The Wonder of Calculation (Karana Kutuhala)

The term for this is "equal." If the anomaly kendra: the angular distance of a planet from its apogee is within the six signs starting from Aries the first 180 degrees of the orbit, the resulting correction is added to the mean planet to find the corrected-mean position; if the anomaly is in the six signs starting from Libra 180 to 360 degrees, the correction is subtracted. ॥ 3 ॥

Calculating the Base and Perpendicular Components—

If the anomaly is less than three signs 90 degrees, it is the base bhujā: the sine-related arc. If it is greater than three signs, subtract it from six signs 180 degrees. If it is greater than six signs, subtract six signs from it. If it is greater than nine signs, subtract it from twelve signs the full 360-degree circle. To find the perpendicular koṭi: the cosine-related arc, subtract the base from three signs 90 degrees. ॥ 4 ॥

Sumatiharṣa’s Commentary:—

If the anomaly is less than three signs, that value itself is the base. If the anomaly is above three signs, then six signs minus that value is the base. If the anomaly is more than six signs, then the anomaly minus six signs is the base. For an anomaly greater than nine signs, subtract the anomaly from twelve signs to get the base. Furthermore, three signs minus the base shall be the perpendicular. ॥ 4 ॥

Correction of the Apogees of Mars and Others:—

For the fast anomaly śīghra-kendra: an anomaly used to account for the Earth's own motion of Mars and Venus, take the smaller of the "elapsed" or "remaining" minutes of the current quadrant and divide by four hundred original: 'khakha-veda' — kha (0), kha (0), veda (4) = 400. The resulting degrees should be subtracted from or added to the apogee of Mars when the anomaly is in the signs starting with Cancer or Capricorn respectively, to make it "true." ॥ 5 ॥
The "other" true value for the Son of the Earth Mars is found by subtracting one-third of the resulting degrees. ॥ 5 1/2 ॥

Sumatiharṣa’s Commentary:—

Regarding the calculation of the fast anomaly of Mars: one should take the fast anomaly derived from the corrected-mean position. As stated in the Siddhāntasiromaṇi: "The corrected-mean position is calculated first, then the fast anomaly." Also in the Narapati: "The result of the apogee should be calculated as before... the apogee corrected by that becomes perfectly true."

This same method was demonstrated by Lakṣmīdāsa, Mṛgāṅka, and others. Therefore, the opinion of the ancient teachers—that one first calculates the result using the mean apogee and then calculates it again from the true apogee—is also our preferred view.

Now, the fast anomaly is divided into quadrants of three signs each. One looks at the "elapsed" portion of the quadrant or the "remaining" portion (the quadrant minus the elapsed amount). Whichever is smaller is converted into minutes and divided by 400. The result in degrees and minutes is the correction. If the fast anomaly of Mars is in the six signs beginning with Cancer, the correction is subtracted from Mars's apogee; if it is in the six signs beginning with Capricorn, it is added. This results in the "true" apogee of Mars. Furthermore, by subtracting one-third of this correction from the result, the "other" true value for Mars is established. ॥ 5, 5 1/2 ॥

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Sudhākara’s Commentary:—

This is clear. Here is the mathematical proof: The maximum "fast-results" for Mars and the others are calculated based on a radius of 120; these are the values traditionally taught.

Now, the formula for the correction related to Mars's apogee is:
= (20 × degrees × 2) / (75 × 3)
= (20 × degrees × 60 × 2) / (75 × 3 × 60)
= (20 × 2 × minutes of arc) / (75 × 3 × 60)
= (2 × minutes of arc) / (85 × 3 × 3) = (2 × minutes of arc) / 765, which is approximately (minutes of arc) / 383.
For simplicity, the text uses (minutes of arc) / 400. Thus, the root text is justified. Similarly, the actual circumference is derived from the calculated degrees... [and] all is consistent with what has been stated. ॥ 5 1/2 ॥

Determining the Sines—

Twenty-one, twenty, nineteen, seventeen, fifteen, twelve, nine, five, and two. ॥ 6 ॥
These are the sine-differences jyākhaṇḍa: segments used to calculate the sine of an angle in terms of degrees for every ten degrees. The "passed" segments are summed, then the remaining degrees are multiplied by the "current" segment and divided by ten original: 'kh-endu' — kha (0), indu (1) = 10. Adding this to the sum of the passed segments gives the total sine. ॥ 7 ॥

Sumatiharṣa’s Commentary:—

The nine segments are: 21, 20, 19, 17, 15, 12, 9, 5, and 2.

The first row shows the individual differences; the second row shows the cumulative sine totals for every 10 degrees up to 90.

These are the nine sine-segments. To find the sine of a desired angle, convert the angle into degrees and divide by ten. The quotient gives the number of "passed" segments. The remainder, in degrees and minutes, is multiplied by the next segment (the "current" segment), divided by ten, and added to the sum of the previously passed segments. This total is the sine. If the degrees are less than ten, the first segment is used for the calculation. ॥ 7 ॥