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.

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*.

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.

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.

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.

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.

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

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.

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*.

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.

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.

Kuriyama, Shigeshi. 1999. *The Expressiveness of the Body and the Divergence
of Greek and Chinese Medicine*. New York: Zone Books.