Moonlight of Mathematics, Part 1 (1936) - Page 15

The cube of the first part, and the cube of the

last adjacent part of the quantity;

or, the cube may be found by the last term [of a sequence]

starting with one, multiplied by three and increased by one. ॥ 22 ॥

This verse offers alternative conceptual ways to understand a cube. One method relates to the sum of a mathematical series, while the other refers to the algebraic identity of $(a+b)^3$.

A number multiplied by three and then by its two

parts, and added to the sum of the cubes of those parts;

or, the cube of the root is the square of the quantity

multiplied by the quantity itself. ॥ 23 ॥

Here, the author defines the cube as both an algebraic sum—$3ab(a+b) + a^3 + b^3$—and by its simplest definition: a number multiplied by its own square ($x \cdot x^2 = x^3$).

Example.

Tell me quickly the cubes of the numbers from one to nine

individually; and also of eighteen, and of the square of six,

and of sixty-one, and of five-hundred and seven. ॥ 4 ॥

The "square of six" is 36. "Sixty-one" refers to the number 61. These examples are designed to test the student's ability to apply the cubing rules to increasingly large numbers.

Setting down the numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 36, 61, 507.

The resulting cubes are: 1, 8, 27, 64, 125, 216, 343, 512, 729,

5,832, 46,656, 226,981, 130,323,843.

Thus ends Cubing.

Now, the Rule for the Cube Root.

Mark the first digit as "cube," and the next two as "non-cube."

Subtract the [nearest] cube from the last "cube" digit.

Having set down the root [elsewhere], divide the next digit

by three times the square of that root. ॥ 24 ॥

The Ghanamūla (cube root) method involves dividing the number into groups of three digits. The first digit of each group is the "cube" place (ghana), and the others are "non-cube" (aghana). This is the start of a step-by-step long-division style algorithm for roots.