In India, mathematics and astronomy always went together in the ancient past. Many new discoveries in mathematics had their origins in astronomical problems. Aryabhateeya (499 CE) of Aryabhata is the first discovered text on mathematical astronomy in India and has four chapters, one of which is completely devoted to mathematics. It has results in arithmetic, algebra, geometry and trigonometry. Trigonometry in its modern form has its origins in India. For computations, the Indian jyaa is far more convenient than the Greek chord (Pictures 1 and 2). Actually, the word ‘sine’ is linked to jyaa; jeevaa is another word for jyaa. This was read as jayb in Arabic that means ‘pocket’ or ‘fold’, which was translated into Latin as ‘sinus’. This later became ‘sine', which we now use.
As Aryabhateeya is very brief, Bhaskara-I wrote a commentary on it in 629 CE that explains its methods. His contemporary Brahmagupta was a brilliant mathematician-astronomer who composed Braahmasphutasiddhaanta in 628 CE. It has many path-breaking results in algebra, geometry and trigonometry, apart from astronomy. One important problem in algebra that he considered was the vargaprakrti, or quadratic indeterminate equations of the form x²− Dy² = 1, where D is a given integer, and one is seeking integer solutions for x and y. Brahmagupta gives a method to find infinitely many solutions of this when one solution is known, using his ‘bhaavanaa’ procedure.
Such a procedure is called the ‘composition law’ in modern mathematics, and is a very important principle used in many contexts. This and a cakravaala algorithm were used by Bhaaskara-II to solve the vargaprakrti problem in his Beejaganita (1150 CE). Actually, this had been discovered by Jayadeva earlier (before 1050 CE). Bhaskara-II explicitly solves it for D=61, for which the smallest solution is given by x=1,76,63,19,049 and y=22,61,53,980. Bhaskara-II’s Leelaavati, Beejaganita and Siddhaantasiromani influenced the course of mathematics and astronomy for centuries to come in India. Sridhara (8th century), Mahavira (9th century) and Sripati (11th century) were very important mathematicians between the times of Brahmagupta and Bhaskara-II.
Many Islamic/Arab works, including Al-Khwarizmi’s Kitab al-hisab al-hind that was on the Indian methods in arithmetic, were influenced by Indian works. They in turn influenced European maths later. Indian mathematics lays emphasis on computations based on rules, or is algorithmic. The word ‘algorithm’ is derived from Al-Khwarizmi, and hence has associations with India.
There is a general impression that there was no progress in mathematics in India after Bhaskara-II. This is not at all true. Narayana Pandita, a 14th century mathematician, made significant contributions, particularly in combinatorics, magic squares and other topics. More importantly, the Kerala school of mathematician-astronomers, beginning with Madhava of Sangamagraama (1340-1420) took the first significant steps in calculus, before this subject developed in Europe in the 17th century. Madhava gave the infinite series for ∏ (pi, the ratio of the circumference and the diameter of any circle), which is expressed as follows:
Vyaase vaaridhinihate ruupahrte vyaasasaagaraabhihate| Trisaraadi vishamasankhyaabhaktam runamsvam prthak kramaat kuryaat|| [“The diameter (vyaasa) multiplied by four and divided by unity (is found and stored).Again, the products of the diameter and four are divided by odd numbers like three and five, and the results are subtracted and added in order (to the original result) .”] So, in modern notation, (∏/4) = 1− (1/3) + (1/5) − (1/7 ) + ... . This series is ascribed to Gregory and Leibniz (17th century), who came three centuries later! Here, the ‘bhuutasankhyaa’ system for representing numbers is used, which employs bhuutas (beings) that have the potential to connote numbers: prthvi, ruupa = 1, netra = 2, guna = 3, veda, vaaridhi = 4, bha= 27, etc. (Later, we will use the fact that while decoding the number, the digits are written from right to left).
Madhava is not content with merely stating this momentous result, which is not a very efficient method for computing ∏ (pi) accurately. The verse quoted above also goes on to give what are known as “end-correction” terms that can be used to find very accurate values of ∏, without using too many terms. Using only 50 terms and an appropriate end-correction term, Madhava found a value of ∏ that is very accurate.
It is expressed in the following verse: Vibudhanetragajaahihutaasanatrigunavedabhavaaranabaahavah | Navanikharvamite vrtivistare paridhimaanamidam jagadurbudhaah|| “Wise men say that if the diameter happens to be 9×1011 (nava-nikharva) units, then the measure of the circumference is 28,27,43,33,88,233 (vibudha … baahavah) units.” The first half of the verse expresses a number using the ‘bhuutasankhyaa’ system: vibudha = 33, netra = 2, gaja = 8, ahi = 8, hutaasana = 3, tri = 3, guna = 3, veda = 4, bha = 27, vaarana = 8, and baahu = 2. Then, the value of ∏ given by the above verse is ∏ = 28,27,43,33,88,233/(9 × 1011) = 3.141592653592, which is correct to 11 decimal places.
The infinite series for the sine and cosine functions were also given by Madhava. Ganitayuktibhaashaa of Jyeshthadeva (c. 1530 CE) gives the proofs for all the infinite series, as well as efficient ways for obtaining very accurate values of ∏. This book can indeed be called the first textbook of calculus.
M S Sriram
Theoretical Physicist & President, Prof. K.V. Sarma Research Foundation
(Email address: sriram.physics@gmail.com)