INTRODUCTORY

The present volume contains three prarupanas, namely, Kshetra, Sparsana and Kala , out of the eight prarupanas of Jivatthana, of which two, namely, Sat and Dravya-pramana have already been published in the previous three volumes, while the last three namely, Antara, Bhava and Alpa- bahutva are going to be included in the next volume.

The kshetra prarupana contains 92 sutras and concerns itself with the determination of the volume of space that living beings occupy under the various conditions of life and existence. The Sutras confine themselves to the treatment of the subject under the usual fourteen spiritual stages (Gunastbanas ) and the fourteen soul-quests (Margna-sthanas ) But the commentator introduces ten other conditions of life which have to be taken into consideration. These fall under three main classes, namely, the place of habitation of the beings (Svasthana), their expansion (Samudghata ) and their journey for rebirth (Upapada) The first of these includes the usual place of habitation (Svasthana-Svasthana) and Places of occasional visits            (Viharvat-svasthan) The expansion of the soul-substance beyond its usual volume (Samudghata) may be due to pain (vedana) or passion (Kashaya) or for a temporary transformation of personality (vikriya), or for a visit to the next place of birth just before death (Maranantika) or by effulgence of luster for evil or good (Taijasa), or for reaching a learned person for the removal of a doubt in knowledge in the case of saints (Aharaka) , or for getting rid of the remnant karmic bonds in the case of an all-knowing saint (Kevali-samudghata). Thus, the commentator calculates the volume of space occupied by the living beings in these ten different conditions under the different spiritual stages and soul-quests.

The spatial units adopted for these measurements are five, namely, (1) the entire universe (Sarva-loka), (2) the lower universe (Adholoka), (3) the upper universe (Urdhva-loka ), (4) the middle world (Madhyaloka). And (5) the human world (Manusa-loka). To make these standards definite and precise, the commentator divides the limitless space into two, namely, the Alokakasa which is pure void and limitless, and the Lokakasa which is situated in the middle of the former, where life and matter subsist and which is limited. It is this Lokakasa which has been adopted as the largest measure in the treatment of volumes.  As regards the shape and volume of this universe, the commentator is confronted with two divergent views, According to one view it is in the form of three conical frusta with a common circular section in the middle, while according to the other view it is in the form of three frusta of pyramids with a common rectangular base in the middle. Virasena with his philosophic insight, discriminating genius and mathematical skill ultimately rejects the former view and adopts the latter. His conclusions are that the entire universe (Lokakasa) has a total height of 14 rajjus and is in its volume 73=343 cubic rajjus, consisting of the lower universe which is 196 cubic rajjus and the upper universe which is 147 cubic rajjus. Between the lower and the upper universe is the rectangular section called the middle world which is 1x7=7 square rajjus and which contains in its middle the human world which is a circular area of 45 lakhs of yojanas in diameter. The rajju is thus the standard unit of this spatial measurement and it is only determined as innumerable yojanas long, equal to the smaller side, and 1 of the larger side of the rectangular middle world, 1 of the height of the lower or

7                                                                           7

upper world and 1 of the total height of the entire universe. This discussion as well

     14

as similar others bring to light several geometrical problems that confronted our ancient thinkers, and their solutions throw a considerable light upon the evolution of mathematical professes and theories in this country. We have tried to illustrate some of these by twenty diagrams in addition to a large number of examples.

Under the Sparsana-prarupana which contains 185 sutras, we find the volumes of space similarly considered from the point of view of the past as well as the future status of those beings, in addition to the present to which Kshetra-prarupana confines itself. The question here is the volume of space which beings of different spiritual stages and soul quests ever happen to touch under one of the ten conditions mentioned above, In this connection the determination of the number of heavenly luminaries shining above the innumerable islands and seas gives rise to a number of interesting mathematical exercises, (see pp 150-161 of the text).

In the kala-prarupana which contains 342 Sutras, the consideration is of the minimum and maximum periods of time spent by the souls, singly or in aggregates, in the various spiritual stages and soul-quests. The smallest period of time comprehended is an instant (Samaya) of which innumerable are included in an avali and a breath (prana) which is equal to 2880 of a second (see Vol. III, Introduction p, 34),                                                       3773

The series of periods of time rises on to a Muhurta (48 Minutes), a day, a fortnight a month, a year, a yuga, a purvanga, a purva and so on to a palyopama and a sagaropama and ultimately to an Utsaripini and Avasarpini which constitute a kalpa. The longest period of time conceived and denominated is a pudgala – parivartana (for which see p. 330 text and explanatory note).

In interpreting the mathematical part of these texts I again received very valuable assistance from my colleague Mr. K. D. Panday, professor of Mathematics in king Edward College, Amraoti. Without his help here, as in the previous volume, it would have been almost an impossible task for me to explain adequately the mathematical portions. As I mentioned in the previous volume, Dr, Avadhesh Narain Singh, professor of Mathematics, in the Lucknow University and author of the History of Hindu Mathematics, has taken a keen interest in the mathematical contents of these texts. He has now studied the mathematical portions of the III volume and has obliged me by writing out a dissertation on the mathematical contents of that volume, The same is being published here under the caption ‘Mathematics of Dhavala’. It is expected that he would continue his valuable study of these texts and the readers might look forward to a very interesting note on the geometrics of the present volume in the volume to be issued next.

Another topic dealt with in the Hindi Introduction of this volume is an answer to the objection raised in a certain quarter that Jaina traditions prohibit the study of these sacred Texts by laymen, and therefore these texts should neither be published in a printed form, nor should they be taught in Jaina Pathashalas, nor should they be allowed to be read any where by any body except by the Jaina ascetics. A critical examination of all the traditions bearing on this subject shows that an injunction against the study of siddhanta by the laymen is fond in a few books dealing with the duties of Jaina house-holders. But all these books are found to have been written by a few obscure and insignificant writers belonging to a period subsequent to the 12^th century A.D. Again, they either do not make clear what is meant by siddhanta, or explain it in a manner so as to make the present texts, as well as other available books, fall outside the sphere of Siddhanta. The injunction is, moreover, in direct conflict with the statements of the most ancient and authoritative Jaina writers who have strongly recommended the study of the Jaina texts of the highest kind by all, laymen as well as ascetics, The author of the Dhavala himself lays down in clear and unmistakable terms at every step of his commentary that the sutras as well as the commentary are so designed as to be useful to all mankind, dull as well as intelligent. The tradition is thus found to be a very late one invented by some man of narrow outlook and small brain during the age of decadence and it is altogether incompatible with the whole spirit and ideology of Jainism and with the clear and definite recommendations of all other writers of far greater importance and authority.

A number of queries concerning the meaning and significance of certain statements in the previous volumes have also been answered in the Hindi Introduction.

 

MATHEMATICS OF DHAVALA

Introductory Remarks

It has been known that in India the study of Ganita – arithmetic, algebra, menstruation etc. - was carried on at a very early date, It is also well known that the ancient Indian mathematicians made substantial and solid contributions to mathematics, In fact they were the originators of modern arithmetic and algebra. We have been accustomed to think that amongst the vast population of India only the Hindus studied mathematics and were interested in the subject, and that the other sections of the population of India, e.g. the Bhuddhists and the Jainas, did not pay much attention to it. This view has been held by scholars, because mathematical works written by Buddhist or Jaina mathematicians had been unknown until quite recently. A study of the Jaina canonical works, however, reveals that mathematics was held in high esteem by the Jainas. In fact, the knowledge of mathematics and astronomy was considered to be one of the principal accomplishment of the Jaina ascetics.

We know now that the Jainas had a school of mathematics in South India, and at least one work-the Ganita-sara-samgraha by Mahaviracharya-of this school was in many ways superior to any other existing work of that time. Mahaviracharya wrote in 850 A.D. and his work although similar in general outline to the works of the Hindu mathematicians like Brahmagupta, Sridharacharya, Bhaskara and others is entirely different in details, e.g. the problems in the Ganita-sara-samgraha are almost all different from those in the other works.

From the mathematical literature available at present we can say that important schools of mathematics flourished at Pataliputra {Patna} Ujjain, Mysore, Malabar, and probably also at Benares, Taxila and some other places. Until further evidence is available, it is not possible to say precisely what the relation between these schools was. At the same time, we find that works coming from the different schools resemble each other in their general outline, although they differ in details. This shows that there was intercommunication between the various schools – that scholars and students traveled from one school to another, and that discoveries made at one place were soon communicated throughout the length and breadth of India.

It seems that the spread of Buddhism and Jainism, gave an Jainism  an impetus to the study of the various sciences and arts. The religious literature of India in general and of Buddhism and Jainism in particular is full of big numbers. The use of big numbers necessitated the development of a simple symbolism for writing those numbers, and

1.    Cf. Bhagavati-sutra with the commentary of Abhayadeva suri edited by Agamodayasamiti of Mehesana, 1919, Sutra 90; English translation by Jacobi of the Uttaradhyayana – sutra, Oxford, 1895, ch. 7,8,38. Has been responsible for the invention of the decimal place value notation, It is now established beyond doubt that the place value system of notation was invented in India about the beginning of the Christain Era-the brightest period of Buddhism and Jainism. The new notation was an instrument of great power and accelerated the development of mathematics from the crude Vedic stage – as found in the Sulba sutras – to the finished stage of the fifth century – as found in the works of Aryabhata and Varahamihira.

One very significant fact which has escaped the notice of historians of mathematics is the following : whilst the general literature of the Hindus, the Buddhists, and the Jainas is continuous from the third or the fourth century B.C. right up to the middle ages, in the sense that works representing each century are found. There is a gap in the mathematical literature. In fact there is hardly any mathematical text earlier than the Aryabhatiya which was composed in 499 A.D. The only exception is a fragmentary manuscript known as the Bakhshali manuscript, which probably belongs to the second or the third century A.D. This manuscript, however, fails to give us any detailed information regarding the state of mathematical knowledge at the time of its composition for the reason that is not strictly etc. It is of the nature of notes on some selected mathematical problems. All that we can infer from the manuscript is that the place value numerals as well as the fundamental operations of arithmetic with them were well known, and that some types of problems treated by later mathematicians were also known.

It has already been pointed out that mathematics as found in the Arya-bhatiya is highly developed, for we find it in a treatment of the entire elementary arithmetic of today including the rules of proportion, interest, barter and exchange, and of algebra up to the solution of the simple and the quadratic equations, simple indeterminate equations etc. The question arises _ Did Aryabhata borrow from some foreign source or is the material contained in the Aryabhatiya indigenous and of Indian origin? Aryabhata writes : -

“ Having paid reverence to Brahman, the Earth, the Moon, Mercury, Venus, the sun, Mars, Jupiter, Saturn, and the asterisms, Aryabhata sets forth the science which is honored here at Kusumapura. ‘This shows that he did not borrow from a foreign source. The study of the history of mathematics in other countries leads to the same conclusion, for the mathematics of the Aryabhatiya was far in advance of what was known at that time in any other country of the world. The possibility of borrowing from some foreign source having been ruled out, the question arises : How is it that practically no mathematical work anterior to that of Aryabhata is available ? The explanation is simple enough. The place value system of notation was invented some time about the beginning of the Christian Era. It must have taken four or five hundred years to come into general use. Aryabhata’s works seems to be the first good text book employing the new arithmetic of the place value numerals. Works anther to Aryabhata’s either used the old type of numerals or were not good enough to stand

1. Aryabhatiya,ii,1.

the test of time. I think that Aryabhata’s great popularity as a mathematician was, in a great measure, due to his being the first to write a good text book employing the place value numerals. Aryabhata was responsible for driving out and killing all previous text books. This explains why we get a series of works from 499 A.D. onwards while no works belonging to earlier times are available.

Thus, we have practically no material to trace the development and growth of mathematics in India before 500 A.D. It becomes a question of paramount importance to hunt and trace out works which may give information regarding the knowledge of mathematics in India anterior to Aryabhata. Mathematical works having been lost, we have to scan and analyze Hindu, Buddhist and Jaina literatures in general, and their religious literatures in particular, to find what material we can in order to reconstruct the history of mathematics in India before 500 A.D. In several of the puranas we have portions dealing with mathematics and astronomy. Likewise, in most of the Jaina canonical works there is to be found some mathematical or astronomical material. This material represents the traditional mathematics of India, and such material is generally about three to four centuries older than the age of the work in which it is contained. Thus, if we examine a religious or philosophical work written in the period 400 to 800 A.D. It’s mathematical content will belong to A.D. to 400 A.D.

It is in the light of the above remarks that we regard the discovery of the Dhavala, a commentary on the Satkhandagama, written in the beginning of the ninth century as very important. Mr. H. L. Jaina has placed scholars under a permanent debt of gratitude by editing the work and getting it published.

The Jaina school of mathematics.

Since the discovery and publication of the Ganita-sara-samgraha by Rangacharyn, in 1912, scholars have suspected the existence of a school of mathematics run exclusively by Jaina Scholars. A recent study of some of the Jaina canonical works has brought to light various references to Jainas mathematicians and mathematical works. The religious literature of the Jainas is classified into four groups, called anuyoga, meaning “the exposition of the principles (of Jainism). One of them is called Karananuyoga or ganitanuyoga, i.e. the exposition of the principles dependent upon mathematics. This shows the high position accorded to mathematics in Jaina religion and philosophy.

Although the names of several Jaina mathematicians are known, their works have been lost. The earliest among them is Bhadrabahu who died in 278 B.C. He is known to be the author of two astronomical works : (I) a commentary on the Suryaprajnapti and (ii) an original work called the Bhadrabahavi Samhita. He is

……………………

1.    see the Introduction by D. E, smith to the Ganita-sara-samgraha ed. By Rangacharya Madras, 1912.

2.    B. Datta: The Jaina school of Mathematics, Bulletin, Cal, Math. Soc, vol. XxxI (1929), pp. 115-145.

Mentioned by Malayagiri (c, 1150) in his commentary on the Suryaprajnapti, and has been quoted by Bhattotpala (966) Another Jaina astronomer of the name of siddhasena has been quoted by Varahamihira (505) and Bhattotpala. Mathematical quotations in Ardha-magadhi and Prakrit are met with in several works. The Dhavala contains a large number of such quotations. These quotations will be considered at their proper places, but it must be noted here that they prove beyond doubt the existence of mathematical works written by Jaina scholars which are now lost 2. Works written by Jaina scholars under the little of Kshetra-samasa and Karana-bhavana dealt with mathematics, but no such works are available to us now, Our knowledge of Jaina mathematics which is of an extremely fragmentary character is gleaned from a non-mathematical works such as Sthananga-sutra, Tattvarthadhigama-sutra-bhasya of  Umasvati, Suryaprajnapti, Anuyogadvara-sutra, Triloka Prajnapti, Trilokasara, etc.

To these may now be added the Dhavala.

 

The importance of the Dhavala.

The Dhavala was written by virasena in the beginning of the ninth century.

Virasena was a philosopher and religious divine. He certainly was not a mathematician. The mathematical material contained in the Dhavala may therefore be attributed to the previous writers, especially to the previous commentators of whom five have been mentioned by Indranandi in the srutavatara. These commentators were kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva, of whom the first flourished about 200 A.D. and the last about 600 A.D. Most of the mathematical material in the Dhavala may therefore be taken to belong to the period 200 to 600 A.D. Thus, the Dhavala becomes a work of first rate importance to the historian of Indian mathematics, as it supplies information about the darkest period of the history of Indian Mathematics-the period preceding the fifth century A.D. The view that the mathematical material in the Dhavala belongs to the period before 500 A.D. is corroborated by detailed study. For instance, many of the processes described in the Dhavala are not found in any known mathematical work. Furthermore, there is a certain imperfection which, one acquainted with the later Indian mathematical works, can easily discern. The mathematics in the Dhavala lacks the finish and the refinement of the Aryabhatiya and later works.

 

Mathematical Content of the Dhavala

Numbers and Notation – The author of the Dhavala is fully conversant with the place value system of notation. Evidence of this is to found everywhere. We quote some methods of expressing numbers taken from quotations given in the Dhavala –

…………………..

1.    Bruhat Samhita, ed, by S. Dvivedi, Benares, 1885, p. 226.

2.    Silanka in his commentary on the sutrakrtanga sutra, Smayadhyayana, Anuyogadvara severs 28, quote three rules regarding permutation and combinations. These rules are apparently taken from some Jaina mathematical work.

(i)               79999998 is expressed as a number which has 7 in the beginning , 8 at the end, and 9 repeated six times in between.

(ii)            46666664 is expressed as sixty-four, six hundreds, sixty-six thousands sixty-six hundred-thousands, and four kotis2.

(iii)          22799498 is expressed as two kotis, twenty-seven, ninety-nine thousands four and ninety-eight 3.

The method used in (I) is found elsewhere also in Jaina literature and at some places, in the Ganita-sara-samgraha4. It shows familiarity with the place value notation, In (ii) the smaller denominations are expressed first. This is not in accordance with the general practice current in Sanskrit literature. Likewise, the scale of notation is hundred and not ten as is generally found in Sanskrit literature5. In pali and prakrit, however, the scale of hundred is generally used. In (iii) the highest denomination is expressed first. Quotations (ii) and (iii) are evidently from different sources.

Big numbers – It is well known that big numbers occur frequently in Jaina literature. In the Dhavala also the various kinds of jiva-rasi, dravya-pramana etc. are discussed. The biggest number that is definitely stated is the number of developable human souls. In the Dhavala it is stated to lie between the sixth-square of two and the seventh square of two; or to be more precise, between koti-koti-koti and koti-koti-koti-koti, i.e.

6                                        7

2                                         2

between             2                           and             2

and more definitely, between (1,00,00,000)3 and (1,00,00,000)4

The actual number of such souls know from other works is 79,22,81,62,51,42,64,33,75,93,54,39,50,336. This number occupies twenty-nine notational places. It has the same number of notational places as (1,0000,000)4 but is greater. This is known to the author of Dhavala who calculates the area of the world inhabited by men and shows that the larger number of men can not be contained in it, and hence that view was wrong.

The Fundamental Operations – Mention is found of all the fundamental operations-addition, subtraction, division, multiplication, the extraction of square and cube-roots, the raising of numbers to be given powers, etc. These operations are

……………………..

1.    Dhavala III, p. 98, quoted verse 51. Cf. Gommata-sara, Jiva kanda, p. 633.

2.    Dhavala III, p. 99, quoted verse 52.

3.    Dhavala III, p. 100, quoted verse 53.

4.    Cf Ganita-sara-samgraha. I, 27. See also History of Hindu Mathematics by Datta and singh, Vol. I. Lahore, 1935, p. 16.

5.    Datta and singh, 1, c, p. 14.

6.    Dhavala III, p. 253.

7.    Cf Gommatasara, Jivakanda S.B.J. serices, p. 104.

Mentioned both with respect to integers and fractions. The theory of indices as described in the Dhavala is somewhat different from what is found in the mathematical works. This theory is certainly primitive and is earlier than 500 A.D. The fundamental ideas seem to be those of (I) the square, (ii) the cube, (iii) the successive square, (iv) the successive cube (v) the raising of a number to its own power, (vi) the square-root (vii) the cube-root (viii) the successive square-root, (ix) the successive cube-root etc. All other powers are expressed in terms of the above. For example, as/2 is expressed as the first square-root of the cube of a,a9 is expressed as the cube of the cube of a, a6 is expressed as the square of the cube or the cube of the square of a, etc. The successive squares and square-roots are as below-

1^st square of a means                     (a)2 = a2

2^nd square of a means                    (a2)2 = a4 = a2

3^rd square of a means           a2

…….                                                 …….

Nth square of a means                  a2

Similarly,

1^st square-root of a means            a1/2

2^nd square-root of a means a1/2

3^rd square-root of a means           a1/2

……….                                            ………

nth square-root of a means a1/2n

Vargita-samvargita- The technical term vargita-samvargita has been used for the raising of a number to its own power. For instance, nn is the vargita-samvargita of n. In connection with this the Dhavala mentions an operation called Viralana-deya- “spread and give.” The viralana (spreading) of a number means the separating of the number into its unities, i.e. the Viralana of n is-

1 1 1 1 1…….. n times.

Deya (giving) means the substitution of n in the place of 1 everywhere in the above. The vargita-samvargita of n is obtained by multiplying together the n’s obtained by the Viralana-deya. The result is the first vargita-samvargita of n, i. e,

1^st vargita-samvargita of n is                n

The application of the process of Viralana-deya once again, i.e, to n. gives the

2^nd vargita-samvargita of n                   (n)nn

A further application of the same procedure gives the-

…………………….

1.    Dhavala Vol. III, p. 53.

3^rd vargita-samvargita of n          {(nⁿ)nⁿ }  {(nⁿ}nⁿ}

The Dhavala does not contemplate the application of the above more than thrice.

The third Vargita-samvargita has been used very often 1 in connection with the theory of very large or infinite numbers. That the process yields. Very big numbers can be seen from the fact that the 3^rd vargita-samvargita of 2 is 256.

The laws of indices – From the above description it is obvious that the author of the Dhavala was fully conversant with the laws of indices, viz.

(i)               a^m aⁿ = a^{m + n}

(ii)            a^m/aⁿ = a ^m-n

(iii)          (a^m)ⁿ = a^mn

Instances of the use of the above laws are numerous. To quote one interesting case, 2 it is stated that the 7^th varga of 2 divided by the 6^th varga of 2 gives the 6^th varga of That is-

2²⁷/2²⁶ = 2²⁶

The operations of depletion and mediation were considered important when the place value numerals were unknown. There is no trace of these operations in the Indian mathematical works. But these processes were considered to be important by the Egyptians and the Greeks and were recognized as such in their works on arithmetic. The Dhavala contains traces of these operations. The consideration of the successive squares of 2 or other numbers was certainly inspired by the operation of depletion which must have been current in India before the advent of the place value numerals. Similarly, there are traces of the method of mediation. In the Dhavala we find generalization of this operation into a theory of logarithms to the base 2,3,4, etc.

Logarithms – The following terms have been defined in the Dhavala 3

(i)               Ardhaccheda of a number is equal to the number of times that it can be halved. Thus, the ardhaccheda of 2m = m. Denoting ardhaccheda by the abbreviation Ac, we can write in modern notation-

Ac of x (or Ac x) = log x, where the logarithm is to the base 2.

(ii)            Vargasalaka of a number is the ardhaccheda of the ardhaccheda of that number, i.e.,

Vargasalaka of x = Vs x = Ac Ac x = log log x, where the logarithm is to base two.

(iii)    4 Trkaccheda of a number is equal to the number-of times that it can be divided by 3. Thus-

…………………….

1.    Dhavala III, p.20 ff. 2.ibid p. 253 ff. 3. Ibid p. 21 ff. 4. Ibid p. 56.

Trkaccheda of x = Tc x = log 3 x, where the logarithm is to the base 3.

(iv)¹ Chaturthaccheda of a number is the number of times that it can be divided by 4. Thus –

Chaturtha-cched of x = Cc x = Log 4x, where the logarithm is to the base 4.

The following results regarding logarithms have been used in the Dhavala:

(1)² Loge (m/n) = log m – log n.

(2)      Loge (m/n) = Log m = log n.

(3)3      Loge m = m, where the logarithm is to the base 2.

(4)4       Loge (xx) 2 = 2x log x.

(5)5       Loge log (xx)2 = log x = 1 + log log x.

(for the left side = log (2x logx)

= log x + log 2 + log log x

= log x + 1 + log log x.

as log 2 to the base 2 is 1.)

(6)6      Log (xx) ^xx = x^x log ^xx

(7)            Let be any number, then-

1^st vargita-samvargita of a = a^a B [say]

2^nd vargita-samvargita of a = B^b = y [say]

3^rd vargita-samvargita of a = y^y = D [say]

The Dhavala gives the following results 7 –

(i)Log B = a log a

(ii)Log log B = log a + log log a

(iii)log y = B log B

log log y = log B + log log B

= log a + log log a + a log a

(vi) log D = y log y

(vi) log log D = log y + log log y

and so on

(8)⁸ Log log D < B²

This inequality gives the inequality –

B log B + log B + log log B < B²

………………..

1. ibidp. 56.         2. Ibid p. 60.       3. Ibid p. 55.       4. Ibid p. 21 ff.  5.1.c.

6. 1.c. It should be mentioned here that nowhere in the text are these logarithms restricted to be integral. The number x is any number. Xx is the first vargita.

Samvargita rasi and (xx)xx is the second vargita-samvargita rasi.

7. Dhavala III, p. 2I-24.                                                       8. Ibid p. 24.

Fractions – Besides, the fundamental arithmetical operations with fractions knowledge of which has been assumed in the Dhavala, we find a number of interesting formula relating to fractions, which are not found in any known mathematical work. Amongst these may be mentioned the following : -

[1]1      n²                                             n

n + (n/p)    =       n       =       p + 1

[2]2 Let a number m be divided by the divisors d and d, and let q and q be the

Quotient (or the fractions). The following formula gives the result when m is divided by d + d’ –

m                                 q’

d + d          =       (q/q) + 1

q

or      =       1 + (q/q’)

[3] 3 If m                      m

d       =       q and  d     =       q’ then –

[4] 4 If a

b       =       q, then –

a

b + b              q

n       =       q – n +1

and       a

b – b                                   q

n                 =       q +         n-1

[5]  5   If   a

b                 =       q, then –

a                               q

b + c =       q -      b

c     +  1

and         a           =       q +         q

b – c                              b

c   - 1

[6] 5   If a                                 a

b                 =       q, and   b   = q + c, then –

…………………..

1.    Dhavala p. 46.   2.ibid p. 46

3       ibid p. 47, quoted verse 27.

4       ibid p. 46, quoted verse 24.

5       Ibid p. 46, quoted verse 24.

6       Ibid p. 46, quoted verse 25.

 

b = b                b

q

c + 1,

and if  a

b       =       q-c, then –

b       = b +      b

q

c       -  1

[7^{] 1If  a}                             a

b       =       q, and     b is another fraction, then –

a                 a

b       -        b       =       q ( b – b )

b

[8]  ²  If  a                          a

b       =       q, and  b + x         =     q –c, then-

[9]  ³   If a

b       =       q, and        b

b – x           =       q + c, then

x =   bc

q + c

[10]  ⁴  If a                    a

b       = q, and    b + x            = q, then –

q’ = q -       qc

b + c

[11] ⁵     If  a                                     a

b       =       q, and               b – c             =  q, then –

The above results are all found in quotations given in the Dhavala. They are not found in any known mathematical work. The quotations are from Ardha-Magadhi or prakrit works. The presumption is that they are taken from Jaina works on mathematics or from previous commentaries. They do not represent any essential arithmetical operation. They are relics of an age when division was considered a difficult and tedious operation. These rules certainly belong to an age when the place value notation was not in common use for arithmetical operations.

The rule of three – The rule of three is mentioned and used at several

…………………..

1.    ibid p. 46, quoted verse 28.

2.    Ibid p. 48, quoted verse 29.

3.    Ibid p, 49. Quoted verse 30.

4.    Ibid p. 49, quoted verse 31.

5.    Ibid p, 49, quoted verse 32.

Places the technical terms in connection with the process are phala, iccha and pramana, the same as found in the known mathematical works. This suggests that the rule of three was known and used in India even before the invention of the place-value notation.

THE INFINITE

Use of big numbers – The word infinite used in various senses is found in the literature of all ancient peoples. A correct definition and appreciation of the idea, however, came much later. It is natural that the correct definition was evolved by people who used big numbers, or were accustomed to such numbers in their philosophy. The following will show that in India the Jaina philosophers succeeded in classifying the various notions connected with the term infinite, and in evolving the correct definition of the numerical infinite.

The evolution of suitable notation for expressing big numbers as well as of the idea of the infinite arise when abstract reasoning and thinking reach a certain high standard. In Europe, Archimedes tried to estimate the number of sand particles on the sea-shore and the Greek philosophers speculated about the infinite and the limit. They, however, did not possess suitable symbols for the expression of big numbers. In India, the Hindu, Jaina and Budhist philosophers used very big numbers and evolved suitable symbolism for the purpose. In particular, the Jainas tried to form an estimate of all living beings in the Universe, of time instants, of locations [points or places] in the Universe and so on.

Three methods of expressing big numbers were employed : -

(1) The place-value notation using the scale of ten. In this connection it may be noted that number-names based on the scale of ten were coined to express numbers as large as 10

(2) The law of indices ( varga-samvarga) was employed to give compact.

Expressions for big numbers, e. g. –

(i)               (2)² = 4,

(ii)            (2²)2² = 4 = 256

{(2²) 2²}

(iii) { (2²) 2² }               = 256         ²⁵⁶     is called the third Vargita-samvargita of 2. This number is greater than number of protons and electrons in the Universe-

………………..

1. See, for example, Dhavala III, p. 69 and 100 etc.

(3) The logarithm (ardhaccheda) or the logarithm of a logarithm

(ardhaccheda-salaka) was used to reduce the consideration of big numbers to those of smaller ones e.g.-

(i)               Log 2 = 2

(ii)            Log Log2 4 = 3,

(iii)          Log2 Log2 256 = 11.

It is no wonder to find that today we take recourse to one or the other of the above three methods of expressing numbers. The decimal place-value notation has become the common property of all nations. Logarithms are used whenever calculations with big numbers have to be made. Instances of the use of the law of indices to express magnitudes in modern physics is common. For instance, the number of protons in the Universe has been calculated and expressed as –

136.2

And ‘Skewes’ number which gives information regarding the distribution of primes is expressed in the form –

34

10

10

10

All the above methods of expressing numbers have been used in the Dhavala. It follows that the methods were commonly known before the seventh century A.D. in India.

……………….

1.    The number 136.2 256 expressed in the decimal notation is 15, 747, 724, 136, 275, 002, 577, 605, 653, 961, 181, 555, 468, 044, 717, 914, 572, 116, 709, 366, 231, 425, 076, 185, 631, 03, 296.

It will be observed that the third vargita-samvargita of 2, i.e. 256. 256 is greater than the number of protons in the Universe. If we imagine the entire Universe as a chess-board, and the protons in it as chessmen, and if we agree to call any interchange in the position of two protons a move in this cosmic game, then the total number of possible moves would be the number –

34

10

10

10

This number is also connected with the theory of the distribution of primes.

Classification of the infinite. The Dhavala gives a classification of the infinite. The term infinity has been used in literature in several senses. The Jaina classification takes into account all these. Accordintg to it there are eleven kinds of infinity as follows : -

(1) Namanata – Infinite in name. An aggregate of objects which may or may not really be infinite might be called as such in ordinary conversation, or by or for ignorant persons, or in literature to denote greatness. In such a  context the term infinite means infinite in name only, i. e, Namananta.

(2) Sthapanananta – Attributed or associated infinity. This too is not the real infinite. The term is used in case infinity is attributed to or associated with some object.

(3) Dravyananta – Infinite in relation to knowledge which is not used. This term is used for persons who have knowledge of the infinite, but not for the time being use that knowledge.

(4) Gananananta – The numerical infinite. This term is used for the actual infinite as used in mathematics.

(5)  Apradesikananta – Dimensionless, i.e. infinitely small.

(6) Ekananta – one directional infinity. It is the infinite as observed by looking in one direction in along a straight line.

(7) Ubhayananta – Two directional infinity .This is illustrated by a line continued to infinity in both directions.

(8) Vistarananta – Two dimensional or superficial infinity. This means an infinite plane area.

(9) Sarvananta – Spatial infinity. This signifies the three dimensional infinity i. e. the infinite space.

(10) Bhavananta – Infinite in relation to knowledge which is utilised. This term is used for a person who has knowledge of the infinite, and who uses that knowledge.

(11) Saswatanata – Everlasting or indestructible.

The above classification is a comprehensive one, including all senses in which the term ananta is used in Jaina literature.

Gananananta (numerical infinite)

The Dhavala clearly lays down that, in the subject-matter under discussion, by the term ananta (infinite) we always mean the numerical infinite, and not any of the

……………….

1.    Dhavala III, p. 11-16.

2.    Ibid p. 16.

Other infinities enumerated above. For, in the other kinds of infinity “the idea of enumeration is not found” 1. It has also been stated that the “numerical infinite is describable at great length and is simpler.” This statement probably means that in Jaina literature ananta (infinite) was defined more thoroughly by different writers and had become commonly used and understood. The Dhavala, however, does not contain a definition of ananta. On the other hand, operations on and with the ananta are frequently mentioned along with numbers called samkhyata and asamkhyata.

The number samkhyata, asamkhyata and ananta have been used in Jaina literature from the earliest known times, but it seems that they did not always carry the same meaning. In the earlier works ananta was certainly used in the sense of infinity as we define it now, but in the later works anantananta takes the place of ananta. For example , according to the Trilokasara a work written in the 10^th century by Nemicandra, Parita-ananta, Yuktananta and even Joghanya-anantananta is a very big number, but is finite. According to this work, numbers may be divided into three broad classes : -

(i)               Samkhyata, which we shall denote by – s,

(ii)            Asamkhyata, which we shall denote by – a,

(iii)          Ananta, which we shall denote by – A.

The above three kinds of numbers are further sub-divided into  three classes as below : -

I.                  Samkhyata (numerable) numbers are of three kinds :

(i) Jaghanya-samkhyata (smallest numerable) which we shall denote by sj;

(ii)Madhyama-samkhyata (intermediate numerable) which we shall denote by – sm;

(iii) Utkrsta-samkhyata (the highest numerable) which we shall denote by-su.

II. Asamkhyata (un-numerable) numbers are divided into three classes : -

(i) Parita-asamkhyata (first order innumerable) which we shall denote by –ap

(ii) Yukta-asamkhyata (medium innumerable) which we shall denote by-ay;

(iii) Asamkhyata-asamkhyata (innumerably -innumerable) which we shall denote by – aa.

Each of the above three classes is further sub-divided into three classes, Viz. Jaghanya (smallest), Madhyama (intermediate) and Utkrsta (highest). Thus, we have the following numbers included under Asamkhyata : -

1. Jaghanya-parita-asamkhyata                              …apj

2. Madhyama-parita-asamkhyata                           …apm

3. Utkrsta-parita-asamkhyata                                  …apu

 

1. Jaghanya-yukta-asamkhyata                               … ayj

2. Madhyama – yukta-asamkhyata                         … aym

3. Utkrsta-yukta-asamkhyata                                  … ayu

 

1. Jaghanya-asamkhyata-asamkhyata                   … aaj

2. Madhyama-asamkhyata-asamkhyata                … aam

3. Utkrsta-asamkhyata-asamkhyata                       … aau

II.               Ananta, which we denote by A, is divided into three classes-

(i)               parita-Ananta (first order infinite) which we shall denote by – Ap;

(ii)            Yukta-Ananta (medium infinite) which we shall denote-Ay;

(iii)          Ananta-Ananta (infinitely infinite) which we shall denote by – AA.

As in the case of the asamkhyata numbers, each of these is further subdivided into three classes- Jaghanya, Madhyama and Utkrasta – so that we have the following numbers in the Ananta class –

1. Jaghanya-parita-ananta                             … Apj

2. Madhyama-parita-ananta                                    … Apm

3. Utkrsta-parita-ananta                                          …Apu

 

1. Jaghanya-yukta-ananta                                       … Ayj

2. Madhyama-yukta-ananta                                     … Aym

3. Utkrsta-yukta-ananta                                           … Ayu

 

1.Jaghanya-ananta-ananta                             … AAj

2.Madhyama-ananta-ananta                          … AAm

3.Utkrst-ananta-ananta                                  … Aau

Numerical value of the Samkhyata – According to all Jaina authorities, the JaGhanya-samkhyata is the number 2 being, according to them, the smallest number that represents multiplicity. Unity was not counted as a member of the aggregate of Samkhyata numbers. Madhayama-samkhyata includes all numbers between 2 and the Utkrsta-samkhyata (the highest numerable) su, which itself is the number immediately preceding the Jaghanya-parita-asamkhyata apj, i. e.,

Su = apj-1.

And apj is defined in the Trilokasara as follows : -

According to Jaina cosmology the Universe is composed of alternate rings of land and water whose boundaries are concentric circles with increasing radii. The

………………….

1. See, Triloka-sara, 35.width of any ring, whether land or water, is double that of the preceding ring. The central core (i. e, the initial circle) is of 100,000 yojanas in diameter and is called Jambudvipa.

Consider four cylindrical pits each of 100,000 yojanas in diameter and 1,000 yojanas deep. Call these A1, B1, C1, and D1 is filled with rape-seeds and further rape-seeds are piled over it in the form of a conical heap, the topmost layer consisting of one seed. The total number of seeds required for the operation is-

For the cylinder : 19791209299968.10

For the superincumbent cone : 17992008454516363636363636363636363636363636363636

The total number of seeds is 199711293845131636363636363636363636363636363636

We shall call the process described above by the term “overfilling” a cylinder with rape-seeds.

Now, take the seeds from the above-filled pit and drop them, beginning from Jambudvipa, one on each concentric ring of land or water of the Universe. The number of seeds being even, the last seed would fall on a ring of water. Let one rape-seed be put in B1 to denote the end of this operation.

Now, imagine a cylinder with the diameter of the boundary of the ring of water into which the last rape-seed was dropped in the above operation, and 1000 yojanas deep. Call this cylinder A2. Imagine A2 to be overfilled with rape-seeds. Drop the seeds, beginning after the last ring of water attained in the previous operation successively on the rings of land and water. This second dropping of seeds will lead to a ring of water on which the last seed is dropped.

Place one more seed in B1 to denote the end of this operation.

Imagine now a cylinder with diameter that of the last ring of water attained above, and 1000 Yojanas deep. Call this cylinder A3. Let A3 be over-filled with rape-seeds and let these seeds be dropped on the rings of land and water as before, and let at the end of the process a seed be dropped in B1.

Imagine the above process continued till B1 is overfilled. The above process leads to cylinders of increasing volumes :

A1, A2, …………A1………….

Let A be the last cylinder obtained when B1 is over-full.

Now, begin with A as the first over-full pit and continue the above process dropping one rape-seed on each ring of land and water, beginning after the water ring into which the last seed in the previous operation was dropped. Then drop one seed in C1 Continue the process till C1 is over-filled. Let A” be the last cylinder obtained by the above process. Then begin with A” and proceeding as before over-fill D1. Let A” be the last pit obtained at the termination of this operation.

The, the Jaghanya-parita-asamkhyata, apj, is equal to the number of rapeseeds cotained in A” And Utkrsta-samkhyata = su = apj – 1.

Remarks : - The central idea in dividing numbers into three classes seems to be this : - The extent to which numeration, i.e., counting, can proceed depends on the number – names available in the language or on other methods of expressing numbers. In order, therefore, to extend the bound of numbers which may be counted or expressed in speech, a long series of names of numerical denominations, based primarily on the scale of ten, was coined in India. The Hindus contented themselves with eighteen denominations by the help of which numbers upto 10 could be expressed in speech. Numbers greater than 10 could be expressed by repetition, as we do now when we say million million, etc. But it was realised that repetition was cumbersome. The Buddhists and the Jainas who needed numbers much bigger than 10 in their philosophy and cosmology coined denominational names for still greater numbers. We do not possess Jaina denominational names, but the following series of denominational

……………….

1.    The Jainas possess in their old literature a list of names denoting long periods of time with the year as the unit. The series is as follows : -

 

This list is found in the Triloka-prajnapti [4^th-6^th  cent], Harivamsa-purana (8^th cent.) and Rajavarttika [8^th cent] with a few variations in the names only. According to a statement found in Triloka-prajnapti, the value of Acalapra is obtainable by multiplying 31 times 84 i. e.

Acalapra = 84,

And that the value will lead us to 90 decimal places. According to Logarithmic tables, however, 84 gives us only sixty decimal places of notation. (see Dhavala III, introduction and footnote, p. 34 ) – Editor.

Names which is of Buddhist is interesting: -

 

It will be observed that in the above series asamkhyeya is the last denomination. This probably implies that numbers beyond the asamkhyeya are beyond numeration i.e, innumerable.

The value of asamkhyeya must have varied from time to time, Nemichandra’s asamkhyata is certainly different from the asamkheya defined above, which is 10

Asamkhyata – As already mentioned, the asamkhyata numbers are divided into three broad classes, and each of these again into three sub-classes. Using the notation given above, we have, according to Nemichandra-

Jaghanya-parita-asamkhyata               (apj) is  = su + 1;

Madhyama-parita-asamkhyata            (apm)    > apj, but < apu,

Utkrsta-parite-asamkhyata                   (apu)     = ayj – 1,

Where –

Jaghanya-yukta-asamkhyata               (ayj)     = (apj) apj;

Madhyama-yukta-asamkhyata             (aam) is  > ayj,but < ayu,

Utkrsta-asamkhyata-asamkhyata        (aau)            = apj – 1;

Where –

Apj stands for Jaghanya-parita-ananta

Ananta – The numbers of the ananta class are as follows: -

Jaghanya-parita-ananta [Apj] is obtained as below: -

Let –

{[aaj] [aaj] } ]

[{[aaj] [aaj] }

{[aaj] [aaj] } ]

B = [{[aaj] [aaj] }

Let C = B + six dravyas

{(cc)cc}

Let D = { (cc) cc }                          + for aggregates

 

Then,                                                                            {(DD)DD}

Jaghanya parita-ananta [Apj] ={ (DD) DD}

Madhyama-parita-ananta [Apm] is > Apj, but < Apu;

Utkrsta-parita-ananta         [Apu] = Ayi – 1;

Where –

Jaghanya-yukta-ananta     [Ayj] = (apj) (api)

Madhyama-yukta-ananta [Aym] is > Ayj, but < Ayu;

Utkrsta-yukta-ananta         [Ayu] = Aaj – 1,

Where-

Jaghanya-ananta-ananta   [Aaj] = (Ayj) 2

Madhyama-ananta-ananta [Aam] is > Aaj, but < Aau;

Where –

Aau stands for Utkrsta-ananta-ananta, which according to Nemichandra, is obtained as follows: -

Let –

{(Aaj) Aaj}]

[{(Aaj) Aaj}

{(Aaj) Aaj}]

x = [{(Aaj) Aaj }

{(xx) xx}

y = { (xx) xx}                + two rasis4

……………….

1.    The six dravyas are the spatial points of : 1) Dharma, 2) Adharma, 3) One Jiva 4) Lokakasa, 5) apratisthita (vegetable souls) and 6) Pratisthita (vegetable souls.)

2.     The four aggregates are: 1) instants of a kalpa, 2) spatial units of the Universe, 3) anubhagabandha-adhyavasaya-sthana, and 4) avibhaga praticcheda of Yoga.

3.     These are: 1) siddha, 2) sadharana-vanaspati-nigoda, 3) vanaspati, 4) pudgala 5) vyavahara kala, and 6) alokakasa.

4.     These are: 1) Dharma dravya, 2) adharma dravya, (aguru-laghu-guna-avibhaga praticcheda of both,)

{(yy) yy}

Z = {(yy) yy}

Now, the aggregate known as kevalajnana is greater than z, and

Aau = Kevalajnana –z +z

= Kevalajnana

Remarks – From the above it follows that –

[I] Jaghanya-parita-ananta [apj] is not infinite unless one or more

e of the six dravyas or the one of the four aggregates, which have been added to obtain it, is infinite.

[ii] Utkrsta-ananta-ananta [Aau] is equivalent to the aggregate called Kevalajnana. The description above seems to imply that the utkrsta-ananta-ananta can not be reached by any arithmetical operation, however, far it may be carried. In fact, it is greater than any number z which can be reached by arithmetical operations. It seems to me, therefore, that kevalajnana is infinite, and hence that utkrsta-ananta-ananata is infinite.

Thus, the description found in the Trilokasara leaves us in doubt as to whether any of the three classes of parita-ananta and the three classes of yukta-ananta and the jaghanya-ananta-ananta is actually infinity or not in as much as they are all said to be the multiples of asamkhyata and even the aggregates that have been added are also asamkhyata only. But the Ananta of the Dhavala is actual infinity, for it is clearly stated that “ a number which can be exhausted by subtraction cannot be called ananta” It is further stated in the Dhavala that by ananta-ananta is always meant the madhyama-ananta-ananta. So the madhyama-ananata-ananta, according to the Dhavala, is infinite.

The following method of comparing two aggregates given in the Dhavala is very interesting. Place on one side the aggregate of all the past Avasarpinis and Utsarpinis (i.e. the time-instants in a kalpa, which are supposed to form a continuum and are consequently infinite) and on the other the aggregate of Mthadrsti jiva-rasi. Then taking one element of the one aggregate and a corresponding element from the other, discard them both. Proceeding in this manner the first aggregate is exhausted, whilst the other is not. The Dhavala therefore, concludes that the aggregate of mithyadrsti-rasi is greater than that of all the past time – instants.

The above is nothing but the method of one-to-one correspondence which forms the basis of the modern theory of infinite cardinals. It may be argued that the method is applicable to the comparison of finite cardinals also, and so was taken recourse to for comparing two very big finite aggregates, so big that their elements could not be counted in terms of any known numerical denomination. This view-point is further supported by the fact that the Jaina works fix the duration of a time-instant, and so the number of time-instants in a kalpa (Avasarpini and Utsarpini) must be

………………….

1.    Dhavala III, p. 25. 2. Ibid p. 28. 3. Ibid p. 28.

Finite, as the kalpa itself is not an infinite interval of time. According to this latter view the Jaghanya-parita-ananta (which according to definition is greater than the aggregate of time instants) is finite.

As already pointed out, the method of one-to-one correspondence has proved to be the most powerful tool for the study of infinite cardinals, and the discovery and first use of the principle must be ascribed to the Jainas.

In the above classification of numbers I see a primitive attempt to evolve a theory of infinite cardinal numbers. But there are some serious defects in the theory. There defects would lead to contradictions. One of these is the assumption of the existence of the number c-1, where is infinite and a limiting number of a class. On the other hand, the Jaina conception that the vargita-samvargita of a cardinal c (I.e. cc) would lead to a new number is justifiable. If it is true that the creation of the numbers of the ananta class anticipated to some extent, the modern theory of infinite cardinals. Any such attempt at such an early age and stage in the growth of mathematics was bound to be a failure. The wonder is that the attempt was made at all.

The existence of several kinds of infinity was first demonstrated by George cantor about the middle of the nineteenth century. He gave a theory of transfinite numbers. Cantor’s researches in the domain of infinite aggregates, have provided a sound basis for mathematics, a powerful tool for research, and a language for correctly expressing the most abstruse mathematical ideas. The theory of transfinite numbers however, is at present in an elementary stage. We do not as yet possess a calculus of these numbers, and so have not been able to bring them effectively in mathematical analysis.

A.N. singh, D. Sc.,

Lucknow University.

INDEX

(Owing to deficiency of types, proper diacritical marks could not be used in the Mathematics of Dhavala. The following index will be helpful in reading the Sanskrit and Prakrit technical terms correctly.)