Acarya Yativrsabha and His Mathematical Contributions
ABSTRACT
In the Jaina School, two systems have been developed on profound basis of mathematics. One of Karma syatem and the other is the cosmological system. Yativrsabha is credited with compilation of both the system theories. The Karma system theory is known to be in pulverised verses from the Kasaya Pahuda of Gunadharacarya (c. Ist century A.D.) containing 233 verses. The pulverised verses are about 7009 in number, said to be very profound and having endless implications.
The cosmological system theory, on the other hand, contains 5677 verses and called the Tiloyapannatti. The Kasaya Pahuda deals with biophysical phenomena in nature between the ralation of bios and matter, describing the cirumstances and conditions or controls, which enable certain bios to get their Karmic bond annihilated.
As this phenomena is so much complicted that the mathematics required for its depiction was modeled as postuniversal measure system, the foundations of which are found to have laid in the Tiloyapannatti.
Thus the Tiloyapannatti, measuring information about the three universes, is classified in the Karnanuyoga group of texts in which mathematical operations aid to their study.This work is in Prakrta language.
It deals with cosmography and cosmology including certains topics of religioscientific and cultural interest. The mathematical contents of this work have already been published by Shastri, N.C., Jain, L.C., Saraswati, T.A. Anupam Jain, Gupta, R.C. and Shivakumar, N.
Credit of some more works like Kammapavadi Curni, Sataka Vurni, Sittari Curnin and Karana Sutra is given to Yativrsabha but none of it is available at present. Shastri and Saraswati and some other scholars are of the opinion that Yativrsabha belongs to 2nd century A.D. while another group which include Jain, L.C. place him during 473-609 A.D.
In the present paper we discuss the mathematical contributions of Yativrsabha on the following points.
| 1. Measurement System | 2. Number System |
| 3. Symbolism | 4. Geometry |
| 5. Solid Geometry | 6. Series |
| 7. Logarithms |
Jaina Literature is very vast and varied. It contains many types of knowledge. Mathematics is an integral part of Jaina literature. It is used to explain cosmological details and Karma Throty Theory. Jainacaryas were never internded for the development of mathematics but they used it as a tool to give authenticity and accuracy of philosophical details. Due to it mathematics envolved in Jaina literature so deeply that without proper knowledge of mathematics one cannot understand the many Jaina cannons accurately. Therefore some acaryas created some purely mathematical texts for the purpose of mathematics teaching. In this way the mathematical material found in Jaina literature can be classified into two grops.
I. Canonical Class— It includes the Mathematics found in Jaina philoisophical texts. These texts belong to the Karnanuyoga and Dravyanuyoga group of Jaina literature. The mathematical material found in Pancastikaya. Tiloyapannatti, Dhavala, Jambudivapannatti Samgaho, Trilokasara etc. Come under this class.
II. Non-canonical Class— It includes the mathematics found in purely mathematical texts like Patiganitasara (Trisatika) of Sridhara (799 A.D.). Ganitasarasamgraha of Mahaviracarya (850 A.D.), Ganita Tilaka commentary of Simhatilakasuri (13th century A.D.) etc. Dipak Jadhav named it exclusive class.Personal discussion in Now. -03.
Life — Acarya Yativrsabha who is a great Jainacarya of Digambara Jaina tradition composed many texts. Not much is known about the life and work of Yativrsabha. He was disciple of the ascetics Aryamanksu and Nagahasti. Aryamanksu had apravahyamana source material while the later had pravahyamana source material.
Yativrsabha was taught by both ascetics and is said to be Sisya (Disciple-scholar) of the former and antevasi (resident scholar) of the later.N. C. Shastri, Tirthankara Mahavira aura Unaki Acarya Parampara, Sagar, 1974, Vol, 2
In the Srutavatara of Indranandi, Aryamanksu and Nagahasti are mentioned as disciles of Gunadharacarya who has compiled kasaya Pahuda Sutta. Therefore from this fact and on the basis of study of several portions of Tiloyapannatti, N.C. Shastri concluded that the period of Yativrsabha should be about 176 century A.D. He said that the reference of later dynastie and events which are found in Tiloyapannatti (T.P.) seems to be added by any later Acarya during the course of editing or copying it.Ibid
A well known scholar of Jaina mathematics L.C. Jain writes that after a long discussion of various facts and the mention of old ancient Prakrit texts by Yativrsabha as Aggayaniya (Maggayani), Saggayani, Ditthivada, Parikarma, Mulayara, Loyavinicchaya and Loganni, A.N. Upadhye and H.L. Jain confirmed the existence of the author, Yativrsabha as flourishing later than Gunadhara, Aryamakshu, Nagahasti, Kundakunda,
Sarvanandi and Kalkin (473 A.D.), and earlier than Virasena (816 A.D.) and possibly also Jinabhadra Ksamasramana (609 A.D.).L. C. Jain, Mathematical Content of Digambara Jaina Texts of Karnanuyoga group, Kundakunda Jnanapitha, Indore, 2003, Vol.-1. Introduction, P. 9. Thus his period may be in between these dates from 473 A.D. to 609 A.D.4 But a senior scholar of History of Indian Mathematics, T.A. Saraswati rightly concluded that Tiloyapannatti is perhaps later reduction of much earlier work.T. A. Saraswati, Grometry of Ancient and Medieval India, M.L.B.D., Delhi, 1979, p.X
Works of Yativrsabha
| I.Tiloyapannatti | II.Kasayapahuda Curni |
Above two are undoubtedly the work of Yativrsabha but the credit of following four works is also given to him.
| I.Kammapayadi Curni | II.Sataka Curni |
| III.Sittari Curni | IV.Karna sutra |
Presently none of them is available.L.C. Jain and Anupam Jain, Philosopher Mathematicians, D.J.JI.C.R. Hastinapur, 1985, p.13. The following characterstics of Yativrsabha have been stated by N.C. Shsatri.
1.Yativrsabha had the knowledge of eight Karma Pravada.
2.From authority of Nandisutra, he could be established to have the knowledge of Karma Prakrti as well.
3.He was disciple of the Aryamanksu and Nagahasti.
4.He was not only a spiritual ascetic but also a grreat scholar.
5.There are difference of opinion between Bhutabali and Yarivrsabha which is clear from the study of Dhavala and Jayadhavala.
6.Yativrsabha is as great as Bhutabali in view of greatness of their personality. Their opinions are recognised universally.
7.Yativrsabha has reflected the maxim style (paddhati) in his Curni maxims.
8.The Curni maxims have been composed for the assimilation of traditionally prevalent knowledge.
9.Yativrsabha had the knowledge of Agama. Yet he achieved all the learning of prevalent teaching style in all traditions and made use of his fine talents in composing Curni Sutras.
Contents of Kasayapahuda Curni
This contains the following 15 chapters—
1.Prakrti-vibhakti (configuration-analysis)
2.Sthiti-vibhakti (life time analysis)
3.Anubhaga-vibhakti (energy analysis)
4.Pradesa-vibhakti (point analysis)
5.Bandhaka (binder)
6.Vedaka (feeler of pathos)
7.Upayoga (role)
8.Catuh-sthana (quadrupier-station)
9.Vyanjana (synonym)
10.Darsanmohasbamana (Subsidence of vision-charm)
11.Darsanlohaksapana (annihilation of vision-charm)
12.Samyamasamyama Labhdi (inhibition-non-inhibition attainment)
13.Samyama Labdhi (inhibition attainment)
14.Cairtramohopasaman (subsidence of disposition charm)
15.Caritramohaksapana (annihilation of disposition charm)
These chapter contain a high class mathematics related with Karma Quantum system theory. For detaila one can refer the works of L.C. Jain published by Kundakunda Jnanapitha, Indore under the title `Mathematical Contents of Digambara Jaina Texts of Karnanuyoga group’ and papers appeared in I. J.H.S. and other books.Anupam Jain, Survey of the work done in the field of Jaina Mathematics Tulsi Prajna (Landnun), 11 (1-3), 1983
Contents of the Tiloyapannatti
The following chapters have been detailed in Tiloyapannatti—
1.Samanya Loka Swarupa (General nature of universe)
2.Naraka Loka (Helish universe)
3.Bhavanavasi Loka (Bhavanavasi universe)
4.Manusya Loka (Human universe)
5.Tiryak Loka (Sub-human universe)
6.Vyantara Loka (Vyantara universe)
7.Sura Loka (Heaven universe)
8.Jyotirloka (Astro universe)
9.Siddha Loka (Accomplishment universe)
The first chapter deals with various types of measuresT.P., 1.91-1.132., the ultimate units are abstract and correlated through set-theoretic measures, mensuration of the universe in various topologically deformed shapes, but equivalent in volumeT.P., 1, 149-1.233.. Volumes of figure enveloping the universe have been worked out for air and vapours.T.P., 269 et seq.
The residence holes of the hellish bios form series and their total numbers are calculated through given formulaeT.P. 2.26-2.158..13 Some set-theoratic measures are given through points contained in stretches as products of the Jagasreni (word-line) and roots of Ghanagula (finger-cubed).T.P., 2.195-2.196.14 Formula for the sum of geometric progression has been applied.T.P., 3.80 et seq.
Mensuration formulae for circle, circumference, arc, chord, area etc. are given and calculated out of Jambudvipa and its regions etc.T.P., 4.9, 4.182.
Measure units of time, from an instant numerate, innumerable and various types of infinites are constructed axiomatically and insertion of known existential sets.T.P., 4.285 et seq. Here the process of vargitasamvargita (square piling) has been used and new generation of infinites greater than previous one generating infinite sets.
Measurement of the meru in frustums of cones are given.T.P., 4.1780 et seq
At the time of Vardhamana Mahavira, 18 languages and 700 dialects have been mentioned to be prevalentT.P., 4.901 There are Karmic conditions mentioned for tele-touch, tele-odour, tele-taste, tele-audio and tele-vision miraclesT.P., 4.987-4.997
Comparative measurements of areas of successively doubling circles and rings are given through formulaeT.P., 4.2525-4.2763 Measures of increased areas of circular islands and oceans in successive increased diametrs are compared through formulaeT.P., 5.277.
Symbolism for measures of sets of firebodied bios etc. is used through construction process which is set-theoreticT.P., 5.280. This has been called number representation at the end of the detailed description.
Comparability is given for various sensed-bios through manipulation of symbolismT.P., 5.314-5.315.
Seven chapters is on astronomy. Description is more or less the same as described in fourth chapter of Trilokasara.
Some efforts to study the mathematical contribution of Yativrsabha and his great book Tiloyapannatti have been made. Some of them are followings.
1.Anupam Jain, Ganita ke Vikasa mem Jainacaryon ka Yogadana (in Hindi), M. Phil Project Report, Meerut University, 1980.
2.Anupam Jain, Survey of the Work done in the field of Jaina Mathematics, Tulsi Prajna (Ladnum), 11 (1-3), 1983, p. 15-27
3.Anupam Jain, Darsanika Ganitaja-Acarya Yativrsabha (in Hindi), Arhat Vacana (Indore), 1(2), 1988, 17-24.
4.Anupam Jain, Darsanika Ganitajna-Acarya Yativrsabha ki kucha Ganitiya Nirupanayen (in Hindi), Pt. Jaganmohanlal Felication Volume, (Rewa), 1989, 310-313.
5.L. C. Jain, Tiloyapannati Ka Ganita (in Hindi), included in Jambudiva- papannatti samgaho, Sholapur, 1958.
6.L. C. Jain, Tiloyapannati evam uska Ganita (in Hindi), Volume 1, 49-58 and Volume 2, 6-36, 1994.
7.L. C. Jain, Tiloyapannatti Kc caturthadhikara ka Ganita (in Hindi) Volume 2, Introduction. 1997.
8.L. C. Jain, Tiloyapannatti ke Pancave aur Satave Mahadhikara ka Ganita, Volume 3. Tijara edition, 1997, 35-35.
9.L. C. Jain, Mathematical Content of Digambara Jaina Texts of Karnanuyoga Group, Volume 1 and 2, Kundakunda Jnanapitha, Indore,2003.
10.L. C. Jain & Anupam Jain, Philosopher Mathematicians, D.J.I.C.R. Hastinapur, 1985.
11.N. C. Shastri, Tiloyapannatti Mem Sredhi Vyavahara Ganita Sambandhi Dasa Sutra ki Utpatti, Jaina Siddhanta Bhaskara (Arrah), 22 (2), 1953. pp. 42-50.
12.R. C. Gupta, Jambudvipa ke Ksetron evam Parvaton ke Ksetraphalon ki Ganana (in Hindi), Tiloyapannatti, Vol. 3, Kota, 48-49.
13.T. A. Saraswati, The Mathematics in First Four Mahadhikaras of Trilokaprajnapti, Journal of Ganganatha Singh Jha Research Institure (Ranchi), 8, 27-50.
Inspite of these valuable efforts, we fail to expose the overall impact of Yatibrsabha on Indian Mathematics. It is also true that we can not make justice with yativrsabha as a Mathematician.
During the course of our study we have found that the mathematical content available in Tiloyapannatti can be classified in the following headings.
1. Measutement System
There arise the necessity of least measuring unit to measure the universe so there defines various units of Kala to understand the relation between various measuring units and Jiva-rasi. The part of the matter which is capable in every way is said to be Skandha and its half part is said to be Desa and again half is Pradesa.
The infinitely small part of the individual part of the matter which cannot be subdivided further is paramanu. Hence it is more fine and different from the present definition of paramanu in science.
The measurement system defined in another way is detailed below—
| Infinitely small part of Dravya | 1 Paramanu |
| Ananta Paramanu | 1 Avasannasanna |
| 8 Avasannasanna | 1 Sannasanna |
| 8 Sannasanna | 1 Trutarenu |
| 8 Trutarenu | 1 Trasarenu |
| 8 Trasarenu | 1 Ratharenu |
| 8 Ratharenu | 1 Balagra of uttama-bhogabhumi |
| 8 Balagra of Uttama bhogabhumi | 1 Balagra of madhyama-bhogabhumi |
| 8 Balagra of madhyam bhogabhumi | 1 Balagra of jaghanya-bhogabhumi |
| 8 Balagra of jaghanya bhogabhumi | 1 Liksha |
| 8 Liksha | 1 Jnu |
| 8 Jnu | 1 Yava |
| 8 Yava | 1 Utsedhangula |
| 500 Utsedhangula | 1 Pramanangula |
| 6 Atmangula | 1 Pada |
| 2 Pada | 1 Vitasti |
| 2 Vitasti | 1 Hatha |
| 2 Hatha | 1 Rikku |
| 2 Rikku | 1 Danda or Dhanusa or Musala |
| 200 Danda |
1 Krosa |
| 4 Krosa | 1 Yojana |
He also defines the following various Rasi according to Angula.
| Partarangula | Suchyangula2 |
| Ghanangula | Suchyangula3 |
| Jagata Pratana | Jagasreni-2 |
| Loka | Jagasreni-3 |
All the above are measures of Ksetra. Yativrsabha also prepared the list of Kalamana Samaya is the smallest unit of time which is defined in a unique way.
Starting from samaya, the smallest unit of time which is countable is defined under the name Acalatma-
| 1 Acalatma | (84)31 x (10)90 years |
The use of zero is remarkable here in 2nd century A.D. Under the catagory of uncountable number, playopama, sagropama etc. are defined, some of them are given here.
| 1 Vyavaharapalya | 19/24 x43 x (2000)3 (4)3 (24) (500)3 (8)21 |
| 1 Vyavaharapalyopama | 100 Vyavaharapalya |
| 1 Uddharapalya | Vyavaharapalya x uncontable crore year |
Similarly Addha palya, Vyavahara playopama, Uddhara sagaropama etc. are also defined.
2. Number System
To explain different philosophical terms, the numbers are classified in 21 ways.
The distinct feature of this classification is the establishment of infinity greater than infinity. Not only definition and grading but the examples are also available.
3. Symbolism
Many symbols are used in Tiloyapannatti to express several things. Some `time‘ symbols change their meaning. It is a drawback but generally symbols have the following meaning (Which are translated into English in some cases) shown against their names.
| Minus | efj |
| Root | cet |
| Jagasreni | |
| Jagapratara | |
| Ghanaloka | |
| Rajju | j |
| Palya | he |
| Suchyangula and Utsedhangula | 2 |
| Avali | 2 |
| Pratarangula | 4 |
| Ghanangula | 6 |
| Multiplication | 1 |
4. Geometry
Tiloyapannatti is a leading and oldest text of Karnanuyoga section. It is very important from the geometrical point of view. Details of the geometrical formulae can be traced in any article mentioned earlierAnupam Jain, Darsanika Ganitajna-Acarya Yativrsabha, Arhat Vacana, Indore, 1 (2) 1988, pp. 17-24.. In the words of Saraswati, First four Mahadhikaras of Tiloyapannatti form a store house of mathematical formulae.27
f circumference of the circle is P, Chord of the circle is C, Arc of the segment of circle is s, Height of the segment of circle is h, Radious of the circle is r, Diameter of the circle is d and Area of the circle is a, then.
| 1.Circumference of the circle P | |
| 2.Square of the chord of 1/4 of circle | 2 r2 |
| 3.Chord of the circle C | |
| 4.Arc of the Segment of the circle S | |
| 5.Height of the segment of circle h | |
| 6.Area of the Segment of the circle a |
Yativrsabha used Jaina Value of , is & 3 (Approx.).
5. Solid Geometry
In the process of finding the volume of universe and other volumes, we find many formulas of Soild Geometry. Some of them like volume of cylinder are folowing-Here I am
quoting a unique formulae for the
Volume of the conch like figure
| 1.Volume of the right circular cylinder | |
| 2.Voume of the frustum of Prism | Area of base x Height |
| & Area of the base | x Perpendicular distance between to |
6. Series
Subject matter of A. P. and G. P. has been discussed in detail in Tiloyapannatti. At several places we get the formula and example for finding first term, sum of n terms, common difference etc. These have been discussed by Saraswati & L.C. Jain in detail.
7. Logarithms
The use of logarithm has been made to corelate the time and space units
| F = F | Plog 2P |
| where F | Finger point set suchyanhula, |
| p | Addha palya (palya samay rasi), |
| L | [ F3] log2P here |
| L | Jagasreni |
| A | Proper innumerate. |
Many other material of Mathematicians interest is available but due to limitation of space 1 am not quoting them.
In short, Mathematical Contributions of Yativrsabha is very outstanding and useful.
Anupam Jain and N. Shiva Kumar
Arhant Vachan-April-Sep.-2005-Page31-42