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
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 12th
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
From the mathematical literature available at present we can say that
important schools of mathematics flourished at Pataliputra {
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
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,
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. Aryabhatas works seems to be the first good text book
employing the new arithmetic of the place value numerals. Works anther to
Aryabhatas 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 Aryabhatas 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
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-
1st square of a means (a)2 = a2
2nd square of a means (a2)2 = a4 =
a2
3rd square of a means a2
.
.
Nth square of a means a2
Similarly,
1st square-root of a means a1/2
2nd square-root of a means a1/2
3rd 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 ns obtained by the Viralana-deya. The
result is the first vargita-samvargita of n, i. e,
1st vargita-samvargita of n is n
The application of the process of Viralana-deya once again,
i.e, to n. gives the
2nd vargita-samvargita of n (n)nn
A further application of the same procedure gives the-
.
1.
Dhavala Vol. III, p. 53.
3rd 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 3rd 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)
am an = am
+ n
(ii)
am/an = a m-n
(iii)
(am)n = amn
Instances of the use of the above laws are numerous. To quote
one interesting case, 2 it is stated that the 7th varga of 2 divided
by the 6th varga of 2 gives the 6th 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
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 = xx
log xx
(7)
Let be any number, then-
1st vargita-samvargita of a = aa
B [say]
2nd vargita-samvargita of a = Bb = y [say]
3rd vargita-samvargita of a = yy = 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
..