History of Science in South Asia Jaina Thoughts on Unity Not Being a Number

At one time, the Jainas in India and the Greeks in abroad held that unity was not a number. This paper provides an insight for the first time into the thoughts offered by the Jainas as to why unity was not a number for them.

Āryarakṣita, Hemacandra, and Vinayavijaya Gaṇī belong to the Śvetāmbara sect of Jainism while Akalaṅka, Vīrasena, Nemicandra, and Mādhavacandra Traividya belong to its Digambara sect.All of their above treatises belong to the canonical class of the Jaina school of Indian mathematics. 3his paper aims at understanding why unity was not a number for the Jainas.It will provide an insight for the first time into their thoughts on this issue.Beyond this, it has three further purposes that are explored in section three (pp.212 ff.below).

E X P R E S S I O N S O N U N IT Y BY T H E JA I NA S
T HE ANUYOGADVĀRA SŪTRA SAYS: [1] se kiṃ taṃ gaṇaṇāsaṃkhā?ekko gaṇaṇaṃ na uveti, duppabhitisaṃkhā| 4 What is number‹-measure› as counting (gaṇaṇāsaṃkhā, Skt.gaṇanāsaṅkhyā)?Unity (ekka) is not for counting (gaṇaṇa, Skt.gaṇana); two, etc. (duppabhiti, Skt.dviprabhṛti) ‹i.e., from two onwards› are numbers (saṃkhās, Skt.saṅkhyās)." This contains three statements.The first is a question.The last two are the answer to it.B. B. Datta is the first historian of mathematics to have brought the second statement to our notice.He infers from it that "the Jainas do not consider unity a number". 5Ganitanand paid attention to three of them.His interpretation for the second statement is that "unity does not admit of numeration." 6īrasena says: [2] eyādīya gaṇaṇā doādīyā vi jāṇa saṃkhe tti| tīyādīṇaṃ ṇiyamā kadi tti saṇṇā du boddhavvā|| 7 In similar words, Nemicandra says: 3 On the basis of theorization the Jaina school of Indian mathematics is divided into the canonical class and the exclusive class.
The treatises of the canonical class contain mathematics along with discussion on Jaina canons.The object of the canonical class was to demonstrate canonical thoughts including on karma and cosmos using mathematics.
For details regarding the canonical class, see "Thoughts are that unity (eya, Skt.eka), etc. are for reckoning (gaṇaṇā, Skt.gaṇanā), two, etc. are numbers (saṃkhās, Skt.saṅkhyās), and three, etc. (tīyādī, Skt.tryādi) are, by rule, the names (saṇṇās, Skt.sañjñās) of growing (kadi, Skt.kṛti)".9 In this paper, we will focus only on unity.However, it will be interesting to know what growing (kṛti) is.Mādhavacandra Traividya (c.982 CE) writes that number, say , is growing (kṛti), if  2 >  and ( 2 − ) 2 >  2 .Since 1 does not pass the preliminary part of the test and vanishes while appearing for the main part, it is no-growing (nokṛti).Since 2 passes the preliminary part but does not pass the main part, it is an "inexpressible growing" (avaktavya kṛti).Since numbers from 3 onwards pass the complete test, each of them is a growing (kṛti). 10rior to Mādhavacandra Traividya, Vīrasena (816 CE) also referred to those three categories of growing (kṛti). 11Vinayavijaya says: [4] naikastu gaṇanāṃ bhajet 12 "Unity (eka) does not render service to counting (gaṇanā)." Āryarakṣita is ascribed authorship of the Anuyogadvāra Sūtra. 13He classified Jaina literature into four disciplines (anuyogas) 592 years after Lord Mahāvīra attained the bliss of liberation. 14J. P. Jain assigns him to c. 75 CE. 15R. S. Shah is of the opinion that material contained in the Anuyogadvāra Sūtra pertains to post 300 BCE. 16Muni Punyavijaya, Dalsukh Malvania and Amritlal Mohanlal Bhojak consider it to be a work of the second century CE and emphasizes that it cannot be placed after 300 CE. 17 Alessandra Petrocchi refers to it as being of approximately fourth century CE. 18 On the basis of these dates, although divergent, regarding the Anuyogadvāra Sūtra and the dates of the other above six treatises it can be said for certain that the Anuyogadvāra Sūtra appears to be the first treatise in which a Jaina author did not consider unity a number.

D I S C U S S IO N
T HE NOTION OF MEASURE was central to the overall Jaina intellectual enterprise.
The terms adopted by Jaina authors for "measure" are pamāṇa (Skt.pramāṇa) or māṇa (Skt.māna).Their classification of measure is broad. 19We shall explore it to the extent required to show how and why gaṇaṇa which occurs in [1] and gaṇanā which occurs in [4] are different from gaṇaṇā which occurs in both [2] and [3] and to explain gaṇaṇāsaṃkhā, which occurs in [1].This is the first of the three purposes of this paper.This kind of exploration will help us to justify why we have adopted two interpretations of gaṇaṇā (Skt.gaṇanā) or gaṇaṇa (Skt.gaṇana).One is "counting."See translations offered for [1] and [4].The other is "reckoning."See the translation jointly offered for [2] and [3].
In the classification of measure according to the Anuyogadvāra Sūtra, reckoning-measure and number-measure as counting are of interest to us.See 1.2.4 and 4.3.7 in Table 1.Similarly, reckoning and number-measure are of our interest in the classification of measure according to the Tattvārthavārtika of Akalaṅka (seventh century CE).See 1.4 and 2.1.1 in the first section of Table 2.The term used by Nemicandra (c.981 CE) in the Trilokasāra to describe reckoning is reckoning-measure.And the term used by him to describe number-measure is number (saṃkhā, Skt.saṅkhyā).See 1.4 and 2.1.1 in the second section of Table 2.
HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 209-231 are single numbers while those that contain  are different closed intervals. 37or example,   = 2 and   is 3,   − 1. 38The others are similar.This kind of number system or Jaina theory of numbers was founded, developed, and applied only in the canonical class of the Jaina school of Indian mathematics, i.e., in the treatises on Jaina canons which includes Karma theory and cosmography.
According to Akalaṅka, these twenty-one folds are number-measure (saṅkhyāpramāṇa). 39The same is according to Nemicandra. 40In order to define   , he states, at the beginning of describing the number-measure (saṃkhāpamāṇa, Skt.saṅkhyāpramāṇa) system of twenty-one folds, that the first of the four defined pits 41 is filled with mustards starting from two. 42In order to explain why the filling starts from two, Nemicandra refers to [3], which means that unity is not a number or number-measure.In the middle of a detailed discussion on "explanation of counting and growing" (gaṇana-kṛti-prarūpaṇā), 43 Vīrasena writes, in his Dhavalā, in order to support his discussion, that [2] has also been said.On the other hand, these twenty-one folds minus  iu or the first twenty folds are number ‹-measure› as counting (gaṇaṇāsaṃkhā, Skt.gaṇanāsaṅkhyā) according to the Anuyogadvāra Sūtra. 44The Anuyogadvāra Sūtra starts describing "number-measure as counting" of twenty folds right from [1].By means of [1] it states that unity is not for counting.Vinayavijaya Gaṇī refers to [4] in the early part of his description on number-measure of twenty-one folds. 45On the basis of the above facts it can be inferred that it was a founding and integral part of this system that unity was not a number.
Observing Table 1, we find that there are eight kinds of number-measure according to the Anuyogadvāra Sūtra.Since number as counting (gaṇaṇāsaṃkhā, Skt.gaṇanāsaṅkhyā) is the seventh of them, it is essentially number-measure as counting (gaṇaṇāsaṃkhappamāṇa, Skt.gaṇanāsaṅkhyāpramāṇa).See Table 1.That is why I interpret the term gaṇaṇāsaṃkhā, which also occurs in [1], as "number-measure 37 I have employed terms like "single number" and "closed interval" arbitrarily.Number-measure system of the Jainas is incomparable in history of world mathematics.It is not yet fully studied by modern scholars.
For its initial understanding see Datta 1929: 140-142; A. N. Singh  1942: 14-20; and N. Singh 1991: 209-230.38 Instead of   = 2 we can write (  ) = 2 where (  ) stands for "number of elements in   " and This kind of expression of   may be agreed with and appreciated.

HISTORY OF SCIENCE IN SOUTH
On the basis of the above entire exploration we can say that the Jainas employed numbers for measurement under two different ideas.One was idea of reckoning and the other was idea of counting.Unity was acceptable to them as "reckoning-measure" but was not acceptable as "number-measure as counting."

WHY WAS UNITY NOT EMPLOYED TO COUNT A UNIT?
Apart from the above, it is essential to understand and explain why unity was not employed by the Jainas to count a unit when it is single and that they held that unity corresponded to unit.This is the second purpose of this paper.Since we do not find any direct material in their treatises, we will have to search for some clue or clues in their discussion on their ontology, cosmography and karma theory, that can help us answer this question.
In this regard, we find the following example followed by a comment as well, offered by Hemacandra (1088-1172 CE) on the second statement of [1].
[6] … yata ekasmin ghaṭādau dṛṣṭe ghaṭādī vastvidaṃ tiṣṭhatītyevameva prāyaḥ pratītirutpadyate, naikasaṅkhyāviṣayatvena, athavā ādānasamarpaṇādivyavahārakāle ekaṃ vastu prāyo na kaścidgaṇayatyato'saṃvyavahāryatvādalpatvādvā naiko gaṇanasaṅkhyāmavatarati, … 48 "When an object like a pot is seen, what one realizes is only a pot and not its number; or it may be due to the fact that in ordinary dealings only one thing, if given or taken, is mostly not "taken into account" ‹← "counted" (gaṇaya, i.This is an English translation offered by H. R. Kapadia for [6].I am responsible for content inserted into the angular brackets.I suggest replacing "taken into account" by "counted" so that Kapadia's translation can fully and literally accords with the subject of this paper.My suggestion is supported by the term "gaṇaya, i.e, gaṇana" occurred in [6].However, the term "taken into account" interpreted by Kapadia has the same sense that the term "counted" has when the complete sentence containing either of them is read.Figure 2: Now, on the basis of Hemacandra's example and that "counting" means "the act of grouping together," we can explain how and why the Jainas did not count a unit when it is single.Hemacandra's example is associated with Figure 2. Let us first see Figure 1.A, B, C, and D are pots.Since they are many, we shall have to group them together in order to know how many they are.If we first see A, we shall later see B, C, and D. Each, taken individually, is a unit.Since we have first seen A, we shall start counting from B, not from A at all, as A cannot be grouped together with itself.Counting up to B will be like this: 1 A, 1 B or 2 {A, B}.Similarly, up to C: 1 A, 1 B, 1 C or 3 {A, B, C}; and up to D: 1 A, 1 B, 1 C, 1 D or 4 {A, B, C, D}.Since total number of units in Figure 1 is 4, its "number-measure as counting" or "numeric value of its measure that has come through counting" is 4. Let us now consider Figure 2.There is pot.It is a unit.Since it is single or there is no other unit, we cannot perform the act of grouping together.Since we have not counted any unit, no number is required to denote measure of single unit.That is why Hemacandra says that "when an object like a pot is seen, what one realizes is only a pot and not its number."Thus, [6] posits that the Jainas did not count unit when it was alone, i.e., single.Now, we shall corroborate that unity corresponds to unit for the Jainas although they did not employ the former to measure the latter when alone.From Table 3 we can understand that the expression "knowledge of a subtle groupsouled vegetable kingdom" mentioned in the Tattvārthavārtika and the expression "knowledge that a non-developable subtle group-souled vegetable kingdom possesses" mentioned in the Trilokasāra for the lowest measure of knowledge refer to are one and the same entity.Traividya for it is "knowledge that an absolutely non-developable subtle groupsouled vegetable kingdom possesses" (sūkṣmanigodalabdhyaparyāptakeṣujñāna). 50imilarly, the term coined for the highest measure of knowledge is omniscience, which the Tattvārthavārtika and the Trilokasāra refer to as "knowledge that Kevali possesses" (i.e., "perfect knowledge") and "knowledge that Jinendra possesses" respectively.See Table 3.
We are able to see from Table 3 that the expressions "ultimate-particle" (paramāṇu), "space-point" (pradeśa), and "infinitesimal fraction of time" (samaya) refer not only to the indivisible part of matter, space, and time respectively but also to their respective lowest measures.They must have been arrived at using an idea of indivisibility relating to matter, space, and time respectively.But the same is not the case with existence (bhāva).On the ground that unity has not been placed before the lowest measure of knowledge, i.e., before "knowledge that an absolutely non-developable subtle group-souled vegetable kingdom possesses," by either of Akalaṅka and Nemicandra while it has been prefixed with each of " ultimate-particle," "space-point," and "infinitesimal fraction of time," I conclude that the "knowledge that an absolutely non-developable subtle group-souled vegetable kingdom possesses" is not an indivisible part of knowledge.This is very important for the following discussion.
Before we proceed we would like to know what existence (bhāva) and knowledge are.Bhavanaṃ bhāva means "to be is existence (bhāva, state or condition)."Existence is an attribute of an entity.Entity is of two kinds.One is the living and the other is the non-living.Attributes of the latter are colour (varṇa), etc. while those of the former are knowledge (jñāna), conation (darśana), and "conscious attentiveness" or attention (upayoga). 51J. L. Jaini writes that knowledge is the essence of soul.There is no soul without knowledge.There is no knowledge or knowability without soul. 52owledge was measured by the Jainas using their number-measure system.Jaini states its importance in the following words.
50 For Traividya's explanation see TriSā, under vv.11-12, p. 13.For English translation of sūkṣmanigodalabdhyaparyāptakeṣu see GoS-āJīKā, vv.51-117, pp.51-83 and vv.299-464,  pp.175-238, especially p. 186 There are two ways known to us of having a very rough and remote Idea of Omniscience.One is by considering the extent of early Jaina sacred literature which is mostly lost to-day; and the other and even a better one is by considering the Jaina theory of numbers ‹(i.e., number-measure system of twenty-one folds)›. 53 the end of the description of the number-measure system, the Trilokasāra lets us know that scriptural knowledge, clairvoyance, and omniscience are numerable, innumerable, and infinite respectively. 54Prior to this information it states that "indivisible corresponding-sections" (avibhāga praticchedas) of omniscience are  iu . 55J. L. Jaini writes in simple terms that the number of units (avibhāga praticchedas) of perfect knowledge (kevalajñāna) is  iu . 56ow we are able to form the following opinions.
The "indivisible corresponding-section" of knowledge is its unit. iu units of knowledge form omniscience. "Indivisible corresponding-section" seems to have been conceived by applying the idea of indivisibility to knowledge as the term "indivisible" (avibhāga) in the expression suggests.Since it is a unit of knowledge, Akalaṅka and Nemicandra have not prefixed one with "knowledge of a subtle groupsouled vegetable kingdom" and "knowledge that a non-developable subtle group-souled vegetable kingdom possesses" respectively. 57Since "knowledge that an absolutely non-developable subtle group-souled vegetable kingdom possesses" is the lowest measure of knowledge, the number of units in it must be   .In other words, "an absolutely non-developable subtle group-souled vegetable kingdom" possesses only two "indivisible corresponding-sections" of knowledge.Since no knowledge is lower in measure than the "knowledge that a non-developable subtle group-souled vegetable kingdom possesses," we can be allowed to assume that one "indivisible corresponding-section of knowledge" alone is not possessed by any soul or group-souled.That may have been the reason that one "indivisible corresponding-section of knowledge" could not be the lowest measure of knowledge.One which is prefixed with each of "ultimate-particle," "space-point," and "infinitesimal fraction of time" in Table 3 is in the capacity of reckoning-measure, not in that of number-measure at all.Now, on the basis of that concept of mathematics is applied where it fits into, we can deduce that unity corresponds to unit of any sort for the Jainas as one "indivisible corresponding-section of knowledge" corresponds to unit of knowledge.

ANCIENT GREEK APPROACHES
The third and last purpose of this paper is to take stock of some thoughts offered by the ancient Greeks.Following the Egyptian view, Thales (c.600 BCE) defined number as "a collection of units."The Pythagoreans made number out of one.Some of them defined it as "a progression of multitude beginning from a unit and a regression ending in it". 58From their doctrine, Aristotle observed that the one was reasonably regarded as not being a number, "because a measure is not the things measured, but the measure or the one is the beginning (or principle) of number". 59He defined number as a "multitude of units" or a "multitude of indivisibles" or "several ones" or a "multitude of measures". 60He asserted that "number is the principle both as matter for things and as constituting their attributes and permanent states". 61In this way, he justified his teacher Plato (c.380 BCE), who had already regarded unity as different from number. 62Heath writes that, by arithmetic Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers.He does not, however, ignore the art of calculation (arithmetic in our sense); he speaks of number and calculation and observes that "the art of calculation (λογιστική) and arithmetic (άριθμητική) are both concerned with number;" … But the art of calculation (λογιστική) is only preparatory to the true science; those who are to govern the city are to get a grasp of λογιστική, not in the popular sense with a view to use in trade, but only for the purpose of knowledge, until they are able to contemplate the nature of number in itself by thought alone.This distinction between άριθμητική (the theory of numbers) and λογιστική (the art of calculation) was a fundamental one in Greek mathematics. 63clid (c.300 BCE) also believed in a similar doctrine when he defined the unit as "that by virtue of which each existing thing is said to be one" and number as "the multitude made up of units". 64Another notion the ancient Greeks held was that unity, like a point, is incapable of division. 65Nicomachus (c. 100 CE) defined number as "a flow of quantity made up of units". 66ntil modern times the view that unity was not a number prevailed in Europe.Boethius (sixth century CE) propagated this view among medieval writers such as al-Khwārizmī (c.825 CE), Psellus (c.1075 CE), Savasorda (c.1100 CE), Johannes Hispalensis (c.1140 CE), and Rollandus (c.1425 CE). 67Not only these writers, but most of the authors, such as Pacioli (c.1494 CE), J. Köbel (c.1514 CE), Tzwivel (1505 CE), Humphrey Baker (c.1568 CE), and many others also, of the early printed books excluded unity from the number field. 68The first printed book on arithmetic by an unknown author in the Venetian dialect and published on December 10, 1478 CE at Treviso, clearly states that, number is a multitude brought together or assembled from several units, and always from two at least, as in the case of 2, which is the first and the smallest number.One is not called a number but the source of number. 69ker writes in his book The Well-spring of Sciences that, an vnitie is no number but the beginning and original of number. 70t Smith writes, it is not probable that Nicomachus (c.100) intended to exclude unity from the number field in general, but only from the domain of polygonal numbers.It may have been a misinterpretation of the passage from Nicomachus that led Boethius to add the great authority of his name to the view that one is not a number.Even before his time the belief seems to have prevailed, as in the case of Victorius (475) and Capella (c.460), although neither of these writers makes the direct assertion. 71en in more recent times some writers have not considered unity to be a number.For example, George Baron (1769-1818 CE), the founder and editor-inchief of the Mathematical Correspondent, categorically stated that, numbers are composed of units, but a unit is not a number; if a book be said to consist of leaves, it is plain that a leaf is not a book. 72 the sixteenth century, thinkers in Europe started to oppose the view that unity is not a number.Hylles (1592), speaking of "an vnit or an integer…," was rather afraid to take a definite stand in the matter, but said that, 67 Smith 1958: 27.68 Smith 1958: 28.69 Smith 1929: 1-3.70 Jackson 1906: 30.  the latter writers, as namely Peter Ramus, and such as have written since his time, affirme not only that an vnite or one, is a number, but also that euery fraction or parte of an vnite, is a number.… 73 Simon Stevin (1585) found it necessary to correct this popular view that unity is not a number.After reviewing the various arguments which history had handed down, he argued that, (i) a part is of the same nature as the whole, and hence that unity, which is part of a collection of units, is a number, and (ii) if from a number there is subtracted no number, the given number remains; but if from 3 we take 1, 3 does not remain; hence 1 is not no number. 74 the end of the century it was recognized due to those thinkers that the ancient view on unity was too narrow.Among them Stevin was the first prominent writer to clearly assert that unity is a number. 75ow we can say that the logistic 76 (λογιστικήor the art of calculation) of the Greeks seems to be somewhat like the reckoning-measure of the Jainas.The arithmetic (αριθμητικήor the theory of numbers) of the Greeks is said to have been more abstract than geometry. 77It appears to be somewhat similar and somewhat dissimilar to the number-measure of the Jainas.Similarities between them are that knowledge.From Hylles' quote it is that "euery fraction or parte of an vnite is a number" but the unit was indivisible for both of the Greeks 78 and the Jainas. 794 CO NC LU D I NG RE M A R K S T HE CONCEPT OF NUMBER-MEASURE developed by the Jainas was essentially "number-measure as counting."Keeping this in view, they developed system of "number-measure as counting," both from   to  im and from   to  iu , to measure the magnitude of total units that they grouped together.The idea of indivisibility enabled them to allow unity to correspond to a unit while the idea of counting, i.e., "grouping together" did not allow them to count a unit when it was alone.For them, counting was prior to measuring.That is why they could not employ unity to measure a unit when it was alone.Similarly, certainly prior to the Jainas, the Greeks did not measure a unit when it was alone, using unity, as for them number meant "multitude" or it was "a collection of units."Since, for them, unity was not a collection, it was not considered a number.

INSIDE INDIA
Outside India, the ancient Greek thoughts regarding unity, first due to the Greeks themselves and later due to the thinkers in Europe and elsewhere, lasted for almost 2000 years.Mathematicians and philosophers continued to argue over whether unity was a number.On the other hand, Jaina thoughts on unity, like those on figurate numbers, 80 logarithms, 81 raising a number to its own power, 82 number-measure and so forth, remained confined to the canonical class of the Jaina school of Indian mathematics.To make the importance of the thoughts offered by the canonical class on unity very clear to the non-Jaina thinkers in India was only a remote possibility; even its exclusive class 83 that includes Śrīdhara (c.799 CE), Mahāvīra (850 CE), Rājāditya (twelfth century CE), Ṭhakkara Pherū (c.1265-c.1330 CE), never referred to the idea that unity was not a number.A plausible reason for this seems to have been that the exclusive class did not find any area of application of those thoughts for public interest.Moreover, the canonical class placed its thoughts about unity in the category of post-worldly measure (lokottaramāna, measure which is not common in ordinary 78 See "Another notion … incapable of division." in the section "Ancient Greek Approaches" above (p.219).79 See Table 3 and the discussion in the section "Why was Unity not Employed…" above (pp.215-218).80 Jadhav 2009: 35-55.81 Jadhav 2002: 261-267; 2003: 53-73.82 Jadhav 2008: 139-149.83 The treatises of the exclusive class of the Jaina school of Indian mathematics are composed exclusively on mathematics.The object of the exclusive class was to provide mathematics education to the contemporary civil life.For details regarding the exclusive class, see Jadhav 2017: 316-331.
HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 209-231 life) and it not only drew a clear line of demarcation between reckoning-measure and number-measure but also had been getting that line brought through its treatises into the notice of its followers and learners to come.

GREEK AND JAINA APPROACHES
Ancient Greek thoughts on unity go back, as we have seen, to at least 600 BCE.It cannot be said with certainty how old the Jaina thoughts on unity are.But it can be said for certain that they developed prior to the division of the Jaina organization since they had developed before the composition of Anuyogadvāra Sūtra and both the Digambaras and the Śvetambaras had held that unity was not a number.The Jaina organization is said to have officially split into Digambara and Śvetambara sects by the first century CE. 84 If the chronological order of the development of the Greek and Jaina thoughts on unity and the similarities between the arithmetic of the Greeks and the number-measure system of the Jainas are kept in view, it may be said that the thoughts on unity might have been transmitted from the Greeks to the Jainas.On the other hand, if the dissimilarities between the arithmetic of the Greeks and the number-measure system of the Jainas are kept in view, any possibility of transmission of the thoughts on unity does not arise.But this opinion may be rejected on the ground that indirect transmission can account for bits and pieces of thoughts while other aspects might have substantially changed.The lack of concrete evidence of transmission, such as Greek loanwords in Prakrit and Sanskrit texts or vice versa, must surely lead us to conclude, at least prima facie, that such transmission did not occur and that these ideas arose independently in the Greek and Jaina cultures.

FUTURE DIRECTIONS
Many more clues, apart from those that helped in this paper to explore Jaina thoughts on unity, can be found, if searched for, in the treatises of the Jainas on their canonical thoughts including those on ontology, cosmography and Karma theory, which can enlighten us further and can inform us about other aspects of their thoughts on unity.For example, those clues may be no-growing (nokṛti), space-point (pradeśa), their number-measure system itself, etc.

ACK NOW L E D G E M E N T S
E XCEPT FOR A FEW CHANGES including its title this paper was presented in Na- tional Symposium on Jaina Mathematics, held at Kundakunda Jñānapīṭha, 84 Specifically by the year 79 CE or some date between 80 CE and 82 CE.See Banerji   2004: 170;  Basham 1986: 291;  Kumar  1997: 47; Schubring 2000: 50.HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 209-231 Indore (India) during December 22-23, 2019.I take this opportunity to thank the organizers, including Prof. Anupam Jain, of the symposium for inviting me.I would like to place on record my thanks to Prof. Peter Flugel (London) for taking an interest in this paper and encouraging me.I wish to thank the anonymous referees for their constructive comments and suggestions offered to improve this paper.I am highly grateful to Prof. Dominik Wujastyk, the editor of this journal, for all his help making this paper into a publication.

Table 2 :
The classification 87 of measure according to the Tattvārthavārtika and the Trilokasāra