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 Dravyapramana 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
soulquests (Margnasthanas ) 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 (SvasthanaSvasthana) and
Places of occasional visits (Viharvatsvasthan)
The expansion of the soulsubstance 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 allknowing saint (Kevalisamudghata). Thus, the commentator
calculates the volume of space occupied by the living beings in these ten
different conditions under the different spiritual stages and soulquests.
The spatial units adopted for these measurements are five,
namely, (1) the entire universe (Sarvaloka), (2) the lower universe
(Adholoka), (3) the upper universe (Urdhvaloka ), (4)
the middle world (Madhyaloka). And (5) the human world (Manusaloka).
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 Sparsanaprarupana 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 Kshetraprarupana 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 150161 of the text).
In the kalaprarupana 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 soulquests. 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 householders.
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
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. Bhagavatisutra 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
Aryabhatiya 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 Ganitasarasamgraha 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 Ganitasarasamgraha ed. By
Rangacharya Madras, 1912.
2.
B. Datta: The Jaina
school of Mathematics, Bulletin, Cal, Math. Soc, vol. XxxI
(1929), pp. 115145.
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 Ardhamagadhi
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 Kshetrasamasa and Karanabhavana
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 nonmathematical works such as Sthanangasutra,
Tattvarthadhigamasutrabhasya of
Umasvati, Suryaprajnapti, Anuyogadvarasutra, 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 Mathematicsthe 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 sixtyfour,
six hundreds, sixtysix thousands sixtysix hundredthousands, and four kotis2.
(iii)
22799498 is expressed as two kotis,
twentyseven, ninetynine thousands four and ninetyeight 3.
The method used in (I) is found elsewhere also in Jaina
literature and at some places, in the Ganitasarasamgraha4. 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 jivarasi, dravyapramana 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 sixthsquare of two and the seventh
square of two; or to be more precise, between kotikotikoti and
kotikotikotikoti, 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 twentynine
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
operationsaddition, subtraction, division, multiplication, the extraction of
square and cuberoots, the raising of numbers to be given powers, etc. These
operations are
..
1.
Dhavala III, p. 98, quoted verse 51.
Cf. Gommatasara, Jiva kanda, p. 633.
2.
Dhavala III, p. 99, quoted verse 52.
3.
Dhavala III, p. 100, quoted verse 53.
4.
Cf Ganitasarasamgraha. 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
squareroot (vii) the cuberoot (viii) the successive squareroot, (ix) the
successive cuberoot etc. All other powers are expressed in terms of the above.
For example, as/2 is expressed as the first squareroot 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 squareroots 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} squareroot of a means a1/2
2^{nd} squareroot of a means a1/2
3^{rd} squareroot of a means a1/2
.
nth squareroot of a means a1/2n
Vargitasamvargita The technical term vargitasamvargita has
been used for the raising of a number to its own power. For instance, nn is the
vargitasamvargita of n. In connection with this the Dhavala mentions an
operation called Viralanadeya 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 vargitasamvargita of n is obtained by multiplying
together the ns obtained by the Viralanadeya. The
result is the first vargitasamvargita of n, i. e,
1^{st} vargitasamvargita of n is n
The application of the process of Viralanadeya once again,
i.e, to n. gives the
2^{nd} vargitasamvargita of n (n)nn
A further application of the same procedure gives the
.
1.
Dhavala Vol. III, p. 53.
3^{rd} vargitasamvargita of n {(n^{n})n^{n} } {(n^{n}}n^{n}}
The Dhavala does not contemplate the application of the above
more than thrice.
The third Vargitasamvargita 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} vargitasamvargita 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} a^{n} = a^{m}
^{+ n}
(ii)
a^{m}/a^{n }= a ^{mn}
(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
2^{27}/2^{26} = 2^{26}
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 numberof 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
Chaturthacched 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} vargitasamvargita of a = a^{a}
B [say]
2^{nd} vargitasamvargita of a = B^{b} = y [say]
3^{rd} vargitasamvargita 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 < B^{2}
This inequality gives the inequality
B log B + log B + log log B < B^{2}
..
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 vargitasamvargita rasi.
7. Dhavala III, p. 2I24. 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 n^{2}
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 + n1
[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 = qc, 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 ArdhaMagadhi 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
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
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
Three methods of expressing big numbers were employed : 
(1)
The placevalue notation using the scale
of ten. In this connection it may be noted that numbernames based on the scale
of ten were coined to express numbers as large as 10
(2)
The law of indices (
vargasamvarga) was employed to give compact.
Expressions for big numbers, e. g.
(i)
(2)^{2} = 4,
(ii)
(2^{2})2^{2} = 4 =
256
{(2^{2}) 2^{2}}
(iii) { (2^{2}) 2^{2}
} = 256 ^{256} is called the third Vargitasamvargita 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
(ardhacchedasalaka) 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 placevalue
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
.
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 vargitasamvargita of 2, i.e.
256. 256 is greater than the number of protons in the
Universe. If we imagine the entire Universe as a chessboard, 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 subjectmatter under
discussion, by the term ananta (infinite) we always mean the numerical
infinite, and not any of the
.
1.
Dhavala III, p. 1116.
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, Paritaananta,
Yuktananta and even Joghanyaanantananta
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 subdivided into three classes as
below : 
I.
Samkhyata (numerable) numbers are of
three kinds :
(i) Jaghanyasamkhyata (smallest numerable) which we shall denote by sj;
(ii)Madhyamasamkhyata (intermediate numerable) which we shall denote by
sm;
(iii) Utkrstasamkhyata (the highest numerable) which we shall denote
bysu.
II. Asamkhyata (unnumerable) numbers are divided into three classes : 
(i) Paritaasamkhyata (first order innumerable) which we
shall denote by ap
(ii) Yuktaasamkhyata (medium innumerable) which we shall
denote byay;
(iii) Asamkhyataasamkhyata (innumerably innumerable) which
we shall denote by aa.
Each of the above three classes is further subdivided into
three classes, Viz. Jaghanya (smallest), Madhyama (intermediate) and Utkrsta
(highest). Thus, we have the following numbers included under Asamkhyata : 
1. Jaghanyaparitaasamkhyata apj
2. Madhyamaparitaasamkhyata apm
3. Utkrstaparitaasamkhyata apu
1. Jaghanyayuktaasamkhyata ayj
2. Madhyama yuktaasamkhyata aym
3. Utkrstayuktaasamkhyata
ayu
1. Jaghanyaasamkhyataasamkhyata aaj
2. Madhyamaasamkhyataasamkhyata aam
3. Utkrstaasamkhyataasamkhyata aau
II.
Ananta, which we denote by A, is
divided into three classes
(i)
paritaAnanta (first order infinite)
which we shall denote by Ap;
(ii)
YuktaAnanta (medium infinite) which
we shall denoteAy;
(iii)
AnantaAnanta (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. Jaghanyaparitaananta
Apj
2. Madhyamaparitaananta
Apm
3. Utkrstaparitaananta
Apu
1. Jaghanyayuktaananta
Ayj
2. Madhyamayuktaananta
Aym
3. Utkrstayuktaananta
Ayu
1.Jaghanyaanantaananta
AAj
2.Madhyamaanantaananta
AAm
3.Utkrstanantaananta
Aau
Numerical value of the Samkhyata According to all Jaina authorities,
the JaGhanyasamkhyata 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. Madhayamasamkhyata includes all numbers
between 2 and the Utkrstasamkhyata (the highest numerable) su, which itself is
the number immediately preceding the Jaghanyaparitaasamkhyata apj, i. e.,
Su = apj1.
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, Trilokasara, 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
rapeseeds and further rapeseeds 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 rapeseeds.
Now, take the seeds from the abovefilled 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 rapeseed 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 rapeseed was dropped in the above
operation, and 1000 yojanas deep. Call this cylinder A2. Imagine A2 to be
overfilled with rapeseeds. 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
overfilled with rapeseeds 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 overfull.
Now, begin with A as the first overfull pit and continue the
above process dropping one rapeseed 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 overfilled. Let A be the last cylinder obtained by the above
process. Then begin with A and proceeding as before overfill D1. Let A be
the last pit obtained at the termination of this operation.
The, the Jaghanyaparitaasamkhyata, apj, is equal to the
number of rapeseeds cotained in A And Utkrstasamkhyata = 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
.
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 : 
1 
Varsa () = 1 year 
2 
Yuga () = 5 year 
3 
purvanga ()
= 84 Lakhs of years 
4 
purva () = 84 Lakhs of purvangas 
5 
Nayutanga (ֵӐ)
= 84 purvas 
6 
Nayuta (ֵ) = 84 Lakhs of Kumudangas 
7 
Kumudanga (Ӑ
) = 84 Nayutas 
8 
Kumud () = 84 Lakhs of Nalinangas 
9 
Padmanga (֩Ӑ) = 84 kumudas 
10 
Padma (֩ ) = 84 Lakhs of padmangas 
11 
Nalinanga (֭Ӑ
) = 84 padmas 
12 
Nalina (֭ ) = 84 Lakhs of Nalinangas 
13 
Kamalanga (ֻӐ
) = 84 Nalinas 
14 
Kamala (ֻ ) = 84 Lakhs of kamalangas kalpa 
15 
Trutitanga (יӐ)
= 84 Kamalas 
16 
Trutita (י) = 84 Lakhs of
Trutitangas 
17 
Atatanga (Ӑ) = 84 Trutitas 
18 
Atata () = 84 Lakhs of Aatatangas 
19 
Amamanga (ִӐ)
= 84 Atatas 
20 
Amama (ִ) = 84 Lakhs of Amamangas 
21 
Hahanga (Ӑ)
= 84 Amamas 
22 
Haha () =84 Lakhs of Hahangas 
23 
Huhanga (Ӑ)
= 84 Hahas 
24 
Huhu () = 84 Lakhs of Huhangas 
25 
Latanga (֟Ӑ) = 84 Huhus 
26 
Lata (֟) = 84 Lakhs of
Lalitangas 
27 
Mahalatanga (ֻ֟Ӑ)
= 84 Latas 
28 
Mahalata (ֻ֟)
= 84 Lakhsof Mahalatngs 
29 
Srikalpa (ߍ) = 84 ; Mahalatas 
30 
Hastaprahelita (ß֯֟)
= 84Lakhs of sri 
31 
Acalapra (ֻ֯) =84Lakhs of Hastaprahelita 
This list is found in the Trilokaprajnapti [4^{th}6^{th} cent],
Harivamsapurana (8^{th} cent.) and Rajavarttika [8^{th} cent]
with a few variations in the names only. According to a statement found in
Trilokaprajnapti, 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:

1 
Eka = 1 
2 
dasa = 10 
3 
sata = 100 
4 
sahassa = 1,000 
5 
dasa sahassa = 10,000 
6 
sata sahassa = 100,000 
7 
dasasatasahassa = 1,000,000 
8 
koti = 10,000,000 
9 
pakoti = (10,000,000)2 
10 
kotippakoti = (10,000,000)3 
11 
nahuta = (10,000,000)4 
12 
ninnahuta =
(10,000,000)5 
13 
akhobhini =
(10,000,000)6 
14 
bindu = (10,000,000)7 
15 
abbuda = (10,000,000)8 
16 
niraabbuda = (10,000,000)9 
17 
ahaha =
(10,000,000)10 
18 
ababa =
(10,000,000)11 
19 
atata =
(10,000,000)12 
20 
sogandhika =
(10,000,000)13 
21 
uppala = (
10,000,000)14 
22 
kumuda = (10,000,000)15 
23 
pundarika = (10,000,000)16 
24 
paduma = (10,000,000)17 
25 
dathana = (10,000,000)18 
26 
mahakathana = (10,000,000)19 
27 
asamkhyeya = (10,000,000)20 
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,
Nemichandras 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 subclasses. Using the notation given above, we have,
according to Nemichandra
Jaghanyaparitaasamkhyata (apj)
is = su + 1;
Madhyamaparitaasamkhyata (apm) > apj, but < apu,
Utkrstapariteasamkhyata (apu) = ayj 1,
Where
Jaghanyayuktaasamkhyata (ayj) =
(apj) apj;
Madhyamayuktaasamkhyata (aam)
is > ayj,but
< ayu,
Utkrstaasamkhyataasamkhyata (aau) = apj 1;
Where
Apj stands for Jaghanyaparitaananta
Ananta The numbers of the ananta class are as follows: 
Jaghanyaparitaananta [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 paritaananta [Apj] ={ (DD) DD}
Madhyamaparitaananta [Apm] is > Apj, but < Apu;
Utkrstaparitaananta [Apu]
= Ayi 1;
Where
Jaghanyayuktaananta [Ayj] = (apj) (api)
Madhyamayuktaananta [Aym] is > Ayj, but < Ayu;
Utkrstayuktaananta [Ayu] =
Aaj 1,
Where
Jaghanyaanantaananta [Aaj] = (Ayj) 2
Madhyamaanantaananta [Aam] is > Aaj, but < Aau;
Where
Aau stands for Utkrstaanantaananta,
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)
anubhagabandhaadhyavasayasthana, and 4) avibhaga praticcheda of Yoga.
3.
These are: 1) siddha, 2) sadharanavanaspatinigoda, 3) vanaspati, 4) pudgala 5)
vyavahara kala, and 6) alokakasa.
4.
These are: 1) Dharma dravya, 2)
adharma dravya, (agurulaghugunaavibhaga 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] Jaghanyaparitaananta [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] Utkrstaanantaananta [Aau] is equivalent
to the aggregate called Kevalajnana. The
description above seems to imply that the utkrstaanantaananta 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 utkrstaanantaananata is infinite.
Thus, the description found in the Trilokasara leaves us in doubt as to
whether any of the three classes of paritaananta and the three classes of
yuktaananta and the jaghanyaanantaananta 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 anantaananta is always meant the
madhyamaanantaananta. So the madhyamaananataananta, 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 timeinstants in a kalpa, which are supposed to form a
continuum and are consequently infinite) and on the other the aggregate of
Mthadrsti jivarasi. 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 mithyadrstirasi is greater
than that of all the past time instants.
The above is nothing but the method of onetoone 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 viewpoint is further supported by the fact that the Jaina
works fix the duration of a timeinstant, and so the number of timeinstants 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 Jaghanyaparitaananta (which according to
definition is greater than the aggregate of time instants) is finite.
As already pointed out, the method of onetoone 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 c1, where is
infinite and a limiting number of a class. On the other hand, the Jaina
conception that the vargitasamvargita 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. Cantors 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.,
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.)
Ababa (ֲ)
xviii 
Bhadrabahu (֦)
iii 
Abbuda (,
sk, ) xviii 
Bhagavatisutra (־֟)
I fn 
Abhayadeva suri
(ֵ֤) I fn 
Bhaskara (Í)
i 
Acalapra (ֻ֯)
xvii fn 
Bhattotpala (ֻּ)
iv 
Adharma (ִ )
xix fn 
Bhavananta (־֭֭)
xiii 
Agamodaya samiti (ִ
״ן) I fn 
Bindu (ײ֭)
xviii 
Agurulaghuguna
(֑ ) xix fn 
Brahmagupta (ɐ㯟)
I,ii 
Ahaha ()
xviii 
Brhat Samhita ()
iv fn 
Akhobhini (׳֭
, sk xviii 
Chaturthachheda (֟)
viii 
Alokakasa (֍ֿ)
xix fn 
Dasa (, sk
) xviii 
Amama (ִ)
xvii fn 
Deya () vi 
Amamanga (ִӐ)
xvii fn 
Dharma (ִ)
xix fn 
Ananta ( ֭)
xiv, xv etc. 
Dhavala (־ֻ)
iii,iv,etc. 
Anantananta (֭֭֭)
xiv etc. 
Dravyananta (֭֭)
xiii 
Anubhagabandhaadhyasaya.sthana (ֲӬ־ֵã֭)
xix fn 
Dravya pramana (֯ϴ֝)
v 
Anuyoga ()
iiii 
Eka () xviii 
Anuyogadvarasutra (ָ֫)
iv 
Ekananta (֭֭)
xiii 
Apradesikananta (Ϥ֍֭)
xiii 
Ganita (ם֟) i 
Apratisthita (ן)
xix fn 
Ganananata (֭֭֭֝)
xiii 
Arddhaccheda (ִ)
vii,xii 
Ganitanuyoga (ם֭֟)
iii 
Ardhamagadhi (֬)
iv x 
Ganitasarasamgraha (םָ֟ 
Ӑ)I, iii,v, 
Aryabhyata (ֵ֙)
ii, iii 
Gommatasara (괴ָ֙)
v fn 
Aryabhattya (ֵ֙ߵ)
ii, iv 
Haha () xvii fn 
Asamkhyat (ӏ֟)
xiv, xvii 
Hahanga
(Ӑ) xvii fn 
Atata ()
xvii fn, xviii 
Harivamsapurana (ӿ֯֝)
xvii fn 
Atatanga (Ӑ)
xvii fn 
Hastaprahelita (ß֯֟)
xvii fn 
Avibhagapratichheda (ֳ
ן֓) Xix fn 
Huhanga (Ӑ)
xvii fn 
Avasarpini (ׯ)
xx, xxi 
Huhu ()
xvii fn 
Bappadeva (֤֯)
iv 
Ichha () xi 

Indranandi (Ӧۭ)
iv 
Bhadrabahavi Samhita (֦
)iv 
Jaghanya (֑֭)
xiv,xv,xvii 
Jaghanyaparitaananta (֑ ָߟ֭) xv, xviii etc. 
Jaghanyaanantananta (֑֭
ˆ֭֭֭) xiv,xv,xix 
Jaghanyaparitaasamkhyata (ָ֑ߟ֟)Xv,xviii,etc. 
Jaghanyaasamkhyataasamkhyata (֑֭ӏ֟)
xv,xviii etc 
Jaghanyayuktaananta
(֑֭㌟֭)xvxix 
(Madhyamayuktaasamkhyata (ִ֬㌟
ӏ֟ ) xv, xviii etc. 
Jaghanyaparitaasamkhyata(ָ֑ߟ֟ xv,xviii etc. 
Mahakathana (֍֭) xviii

Jambudvipa
(ִ߯) xvi 
Mahalata
(ֻ֟) xvii fn 
Jiva (߾)
xix fn 
Mahalatanga (ֻ֟Ӑ)
xvii fn 
Jivakanda
(߾֍֝) v fn 
Mahaviracarya
(־߸ֵ֓) i 
Jivarasi
(߾ָ) v 
Malabar
(ֲָֻ) iv 
Kalpa ()
xix fn, xx,xxi 
Malayagiri
(ֵֻא) iv 
Kamala (ֻ)
xvii fn 
Mithyadrsti Jivarasi
(״֣֥ ߾ָ) xx 
Kamalanga
(ֻӐ) xvii fn 



Karanabhavana
(ֳ־֭) iv 
Nahuta ()
xviii 
Karananuyoga
(֭) iii 
Nalina (֭)
xvii fn 
Kathana (֭)
xviii 
Nalinanga
(֭Ӑ) xvii fn 
Kevalajnana
(ֻ֖֭) xx 
Namananta
(ִ֭֭) xiii 
Koti (י)
v, xviii 
Nayuta (ֵ)
xvii fn 
Kotippakoti
(י֍י) xviii 
Nayutanga
(ֵӐ) xvii fn 
Ksetrasamasa (
ִ) iv 
Nemicandra
(״֭֓) xiv,xviii,xix 
Kumuda
() xvii fn, xviii 
Ninnahuta
(֮, sk ֝) xviii 
Kumudanga
(Ӑ) xvii fn 
Nirabbuda
(ָ, sk ָ)xviii 
Kundakunda
() iv 
padma (֩)
xvii fn 
Kusumapura
(֯) ii 
Padmanga (֩Ӑ)
xvii fn 
Lata (֟)
xvii fn 
Paduma (֤,
sk ֩) xviii 
Latanga
(֟Ӑ) xvii fn 
Pakoti
(֍י, sk ύי) xviii 
Lokakasa
(֍ֿ) xix fn 
Pali (ֻ)
v 
Madhyamaanantaananta (ִ֬
֭֭) xv, xix 
paritaananta
(ָߟ֭) xiv 
Madhyamaasamkhyataasamkhyata(ִ֬ӏ֟ӏ֟)
xv,xviii etc. 
Pataliputra
(֙֯) i 
Madhyamayuktaananta
(ִ֬㌟֭) 
Phala () xi 
Sahassa (, sk á) xviii 
Prakrit
(֍) iv, v, x 
Samantabhadra
(ֳִ֦֭) iv 
Pramana
(ϴ֝) xi 
Samkhyata
(ӏ֟) xiv, xv 
Pratisthita (ן)
xix 
Sarvananta
(־֭) xiii 
Pudgala
(ֻ) xix fn 
Saswatananta
(֤֭֭֟) xiii 
Pundarika
(㝛ߍ) xviii 
Sata (֟,
sk ֟) xviii 
Purana
(֝) iii 
Sathandagama
(֙ӛִ) iii 
Purva ()
xvii fn 
Shamakunda
(ִ֍) iv 
Purvanga
() xvii fn 
Siddha
(֬) xix fn 
Rajavarttika
(֕־ן) xvii fn 
Siddhasena (֬)
iv 
Rangacarya (ֵ֓)
ii 
Silanka
(Ӎ) iv fn 
SadharanaVanaspatinigoda
(ָ֬,֭ïן ) Xix
fn3 
Sogandhika
(֍, sk ۭ֍) xviii 
Uppala
(ֻ, sk ֻ) xviii 
Smayadhyayana
(ôֵֵ֭֬) iv fn 
Utkrstaanantaananta
(֭ ֭) 
Sridharacarya
(ֵָ߬֓) I, ii 
Utkrstaasamkhyataasamkhyata (ӏ֟ӏ֟) xv,xvii
etc. 
Srikalpa (
ߍ) xvii fn 
Utkrstaparitaananta
(ָߟ֭) 
Srutavatara (־ָ֟) Iv 
Utkrstaparitaasamkhyata (ָߟӏ֟) xv, xviii etc. 
Sthanangasutra
(ã֭Ӑ ) iv 
Utkrstayuktaananta
(㌟֭)xv,xix

Sthapanananta
(ã֭֭֭֯) xiii 
Utkrstayuktaasamkhyata (㌟ӏ֟) xv,
xviii etc. 
Sulbasutra
(㻲) ii 
Utsarpini
(ׯ) xx, xxi 
Suryaprajnapti
(ϖۯ) iv 
Uttaradhyayana sutra
(ֵָ֭֬) I fn 
Sutrakratanga sutra
(֍Ӑ ) iv fn 
Vanaspati
(֭ïן) xix fn 
Tathvarthadhigamasutrabhasya 
Varahamihira (ָ״) ii, iv

Taxila
(ցֻ) I 
Varga
() vi 
Trilokaprajnapti
(סֻϖۯ) xv, xvii fn 
Vargasamvarga
(Ӿ) xi 
Trilokasara
(סָֻ) iv,xiv,xv,xx 
VargaSalaka
(ֻ֍) vii 
Trikachheda
(ס֍) vii 
Vargitasamvargita (אӾא) vi vii,viii,xi 
Trutita
(י) xvii fn 
Varsa
() xviii fn 
Trutitanga
(יӐ) xvii fn 
Viralana
(ָ֭) vi 
Tumbulura
(㴲) iv 
Viralanadeya
(ָ֭  ) vi 
Ubhayananta
(ֵ֭֭) xiii 
Virasena
(߸) iv 

Vistarananta
(ßָ֭֭) xiii 
Umasvati
(þן ) iv 
Vyavaharakala
(־ָ ֻ) xix fn 

Yoga
() xix fn 

Yojana (֭) xv 

Yuga ()
xvii fn 

Yukta
(㌟) xiv, xv 

Yuktananta
(㌟֭֭) xiv 