Sthananga Sutra | Jain Mathematics

As per the Śvetāmbara belief, Sthananga Sutra forms part of the first eleven Angas of the Jaina Canon which have survived despite the bad effects of this Hundavasarpini kala. This is the reason why, under the leadership of Devardhigani Ksamasramana, the eleven Angas of the Svetambara canon were formalised and reduced to writing. This took place at Valabhi 993 years after Māhavīra's nirvana. (466 CE). In the vacana held at Valabhi, in Gujarat, the Sthananga Sutra was finalised and redacted. The language used is Ardhamagadhi Prakrit. The mula sutras of the Sthananga Sutra are difficult to understand without the help of a commentary or tika. Hence, in the 11th century CE, Abhayadevasuri wrote a comprehensive Sanskrit gloss on the Sthananga Sutra.

Description

The Sthānāngasūtra is known in Prakrit as the Thanam. The word thanam denotes quantum. Hence, the style of the Sthananga Sutra is unique. It is divided into ten chapters, and each chapter enumerates certain topics according to their numbers. Each chapter is titled as a Thana. (Sanskrit: Sthānā) This āgama defines and catalogues the main substances of the Jain metaphysics. Diverse topics such as the Dharmakathanuyoga, Carananuyoga, Karananuyoga and Dravyanuyoga are covered. While the focus is on Karananuyoga, this unique āgama serves as a huge anthology to all branches of Jaina knowledge.

Because all topics, terms and things are thought of as fitting well with number one, number two, and so on, up to number ten, and because they are listed accordingly, the word "sthāna" in the titles of the ten chapters as well as in the title of our work means "place". The Sthānāngasūtra is an anga-text in which "terms and things" are listed in their "right place". Sthānānga maybe considered as a memory aid for an ācārya, so that he might not forget the varied subject matters he wants to teach. With this work he has a kind of guideline for his lessons at hand and can easily reply to questions asked by his disciples.

Importance of Sthānāngasūtra can be gauged from the fact that Vyavahāra Chedasūtra (10, 20-34) mentions that it is suitable to be studied only by those ascetics, who have at least eight years standing in monkhood. Further more it is stated that only a monk who knows the Sthānānga by heart may attain the position of an ācārya, which, entitles him to supervise the monks and nuns in regard to their conduct and study.

Authorship

The first sūtra in the Sthānānga goes as follows: sūyam me āusam tenam Bhagavayā evam akkhāyam - "I have heard, o Long-Lived one, that the Venerable (i.e. Mahāvīra) has said thus." From this it can be gauged that as per the tradition it was recited by ganadhara Sudharman, the fifth direct disciple of Mahāvīra, to his disciple Jambūsvāmin.

Contribution to Mathematics

Sthananga Sutra lists the topics which made up the at the trom the time of 2nd century BCE onwards. 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 as .:- the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. It also gives classifications of five types of infinities.

The topics of mathematics, according to the Sthananga-sutra (sutra 747) are ten in numbers:

• Parikarma (four fundamental operations),


• Vyavahara (subjects of treatment),


• Rajju (geometry),


• Rashi (mensuration of solid bodies),


• Kalasavarna (fractions),


• Yavat-tavat (simple equation),


• Varga (quadratic equation),


• Ghana (cubic equation),


• Varga-varga (biquadratic equation) and


• Vikalpa (permutation and combination).

However, the historians of mathematics differ in explaining some of the terms from the commentator, Abhayadeva Suri (1050 AD).

Jain Arithmetic

-Dr. Mahavir Raj Gelra

The beginning of the studies of mathematics in India is regarded a thousand years before Christian era. This was the period when the Jain knowledge prophesied by the Lord Parshwanath was flourishing. Jain arithmetic finds its parallels in Vedic mathematics. During Vedic period, the sacrificial altars were constructed in various safe geometrical shapes to confine the sacred fire. The shapes of the pits used to be in geometrical segments, e.g., triangle, quadrilateral, oval, spherical, circular etc. This is indicative of rich wealth of geometric knowledge prevailing then.

Jain mathematics has been extensively used in the explanations and discussions on the six categories of matter existing in the Lok. Jain Arithmetic has two basic branches -

(i) Geometry - Jains developed the basic geometry and used it to explain the shape and extent of the universe, its centre (ruchak-pradesh), heaven, hell, etc. Besides curved directions. Krishna-rajji (black-hole equivalents) etc. are also described using geometry. In Sthanang and Uttaradhayyan Sutra, five basic shapes (sansthan) are described -

1. Sphere
2. Circle
3. Triangle
4. Quadrilateral
5. Rectangle

In Jain writings, the reference to triangular, rectangular and hexagonal 'earths' are described in the descriptions of Krishna-rajji. These shapes are described in detail in the relevant chapters of this book. However, it is necessary to mention here that the geometry of Vedic and Jain origins are quite similar and seem to have their genesis in the Indian mathematics.

(ii) Arithmetic - In the chapters of time, speed, karma etc., basic mathematical quantities of numerate, innumerate and infinity find wide spread mention. With the help of quantitative analysis, Jains have calculated the distances, time, speed, life-span of animates, etc.

Units of measurement

In the studies of six basic mattereals of the universe in the Jain literature, classifications into the minimal, medium and maximal have been done while applying the mathematics of the numerate, the innumerate and the infinite. First of all, we shall understand the finest parts of the six substances from the following table, before going in for detailed mathematical discussion, as they form the basis of development of mathematics -

Jain laureates have presented the quantitative analysis of six forms of substances (dravya) using certain basic indivisible units as tabulated below -

S.No.	Substance	Unit
1.	Pudgal (Particle/Matter)	Parmanu (Atom)
2.	Kaal (Time)	Samay
3.	Dharma, Adharma, Aakash, Jiva	Pradesh

1. Parmanu

It is the smallest unit used to describe the matter or substance. In this entire universe (Lok), pudgals are classified in only two types -

• Microphysical or massless (Sukshma)
• Macro or massive (Sthula)

These two pudgals manifest distinct physical properties as below -

Micro (Sukshma)	Macro (Sthula)
Massless	With Mass
Motion Unhindered by the presence of matter	Motion obstructed by the presence of other macro particles
Speed beyond that of light is achievable	Speed limited to that of light

Describing the properties of waves and particles, famous scientist, Heisenberg had stated that these two did not follow same set of physical laws. This commonality between the science and Jain Agamas is amazing as the two are separated more than two thousand years on the time scale.

Micro and macro pudgals are convertible into each other but once converted, their properties change to such an extent that the physical laws applicable to micro state are no longer valid in the macro state and vice-versa. Jains, therefore, insist that the micro pudgals (also known as Nishchay Parmanu)constituents an intangible world perceptible only by the intelligence, and simultaneously, macro pudgals (also known as Vyavhar Parmanu) constitute a tangible world observable by our sensory organs (Indriya). In micro form, pudgals remain weight less and travel unrestricted in the space. In this state, the pudgal is understood to be in the form of pure energy. Light, temperature, gravity, magnetism, electrostatic bonds etc., are manifestations of micro world. As described in Anuyog Dwara, infinite micro pudgals integrate to form a macro pudgal. This basic entity is called Vyavhar Parmanu. Therefore, the Jain literature gives us following classification of basic building blocks of the universe-

1. Sukshma or Nishchay (Deterministic) Parmanu
2. Sthula or Vyavhar (Behavioural) Parmanu

Vyavhar Parmanu is a specially coined definition in the sutras of Anuyog Dwara. This is because the human behaviour (Vyavhar) is entirely dependent on the macro-pudgals as the latter alone comes within the realms of our sensory perception. As discussed earlier, macro pudgals cannot travel unrestricted. In other words, they keep on interacting with other particles during motion. This property of interaction makes the macro pudgals tangible and perceptible by our sensory organs. We have no measurements available for micro or sukshma parmanu as they are rendered ethereal or intangible. On the other hand, Jain literature has full set of units to describe the ascending order of complexity of Sthula or Vyavhar Parmanu -

∞ Nishchay Parmanu = 1 Vyavhar Parmanu
(Both Nishya and Vyavhar Parmanu are beyond the realm of sensory perception)
8 Vyavhar Parmanu = 1 TrusParmanu
(From Trus Parmanu onwards, Sthula substances are within the domain of sensory perception)
8² Vyavhar Parmanu = 1 RathParmanu
8³ Vyavhar Parmanu = 1 Balagra
8⁴ Vyavhar Parmanu = 1 Liksha
8⁵ Vyavhar Parmanu = 1 Uka
8⁶ Vyavhar Parmanu = 1 Yav
8⁷ Vyavhar Parmanu = 1 Angul
-------------------------------------------------
24 Angul = 1 Hath
4 Hat = 1 Dhanushya
2000 Dhanushya = 1 Guvyut
4Guvyut = 1 Yojan

1.1. Significance of Numeral Eight (8)

In above table there is striking importance of numeral eight. We can notice the use of multiples of eight up to the measurement of Angul, after which the units assume different multiples. This must have been done with a definite purpose. If we bifurcate the dimensions, we have a clear demarcation -

1. Measurements before Angul
2. Measurements after Angul

If we put all the pieces of mathematical jigsaw puzzle together, a clear picture emerges. In the Jain canonical texts, geographical extent and relative positions of heavenly bodies (cosmology) are described in the units of Angul and beyond. This indicates that the unit Angul is utilized for linear, single dimensional measurements. Whereas, all the units smaller than Angul are indicative of Volume of the particle as a whole. This is inferred from the fact that the Jain mathematics considers 2 as the smallest number. Numeral 8 is derived from 23. Therefore, increment in the multiples of 8 suggests that the units are indicative of increasing volume. Writing in equation form -

2³ = 2x2x2 = 8

This is clearly a three dimensional measurement. The only possible explanation to this demarcation in units could be that at infinitesimal and minute levels, individual linear dimension has no significance as the particles retain their spherical shapes which can be better described in volume terms rather than length.

1.2. Electron, Proton and Quark

Physicists of the current generation find themselves back at the square one as far as identification of tiniest particle is concerned. Scientists are still puzzled by the behavioural observation of infmitesimally small particles. Initially, an atom was considered as the smallest building block. But, soon electrons, protons and neutrons were discovered. Later on quarks were experimentally detected during the transitional phases but they were not found to exist independently. Recently, a new particle comprising five quarks has been identified by the scientists, which they believe existed since the time of big-bang. However, the tiniest particle is so enigmatic that its discovery still looks elusive.

Jain Literature can provide a helping hand to the modern scientists in this subject. Mahapragya writes that the Sukshma (Nishchay) Pudgal, as described in the Jain Agams, is indivisible, indestructible, and imperishable basic constituent of the substance. This leads to very important conclusion -

If a particle breaks to disappear as energy (Sukshma Pudgal), it can be treated as the smallest particle (Sthula Pudgal) of the universe. Scientists have so far been able to find small particles which disintegrate further into smaller particles but have not been able to isolate such a particle which when broken disappears entirely in the form of energy. The day we can find such a particle, we can surely claim to have found the basic building block of this universe. Since all our efforts to break particles down the smallest one have not yielded results so far, we must now attempt an alternative method. If we can concentrate the energy to a miniscule space it will integrate to result in a particle which will be the smallest particle.

2. Kaal (Time)

We have, so far, endeavoured to know the smallest particle of substance. In Jain belief, Time is an independent entity. What is the smallest unit of time? All the activities of Pudgals are space-time related. Accordingly, the time factor in the micro (sukshma) world is 'samay'. Infinite such 'samay' constitute one 'avalika'. Avalika is the smallest unit of time in the macro (sthula) world.

We have seen earlier that -
∞ Nishchay Parmanu = 1 Vyavhar Parmanu
Similarly,
∞ samay = 1 avlika

The factor of infinity (∞) in both these equations suggests that there exists a quantum jump from micro to macro level.

As in the case of smallest particle, scientists are still searching an activity or phenomenon which is accomplished in the smallest period of time. They have found visible light, x-rays and gamma rays in which the wavelengths are as low as a millionth of a centimeter. This means the wave activities are taking place at the Nano- and Pico-second (one-millionth of micro second) scale. In a science magazine, 'Nature', some Austrian scientists have claimed to observe fastest ever happening in which the event is said to happen in one-hundredth of an 'ato-second'. The 'ato-second' is so small a unit that to bring it at par with a second will take 30 million years. Scientists employed the motion of electrons to measure this event. Researchers excited the electrons with the help of far ultra-violet light beam. According to the Professor Farank Cruise of Vine Technique University, some electrons were accelerated to such an extent that they detached from the parent atom permanently. These electrons were topographically photographed by Fucycle Laser. These photographs revealed the activities taking place in the time frame of one-hundredth of an 'ato-second'. This research has paved the way to manufacture highly accurate clocks. At present the accurate clocks are working at microwave frequencies, with the advent of ato-second phenomenon future clocks of extremely high accuracy and stability will work on optical frequencies obtained from lasers.

Again, the ancient Jain literature seems to be quite in consonance with the modern science. As mentioned earlier the Jains have stipulated one 'samay' as the time taken by the activities of massless sukshma pudgal. We have christened this unit as 'Timon'.

We shall now ponder upon the units of space. Up till now we have discovered that the particles and time-scales of micro and macro levels are different and the two can become mutually equivalent only by using the factor infinity. Can the space too, be described in micro and macro units? Jain Agams have explained the parmanu-samay-akash as interrelated entities and one can be defined with the help of another.

3. Aakash - Parmanu

After having known the units of the substance and time, parmanu and samay respectively, it is necessary to know the units of space. Is there any equivalence in the minima and maxima of pudgal, time and space respectively?

It is interesting to study the Jain canonical literature where the examples are given concerning the units of space.

1. It has been mentioned that a theoretical unit 'aakash-pradesh' is the space occupied by one parmanu (dion). It must be observed here that the dion is the smallest massless derivative of the pudgal.
2. A second mention of 'aakash-pradesh' which seems to be more practical is the space occupied by infinite parmanus bundled as a shukshma pudgal (quadon).

These two statements sound paradoxical and are being keenly examined by Mahapragya as follows:

"The two mentions of 'aakash-pradesh' actually differentiate the micro and the macro worlds. As the dion (parmanu) acquires practical utility only after infinite of them consolidate to form a quadon, the space occupied by a parmanu is of limited use when we discuss the shukshma or the micro world only. The real space co-ordinates are constructed only by the unit-space occupied by the packet of infinite parmanus forming a quadon (skandh)."

4. Aakash-Kaal

'A dion (paramanu), if moves with slowest speed, travels from one space unit (Akash Pradesh) to the adjacent one only. On the other hand, if it travels at its fastest speed, it gets transferred from one end of the universe to the other (a distance of 14 Rajju or innumerate Yojana).' This statement of Jain Agams actually constitutes the Theory of Relativity. The space-time (Aakash-Kaal) linearity is affected by the speed of the object. Einstein proved from his mathematical calculations that an astronaut who travels to a distant star at a speed of 70% of the speed of light, he will not only experience the slow passage of time, but will also experience the distance being lowered by virtue of his high speed.

We have dions (paramanu) and octons (Sthula Pudgal) as far as particles are concerned; we have samay (unit of finer time) and Avlika ( unit of real time) but, no description is available as micro space point and macro space point. In an indirect mention, however, in the Acharanga Niryukti the difference in space units and paramanu-kaal is highlighted. It says that if the space points contained in a finger-width of space are exhausted by taking out one space point in each consecutive instants, it will take innumerate ascending and descending periods of time to evacuate that region. This comparison of region with time manifests the nature of space. Hence the finger width measure of space can be called as macro space and the space point can be called as part of the micro-space.

With Jain standpoint it is proper to recognize that the separate units are requisite for micro and macro domains. These units cannot be interchangeably applied. An innumerate number of dions form a practically mentionable and usable entity in the macro domain. In conclusion, we derive following postulates for matter, time and space -

1. Almost infinite dions combine to form an atom. This implies that a finite combination of dions can form quadons (skandh) but even then they are not useful in practical or macro world. It is the distinctiveness of Jain philosophy which states that only a collection infinite dions is of practical utility. This collection, named as octon makes the basic building block of the macro domain. The analogy between the octons of Jain philosophy and the quanta of modern science is quite remarkable. Modern science has developed its quantum mechanics on the basic principle that the energy is transacted in the forms of packets called quanta.
2. Like matter, practical unit of kaal is Avalika which is said to be constituted by the elapse of innumerate samay (the smallest unit of time in micro domain). Here again, any finite accumulation of samay cannot result in Avalika. These findings caution us that any evaluation of event is possible only after ascertaining the domain of the event - micro or macro. Until then, the results could be erroneous. A massive factor of innumerate (near infinity) is responsible for this stark difference.
3. Similar explanations are forwarded for the defining the space units. Matter and soul travel in definite pathways in the space. Seven such pathways are mentioned in the Bhagwati Sutra-

Track	Shape
Straight Line (Rizu-Ayat)
Right Angle (Ektovakra)
Double Right Angle (Dvitovakra)
Single Split (Ektokhaha)
Double Split (Dvitakhaha)
Circular (Chakrawala)
Semi-circular (Ardhachakrawala)

Entire universe (lok), including the irregular discontinuities at the boundaries can be traversed by the combination of these tracks. This, however, must be kept in mind that the space changes its characteristics with the speed of travel. We can only employ space co-ordinates to assign the direction and trajectory of motion of matter particles (like dions, quadons and octons) and souls.

Highlight of these discussions about the ancient Jain literature is that the six fundamental entities (mattereals) are to be described separately in the micro and macro domains.

Jaina mathematics

Article by: J J O'Connor and E F Robertson

It is a little hard to define Jaina mathematics. Jainism is a religion and philosophy which was founded in India around the 6th 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 5th century BC up to the 18th 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 4th 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 2588 years. Note that 2588 is a very large number!

2588 = 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 rq (taken to be the radius of the earth) and having a fixed height h. The number nq = f(rq) is the number of very tiny white mustard seeds that can be placed in this container. Next, r1 = g(rq) is defined by a complicated recursive subprocedure, and then as before a new larger number n1 = f(r1) 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 296.

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)

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