficients and values. Saturn's position calculation also follows this model, but the coefficients and bonuses are changed. Finally, the calculation of the lunar node also uses multiplication and addition, but the coefficients and values involved are different from those of the aforementioned planets. These calculation methods reflect a deep understanding and precise measurement of the laws of planetary motion in ancient astronomy.

Section 32: Special Methods for Planetary Calculation

Section 32 elaborates on a special method for calculating the planetary ecliptic longitude. Firstly, multiply the number of years by the corresponding multiplier, and then perform carry conversion on the product. During the conversion process, it is necessary to divide each digit by its corresponding divisor to obtain the quotient and remainder. The unit of commerce is its previous bit, and the remainder is carried up to the previous bit. Through this method, the amount of motion of a planet over a certain period of time can be calculated. Taking Mars as an example, after 100 years, its ecliptic longitude is 2 houses, 0 degrees, 14 minutes, and 15 seconds, and the final result is obtained by adding the following numbers.

Section 33: Another Method for Calculating Mars

Section 33 introduces another method for calculating the ecliptic longitude of Mars, which calculates the ecliptic longitude of Mars by taking half of the sun and a fixed value for Mars. The article mentions the number of days in a solar eclipse and the number of days in a Martian eclipse, and obtains the continuous fraction of the ratio of the two through the division of twists and turns. The author points out that the Mars motion can be obtained by multiplying the solar motion by the ratio of the number of days per week, without the need for additional correction. At the same time, the author questions the statement of "adding a fixed value for Mars" in the scripture and compares it with the method of finding Jupiter and Saturn in section 35, finding no similar expression. Dvivedin's comments did not provide an explanation for this issue.

Another method for calculating Mercury, Jupiter, Saturn, Nodes, and Venus in Sections 34-36

Elaborated in detail the calculation methods of the motion of Mercury, Jupiter, Saturn, the Node, and Venus by Brahmagupta in the Brahma Revised System. By using specific mathematical formulas and continuous fraction approximation, calculate the fast circular high points of each planet. The article mentions the relative position relationship between the Sun, Mercury, and Venus, as well as how to calculate the positions of other planets based on the known position of the Sun. At the same time, the article also points out some doubts and possible errors in the calculation process and literature transmission.