Aryabhatiya is a Sanskrit astronomical treatise which is a masterpiece and one of the few known surviving work of the 5th Century Indian Mathematical Genius, Aryabhata. Only three books by Aryabhata still exist today although there is some question as to the authenticity of the third book.
Another book written by Aryabhata is the Arya-siddhanta. This book has been lost and is only known because of the work by other mathematicians. A mathematician working at the same time as Aryabhata mentioned Aryabhata’s work as have later mathematicians such as Bhaskara I and Brahmagupta. The Arya-siddhanta looked at astronomy.
A third book that may have been written by Aryabhata is an Arabic translation that the translator claims was written by Aryabhata but the name of the work in Sanskrit (the language Aryabhata wrote in) is not known. In Arabic, the name of the book is either Al-ntf or Al-nanf.
The Aryabhatiya is the only book that comes directly from Aryabhata. He wrote the book in verse and there are 108 verses as well as 13 introductory verses. The name of the book came from later commentators and there is a question as to whether Aryabhata actually named the book. Other names given to the book is Ashmakatantra (the treatise from the Ashmaka) as well as Arya-shatas-ashta. Arya-shatas-ashta means Aryabhata’s 108 which refers to the number of verses in the book.
The verses are written in the same style as sutra literature (religious texts) and is very terse. Most of the explanations around Aryabhata’s work comes from his commentators instead of from him. The main commentators of his work were Bhaskara I, who is considered the main scholar of the astronomical school set up by Aryabhata, and Nilakantha Somayaji, who wrote about Aryabhata’s work in the book Aryabhatiya Bhasya in 1465.
An example of Aryabhata’s writing style can be seen in the verse on pi. Aryabhata wrote that to calculate pi, one needed to “Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached.”
Aryabhata broke the text up into four chapters (called padas). The four chapters are called Gitikapada, Ganitapada, Kalakriyapada, and Golapada.
The first chapter (Gitikapada) consists of 13 verses and looks at large units of time. Aryabhata develops a cosmology that is different from earlier writers and also has a table of sines. In this book, Aryabhata calculated the number of years in a mahayuga. The Hindu religion breaks up time into one thousand cycles. Each cycle is called a mahayuga and Aryabhata claimed that one mahayuga was equal to 4.32 million years.
The second chapter (Ganitapada) consists of 33 verses and looks at calculating the volume and area of different geometrical figures. The chapter also looked at geometric progressions as well as arithmetic. In this chapter, Aryabhata also looked at shadows and sundials, particularly gnomons, the part of a sundial that is used to cast a shadow. The chapter also covers various mathematical equations, such as indeterminate equations, quadratic equations, and simultaneous equations.
The third chapter (Kalakriyapada) consists of 25 verses and looks at various units of time as well as methods to calculate the position of the planets for any given day. The chapter highlights a seven-day week and gives names for the days. The chapter also looked at calculations around the intercalary month. An intercalary month is just an extra month that is used to ensure the seasons keep track with the months.
The last chapter (Golapada) consists of 50 verses and examines properties of celestial sphere. A celestial sphere is an imaginary sphere where the observer is in the center of the sphere and all other celestial objects lie on the sphere. In this chapter, Aryabhata also looks at what is known as the celestial equator, the imaginary line that circles around the celestial sphere—it is basically just Earth’s equator projected into space. Aryabhata also looks at the causes of night and day, the shape of the earth, and zodiac signs.
Aryabhata made a number of innovations in his work in mathematics. His work was basically unknown in the West for hundreds of years after his death. It wasn’t until his work had been translated into Latin during the early 1200s that his work became known to Western mathematicians. Once it was translated, it became very influential and many of Aryabhata’s ideas were utilized by European mathematicians. Aryabhata’s work was well known within Islamic mathematical circles and was helpful during the Islamic Golden Age.
As it seems with most Ancient mathematicians, Aryabhata was also interested in calculating a value for pi. He came to an approximation of pi that was correct to five numbers. In the verse where Aryabhata gives an equation for calculating pi (shown earlier), Aryabhata used the words “can be approached.” One commentator on Aryabhata’s work (Nilakantha Somayaji) stated that the use of the word “approached” means that Aryabhata recognized that his calculation only gave an approximation of pi. Somayaji also believes that the words Aryabhata used shows that Aryabhata recognized that the value of pi is an irrational number (i.e., the number never ends nor repeats itself). This was not proven in Europe until 1761.
Place-Value System and Zero
Aryabhata used a place-value system in his calculation. The place-value system was first seen in the Bakhshali Manuscript, a manuscript of Indian mathematics that was written on birch bark in the third century. The manuscript was found near the village of Bakhshali, which is the reason for the name of the manuscript.
Aryabhata did not use the Brahmi symbol for zero in his work but a French mathematician believed that Aryabhata knew of zero and that its use was implicit in his work. Actually, Aryabhata did not use any Brahmi numerals, which were the number symbols that Indian mathematicians were using at the time. Aryabhata followed the Sanskrit tradition of using letters as numbers. This allowed Aryabhata to write his quantities, such as the table of sines, as a mnemonic (an aid to remembering something).
Aryabhata was interested in finding Diophantine equations. A Diophantine equation is an equation that has more than one unknown integer (a number that is not a fraction). A simple Diophantine equation would be “ax + by = c”. In this equation a, b, and c are given integers and the x and y integers are unknown.
Aryabhata came up with a method of solving Diophantine equations that is the standard method for solving first order Diophantine equations today. The algorithm is often called the Aryabhata algorithm.
Aryabhata did a lot of work in astronomy as well as mathematics although he is more recognized for his work in mathematics than astronomy; although, his work in astronomy has been influential in the Arab world.
Aryabhata used the Audayaka system in astronomy. In this system, the days begin at dawn on the equator. In one of his astronomical writings that has been lost (but partially reconstructed based on comments in Brahmagupta’s Khandakhadyaka), Aryabhata proposed a model in which the days began at midnight (the ardha-ratrika system).
The Solar System
Aryabhata’s solar system was a geocentric model. This means that the Earth is the center of the solar system.
According to Aryabhata, the sun and the moon move in epicycles (small circles) that orbit around the Earth. The order of the planets from the Earth according to distance are the moon, followed by Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. After Saturn, there are asterisms, which are groups of stars that are highly visible in the night sky.
Size and Rotation of the Earth
Aryabhata was able to calculate a value for the size of the Earth that was only 0.2 percent smaller than the Earth’s actual size. Aryabhata calculated that the circumference of the Earth was 39,968 kilometers whereas the actual circumference size is 40,075 kilometers.
Aryabhata believed that the Earth rotated on its own axis and that the movement of the stars was a result of this rotation. When converted to modern units of time, Aryabhata’s calculation of the time it takes for the Earth to rotate on its axis is almost exactly the same as it has been measured today. He calculated the rotation as taking 23 hours, 56 minutes and 4.1 seconds. The modern calculation is 23 hours, 56 minutes and 4.091 seconds.
In calculating the length of the year, Aryabhata is only off by three minutes and twenty seconds. This was most likely the most accurate calculation at the time.
Aryabhata, contrary to popular belief at the time, stated that the moon and the planets shone because of sunlight reflecting off of them. He stated that a lunar eclipse occurred when the moon moved into Earth’s shadow. Aryabhata also looked at the size of the Earth’s shadow and how to calculate the length of a lunar eclipse.
The work done by Aryabhata on calculations concerning calendars are still in use today. They are used to fix the Hindu calendar (Panchanga) and were even used in the Islamic world as the basis for the Jalali calendar.
The works of Aryabhata claimed him name and fame from various famous entities in the medieval era as his works formed the base of numerous complex calculations and derivations and truly, it was he who made India proud with his treatises and doctrines related to Mathematics and Astronomy , hence, again portraying the rich educational and cultural traits of the Ancient civilizations of India.