Lilavati (Bhaskaracharya) (1800) - Page 5

Of this addition 1, having made half of the coefficient original: "guṇārddhaṃ". The number that multiplies the square root. i into 9 49, its square root [is taken]. With half the coefficient 3 or of the square joined [making] 6; then reduced by the denominator [becoming] 4, and squared, the result is the measure of the flock of swans: 16.

Now, an Example original: "udāharaṇaṃ" on the Halving of the Sum of Roots original: "mūlayuterddale":

Suppose some wealth original: "dhanakāḥ" is joined with nine times its own square root, and the result is two hundred and forty. Let that quantity be declared original: "rāśirnigadyatāṃ". 59.

Statement original: "nyāsaḥ". The formal layout of the mathematical problem.:
Square root coefficient: 9.
Visible quantity original: "dṛśyaṃ". The constant or known integer in the equation.: 240.

By the method previously mentioned, the quantity produced is 289.

Another Example:

Ten times the square root of a flock of swans original: "haṃsakula" flew to Lake Manasa at the arrival of the clouds the monsoon season. One-eighth of the flock went to the forest of Tapasvini. A pair of swans was seen engaged in the sport of play in water adorned with young lotus stalks. Oh girl, tell the total number of the flock! 60.

Statement original: "nyāsaḥ":
Square root coefficient: 10.
Fraction: 1/8.
Visible quantity: 2 Note: The OCR says 6, but the verse mentions a "pair" (yugata), and standard versions of this problem usually result in 144..

By the rule "that which is half, here diminished or increased" and so forth: The root coefficient and the visible quantity are divided by one minus the fraction 9/8 This is a technical step to normalize the equation. The root coefficient becomes 5 and the visible quantity becomes 48. From these, by the rule "the square root multiplied by the coefficient of the quantity diminished or increased" original: "guṇaghnamūlonayutasya" etc., the measure of the flock of swans is found to be 144.

Another Example:

Arjuna original: "pārthaḥ", a title for the hero Arjuna from the Mahabharata epic., in order to slay Karna Arjuna's great rival in the epic battle., unleashed a multitude of arrows in his fury in battle. With half of them, he parried his opponent's arrows; with four times the square root...