Aryabhata (476 CE – 550 CE) was the first Hindu mathematician and astronomers from India. He wrote couple of treatise about mathematics and astronomy. Some of them were lost. His most famous works Aryabhatiya completed in 499 CE and the Arya-Siddhanta. Aryabhatiya consists of 108 verses, in which Aryabhata wrote about the mathematics and astronomy at the age of 23 in 499 CE. He was born in India at Asmaka or Kusumapura in 476 CE. There is no clear evidence of the place of birth (Indian Streams Research General, September 2012). Aryabhata studied in Kusumapura and stayed there for some time. The evidences from Hindu, Buddhist tradition, and Bhaskara I (629 CE) recognize Kusumapura as Pataliputra, currently known as Patna. Aryabhata was the head of an institution at Kusumapura. The University of Nalanda was in Pataliputra at the time. This university had an astronomical observatory that forces the belief that Aryabhata was the head of the Nalanda University. Aryabhata set up an observatory at the Sun temple in Taregana, Bihar (Aryabhata – Indian Mathematician).

Aryabhatiya deals with mathematics and astronomy. That consists of an introduction containing astronomical tables and Aryabhata’s system of phonemic number notation. This work consists of three sections: Ganita (means mathematics), Kala-kriya (means Time calculations), and Gola (means Sphere). Ganita covers decimal number system, algorithms for square and cubic roots, geometric measurements, the algorithm for Pi, tables of sines using Pythagorean Theorem, quadratic equations, proportions, and the solution of linear equations. This discusses the Aryabhata’s method to solve the mathematical problem, Kuttaka (means pulverizer) also known as Aryabhata’s algorithm. This algorithm suggests breaking a problem in smaller fractions. Kala-kriya speaks about astronomy. It is about treating planetary motion and include the definition of various units for time, eccentric, epicyclic planetary motion modes, longitude, and latitude.

Gola discusses the plane trigonometry to spherical geometry. It also has prediction of solar and lunar eclipses and explicit statement about westward motion of stars because of the spherical rotation of the Earth about its axis (Indian Streams Research General, September 2012). The Arya-siddhanta was the work on astronomical computations. Surya Siddhanta was the base of this work and considered the start of the day at the midnight, as opposed to sunrise according to Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular, and circular (dhanur-yantra/chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.

Bakhshali Manuscript discussed the place-value system first in the 3rd century. Georges Ifrah, the mathematician from France, acknowledged that awareness of zero by Aryabhata in place-value system because of a place holder for the powers of 10 with null coefficients. Instead of using Brahmi numerals Aryabhata continued the tradition from Vedic times by using letters of the alphabet for denoting numbers, expressing quantities, such as the table of sines in a mnemonic form (Indian Streams Research General, September 2012).

The Surya Siddhanta laid foundational rules to determine the true motions of the luminaries and introduced the sine, cosine trigonometric functions. Aryabhata devised the formulae for calculating the area of triangle and circle. He also devised the same for pyramid and sphere. Formulae for triangle and circle were correct. Most historians claimed that formulae for sphere and pyramid were incorrect. He created a table of sines and versine with formula sin (n+1) x – sin nx = sin (n-1) x – (1/225) sin nx versin= 1 – cosine

Aryabhata’s definition of jya (sine), kojya (cosine), urkrama-jya (versine), and otkram-jya (inverse sine) influence the trigonometry (Indian Streams Research General, September 2012). Aryabhata concluded that the approximation for pi ([pic]) is irrational. In Ganitapada he gave the formula for the ratio of circumference to the diameter as ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures (Aryabhata – Indian Mathematician). The speculation was that Aryabhata used ‘āsanna’ (means approaching), to mean that not only is this approximation but also that the value is irrational.

This shows quite a sophisticated insight from him because Lambert proved the irrationality of pi in Europe only in 1761. Bhaskara’s commentary on Aryabhatiya discusses the topic known as Diophantine equations, e.g., integer solutions to the equations that have the form ax+by = c. That formula to find value of N stated as N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. Vedic text Sulba Sutras discussed these notoriously difficult diophantine equations. Aryabhata provided rules of algebra in the Aryabhatia and those are as follows: and 13 + 23 +…+n3= (1+2+…+n) 2

In some texts, Aryabhata seems to ascribe the apparent motions of the heavens to the Earth’s rotation. He believed that the planet’s orbits as elliptical rather than circular. Aryabhata correctly insisted that the earth rotates about its axis daily and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view in other parts of the world that the sky rotated. The first chapter of the Aryabhatiya indicated this, where he gives the number of rotations of the earth in a yuga, and made more explicit in his gola chapter (A He used analogy of movement of boat going forward. During this movement person feels an unmoving object going in opposite direction than the boat. With this analogy he discussed the appearance of unmoving stars going uniformly westward. The cause of rising and setting is that the sphere of the stars together with the planets apparently turns due west at the equator, constantly pushed by the cosmic wind.

Aryabhata described a geocentric model of the solar system, in which he mentioned that the Sun and Moon in turn revolve around the Earth. He calculated the positions and periods of the planets with respect to uniformly moving points. He stated that speed at which Mercury, Venus, and Sun move around the Earth is identical and is different from the specific speed of Mars, Jupiter, and Saturn. He represented each planet’s motion through the zodiac. Most historians of astronomy expressed that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy. Historians saw another element in Aryabhata’s model, the śīghrocca, the basic planetary period in relation to the Sun as a sign of an underlying heliocentric model. He explained solar and lunar eclipses. He stated that the Moon and planets shine by reflected sunlight and explained eclipses in terms of shadows cast by and falling on Earth. His theory explained the lunar eclipse occurs when the moon enters into the Earth’s shadow and discussed the length the size and extent of the Earth’s shadow. He provided the computation and the size of the eclipsed part during an eclipse.

Later Indian astronomers improved on the calculations, but Aryabhata’s methods provided the core. Aryabhata calculated the sidereal rotation as 23 hours, 56 minutes, and 4.1 seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, six hours, 12 minutes, and 30 seconds is an error of three minutes and 20 seconds over the length of a year (Indian Streams Research General, September 2012). Aryabhata’s work influenced the Indian astronomical tradition and several neighboring cultures through translations. His work as translated in Arabic during the Islamic Golden Age (c. 820 CE).

Al-Khwarizmi cited some of his results and in the 10th century Al-Biruni stated that Aryabhata’s followers believed that the Earth rotated on its axis. Aryabhata’s astronomical calculation methods were also very influential. Islamic world widely used the trigonometric tables to compute many Arabic astronomical tables (zijes). Calendric calculations devised by Aryabhata and his followers contributed the practical purposes of fixing the Panchangam (the Hindu calendar). Other cultures used this for forming the calendar systems.

India honored Aryabhata by naming India’s first satellite as Aryabhata. An Institute for conducting research in astronomy, astrophysics, and atmospheric sciences is the Aryabhatta Research Institute of Observational Sciences (ARIOS) near Nainital, India. Indian authorities named the inter-school math competition as ‘Aryabhata Maths Competition’, as is Bacillus Aryabhata, a species of bacteria discovered by ISRO scientists in 2009.

References

Indian Streams Research General: Avhale, P. S; Waghmare, R. V.; Kolhe, S. B. Indian Streams Research Journal. Sep2012, Vol. 2 Issue 8, Special section p1-5. 5p. Retrieved from https://ehis.ebscohost.com/eds/detail?vid=2&hid=117&sid=d84c9078-6d85-4131-9209-e44cdb4cba58%40sessionmgr110&bdata=JnNpdGU9ZWRzLWxpdmU%3d#db=a9h&AN=82351338