Pramāṇa · Search · About · API

A Concise History of Science in India c7.s3: Jaina Mathematics

A Concise History of Science in India c7.s3

Language
English

Contents

Jaina Mathematics

c7.s3.1

Despite great interest attached to mathematics by the Jainas of ancient India, very little of their mathematical effort has unfortunately survived to this day. We have of course the mid-ninth century mathematical classic *Gaṇita-sāra-saṃgraha* by Mahāvīra (*c.* A.D. 850), but no such comparable work belonging to the pre-Christian centuries is extant. We only have a few fragments or insertions in canonical or other types of non-mathematical literature from which to judge of their early achievements. That a good body of mathematical literature must have existed at one time seems indicated in Mahāvīra's own statement in which he described himself as a mere compiler from the great ocean of the knowledge of mathematics from which long lines of holy sages had skilfully gathered many precious mathematical gems. About the first century B.C. the *Sthānāṅga-sūtra*, a Jaina canonical work, listed several mathematical topics which used to be cultivated at that time. These topics are: *saṃkhyāyana* (science of numbers), *parikarma* (fundamental operations), *vyavahāra* (subjects of treatment), *rajju* (geometry, like *śulba*), *rāśi* (heap, solid mensuration), *kalāsavarṇa* (fractions), *yāvat-tāvat* (equations, algebra), *varga* (square, quadratic equations), *ghana* (cube, cubic equations), *varga-varga* (biquadratic equations) and *vikalpa* (permutations and combinations). The Sanskrit version of the above enumeration in Pali runs as follows:

c7.s3.2

*parikarma rajjuḥ rāśiḥ vyavahārastathā kalāsavarṇaśca* |
*pudgalāḥ yāvattāvat bhavanti ghana ghanamūlaṁ vargaḥ vargamūlam* ||

c7.s3.3

These subjects and the various technical terms used by the Jainas passed later on into the mathematical works of scholars irrespective of their religious beliefs and adherences. It is thus quite reasonable to believe that in the period intervening the literature of the *Brāhmaṇas* and the *Sūtras* of the various Vedic schools and the period of specialization and Siddhāntic astronomy from about the fourth or fifth century A.D. the Jaina mathematicians played a significant role. The disappearance of their works might be due to (*a*) supersession of their simple processes and methods by better and more sophisticated ones due to Āryabhaṭa, Brahmagupta, Mahāvīra, Śrīpati, Śrīdhara, Bhāskara and others and (*b*) progressive deterioration of the culture of mathematics in their religious order.

c7.s3.4

*Jaina Mathematical Sources*

c7.s3.5

As to early Jaina mathematical sources, we have already referred to the *Sūryaprajñapti* and the *Candraprajñapti* which, although astronomical works, contain useful information about mathematics as well. Like the *Sthānāṅga-sūtra*, we have stray references of mathematical and astronomical importance in canonical works like the *Bhagavatī-sūtra* (first century B.C.), the *Uttarādhyāyana-sūtra* (first century A.D.) and the *Anuyogadvāra-sūtra* (first century A.D.). As to persons by name, mention should be made of Bhadrabāhu (d. 298 B.C.), who had the unusual gift of reproducing from memory the entire canonical literature of the Jainas and was befittingly called a *śrutakevalin*. He wrote a commentary on the *Sūryaprajñapti*, as we know from Malayagiri and possibly a compendious *Saṁhitā* known after his name. Bühler noticed a work entitled *Bhadrabāhavī Saṁhitā*, but it has not been established that it was the work of Bhadrabāhu we are considering. Umāsvāti (c. first century A.D.), the reputed Jaina metaphysician and author of *Tattvārthādhigama-sūtra-bhāṣya* was, of course, no mathematician but referred to mathematical formulae in his metaphors, which indicate that such mathematical topics must have been widely current at least among the intelligentsia of his time. Both Bhadrabāhu and Umāsvāti hailed from Kusumapura near Patna. Umāsvāti was also the author of a *Kṣetrasamāsa* (collection of places) or *Jambūdvīpasamāsa*. This title belongs generally to a class of works popular among the Jainas, dealing with geography and mensuration. The cosmographico-geographical details, the elaborate specification of the dimensions of the different *dvīpas* and fantastic cosmological theories given in this class of works conceal at the same time important mathematical information for this early formative period. Finally, we have the name of Siddhasena, the Jaina in Varāhamihira's references, from which commentator Bhaṭṭotpala quoted a number of passages which show Siddhasena's interests in astronomy and mathematics.

Theory of Number

c7.s3.t1.1

We have already mentioned the Jaina interest in the enumeration of large numbers. By the conception of *śīrṣaprahelikā* they suggested a number of the order of 8,400,000²⁸. The Jaina terminology uses the word *sthāna* for the decimal place. As to names beyond thousand (*sahasra*), they prefer to proceed as follows: tens of thousands, hundreds of thousands, tens of thousands of thousands (*koṭi*), tens of *koṭi*, hundreds of *koṭi* and so on. In the Jaina system unity is not regarded as a number—*eko gaṇanāsaṃkhyā na upeti*. Numbers are classified into *oja* (odd) and *yugma* (even) as in the *Brāhmaṇa* system.

c7.s3.t1.2

The whole range of numbers is again divided into three groups: *saṃkhyeya* (numerable), *asaṃkhyeya* (innumerable) and *ananta* (infinite). Each group is subdivided into three orders, e.g. the first group (numerable)—lowest, highest and intermediate; the second group (innumerable)—nearly innumerable, truly innumerable and innumerably innumerable; the third group (infinite)—nearly infinite, truly infinite and infinitely infinite. To form an idea of the highest numerable (*utkṛṣṭa saṃkhyeya*) of the first group, one is advised to count one by one white mustard seeds while filling with them a trough as large as the size of the Jambudvīpa of diameter 100,000 *yojana* and circumference 316,227 *yojana* 3 *gavyuti* 128 *dhanu* 13½ *aṅgula* and a little over.a In the same manner, one is to fill up with same seeds other troughs formed by the various oceans and islands mentioned in the Jaina cosmology, and count them one by one. The total number of mustard seeds will still be less than the highest numerable. All this is stated in the *Anuyogadvāra-sūtra*, which then continues to enumerate the number-groups of higher orders. If *N* be the highest numerable number as defined above, the higher groups may be represented as follows:

c7.s3.t1.3

{l} (N+1)\phantom{¹}\ \dots\ [(N+1)²\ -1] / (N+1)²\ \dots\ [(N+1)⁴\ -1] / (N+1)⁴\ \dots\ [(N+1)⁸\ -1] / (N+1)⁸\ \dots\ [(N+1)¹⁶-1] / (N+1)¹⁶\dots\ [(N+1)³²-1] / (N+1)³²\dots

c7.s3.t1.4

The highest numerable number of the Jainas reminds us of the Alef-zero of modern mathematics, and the Jaina imagination clearly went much further than that.

c7.s3.t1.5

The Jainas are strongly believed to have possessed some kind of numeral symbols although it is not possible to say what these forms were. The *Samavāyāṅga-sūtra* (fourth century B.C.) and the *Prajñāpanā-sūtra* (second century B.C.) mention *aṅkalipi* and *gaṇitalipi* in connection with a list enumerating the different written characters. The two names suggest that in all probability different numeral forms were in use for different purposes, *aṅkalipi* being used in engraving and *gaṇitalipi* in ordinary writing.

c7.s3.t1.6

a Datta (2), pp. 140–41.

c7.s3.t1.7

*Fundamental Operations, Factors*

c7.s3.t1.8

Umāsvāti mentions two methods of multiplication and division of which one is the ordinary method now generally followed and the other is a shorter operation carried out in successive stages by factors. The method of multiplication by factors reappears in the works of Brahmagupta and later authors. Śrīdhara, in his *Triśatikā*, develops the method of division by factors. The Jainas are found to have large and complicated arithmetical factors. One interesting feature is their frequent resort to approximation in dealing with mixed number. Whenever the fractional part is less than 1, it is neglected, but, when greater than ½, it is replaced by unity. Thus 315089 is used in place of 315089 218079/630178 and 318315 in place of 318314 558404/630628. To indicate such approximation, expressions like *kiñcidviśeṣādhika* (a little more), *kiñcidviśeṣoṇa* (a little less) were used.

c7.s3.t1.9

*Mensuration*

c7.s3.t1.10

*Rajju*, as we have seen, was the Jaina name for geometry or mensuration, in which they closely followed the Vedic *śulbavids*. Of the many geometrical terms met with in their literature, the following may be mentioned: *sama-cakravāla*, *vṛtta* (circle), *vyāsārdha*, *viṣkambhārdha* (semidiameter, radius), *jīva* (chord), *dhanupṛṣṭha* (arc), *sama-caturasra* (square), *sama-catuṣkoṇa* (even parallelogram), *caturasra* (quadrilateral), *āyata* (rectangle), *tryasra* (triangle), *viṣama-cakravāla*, *parimaṇḍala* (ellipse), *pratara* (plane), *ghana* (solid), *ghana-tryasra* (triangular pyramid), *ghana caturasra* (cube), *ghanāyata* (rectangular parallelepiped), *ghana vṛtta* (sphere), and so on.

c7.s3.t1.11

A number of formulae concerning the mensuration of the circle were recorded by Umāsvāti in his *Tattvārthādhigama-sūtra-bhāṣya* and *Jambūdvīpasamāsa*. These formulae are:

c7.s3.t1.12

For a circle of diameter *d*, circumference *p* and area α,

c7.s3.t1.13

$p=√10.d$ $α=¼pd.$

c7.s3.t1.14

For the segment of a circle of arc *a* less than a semi-circle, chord *c*, sagitta or arrow *s* and diameter *d*,

c7.s3.t1.15

$c=√(4s(d-s))$ $s=½(d-√(d²-c²))$ $a=√(6s²+c²)$ $d=1/s(s²+(c²)/4).$

c7.s3.t1.16

The relation of the arc *a* with respect to *s* and *c* is found in the works of Mahāvīra and Āryabhaṭa II. They have also given alternative formulae varying only slightly in the numerical coefficient of *s*². Heron of Alexandria (third century A.D.) gives the relation as

c7.s3.t1.17

$a = √(4s²+c²) + s/4.$

c7.s3.t1.18

Shen Kua, in his *Meng Chhi Pi Than* (A.D. 1086), gives the formula for the sagitta as

c7.s3.t1.19

$a = c + (2s²)/d.$

c7.s3.t1.20

The above-mentioned formulae, though found in a work by Umāsvāti, do not appear to have been discovered by him because some of these formulae were most likely utilized in earlier texts (*Sūryaprajñapti*, *Jambūdvīpa-prajñapti*) in the computations of the circumference of the Jambūdvīpa from its given diameter of 100,000 *yojana*, the linear dimensions of the seven parts in which Jambūdvīpa is divided, the dimensions of *Bhāratavarṣa* forming the southernmost segment of this *Dvīpa* and so on.

c7.s3.t1.21

*Value of π*

c7.s3.t1.22

In the relation given above of the circumference of a circle to its diameter, the value of π is shown to be √10. The astronomical tract *Sūrya-prajñapti* uses this value as also the less accurate one of 3. In the *Uttarādhyāyana-sūtra*, it is stated to be a little more than 3. But all medieval Jaina works systematically used √10 as the value of π in spite of a more accurate value having by then come into general use in all Brāhmaṇa astronomical and mathematical works.

c7.s3.t1.23

*Laws of Indices*

c7.s3.t1.24

The power series found in the Jaina works are limited to successive squares or square roots, e.g. a², (a²)², (a⁴)², a2n, (a½), (a½)½, (a¼)½, (a⅛)½, (a1/n)½. In other words,

c7.s3.t1.25

| first | square of *a* means a² |
| second | square of *a* means a2 × 2 |
| third | square of *a* means a2 × 2 × 2 |
| *n*th | square of *a* means a2 × 2 × 2 × to n term. |

c7.s3.t1.26

In the *Anuyogadvāra-sūtra*, the world population is given to be a number obtainable by multiplying the sixth square of two by the fifth square, or a number which can be divided by two 96 times; it occupies twenty-nine places, or, according to another interpretation, lies between the twenty-fourth and the thirty-second place. According to the definition of the square terms, sixth square means 22⁶ or 2⁶⁴ and fifth square means 22⁵ or 2³². Therefore,

c7.s3.t1.27

$N = 2⁶⁴ × 2³² = 2⁹⁶ = 79,228,162,514,264,337,593,543,950,336.$

c7.s3.t1.28

Notice that the figure obtained occupies 29 places and is divisible by two 96 times. Here we have a clear indication of the application of the following laws of indices: $am × an = am+n$ $(am)n = amn$

c7.s3.t1.29

*Permutations and Combinations*

c7.s3.t1.30

Statements of results presumably arrived at by the methods of permutations and combinations appear quite early in the Jaina literature. In the *Bhagavatī-sūtra*, *Anuyogadvāra-sūtra* and the *Jambūdvīpaprajñapti*, possible numbers of combinations out of *n* fundamental categories taken one at a time (*ekaka-saṃyoga*), two at a time (*dvika-saṃyoga*), three at a time (*trika-saṃyoga*) have been speculated upon, and possible number of selections out of a number of males, females and eunuchs, all calculations of various groups to be formed out of different senses, etc., have been given correctly. In the *Anuyogadvāra-sūtra*, the number of permutations of six things is given by 1 × 2 × 3 × 4 × 5 × 6 with further direction of deducting 2 to give the result less the direct and reverse orders. An English translation of the passage runs as follows:

c7.s3.t1.31

‘What is direct order? Dharmāstikāya, Adharmāstikāya, Ākāśāstikāya, Jīvāstikāya, Pudgalāstikāya and Samaya—this is the direct order. What is the reverse order? (Read the above from Samaya backwards). What is Ananupūrvī (mixed) order? In this series, the first term is unity, the common increase unity and the number of terms six. Multiply the terms one after the other and deduct two. This is the Ananupūrvī.’a

c7.s3.t1.32

Śīlāṅka, a Jaina commentator of the ninth century A.D., reproduces from some ancient mathematical texts rules for permutations and combinations, with his own illustrations. One quotation, curiously enough, is in Ardhmāagadhī. The rules are:

c7.s3.t1.33

‘Beginning with unity up to the number of terms, multiply continually the (natural) numbers. That should give the result as per calculations of permutations and combinations.’

c7.s3.t1.34

That is $N = 1 . 2 . 3 . . . (n-1)n.$

c7.s3.t1.35

Another rule for finding the actual spread (*prastārānayanopāya*) runs as follows:

c7.s3.t1.36

‘Divide the total number of permutations by the last term and the quotient by the next. These are to be placed one after the other by the side of the initial term in the calculations of permutations and combinations.’

c7.s3.t1.37

a *Anu. Sū.*, 97; see also Chakravarti (G.), pp. 79–88.

c7.s3.t1.38

The explanation of the rule after Śīlāṅka is as follows:

c7.s3.t1.39

(1) The total number of permutations of *r* things taken all at a time $= 1 . 2 (r-1)r = r!$

c7.s3.t1.40

(2) The total number of permutations having a particular thing a_r for its initial digit = (r!)/r = (r-1)!

c7.s3.t1.41

(3) The total number of permutations in which a_r-1 will be next to a_r in (2) = ((r-1)!)/(r-1) = (r-2)!

c7.s3.t1.42

Proceeding in this way, different permutations of things can be found out.

c7.s3.t1.43

It must be noted, however, that meter and prosody offered a much more fertile field for the exploration of this branch of mathematics and the *chanda* (meter) specialists naturally enough led the way as already noticed before.

Cites (12)