# Aryabhata II

### Quick Info

Born
India
Died
India

Summary
Aryabhata II was an Indian mathematician who wrote about astronomy as well as geometry. He constructed tables of sines accurate up to about 5 figures.

### Biography

Essentially nothing is known of the life of Aryabhata II. Historians have argued about his date and have come up with many different theories. In [1] Pingree gives the date for his main publications as being between 950 and 1100. This is deduced from the usual arguments such as which authors Aryabhata II refers to and which refer to him. G R Kaye argued in 1910 that Aryabhata II lived before al-Biruni but Datta [2] in 1926 showed that these dates were too early.

The article [3] argues for a date of about 950 for Aryabhata II's main work, the Mahasiddhanta, but R Billiard has proposed a date for Aryabhata II in the sixteenth century. Most modern historians, however, consider the most likely dates for his main work as around 950 and we have given very approximate dates for his birth and death based on this hypothesis. See [7] for a fairly recent discussion of this topic.

The most famous work by Aryabhata II is the Mahasiddhanta which consists of eighteen chapters. The treatise is written in Sanskrit verse and the first twelve chapters form a treatise on mathematical astronomy covering the usual topics that Indian mathematicians worked on during this period. The topics included in these twelve chapters are: the longitudes of the planets, eclipses of the sun and moon, the projection of eclipses, the lunar crescent, the rising and setting of the planets, conjunctions of the planets with each other and with the stars.

The remaining six chapters of the Mahasiddhanta form a separate part entitled On the sphere. It discusses topics such as geometry, geography and algebra with applications to the longitudes of the planets.

In Mahasiddhanta Aryabhata II gives in about twenty verses detailed rules to solve the indeterminate equation: $by = ax + c$. The rules apply in a number of different cases such as when $c$ is positive, when $c$ is negative, when the number of the quotients of the mutual divisions is even, when this number of quotients is odd, etc. Details of Aryabhata II's method are given in [6].

Aryabhata II also gave a method to calculate the cube root of a number, but his method was not new, being based on that given many years earlier by Aryabhata I, see for example [5].

Aryabhata II constructed a sine table correct up to five decimal places when measured in decimal parts of the radius, see [4]. Indian mathematicians were very interested in giving accurate sine tables since they were used to calculate the planetary positions as accurately as possible.

### References (show)

1. D Pingree, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. B Datta, Two Aryabhatas of al-Biruni, Bull. Calcutta Math. Soc. 17 (1926), 59-74.
3. T Hayashi, T Kusuba and M Yano, Indian values for π derived from Aryabhata's value, Historia Sci. No. 37 (1989), 1-16.
4. S K Jha and V N Jha, Computation of sine-table based on the Mahasiddhanta of Aryabhata II, J. Bihar Math. Soc. 14 (1991), 9-17.
5. V N Jha, Aryabhata II's method for finding cube root of a number, Ganita Bharati 19 (1-4) (1997), 60-68.
6. V N Jha, Indeterminate analysis in the context of the Mahasiddhanta of Aryabhata II, Indian J. Hist. Sci. 29 (4) (1994), 565-578.
7. D Pingree, On the date of the Mahasiddhanta of the second Aryabhata, Ganita Bharati 14 (1-4) (1992), 55-56.

### Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update November 2000