Aryabhata's Contributions to Mathematics
Aryabhata's contributions to mathematics are noteworthy. He proposed formulas for calculating the areas of triangles and circles, which proved accurate. The Gupta emperor Chandragupta II appointed him head of the university for his extraordinary work. He introduced the concept of an infinite series for the value of pi. He calculated the value of pi as 62832/20000, which was remarkably accurate.
Aryabhata was one of the pioneering mathematicians who introduced the "jya" (sine) table, presented as a verse by incrementing each unit to 225 minutes or 3 degrees 45 minutes. He used alphabetical codes to define the progression series. Using Aryabhata's table, calculating the value of sine 30 (corresponding to the sine of a half-angle) as 1719/3438 = 0.5 yields an accurate result. His alphabetical codes are commonly known as the Aryabhata cipher.
Works Edited by Aryabhata
Aryabhata wrote several works on mathematics and astronomy, some of which are lost. However, many of his works are still used today, such as the Aryabhatiya.
Aryabhatiya
Aryabhatiya is a mathematical work by Aryabhata, providing a comprehensive description of arithmetic, algebra, and trigonometry. It includes continued fractions, quadratic equations, sine tables, the summation of power series, and more. Aryabhata's work is primarily described in this text [Aryabhatiya]. The name was given not by Aryabhata, but by later scholars.
Aryabhata's disciple Bhaskara I referred to this work as "Ashmaka-tantra" [a treatise from Ashmaka]. It is also commonly called Aryabhata-shat-ashta [Aryabhata's 108] as it includes 108 verses. It is written in a highly concise form, with each line representing ancient mathematical principles. The work is divided into four chapters or sections, comprising 108 verses and 13 introductory verses.
Verse Section [13 verses]
Mathematics Section [33 verses]
Time Calculation Section [25 verses]
Sphere Section [50 verses]
Arya-Siddhānta
This work by Aryabhata is not fully available. However, it describes the use of various astronomical instruments, such as the sūktī, shadow instrument, cylindrical rod, umbrella-shaped instrument, water clock, angle-measuring instrument, semi-circular/spherical instrument, etc. It includes principles of solar theory, emphasizing the calculations of midnight and other astronomical events.
