Moonlight of Mathematics, Part 1

Rule

The square root of the sum of the square of the difference

and four times the product is the sum.

The square root of the square of the sum diminished by

four times the product is the difference. || 35 ||

These rules describe the algebraic identities $(x + y) = \sqrt{(x - y)^2 + 4xy}$ and $(x - y) = \sqrt{(x + y)^2 - 4xy}$. This allows a mathematician to find the sum of two numbers if only their product and difference are known, or vice versa.

Example

Of which two quantities is the product sixty and the difference seven?

What is their sum? Tell me the difference from the sum

if you understand the method of concurrence. || 15 ||

Statement. The product of the two quantities is 60. The difference of the quantities is 7. The square of the difference is 49.
By this rule, added to 60 times four (which is 240), the sum [289] has a square root of 17.
Thus, the sum is 17. By the operation “the sum placed in two positions—”original: "yogo dviṣṭha—" This refers to the rule of concurrence (saṅkramaṇa) where the sum and difference are added and subtracted, then divided by two: $(17+7)/2=12$ and $(17-7)/2=5$. the two quantities produced are 12 and 5.

Rule

The square of the difference of two quantities

plus twice their product is equal to the sum of their squares.

The square root of that [sum of squares] added to twice the product

becomes the sum of the quantities. || 36 ||

This provides the identity $x^2 + y^2 = (x - y)^2 + 2xy$, and subsequently $(x + y) = \sqrt{(x^2 + y^2) + 2xy}$.

Example

Friend, if the difference of two quantities is five

and their product is three hundred,

tell me the sum of their squares, the square of their sum,

and the sum itself, if you know. || 16 ||

Statement. The difference of the two quantities is 5. The product of the quantities is 300. With the square of the difference, 25,
added to twice the product, 600, the result is 625, which is the sum of the squaresoriginal: "vargayogaḥ." The sum of $x^2 + y^2$.. To the sum of the squares,
625, twice the product, 600, is added, making the square of the sum 1225. Its square root [is 35]...

1. The proof here is clear following the method of BhāskaraA famous 12th-century Indian mathematician.: "The difference between four times the product and the square of the sum," and so on.
2. The proof here: the two quantities are assumed to be the unknownoriginal: "yā" and "kā," abbreviations for "yāvat-tāvat" (as many as) and "kālaka" (black), the standard variables in Indian algebra equivalent to $x$ and $y$., then according to the statement of the problem...