The Wonder of Calculation (Karana Kutuhala)

The logic for all these calculations is to be understood as follows: after performing the division of the sine's multiplier by ten, the remainder is indeed clear. ॥ 10 ॥

Finding the True Rising from the Mean Sun’s Rising—

The Sun’s Equation original: "bhānoḥ phalaṃ"; the correction needed to move from mean to true position is multiplied by twenty-seven and applied to the Mean Moon; the Sun’s increase or decrease is likewise applied. ॥ 10 ½ ॥

Sumatiharṣa’s Commentary:—
Regarding the phrase "of the Sun": divide the sine by five hundred and fifty 550. The result is the Sun’s Equation. When this is divided by twenty-seven, the resulting degrees and minutes should be added to or subtracted from the Mean Moon. If the Sun's Equation is positive, add it to the Mean Moon; if it is negative, subtract it. Then the Moon becomes "corrected." This is not done for the other planets because the difference would be negligible. ॥ 10 ½ ॥

Sudhākara’s Commentary:—
Here is the mathematical proof: If the equatorial rising nirakṣodaya: the time it takes for a sign to rise at the Earth's equator is obtained for thirty degrees, what then for the degrees of the Sun’s Equation? Furthermore, if the Moon’s motion in degrees is obtained for thirty-six hundred water-measures pala: a unit of time equal to 24 seconds, what is obtained by the previously calculated time-units? The form of this correction in degrees is as follows:

Since mean values are used everywhere here, the equatorial rising is taken as equal to thirty hundred 3000 palas. Similarly, the Moon’s motion in minutes is taken as eight hundred 800 for the sake of simplicity. Therefore, the rearranged form of the correction is:

Thus, it is proven. ॥ 10 ½ ॥

The Equation of Motion for the Sun and Other Planets—

For the Sun, multiply the current sine segment bhogya-khaṇḍa: the specific segment of the sine table the planet is currently in by twenty and divide by nine hundred; for the Moon, multiply by thirteen and divide by four. ॥ 11 ॥

For Mars and Mercury, multiply by two and divide by seven. For Jupiter and the others, divide by fifty, twelve, and one hundred and twenty respectively. This is the Equation of Motion; it is added or subtracted starting from the signs of Cancer or Capricorn respectively. ॥ 12 ॥

Sumatiharṣa’s Commentary:—
For whichever planet one seeks the "Equation of Daily Motion" bhukti-phala, if one is calculating for the Sun, take the current sine segment of its "base" bhuja-jyā: the sine of the anomaly, divide it by nine, and the resulting minutes are the Sun’s Equation of Motion. Similarly, for the Moon, take its current sine segment, multiply by thirteen, and divide by four. For Mars, multiply by two and divide by seven.

This result is the Equation of Motion for Mars. For Mercury, use the segment as is divided by 7, as per the previous rule. For Jupiter, take its current sine segment and divide by fifty. For Venus, divide the segment by twelve. For Saturn, divide the current sine segment by one hundred and twenty.

Each respective Equation of Motion is added to the planet’s mean daily motion when the "Slow Anomaly" manda-kendra: the angular distance from the planet's apogee is in the half-circle starting with Cancer. It is subtracted when the anomaly is in the half-circle starting with Capricorn. When this is done, the daily motions of the Sun and Moon become "True" sphuṭa; for Mars and the others, they are called "Slow-Corrected" manda-sphuṭa: corrected for the first anomaly but not yet for the second. ॥ 12 ॥

Sudhākara’s Commentary:— Now, for the calculation of the Equation of Motion, the proportion is: If the sine segment is obtained for ten degrees, what is obtained for the motion of the Slow Anomaly in degrees? This is the difference between the sines of today and tomorrow. The resulting Equation of Motion in minutes for the Sun is:

The proof for all the other planets should be understood in the same way, making the entire system faultless.

Calculating the Equation of Conjunction for Mars and Other Planets—

The "perpendicular-sine" koṭijyā: the cosine produced from the "Variable Anomaly" cala-kendra: the anomaly related to the planet's orbit around the Sun is multiplied by its own specific multiplier the planet's specific constant and then doubled. This is added to or subtracted from sixty.

When the anomaly is in the half-circle starting with Aries or Capricorn, add the square of the multiplier to fourteen thousand four hundred. The square root of that sum is the "Hypotenuse" śravaṇa: the distance from the Earth to the planet. The "base-sine" dorjyā multiplied by the planet's constant and divided by the Hypotenuse gives a value; the arc of that value is the "Fast Correction" śīghra-phala. This should be added to or subtracted from the Slow-Corrected planet to find its True position. ॥ 13 ॥

Sumatiharṣa’s Commentary:—
Subtract the Slow-Corrected position from the "Fast Apex" śīghrocca: the point of conjunction; the remainder is the "Variable Anomaly." Find the cosine koṭijyā of that anomaly. Multiply it by the planet's specific multiplier, then multiply by two and divide by sixty. Add this to sixty. Then take the square of the planet's multiplier and add it to 14,400. Subtract the doubled cosine from this when the anomaly is in the Cancer half-circle. The square root of this—found by the method "Subtract from the last odd place" described in the Līlāvatī the famous mathematical treatise by Bhāskara II—is known as the "Hypotenuse" karṇa.

Then, the sine dorjyā of the Variable Anomaly is multiplied by the multiplier and divided by the Hypotenuse. Find the arc of this result using the previously mentioned sine segments. This arc is the "Fast Correction." It is added to or subtracted from the Slow-Corrected planet based on whether the anomaly is in the half starting with Aries or Libra. In the Aries half-circle, it is added; in the Libra half-circle, it is subtracted. Then the planet becomes "True" sphuṭa. ॥ 13 ॥