## Jaina Mathematics

^{th}century BC. To a certain extent it began to replace the Vedic religions which, with their sacrificial procedures, had given rise to the mathematics of building altars. The mathematics of the Vedic religions is described in the article Indian Sulbasutras.Now we could use the term Jaina mathematics to describe mathematics done by those following Jainism and indeed this would then refer to a part of mathematics done on the Indian subcontinent from the founding of Jainism up to modern times. Indeed this is fair and some of the articles in the references refer to fairly modern mathematics. For example in [16] Jha looks at the contributions of Jainas from the 5

^{th}century BC up to the 18

^{th}century AD.This article will concentrate on the period after the founding of Jainism up to around the time of Aryabhata in around 500 AD. The reason for taking this time interval is that until recently this was thought to be a time when there was little mathematical activity in India. Aryabhata‘s work was seen as the beginning of a new classical period for Indian mathematics and indeed this is fair. Yet Aryabhata did not work in mathematical isolation and as well as being seen as the person who brought in a new era of mathematical investigation in India, more recent research has shown that there is a case for seeing him also as representing the end-product of a mathematical period of which relatively little is known. This is the period we shall refer to as the period of Jaina mathematics.

There were mathematical texts from this period yet they have received little attention from historians until recent times. Texts, such as the *Surya Prajnapti* which is thought to be around the 4^{th} century BC and the *Jambudvipa Prajnapti* from around the same period, have recently received attention through the study of later commentaries. The *Bhagabati Sutra* dates from around 300 BC and contains interesting information on combinations. From about the second century BC is the *Sthananga Sutra* which is particularly interesting in that it lists the topics which made up the mathematics studied at the time. In fact this list of topics sets the scene for the areas of study for a long time to come in the Indian subcontinent. The topics are listed in [2] as:-

… the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

The ideas of the mathematical infinite in Jaina mathematics is very interesting indeed and they evolve largely due to the Jaina’s cosmological ideas. In Jaina cosmology time is thought of as eternal and without form. The world is infinite, it was never created and has always existed. Space pervades everything and is without form. All the objects of the universe exist in space which is divided into the space of the universe and the space of the non-universe. There is a central region of the universe in which all living beings, including men, animals, gods and devils, live. Above this central region is the upper world which is itself divided into two parts. Below the central region is the lower world which is divided into seven tiers. This led to the work described in [3] on a mathematical topic in the Jaina work, *Tiloyapannatti* by Yativrsabha. A circle is divided by parallel lines into regions of prescribed widths. The lengths of the boundary chords and the areas of the regions are given, based on stated rules.This cosmology has strongly influenced Jaina mathematics in many ways and has been a motivating factor in the development of mathematical ideas of the infinite which were not considered again until the time of Cantor. The Jaina cosmology contained a time period of 2^{588} years. Note that 2^{588} is a very large number!

^{588}= 1013 065324 433836 171511 818326 096474 890383 898005 918563 696288 002277 756507 034036 354527 929615 978746 851512 277392 062160 962106 733983 191180 520452 956027 069051 297354 415786 421338 721071 661056.

So what are the Jaina ideas of the infinite. There was a fascination with large numbers in Indian thought over a long period and this again almost required them to consider infinitely large measures. The first point worth making is that they had different infinite measures which they did not define in a rigorous mathematical fashion, but nevertheless are quite sophisticated. The paper [6] describes the way that the first unenumerable number was constructed using effectively a recursive construction.The Jaina construction begins with a cylindrical container of very large radius *r*^{q} (taken to be the radius of the earth) and having a fixed height *h*. The number *n*^{q} = *f*(*r*^{q}) is the number of very tiny white mustard seeds that can be placed in this container. Next, *r*_{1} = *g*(*r*^{q}) is defined by a complicated recursive subprocedure, and then as before a new larger number *n*_{1} = *f*(*r*_{1}) is defined. The text the *Anuyoga Dwara Sutra* then states:-

Still the highest enumerable number has not been attained.

The whole procedure is repeated, yielding a truly huge number which is called jaghanya- parita- asamkhyata meaning “unenumerable of low enhanced order”. Continuing the process yields the smallest unenumerable number.Jaina mathematics recognised five different types of infinity [2]:-

… infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite.

The *Anuyoga Dwara Sutra* contains other remarkable numerical speculations by the Jainas. For example several times in the work the number of human beings that ever existed is given as 2^{96}.By the second century AD the Jaina had produced a theory of sets. In *Satkhandagama* various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets.

Permutations and combinations are used in the *Sthananga Sutra.* In the *Bhagabati Sutra* rules are given for the number of permutations of 1 selected from n, 2 from n, and 3 from n. Similarly rules are given for the number of combinations of 1 from n, 2 from n, and 3 from n. Numbers are calculated in the cases where n = 2, 3 and 4. The author then says that one can compute the numbers in the same way for larger n. He writes:-

In this way,5,6,7, …,10, etc. or an enumerable, unenumerable or infinite number of may be specified. Taking one at a time, two at a time, … ten at a time, as the number of combinations are formed they must all be worked out.

Interestingly here too there is the suggestion that the arithmetic can be extended to various infinite numbers. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion was noted. In a commentary on this third century work in the tenth century, Pascal‘s triangle appears in order to give the coefficients of the binomial expansion.Another concept which the Jainas seem to have gone at least some way towards understanding was that of the logarithm. They had begun to understand the laws of indices. For example the *Anuyoga Dwara Sutra* states:-

The first square root multiplied by the second square root is the cube of the second square root.

The second square root was the fourth root of a number. This therefore is the formula

*a*).(√√

*a*) = (√√

*a*)

^{3}.

Again the *Anuyoga Dwara Sutra* states:-

… the second square root multiplied by the third square root is the cube of the third square root.

The third square root was the eighth root of a number. This therefore is the formula

*a*).(√√√

*a*) = (√√√

*a*)

^{3}.

Some historians studying these works believe that they see evidence for the Jainas having developed logarithms to base 2.The value of π in Jaina mathematics has been a topic of a number of research papers, see for example [4], [5], [7], and [17]. As with much research into Indian mathematics there is interest in whether the Indians took their ideas from the Greeks. The approximation π = √10 seems one which was frequently used by the Jainas.

Finally let us comment on the Jaina’s astronomy. This was not very advanced. It was not until the works of Aryabhata that the Greek ideas of epicycles entered Indian astronomy. Before the Jaina period the ideas of eclipses were based on a demon called Rahu which devoured or captured the Moon or the Sun causing their eclipse. The Jaina school assumed the existence of two demons Rahu, the Dhruva Rahu which causes the phases of the Moon and the Parva Rahu which has irregular celestial motion in all directions and causes an eclipse by covering the Moon or Sun or their light. The author of [23] points out that, according to the Jaina school, the greatest possible number of eclipses in a year is four.

Despite this some of the astronomical measurements were fairly good. The data in the *Surya Prajnapti* implies a synodic lunar month equal to 29 plus ^{16}/_{31} days; the correct value being nearly 29.5305888. There has been considerable interest in examining the data presented in these Jaina texts to see if the data originated from other sources. For example in the *Surya Prajnapti* data exists which implies a ratio of 3:2 for the maximum to the minimum length of daylight. Now this is not true for India but is true for Babylonia which makes some historians believe that the data in the *Surya Prajnapti* is not of Indian origin but is Babylonian. However, in [22] Sharma and Lishk present an alternative hypothesis which would allow the data to be of Indian origin. One has to say that their suggestion that 3:2 might be the ratio of the amounts of water to be poured into the water-clock on the longest and shortest days seems less than totally convincing.

**References (23 books/articles)**

**Article by:** *J J O’Connor* and *E F Robertson*