Aryabhata lived between 476 – 550 AD , 1550 years ago, and has been credited as the first major mathematician-astronomer from the Indian classical age.There was some confusion about his name was Aryabhatta or Aryabhata. There are numerous references in many astronomical book calling him Aryabhata and very few Aryabhatta. Hence it is generally agreed by scholars that he was Aryabhata.

Aryabhata was in 476AD in Pataliputra, present day Patna, Bihar. Even his place of birth had been questioned. There is a hypothesis that he was native of Kodungallur, ancient Kerala as many references to him were found in books published in ancient kerala.

While his birth place is under question, scholars say it is certain that he went to Kusumapura which is later identified as Pataliputra. There are verses saying that Aryabhata was head of kulapa or institution in Pataliputra. As the Nalanda university in Pataliputra had a astronomical observatory, it is assumed that he was head of the astronomy observatory. He also set up another observatory at the Sun Temple in Taregana, Bihar.

Aryabhata is the author of several treatises on mathematics and astronomy. Unfortunately, many of his works are lost and we are lucky that Aryabhatiya, a book on mathematics & astronomy has survived.

The mathematics section of the Aryabhatia covers wide ranging topics such as algebra, trigonometry, fractions, quadratics equations, sum of power series and also Sine tables.

Through references in Aryabhata’s contemporary Varahamihira and later mathematicians such as Bhaskara I and Brahmagupta, we know about Arya-Siddhanta, book on astronomical computations. This book seems to be a Sanskrit summary of Greek and Mesopotamian theories in astronomy and mathematics. This uses midnight day as the start of the day versus sunrise used in Aryabhatia. This book had descriptions on many astronomical instruments such as Shanku-yantra (sun dial), Chhaya-yantra to measure shadows, dhanur-yantra, chakra-yantra, yasti-yantra and even chhatri yantra an umbrella based devices used for astronomical measurements.

References to a third book as the original source is found in the Arabic transition book called Al-nanf. This Blog we will talk about few interesting things in Aryabhatia and let the interested readers research on the other books.

Aryabhatia is written in the typical of sutra literature, in which each line is a summary or a bookmark for a complex system or solution. The detailed explanations are done by the commentators. Aryabhatia consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

Gitikapada (13 verses) covers extremely large units of time such as Kalpa (Period of time since universe creation), manvantra (life span of Manu the original creators of humans) and yuga (circle of creation and destruction).

Interesting information about current Kaliyug. This Yug will last for 432,000 years (1,200 divine years). Kali Yuga began 5,124 years ago and has 426,876 years left as of 2023 CE. Kali Yuga will end in the year 428,899 CE. References to the relative nature of time for example divine years are different from human years is very surprising. This concept of relative nature of time with respect to speed of light is popularized by Albert Einstein inly in 1905 AD but was used extensively by many scholars including Aryabhata. Gitikapada also has a table of Jya, what we can as Sine Tables. This table is described in a single verse.

Ganitapada contains 33 verses. This pada covers measurements of 2D, 3D objects, kuttaka-quadratic and indeterminate equations and measurements of shadows. Kalakriyapada contains 25 verses. Methods for determining positions of planets, different units of time, seven day week with names for the days. Kalakriyapada contains 50 verses. This pada covers celestial sphere features such as ecliptic, celestial equator, shape of earth, explanations for day & night and also rising zodiac signs i.e starts on the horizon.

I will cover few snippets from this book to show to you the depth and breadth of Aryabhata’s knowledge in mathematics and astronomy. Interested readers can research lot more material available in public domain.

Let us start with Mathematics.

Even though Aryabhata did not explicitly use the Zero symbol, he mentions place value system with a power of 10 which is the foundation for the current decimal system. Experts believe that zero was implicit in the Aryrabhata’s place value system.

Aryabhata gave a formulae for Pi -π. He used a 10000 unit radius circle or 20000 units diameter circle in the formulae. In Ganitapada he writes

“Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.” Using this formula, the circumference is 62832. We can calculate value of π as 62832/20000 = 3.1416 accurate up to 3 decimal places.

In addition, the term used by Aryabhata Asanna can be interpreted as approaching or approximation or Irrational. Considering that irrationality of π was proven only in 1761 in Europe, this interpretation could be a remarkable achievement by Aryabhata.

Aryabhata gives the formula for calculating area of a square in Ganitapada. He says “for a triangle, the result of a perpendicular with the half-side is the area.”

This matches with the formula area = height * base/2.

Another interesting topic we find in Aryabhatia is discussions on Sine and Cosine. He called it ardha- jya which got cut to Jya. Sine table is listedin the book. There is an interesting story of how Jya got wrongly translated to Arabic and finally we ended up with name Sine and Cosine.

Aryabhata started a new way of solving algebraic problems. He called it kuttaka-ganita. Basically, break the problem in to smaller pieces and recursively solve the bigger problem. Aryabhata used this method to solve indeterminate equations. This principle of breaking the problem in to smaller ones and solving them one by one is now a popular business best practice.

Aryabhata also provided formulae for sum of series of squares or cubes. His verse is “The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes.”.

For example N = 1 ^{2} + 2 ^{2} +3 ^{2} +4 ^{2} ; Number of terms = 4 ; N = ( 4*5*9)/6 = 30.

N= 1 ^{3} +2 ^{3} +3 ^{3} +4 ^{3} ; Sum of the series = 1+2+3+4 = 10. 10 ^{2} = 100

Let us now look at few slokas in Astronomy Aryabhata’s system is based on Udayakala-sunrise and was called Audayaka system. Aryabhata also discussed midnight based system in other books which are now lost.

Aryabhata correctly predicted that Earth rotates on its axis which is the reason for day and night and also apparent movement of stars. He said “In the same way that someone in a boat going forward sees an unmoving object going backward, so someone on the equator sees the unmoving stars going uniformly westward”.

The Aryabhatia slokas are based on Geocentirc model where Earth was the centre and all the other planets and stars rotated around the earth. He postulated that planets have two epicycles one manda (slow) and other sighra (fast) and may have understood the elliptical nature of the motions.

He postulated the order of the planets in terms of distance from earth as the Moon, Mercury, Venus, the Sun, Mars, Jupiter.

He correctly predicted that moon and other planets shine by reflected light from the Sun. He applied this logic and explained the solar and lunar eclipses as shadows cast when certain alignment of earth, moon and Sun happens. In Gola 37-48, he explains that lunar eclipse happens occurs when moon enters the earths shadow and also provided the computations about the duration of the eclipse and also size of the eclipsed part. Guillaume Le Gentil, a 18 th century scientists measured the eclipse and was long by 68 seconds while Aryabhata’s calculations were short by only 41 seconds.

Aryabhata calculated the day as 23 Hours, 56 minutes and 4.1 seconds. The modern value is 23:56:4.091. Aryabhata calculated a year as having 365 days, 6 hours, 12 minutes and 30 seconds. This differs with modern value by 3 minutes and 20 seconds!.

We all know now that Geocentric model was incorrect and Heliocentric model with fixed Sun with other planets rotating the Sun is the correct model. There are lots of debates on whether Aryabhatas various calculations are internally based on Heliocentric model.

Aryabhata is recognized as the symbol of Indian Astronomy. The first Indian satellite launched in space bears his name. Various astronomical laboratories are also named after him.

Hope you find this blog interesting and you understand more about scientific thinking in early India.

References. Wikipedia, Brittanica and other public domain sources.