Ancient Jaina Mathematics: an Introduction

by D.P. Agrawal

The origins of Buddhism and Jainism can be placed around the middle of the first millennium BC. Both Jainism and Buddhism were basically rebellions against the rituals and sacrifices of the earlier Brahmanical religions. Somehow the Buddhists seem to have specialised in medicine and the Jainas in maths. Kuriyama says that surgery and physical Ayurveda became two separate traditions, surgery being more important amongst the Buddhists, who… are less hung up about ritual purity and contact with taboo bodily products such as blood. While in Jainism, the founder of the sect Mahavira himself has been claimed as a mathematician. During the period of the Brahmanas, the maths served the main purpose of rituals. The credit for giving maths the form of an abstract discipline goes to the Jainas.

Jaina mathematics is one of the least understood chapters of Indian science, mainly because of the scarcity of the extant original works. For example, the Jainas recognized five different kinds of infinity. They were the first to conceive of transfinite numbers, a concept, which was brought to Europe by Cantor in the late 19th century. The two thousand year old Jaina literature may hold valuable clues to the very nature of mathematics. This is one area where further research could prove very fruitful.

Joseph's book, The Crest of the Peacock, makes a delightful reading and is a powerful book against the Eurocentric History of Science and Technology. And precisely for this reason it has been criticised by western scholars. The following introduction to Jaina maths is mainly based on Joseph's work and A Concise History of Science in India edited by Bose et al.

Unfortunately sources of information on Jaina mathematics are scarce. A number of Jaina texts of mathematical importance have yet to be studied. Surya prajnapti, Jambu Dwipa Prajnapti, Sthananga sutra, Uttaradhyayana sutra, Bhagwati sutra and Anuyoga Dwara sutra are the oldest canonical literature. The first two works are datable to the third or fourth century BC; the others are at least two centuries later.

Basically their religious literature is classified into four groups, called "anuyoga"(the exposition of the principles of Janisim). Ganitanuyoga (the exposition of the principles of mathematics) was one of them. In the period between the end of the Brahamanas and beginning of the Siddhantic astronomy (c. 4th Century AD), the Jaina mathematics played a significant role.

Available Sources The only treatise on arithmetic by a Jaina scholar, which is available at present, is the Ganita-sara-samgraha of Mahavira (c. AD 850). The author of the Ganita-sara-samgraha held that the great Mahavira, the founder of Jaina religion, was himself a mathematician.

In the history of the Jaina religion Bhadrabahu (c. Died 298 BC), a very prominent personage, is reputed as the last of the Srutakevalin, because of his phenomenal memory, which enabled him reproduce the entire Jaina canonical literature. Bhadrabahu is also known to be the author of the two astronomical works: (1) a commentary on the Suryaprajnapti and (2) an original work called the Bhadrabahavi-samhita. None of these works is available at present. Buhler found a work by the name of the Bhadrabahavi-samhita, but modern scholars have suspected its authenticity on the ground that:

(1) It is of the same character as the other Samhitas; (2) It has not been mentioned by Varahamihira (AD 505) who has referred to many anterior writers; (3) It gives the date of its last redaction as AD 511.

Umasvati was a reputed Jaina metaphysician, but not a mathematician, though he did refer to mathematical formulae. Siddhasena was another mathematician who has been referred to by Varahamihira also. However, from the specific treatises on mathematics we can get a lot of information about the Jainas' knowledge of mathematics from various Ardhamagadhi religious and secular books. Some valuable information as regards the knowledge amongst the early Jainas is expected to be found in the Ksetrasamasa (collection of the places) and Karanabhavana. Jinabhadra Gani (AD 550) wrote two works of the same class: a bigger one, called Brhata ksetrasamasa and a smaller one called Laghu ksetrasamasa.

Topics of MathematicsAccording to the Sthananga-Sutra (c. First Century BC) the main themes for discussion in mathematics are ten in number: parikarama (fundamental operation), vyavahara (subjects of treatment), rajju (geometry), rasi (heap, mensuration of solid bodies), kalasavarna (fractions), yavat-tavat (simple equation), varga (quadratic equations), ghana (cubic equations), varga-varga (biquadratic equations) and vikalpa (permutations and combinations).
Abhayadeva surely thinks that varga, ghana and varga-varga refer respectively to the rules for finding the square, cube and fourth power of a number. But in Hindu mathematics from the earliest times squaring and cubing are considered as fundamental operations and as such they are covered by the terms parikarma. Abhayadeva Suri held that yavat-tavat refers to multiplication or to the summation of the series (samkalita). The early Jainas attached great importance to the subject of permutations and combinations (vikalpa).

The term yavat-tavat entered into the Hindu mathematics more than five centuries before the Greek Diophantus as the symbol for the unknown. The Greek Diophantus suggested that it is connected with the definition of the unknown quantity as "containing an indeterminate or undefined multitudes of units." The ancient work Curni defines the term parikarma as referring to those fundamental operations (sixteen in number) of mathematics as will befit a student to enter into the rest and the real portion of the science.

Certain Mensuration FormulaeIn the Tattvarthadhigama-sutra-bhasya of Umasvati, there is also an incidental reference to two methods of multiplication and division. The multiplication by factor has been mentioned from Brahmagupta and the division by factor is found in the Trisatika of Sridhara. Umasvati is famous as one of the greatest metaphysicians of India and he is held in high esteem equally by the two main sections of the Jainas. He lived about 150 BC.

Schools of MathematicsThe culture of mathematics and astronomy survived in the School of Mathematics at Kusumapura (in Bihar), up to the end of the fifth century of the Christian era, while the school had begun near about the beginning of the Christian era. The famous Jaina saint Bhadrabahu (author of two astronomical works, a commentary on the Suryaprajnapti and the Bhadrabahavi-samhita) lived at Kusumapura. Two other important and well-known centres of mathematical studies in ancient India were Ujjain and Mysore. The Ujjain school included the greatest of Indian astronomers Brahmagupta and the mathematician Bhaskaracarya, while the southern school of Mysore had its representative in Mahaviracarya.

Suryaprajnapti (400BC) and other early Jaina Sutras give the length of the diameter and circumference of certain circular bodies. The formula for the arc of a segment less than a semicircle reappears in the Ganita-sara-samgraha of Mahavira and the Mahasiddhanta of Aryabhata II (AD 950). The Greek Heron of Alexandria takes the circumference of the segment less than a semicircle. In the Uttaradhyayana-sutra, the circumference is stated roughly to be a little over three times its diameter.

Permutations and CombinationsThe early Jainas seem to have a great liking for the subject of combinations and permutations. A permutation is a particular way of ordering some or all of a given number of items. Therefore the number of ways of arranging them gives the number of permutations, which can be formed from a group of unlike items. A combination is a selection from some or all of a number of items, unlike permutations, the other is not taken into account. Therefore the number of ways of selecting them gives the number of combinations, which can be formed into a group of unlike items. Permutations and combinations were favourite topics of study among the Jainas. In the Bhagawati sutra are set forth simple problems such as finding the number of combinations that can be obtained from a given number of fundamental philosophical categories taken one at a time, two at a time, three at a time or more at a time. The Jaina commentator Silanka has quoted three rules regarding permutations and combinations, two of them are in Sanskrit verse and the other is most interestingly in Ardhamagadhi verse.

Law of IndicesThe law of indices cannot be formulated precisely. But there are some indications that the Jainas were aware of the existence of these laws.

Theory of NumbersLike the Vedic mathematicians, the Jainas had an interest in the enumeration of very large numbers, which was intimately tied up with their philosophy of time and space. All numbers were classified into three groups enumerable, innumerable and infinite, each of which was in turn sub-divided into three orders. The Jainas also classify numbers into odd and even categories. The Jainas could conceive of such huge units of time as 756X 1011 X 8,400,00028 days, which was termed Sirsaprahelika.

Certain Technical TermsIn the Jaina literature the modern geometrical term semi-diameter was found in the writings of Umasvati who calls it vyasardha or viskambhardha. The terms jiva for the chord of a segment of a circle and dhanuprstha for its arc occur in several early canonical works. The numeral symbols were written in two forms: ankalipi and ganitalipi.

GeometryThe term rajju was used in two different senses by the Jaina theorists. In cosmology it was frequently used as a measure of length of about 3.4 X 1021. But in a general sense the Jainas used this term for geometry or mensuration, in which they followed the Vedic sulbasutras. A variety of geometric terms were known to them: sama-cakravala, vratta (circle), jiva ( arc), parimandala (ellipse), ghana vratta (sphere) etc. They had derived the value of pi as root of 10.
As mentioned before, the Jainas recognized five different kinds of infinity: infinity in one direction; infinity in two directions; infinity in area; infinity everywhere; and infinity perpetually. This is quite a revolutionary concept, as the Jainas were the first to discard the idea that all infinities were same or equal, an idea prevalent in Europe till the late 19th Century.

The highest enumerable number (ie, N) of the Jainas corresponds to another concept developed by Cantor, aleph-null, also called the first transfinite number. In their theory of sets, the Jainas further distinguished two basic types of transfinite number. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.

This brief glimpse into the Jaina maths clearly shows that mostly this is an uncharted area where a lot of research needs to be done. Two thousand year old Jaina mathematics may hold clues to the very nature of the foundations of mathematics: there lies its importance, and the challenge.

Main Sources:
Joseph, George Gheverghese. 1994. The Crest of the Peacock: Non-European Roots of Mathematics. London: Penguin Books.
Sen, S.N. 1971. Mathematics. In A Concise History of Science in India (Eds.) D. M. Bose, S. N. Sen and B.V. Subbarayappa. New Delhi: Indian National Science Academy. Pp. 136-212.
Reference:Kuriyama, Shigeshi. 1999. The Expressiveness of the Body and the Divergence of Greek and Chinese Medicine. New York: Zone Books.

Mahaviracharya: 9th Century Mathematician

Prof. Yashwant Maliya

The Jains have always been very interested in mathematics. One distinguished Jain mathematician monk was Mahaviracharya who wrote "Ganita-sara-samgraha" in 850 AD during the reign of the great Rashtrakuta king Amoghavarsha. Amoghavarsha had become a Jain monk in the later part of his life. His capital was in Manyakheta in modern Karnataka.

Some of the interesting things in Ganita-sara-samgraha are:

A naming scheme for numbers from 10 up to 10^24, which are eka, dasha, ... mahakshobha.
Formulas for obtaining cubes of sums.

He was the first person to mention that no real square roots of negative numbers can exist. The imaginary numbers were not identified until 1847 by Cauchy in Europe!

He gave techniques for least common denominators. It was used in Europe not before the 15th century.

He gave techniques for Combinations (n choose r). It was later invented in Europe in 1634.
He discussed techniques for solving linear, quadratic as well higher order equations.
He studied several arithmatic and geometric series.

He gave techniques for calculating areas and volumes. No personal information about Mahaviracharya is available.

This is based on an article recently published in Vishva Viveka (a Hindi Quarterly published from New Orleans, LA, USA) by Prof. S.C. Agrawal and Dr. Anupam Jain. As you will recall, Dr. Anupam Jain is the editor of a Jain research journal "Arhata Vachan".

There is an annotated Hindi translation on Ganita-Sara-Samgraha by Prof. L.C. Jain.

Yashwant K. Malaiya

References for Jaina mathematics

Books
L C Jain, The Tao of Jaina sciences (Delhi, 1992).
G G Joseph, The crest of the peacock (London, 1991).
Articles:
R C Gupta, Chords and areas of Jambudvipa regions in Jaina cosmography, Ganita Bharati 9 (1-4) (1987), no. 1-4, 51-53.

R C Gupta, Madhavacandra's and other octagonal derivations of the Jaina value π = √10, Indian J. Hist. Sci. 21 (2) (1986), 131-139.

R C Gupta, On some rules from Jaina mathematics, Ganita Bharati 11 (1-4) (1989), 18-26.

R C Gupta, The first unenumerable number in Jaina mathematics, Ganita Bharati 14 (1-4) (1992), 11-24.

R C Gupta, Circumference of the Jambudvipa in Jaina cosmography, Indian J. History Sci. 10 (1) (1975), 38-46.

R C Gupta, Errata: "Chords and areas of Jambudvipa regions in Jaina cosmography", Ganita Bharati 10 (1-4) (1988), 124.

A Jain, Some unknown Jaina mathematical works (Hindi), Ganita Bharati 4 (1-2) (1982), 61-71.

L C Jain, On certain mathematical topics of the Dhavala texts (the Jaina School of Mathematics), Indian J. History Sci. 11 (2) (1976), 85-111.

L C Jain, System theory in Jaina school of mathematics, Indian J. Hist. Sci. 14 (1) (1979), 31-65.

L C Jain and Km Meena Jain, System theory in Jaina school of mathematics. II, Indian J. Hist. Sci. 24 (3) (1989), 163-180.

L C Jain and Km Prabha Jain, Certain special features on the ancient Jaina calendar, Indian J. Hist. Sci. 30 (2-4) (1995), 103-131.

L C Jain and Km Prabha Jain, Constant-set (dhruva-rasi) technique in Jaina school of astronomy, Indian J. Hist. Sci. 28 (4) (1993), 303-308.

L C Jain and R K Trivedi, Todaramala of Jaipur (a Jaina philosopher- mathematician), Indian J. Hist. Sci. 22 (4) (1987), 359-371.

P Jha, Contributions of the Jainas to astronomy and mathematics, Math. Ed. (Siwan) 18 (3) (1984), 98-107.

S K Jha and M Jha, A study of the value of π known to ancient Hindu & Jaina mathematicians, J. Bihar Math. Soc. 13 (1990), 38-44.

S S Lishk and S D Sharma, Role of pre-Aryabhata Jaina school of astronomy in the development of Siddhantic astronomy, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, New Delhi, 1976, Indian J. Hist. Sci. 12 (2) (1977), 106-113.

S S Lishk and S D Sharma, Season determination through the science of sciatherics in Jaina School of Astronomy, Indian J. Hist. Sci. 12 (1) (1977), 33-44.

S S Lishk and S D Sharma, Zodiacal circumference as graduated in Jaina astronomy, Indian J. Hist. Sci. 14 (1) (1979), 1-15.

I Schneider, The contributions of the sceptic philosophers Arcesilas and Carneades to the development of an inductive logic compared with the Jaina-logic, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, New Delhi, 1976, Indian J. Hist. Sci. 12 (2) (1977), 173-180.

S D Sharma, and S S Lishk, Length of the day in Jaina astronomy, Centaurus 22 (3) (1978/79), 165-176.

J C Sikdar, Eclipses of the Sun and Moon according to Jaina astronomy, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, New Delhi, 1976, Indian J. Hist. Sci. 12 (2) (1977), 127-136.