Moonlight of Mathematics, Part 1 (1936) - Page 21

Example.

O learned one, if you understand the rule of concurrenceoriginal: "saṅkrāmaṃ." A method for solving two linear equations with two variables, specifically when given the sum (yoga) and difference (viora or antara)., tell me quickly:
What are the two quantities whose sum is sixty-three,
and whose difference is nine? ॥ 13 ॥

Statementoriginal: "nyāsaḥ." The formal layout of the problem's values.. The sum is 63. The difference is 9. The sum of these two is 72. The difference between these two is 54.
Half of these results gives the two quantities: 36 and 27.
The rule of concurrence states: $(Sum + Difference) / 2 = Larger Number$ and $(Sum - Difference) / 2 = Smaller Number$. Here: $(63+9)/2 = 36$ and $(63-9)/2 = 27$.

Another rule for Concurrence.

The difference of the squares of two quantities,
divided by their difference, results in their sum.
Divided by their sum, it results in their difference.
From these two, the quantities are found by concurrence. ॥ 32 ॥

Example.

O friend, for which two quantities is four hundred seen
as the difference of their squares?
Tell me those two if you know, whether their difference
is eight, or their sum is one hundred. ॥ 14 ॥

Statement. The difference of the squares of the two quantities is 400. Their difference is 8. The difference of the squares (400) divided by the difference of the quantities (8) results in the sum of the quantities: 50. Following the rule "the sum placed in two positions," the two quantities are produced: 29 and 21.
Calculated as: $(50+8)/2 = 29$ and $(50-8)/2 = 21$.

Another Statement. The difference of the squares is 400. The sum of the quantities is 100. The difference of the squares divided by the sum results in the difference: 4. Following the rule "the sum placed in two positions," the two quantities are produced: 52 and 48.
Calculated as: $(100+4)/2 = 52$ and $(100-4)/2 = 48$.

Another rule for Concurrence.

From twice the sum of the squares,
subtract the square of the difference;
the square root of that result is the sum. ॥ 33 ॥

Example.

The sum of the squares is one hundred for two quantities
whose difference is measured as two.
Tell me those two quantities quickly, if you know mathematics.

1. original: "vargāntaraṃ rāśiviyogabhaktam..." "The difference of squares divided by the difference of the quantities," and so on, is exactly as stated by BhāskaraBhāskara II (c. 1114–1185), the famous Indian mathematician and author of the Līlāvatī..
2. Here is the derivationoriginal: "upapattiḥ." A logical proof or demonstration of how the formula works.. Let the two quantities be 'so much' and 'black' original: "ya, ka." Abbreviations for "yāvat-tāvat" (unknown 1) and "kālaka" (unknown 2), standard variables in Indian algebra.. The sum of their squares is...