The Wonder of Calculation (Karana Kutuhala)

Determining the Arc—

Subtract the segments from the remainder multiplied by ten; the unpurified quotient obtained shall be the arc in degrees and so on. This should be added to ten multiplied by the number of segments already subtracted. With the segments reversed, the reverse arc and sine are also found. ॥ 8 ॥

Sumatiharṣa’s Commentary:—

To find the arc dhanu: the angular measure or inverse sine of a given sine jyā: the chord-half used in Indian trigonometry, one begins with the first segment and subtracts as many segments as possible. From the remainder, multiplied by ten and divided by the "unpurified" segment the next segment in the table that was too large to be fully subtracted, the resulting degrees are added to the number of fully subtracted segments (each representing ten degrees). This gives the arc in degrees.

Now, if the segments—25, 5, 12, 15, 17, 19, 20, 21—are used in reverse order, the "inverse arc" and sine can be found through the same method. However, there is no practical need for that here; it is mentioned only in passing.

Furthermore, because of the small margin of error and for the sake of simplicity, even though the teacher Bhāskara II has already spoken, the clarification of the "current sine difference" bhogya-khaṇḍa: the segment of the sine table currently being traversed is found in the Siddhānta Śiromaṇi Bhāskara II's foundational theoretical work, written around 1150 CE:—

"The difference between the past and future segments, multiplied by the remaining degrees and divided by twenty, is added to or subtracted from half the sum of the past and future segments. This results in the 'clear' or corrected future segment to be used in calculating the sine or its inverse."

For example, when calculating the sine for 24 degrees: dividing 24 by ten gives a quotient of 2 and a remainder of 4. The "past" segment is 20 and the "future" segment is 19. From their sum (39), half is 19;30. Because this is a standard sine calculation, subtracting the correction yields 19;14 as the "clear" future segment. Using this corrected segment in the calculation, the sine is found to be 48;43. This should be known as the sine of the maximum declination the tilt of the Earth's axis, approximately 24 degrees in this system.

Now, for finding the arc from the sine, the method for correcting the future segment according to the Siddhāntas is:—

"Subtract the segments from the remainder; multiply by the difference between the past and future segments, and divide by the future segment. The result is added or subtracted to the future/past segment to find the corrected future segment for the purpose of determining the arc."

By this method, the difference between past and future is 1. Multiplied by the remaining degrees 4, it is 4. Divided by twenty, it is 0;12. Following the rule "The result is added or subtracted...", we find: for the sine of the maximum declination 48;43, the subtracted segments total 29;20, leaving a remainder of 7;43. Half of this is 3;52. The difference between past and future (1) is multiplied by 3;52 and divided by the future segment (19), yielding 0;12. This is subtracted from half the sum of past and future (19;30), resulting in 19;18 as the "clear" future segment. Using this in the rule for the "unpurified quotient," the arc is found to be 24 degrees. This refined method should be used by those desiring high precision. ॥ 8 ॥

Sudhākara:—The explanation for clarity and the logic are simple.

Calculating the Equation of Center—

The sine of the mean anomaly of the Sun and other planets should be multiplied by two and then divided by: 550 (for the Sun); 238 (for the Moon); 107 (for Mars); 198 (for Mercury); 228 (for Jupiter); 784 (for Venus); and 157 (for Saturn). ॥ 9 ॥
The resulting degrees are added to or subtracted from the planet's position depending on the anomaly. For the Sun and Moon, this makes them "clear" true/accurate; for the others (Mars, etc.), they become "corrected-mean" positions. ॥ 10 ॥

Sumatiharṣa’s Commentary:—

"The Sun and others..." One must calculate the base-sine bhuja-jyā of the mean anomaly manda-kendra: the angular distance from the planet's apogee of the Sun and other planets. This sine is multiplied by two (which is the same as multiplying by ten and dividing by five).

If it is for the Sun, divide by 550. The resulting correction in degrees, etc., is 1;34;32. When this is applied to the Mean Sun (1s 1° 33' 26"), the Sun corrected by the Equation of Center becomes 1s 3° 7' 58". From this, the correction for terrestrial latitude cara: ascensional difference is subtracted (86), resulting in the True Sun: 1s 3° 6' 32".

If that sine is for the Moon, divide by 238.
For Mars Bhauma, divide by 107.
For Mercury Budha, divide by 198.
For Jupiter Guru, divide by 228.
For Venus Śukra, divide by 784.
For Saturn Śani, divide by 157.

These degree-corrections are added or subtracted based on the anomaly: if the anomaly is in the half-circle starting with Aries (0–180°), the correction is added; if in the half-circle starting with Libra (180–360°), it is subtracted from the mean planet. By doing this, the Sun and Moon become "clear" (their true positions). For the others, like Mars, they are called "corrected-mean" manda-sphuṭa because they still require the second correction (the Equation of Conjunction).

The maximum Equation of Center parama-mandaphala in minutes for each is:
Sun: 130;50; Moon: 302;31; Mars: 672;54; Mercury: 362;10; Jupiter: 315;43; Venus: 110;0; Saturn: 458;33. ॥ 10 ॥

Sudhākara’s Commentary:—

This is for the sake of clarity. Here is the mathematical logic. Having brought the maximum equations of center according to the rules of the Siddhāntas, a proportion is made. For example, for the Sun: if for a base-sine of 120, the maximum correction in degrees is obtained, then for the desired sine, what is obtained?
The Sun's Equation of Center = $\frac{\text{sine} \times 130}{120 \times 60} = \frac{\text{sine} \times 10}{554}$
Here, the equation is $\frac{120}{60}$. Because this is slightly more, a simplified integer is used for the divisor to provide a very close approximation...