The Wonder of Calculation (Karana Kutuhala)

Sudhākara:—

This is for the sake of clarity. The logical proof upapatti: mathematical derivation or rationale is as follows: By multiplying by 12,000 original: 'khābhrakhārkair' — kha (0), abhra (0), kha (0), arka (12) and so on, the result in seconds of arc vikalā: 1/3600th of a degree is obtained. When this is corrected by the difference between the actual observed planet and the planet calculated from the day-count, the true position is revealed. For example, to understand the correction factor bījakarma: a correction applied to mean motion to align it with observation for the Moon in arc-seconds, first, the daily motion in degrees is taken as $\frac{1400080}{106257}$. Its continued fraction expansion vitatamāna: a method of successive approximations is:

$13 + \frac{1}{9 + \frac{1}{5 + \frac{1}{1 + \frac{1}{9 + \frac{1}{29 + \frac{1}{1 + \frac{1}{2 + \frac{1}{1 + \frac{1}{3 + \frac{1}{1 + \frac{1}{3 + \frac{1}{6}}}}}}}}}}}}}$

From this, the approximate values These are the 'convergents' of the fraction are: $\frac{13}{1}, \frac{66}{5}, \frac{79}{6}, \frac{224}{17},$ and so on.

In calculating the planet's position from the day-count, the Teacher referring to Bhāskara II, the author adopted this value: $\frac{224}{17}$. This can be expressed in another form as $\frac{224}{17} = 13 - \frac{14}{17}$. By calculating the difference between this and the true value, a second small fraction $\frac{1}{8600}$ was taken due to the minimal discrepancy. From this, the Moon’s completed cycles bhagaṇa: a full 360-degree revolution through the zodiac and remaining parts over one solar year are determined as: $13\text{ cycles, } 4\text{ signs, } 12^\circ, 46', 30'', 46''', 20''''$.
The true Moon in cycles and parts is: $\frac{57753300000}{4210000000} = \frac{192511}{14400} =$

$13\text{ cycles, } 4\text{ signs, } 12^\circ, 46', 30'', 0''', 0''''$. The difference between these two in arc-seconds is a positive (additive) $1, 29, 13, 40$ represented in sexagesimal parts: seconds, thirds, fourths. Likewise, the correction factor bījakarma in arc-seconds for one year is negative (subtractive) $1130$. The difference between these is a negative $0, 0, 46, 20$, which equals $\frac{139}{3 \times 60 \times 60} = \frac{1}{78}$ for one year. For any desired year, this is $\frac{\text{Years}}{78}$. Thus, the correction for the Moon is proven. In this way, the corrections for all planets should be understood.

Conclusion of the Chapter on Mean Motion:—

Thus ends the calculation of the positions of the heavenly bodies original: 'nabhoga-madhya-sādhanam' — literally 'the accomplishment of the middle of the sky-goers'
in this Karaṇakutūhala (The Delight of Calculation) composed by Bhāskara,
a work beloved by those of refined intellect. || 1 ||

Sudhākara:—

The meaning is clear.

By the son of Śrī Kṛṣṇālu, in this "Ornament of Explanation" Vāsanāvibhūṣaṇa,
the excellent explanation of the mathematics of mean planetary positions
has been completed, endowed with fine sayings and logical proofs.
Thus concludes the First Chapter on Mean Motion in the Vāsanāvibhūṣaṇa commentary on the Karaṇakutūhala. || 1 ||

This is the first chapter, "Mean Motion," from the two commentaries by Sumatiharṣa and Sudhākara Dvivedī, edited by Satyendra Mishra.