Moonlight of Mathematics, Part 1 (1936) - Page 14

Now, the Rule for the Square Root.

Distinguish the digits into "odd" and "even" places, starting from the last.

Subtract the [nearest] square from the last odd digit. By twice the

resulting root, divide the next even digit.

Then, subtract the square of that quotient from the next odd digit. ॥ 19 ॥

Place the doubled quotient in the row [of the root],

and after shifting those digits, repeat the process.

In this way, half of the final row becomes the square root. ॥ 20 ॥

This describes the traditional Indian method of extracting square roots digit by digit. Digits are marked as "odd" (units, hundreds, etc.) and "even" (tens, thousands, etc.). The root is built up through a cycle of subtraction and division.

Example.

Setting down the previous squares to find their roots: 1, 4, 9, 16, 25, 36, 49, 64, 81, 289, 104,976. The roots obtained in order are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 324.

Thus ends the Square Root.

Now, the Three Verses on the Method for Cubing.

The product of three identical numbers is a Cube.

Place the cube of the last digit; then the squares of the last and preceding digits,

multiplied by three and by each other,

placed in successive places and added together. ॥ 21 ॥

This rule provides the algorithmic equivalent of the algebraic expansion $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ for multi-digit numbers.

This follows by reversing the verse: "The difference between four times the product and the square of the sum." It is demonstrated as follows: Let two parts of any quantity be represented by 'x' original: "yā." short for "yāvat-tāvat," the first unknown variable and 'y' original: "kā." short for "kālaka," the second unknown variable. Then, the square of the quantity = (x + y)² = x² + 2xy + y². This is equal to: x² + 2xy - 4xy + y² + 4xy = (x - y)² + 4xy. This insertion provides a geometric or algebraic proof for the alternative squaring method mentioned in Verse 18 on the previous page.

Vargamūla: Square root, Ghana: Cube, Sūtra: A concise rule or formula, Udāharaṇa: Example, Rāśi: A quantity or number, Khaṇḍadvaya: Two parts or segments, Yāvat-tāvat / Kālaka: Traditional algebraic variables equivalent to x and y