Moonlight of Mathematics, Part 1

Therefore, the sum is 15. The difference mentioned was 5. By the operation “the sum placed in two positions,”original: "yogo dviṣṭha." This refers back to the rule where the sum and difference are added and subtracted, then divided by two to find the two individual numbers. the two quantities produced are 10 and 5.

Rule

From twice the sum of the squares,

diminished by the square of the sum, the square root is the difference.

This follows the algebraic identity: $\sqrt{2(x^2 + y^2) - (x + y)^2} = x - y$.

Example

The sum of the squares of two quantities is one hundred.

The sum of the quantities is fourteen.

Tell me those two quantities quickly, if you know the method of concurrence. || 17 ||

Statement. Sum of squares = 100. Sum of the quantities = 14. Here too, from twice the sum of squares (200), diminished by the square of the sum of the quantities (196), the remainder is 4; its square root produced is the difference, 2.
By the rule “the sum placed in two positions,” Sum (14) + Diff (2) = 16, divided by 2 = 8. Sum (14) - Diff (2) = 12, divided by 2 = 6. the two quantities produced are 8 and 6. The rest of this is useful for geometry; I will explain it in that section.

Now, a rule on the collection of classes:

Just as the statement of the problem describes,

a known value should be assumed and then multiplied, || 17 ||

divided, added, or subtracted,

performing operations such as proportion.

The result produced from the assumed value

then becomes the divisor for the visible quantity. || 18 ||

This describes the "Rule of Supposition" or "Rule of False Position." One assumes a convenient number (usually 1), performs all the operations described in the word problem, and then uses the ratio between the expected result and the "false" result to find the true answer.

The following is an algebraic proof using abbreviations: 'Diff' (aṃ) for difference, 'X' (yā) for the first unknown, 'Y' (kā) for the second unknown, and 'Prod' (ghā) for product.

Diff = X - Y } Diff² + 2(Prod) = (X - Y)² + 2XY =
Prod = XY } X² + Y².

Then X² + Y² + 2XY = (X + Y)² = Diff² + 4(Prod) = Sum².
All this is exactly as stated by Bhāskara. Referring to the 12th-century mathematician Bhāskara II.

1. For the proof here, see Rule 33.
2. The phrase "assume a desired quantity according to the statement" etc. is indeed stated by Bhāskara.