Development of Mathematics & Jainism

By Dr. B. S. Jain

Importance of Mathematics in Jain Religion : Jain of ancient India attached great importance to the study of Mathematics and this subject was regarded as an integral part of their religion. The knowledge of Samkhyana (the science of numbers, meaning arithmetic. and astronomy) us stated to be one of the proper time and place for religious ceremonies.

According to Jains, a child should be taught firstly writing, then arithmetic as most important of the seventy two sciences or arts. According to the Jaina legend, their first Tirthankar Rishabhanath, taught the Brahmi script to his daughter Brahmi, and mathematics to his other daughter Sundari. The sacred literature of the Jainas is called Siddhanta or Agama and is very ancient. Jainas evolved their own theories and made notable contribution to the science of medicine, mathematics, physics, astronomy, cosmology, the structure of matter and energy, the fundamental structure of living beings, the concept of space and time, and the theory of relativity.

The Indian name for mathematics is Ganita. It literally means the science of calculation or computation, Ganita-Sar-Samgraha (GSS) of Mahaviracarya (850 A.D.) is the only treatise on arithmetic and algebra, by a Jain scholar, that is available at present. Suryaprajnapti and the Chandraprajnapti are two astronomical treatises. The other mathematical treatises by the early Jainas have been lost. The author of GSS has always held Bhagwan Mahaveera, to have been a great mathematician. 

Amongst the religious works of the Jainas, that are important from the view point of mathematics are :

1. Suryaprajnapti
2. Jambudvipaprajnapti
3. Sthananga Sutra
4. Uttradhayana Sutra
5. Bhagwati Sutra
6. Anuyoga-dvara Sutra

Kusumpura School of Mathematics :
In the Sulba Sutra period (750 B.C. to 400 A.D.) three existed three important schools of mathematics :
i) The Kusumpura or patliputra School near modern Patna. Bhadrabahu (4th cent. B.C.) and
 Umaswati (2nd cent. B.C.) belonged to this school.
ii) The Ujjan School
 Brahmagupta (7th cent. A.D.) and Bhaskaracarya (12th cent. A.D.) belonged to this school. 
iii) The Mysore School
 Mahaviracarya (9th cent. A.D.) or briefly Mahaveera belonged to this school.

There was a close contact between the three schools and the mathematicians of one school visited the other schools frequently.

The Kusumpura School in Bhihar (ancient Magadha) was a great centre of learning. The famous University of Nalanda was situated in modern Patna and was a centre of Jaina scholars in ancient times. The culture of mathematics and astronomy survived in this school upto the end of the 5th cent. of the Christian era when flourished the famous algebraist Aryabhata (476 A.D.) who made many innovations in Hindu astronomy. Aryabhata was the Kulpati of the university of Nalanda. He was unanimously acknowledged by the later Indian mathematicians as father of the Hindu Algebra. The influence of this school continued unabated for several centuries after Aryabhata. 

Bhadrabahu came down from Bihar (Magadha) in 4th cent. B.C. and settled down at Sravanabelgola in the Mysore State. On his way he passed through Ujjain and halted there for some time. He was one of the great preceptors of the Jainas and at the same time an astronomer and a mathematician too. He could reproduce from memory the entire canonical literature of the Jainas and was befittingly called a Srutakevalin. Bhadrabahu is the author of two astronomical works :

i) A commentary of the Surya Prajnapti (500B.C.), and
ii) An original work called the Bhadra Bahavi Samihita.

Umaswati was a Jaina metaphysician of great trpute. According to Swetambar Jainas, he was born at a place called Nyagrodhika and lived in the city of Kusumpura in about 150B.C. According to this sect, his name is said to be a combination of the names of his parents, the father Swati and the mother Uma. But Digamber Jains' version is that his name was Umaswami and not Umaswati.

The earliest commentator of Umaswati is Siddhasena Gani or Dicakara who lived in 56 B.C.

Tattvartha-dhigama Sutra-Bhashya. It is an important work of Umaswati. In this text, an attempt has been made to explain the nature of things and the authority of this work is acknowledged both by the Swetambaras and the Digambaras. Umaswati was also the author of another work known as Ksetra-Samasa (collection of places). This work is also known as Jambudvipa Samas and Karana-Bhavana are two classes of works that give in a nutshell the mathematical calculations employed in Jaina cannonical works. The earliest Ksetra-Samas was by Umaswati. It is noteworthy that Umaswati was not a mathematician. The mathematical results and formulae as quoted in his work, it seems, were taken from some treatise on mathematics known at that time.

Topics in Mathematics
According to Sthanaga Sutra (before 300 B.C.), the topics of discussion in mathematics are ten in number :

 i) Parikarma - (fundamental Operations)
ii) Vyavahara - (subjects of treatment)
iii) Rajju - ('rope' meaning geometry)
iv) Rasi - ('heap' meaning menstruation of solid bodies)
v) Kala Savarnama - (Fraction)
vi) Yavat-tavat - ('as many as' meaning simple equations)
vii) Varga - ('square' meaning quadratic equations)
viii) Ghana - ('cube' meaning cubic equations)
ix) Varga-varga - ('biquadratic equations')
x) Vikalpa or Bhog - ('permutations and combinations')

Tattvartha-Dhigma Sutra-Bhashya Of Umaswati : In a reference has been made of two methods of multiplication and division. In one method, the respective operations are carried out with the two numbers considered as a whole. In the second method, the operations are carried on in successive stages by the factors, one after another, of the multiplier and the divisor. The former method is our ordinary method, and the later is a shorter and a simpler one. The method of multiplication by factors has been mentioned by all the Indian mathematicians from Brahmagupta (7th cent. A.D.) onwards. The division by factors is found in Trisatika of Sridhara (8th cent. A. D.). This method reached Italy in the middle ages through the Arabs and was called the 'Modo per rekeigo'.

Ganita-Sara-Samgraha of Mahaviracarya (850 A.D.) : Mathaviracarya (briefly Mahavira) was the most celvbrated Jain mathematician of the mid-ninth century. His great work, Ganita-Sara-Smgraha (GSS) was an important link in the continuous Chain of Indian mathematicial texts, which occupied a place of pride, particularly in South India. Raja-Raja Nerendra of Rajamahendry got it translated into Telgu by one Pavuturi Mallana in the 11th century A. D. Mahaveera occupied a pivotal position between his predecessors (Aryabhata I, Bhaskaracarya I and Brahmagupta) and successors (Sridhara, Aryabhata II and Bhaskaracarya II).

The GSS consists of nine chapters like the Bijaganita of Bhaskaracarya II. It deals with operations with numbers except those of addition and subtraction which are taken for granted; squaring and cubing; extraction of square-roots and cube-roots; summation of arithmetic and geometric series; fractions; mensuration and algebra including quadratic and indeterminate equations. Twenty-four notational places are mentioned, commencing with the unit's place and ending with the place called maha-ksobha, and the value of each succeeding place is taken to be ten times the value of the immediately preceding place.

In the treatment of fractions, Mahaveera seems to be the first among the Indian mathematicians who used the method of least common multiple (L.C.M.) to shorten the process. This is called niruddha. Mahaveera knew that a quadratic equation had two roots. This has been substantiated by problems given in his work and the rules given therein for solving quadratic equations. Mahaveera called the process of summation of series, from which the first few terms are omitted, as Vyutkalita, and has given all the formulae for geometric progression (G. P.) thus earning for himself a prominent position in this respect.

In keeping with the traditions of those days, many topics on algebra and geometry have been discussed in the GSS. Mahaveera's work on 'rational triangles and quadrilaterals' contains many other problems of similar nature, and a number of illustrative examples are given therein. But it is noteworthy that his investigations in this particular field have certain remarkable features, and they deserve a special consideration for the following two reasons :

He treated certain problems, on rational triangles and quadrilaterals, which are not found in the work of any anterior mathematician e.g. problems on right triangles involving areas and sides, rational triangles and quadrilaterals having a given area or circum-diameter, pair of isosceles triangles etc; (ii) in the treatment of other common problems, Mahaveera introduced modifications, improvements or generalizations upon the works of his predecessors, particularly of Brahmagupta (6th cent. A.D.)

It may be remarked here that the credit, which Mahaveera rightly deserves for his discovery of certain methods for the solution of rational triangles and quadrilaterals has gone almost unnoticed by historians of ancient mathematics, like L. E. Dickson.

Mahaveera, by his protracted achievements in several branches of Mathematics, has a distinct position and his contributions stimulated the growth of mathematics.

Delhi University

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Source : Article From 'Sixth World Jain Conference' ( 1995) Souvenir

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