Cube root in Āryabhaṭīyam

Introduction

Here we will be knowing how to find the cube root as mentioned in Āryabhaṭīyam.

Verse

अघनाद् भजेद् द्वितीयात् त्रिगुणेन घनस्य मूलवर्गेण ।

वर्गस्त्रिपूर्वगुणितः शोध्यः प्रथमाद् घनश्च घनात् ॥

Translation

Digits starting from the left till the cube place to be subtracted from the maximum cube value. Divide the second non cube place by thrice the square of the cube root. Subtract the thrice the cube root multiplied by square of the quotient from the first non cube place. Subtract the cube of the quotient from the next cube place. This process to be repeated till the last digit.^[1]

Example:Cube of 1771561

Starting from unit place mark Ghana-sthāna (cube) (G) ,Prathama- Aghana-sthāna (non-cube) (A₁) , Dvitīya- Aghana-sthāna (non-cube) (A₂) , Ghana-sthāna (cube) (G) respectively.


G	A₂	A₁	G	A₂	A₁	G
1	7	7	1	5	6	1


	G	A₂	A₁	G	A₂	A₁	G	Step details	Cube Root
	1	7	7	1	5	6	1
-1³	1							Subtract the maximum possible cube (1 = 1³) from the left most Ghana digit (1) . Cube root of the number (1) is 1 which will be the first digit of the cube root of the required number. Write this number in the Cube root column.	1
÷ 3 X 1² = 3 )	0	7	(2					Bring down the next digit from the dvitīya-aghana place (7) and place it to the right of the remainder (0). Now the number is 7 and divide this by thrice the square of the cube root obtained so far (1) = 3 X 1² = 3.
		6	(2					Subtract the above number from the maximum possible number 3 X 2 = 6. Here the quotient is 2.
		1	7					Bring down the next digit from the prathama-aghana place (7) and place it to the right of the remainder (1). Now the number is 17.
-3 X 1 X 2² = -12		1	2					Deduct thrice the cube root obtained till now multiplied by square of the previous quotient (2) = 3 X 1 X 2² = 12.
-3 X 1 X 2² = -12			5	1				Bring down the next digit from the ghana place (1) and place it to the right of the remainder (5). Now the number is 51.
-2³				8				Subtract the cube of the previous quotient (2). Write this quotient next to the cube root obtained till now (1) in the Cube root column.	1 2
÷ 3 X 12² = 432			4	3	5	(1		Bring down the next digit from the dvitīya-aghana place (5) and place it to the right of the remainder (43). Now the number is 435 and divide this by thrice the square of the cube root obtained so far (12) = 3 X 12² = 432.
			4	3	2	(1		Subtract the above number from the maximum possible number 432 X 1 = 432 Here the quotient is 1
					3	6		Bring down the next digit from the prathama-aghana place (6) and place it to the right of the remainder (3). Now the number is 36.
-3 X 12 X 1² = -36					3	6		Deduct thrice the cube root obtained till now multiplied by square of the previous quotient (1) = 3 X 12 X 1² = 36.
						0	1	Bring down the next digit from the ghana place (1) and place it to the right of the remainder (0). Now the number is 1.
-1³ = - 1							1	Subtract the cube of previous quotient (1). Write this quotient next to the cube root obtained till now (12) in the Cube root column.	1 2 1
							0

As the remainder is zero , the cube root is exact.

Cube root of 1771561 = 121

Example:Cube of 12167

Starting from unit place mark Ghana-sthāna (cube) (G) ,Prathama- Aghana-sthāna (non-cube) (A₁) , Dvitīya- Aghana-sthāna (non-cube) (A₂) , Ghana-sthāna (cube) (G) respectively.


A₁	G	A₂	A₁	G
1	2	1	6	7


	A₁	G	A₂	A₁	G	Step details	Cube Root
	1	2	1	6	7
-2³		8				Subtract the maximum possible cube (8 = 2³) from the digits till left most Ghana digit (12) . Cube root of the number (8) is 2 which will be the first digit of the cube root of the required number. Write this number in the Cube root column.	2
÷ 3 X 2² = 12	12)	4	1	(3		Bring down the next digit from the dvitīya-aghana place (1) and place it to the right of the remainder (4). Now the number is 41 and divide this by thrice the square of the cube root obtained so far (2) = 3 X 2² = 12.
	12)	3	6	(3		Subtract the above number from the maximum possible number 12 X 3 = 36. Here the quotient is 3
			5	6		Bring down the next digit from the prathama-aghana place (6) and place it to the right of the remainder (5). Now the number is 56.
-3 X 2 X 3² = -54			5	4		Deduct thrice the cube root obtained till now multiplied by square of the previous quotient (32) = 3 X 2 X 3² = 54.
-3 X 2 X 3² = -54				2	7	Bring down the next digit from the ghana place (7) and place it to the right of the remainder (2). Now the number is 27.
-3³				2	7	Subtract the cube of the previous quotient (3). Write this quotient next to the cube root obtained till now (2) in the Cube root column.	2 3
					0

As the remainder is zero , the cube root is exact.

Cube root of 12167 = 23

References

↑ Āryabhaṭīyam (Gaṇitapādaḥ) (in Saṃskṛta). Delhi: Samskrit Promotion Foundation. 2023. pp. 15–21.{{cite book}}: CS1 maint: unrecognized language (link)

[1] Āryabhaṭīyam (Gaṇitapādaḥ) (in Saṃskṛta). Delhi: Samskrit Promotion Foundation. 2023. pp. 15–21.{{cite book}}: CS1 maint: unrecognized language (link)

[1]