# Ancient Jaina Mathematics

## 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 Mathematics

According 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 Formulae

In 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 Mathematics

The 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 Combinations

The 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 Indices

The 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 Numbers

Like 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 10^{11} X 8,400,000^{28}days, which was termed Sirsaprahelika.

#### Certain Technical Terms

In 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*.

#### Geometry

The 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 10^{21}. 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.