Cubes in Līlāvatī

Introduction

Here we will know how to find the cube of a number as mentioned in Līlāvatī.

Verse 24 :

समत्रिघातश्च घनः प्रदिष्टः

स्थाप्यो घनोऽन्त्यस्य ततोऽन्त्यवर्गः ।

आदित्रिनिघ्नस्तत आदिवर्ग:

त्र्यन्त्याहतोऽथादिघनश्च सर्वे ॥ 24 ॥

Translation

Cube of a given number is its product with itself thrice over. If we want to find the cube of a number of two digits, say, 10a + b, write a³ first. Below this, write 3a² b by shifting this result one place to the right. Below this write 3ab² after shifting it one place to the right. Below this write b³ after shifting it one place to the right. Add all the results, and the result is the cube. This procedure can be modified by starting from b but then each time the shifting should be made to the left. If there are more than two digits, then find the cube of the two digits at the extreme left and continue with the procedure given above.

Example: Cube of 27

27 = 10 X 2 + 7 which is the form 10a + b where a = 2 and b = 7

a3 = 23	8							8
3a2b = 3 X 22X 7	8	4			Shift this one place to the right			8	4
3ab2 = 3 X 2 X 72	2	9	4		Shift this one place to the right			2	9	4
b3 = 73	3	4	3						3	4	3
							1	9	6	8	3

Answer : 27³ = 19683

Example: Cube of 125

125 = 10 X 12 + 5 which is the form 10a + b where a = 12 and b = 5

a3 = 123 (Refer the calculation below)	1	7	2	8		1	7	2	8
3a2b = 3 X 122X 5	2	1	6	0	Shift this one place to the right		2	1	6	0
3ab2 = 3 X 12 X 52	9	0	0		Shift this one place to the right				9	0	0
b3 = 53	1	2	5		Shift this one place to the right					1	2	5
						1	9	5	3	1	2	5

Let us find 12³

12 = 10 X 1 + 2 which is the form 10a + b where a = 1 and b = 2

a3 = 13	1			1
3a2b = 3 X 12X 2	6		Shift this one place to the right		6
3ab2 = 3 X 1 X 22	1	2	Shift this one place to the right		1	2
b3 = 23	8		Shift this one place to the right				8
				1	7	2	8

12³ = 1728

Answer : 125³ = 1953125

See Also

लीलावती में घन