Jinamanjari 1999 04 No 19

Author(s): Jinamanjari
Publisher: Canada Bramhi Jain Society Publication
Catalog link: https://jainqq.org/explore/524019/1

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International Journal of Contemporary Jaina Reflection
JINAMANJARI
Volume 19 Number 1 April 1999 ISSN 1188-2287
THIS ISSUE EXPLORES
JAINA MATHEMATICS
Theme Guest Editor

Dr. Padmavatamma Professor of Mathematics University of Mysore, India
A Bi-annual Publication of Brümbi Juin Society Est. 1988 a non-profir tax-exempt organization United States of America and Canada

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A NEW EDITION
OF
GANITASĀRA SANGRAHA OF ĀCĀRYA MAHĀVIRA
An Ancient Jaina Mathematics Text in Sanskrit
In Three Languages ENGLISH, HINDI AND KANNADA
The First Edition of the Text in English was Published in 1912
With An Introduction By
Professor D. E. Smith University of Columbia
The New Edition In Three Languages GANITASĀRA SANGRAHA OF ĀCĀRYA MAHAVIRA
Editor: Dr. Padmavatamma
Professor of Mathematics University of Mysore, India
IS
PUBLISHED NOW
Under the Patronage Of Rev. Devendrakirti Bhattāraka
For
Sri Siddhantakirti Granthamāla
Jain Matha of Humcha Ilonibuja - 577436, Karnataka, India

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JINAMANJARI
for the expansion and diffusion of Jaina knwledge and reflection
a bi-annual journal published every April and October
Editor-in-Chief Dr. S.A. Bhuvanendra Kumar Associate Editor
S.N. Prakash
Production
Siri P. Kumar
Mikal A. Radford
Publication Council President
Dr. Dilip K. Bobra Tempe, AZ
Members
Dr. Pradip Morbia Port Neches, TX Mr. Jitendra A. Shah
Edmonton, AB
Prof. S.A.S. Kumar Bangalore, India
Dr. C.K. Khasgiwala Andover, MA
Dr. Jagat S. Mehta
Rochester, NY
Dr. Mahendra Varia Martin, KY
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Dr. Narendra Hadpawat
Woodmere, NY
Dr. Pushpa R. Jain Bluefield, VA
Papers must accompany with notes and references separate from the main text Send to the Editot
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Copyright of Articles Reserved

2 Introduction to Jaina Mathematics - Dr. R.C. Gupta
8 Mathematical Philosophy in Jaina Thought
- Drs. L.C Jain and Padmavatamma
IN THIS ISSUE
13 Asti And Syad In Jaina Thought
- Dr. Mahendra Jain.
18 Mathematical Contents Of Jaina Texts Drs. L.C Jain and Padmavatamma
24 The Ganitasära Of Thakkara Pheru Dr. S. Rajesvara Sarma
·
-
27 A Glimpse of Geometry of Certain Figures Drs. V. Mishra and S.L. Singh
39 Zero In Place Value System of Jain Mathematics Drs. L.C Jain and Padmavatamma

58 Satprarūpaņa Sūtra
Dr. N.L. Jain
78 Book Reviews
- Mikal Austin Radford
- Ashij J. Kumar

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Jinamañjari, Volume 19, No.1, April 1999
INTRODUCTION TO JAINA MATHEMATICS
Dr. R.C. Gupta. Professor Emeritus, Jhansi, India
Although declared as unorthodox by some, the Jaina system of philosophy is unique and has its own characteristics which it place high among the various systems of not only India, but of the whole world. One such characteristic is the doctrine of anekantavda or the theory of manifold-ness. In fact no history of human thoughts and ideas will be complete without this Jaina contribution. The remarkable thing is that the Jaina thoughts seem to be as lively in the current time as they were in the past.
Ours is an era of science and technology in which mathematics plays a significant role as a powerful tool. Without forming a large number of equations and solving them accurately, a rocket or satellite cannot be launched or orbited successfully and provide communication services.
The importance of mathematics was recognised in India. In the Jaina school, mathematics played an important role in a well-rounded philosophical education. Rşabha, the first Tirthankara, was known to have taught arithmetic to his eldest son, Bharata, according to the tradition. Much more use of mathematics was made in Jaina philosophy and cosmology than in any other system. According to an ancient fourfold classification of the Jaina canonical literature, the Ganitānuyoga or Karaṇānuyoga is just one excellent examples. From the famous Jaina mathematician Mahāvīrācārya's Ganitasāra Sangraha (ca. 850 C.E.) we find the following description of the universal utility of mathematics:
In all transactions which relate to worldly, Vedic, or other similar religious affairs, calculation is of most use. In the science of love, in economics, in music and in drama, in the science of cooking, in medical science, in architecture, in prosody, poetics and poetry, in logic and grammar, and in relation to all that constitutes the peculiar value of the arts, the mathematics is held in most high esteem. In relation to the movements of the sun and other planets, in eclipses, in conjunction of the planets, in problems related to direction, position, and time, in the moon's phases, indeed in all these, the use of mathematics is most accepted. The number, the diameter, and the perimeter of islands, oceans, and mountains; the dimensions of the

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habitations and halls belonging to the inhabitants of the world, between the worlds, of the jyotrloka, of the world of gods and of hell-dwellers and other miscellaneous measurements etc., -- all these are known through mathematics. The configuration of living beings, the span of their lives, their eight attributes and the like, their journeys and dwelling together, etc., are all dependent upon mathematics. What is the use of saying much? Whatever there is in all the three worlds with living or moving and non-moving beings cannot be comprehended without mathematics.
It was J.W.L. Glaisner who, while giving his Presidential Address to the British Association for the Advancement of Science, said in 1890 that "no subject loses more than mathematics by any attempt to dissociate it from its history." This is more so in the case of India with its continuous tradition of at least 5000 years in philosophy, literature and the sciences.
Unfortunately, India has not been given due place and credit in the writings on the history of world mathematics. One reason for this lapse is ignorance, and the other is the difficulty of access to sources both primary and secondary.
During recent years, much work has been done in the field of history of the exact sciences of India (including the Jaina contributions). As a result, a number of new findings and discoveries have been made and a vast amount of material has been published. In this regard, an academic journal -- Ganita Bhārati (ISSN 0970-0307) was launched in 1979.
The work in the field of the Jaina exact sciences has a number of forums and institutions; guidance of the Jaina monks, who are invariably great scholars themselves. However, the number of truly devoted research workers in the field of Jaina mathematics is relatively small. Some of the reasons for this situation are caused by difficulties in understanding the ancient languages and their complex terminologies, the historical as well as the scientific methodology, and other technicalities. There are hardly any facilities in India for formal training in the history of science through various sources, and therefore, scholars who are expert in reading and understanding ancient manuscripts and who are also simultaneously learned in ancient as well as modern mathematical sciences are quite meagre in number.
When the study and investigation in the history of the oriental sciences was first started, they were mostly undertaken by Orientalists. However, it is now accepted that it is more desirable that the history of mathematical sciences de investigated by the mathematicians themselves.

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In order to do this, it is necessary to become experts in both linguistic and historical methodology.
A unique scholar of Jaina Mathematics in the present is Professor Laxmi Chandra Jain (b.1926). His vast knowledge of ancient Jaina sources, and long experience has made him a great authority of the Jaina exact sciences. His extensive research investigation into that part of Jaina Mathematics called lokottara ganita (post-worldly mathematics) has shed light on several hitherto unknown aspects in terms of modern mathematical language and notation. Special mention should be made of his research project the Labdhisāra of Nemicandra Siddhānta-Cakravartī (c. 1000 C.E.), which was successfully completed in four volumes in 1987. The text is an advanced theory of the Jaina Karma system.
According to ancient Indian cosmography -- whether Vedic, Buddhist, or Jaina -- there is a series of successive concentric rings (valays) of lands and seas with Mount Meru (a sort of celestial axis) standing in the centre. But while the widths of these rings are successively halved in Buddhist descriptions, they are said to be successively doubled in the Jaina cosmography as we go further away from the centre of Mount Meru or Sumeru. In this particular case, if D is the diameter of the Jambūdvipa, then the width of the nth ring (whether sea or land) surrounding it, is given in the following formula:
Wn = 2n D
Then
1,= 2 W,-3D
Now the Tiloyapannattī defines the khandas (say Kn) of any ring by the relation of the following:
Ky=(22,- 12) / D2
It can be easily seen that
K1 = (20 - 1) 24- 1 = P, for example.
Vīrasenācāry (early 9th century), in his Dhavalā, has given not only Kn, but also the values of Pn up to n= 7. The interesting thing to note is that the set of numbers Pn also includes the so-called perfect numbers.

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Again, since the boundaries of the Jaina cosmographic islands and seas are found to be circular, the geometry of a circle plays an important role in Jaina laukilak ganita (worldly or practical mathematics). But a proper and correct understanding of the various canonical values of various cosmographic lengths and areas requires an insight into the ancient methods of computation which were accessible and followed.
For instance, in modern mathematics we calculate the paridhi (circumference) of a circle of diameter dby using the formula:
p= πd
where the well-known constant n (now known to be a transcendental number) can be taken to as many decimal places as we need or want. But the approach of the ancient Jaina school was somewhat different. For obtaining results of these ancient calculations, we must not only know the type of approximation of n , which was used implicitly or explicitly, but also the manner in which it was used.
For example, the Tiloyapannatti gives the length of the circumference of the Jambū Island -- the diameter of which is a lakh yojanas (100,000) to a very fine unit called avasannāsanna skandha where one angula is equal to 812 of these fine units. This canonical value can not, however, be obtained by simply or directly employing,
ne
✓ 10
to any desired places or degree of accuracy -- although it is this very approximation of whose equivalent was used by the Jainas in this connection. However, a peculiar use of the rule given by
p= V(100),
and then applying the formula
(a2 + x) = a + (x/2a)
for extracting the square-root properly, and finally reducing the result to the desired sub-units, can lead us to a value of circumference p which tallies exactly with the ancient canonical measure found in the text, the

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Tiloyapanṇatti. The question of pramāņa versus utsedha angula does not seem to be involved here.
Similarly, the ancient canonical values of the areas of the various regions of the Jambu Island can be obtained by a proper and intelligent use of the rule
A = √10 (ch/42)
Where c is the chord (jīvā) and h is the heights of the segment (isu) of a circle. The above rule implies the formula
А= лch/4
for the area of a segment of a circle. Had Mahāvīrācārya used this typical traditional formula in his treatment of an ellipse as a double segment, he would have hit upon the true-modern formula a ab for the area of an ellipse (where a and b are the semi-axes).
One of the most original contributions of the Jaina School of Mathematics is the concept of getting vargita-samṁvargita of any quantity to various orders. It leads to formation of very quickly increasing sequences. The (first) vargita-samvargita of x is defined by xx and is denoted by x]
x=xxy, say.
The vargita-samṁvargita of the first vargita-samvargita of x is called the second vargita-samvargita of x. That is, the vargita-samvargita of y will be the second vargita-samvargita of x. It is denoted by xP. Therefore,
Thus
xP = yy = (xx) (xx)

Similarly, the vargita-samvargita of y will be called the third vargita-samvargita of x. And so on. For example, with x = 2 we have the following
= z say.
27=22 = 4
22 = 44 = 256
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273 = (256)256
Using the above notation, the writer of these lines was able to obtain a good lower bound for the jaghanya parīta asaṁkhyāta (minimal limited innumerable). It can be expressed as
log [kN]/(log 8)
where the integer next to N is given by
7.9656 x 10135 (nearly)
and kis great than
45
121010
It should be noted that this value of k is better than that of ga obtained by Muni Mahendra Kumar II in his Visvaprahelikā (Bombay, 1969), and of course, a part of the mathematics of the Jaina cosmos that requires a strenuous imagination. U

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Jinamañjari, Volume 19, No. 1, April 1999
Mathematical Philosophy In Jaina Thought
Dr. L.C. Jain, Jabalpur, India Dr. Padmavatamma, University of Mysore, India
"he words or the languages, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be voluntarily reproduced and combined."
- Albert Einstein, Ideas and Opinions, Calcutta, 1979.
The words, "Mathematical Philosophy," seem to have originated with Bertrand Russell (1872 - 1970), author of the Principia Mathematica. It was co-authored with Alfred Whitehead. Russell, who. created "Russell paradox" in relation to the fringes of the set theory of the infinities as propounded by Georg Cantor (1845 -1918). According to Cantor, the early Greek geometers passing from the empirical rules of Egyptian land-surveying to the general propositions by which those rules were found to be justifiable, became engaged in -- especially after dealing with Euclid's axioms and postulates -- a new type of existential thought, mathematical philosophy.
More recently it has been observed that the early Jaina School of Mathematics was also engaged in a similar pursuit as is demonstrated in the mathematical philosophy of Karma theory with its mathematical discourse of symbolism contained in the Purva (ca. 2nd-5th century C.E.) texts that were in the hands of the Digambaras.
Pandita Todaramalla of Jaipur (c. 1721-61) was the last worker in this particular field. Due to his great accomplishment, we now have a guide to the non-universal mathematics of Jaina Karma theory contained in such works like the Dhavalas of Virasenacarya (begun in the year 792 and completed on the 8th of October 816 C.E), the Gommatasara, the Trilokasara of Nemicandra Siddhantacakravarti (ca. 955-985 C.E.), the Tilyapannatti, and works credited to Madhavacandra Traividya (c.12th century) and Kesava Varni (c. 13th century). The mathematical philosophy of Karma theory with its mathematical discourse of symbolism was extensively researched -- Project of Labdhisara at the Indian National Science Academy, New Delhi, 1984 through 1987.
It was already felt by Boole, Frege and Russell that deeper realms of philosophy could be approached only through words or symbols which could express the propositions between the truth and

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untruth. The parallel to this is the concept of syadvada in the Jaina philosophy. Thus the status of an object being relative to different points of view, a single proposition about its state marred the prospects of its description in various aspects in the old philosophies.
The Jaina philosophy, however, was free from this mono-ended pursuit and it followed a poly-endedness. This led to the existential and constructive spheres of the innumerate and the infinities in a proper and simple way through its set theory (rasi siddhanta). The secret of the mathematical philosophy in the Jaina School thus lay in their attempt to give a new shape to the expressions in logic and intuition. This was achieved with the word "syatin the course of the parikarmastaka, and not only among the finite sets, but, also for the innumerate and infinite sets of various comparabilities.
Today the problem of the comparability is still unsolved in the modern set theory of infinities of various types. In the Jaina set theory there are not only the constant sets but also the variable sets scaling the infinities of Jaina Karma theory through constructions and other analytical methods. The various types of units, measures and calculations between them were needed in their Karma system and cybernetics which was an aggregate of various sub-systems and groups of operations to annihilate the Karma state matrix. Therefore, the School had its own formalised symbolism and symbolic logic, that much like Russell later, became a Karma theory via mathematics.
The Innumerate in Jaina Set Theory: A Philosophical Sub-System
Cantor's theory of sets, had to face the contradictions, antinomies, and inconsistencies as any theory has to face for its survival. His sets included, not of the philosophies, but of proper characteristics that could prove that a set, though infinite, could be greater than another set, as well as that it could be constructed thorough the principle generalised induction. Comparability between infinite set began a new arena of research that went beyond the old philosophical domain in which there was no place to compare infinities (e.g., improper mathematical infinities for their smallness or greatness).
With such a new prospect of the infinities, the Jaina Karma philosophy took a new form. Through various sequences ranging from unity to the supreme sets of omniscience (kevala jnana), the Jainas located the terms of various types of sets involved in the calculations of annihilation of the perpetual karmic cycle of births and deaths. They filled the gaps between such sets as those which had the number of members as numerate, innumerate, and infinite.

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The Jaina School of mathematical philosophy, therefore, took a positivistic approach to introducing the innumerate and the infinite. They were meant to explain the endless processes from ab aeterno to ad infinitum, the relations between various sets involved in the realities of various life types. They had to find, mathematically, a path to perpetual immortality in which there was neither births, rebirths, or the agonies They were looking for a way to achieve the perpetual bliss, infinite power and total knowledge.
To do this, they created the indivisible system of units, as the indivisible instant (samaya) and the indivisible space (pradesa), this is somewhat similar to the problem of the Eleatic School's, and Zeno's, paradoxes. As most of us know Zeno presented a series of mathematical paradoxes that have baffled mathematicians and philosophers until Russell developed his theory of infinite regression ( innumerate regression). Prior to Russell both the Greeks, and those who followed, were obliged to leave most discourse on infinities to a simple statement of, "as small as we please and as great as we please." For example, we find the following -- which was considered to be a truism by Socrates rather than a series of paradoxes -- from Zeno (a student of Parmenides, 5th B.C.E.):
1. Dichotomy: There is no motion, for whatever is transformed into motion, it will be required to reach the middle (of the distance) before it reaches the end (and for reaching that half-way point it will have to reach half of the half-way point, and so on ad infinitum).
2. The Arrow: Zeno states, every object is either at rest or in motion when it occupies the space equal to its own. That object is in that space now (this instant) and always. The moving arrow, therefore, is at rest (and not moving). This is a paradox.
There are two more paradoxes of Zeno presented in the Jowett's translation of The Dialogues of Plato (Vol.2.) The two cases outlined above could not be explained away without the innumerate processes in the nature of motion of physical objects. Unfortunately, this did not allow for division ad infinitum. Such sequences which could have a finite sum may come under the sets with innumerate members. According to Socrates, however, Zeno's paradoxes were not directed against the Pythagorean Schools because they dealt with ultimate units. The Jaina School also dealt with Karma theory through the theory of ultimate units as we have seen already. Even the phases of the

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bios are dealt with by a measure in terms of the indivisiblecorresponding-sections (avibhagi-praticchedas), which calculates the emotions in terms of the tetrads in the ultimate particles of matter bound as karma paramanus. These have the configuration (prkrti), mass number (pradesas), life-time (sthiti) and energy-level (anubhaga). As for the innumerate number, it plays a role in between the finite and infinite.
Paradoxes of Cantor's Infinite Sets and Jaina Set Theory
Let us have a look at the paradox of Cantor's Set theory when it was in its inception. Hausdorff states, "It is to the undying credit of Georg Cantor that, in the face of conflict, both internal and external against apparent paradoxes, popular prejudices, and philosophical dicta -- infinitum actu non datur -- there is no actual infinite. Even in the face of doubts that had been raised by the very greatest mathematicians, he dared this step into the realm of the infinite (Set Theory, 1962, New York,
p.11)."
In 1901 Bertrand Russell discovered that a contradiction could be derived from the axiom of abstraction (which was one of the basis of Cantor's Set theory). He considered the set of all things which have the property of not being members of themselves. The paradox can be related through the barber's paradox in the following story. There is a barber in a village who shaves all those who do not shave themselves. The problem is, who is to shave the barber. Such a set is contradictory to its very existence, but in the Jaina Karma theory the set of indivisiblecorresponding-sections of omniscience could have as its constituent member as the set itself. In the physical nature of things we have to set a limit even to the measure of the greatest infinite set.
Russell's paradox is called the logic of the mathematical paradox because it arises from purely mathematical constructions. The barber's paradox may be called the linguistic or the semantical one. Russell's paradox was introduced to show that the obvious, direct axiomatisation of intuitive set theory is inconsistent. The set of all things automatically leads to an infinite set and perhaps to the greatest set. Could this set be a member of itself?
In the Jaina theory of Karma, sets are constructed which have real existence, otherwise the constructs are refuted. Similarly, whenever occasion arises to calculate terms, one gets terms beyond a limit which are avoided as inconsistent. Take for example, the set of the omniscience of all the accomplished souls. This set will have only one value and that will be the omniscience itself. This solves Russell's paradox. However, it was unfortunate for the creator of set theory, Georg Cantor, whose
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foundational edifice fell before him. Attempts to revise the foundation of mathematics were soon at hand, and various schools arose in Europe having a school of logistics, school of intuitionism, and a school of fornalism. O
REFERENCES
Heath, T, Greek History of Mathematics. Vol. 1. Oxford, 1921. Jowett, B, The Dialogues of Plato. Vol. 2. Oxford, 1953.. Hausdorff, F., Set Theory. New York., 1962. Jain, L.C. "Divergent Sequences Locating Transfinite Sets in the Trilokasara."
IJHS (14)1., 1979.
The Tao of Jaina Sciences. Delhi. (Cf. also projects as INSA), 1992. Wilder, R., The Foundations of Mathematics. New York., 1952.
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Jinamanñari, Volume 19, No. 1, April 1999
ASTI AND SYĀD IN JAINA THOUGHT
Dr. Mahendra Kumar Jain, Newark, DE. USA
Prescription of Five Precepts, which are known as anuvratas or mahāvratas, emphasize primarily on the personal conduct and social and societal behavior that is valid irrespective of the time and place. Remarkably the emphasis and the motive for such a conduct is neither based on divine insights nor on not-verifiable knowledge. The Jaina precept of ahimsa is an example of par excellence to cite, as it derives its force from an appreciation of deeper patterns of human behavior and aspirations articulated by ahimsa paramo dharma - “live and let live." It is not just about survival but far beyond the survival instinct. Interpretations of what is an acceptable level of violence remains a matter of compromises or adjustments in the moral and humane living. Debate about the rationale for such compromises or adjustments sets the intellectual tone for the Jaina doctrine of karma.
To establish asti - what is, only verifiable knowledge is paramount in Jainism. A prerequisite to have verifiable knowledge therefore is through truthful conduct, impressively implicated in the question of gañadharas to Lord Mahāvīra:
Q. Can ātmaswarupa - inner form be known from gvān or from agyān?
A. Certainly it is not possible without gyān, however, to see the whole it is also necessary to know agvān.
This insight underlines and highlights not only a useful guidance for personal conduct, but also triggers an impetus for the anekānta doctrine which entertains alternatives essential to reduce the level of agyān, that is what is not known. Logic syllogisms developed for this purpose formalize the thought process for arriving at a valid inference and reduce the level of doubt. This is both critical and important to develop good attitude, to good decision-making, and for evaluating liabilities of the knowledge base. Liabilities in the use of gyān for inference come from not knowing what is not known. This is where
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every inference is subject to ‘syād.' As formulated by Mallisena (c. 1290 C.E.) in his Srādvūda Manjari such doubt is intrinsic in any valid inference. While to know certain facts required to develop an (partial) understanding of what it is - āsti, it is a case of never know all the necessary facts to elaborate what it is not - nästi. Obviously the "whole,' alluded to in the answer of Mahāvīra, is the sum total of is and is not, in the sense of a system that is described by Set Theory.
Logical syllogisms are based on intrinsically rigorous mathematical methods that assure internal consistency and completeness. Sensory system with intrinsic limitations and the linearity of conventional language fragments knowledge, as with the conceptualization of a whole elephant by the knowledge of its body parts not only requires the knowledge of its morphology, habitat, social surrounding, and the course of its personal history and genealogy. More over it is always colored by personal experiences and not necessarily by a set of facts, rules and laws to organize the observable. Hence, the piecing together of the available fragmented information is postulated by the doctrine of anekantavada by Mahāvīra.
What is not known falls in the realm of agyān, and it is clearly distinguishable from the wrong, unverified and contradictory knowledge which could be categorized as mithyāgyān. Later Jain saint scholar philosophers such as Samantabhadra, Siddhasena, Mallavādi and Jinabhadragani articulated and elaborated that doubt is worthy of deeper intellectual inquiry with a true value in relation between mithyāgyān and doubt.
Like asti and năsti, syād (doubt) and contradictions have deeper intellectual roots that go back at least 3000 years. The Aristotelian inference syllogisms based on two states - true and false - have been developed in the form of Boolean Algebra, which is the foundation for modern computer science. In an attempt to explore the truth inherent in doubt, Prof. G.N. Ramachandran of the Institute of Science in Bangalore India has developed a second and third order Boolean Matrix Algebra (Mathematical Philosophy Report #79,1990). This approach applied to understand the deeper structure of syādvāda and saptabhangi syllogisms provides insights.
In the Indian tradition, the inference schema is illustrated by examples such as the following one:
There is fire on the hill (based on the fact that) There is smoke on the hill (the rationale is that)
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Wherever there is smoke there is fire (is it really true?) as in kitchen tatha there is fire on the hill (the assertions now hypothesis)
The purpose of this syllogism is to reduce the level of doubt. The past experience of the fire in a kitchen is used to infer a current unknown event. The initial assertion is rationalized by the connective tathat, which means 'based on the previous facts it follows.' It is obvious that without such a formal secular syllogism for arriving at a valid inference it would be impossible to coexist, communicate and develop democratic institutions including science.
The Greek syllogism attributed to Aristotle relies on a nearabsolute knowledge that 'all humans are mortals. Since 'Socrates is a human,' tathat 'Socrates is mortal.' A fact that is not often appreciated; and at least not overtly recognized by Greek syllogism is the acknowledgment of the liabilities intrinsic in the inference. i.e. inference is as good as the knowledge on which it is based. Similarly the possibility that there could be smoke without a fire is left open in Nyāya. And therefore it becomes the basis for the principle of syādvāda.
Syādvāda is not a figure of speech as meant in the common usage of the word 'perhaps.' nor it is a rhetorical devise. Similarly, the doubt intrinsic in the inference does not come from unverified knowledge, as:
- it cannot be entertained the possibility that what appears to be smoke could be a rain cloud or a dust storm;
- Contradictions where true or false coexist cannot be considered;
- Not concerned with what is ‘unknowable, or “cannot be fathomed.'
Such themes are dealt by mystics who might even surmise that “knowing nothing is the reason to doubt that one knows everything.' Similarly, syād does not arise from the momentary, probabilistic or average character of what 'it is,' or nor does it relate to fuzzy-logic. In order to identify the logical basis for the origin of doubt, formalisms to express doubt as uncertainty with statistical significance have been developed. Such approaches are of course useless for reconstructing reality from a set of events. Therefore syād is the deterministic statement about the doubt intrinsic in an inference based on two (or more) events. It represents the kind of doubt that is always present in any scientific statement and it is necessary for further inquiry to reduce the level of
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doubt. For example, if only one of the two women is pregnant.' the probability of cach being pregnant is 50%; however the assertion that each of them is 50% pregnant is meaningless
The Aristotelian binary logic of 'true' or 'false' (is or is not) has set foundations of Boolean Algebra, which forms the conceptual basis for all computer operations based on the binary states, 0 and 1, as intrinsic in the set theory. For a set of A and B, it can be said ‘not A is B' or 'not B is A.' This is because the universe of the A+B set is a closed universe:
i.e. if we know which one of the two women is pregnant, by implication we also know which one is not.
The usefulness of a syllogism comes from the implication. More situations develop under conditions where all the elements of a set are not defined, or their relationship is governed by nonsymmetry of implication. For example, if P implies Q, it does not necessarily mean implies P. Consider the verb 'implies' which is also used as a logical connective. For example, in the inference schema, if smoke always implicd fire, and if fire always implied smoke, it will be closed argument where fire is equivalent to smoke, and therefore there is no need for inference.. It will be self evident and redundant statement (tautology). On the other hand, consider the statement from a person who says "what I say is not true." Is it really true or false? This question has baffled Western logicians ever since Aristotle. But according to syād principle, it is obviously a contradictory statement and therefore not worth a discourse.
Genesis of doubt in an implication can best be appreciated by the following statement:
A or B implies C. If we know either A or B, we know C.
However, by knowing that C is true, one does not necessarily know A or B except for the fact that at least one of the two is true. If both A and C are true, B can be true or false resulting in the outcome of the doubtful inference. If true and false states are presented in notation, they may be symbolized in the following form:
True and false represent (1, 0) and (0, 1) respectively. Doubt and Contradiction represent (1, 1) and (0,0) respectively.
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Many such situations may be routinely encountered, and therefore additional and independent knowledge about the system is needed in order to resolve such doubts. Prof. Ramachandran as noted earlier has shown that this representation of doubt has numerous advantages and it can be put to use in computer algorithms. As such the evaluation of truth functionality of three statements about a system can be adequately represented by three bits - true, false and inexpressible - which predict seven states of truth. The eighth state which represents 'neither is, nor is not, nor inexpressible' (0,0,0) is a contradictory and therefore must be left out for the consideration of mystics.
The seven states of truth - the spatabhangi example from the Syadvāda Mañjari readily illustrates the physical significance of these syllogistic states:
syād it is
(1,0,0) ... it is not
(0,0,1) it is inexpressible (0,1,0) ... is, is not
(1,0,1) ... is, is inexpressible (1,1,0) ... is not, is inexpressible (0,1,1) ... is, is not, is inexpressible (1,1,1)
In short these secular thoughts represented by notation impressively explicit the Syādvāda and Saptabhangi principle being not only consistent with each other but they are extensions of binary Boolean algebra. Inference syllogisms of higher order are obviously widely accepted for intellectual discourse but their mathematical foundations are not established. Interestingly, these syllogisms have intellectual continuity that emphasizes attempts to develop a secular basis for the elaboration of the deeper forms and inner structures of thought and for the interpretation of content and physical significance. Prof. Ramachandran has shown that the four states resulting from two elements generate foundations for the prepositional logic, and it appears that the eight states resulting from three elements could provide a basis for the predicate logic.

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Jinamañjari, Volume 19, No. 1, April 1999
Mathematical Contents Of Jaina Texts Dr. L.C. Jain, Jabalpur, India Dr. Padmavathamma, Mysore University, India
Certain mathematical contents in Prakrit texts of the Jainas are explicated in regard to the view of symbolic mathematics employed in the Karnataka Vrtti (commentary) on the Gommaṭasāra and the Labdhisāra, as enumerated by Pandit Toḍaramala (c.1721-61 C.E.). This centralizes the karma theory in the Jaina School. The cosmological theory contains both astronomical and geographical mathematics with deeper in approach to the setting of the mathematical background of a model.
Jaina technical terms in Prakrit texts of the Karanaṇānuyoga or the Dravyanunyoga, import mathematical significance - philosophy tinged with mathematics. As Bertrand Russell has in his Introduction to Mathematical Philosophy, the Jaina mathematical philosophy appears in the texts for the first time. The Karṇāṭaka Vrtti of the Gommaṭasāra carries this mathematical philosophy in detailing the Jaina karma system in symbolic forms, arithmetical, algebraical and geometrical.1
Mahāvirācāraya,2 the author of the Gaṇitasāra Saṁgraha, did collect the mathematical material from the Jaina source material and he goes further in stating that whatever else is to be said may be seen in the Āgama. Sridharācāraya is still controversial, however, in the history of mathematics.3
The Parikarma, a commentary work of Kundakundācāraya and the Jyotiṣapatala of Mahaviracāraya are not yet available. Also, reported commentary works of celebrated Tumbulūrācāraya and Samantabhadrācāraya are not available. These could have traced and detailed the algebraic symbolism of the Karnataka Vrtti of Kesava Varņi. There are several problems in the history of mathematics and science regarding the source in India, as talents of scholars of Jainism and Prakrit have not been channeled towards at university level. Only recently, university of Bhopal in India has a bold step in this direction by offering a course in the department of religion and culture.
There are problems regarding the origination of and motivation of a paradigm shift in the terminology and usage of symbols. In the Prakrit texts, we find the logical and philosophico-mathematical terms

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as in other philosophies - but the mathematical manipulation through symbols is a peculiarity of the Digambara Jaina School. Similar achievement in southern India appears - perhaps round about the period of Kundakundācārya, when writing of the scripture was in full swing after the compilation of the Satkhaṇḍāgama and the Kasayapadhuda texts.5 Round about this period we find certain revolutionary events which speak of the mathematical talents of some genius.
Mathematical Terms, Symbols, and Events Of Early Common Era Without going into their controversial details, but leaving it up to the scholars to solve the problem of their source on the basis of an indispensable necessity, let us shall relate the events.
Zero in the place value system was needed by the Jaina School, We also find the place value system employed in the addition of the factor as well as in their subtraction - in the Karṇāṭaka Vrtti. Zero was adapted in the writing of the Mahabandha to fill up the gaps and so on. The Jain calendar records a precession in the Vedāṁga Jyotiṣa calendar during this period, and the establishment of Vikrama Samvat in India. Perhaps this was the era when various texts quoted by Virsenācārya were compiled for mathematical imports of the karma philosophy. For example, the Varangā sūtra, the Vedanākṣetravidhāna, the Khettaniogaddāra, the Pariyama, the Kālavihāno, and so on were some of the mathematical texts which could survive against the time.6
The cosmological texts including astronomy and geography, e.g., the Tiloyapaṇṇatti, the Suryaprajñapti, the Candapanṇatti, and so on, did not only depict calendrical details as the Vedamga Jyotisa but there was also a unified astronomical theory, set in a mathematical universe.7 When several processes are depicted through a single manipulation it becomes a unified theory which is regarded as simple. The Greeks split it through the epicycles for finer calculations. Einstein gave a unified theory. Now there is an attempt for a theory of everything (TOE) on physics. The Jaina School tried to give such a theory of everything for the biological phenomena through the mathematical theory of karma. The question is whether we should computerise such a theory and prepare files in the software to execute programmes as if happening even in astrology. What could be the results for the benefit of society or a nation? Astronomical programmes will be found to be simpler. Roger Billiard has already computerised the Yuga system of Indian astronomy leaving the Prakrit version of a calendar. Success in building up various programmes in the karma theory will depend on how we are able to form states, inputs and outputs from the mathematical data
1

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furnished in the commentaries of the Gommaṭasāra and the Labdhisāra through C++ language.
Let us have a look at the mathematical material in these texts which could be helpful in computerisation. The simile and number measures (upamā and samkhyā pramāna) are finite and transfinite cardinals and ordinals of various types of sets (rāsis). There are fourteen types of sequences (dhārās) in the Trilokasara" which locate several types of sets and their measures. Every topic in the karma theory deals with the minimum (jaghanya) and the maximum (utkrstṛṣṭa) fixing the domains and ranges between which the computer is to work. The eight operations called the parikarmāṣṭaka not only deal with the finite quantities but also transfinite quantities as well as the fixed and variable sets. The trikona yantra (triangular matrix) can be given several programmes for the variability of the measures of the mass number (pradesas), configurations (prakrtis), energy levels (anubhāgas) and the lifetimes (sthitis) of the karma ultimate particles (paramāņus). The Labdhisara depicts these variations in a symbolic way of mathematics. 10 The equations and inequality relations given in this way may pave the way to a more complex manipulation of the problems posed in the modern set-up of the karmic data.
Before we give the measurable terms it will not be out of place to suggest that the vast mathematical data could be arranged in a computer file in a graded manner. From the lowest value we go to the largest value in a certain programme and these could be coded in one of the computer's high level languages such as Fortran, Mathematica or C++. The controllable and observable situations are defined in terms of the control (guna) and reachable (mārgaṇā) stations (sthānas). Thus the computer could be helpful in showing the time-dependent and timeindependent phenomenology of the karma theory of the Prakrit texts referred to above.
We now relate only the terms of the Prakrit texts which denote a measure which could be calculated to give a rough or fine gradation or topology. One should note that a variable measure is given in an algebraic way, set theoretic in approach. Its measure is therefore given between its minimum and maximum values and talents are required to approach a proper and suitable value. This could be approximate also as is found in several places of the Labdhisara or the Suryaprajñapti, Tiloyapanṇatti or the texts of the Gaṇitanuyoga. Surely, this is based on probability.
Datta had collected some terms and he tried to give their interpretation admitting that his attempt was premature. 11 The Dharvalā

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texts and the Karṇāṭaka Vṛtti were not before him. Yet one has to delve deeply into the theory also for showing the origination and motivation of the terminology. He dealt with their definitions and their historical importance. Dr. A.N. Singh also attempted the same thing while he contributed articles on the mathematics of the Dharvalā and other texts. 12 However, a study of the heirarchy of various topics is needed for fruitful results and recognition.
In this brief article we give certain mathematical terms out of which the asterisk marked will be those whose measure could be rather ascertained (translated into Hindi from their Prakrit version): Parikarma (eight operations -- pratutpanna, bhāgahāra, varga, vargamula, ghana, ghanmūla, sankalita and vyutkalita); Rāšis (set-synonyms like samūha, pinda, puñja, sampāta, abhinna, etc.); Jīva rāśi*; Ajīva rāsi*; salākā (counting rod); samkhyāta*; asaṁkhyāta*; Ananta*; Angula (finger)*; Jagasreni (world line)*; Loka*; Palya*; Sāgara*; Muhūrta*; Antarmuhurta*; Samaya*; Pradeśa*; Varga*; Vargaṇā*; Spardhaka*; Guṇahāni*; Nāṇā Gunahāni*; Anyonyabhyasta rāsi*: Samayaprabaddha*; Pudgala paramānu rāśi*; Ākāśa pradeśa rāśi*; Kāla samaya rāsi*; Kevalajñāna rāśi*; Kalpa*; Avali*; Rajju*; Yojana*; Kalāsavarna; Yāvat-tāvat; Addhā; Uddhāra; Vyavahāra; Ardhaccheda; Trikaccheda; Vargasalākā; Vargita-saṁvargita; Vikalpa; Bhamga; Samdrsti; Gaṇana Sthanā; Oja and Yugma rasis; Guṇaśareņi; Sarvadhana; Gaccha; Mukha; Madhyadhana; Ādidhana; Uttaradhana; Dhārās; Alpabahutva; Utsedha; Dhanuṣa; Bāṇa; Viṣkambha; Kṣetraphala; and so on.
13
Various terms of the Labdhisara have been defined by Todaramala in the Arthasamdṛṣṭi Adhikāra of his Samyakjñāna Candrikā commentary. These terms like the Apakarṣaṇa, Utkarṣaṇa, etc., give operational details in the theory of karma.
The relations between various entities have been given through several formulae, both in the karma theory and the cosmological theory. These formulae can be seen in a collected form in the project work on the Labdhisära assigned by the Indian National Science Academy.14 For astronomical formulae, some may be seen in Jaina Astronomy, published doctoral thesis of Lishk. 15 For cosmological formulae, one can see the "Mathematics of the Tiloyapanṇattī." "16
Concluding Remarks
The appearance of the Ganitasara Samgraha of Mahāvīrācarya in 1912 gave the first indication of the Jaina School of Mathematics in South India. It was a full book on practical mathematics. He was the

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first mathematician in the world to recognise the imaginary qualities. Most of his formulae may be seen in other forms in the Digambara Jaina texts on the karma theory. Formulae given in the commentary of the Süryaprajñupti deserve special attention. The mathematics of the Medieval period may also be seen in the works on astronomy and astrology which still await Hindi and English translation. These may be found in the Digambara and Śvetāmbara Grantha Bhaņdāra. I
REFERENCES AND ENDNOTES
1. Gommatasāra of Namicandra Siddhāntacakavartī, Vols. 1-4, Bhartiyajñana Pitha, New Delhi, 1978-81. 2. The Ganitasāra saņgraha of Mahāvīrācarya, ed. and trans. by L.C. Jain, Sholapur, 1963. 3. A recent short article in the Ganita Bhārati, Vol. 9 (1987), numbers 1-4, p. 54-56, by Ganitanand, Ranchi, has appeared on the date of Śrīdhara. His remarks are worth mentioning here. S.B. Dixit (1896) had found a reference to Srīdhara by name in an old manuscriopt of Mahāvīra's Gañitasāra samgraha (ca. 85), and so put the former before the latter, Royal Asiatic Society, Bombay Rs. 230 of GSS also ends with the words (ABORI? Vol. 31, p. 268).
The similarity of several rules and of many other features between the works of Sridhara and Mahāvīra is accepted by scholars. Both may have drawn from a third and common source which is not known or likely to be known. But most of the scholars considered Mahāvīra as a borrower. (He himself named his work as a "collection:.) The date circa 799 C.E. was assigned to Sridhara by N.C. Jain, by equating him to the Jaina author of Joytirjñānavidhi (799). To reconcile the salutations "Sivam" and "Jinam" of the different manuscripts, it has been suggested that the same Sridhara, after writing mathematical works, may have become a Jaina toward the end of his life.
The above note also gives the opinion of B. Dutta and A.N. Singh that 799 C.E. is the probable date of Sridhara. It appears that the common source material for both of the above mathematicians has been the Kasāyapāhuda and the Satkhandāgama and their commentaries which might have been be them. As the Medieval Jaina writers had been writing Jina and Siva for the same deity, some scribe might have had it changed under certain unknown circumstances. It does not seem possible that Sridhara could have availed the opportunity of the Jaina source material as a non-Jaina, and he must have compiled the work as a Jaina. It also seems possible that under certain circumstances he might have adopted Saivism but whether he wrote two such manuscripts after his conversion is doubtful. Thus, looking into the needs of the Digambara Jaina School of Mathematics in South India for their theory of karma, it seems now most probable that both took help from the same source material of the south, and both were Jainas in the Digambara Jaina Schools of Mathematics. For this purpose of convincing argument one may see the project
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work on the Labdhisăra of Namicandra Siddhantacakravartī, Indian National Science Academy, 1984-87, by L.C. Jain.
4. Mention has been made by N.C. Jain while he was at Arrah Jaina Siddhanta Bhavana, and this manuscript is not available now.
5. These texts are in several volumes and have gone out of print. New editions of the former are now coming out in the press. Satkhaṇḍāgama of Acārya Puspadanta and Bhūtabali, Books 1-16, Amaroati, Vidisha, 1939-1959. Cf. also Kasaya Pahūda of Gunabhadracārya, along with the Jayadhavalā commentary of Virsenācārya and Jinasenācārya, Vols. 1-13, and the following Mathura, 1944-....
6. For the texts of the Svetambara Jaina School, cf. the exhaustive article, The Jaina School of Mathematics, by B.B. Dutta, Bul. Cal. Math. Soc., Vol. xxi, No. 2, 1929, pp 115-145.
7. For details, see the "Jaina Astronomy" by Dr. S.S. Lishk, (1978), Doctoral Thesis approved by the (Patiala) Punjabi University, 1987, Vidyasagara Publications, Delhi. Cf. also, Jain, L.C., "On the Spiro-Elliptic Motion of the Sun Implicit in the Tiloyapanṇatti", IJHS, Vol. 13, No. 1, 1978, pp. 42-49.
8. Jain, L.C., System Theory in the Jaina School of Mathematics, IJHS, Vol. 14, No. 1, 1979, pp. 29-63.
9. The Trilokasara of Nemicandra Siddhantacakravarti, Sri Mahāvīrji, 1976. Cf. also Jain, L.C., Divergent Sequences Locating Transfinite Sets in Trilokasāra, IJHS, Vol. 12, No. 1, 1977, pp. 57-75.
10. Cf. the project referred to in 4.
11. Cf. the ref. 8.
12. Singh, A.N., Mathematics of Dhavalā, Ṣaṭkhaṇḍāgama, book 4 loc., cit., Amaraoti, 1942, pp. i-xxiv. Datta, B.B., and Singh, A.N., History of Hindu Mathematics, Bombay, 1962.
13. Cf. ref, 4 for details.
14. Cf. Jain, G.R., Cosmology Old and New, Gwalior, 1942.
15. Cf. ref.9.
16. Cf. Jain, L.C., Tiloyapanṇatti ka Ganita, an introduction to the Jambudiva pannatti Samgaho, Sholapur, 1958, pp. 1-109.

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Jinamañjari, Volume 19, No. 1, April 1999
THE GANITASĀRA OF THAKKURA PHERU
Dr. S. Rajeswara Sarma, Aligarh University, India
As John Scott Deyell points out in his Ph. D. dissertation, Living Without Silver: The Monetary History of Early Medieval India submitted to the University of Wisconsin-Madison 1982, the century rulers of Delhi had employed bankers in the minting operations. It is well known that during the period much of the economic activity in the Gujrat-Rajasthan-Delhi region was controlled by the Jains. The members of the Srīmāli caste of the Jainas were specialized in minting and money exchange. Among these members of the Srimālis, Thakkura Pheru stands out as a writer on a wide range of scientific subjects in popular speech. He wrote six scientific works:
1. Vāstusāra on architecture and iconography, 19 September, 1315. 2. Jyotişāra on astrology and astronomy, 1315. 3. Ratnapariksa on gemology, 1315. 4. Gamitsara on arithmetic, 1318. 5. Dhātutpatti on metallurgy and perfumery trade, and 6. Dravyaparīkṣa on assay and money exchange, 1318.
All the works are published in the Thakkura-Pheru-VoiracitaRatnaparikṣādi-Saptagsantasamgrha, ed. Munivijaya, Jodhpur 1961.
About the Author Pheru was born sometime around 1270 C.E. at a place called Kannana which is situated in the modern state of Harayana. Kannana was not only far from the then capital of Delhi but it was the a Jain center of pilgrimage for the Jains. Pheru's father Canda was a prosperous banker, and he had the court title of Thakkura, which suggests that he may have been associated with the treasury at Delhi. Pheru's grandfather Kaliya or Kalasa, although a prosperous banker, did not possess the Thakkura title. The family belonged to the Jaina Kharatara gaccha.
Pheru appears to have presumably completed his formal education in 1291 C.E. and during this year he composed the Karataragaccha-Yugapradhāna-Catupādika, an eulogy to his Pontiffs.
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Later, he joined the treasury of Alauddin Khiji at Delhi, and wrote the Ratnaparikṣā, a manual on gemology, for the instruction of his son Hemapāla in 1315 C.E. In the same year, he wrote the Jyotişāra and the Vāstusāra. During the time of Qutubddin Mubarak Shah, he wrote the Dravyapariksa in 1318 C.E. He held the position(s) successfully during the successive Sultāns - Alauddin (1296-1316), Shihabuddin Umar (1316), Qutubddin (1316-1320) and Ghiyasuddin Tughluq (13201325).
The Work Ganitasāra The text is not dated but must have been written before 1318 C.E. It is more innovative and not so much in the theoretical portions but in the application of arithmetical rules to a wide range of areas. It is common place to say that arithmetic is one of the most practical sciences, its rules being employed by traders, masons, carpenters, tax-collectors and the like for the calculations connected with their professions. The units of measurement and the examples to illustrate arithmetical rules given in the text throw a flood of light on the economic and social conditions of the period.
In the section of solid geometry, Pheru gives the rules for the volumes of domes (gonamța), square and circular towers with spiral stairways in the middle (pāyaseva), towers with fluted columns (munāraya), niches (tāka), staircases (sopāna), bridges (pulabamdha) and so on (III.74.86). The purpose of such rules is to enable the chief mason to calculate the number of bricks or stones needed for the these constructions. Accordingly, Pheru points that to do this calculation more exactly one should first calculate the total volume of the wall-space, subtract from this the volume of the space occupied by the doors and windows and reduce the remainder by the three-twentieths, the latter being the volume of the mortar (III. 70-71). The result when divide by the volume of a single brick yields the number of bricks.
The munāraya is like a circular tower with a spiral stairway in the middle, as far as the inside is concerned. But the difference is the wall contains half triangles and half circles (III.80). The meaning of the cryptic last sentence is that in a horizontal cross-section of the munāraya, the outer circumference consists of alternate triangles and semi-circles. It should be remembered that about a hundred years before this time, Qutubddin Aibak built the Qutab Minar in Delhi and that Alauddin himself started constructing another tower twice as high. Now, the lower story of the Qutab Minar consists of alternately angular and circular

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columns, and it is clear that Pheru is referring here to such a tower with fluted columns.
In another section dealing with cloth (IV.i.18-37). Pheru mentions different kinds of silk, woolen and cotton materials, the rate of shrinkage or loss in washing, cutting and sewing, and the area of cloth needed to make various types of tents. There is a last section (IV.1.1-17) listing the average yields of grains, pulses, etc. per bighā, the average yield of mollasses and brown sugar per maund of sugarcane, the amount of clarified butter that can be obtained from cow's and buffalo's milk and so on. His rule of converting Vikrama dates into Hirji dates and vice versa (IV.1.17) which is probably the first such rule to be formulated in India. Though all these are not germane to arithmetic as such but Pheru is adapting arithmetic to suit the needs of a variety of professions. O
This is a condensed part of the paper “Thakkura Pheru and the Popularization of Science in India” in Jainthology, Ed. Ganesh Lalwani, Jain Bhavwan, Calcutta 1991.
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Jinamañjari, Volume 19, No.1, April 1999
A Glimpse at the Geometry of Certain Figures Dr. V. Mishra , Sant Longowal Institute Longowal-148106 India Dr. S.L. Singh, Gurukula Kangri Uni.versity, Hardwar-249404 India
There was a noticeable achievement in scientific and quasiscientific compositions during the period of ninth through fifteenth centuries C.E. The paper discusses geometry figures of polygon, ellipse, circular segment and spherical segment presented during this period.
The Polygon 1.1 The Līlāvati (LV) of Bhāskara II (b.1114 C.E.) (1, Part II, pp.206208), see also [5]) prescribes the following rule for finding the length of a side of a regular polygon inscribed in a circle:
त्रिद्वयड्काग्निनथश्चन्द्रैः त्रिबाणष्टयुगाथिः। वेदाग्निबाणखाश्वैश्य खखाथाथरसेः क्रमात् ।।धएघ।। बाणेषुनखबाणोश्च द्विद्विनन्देषुसागरैः।। कुरामदशवेदैश्च वृतव्यासे समाहते ।।धएछ।। खखखाथार्क संथके लथ्यन्ते क्रमशो थुजाः। वृतान्तरलपूर्वाणाँ नवारलान्त पृथक्पृथक्।।धएष।।
(Kșetravyavahāra of LV)
This can be read to mean (cf.[5]: Multiply the diameter of the (given) circle, in order, by (the coefficients) 103923, 84853, 70534, 60000, 52055, 45922, and 41031. On dividing (each of the products just obtained) by 120000, there are obtained the sides respectively of the (equilateral) triangle to the (regular) nonagon (inscribed in the circle) separately.
That is, the length in of a side of regular n-gon is obtained by the formula:
(1.11) In = (1/120000) kn; (d being the diameter of a circle) where seven coefficients kn (3Page #30

Gupta [5] infers that coefficients kn surely indicate the sides of regular polygons inscribed in a circle of a radius 60000 as kn= ln for d 120000.
Commentator Gaṇeśa (ca. 1545 C.E.) (cf.[5]), however, for the first time tries to give the rationale of kn by two methods. The first method, based on the table of sines, is far from being satisfactory. The second method which employs the (so called) Pythagoras theorem, discusses only for n = 3,4,6,8, but refutes that the same technique cannot be applied for other cases. From Figure 1 (see below),
(1.12) Ind sin π/n; (d =2R).
Comparing (1.11) and (1.12),
(1.1.3) kn=120000 sin л /n.
Comparison of modern values of kn's (3 ≤ n ≤9) with those of LV and Kriyākramakarī (= KKK) (see[5]) is given in the following chart:
Modern Calculation
LV
F MYO
kn
k3
K4
k5
k6
k7
k8
k9
103923
84853
70534
60000
52066
45922
41042
103923
84853
70534
60000
52055
45922
41031
KKK
1.2 For the area of a regular polygon, Utpala (ca. 10th century C.E.) in his commentary [3] (see also [9]) on Varahamihira's Bṛhatsaṁhitā gives the following rule:

103922 (103923)
84853
70534
60000
52067
45922
41043
द्विर्न्यस्य परिधेर्वर्गमेकस्मादश्रिजार्धितात ।
लब्धँ सँशोध्य परतो थक्त्वा द्वादशथिः फलम् । 62.73||
This means: Put the square of the perimeter (paridhi) at two places. At one place divide by (square of) half the number of sides. Subtract the quotient (so obtained) from the other (square of the perimeter and (then) divide by twelve. The area (phalam) is:
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An=[p2-p2/(n/2)21/12
or
An = ln (n2 - 4)/12; (p = ln)
Whereas the Ganitasarasangraha (= GSS) of Mahāvīra (ca. 850 C.E.) regarding the area of a general plane polygon describes the relation ([15]), see also [10]) in the following:
रज्जवर्धकृति त्र्शो बाहुविथक्तो निरेकबाहुगुणः । पामश्रवत फल हि विम्बान्तरे चतुर्थांशः||V||.39||
This can be translated as
(1.22) An = (s2/3) (n - 1)/n
wherein s = (sum of the sides' length)/2.
For a regular polygon, s = (nl)/2, and therefore from (1.22) [12]
(1.23) An = ln2(n - 1)n/12
The same rule (1.23) in equivalent form is also found in Ganitakaumudī (GK) (c.1356 C.E.) of Nārāyaṇa Pandita (cd.[10]).
We know from the modern analysis that the exact area (which we shall denote byAon) of a regular polygon is governed by (referring figure 1)
Aon = nlnr/2, i.e.
(1.24) Aon = (n/4)12n cot л /n

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Figure 1. Regular Polygon of n-sides
The degree of accuracy of An given by (1.21) may be accessed by calculating the percentage error En as:
1 An En = [ Aon - 1] x 100 = [
n2 - 4 3ntan (/n - 1] x 100.
Table 1A show the percentage error if one uses the formula (1.21) for different values of n.
Table 1A
E3 = 3.77 E4 = 0 E5 = 1.72 E6 = 2.64
E10 = 3.97 E20 = 4.53 E30 = 4.64 E00 = 4.72
The applicability of An given by (1.23) may also be estimated by computing the percentage of error En below (see Table 1B)
En = [
n-1
3
tan (Tu/n - 1] x 100.
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Table 1B
E3 = 15.47 E4=0 Es=3.13
E6=3.77
Conclusion
We can infer that the results (1.21) and (1.23) give the exact area in the case of a regular 4-gon (i.e., a square) only and approximate ones for all other regular polygons. Moreover, the amount of inexactness is the same for both results in case the sides are large enough. We remark that (1.23) is fairly applicable when the number of sides is near about 20.
E10= -2.52 E20 = 0.31 E30 = 1.60 E00 = 4.72
The Ellipse
Datta and Singh ([2], refer also Gupta [11]) find the word viṣamacakaravāla (unequal circle) applied for ellipse in the Jain text Suryaprajñāpti (ca. 500 B.C.E.), and Gupta [op.cit.] as parimandala in both Buddhist text the Dhammasangani (ca. 400 B.C.E.) and the Jaina text the Bhagvatīsūtra (ca. 350 B.C.E.). Mahāvīra in his GSS uses the term ayatavṛtta (elongated circle) for the same [11]. For an accurate (sūkṣma) values of perimeter P and area A of an ellipse, the GSS states the following rule ([15]], see also [4]):
anrightausymar fæerjunen4gpfayar uå ukfur: 1 व्यास चतुथगिगुणश्चायतवृत्तस्य सूक्ष्मफलम् | | ||.63||
This means (cf. [4]): (The square-root of) the sum of six times the square of the breadth and the square of double the length is the perimeter. (That perimeter) multiplied by a fourth-part of the breadth is the accurate area of the elongated circle.
This is the equivalent to
(2.1)
(2.2)
where a and b are semi-major and semi-minor axes respectively. Using the modern, the exact area is found to be
P = 2 (4a2 = 6b2)1/2
A = b(4a2 = 6b2)1/2

31

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(2.3)
40 = t ab
and the exact perimeter
P.- 41m2 (a’ sin’o + b2 cos?d}"2 do or
1
-
e? Ā
1 32 5?...... (2n - 3) 2 22 42 6? . .(2n)
12 (2n-1ean
(2.4) Po=2
Figure 2: Ellipse
~
acost, tsing)
---ofã--
*
FER(-e)
For very fine approximations of perimeter of ellipse by Ramanujan, refer [11]. The extent to which the perimeter given by (2.1) is valid may very well by judged by computing its percentage of error Ee (see Table 2A):
E, - 18 - 1] x 100 = hotelov . ]x 100 where k = 1-*-- so - 17300 - DE> 10 cm 114 100 ke 1-
32

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Table 2A
E0.1 >0.36 E0.2 > 0.48 E0.3 > -0.10 E0.4 > 1.44 E0.5 > -0.21
E0.6>-0.97 E0.7 > -2.78 E0.8 > -3.66 E0.9 > -6.28
The degree of accuracy of an area specified by (2.2) may also be easily extracted from its percentage error Ee calculated below (see Table 2B)
E. - [À 1] *100 = [V10-Com - 1] x 100
Table 2B
E0.1 = 0.36 E0.2 = 0.56 E0.3 = 2.10 E0.4 = -4.29 E0.5 = -7.20
E0.6 = -10.87 E0.7 = -15.42 E0.8 = -21.0 E0.9 = -27.83
It may be concluded that it is remarkable that the formula (2.1) is applicable for e near 0.1-0.3, 0.5, 0.6 and (2.2) for e in the range 0.1 and 0.2 including vicinities.
The Circular Segment 3.1 Consider a circular arc AB of radius r subtending an angle 20 at the centre. See Figure 3.
The formula (3.11) s = [12 + (1x2 - 4)h2]1/2 with t = 3 is found in the GSS as a practical rule ([6] and [8]). Most Jaina works including the GSS use this formula explicitly with x= V10
33

Page #36

Figure 3: Circular Arc
AL"
TN 2 -->
The validity estimation of (3.11) may be easily inferred from its percentage error Ee (refer Table 3.1):
5-16-14 100 - [1si" 8 * 6.2016 cover * - 12 100
wherein 1 =2r sinõ, h = r(1 -coső), so = 2rð.
Table 3.1
Ep = 0 E15 = 0.11 E30 = 0.40 E45 =0.73 E60 = 0.92 E75 = 0.47 E90 = 0
E105 =-1.50 E120 = - 3.89 E135 = - 7.25 E150 =-11.57 E165 = -16.82 E180 = -22.88
90°. It is worth
Evidently (3.11) is technically applicable of 0° << noticing that this formula is exact when Ő is 90°.
3.2 Tiloyapannatti of Yativrşabha (fl. 473-609 C.E.) uses the following formula [8]:
(3.21) A = [10(1h/4)2]1/2
34

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to find the area of various regions of Jambūdvīpa (Jambu Island). These regions are of the circular segment form.
The formula (3.21) in its general form may be considered as
(3.22) A = tlh/4 = tr?sin) (1 - COSÕV2. However, the true area of Ao is given by the following:
A0= Area of the sector OAMB - Area of the triangle OAB. That is (3.23) Ap=12 (Ó - sinò coső). The percentage of error Ee will furnish the degree of accuracy of the area given by (3.22) (see Table 3.2A):
5 - CÃ - 1] x 100 = [e in
11 x 100 =
asino(1 – cos) 2e - sin cos
mo
- 1] x 100
- 1 x 100
Table 3.2A
Eo = 0 E15 = 17.80 E30 = 16.11 E45 = 13.98 E60 = 10.74 E75 = 6.19 E90 = 0
E105 = -8.29 E120 = - 19.26 E135 = - 13.61 E150 = -51.96 E165 = -80.53 E180 =. 100.00
Notice that he following formula for the area of a circular segment
(3.24)
A = a (1 + h)h/6
appears in various texts, viz., the Ganitapañcavissī, the Trisstikā of Śrīdhara (ca. 8th century C.E.), the GSS, the Mahāsiddhānta of Āryabhața II (ca. 950 C.E.), the Ganitasāra of Thakkura Pherū (ca. 1315 C.E.) and the Pañcaviñsatikā (ca. 1428 C.E.) (see [13]). In polar form, the formula (3.24) may be written as
(3.35) A = 1 r2 (2sino + 1 - COSÕ) (1 - COSÕ}/6,
35

Page #38

and that the corresponding percentage error is given by the following formula (see Table 3.2B)
r
(2 sino + 1- cos )(1-сoso) - 11 x 100
• 600-sin cose)
Table 3.2B
Eo = 0 E15 = - 16.60 E30 = - 12.18 E45 = -8.27 E60 = -4.86 E75 = - 2.04 E90 = 0
E 105 = 0.98 E120 = 0.44 E135 = - 2.32 E150 = -8.22 E165 = - 18.32 E180 = - 33.33
It may be conclusion that the formulae (3.22) and (3.24) furnish the true area only when = 90°. The formula (3.22) is practically applicable when ő is near approximately 0° and 90° whereas (3.24) is technically suitable in and near about 0° <Ó < 120°.
3.3 The Bșhatkşetrasamāsa (cf.[7]) of Jinabhadra Gani (6th century C.E.) uses the following formulae to find the area of the zone of the circle (see Figure 4 also).
Figure 4: Zone of the Circle
DID
(3.31) A1 = Y; (c1 +c2)b.
Clearly it represents the area of the inscribed isosceles trapezium ABCD and is less than the true area and so it can be treated as a rough approximation.
36

Page #39

(3.32) A2 = [/2 (cy2 + c2212b.
The area A2 is less than the circumscribed isosceles trapezium Albicldl. for a plausible derivation of (3.32), refer [7].
The Spherical Segment
The GSS ([14], see also [8]) enunciates the following rule to find the surface area of a spherical segment:
Rugat agerft falu RPJUT: Hlaic
A4 A4 ||||:25||
This means (cf. [8]): Know that one fourth of the circumference multiplied by viskambha gives the area (of the concave and convex surfaces). According to Gupta [8], this can be interpreted as the following:
A=(p/4) (curvilinear breadth) = a ds/4 or (4.1) A=tr2 sino using p = id, d2r sin) and s = 2ro. However, the true surface area is (4.2) Ap=26 th = 21 r2 (1 - Coso) wherein h=r(1 - CosO).
Figure 5: Spherical Segment
tha
to dama
The degree of validity of the surface area given by (4.1) may be estimated just by finding the percentage of error Eo (refer Table 4):
sino
- 1
x 100
50 - [- 1] + 10 = (2, - eston - 1] x 100
-
1
x 100 =
OSA

For Private & Person37Use Only

Page #40

Table 4
Eo = 0
E105 = - 29.69 E15 = -0.44 E120 = - 39.54 E30 = - 2.30
E135 = -51.20 E45 = -5.19
E150= - 56.37 E60 = -9.31
E165 = -81.04 E75 = - 14.71
E180 = - 100.00 E90 = -21.46 Notice that (4.1) is useful for > up to and near 150.0
REFERENCES
1. Apte, D.V. (ed.), Līlavati with the Commentaries of Ganesa and Mahidhara: Two Parts. Poona, 1937. 2. Datta, B. and A.N. Singh., "Hindu Geometry." Indian Journal of the History of Science(15), p. 124, 1980. 3. Dvivedi, S. (ed.), Brahatsamhita of Varahamihira with the Commentary of Utpala, Part II. Varanasi, 1968 [1897). 4. Gupta, R.C., "Mahavīracarya on the Perimeter and Area of an Ellipse." Math. Edu. Siwan) 8, Sec. B, pp. 17-19, 1974
"The Līlavati Rule for Computing Sides of Regular Polygons. "Math. Edu. (Siwan) 9, Sec. B, pp. 25-29, 1975. "Jaina Formulas for the Arc of a Circular Segment." Jaina Journal (January), pp. 89-94, 1979. "Jinabhadra Gani and Segment of a Circle Between the Parallel Chords." Ganita Bharati 7, pp. 25-26, 1985. "On Some Rules Form Jain Mathematics." Ganita Bharati 11, pp. 18-26, 1989. "An Ancient Approximate Rule for the Area of a Polygon." Ganita Bharati 12, pp. 108-112, 1990. "Mahavīracarya's Rule for the Area of a Plane Polygon." Arhat Vacana 4. pp. 45-54., 1992. 1993. "Rectification of Ellipse from Mahavira to Ramanujan." Ganita Bharati 15, pp. 14-40. "Areas of Regular Polygons in Ancient and Medieval Times." Ganita
Bharati 16, pp. 61-65, 1994. 5. Hayashi, T, "Srīdhara Authorship of the Mathematical Treatise
Ganitapancavimsī." Historia Scientiarum, Vol.3-4, pp. 233-250, 1995. 6. Jain, L.C. (ed. and trans.), Ganitasarasangraha (in Hindi). Sholapur, 1963.. 7. Rangacharya, M. (ed. and trans.), Ganitasarasangraha of Mahavīracarya. Madras., 1912.
38

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Jinamñnjari, Volume 19, No.1, April 1999
Zero In Place Value System of Jain Mathematics
Dr. L.C. Jain, Professor Emeritus, Jabalpur, India Dr. Padmavathamma, University of Mysore, India
In view of the knowledge derived from Babylonian mathematical cuneiform texts by Neugebauer, Sachs and others, and the claims of Needham and Ling in the "Science and Civilisation in China," various options have been set forth regarding the use of zero and its adoption in the place value system. In India, such volumes as the Dhavalā, the Jayadhavalā, the Fīvatattvapradipikhā Karnātaka and other commentaries as well as the Tiloyapannatti have brought forth rich mathematico-symbolic material. The karma theory points towards the necessity of a place value system in the Purvagata texts - parts of the Agrāyani and the Jñanapravāda - current in the Digambara tradition. They are required to be studied for further opinions and investigation regarding origin of the zero and its use in place value systems. The model for astronomy in India had to do with the geometry of a circle and a straight line and their extension to cosmological details. The study of Jaina karma system as cybernetics in an abstract form is a unified theory of bios and matter with a mutual dynamic interactions, and the Jaina School of algebra and arithmetic was the outcome of the unique experiences and examinations thereof.
Recent INSA project study on the Jaina School of Mathematics has clearly identified two schools: the Digambara and the Svetāmbara. The former held proficiency in the mathematico-symbolic theory of karma whereas the latter seems to tend towards the preservation of religious tenets, astronomy and astrology. The paper highlights this fact that more symbolic details on the use of zero are available in the Digambara School than in the Svetāmbara tradition, contarary to the findings of Datta, Singh and Kapadia.
Introduction
The study of the mathematical cuneiform texts of the old Babylonian period (1600 B. C. E.) has shown their system of numerationis to be based on place-value notation, a sezagesimal scale. Neugebauer opines that it was transmitted to the Greeks and then to the Indians who contributed to the final step by using the place-value
39

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notation for the smaller decimal units." He also finds that the Arabic form for zero symbol (a little circle with a bar over it and related forms) is simply taken from the Greek astronomical manuscripts, as recognised by F. Woepoke in 1863.2 However, he felt that no definite answer could be given to the question when the zero sign was introduced in Babylonian mathematics, which did not exist before 1500 B.C.E., but was in full use from 300 B.C. E. onwards.3
Needham and Ling find that in the far east, the Shang Oracle bone numeral forms (14001100 B.C.E.) and the method of writing numbers with them were based on the decimal place-value ideas continued in the rod numerals. This suggests the possibility of the discovery of zero in south-east Asia where Indian culture "met the southern zone of the culture of the Chinese." They also opine that ancient systems point to a date well before +500 for the development of place-value and the zero concept, it was never entirely convincing due to uncertain chronology of Indian history and the difficulty of dating literary and epigraphical evidence. Apart from this, there has been an emphasis on the "emptiness" of Tao mysticism and the "void" of Indian philosophy whicg are said to have contributed to the invention of a symbol for sünya (zero). To this, the Jaina annihilation of a karma existential matrix -- perpetuating transmigrations that lead to a zero or nil existence of the world culminating in an eternal life of bliss and unending power -- may be added. In this regard, the role of zero and decimal place-value system may be presented from the recentlypublished Digambara Jaina texts like the Tiloyapannatti, the Chakkaņdāgama,8 the Kasāyapāhuda,9 and the Mahābandha10 of Bhūtabali. Their summary texts by Nemicandracārya in his works - the Trilokasāra, the Gommațasāra, the Libdhisāra, 11 and the Kșapaņasāra along with its commentaries by Mādhvacandra Traividya, Keśavavarņi and Țodaramala may also be referred when required. These texts give an elaborate exposition of a systematic development of mathematical models for geography, astronomy, cosmogony, cosmology and karma theory; as well display the need for systems to express large numbers and set-theoretic manoeuvre.
The INSA research projects -- "Labhdisāra of Nemicandra Siddhānta Cakravarti" (1984-87) and "Mathematical Contents in the Digambara Jaina texts of the Karaņānuyoga Group (1992-96) conducted at Indian National Science Academy have shed more light on the use of zero in the form of a circular symbol and its application in he place-value system of the Digambara Jaina School of Mathematics. 12 Round about 1935, some stray material on zero and its use in the decimal
40

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place-value system were located in the form of big numbers from Śvetambara texts by Datta, 13 Singh 14 and Kapadia. 15 Word-symbol notation was also used by Jinabhadragani.
So far as the Digambara Jaina texts are concerned, Datta and Kapida found the zero and its use in decimal place in greater strength in the Gommatasära of Nemicandra Siddhanta Cakravarti. The Ganitasārasangraha of Mahāvirācārya (9th century C.E.) had already been published in 1912 with its use of zero not only in place value but also in Parikarma. A few years later, Singh took up the challenging task of exploring the mathematical contents of the third and fourth volumes of the Dhavala Commentary on the Satkhandāgama.16 This consisted of the use of zero in a variety of expressions in decimal place value.17 Later, L.C. Jain located similar material in the Tiloyapannatti (c. 5th century Prakrit text on the information about the three worlds or universes).18 He found additional material on the same topic in the ASG and ASL chapters in the Samyak Jñana Candrika - in the Dhumdhari dialect of Rajasthan - a detailed commentary on the Gommațasāra and the Labdhisāra, by Țodaramalla. 19
Use of Zero in the Digambara Jaina School
In the above mentioned texts and commentaries the zero symbol (a circle) has been used as follows:
1. Represents through vowels and in the Katapayādi system.20 2. Stands for a negative sign.21 3. Stands for one-sensed and two sensed bios and so on.22 For example, a three-sensed bio is denoted by 000. 4. Stands for a void - agrhita stage in which a bio does not assimilate karma particles in pudgala parivartana (specific material change).23 5. Stands for filling up the gaps.24 This appears in the writing of the Mahabandha Sutras implying a bridge. Țodaramalla has also expressed such a use in bridging the intervening işekas (cells) in a column of lifetime (sthiti) structure, denoted in the following fashion:
5 1 2 The triangular karma life - a number of karma particles with certain energy level in a specific configuration (prakrti) in every nișeka - is a complicated structure.
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6. Increased through in number through a lower script.25 For example, 65 000 is written as 65 ()
3. The above symbolism is at its supreme in the Karnātaka Vrtti of the Gommațasāra, compiled by Kesava Varņi (c. 13th century C.E.).26
(i) The use of zero as a place value might have preceded or followed the use of other numerals as place values, in order to express a big number, needed in their mathematical philosophy of karma, replacing a cumbersome method. For example, in the Satkhandāgama ! the number of developed human beings is stated as lying between 226 and 227 which is also stated to lie between ((10)7)3 and ((10)7)4 or between Kodākodākodi and Kodākođākodākodi. Singh has already remarked28 how Vīrasenācārya - the author of the Dhavalı - quotes verses representing various styles to express big numbers, apart from the place value with different directional moves. 29
(ii) It is interesting to see how words and numbers were joined to express a number 61, 97, 88, 46, 66, 81, 64, 16, 20, 00, 00, 000 in verse form quoted by Virasenācārya from an earlier ancient work.
(iii) In the Tiloyapannatti (c. 5th century C.E.) the decimal place value notation has been usual as from right to left.30
(iv) In the same text, the denominated time units are developed, stopping for a while at "acalatma" which denotes (84)31 (10)90 years.31 This number has been expressed as (84)/31/90 where 31 stands for the product of 84 to be done 31 times into itself; and 90 stands for zeroes to be placed after the result of the product as place value.
(v) An important application of the place value for subtraction of some desired factors among a set of factors from their product has been used in the Tiloyapannatti,32 the commentaries of the Gommatasāra33 and the Labdhisāra, 54 most frequently for expressing numbers arriving naturally as results in dealing with the systematic karma theory.
Decimal and Notational Places in DVL
The DVL, Books 3 and 4, contain good material on decimal places. The source book CKG contains the following words which signify the decimal places (c. 2nd century C.E.):
42

Page #48

Word coddasa
dasa
sunna
sahassa
terasa
aṭṭhattisa
saya tehattari
ekkavisa
naudi
solasa
aḍyalam
vārasa
kodi lakkha
cauvanna
satthi
sada
culasii
Meaning fourteen
ten

zero
one thousand
thirteen
thirty-eight
hundred seventy-three
twenty-one
ninety
sixteen
forty-eight
twelve
crore
lac
Verse
2
P. 236
DVL P.
p. 34
DVL p. 37
DVL P.
40
p. 36 P. 36
p. 39
Book (volume) Remarks CKG. 1
DVL, p. 67
DVL, p. 67
p. 37
p. 40
36
7
DVL, p. 89
9
2
DVL, p. 92
9
p. 43
3
[Note: Here 59398206 is written in words as pañca koḍio teṇauḍilakkha aṭṭhinaudi sahasṣā chauttaraṁ visadaṁca. This is from left to right in decimals and numerals.]
fifty-four
sixty hundred
eighty-four
coda for
four, dasa for
ten
3
43

projected
in DVL from earlier source
te for three, dasa or rasa for ten attha for eight, tisa for thirty
te for three,
hattari for
seventy
ekka for one, visa
for twenty
so for six, lasa
for ten
ad for eight,
yalam for forty vā for two, rasa for ten
cau for four, vanna for fifty
[Note: On p. 52, the number 4666664 is described beginning with sixtyfour, then towards left six hundred, then sixty-six thousand, then sixtysix hundred thousand, then four crore:
cul for four, asii for eighty

Page #49

causatthi chacca saya chāsaṭṭhi sahassa cava parimānaṁ / chāsatthisaya sahassā koḍi caukkaṁ pamattanaṁ / / 52 //】
[Note: On p. 56 the style is quite different for writing 69999996, which is spoken as six in the beginning, six in the end and nine six times in the middle:
"chakkādi chakkaṁtā channaramajjhā ya saṁjadā savve ("] sattakodisaya seven hundred crore p. 68
7X(10)9
[Note: In verse 45 of CKG, the number of developed illusion visioned human beings is given as above koḍākoḍākoḍie and below koḍākoḍākoḍākoḍie; i.e., above (10) and below (10)
"manusapajjattesu micchāiṭṭhi davva pamāneṇa kevadiyā koḍākoḍākoḍie uvari koḍākoḍākoḍākoḍie heṭṭhado//"]
From the above it appears that the decimal places had to do with the vertical line or horizontal line and zero. It also appears that if dasa, a word for ten, was abbreviated from damda and sunna, where damda stands for a rod and sunna for a zero. Similarly, soda appears for one hundred, abbreviated from sadasa, or a suņṇa added to dasa. Similarly, sahasa, a word for one thousand, appears from ha as haṭṭha for a rod and three "sss" for three zeros. Further, sada sahassa, dasa sahassa could be interpreted for their choice of denomination for one lakh and 10000. One kodi (a crore) also comes under dasa soda sahassa. This could possibly be thought of as their entry into the realms of a decimal place value system.
Notational Places in TPT
The TPTdescribes two fundamental instant sets, the Palya and the Sagara, needed to give measures of various exisential sets, as simile construction sets. In the process of evaluating the Palya35 the number Palya-hair (palya-roma) is a number given by (413452630308203177749512192) (10)18 expressed as a product of 19/24 and the following three rows:
of
SO/96/500/8/8/8/8/8/8/8/8/ SO/96/500/8/8/8/8/8/8/8/8/ SO/96/500/8/8/8/8/8/8/8/8/

44

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where SO stands for three zeros, to be placed on the right after the product. The above number is expressed in the decimal place value style as follows:
atthārasa thānesuṁ sunnāniṁ do navekka do ekkol paņa-ņava-caukka-sattā saya-sattā ekka-tiya-sunnāl/123/N do attha sunna-tia-naha-tiya chakkā donni-paņa caukkānil tiya ekka caukkanim anka kamena palla romassal/124/N
Translation: Eighteen zeros in the last places, two, nine, one, two, one, five, nine, four, seven, seven, seven, one, three, zero, two and four. These are the digits of palya-hair in order. In the following circa 4th-5th century text, the symbols also occur, following the verse. The decimals with words appear as follows:
Verse
Chapter
125 156 168
168 171
179
182
Word sada lakkha sagavisa unavanna coddasa dasa vādāla. terasa tettisa saya pannattasi unadala chappana ekkona vanna causidi satthi
Meaning Symbol hundred
100 lakh
10000 twenty-seven 27 forty-nine 49 fourteen ten forty-two thirteen thirty-three 33 hundred
100 seventy-five thirty-nine fifty-six
56 forty-nine 49 eighty-four sixty
13
186 191
191
75
30
198 198 201 214 229
84
60
285
[Note: The number 31980000 is expressed as twenty less one lakh eighty thousand, tinham kodīņam: Ch. 1, p. 123 (TPT(V))
"egünavīsa-lakkha-asidi-sahassa-tiņham kodīņam"
Thus, the only change in the decimal place for the denomination of sada sahassa has been to lakh in TPT and DVL from the CKG. In other denominations only declensions count.
45

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The notation in decimals may be seen at peak denominations of time in years in Chapter IV of TPT(V). described in VV.294 et seg. They are as follows:
in the This is
Denominations of Notation dasa sada
sahassa
dasa sahassa
lakkha
(pūrvanga) puvvānga (pūrva) puvva
Meaning
Remarks ten
two yugas of four years each hundred
on multiplying by 10 from the
preceding thousand on multiplying by 10 from the
preceding ten thousand on multiplying by 10 from the
preceding lakh
on multiplying by 10 from
the preceding (10)5x84 on multiplying the preceding by 84 [(10)X8412 on squaring the preceding or
7056(10)10 [(10)5x8412884 on multiplying the preceding by 84 [(10)5x84]2X84X84X(1095
on multiplying the preceding by
84(10)5 [(10)5x8412x84x84X(10)5x84
on multiplying the preceding by 84 (10)20(84)6 on multiplying the preceding by
84(10)5 (10)20(84)7 on multiplying the preceding by 84 (10)25(84)8 on multiplying the preceding by
84(10)S
(parvänga) pavvānga (parva) pαννα
nayutanga
nayuta
kumudānga kumuda
The above process of multiplication with 84 and 84(10)5 goes on until mahalata is obtained. Then the process of multiplication by 84 lakh or 84(10) continually gives Śrīkalpa, hastaprahelita and ultimately acalatma whose value is (10)9°(84)51 years. This is to be carried over to maximal numerate (utkyst samkhyāta), of which the process of construction has been very elaborately described by R.C. Gupta, as maximal numerate plus one, or as the first maximal innumerate. 30
It may be noted that for setting the measure of various types of sets in karma theory, there have been various types of constructed numbers as numerate, innumerate and infinite which have not been expressed to dazzling heights without any purpose. Hence it appears that the school needed a system of this decimal type in their theory:37
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Words For Zero
First the words for zero are examined from CKG, DVL and KPS, J DL: Word
Value
Remarks
zero
Meaning avahirijjanti38 making a set zero
by reduction (exhaustion one by one)
attha sunna
viņāsaṇaṭṭham (tattha) avahiranti
antarmuhurta
tending towards muhurta in the limit
of difference tending
to zero
avaharraṭṭhā
seen symbols for division
satditthiņa ditthā values of denominations
khavaga
anantā bhāgassa infinitesimal
anantim bhaga
part of infinites
asamkha bhaga innumerable part
asamkhajjodi bhāga
abhāva
sunnaṭṭhāna
gayana
nabha
for removal of zeros at the end

making a set zero or or exhausted
of time
destroyer or annihilator
innumerable part
absence
zero place
sky
sky
eight
zero
approximate
limit
to infinitesimal
major part limit to
innumerably small part limit to
innumerably small part
nil
zero
zero
47

DVL 1, 2, 3. Concept of exhaustion one by one and
1-1
correspondence, p. 28. Book 3, vide p. 220, also DVL 1, 2, 4, p.
36
CKG 1, 2, 3, p.
27, vide also p.
230, in VV, 1, 2,
3, 4, 1, 1, 35 DVL, 1, 2, 6, p. 69
DVL, p. 37,1, 2, 6, p. 87
DVL, p. 94, 12, 12, p. 94
DVL, p. 121, 1,
2, 15
DVL, p. 63, 1, 2, 16, p. 130
CKG, 1, 2, 17
DVL, 1, 2, 18, p. 160-1
DVL, 1, 2, 19, p.
181, 188
DVL, p. 71: 1, 2,
45
p.225 in place
value

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[Note: The verse depicting place value, from right to left.]
gayaṇṭṭha-naya-kaṣayā causatthi-miyanka-vasu-kharā-davvā/ chāyalā-vasa-ṇabhacda-poyattha-camdo ridū kamaso//71//
Translation: The number is given by taking in order eight zeros, naya (two), kaṣayā (sixteen), sixty-four, mrganka (one), eight, khara (six), dravya (six) forty-six, eight, zero, acala (seven), padartha (nine), moon (one) and ṛtu (six). This number is 61970846668164162(10)8. The verse projected is ancient and earlier to DVL, collected as such..
There is one more verse next, projected in DVL, showing the area of Jambu Island in the place value system, but left to right, in which the word sunna (zero) occurs explicitly two times:
satta nava sunna panca-chattha nava caduekkam ca pańca sunnam cal
jambudivassedam ganitaphalam hodi nadavvā//72//
Translation: The area of the Jambu Island should be known (given by writing numbers as) seven, nine, zero, five, six, nine, four, one, five, zero (in square yojanas). The number is 7905694150.]
It may be noted that zero has not only been used in place value notation, but also defined set theoretically to give the concept of a set having no element or number, by exhausting a set with elements one by one. This is not only for instant-sets (samaya rāšis) but also for pointsets (pradeśa rāśis). For example, "khetteņa veindiya-tiindiyacauvindiya tasseva pajjatta apajjattehi podaramavahirdi angulassa asaṁkhejjadi bhāga vaggapoḍithāena". Translation: Relating to region, two-sensed, three-sensed, four-sensed bios set-cardinals exhaust the universe square (jagapratara) as divided by the square of the innumerate part of linear finger-width (in terms of point-sets).
Word
Sunna
Meaning
zero

Value
zero
Remarks
JDL, 1, p. 33, p. 81, vol. 1
[Note: The verse carries zero here in a placevalue, while giving the number of letters in scriptural (śruta) knowledge (right to left)]
paṁcekka chakka ekka ya du-paṁca ṇava sunna sattatiya satta/ sunna du-caukka satacchacadu cadu aṭṭhekka suda vannāl/33//

48

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Translation: Five, one, six, one, two times five, nine, zero, seven, three, seven, zero, two times four, seven, six, four, four, eight and one (when placed in leftward sequence), give the letters of the scripture. The number is 18446744073709551615.]
zero
suņņa nattha
absence abolish
JDL, vol. 4, V. 22, p. 11 JDL, vol. 6, V. 22, p. 127
zero
Note: The sentence is
kammapadesaānamuvasāmanaā-nikacaņā-ạidhatti karaņāņam visohie vinasapaduppaya ņattham
Translation: In order to abolish (annihilate) the karma particles, the operations of subsidence, preservation and reservation have been mentioned.
Now the TPT and TLS are sought for zero words:
Value
Word sunna
Meaning zero
Remarks TPTTV), 123, Ch.
ritta
vacant
zero in four places
0000
cauthāņesuṁ sunnā. padatthāņe sunnam natthi tudareņu
V. 127. The concept of exhaustion of a set. Ch. 3, V. 54, 86, 88 Ch. 4, V. 53 Ch. 4, V. 55
zero in place of fost
zero in place of tudarenu (scale) sky on placing 90 zeros
ambara naudi unnamgam
(10)90
Ch. 4, V. 59 Ch.4, V. 312
[Thousand is also given as ten hundred, V. 1199.1 [Note: dasa ghana means (10)3 for 1000, V. 1170.]
ņabha
sky gayaņambara sky-sky ayasa-ṇabha sky-sky nabha-nabha sky-sky nahba-ambara- sky-sky-sky gayana
O in unit's place 00 in two last places 00 in two last places 00 in two last places 000 in three last places
Ch. 4, V. 1173 Ch. 4, V. 1171 Ch. 4, V. 1175 Ch. 4, V. 1176 Ch. 4, V. 1176 from left to right
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Note: The verse for example is as follows:
nabha-nabha-ti-cha-ekkekkam anka-kame homti savva
Translation: All debtors were (given by) zero, sky, sky, three, six, one and one (116300), in ordered digits. The total number of these seven classes is given by sky, sky, eight, four, eight and two (2848000) in sequence of digits.
vādigaṇā/
suttagaṇā ṇabha-ambara-gayanaṭṭha-caukka-aḍa donni//1176//
gayaṇa nabha
sunna
tha
sunna
sky-sky indirectness
Note: Here the presence of a Cakravarti in the period of a Tirthankara is shown by 1 and his absence by 0. Total boxes are 34.]
place second
Note: The area of the Videha region as 2969349902/300 361 is shown in the following verse where kha occurs (4th to 5th century C.E):
zero
00 for (10)2 product placing in some of the boxes showing directness and

indirectness between
fordpounders and Cakravartis
zero
Ch. 4, V. 1228 Ch. 4, VV. 12981302
du-kha-nava-cau-tiya-nava-chauṇava-duga joyaṇekka-pattie/ bhāgā tiņņi saya igi-chattiya-haridā videha khatta phalaṁ//
Translation: The area in square yojanas of Videha is given by the number (written from right to left) as two, sky (kha) nine, nine, four, three, nine, nine, six, two yojanas (square) in a line. (To this is added) three hundred divided by one, six, three (or 361).
kham-ṇabha
sky-sky
00 or (10)2
Now we come tothe 10th-11th century work, TLS.
Ch. 4, V. 2407, V. 2807
Ch. 4, V. 2682
31 zeros or (10)31 TLS, V. 21, p. 28
[Note: Here the number 19791209299968(10) is given in place value in words as vitha, nidhi, naga, nava, ravi, nabha, etc., from left to right.]
50

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ņabha-nabha
па
sky-sky alphabet na for zero
place value place value (left to right)
TLS, V. 24, p. 31 katapayadi system denoting zero, cf. ASG, p. 2
[Note: Here the number of hairs in a palya is given.]
gayana sunna
sky zero
zero place value
TLS, V. 309, p. 254 TLS, V. 313, p. 261
Note: The number is in place value from right to left. It is one krosa more than 7905694150 yojanas (square). Example:
chädalā suņņa sattaya uāvaņņam homti meru pahudiņas/ pamcaņņam paridhayaḥ kramanna ankakramenaival/3861/
ņabha
sunna
zero
sky zero in place value TLS, V. 770, p.
610 (right to left) zero in place value JPS, 11, 41, p.
189 (left to right) JPS, 1.135, (right to left) JPS, 10.94, p. 183 (left to right)
[Note: In TPS, the number 6648658 780504 has been the combined area of the Koloda Ocean, but related as (V. 11.47)
chavatthim adadālam atịhatthim sattasidimasidis cal pannasam ca caukkam havadi ya kālodadhisankhāl/491/
Translation: The area of the Koloda Ocean (added with the area of Jambu Island, etc.) is sixty-six, forty-eight, sixty-eight, eighty-seven, eighty, fifty and four (square yojanas).
This is an example of stating the number in decimals, place value, pairing and from left to right. The pairing is also seen in V. 10. 96, p. 183 of the same JPS, where each digit is stated and the 10 in the end is stated as "dasa", and not as 10, from left to right. Thus, combinations of two, three, etc. digits were followed for the specific purpose of expressing numerical measures in different units of time and space in astronomy. Karma theory, however, required several types of units apart from the space and time units, even beyond the theory of matter requirements, as the bios phases were required to be related with particle phases of material (pudgala).
51

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Notational and Decimal Key Words in the Earliest Works
The Kasayapāhuḍa sutta of Gunadharācārya has been regarded as the earliest monumental work on the theory of karma (c. 1st century BC or later), about two hundred years earlier than the Chakkhaṇdāgama of Dharasenācārya's disciples, Puspadanta and Bhutabali (c. 2nd century AD). This text is in Saurseni Prakrit and was produced from the third Pahuda, "Pejjadasa" (Love and Malice) of the tenth Vastu of the fifth Purva, "Jñana Pravada" of the fourth class Purvagata from the Drstivada, the twelfth anga of Jaina scripture. It describes a profound and unified system-theoretic technique founded on the set theoretic approach, gradually developed through symbolic operational details of relations between phases of karmic material particles in the form of asrava (input), sattva (state), bandha (bond), nirjarā (output, etc.) over quantitative maps through matrices (vide LDS).41 The tradition goes back to the South Indian Śravanabelgola hill Candragiri, where ācārya Bhadrabahu-I (c. 4th century BC)42 along with his initiates, Maurya Emperor Candragupta (respective knower of 14 and 10 Purvas) spent twelve years in isolation, possibly for the purpose of recording karma theory through Brahmi and Sundari scripts (for mathematical expressions), some years earlier than Asoka's inscriptions began to in Brahmi.43
appear
Word
dasama (dasa)
sadeaside
panṇārasa
ekkārasa
aṭṭāvisam
sattarasa
solasa
bārasa
visa
ūnavisa
tirasa
Meaning tenth

one hundred and eighty
fifteen
eleven
twenty-eight
seventeen
sixteen
twelve
twenty
nineteen
thirteen
eighteen
zero places (absence)
numerable thousand
Remarks
KPS, V. 1, p. 1, V. 86, p. 611
V. 2, p. 4
V. 2, p. 4
V. 7, p. 8
V. 8, p. 9
V. 27, p. 260
V. 26, p. 261
V. 26, p. 261
V. 26, p. 261
V. 28, p. 263
V. 35, p. 267
aṭṭharasa
sunnaṭṭhāṇā
samkhejja sahassa
[Note: The verse gives a rare type of place value in terms of a variable "numerable" (samkhajja) as follows
V. 45, p. 274
V. 49, p. 277 and the following
V. 114, p. 611
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Page #58

samkhajjā ca manussesu khinamshā sahassasa niyama/ sesäsa khinamshā gadisu niyamā asamkhajjaāl/114/1
Translation: Among the human (male) beings the annihilator serene visioned are numerable-thousand as per rule. In the remaining life-courses they are innumerable as per rule.
vassa sada sahassa samkhāe sadsasahassesu asamkheijesu
hundred thousand numerable years innumerable hundred thousand
V. 131, p. 760 V. 131, p. 760 V. 131, p. 760 V. 131, p. 760
Note: The verse gives the life-time of bonds here:
vassa sada sahassāim tthidisaṁkhāe du mohaniyam tu/ bandhadi ca sadasahassesu asamkhejjesu sesaņi//131/1
Translation: In the inter-state of two instant-effect, (the bios) binds the charmkarma as present transition-proestablisher for life-time of innumerablethousand years, and the remaining karmas are bound for life-times of innumerable-hundred -thousand years.
Note 2: The calculation is further carried on to infinity -- a proper infinite, as if of G. Cantor, detailed further in LDS.
bārasa ņava cha tinni ya kittio homti adha va anamtão/ ekkekkamhikasāu tige tige adhavā anamāol/V. 163, p. 806//
Translation: The tracks of fluorescent anger, etc. affections are twelve, nine, six and three or infinite. In every one of the affections there are three krști (tracts) respectively or there happen to be infinite tracts.
There is one more word gaņānadiyamteņa meaning beyond numeration or meaning innumerable (V. 224, p. 885).Up to this time of Gunadharācārya the verses were pronunciated and then their meanings were lectured (KPS, p.4). Professor A. Cakravarty has placed Kundakundācārya, author of Parikarma, a commentary on the first three sections of the Satkhandāgamā, and it appears, according to the veneration he attained just after Guatma Gandhara, as early as the 1st century B.C. E. By the time of ācārya Bhūtabali, we find in literature, Matābandha, the symbol for zero has been used for fitting up gaps in writing of detailed karma-bond theory (prior to Kundakundācārya) throughout the work. (vide MBD p. 41 et seq.). For example,
navari şiddā-pacalā ogham/ thiņagiddnio nucca 0 anamtanu () jahao amto 01
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Translation: Specifically, that of sleep, deep sleep is as the ogha (i.e., minimal, maximal inter-muhurta), the somnambulation (three), illusion, infinite probonding (four), is minimal inter-muhurta (V. 115, p. 91. MBD).
This treatment shows that by this time zero had not been used as a decimal place value. The commentator, Kundakundācārya, author of Parikarma must have before him CKG, MBD and KPS. This sort of abbreviating the full words and sentences, in which seven volumes of MBD have been written, might have inspired him to start with a line and zero (a danda and sunna) to go ahead of nine digits as ten. This is a conjectural possibility. However, although Parikarma has been quoted by saint scholar Virasenacarya, the text is not extant
Concluding Remarks
The above illustration gives only a part of various logical, mathematical and intuitionistic techniques adopted by the Digambara Jaina School for the expositoon of a scholastic system theory (Karma Siddhanta) which became a topic of discussion by L. Alsdorff. This school claims to be in possession of only small portions of the second and fifth Purvas of the Drstivada knowledge. Thus, it forces a rich source of material on the use of zero and its various applications apart from the decimal place value system.
REFERENCES AND ENDNOTES
1.Neugebeuer, O. The exact Sciences in Antiquity, Providence, 1957, p. 20. 2. Ibid, p. 26.
3. Ibid, p. 27
4. Needham, J. and Ling, W. Science and Civilization in China, Vol. III, Cambridge, 1959, p. 11. Needham finds also that from the time of King Asoka (3rd century) there was a steady development of the forms akin to the HinduArabic numerals, the first three integers being written in just the same way as the Chinese, the first epigraphic evidence for zero in India being in the late 9th century, ibid, p. 10.
5. Cf. ibid, p.10.
6. C.f. ibid, p.12.
7. The Tiloyapanṇatti of Yativṛsabhācārya (C. 5th century AD) Vol. 1, 1943. Vol. 2, 1952, edited by H.L. Jain and A.N. Upadhya, Sholapur (abbr. TPT). 8. The Chakkaṇḍāgama of Puspadanta and Bhutabali Ācāryas (abbr. CKG) (c. 3rd century C.E. with Dhavala commentary of Virasenācārya (c. 9th century C.E.) (abbr. DVL), Vols. 1-16, Amarasti etc., 1939-1959.
9. The Kaṣāyapahuḍa of Gundadharācārya (c. 1st century C.E.) (abbr. KSP) with Jayadhavala commentary of Virasenācārya and Jinasenācārya, Vols. 1-14, edited by Pt. Kailash Chandra, Mathura, 1972.

54

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10. The Mahābandha (abbr. MBD) of c. 2nd century C.E., Vols. 1-7, Kashi, 1947-1948. 11. He belongs to the end of the 10th century C.E. (a) The Trilokasāra with commentary of Madhvacandra Traividya and Visuddhamati Arvikā (abbr. TLS) Shri Mahavirji, 1976. (b) The Gommatasāra, Vols. 1-4 (abbr. GJK and GKK) with Karnataka Vrtti of Keśavavarņi (c. 13th century C.E.) and translation Samyakjñana Candrika commentary of Țodaramalla in Hindi (c. 18th century C.E.), New Delhi, 197881. ASG will denote its Artha Samdrstu. (c) The Labhisāra including the Kșapanasāra along with Dhữndhāri commentary of Țodaramala (abbr. LDS, KPN), Agas, 1960, edited by Pt. Phool Candra Siddānta Shăstri. ASL will denote its Artha Samdsști (symbolism for norms). 12. The first volume of the first project has been published by the S.S.M.K. Jain Trust, Nehru Park, Katni, (M.P.) in 1994. 13. Datta, B.B., Geometry in the Jaina Cosmogrophy, Springer-Verlag: Quellen und Studion B Nr. 14-Datta, 1-5, 1930, P. 1-10. Cf. also Bag, A.K, Symbols for Zero in Mathematical Notation in India, PCAHC, BANC, 1970, 247-254. 14. Singh, A.N. (with Datta, B.B.), History of Hindu Mathematics, Parts I and II, Bombay, 1962. 15.Kapadia, H.R., Introduction to the Ganita Tilaka of Sripati, Baroda Oriental Institute, V-XIX, 1937. Kapidia, H.R., History of Nāgari Numerals, A.B.D.R.S., 19, 1978-89, 386–394. 16. Singh, A.N., Mathematics of Dhavalā-I, in the Dhavalā, Vol. 4, edited by H.L. Jain, et al., Amaravati, V-XXI, 1942. 17. Three examples are: (i) 79999998 is expressed as a number which has 7 at the beginning, 8 at the end, and 9 repeated 6 times in between (V. 51). (ii)
66664 is expressed as sixty-four, six hundreds, sixty-six thousands sixty-six hundred thousands andfour kotis (V. 52). (iii) 22799498 is expr kotis, twenty-seven, ninety-nine thousand four and ninety-eight (V. 53). Singh remarks that quotations like the above found in Jaina works point to the early use of place value notation in India and afford valuable evidence not obtainable from Hindu works. Cf. Jaina Antiquary, Vol. XV, Dec. 1949, no. ii 46-53. 18. Jain, L.C. Mathematics of the Tiloyapaņņațți (Hindi) in the Jambūdvipaņņații Samgraha, Sholapur, 1958, 1-109. Cf. the Tiloyapannatti (TPT) of Yativșsabhācārya, ed. Upadhye, A. Nand Jain, H.L., I (1943), II (1952), Sholapur. Cf. also the Trilokasāra of Nemicandra Siddhāntacakravarti (TLS) and its commentaries by Madhava Candra Traividya and Țodaramalla, Bombay (1910) and Shri Mahavirji (1976). 19. Artha Samdrsți chapters on the Gommatsāra and the Labdhisāra (ASG and ASL) of Nemicandra Siddhāntacakravarti in SIC (Samyak Jnana Candrika) commentary by Țodaramalla, Calcutta, c. 1919. Cf. also recent New Delhi and Agas Publications, the Karnataka Vrtti of the Gommatsara by Keśava Varni (c 13th century C.E. in New Delhi publication (vol 1-4) contains more detailed symbolic material.
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20. Cf. Labdhisāra, INSA project, Vol. I, Kotni, 1994, 68-70. 21. Ibid, p. 81. 22. Cf. Dhavalā, Vol. 10, P. 421 et seq. In the Bakhaşāli manuscript the symbol stands for an unknown term. Datta, B.B., The Bakhasāli Mathematics, BCMS, XXI, 1929, 115-145. 23. Vide 9, p. 93. 24. Ibid, p. 70. 25. Ibid, p. 71. 26. Vide 8. 27. Cf. Dhavalā, Vol. 4, 1942, 1, 2, 14, 51, p. 98. 28. Vide 6 29. The verse is from ibid, Vol. 3, 1, 2, 45, 71, p. 255:
gayanattha kasāya causathi aiyanka vasu kharā davvā/
chāyalā vasu nabhäcala payattha cande ridu kamasell 30. Cf. TPT W. 1176, 2402 for example: The numbers 116300 and 2848000 are given by
ņabha nabha ti cha ekkekkam anka kame honti savve vadi ga sattigāņa nabha ambara, gayanattha caukka ada donnil/1176//
The area of Bharta kşetra, 6021353 294 is given by
361 pañca titi akka duga ņabha chakke ankakkasena joyaņayā/ ekka cha ti harida cau nava duga bhāga bharahakhetta phalam
(Kota edition) 1/2462 // 31. Vide ibid 4.308, p. 178 (Sholapur edition). 32. Vide 7 33. Vide ASG, p. 20 and LDS project, Vol. 1, p. 90. 34. Vide ibid p. 90. 35. Cf. TPT, Ch. 1, 28, p. 113-132. 36. Gupta, R.C., The First Unenumerable Number in Jaina Mathematics, Ganita Bharati 14 (1-4), 1992, 11-24. This number is analysed by him to have been constructed through pits in units, tens and hundreds places. Cf. also TPT, p. 102-103 for an easier analysis by L.C. Jain. 37. Jain, L.C., The Tao of Jaina Sciences, Delhi, 1992, Vol. 1, p. 1-46. Vide also Kapida, H.R., Introduction to Ganita Tilaka of Sripati, Baroda, 1937, p. VLXIX, for information on notational places in GSS, Sthānānga, Jambūdvipa prajñapti, Sürya prajñapti, Anuyogadvāra and Jivasamāsa, etc., as also the Jyotisa karandaka, the Tativartharājavārtika and the Svopajña bhāsya of the Tattvārthadhigamasūtra, wherein the names differ somewhere along with the process of constructions. But in all of them 10 has been used along with 84 for this purpose. It may also be noted that for finding out the value of pi, the number 10 has been used for finding the finer values of pi through difficult processes as may be seen in Gupta, R.C., "Circumference of the Jambūdvipa on Jaina Cosmography", IJHS, 10.1, 1975, 38-44. The originality of manipulation is evident therein. The number 84 is also connected with the generative places

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called yoni or jāti which are 84 lac in number of various types of bios classified in several ways.
38. For example, the verse 1, 2, 34 in CKB is:
asaṁkhejjāhi asappiņi-ussappinihi avahiranti kāleṇal/34//
Translation: Relative to time (the five-sensed subhuman bios set) is exhausted by innumerable-innumerate hypo-serpentine and hyper-serpentine (periods of time or instant-sets). The method of one to one correspondence is described here in DVL, p. 233, Book 3, 1, 2, 35.
39. CKG, V. 1, 2, 7, 9, p. 313, Book 3 (DVL).
40. This is the combinatorial work from 64 alphabets.
41. The details of these operations are given in detail in GKK, II, p. 675 and LDS. The Nikaita operation means that the karma perticles could neither be brought into rise trail (udayavali) nor could be brought with other configuration (prakrti) form nor could be uptracted or downtracted. The nidhatti operation means that the karma particles could neither be brought to the risetrail(udayavali) nor could be transited operation in which the karma particles could not be brought into rise-trail.
42. Edited by H.L. Jain, Siddanta Shastri Curni sutras of Yativṛṣabhācārya (c. 5th century C.E.), Calcutta, 1995, (abbr. KPS).
43. Vide Jinasena, the Harivaṁsāpurāṇa, 9/24, and the Mahāpurāna, 16/103105-109. The Sundari may be taken to be Hīnākṣarī and Brahmi as Ghaṇākṣarī, which were requires to be perfect for the learning of karma theory. Vide, Jain, L. C., Secret of Hinākṣarī and Ghanākṣarī (in the moon-cave of Girinagara), Arhat Vacana, 1 (21, 1988, 11-16).
Vide also Goyal, S.R. Brahmi:An Invention of Early Mauryan Period in the region of Brahmi script, ed. by S. P. Gupta and K.S. Ramacandra, Delhi, 1979. For more details, Cf. Jain, L.C., Arhat Vacana, 2(2), 17-26, 1920, 2(3), 1-2, 1990 and 89, 92, ibid, 4 (1), 1992, 13-20, 5(3), 1993, 155-171.
44. Vide Jain, L.C., Arhat Vacana, 3(3), 1991, 33-91, ibid, 2(1), 1989, 7-12, ibid, 2(4), 1990, 82-92. These are on Kundakundācārya relevant to decimal place value. Even the use of 10 in finding the value of pi as the square root of 10 and its vāsana (rationale) by Madhavacandra Traividya is in support of this conjecture. Cf. Gupta, R.C., IJRS, 21(2), 1986, 131-139.
For the use of place value in the subtraction of factors, vide Jain, L.C., IJHS, 24(3), 1989, 163-180.
45. German scholars, Vol. 1, Varanasi, 1973, 1-5

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Jinamañjari, Volume 19, No. 1, April 1999
SATPRARŪPAŅĀ SŪTRA English Rendition of Hindi Edition by Pandit Kailash Chandra Shastri
Dr. N.L. Jain, Rewa, India Continued from previous issue of Jinamañjari
In conformity with the Jaināgama coming down from ancient times, verse - 2
Etto imesim coddasaņham jivasamāsāņam magganam magganatthadāye Tattha imāņi coddasa ceva thaņāni bhavanti - 2
On this basis of the Āgamās, the fourteen mārgaṇasthānās (stations of investigation) are worth knowing and search into the fourteen jīva-samāsās (spiritual stages) - 2.
Q. Which category of märganasthāna, physical or psychical is intended in this verse ? A. According to the Jain tenets, the psychical type is intended.
Q. How one would learn about it? A. It is learnt from the term “Etāni (imāni)” in the verse, as explained and elaborated by Virasena in his Dhavala commentary. According to it, the directly perceived psychical märganasthāna is intended by the term "imāni" and the physical type is not intended as it is distant and variable with respect to location, time and nature. Thus, the nonomniscient can not have direct perception.
Q. What is the definition of mārganasthāna ? A. It is the method or topic of investigation wherein fourteen mārganasthānas in association with eight anuyogavārās (disquisition doors) are applied to study existence, numeration etc. It is also explained in the Gommatasāra Jivakānda (G)), 'verse 141: "One should know that there are only fourteen märganasthānas; and mārganasthāna is defined as topic of search through or under which the modes of the living beings,+ like the infernal modes etc are observed or indicated through scriptural knowledge."
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+ The Sat-khandagama (SK) is the first ancient Digambara text to describe living beings through spiritual stages for which the author uses the term "jīvasamāsa" ( abridgment or categorization of the living beings under spiritual stages). It seems that there are many ways of such categorization on different bases like senses, embodiments etc. Later, this term was designated as "jīvastāna" and "guṇasthāna" in order to avoid terminological confusion. The term “guṇasthāna" has later developed as quality-based.
Q. What are the fourteen märgaṇasthānas ?
A. Tam jaha - 3, and they are enumerated in verse - 4:
Gai indiye kaye joge vede kasãe ṇāṇe sanjame dansaṇe lessãe bhavie sammatte saņņi āhārae cedi - 4
The living beings are studied under fourteen mārgaṇasthānas, which are: gati (destiny), indriya (senses), kaya (embodiments), yoga (activity), veda (libido), kasāya (passion), jnāna (knowledge), samyama (restraint), darsana (conation), lēśya (aureole), bhavyatva (liberatality), samyaktva (righteousness), sanjni (instinctive) ähāraka (intakership).
Q. Why there is in this verse the use of locative case in each of the terms like destiny?
A. This is meant to denote that the mārgaṇasthānas like destiny etc. are the substratum for the living beings.
Q. Does the case of investigation stand scrutiny here as its four elements investigator, object, method and means being not mentioned, but are required for the study of anything in the world? A. Though the question is valid, the assumption of the absence of elements is not. The elements of investigation are found in the following contests: 1] The investigator is the faithful living being believing in the categories of the living and nonliving beings etc. 2] The object is the living beings and other entities. 3] The methodology is the investigation of destiny etc. which form the substratum for the objects. 4] The means is the teacher.
-
and
Q. Why this verse has described only the investigations and not the other three elements?
A. Since the investigations are invariably related with the other three elements, their inclusion is found within the description of investigation.

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Q. What is the definition of gati ? A. It is the specific mode of the existence of the living beings or souls. Alternatively, it can be said that it is the movement from one state of birth to another state of birth. The Prākrta Pancasangraha (PP) verse 1.58 also defines it this way: “The specific activity or movement of the living being due to the fruition of the physical-making karma is gati. Alternatively, gati is the instrumental cause for moving living beings in four states of infernal etc.”
Q. What is the definition of indriya ? A. It is defined as that agent which is engaged in its own object and which does not engage itself in the objects of other indriyas. In other words, indriya is the master of its own objects. GJ verse 164 also explains it this way: “The indriyas are like Ahamindra (I an lord), who has no classes of lords and servants, but feels as the lord of deities. Indriyas are capable of causing the knowledge about their own objects, independent of other indriyas.”
Q. What is the definition of kāya ? A. It is defined as pudgala in the form of gross body etc. (there are five types of bodies) accumulated due to the action-based propensity of the soul. The PP, verse 1.75 explains it this way: “Know, it is the pudgala in the form of gross body etc. accumulated due to the action-based propensity of the soul. Jain canons speak of six kinds of them, in the form of earth, water, air, fire, plant and mobile, and they form two categories namely, non-mobile and mobile kāyas. The first five kinds belong to the first category and the sixth kind belongs to the latter category.”
Q. What is the definition of yoga? A. It is defined as the generation of potency instrumental in receiving karma due to three-fold propensities of the living being. Alternatively, it is the process of expansion and contraction of the space-points of soul (due to its propensities). It is also said in the PP, verse 1.55: “Jinas have spoken of it as the effort in terms of potency (or energy) instrumental in receiving karma by living being associated with mental, vocal and physical actions on its own. Alternatively, it is defined as motions or vibrations of the space points of the soul."
Q. What is the definition of veda (libido)?
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A. It is the rising of infatuation in the form of normal copulation or sex activity. It is also said in the GJ, verse 272 in this way: “The living being or soul performs many follies due to the normal or premature fruition of karma due to veda. Accordingly, experiences or the feelings due to the fruition of this activity is termed as veda.”
Q. What is the definition of kaşāya (passion)? A. It is the agency which is instrumental in ploughing, tilling and seeding the field of karma which produces various types of grains of pleasure and displeasure. It is also said in the GJ, verse 282 in this way: “Kaşāya is defined as the agency which ploughs the field of karma producing many types of grains of sorrow and happiness and whose boundary is very large and far away (in terms of birth cycles).
Q. What is the definition of jnāna (knowledge) ? A. It is defined as specific energy or capacity which enlightens about true realities and through which living being learns about the reals and realities along with their attributes and modes. The GJ, verse 299 defines it in this way: “It is an agency through which the living being perceives or learns all about all the three-time existing reals, their attributes and modes directly or indirectly.”
Q. What is the definition of sanyama (restraint ) ? A. It is defined as the process of accepting and observing vows and carefulness, control of passions, renunciation of all types of violence and winning over the senses or sense objects. Verse 465 in the GJ also defines it as: “It is the process of accepting five major vows - ahimsa, satya, nonstealing and aparigraha; observance of five carefulness in walking, talking, food-intakes, picking and placing and excretions; control of four passions - anger, pride, deceit and greed; renunciation of mental, vocal and physical weapons of himsa; winning over of five senses of touch, taste, smell, sight and hearing.”
Q. What is the definition of darśana (conation)? A. It is defined as internally oriented vision or enlightenment of consciousness. In contrast, the knowledge is defined as the externally oriented enlightenment of consciousness.
Q. The term cit (consciousness) is defined as the experiencing of the nature of the self-soul in accordance with the karmic destruction-cumsubsidence. In contrast, the term prakāśa (enlightenment) is defined as
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the knowledge of external objects different from the self-soul. Moreover, the knowledge is defined as the medium through which the living being learns the nature of the self and the external objects. Is then the difference between darśana in terms of internal enlightenment and knowledge in terms of external enlightenment is unproven? A. There is difference between jnāna (knowledge) and (darsana) conation, and no enlightenment of objects occurs through knowledge like “is this an earthen pot" or 'this is a fabric” etc.
Q. It should then be admitted on this basis that darśana (conation) perceives the inner and external generality while jnāna perceives the inner and external particularity. A. The particularity without generality is not capable of artha-kriyakāri (casual efficiency). Further, an entity which is incapable of casual efficiency is a non-entity. Hence, the knowledge perceiving a non-entity cannot be called valid. Similarly, the conation perceiving only the generality can also be not valid. All this means that the particular devoid of general and the general devoid of particular is a non-entity. Knowledge perceiving particular devoid of only general, and conation perceiving general devoid of only particular can be admitted as a valid cognition ? So knowledge is that which perceives general-cumparticularized eternal; and conation is that which perceives general-cumparticularized inner self,
Q. If one accepts the above definition of darśana and jnāna, will it not be contradictory with the scriptural definition of darsana, which perceives the entities in general (vide the GJ verse 543)? A. The term general in this definition has been made to indicate the self as it is the common substratum for all the external entities. To confirm this point, the GJ verse 543 has given a term with an adjective meaning in terms of “not perceiving the shape, etc (i.e. details) of entities." It means that the perception of generality irrespective of particularity is called darśana. It is also said in the GJ verse 543 that the Jaina scriptures have defined darśana as that which grasps or perceives the general-cumparticularized external entities in general without individual differentiation points or details.
Q. What is the definition of lesyā (aureole) ? A. It is the mental, vocal, or body propensity painted (or associated) with passions. It means that it is neither propensity passions nor actionbased propensity. However, it should not be taken to mean that there will
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be no lesyä in the detached ones in the eleventh or higher spiritual stages, as the leśyā is activity-prominent rather than passion-prominent. Lesyā is the adjective of activity. It is also said in the GJ verse 489: “The saints knowing about the nature of lesyā have said that it is defined as that medium through which the living being besmears himself with sins and sacred.”
Q. What is the definition of bhavya (liberatable)? A. It is the individual who is capable of attaining liberation from this world. The contrast is the individual who is non-liberatble being incapable of attaining liberation. The GJ verse 558 also states that there are two kinds of liberatables - those who have already attained liberation and those who have the capacity to attain.
Q. What is the definition of samyaktva (righteousness)? A. From the pure or absolute standpoint, it is defined as the faith in the reals and realities with reference to: pacification, desire for salvation, compassion and belief in scripture and rebirth. It is also said in the GJ verse 561: "One who believes in six realities, five existents and nine reals as sermonized by the Enlightened and described in the scripture. Or, it may be the belief in the above through the methods of valid cognition, standpoints and positing.”
Q. What is the definition of sanjni instinctive)? A. The instinct or mind is defined as that which knows or feels well. The sanjni being is that who has the mind (physical or psychical). A non-instinctive does not have any type of mind. Hence the instinctive living being receives education, performs actions and receives or delivers instructions and speaks with the help of mind.
Q. What is the definition of ahāraka (intaker)? A. The living being who acquires mattergic mass suitable for the growth of bodies like gross bodies, etc. is a ahāraka. It is also said in the GJ, verse 665: “That living being who receives spardhaka pudgala (of mattergic supervarifirm) karma suitable for one out of the three bodies of the nature of gross protean and projection and for the growth of speech and mind."
Q. What is the definition of anāhāraka (non-intaker)? A. It is he living being who does not receive mattergic mass for the formation of growth of gross body etc. It is also said in the GJ verse 666
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in this way: “The living beings of [i] all the destinies under transmigration, (ii) potentially omniscient, [iii] static omniscient and (iv] liberated ones, as a rule, are anāhārakas while remaining ones are ahārakas."
To describe anuyogadvaāra for the study of spiritual stages, objects of investigation is found in the following verse:
Edesim ceva coddasanham jivasamāsāņam parūvaṇatthādaye tattha imāni attha aniyogaddārāņi ņāyavvāņi bhavanti - 5 Following eight anuyogadvaāras are worth knowing to enunciate or study of fourteen spiritual stages (as indicated in v.2)
Tam jaha. 6 Eight anuyogadvaāras are described in the follwing v.7
Santaparivaņa,davvapamāṇanugamao, khettänugamo, phosanāņugamo, kālāņugamo, antarāņugamāmo, bhāvāņugāmo, appábahugānungamo cedi - 7* Eight anuyogadvāras are: [I] enunciation of existence [ii] confomatory explanation on numeration of realities (and reals) (iii) conformatory explanation on location [iv] conformatory of explanation on contact [v] conformatory explanation on time [vi] conformatory explanation on interval (vii) conformatory explanation on current mode or disposition and (viii) conformatory explanation on relative numeration.
It is also said in the scriptural verse (no reference given) that [I] the description of the existence of reals and realities is called enunciation of existence sü] the description of the numeration of the categories of existents which have been known to be in existence is explanation on numeration (reals and realities (iii) the description about the current location of the reals and realities is explanation on location [iv] the description of the past and present contact (of reals and realities with each other) is explanation on contact (v) the description of minimum and maximum duration of the existence of realities and reals is explanation on interval (vi) the description of time interval or zero interval about the realities and reals is explanation on interval [vii] the description of changes, transformations, dispositions and current state, etc of the entities in the world is explanation on mode (viii) the term relative numeration referring to entities in the world is self-explanatory.

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Verse - 8 explains the first anuyogadvāra - sat-prarūpaņao:
Santa-parūvaņadāya duviho niddeso-oghena asesena - 8 There are two kinds - [I] general and [ii] particular - 8
From verse - 5 above, the meaning of v.8 should be taken as: There are two types of descriptions - general and particular, describing general enunciation (of realities and reals) in case of the former and particularities in the latter.
* K.C. Shastri seems to have taken the terms anugama and prarūpaņa as synonymous. That is why he has used the term prarūpana while giving the sense of the quoted verses. However, it should be kept in mind that the term anugama clearly means explanation in conformity with the scriptures while the term prarūpana means the excellent examination or explanation. This excellence can arise only when the descriptions are conforming to the scriptures. Thus the idea of conformity with scriptures is automatic in this term. While defining anuyogadvāra, the word conformity should be taken in each case, and they are the methods of studies of any entity.
In general, anuyogadvāras could be applied to the studies on all realities and reals, which the early Jaina texts have employed them for the studies on the living beings. Later Jaina scholar Akalanka employs them to the non-living beings also.
Q. How to define jīvasamāsas (spiritual stages)? A. This term is the substratum in which the living beings are observed to be existing during the bhāva (modal) development.**
** They are described in the Tattvarthasutra and other. According to some scholars, there are six types where they add mixed or conjunctional mode. However, five kinds are popular in the tradition. Though these modes have karmic and non-karmic origins in the scriptures, currently they form important part of psychology in explaining, controlling and improving human behaviour or mental states, such as repression, regression, denial, projection, sublimation, displacement, rationalization, redirection, compensation and dissolution. According to the religious interpretation, better or purer the bhāva better will be the religiosity, happiness and spirituality, and jīvasamāsas are specific steps.
Q. Where do these living beings exist or reside for their purpose ? A. They reside in their qualities or attributes which are developed gradually through jīvasamāsas.
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Q. What are the qualities of the living beings ? A. The term quality here stands for mental dispositions due to karmic or non-karmic causes, and they are of five kinds: [1] audayika (fruitional) [ii] aupasamika (caused by karmic subsidence) (iii) kasayika (caused by karmic destruction) [iv] kasayika-aupasamika (part destruction and part subsidence of karma) and [v] parināmika (inherent). The attributed living being is also designated as the attribute or guņa (quality) because of the concomitant association. It is also said in the GJ verse 8: living beings are named as "quality" or "attribute” by the omniscient.
Verse - 9 describes spiritual stages in general and the first stage in particular:
Oghenna atthi micchāitthi - 9*** In general, there are wrong-faithed living beings - 9
***In Prakrit, the term atthi may have meanings in both singular and plural numbers - is and are, which continue up to verse 23. The term micchāitthi in here and in the followings up to 23 will have expressions in plural number, as well expressed in terms of abstract noun or attribute (wrong faith) or adjective (wrong-faithed). In this context, "attributed” is implied as the living being.
Q. Who is a mithyādraști (wrong-faithed) one ? A. Those living beings who are have predilection towards falsity or irrationality are called wrong-faithed. It is explained also in the GJ verse 17: “The living being has reverse or false predilection due to the experience of wrongfulness arising because of the fruition of the wrong - faith. He does not beieve the true religion in the same way as the man with bilious fever does not taste even the sweet juice.”
Second guṇasthana is described in verse - 10:
Sasana-sammäitthi - 10 In general, there are the sasādanis (who linger on right-faith) -10.
The terms oghena (in general) and atthi (are) are followed here from verse - 9 to get the correct meaning, and are followed up to verse.
Q. What is the definition of sasādani ? A. The opposition to, or offending of right faith with denial or āsādana (lingering) is called sasādani, whose right faith has been
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vanished due to the fruition of the infinite-bonding passion, and has not developed firm bhāva about wrongness.
Q. As there are three types of faiths - [I] right (ii) wrong (iii) ubhayarupa (mixed). Sasādani (lingering right-faithed) is neither mithyadrașt as he does not accumalte karma, nor is he samyakdrști (right-faithed); he is not ubhayarupi (mixed) type. Since there is no fourth kind of rightcum-wrongness, why there should be a category of sasādani in spiritual stage ? A. In asādana there is false predilection, which has two kinds: one arising from infinite-bonding kasāya and the other from karma of wrong faith. The āsādana having infinite-bonding kasāya therefore is wrong-faithed. But there being no wrong predilection arising out of wrong-faith, āsādana is right-faithed rather than wrong-faithed.
Q. How could it be right-faithed when there is wrong predilection arising from wrong-faith? A. It is because of the fact that the living being was right-faithed earlier. This is also has been said in the GJ verse 20: Āsādana living being in his second spiritual stager approaches wrong faith, falling from right faith. It means that his right-faith is vaning but has yet to acquire wrongfaith completely.
The next verse describes the third spiritual stage of samyag-mithyādraști (right-cum-wrong faith):
Samma-micchāitthi - 11 In general, there are the right-cum-wrong-faithed living beings.
Q. What is the definition of samyag-mithyādraști ? A. He is called so for he is right-cum-wrong-faithed as having both types of faiths: right-cum-wrong.
Q. As it is not possible to have both types of faiths at a time in a living being, does third spiritual stage stands scrutiny? A. When it is possible to have right and wrong faiths in a living being in succession, it could also be possible to have both types of faith in him at the same time. We find such right-cum-wrong-faithed who hold that the enlightened ones are also venerable deities, without resigning from the worship of their earlier accepted deities.
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Q. Which is that bhāva, (volition) of the five stated earlier, in this third stage? A. There is this kşayopașamik bhāva (destruction-cum-subsidential).
Q. What is the relationship of this kşayopașamik bhāva with living being crossing from mithyātva (wrong) to samyag-mithyātva (right-cumwrong) gunasthana (spiritual stage) ? A. Samyag-mithyātva is acquired due to [i] udayabhāvi kaşāya (nonfruitional destruction), aupamāsika (subsidence) all-destroying spārdhaka*** (large group of karmic atoms - supervariform) of mithyātva-karma (wrong-karma); and of the karma of right-cum-wrongfaith. Hence the relation.
*** The karmas are not ultimate atoms but are larger groups called varganas (variorums) and still larger varganas are called spardhakas (supervariforms). And the karma may be in the form of either of these two depending upon the intensity with which they are earned.
Q. As there is karmic reap from samyag-mithyātva, why the audayika bhāva (fruitional volition) has been stated in this stage ? A. Just as there occurs a total destruction of samvyaktva (right faith) on account of mithyātva (of wrong faith), there is no total destruction of right faith on account of samyag-mithyātva. Therefore, this third gunasthana is called kşayopaşamika bhāva (destruction-cumsubsidential volition) instead of calling it audavika.
Q. When the reap of samyag-mithyātva (right-cum-wrong faith) does not totally destroy the right faith, why it has been referred as alldestroying (in terms of supervariform)? .. A. Karma generated by samyag-mithyātva only limits the totality of right faith. It is with this point of view that right-cum-wrong faith has been called all-destroying. It is also said in the GJ verse 22: "Just as a mixture of curd and jaggery cannot be separated into constituents, mixed bhāvas in the form of right-cum-wrong faith can not be separated. Hence, mixed volition is the third stage of mixed faiths.”
Following verse describes the fourth guṇasthāna of asanyata (nonrestrained) samyagdraști (right faith):
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Asanjada-sammäitthi - 12 In general, there are asanyata (non-restrained) samyagdraști (rightfaithed) living beings.
Q. What is the definition of asanyata asanyata samyagdraști ? A. A samyagdraști is one who has a right or true belief, vision or predilection; is and is of three kinds [i] kşayak (destructional rightfaithed) who has completely destroyed seven karmic species, and never acquires wrong faith nor doubts scriptures. [ii] vedaka (destruction-cumsubsidential right-faithed) [iii] aupaşamika ( subsidential right-faithed) who may become wrong-faithed, lingering right-faithed or right-cumwrong-faithed.
Q. Of the five bhāvas, which one is there in the fourth gunasthana ? A. The right faith arising out of the destruction of the seven karmic species is kşayak (destructional right-faith). The right faith arising out of subsidence of the same seven species is aupaşamika (subsidential rightfaith). The right-faith experiencer is vedaka-samyaktva-kşayo aupașamika (destruction-cum-subsidential), which arises due to the fruition of the species of righteousness (including right faith) leading to experience of partial destruction of right faith.
Q. Why is adjective asanyata (non-restrained) for the right-faithed? A. This adjective is terminal ending, and indicates that the first three gunasthānas are also non-restrained along with the fourth stage.
Q. Why this adjective asanyata does not indicate its character in the fifth gunasthāna ? A. It does not indicate this point because higher stages have either guņasthānas sanyama-asanyata (restraint-cum-non-restraint) or sanyama (restraint) only. It is also said in the Gj verses 27 and 29 in this way: "Normally the right faithed living being does believe in the sermons of the Jinas, but he may also believe the wrong or contrary instructions from the teachers out of ignorance or without knowing the sense. The right-faithed asanyata living being believes in the sermons of the Jinas."
Next verse describes guṇasthāna of deşavirați (partial restraint or abstinence):
Sanjadā-sanjadā - 13 In general, there are sanyama-asanyata (restrained-cum-nonrestrained living beings).
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Q. What is the definition of sanyama-asanyata ?
A. Those living beings who are asanyata (non-restrained) despite being sanyama (restrained) also.
Q. The sanyama living being cannot be asanyata, and vice versa as these terms are mutually contradictory. Hence, is there the possibility of the existence of this fifth spiritual stage?
A. There is no contrariety in agreeing to the existence of restraint and nonrestraint in a single living being. Because, the factors for their origin are different. The restraint is caused due to the abstinence of violence of mobile beings and the nonrestraint is caused due to non-abstinence of violence of the non-mobile beings. It is also stated in the GJ verse 31: "The living being is said to be rstrained-cum-nonrestrained who is abstained from violence of mobile beings and is non-abstained from violence of the non-mobile beings at the same time despite his faith in the Jinas and their sermons."
Verse14 describes the sixth stage of pramāda (nonvigilant) sanyama (restraint):
Pamatta-sanjadā - 14
In general, there are pramāda sanyama (non-vigilantly restrained) living beings.
Q. What is the definition of the term pramāda sanyama?
A. It is intoxication by prosperity (mada-mad; pra-posperity). In contrast, restrained living beings violence in a proper.
Q. If the sixth stagers are pramādis (non-vigilant) they can not be restrained as they can not know the true nature of their self. If they are restrained, they can not be non-vigilant as the restraint can be there only
when the non-vigilance goes away. Hence, how could there be a
spiritual stage like the sixth one?
are
A. Five sins - himsa, asatya, stealing, nonchastity and parigraha not annihilated by pramāda (non-vigilance), which, however, may cause some flaws or mutilations in the sanyama (restraint).
Q. How it could be ascertained that flawed sanyama pramāda (restrained-nonvigilant) is intended and not the kṣiņa sanyama pramāda (restraint-destroying nonvigilance) in this sixth spiritual stage?

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A. There is no possibility of sanyama (restraint) in the presence of kşiņa sanyama pramāda in the sixth stage and therefore, it ascertains that it is only the flawed pramāda (novigilance) is intended in here.
Q. Out of the five modes, which bhāva (mode) is in this stage? A. There is kşayopaupasamika bhāva (destruction-cum-subsidential) in this stage as sanyama (restraint) is caused due to [i] udayabhavi (nonfruitional) karmic drippings of the currently existing kşiņa kaşāya [ii] subsidence of existing karmic future fruition [ii] result of the gathering
passion.
Q. When sanyama (restraint) is there due to gathering passion, why it is not stated that there is audayika bhāva (fruitional mode or volition) in this stage ? A. Because, sanyama is not caused only as a result of gleaning passion.
Q. What is then the function of the fruition of the karma of gleaning passion here? A. It mutilates sanyama only. It is also said in the GJ verse 33: “The living being is called pramāda sanyama (nonvigilantly restrained) who manifests and non-manifests physically and psychically); conduct oneself and adheres to mahāvratās."
The next verse describes the seventh spiritual stage of guarded sanyama (restraint) among all the types of kşayopaupasamika (destruction-cumsubsidential restraints):
Appamatta-sanjadā -15 In general, there are apramat-sanyat (vigilantly restraint) living beings.
Q. Who are apramat-sanyats ? A. They are those whose restraint is devoid of pramāda (nonvigilance). It is also said in the GJ verse 46: "The living being -- who has destroyed all pramāda; conducts oneself with vows and attributes; and absorbs in meditation (of the third and fourth type), but holds steady without going up or down the spiritual ladder -- is called apramatsanyat."
The next verse describes the eight spiritual stage:
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Apuvvakaraṇa-pavitha-suddhi-sanjadesu atthi uvasama khavā - 16 In general, there are karmic subsider and destroyer living beings through apurvakāraņa-sayana (unprecedented volitions).
purified
Q. What is the definition of apūrvakaraṇa-sayanat ? A. The terms karaṇa here means mental disposition or bhāva and apūrva means as nonexistant or unprecedent. Thus the word apūrvakarana indicates that there are innumerable types of mental dispositions with respect to many living beings varying gradually in each samaya, from the very beginning. However, mental dispositions of the living beings of given time are unusually different from that of the mental dispositions of the living beings of any intended time. Thus apūrvakarana bhāva are those which did not exist at previous times and have unusual character at each time. The living beings with such apūrvakarana bhāva are called apūrvakarana-sayanat (unprecedented volitionally restrained). They could be aupașamika (karmic subsiders) and kșapak (karmic destroyers) because of his tendency towards destruction or subsidence of karma.
Q. In the eigth stage neither there is kşaya (destructional) nor upaşam (subsidence) of karma. Hence how the living beings at this stage could be called aupașamika and kşapak ones? A. As a rule, the living being at eighth stage does destroy or subside the conduct-deluding karma in future. Hence, the living being at this stage has been formally or figuratively called kșapak (derstroyer) or upașamak (invader) of karma.
Q. What is the bhāva at this stage out of the five kinds ? A. There is the destructional volition in kșapak being and there is the aupașamika (subsidential) volition in upașamak being. It is said also in the GJ verses 51, 52 and 54: "The Jinas have said that there is never any similarity in volitions of the living beings existing at different times in the stage of apūrvakaraņa (unpreccedented) bhāva. However, there is both similarity and dissimilarity in bhāvas of the living beings existing at the same time. In this spiritual stage therefore the living beings existing at different times have apūrvakarana bhāva. The living beings with such volitions are always ready for upaşam (subsidence) or kşaya (destructional) sub-species of the mohaniya (deluding) karma.”
The next verse describes the last stage of the spiritual stages involving coarse kaşaya (passions):

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Aniyaṭṭhi-bādara-sāmparāiya-pavittha-suddhi-sanjadesu atthi
uvasama khavā -17*
In general, there are upaşamak (invader) and kṣapak (subsider) of karma living beings who realize greater purity.
* The meaning of the term samparaya is passion and the term badara means gross that lead to the conjoined word gross-passions. The living beings associated with gross-passions could be subsiders or destroyers of karma, at this stage some subsiding some species of the deluding karma and destroying some other species in future. This spiritual stage is of karma subsidential and destructional.
Q. Who is called the aniyatthi?
A. The term nivṛtti in here means dissimilarity and the term aniyatth or anivṛtti will mean the opposite of dissimilarity. This means that there exist living beings with anivṛtti kāraṇa (similar volitions) and nivṛtti (dissimilar volitions) existing at the same time and there are only of the living beings existing at different times. Such a living being is called aniyatthi.
Q. Why there are no separate spiritual stages for upaşamak (subsidence) or kşamak (destructional) types?
A. This is because there is similarity with respect to volitions in both the cases. This is explained in the GJ verses 56-57: "The living beings existing at the same time are mutually different with respect to their shapes of body etc. and there is no difference in their volitions. They are called anivṛtti kāraṇa jiva (similar volitioned living beings) They always have similar volitions every instant of time involving infinite-times volitional purity. These living beings are destroyers of the forest of karma through the fiery flames of very pure meditation."
Next verse describes the last spiritual stage of the kuṣeela (conductstained) monks:
Suhuma-sāmparaiya-paviṭṭha-suddhi-sanjadesu atthi uvasamā
khavā - 18
In general, there are karmic subsider and destroyer living beings purified through volitions involving subtle passions.
Q. What is suhuma-sāmparāiya ?
A. The subtle passions are called suhuma-sāmparāiya and there are both upaşamak (subsidence) or kṣamak (destructional) types among them. In this spiritual stage, the living being which destroys, will destroy and has destroyed many of the karmic species has upaşamak bhāva

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(destructional volition). Similarly, it, which also subsides, will subside and has subsided many of the karmic species in this stage has aupașamika bhāva (subsidential volition). The kşayika bhāva living being possess destructional volition with respect to samyag-darsana. In contrast, the living being on subsidential ladder has both upaşamaka and upaşamak volitions, as he can ascend the subsidential ladder through both the categories of righteousness.
Because of both adjective terms-unprecedented and similar (in v.16 and 17 above), this spiritual stage has altogether different types of volitions in comparison to the earlier stage. It is said also in the GJ, verse 59: “The subtle-passioned living beings with sūkshma-samparāya (subtle nominal desire), which is the first pure contemplation, have apūrvakārana (unpreccedent) spardhaka (large group of karmic atoms - supervariform).”
The next verse describes the last spiritual stage of upaşamaśreņi (subsidential ladder):
(Uvasanta-kasāya-vīyarāya-chadumatthā - 19. In general, there are living beings - upașānta-kaṣāya-vitarāgi ( detached non-omniscient with subsided passions).
Q. What is definition upaşānta-kaṣāya-vitarāgi ? A. One whose passions have gone subsided is called upașānta-kaṣāyi and his detachment destroyed, he is called vitarāgi. The jnānāvarniyia (knowledge-obscuring) and darşanāvarņiya (conation-obscuring) karmas are called chadma (disguise), and the person under these two obscurations is called chadmasta (non-omniscience). With the adjective word 'detached,' vitarāgi. chadmasta is excluded up to the tenth gunastāna. There is the adjective of ‘subsided passions' which excludes the twelfth higher stage individuals.** It is also said in the GJ, verse 61: "The spiritual stage of upașānta-kaṣāya (subsided passions) is the purified bhāva form arising due to the subsidence of mohaniya (deluding) karma in totality, like the muddy water is purified by kataka fruit (nut-plant), or the clean water of the pond in the autumn.'
** The duration of this stage is just 48 minutes. Afterwards, the living being at this stage falls down to lower stages due to the completion of life span or completion of the duration of this stage.
The next verse describes nirgrantha (passionless) spiritual stage:
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Khina-kasāya-viyarāya-chadumatthā - 20.
In general, there are kṣina-kaṣāya-vitarāga-chadmastha jivās (detached non-omniscience with destroyed passions).
A.
Q. What is the definition of kṣiṇa-kaṣāya- vitarăgi-chadmastha jivās? One who is kṣiņa-kaṣāyi (passions destroyed) and the vitarāgi (detached non-omniscient) is called kṣina-kaṣāya- vitarāgi-chadmastha despite being detached due to destroyed passions.
Q. A kṣina-kaṣāyi, no doubt is a vitaragi. Then how the word vitaragi should be understood?
A. The word vitarāgi has been associated with this stage to indicate the fact that this stage refers to the psychical destruction of passions rather than namal - representational or substantive destruction of passions. The Jinas have said that the passionless living being of the psyche stage has become pure like the water kept in a quartz vessel.
Q. Out of five bhāvās, what kind is found in this stage?
A. Prior to attining this stage, there is the total destruction of mohaniya karma. Hence, this stage has kṣiṇa-bhāva (destructional volition). It has been also said in the GJ, verse 62, which has been explained in the precceding explanation.
The next verse describes the thirteenth spiritual stage of sajogi (dynamic) omniscient:
Sajoga-Kevali - 21.
In general, there are dynamically omniscient living beings.
Q. What is the definition of Sajoga-Kevali (dynamically omniscient) ? A. The term kevala here indicates absolute knowledge or omniscience. The absolute knowledge is defined as the non-assisted knowledge which does not require the services of senses, mind and light. The omniscient is the individual who achieves this type of knowledge. The term yoga (activity) is the propensity of mind, speech and body. The individual having these activities is known as sajogi. The term activity is the terminal pointing here, and therefore, it indicates that all other precceding and lower stages including this stage also have activities. This spiritual stage has kṣina-bhāva (destructional volition) due to the destruction of four destructive karma including mohaniya karma. It is also said in the GJ, verses 63 -63: "The eternal scriptures state that the individual is called [a] kevalin, because of his non-assisted infinite knowledge and

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conation, [b] sajogi because of activities or dynamism and (c) Jina because he is devoid of destructive karma whose ignorantial darkness has been totally destroyed due to the rays of the sun of omniscience and who has been designated as supreme soul because of the manifestation of nine absolute prodigies of: destructional donation, gain, enjoyment of consumables and nonconsuimables, infinite potency, righteousness, omniscience, conation and conduct.”
Now the venerable saint scholar Puspadanta states the next verse to describe the last and the fourteenth spiritual stage:+
Ajoga-Kevali - 22. In general, there are ajoga (static) kevali jivās.
Q. What is the definition of ajoga kevali ? A. He is the kevalin with absolute knowledge and whose activity has become static.
Q. Which of the five bhāvās, the ajoga kevali possess ? A. There is the kşayika bhāva in this stage because of the total destruction of the four ghātiya (destructive) karmas* and because of their readiness to destroy the other four aghātiya (non-destructive) karmas.** It is said also in the GJ, verse 65: "The ajoga kevali are those [a] who have enriched themselves with 18, 000 types of good conduct; [b] have completely stopped the influx of new karma; (c) are devoid of new karmic bonds, and (d) are omniscience without activity.” *Agāvaraniya, darśanāvarniya, mohaniya and antarāya. ** Nāma, āyuh, gotra and vedaniya. + Thus there are fourteen spiritual stages leading from the volitioned state to the volitionless stage and from the lowest to the infinite internal energy. All these stages are associated with worldly living of the jivās.
Now Puspadanta describes Siddhā (spiritual stage) after describing the worldly fourteen stages:
Siddhā-cedi - 23. In general, there are the Siddhā jivās.
Q. Who are the Siddhā jivās ? A. They are those who have [a] destroyed all karmas [b] acquired the infinite bliss discarding worldly materials, [c] possessed all the best attributes, and [d] their soul is lesser than their terminal body, and [e] reside at the apex of the universe.
- to continue in forthcoming issues
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Jinamañjari, Volume 19, No.1, April 1999
BOOK REVIEWS
Absent Lord: Ascetics and Kings in Jain Ritual Culture. Lawrence A. Babb. New Delhi, Motilal Banarsidass. 1996. Pp. xi + 244, with glossary and index. ISBN 0-520-203240-0 (Softbound)
In his latest contribution Lawrence Babb creatively ruptures the bubble of analytic structuralism (Bell, Ritual Theory, 1992:219) and imaginatively illustrates how components of the new ethnography can assist his readers to grasp the subtle complexities contained within the "ritual theatre” of the Mūrtipūjak Jaina communities of Jaipur and Ahmedabad. Although this methodological approach to ritual might appear to be a stretch for some, he comes by it honestly and makes an excellent case -- by paraphrasing Milton Singer. According to Babb, there is, however, a notable difference between the "theatre of entertainment" and the "theatre of ritual." . That is, the players are largely their own audience, and their roles and performance are reliant on interactive discourse.
Babb raises two fundamential questions: Who are the performers? And what are their roles in the dramaturgy of their ritual theatre? Human performers are the obvious answer. Babb recognises in chapter 1, it is not just living beings who perform in ritual: objects of worship, but also play an important role. As he states, the dynamics operating within the püjās. are not just a matter of interaction between the lay-community and the living sādhus and sādhvis.. There is also ritual interaction between the living cast of characters and non-living other such as the Tirthankaras (as both liberated soul -- the ascetic ideal - - and mūrti as axis muni of the community) and the Dādāgurus (the icons of the magical monks of antiquity).
According to Babb the worship of the Tirthankaras presents a significant existential problem for those acting within the ritual performance. Unlike the Hindu pūjā in which the worshiper anticipates the recovery of offerings that have been transformed by a divine receiver, the Tirthankaras are fully liberated beings who are no longer present cosmologically. He is quick to anticipate the obvious question: With the mūrti being both a non-living and non-present entity, what posible meaning and expectation lies behind such rituals as the snātra pājā (bathing of the mūrti) or the aștprakāri pājā (eight-fold worship)? He
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finds answers from his Jain informants: Jain layperson does not expect a response from the non-living other, but instead, sees ritual performance as both a self-reflexive act and a "substitute form of world renunciation. .. the principal means of shedding the karmas that impede the soul's liberation.” In this sense, he explains the act of worshipping an ascetic becomes -- itself -- and ascetic act (92)."
Babb from his fieldwork in the temple complex at Mohan Bārī (Khartar gaccha) notes a ritual subculture that venerates the sacred personae of the Dādāgurus. Contrary to the ritual network involving the Tirthankaras, the Dādāgurus are deceased ascetic monks who have the ability -- much like the ritual complex involving a Hindu deity -- to respond to and within a ritual performance. In this context of the Jain veneration for asceticism, however, there is a crucial difference between the ritual theatre of the Dādāgurus and Hindu pūjā. As Babb states, "the Dādāgurus are beings who behave as deities, but because they are ascetics, belong to the category of beings who are (unlike deities) truly worthy of worship (134)." Or as we see in chapter 4, "these figures are Jain ascetics who can be worshipped as gods [and] who will advise their devotees in worldly matters... they provide a method for lay Jains to pursue worldly values through ritual, but ritual of a sort that is nonetheless indisputably Jain (171)."
In conclusion then, what are Babb's readers to make of this Svetāmbara ritual culture? On the surface, particularly in the case of the Dādāgurus, one is forced to ask: is this not a ritual culture that in many ways is similar to that of the Buddhist, Vaişņavas and Saivas? In a penetrating discussion that incorporates theory, tradition and conversations with his informants, Babb recognises that ritual reflexivity within the Jaina context may have "ritual crossovers" with the larger South Asian ritual culture. There is, however, an extremely important component that separates the Jaina tradition from that of "Puştimārg's Krşņa and Raheja's godlings (192)." In the situation of any ritual theatre involving the Tirthankaras, there is both an ideal of asceticism and a non- transactional presence that is absent from the rituals of the Jaina's South Asian counterparts. As for the mixed case of the Dādāgurus, Babb concludes that despite the similarities to the worship of Hindu deities, the object of worship is still a Jain ascetic whose "historical context and supporting ideology are clearly Jain. . . .not a form of "Hinduism" grafted onto a Jain base (193)."
- Mikal A. Radford, a Doctoral student in the Department of Religious Studies and Social Sciences at McMaster University Canada.
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Open Boundaries: Jain Communities and Cultures in Indian History. Editor, John E. Cort. Albany: State University of New York Press. 1998. Pp. vii + 264 with references and index. SUNY Series in Hindu Studies: Wendy Doniger, editor. ISBN 0-7914-3786-8 (Softbound).
In his critical analysis of the relationship between language and literary interpretation during the 1970s, Stanley Fish comme Foucault posits that the interpretive activities of a particular community should never be considered self-contained phenomena operating with a fixed language system. Instead, it should be thought of in terms of being an entity whose capacity to interpret meaning is interwoven within the fabric of an interactive social context (Is There a Text in This Class?).
A parochial community, therefore, derives meaning from the practices and assumptions of its own institutions, and to a varying degree, from those institutions that surround it. From this perspective, Fish continues, the meaning of either an utterance or text "is not a function of the values its words have in a linguistic system that is independent of context; rather, it is because the words are heard as already embedded in a context that they have a meaning (309)."
Pushing past the margins of literary criticism, the contemporary discourse over meaning, identity and context has solidly entered the parlance of ethno-historians, and is what lies at the heart of each of the ten essays that comprise Open Boundaries: Jain Communities and Cultures in Indian History. As John Cort states in his introduction:
A sense of self-identity, whether in terms of the individual person or a social group, in never constructed in isolation, but rather is always a contextualized process, in which the sense of "self" is in dialogue, opposition, or dialectical relationship to a sense of what is "not-self" or "other."...Jains have always been active participants in larger contexts, and that therefore any adequate understanding of the Jains and Jainism must take into account both the larger contexts and the forms of Jain participation in those contexts (1-2).
Though admittedly not a new methodological approach, what makes this text superbly unique is its ground-breaking application to such a wide range of subject material (i.e., philosophy, the use of mantra, narratives, art, ritual, sex, politics, and issues of gender). Beginning with discussions held at the 1991 University of WisconsinMadison Conference on South Asia and a four-day workshop held in the summer of 1993 at Amherst College, it was agreed that for too long the Jaina tradition has been studied in both the West and Indian as either a
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singular, isolated socio-religious entity that has been mistakenly separated from the context of a multiform South Asian landscape, an "essentially marginal, unimportant heterodox group (Buddhist-Jain)," or "a degenerationist model in which a supposedly pure, original ur-Jain doctrine is contrasted with the later impure, degenerated Jainism largely composed of half-understood and ill-digested Hindu influences and accretions (3)."
The intent of Open Boundaries is to correct this type of reductionist analysis. Though the granting of direction and degree of influence of one tradition on the other may vary from article to article, each of the contributors to this project counter the image of a single, marginalised Jaina identity. Instead, through a wealth of methodologies adopted from such sub-disciplines as textual analysis, history, art, economics, anthropology, sociology and ritual studies, this project has pieced together an extraordinary mosaic that portrays the Jainas as a varied, interactive and dynamic community involved in an energetic discourse not only with its own constituents, but with South Asian society as a whole.
In sum, the scholarship of John Cort, Christopher Key Chapple, Paul Dundas, Gary A. Tubb, James Ryan, Michael W. Meister, Lawrence A. Babb, Indira Viswanathan Peterson, Leslie Orr and Richard H. Davis has contributed to a project that is impressive, innovative, and more importantly, accessible to scholars, students of Jainism, and neophytes alike. Though at times the boundaries between what has been labeled orthodox and heterodox tradition can get disoncertingly blurred -- and this tends to be more a problem for those who have come to depend on static categories rather than a problem with the text itself -- this project is a marvelous foundation to what will hopefully become a new direction in both Jaina and South Asian studies. I
- Mikal A. Radford, a Doctoral student in the Department of Religious Studies and Social Sciences at McMaster University Canada.
Jaina Philosophy and Religion. Monk Nyayavijayaji. English Tr. by Nagin J. Shah. New Delhi, Motilal Banarsidass & B.L. Institute of Indology, 1998. pp.437. Price:Rs. 450. ISBN: 81-208-1490-8.
This is a voluminous work - a translation from Gujarati - encopasses the total philosophy of Jain religion in six chapters. The eastern philosophy always looked into the final end from life-cycle, and in this connection Jainism being one of the oldest religions dating back to the pre-advent of the Aryans has been said to be one of the foremost

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structurally organized traditions that originated in the Indian subcontinent. The blissful happiness of mokṣa in Jainism is pivoted on the lives of all living beings and their individual path as envisaged by precepts of ahimsa, truth, karma, conduct and ethics. This theme was furthermore rejuvanated by Lord Parsva in ca.800 B.C.E. and by Lord Mahāvīra in ca. 600 B.C.E. In the later period Lord Buddha, a contemporary of Mahāvīra, set on the same goal sought the end to the misery of samsāra.
The first chapter deals with the to be liberated soul and its surrounding matter. It follows a chapter on the path and process for liberation. The third chapter deals with metaphysico-ethical and spiritual discussion, followed by a chapter which completely deals with the Jaina doctrine of karma and its mechanism. Chapter five: Jaina Logic presents syādvāda doctrine supported by saptabhngi (sevenfold judgement) and naya (standpoint). The final chapter underlines the very nature of Jaina secular philosophical approach quoting and enumerating ancient texts and saint scholars - Haribhadra, Hemacandra, Yasovijaya and Jayasekharasuri.
For students of religion or for ordinary folks, this book is very helpful for it presents a comprehensive Jaina philosophy very authoritatively. This could have its title as Basics of Jaina Philosophy. - Ashij J. Kumar, a Graduate student in the Department of Environmental Sciences at University of Toronto, Canada.
Acārānga: Restoration of Śrutaskandha Chapter One. Editor. Dr. K.R. Chandra. Ahmedabad, Prakrit Jain Vidhya Vikas Fund, 1998.
The Acaranga, first canonical text of the Jainas, has been examined and discussed by Dr. K.R. Chandra, Professor Emeritus of Prakrit, as to its language and character. Conceptualizing and applying the principles of restoration of the original language of ancient Ardhamāgadhi, Chandra has produced this work on the basis of available archaic readings from the manuscripts. Substantiated by evidence, he has produced a glimpse of some phonological and morphological features of the text.
The work is indeed an important contribution to the study of phonology, morphology and orthography of Prakrit language. - Ashij J. Kumar, a Graduate student in the Department of Environmental Sciences at University of Toronto, Canada.

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