Geometry in Ancient and Medieval India c8: Chapter III. Early Jaina Geometry

Geometry in Ancient and Medieval India c8

Language: English

c8.s1 3.5. Solid figures — 9 parts
c8.s2 3.6. The trapezium and the trapezoidal solid — 12 parts

Chapter III. Early Jaina Geometry

c8.2

**3.1.** To the period of mathematical development represented by the *Śulbasūtras*, probably belongs the mathematical knowledge of the Jainas too. Dating their canonical literature is as difficult as the dating of the Vedic literature. For long Vardhamāna Mahāvīra, of about the same period as the Buddha, was held to be the founder of Jainism. But Mahāvīra is actually the last of the *Tīrthaṁkaras* and some of the earlier *Tīrthaṁkaras* and *Cakravartins* like Ṛṣabha and Bharata are well known in Hindu Purāṇic literature also. Hence it is more likely that this dissident faith, revolting against sacrificial killing, was quite an old rival to the Vedic faith or that it had taken root in India even before the Vedic faith. The mathematical knowledge contained in the Jaina religious writings should therefore have been more or less parallel to that in the Vedic literature. Whereas the practical necessity of building the altars turned the sacrificing Vedic tribes to mathematics, to geometrical constructions, the Jainas were impelled to indulge in mathematical calculations by an abstract love of precision. Still the connection between the two is easily noticed. The trapezium, the shape of the *Mahāvedi* of the Vedic religion, which in the speculations of the *Brāhmaṇas* attained cosmic significance, reappears in Jaina cosmography as the shape of the universe, the shape of the mountains, and the shape of the *varṣas* (the continents). Along with this geometrical figure, the circle and its parts also came to assume great importance, since the earth and the orbits of the heavenly bodies were given a circular shape. And so the geometry or rather mensuration of the circle and the trapezium constitutes the geometry of the Jaina religious literature.

c8.3

**3.2.** The ancient works which are classed as *Gaṇita* texts are the *Sūryaprajñapti*, the *Candraprajñapti* and the *Jambudvīpaprajñapti*. Of these the first two cover more or less the same ground — the peculiar configuration and measurements of the paths of the heavenly bodies worked out into the minutest details. The earth stretches flat with the continents and oceans alternating in concentric circles forming a huge Newton's rings pattern, as it were. The sun and the moon move in concentric circles in planes parallel to the earth, the circles having mount Meru as their centre. The diameter of these circles or *maṇḍalas* goes on increasing gradually. Various views about the diameter and circumference of the *maṇḍalas* are enumerated and rejected by the *Sūryaprajñapti*. According to these the ratio of the circumference to the diameter of a circle¹ i.e. π is 3, while the value accepted by the *S. P.* is the nearer approximation √10. This value appears regularly in the Jaina works and even Brahmagupta adopts it.

c8.4

¹ *Sūryaprajñapti* I. 7 and IV. This very rough value of π is met with in ancient Babylonian and Chinese mathematics also. The Egyptian value of (8/9)².4 is very near to the Jaina value of √10.

c8.5

A few other geometrical figures are mentioned:²

c8.6

² *S.P.* I. 8.

c8.7

c8.8

It is interesting to find a distinction being made between the *caturaśra* and *catuṣkoṇa*. The accepted meaning of *asra* or *aśra* or *asri* is a corner and that of *caturaśra* a four-cornered figure which is a quadrilateral. But in the light of the passages in which this distinction occurs *asra* is perhaps to be taken to mean side. Dr. B. B. Datta following Weber translated *samacaturaśra* and *viṣamacaturaśra* as even and oblique square; and *samacatuṣkoṇa* and *viṣamacatuṣkoṇa* as even and oblique parallelogram.³ The occurrence of the word *koṇa* distinguished as equal and unequal, in such an early work as the *Sūryaprajñapti* indicates that the Indians had some conception of angles and that the word *koṇa* appearing in later works is of native origin and not an Indianisation of the Greek *gonia*. That the *viṣavacakravāla* (unequal circle, ellipse) should be distinguished from the *samacakravāla*, the equal circle is also interesting more especially when two perpendicular diameters distinguished as *āyāma* and *viṣkambha* are mentioned.¹ These, of course, are equal in a circle.

c8.9

³ B.B. Datta, The Jaina School of Mathematics, *Bull. of Cal. Math. Soc.* Vol. XXI. 1929.

c8.10

¹ *S.P.I.* 8. In this and other passages where the circumference corresponding to large diameters are calculated with π = √10 the finding of the square root of large numbers is involved. This is not possible unless there was a system of writing numbers with place value and zero.

c8.11

**3.3.** The *Jambudvīpaprajñapti* speaks about the arc (*dhanuḥpṛṣṭha*) and the chord (*jīvā*).² The arc is also often calculated from the chord, from which we have to conclude that the approximate formulas occuring in the other Jaina works were known to the author of the *Jambudvīpaprajñapti*. The authors of the *Sūryaprajñapti* and the *Candraprajñapti* also could not have been unaware of these, since the two works have the same cosmographical details in their background. The *Jyotiṣkaraṇḍaka* which purports to expound the knowledge contained in the *Sūryaprajñapti*³ gives the following approximate formulas.⁴ $c = √(.4h(d-h))$ $a = √(6h²+c²)$ $h = √((a²-c²)/6)$ $c = √(a²-6h²)$ $circumference of a cricle = √(10d²) (= π d)$ $Area of a circle = (circumference × d)/4$ (c is the chord, a the arc, h the height of the segment and d the diameter of the circle)

c8.12

² As in S. II, p. 67 and S. 16, p. 84, *Śrī Jambudvīpaprajñaptyupāṅgam*, Jain Pustakoddhara Fund Series No. 54.

c8.13

³ *Jyotiṣkaraṇḍaka Prābhṛta* 1 Sūtra 1.

c8.14

⁴ *Sūtras* 180, 181, 182, 183 and 184. The verse 180 rendered into Sanskrit runs अवगाहीनं विष्कम्भम् उद्गाहसंगुणं कुर्यात् । चतुर्गुणितस्य मूलं मण्डलक्षेत्रस्य अवगाहः॥ The same word *avagāha* occuring in the definition and as the thing defined makes the sense vague. I have followed the commentator Malayagiri in taking the second *avagāha* to mean *jīvā* or chord. The same verse with the last line changed into ‘सा जीवा सवंखेत्ताणाम्’ is quoted in Bhāskara I’s commentary on the *Āryabhaṭīya*.

c8.15

The *Tattvārthādhigamabhāṣya* (c. 150 B.C.)¹ of Umāsvāti gives, besides these, the formulae.²

c8.16

¹ B.B. Datta, The Jaina School of Mathematics, *Bull. of Cal. Math. Soc.* Vol. XXI. 1929.

c8.17

$h=½ (d-√(()d²-c²$ $and d=(c²+h²)/h$

c8.18

These formulae recur in a number of Jaina religious and semi-religious works, e. g. the *Jambudvīpasamāsa* of Umāsvāti (fourth *Āhnika*), the *Laghusaṁghāyaṇi* of Haribhadra Sūri, which contents itself with the expressions for the circumference and area of a circle (Verse 7).

c8.19

The first of these, which can be derived from the right triangle formed by the half chord, the radius at its end and part of the radius perpendicular to the chord (i.e. from the in figure 1), or from the ', requires thorough familiarity with the properties of the right triangle and of circles and chords, for its formulation. The next three are rough approximations from the formulae for the circumference of a circle.

c8.20

*Fig.* 1

c8.21

For, when the segment is a semicircle c=d and h=d/2 and so a=(√(10d²))/2 can be written as

c8.22

$=√((4d²+6d²)/4)$ $=√(d²+6(d/2)²)=√(c²+6h²)$

c8.23

² *Śrīmadumāvātiviracitam Sabhāṣyatattvārthādhigamasūtram* Ed. by Khubcandraji Siddhantasastri, Raychand Jaina Sastra Mala publication. p. 170.

c8.24

**3.4.** There has been much speculation¹ about the origin of the value √10 for π, found in all the Jaina works as also in Brahmagupta. The explanation offered by Hunrath seems to be the most plausible, since it is based on the above formula for the chord occuring quite early in the Jaina works. Hunrath's explanation as given by Cantor. is :

c8.25

¹ Quite a few theories have been discussed by M. Cantor in his *Vorlesungen Uber Geschichte der Mathematik*, 4th ed., Vol. I, pp. 647-49.

c8.26

The arrow of the segment formed by the side of a regular inscribed hexagon will be h₆ = ½(d - √(d² - (d²)/4)) = d/4(2 - √3) = d/12 taking √3 = 5/3 approximately.

c8.27

Now the side of the regular inscribed polygon of 12 sides will be given by S₁₂² - h₆² + ¼ S₆² (where S₆, S₁₂ are the sides of the hexagon and the 12-sided figure respectively.) = (d/12)² + ¼(d/2)² = (d²)/144 + (d²)/16 = (10d²)/144 Hence its perimeter = 12√((10d²)/144) or (perimeter)² = 10d²

c8.28

If this is equated to the square of the circumference, π comes out neatly as √10.

c8.29

And this value was quite convenient for the Jaina theologian-mathematician, since the islands and oceans had always diameters measured by powers of ten *yojanas*.

3.5. Solid figures

c8.s1.1

The *Prajñāpanopāṅgam* whose author Śyāmārya or Śyāmācārya died in 92 B.C.² says

c8.s1.2

² B. B. Datta, *Jaina School of Maths.*

c8.s1.3

जे सण्ठाणपरिणया ते पञ्चविहा पण्णत्ता तं जहा परिमण्डलसण्ठाणपरिणया वट्टसण्ठाणपरिणया, तं ससण्ठाणपरिणया, चउरंससण्ठाणपरिणया, आयतसण्ठाणपरिणया ।¹

c8.s1.4

¹ ¹ये संस्थानपरिणताः ते पञ्चविधाः प्रज्ञप्ताः, तद्यथा, परिमण्डलसंस्थानपरिणताः, वृत्तसंस्थानपरिणताः, त्र्यस्रसंस्थानपरिणताः, चतुरश्रसंस्थानपरिणताः, आयतसंस्थानपरिणताः — is the Sanskrit rendering.

c8.s1.5

(I.S. 4)

c8.s1.6

“The arrangements (of the atoms) are said to be of five kinds. That is, elliptically arranged, circularly arranged, triangularly arranged, arranged as a square and arranged like a rectangle”.

c8.s1.7

Commenting on this Malayagiri says that all these are of two kinds, solid (*ghana*) and plane (*pratara*). That is, there are elliptical cylinders, spheres, triangular pyramids, cubes and rectangular parallelopipeds. The minimum number of atoms (or shots) necessary to form these geometrical figures, plane and solid, is also given in each case. To corroborate what he has said, the commentator quotes *karaṇagāthās* in Prākṛt to the same effect. The minimum odd and even numbers of *paramāṇus* with the corresponding configuration are shown in the table on p. 67. The *parimaṇḍala*, it is said, cannot be formed with an odd number of *paramāṇus*. The 20 shots if arranged as directed will form a square with rounded corners rather than an ellipse. In fact it is very doubtful if an ellipse is really meant. The *Bhagavatī-sūtra* mentions all these shapes and numbers.² But its commentator Abhayadeva illustrates the *prataraparimaṇḍala* made up of 20 *paramāṇus* by an irregular curvilinear closed figure. Dr. B. B. Datta points out³ that the *Uttarādhyayanasūtra* and the *Aupapātikasūtras* describe the *īṣat prāgbhāra* as resembling in shape an open umbrella and conjectures the figure referred to as the segment of a sphere. The sūtras say that the thickness of this solid, greatest at its middle, decreases toward the edges at a fixed rate. Hence mensuration of spherical segments was also plausibly familiar to the Jainas.

c8.s1.8

² ²24th *Śataka*, 3rd *Uddeśa*, S. 726.

c8.s1.9

³ ³Jaina Mathematics, *Bull. of Cal. Math. Soc.*, Vol. XXI, 1929.

3.6. The trapezium and the trapezoidal solid

c8.s2.1

The universe, according to the Jainas, is in the form of three trapezoidal solids piled one upon another. As such the

c8.s2.2

c8.s2.3

trapezium must have been a familiar geometrical figure in their literature from very early times. In the Jyotiṣkaraṇḍaka (10th Prābhṛta V. 190-91) we find a simple rule for calculating the diameter at any height of a mountain shaped like the frustum of a cone. The rate of increase of the diameter is given by $(diameter at base—diameter at top)/height$ Hence the height at which the diameter is required should be multiplied by this ratio and added to the diameter at the top to get the diameter at that height. The geometrical fact that corresponding elements in similar figures are proportionate, known to the authors of the *Sulbasūtras* and the *Brāhmaṇas*, who increased the size of the *vedis* without altering their shape or the proportions of their parts, is apprehended in a more comprehensive and scientific way here. The *Tiloyapaṇṇatti*, which from the quotations from and the references to it in other works must be an early work, has a fuller treatment of trapezia and trapezoidal solids and a fuller application of the properties of similar figures. But since the extant version of the *Tiloyapaṇṇatti* belongs to a much later period, the work is reserved for treatment in a later chapter.

c8.s2.4

**3.7.** An important branch of Jaina religio-computational literature, viz., the *karaṇa-gāthās*, are either entirely lost or are yet to be brought out of the obscurity of forgotten private libraries. Stanzas from such works are quoted in almost all Jaina commentaries. What is more, even Bhāskara I, the great exponent of Āryabhaṭa I’s mathematics and astronomy, quotes three Prākṛt *gāthās* in his commentary on the *Āryabhaṭīya*. Amongst the mathematicians mentioned by Bhāskara four bear the rather unusual names Maskari, Pūraṇa, Mudgala and Pūtana. The passages in which these names occur are—

c8.s2.5

एतदेकैकस्य ग्रन्थलक्ष्या न मस्करि-पूरण-मुद्गलप्रभृतिभि: आचार्यैर्निबद्धं कृतं, स कथं अनेनाचार्येण अल्पेन ग्रन्थेन शक्यते¹

c8.s2.6

¹ ^1R. 14850 Govt. Or. Mss. Lib. Madras, p. 2.

c8.s2.7

(This has not been composed as a treatise by the *ācāryas* like Maskari, Pūraṇa and Mudgala with even one lakh verses for each (topic). How can this *ācārya* manage it within a short treatise ?)

c8.s2.8

² ^2Ibid, p. 74.

c8.s2.9

गणितविदो मस्करीपूरण-पूतनादयः सर्वेषां क्षेत्राणां फलम् आयतचतुरश्रक्षेत्रे प्रत्याययन्ति¹।

c8.s2.10

¹ Ibid p. 74.

c8.s2.11

(Mathematicians like Maskari, Pūraṇa and Pūtana show the rationale of the areas of all figures in rectangular figures.) One of these names, that of Pūraṇa,² occurs in the *Vyākhyāprajñapti* (*3rd Śataka, Uddeśa 2*). Does this circumstance taken along with the fact that Bhāskara quotes from Jaina works, warrant the assumption that these mathematicians were Jainas? Be that as it may, it is a pity that mathematicians almost on a par, according to Bhāskara, with Āryabhaṭa and responsible for such voluminous outputs in the field of mathematics should remain mere names for us.

c8.s2.12

² One *Purāṇa* is mentioned as one of the sages who surrounded Bhīṣma. (*Mahābhārata, Śāntiparvan, Rājadharmānuśāsana* Ch. 47, 12).

Linked texts (11)

Sūriyapannatti — Ardhamāgadhī Prakrit
Viyāhapannatti — Sudharman, Ardhamāgadhī Prakrit
Joisakaraṃḍaga — Māhārāṣṭrī Prakrit
Uttarajjhayaṇa — Ardhamāgadhī Prakrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Mahābhārata — Vyāsa, Sanskrit
Brāhmaṇa — Sanskrit
Jambuddīvapannatti — Ardhamāgadhī Prakrit
Candapannatti — Ardhamāgadhī Prakrit
Tiloyapaṇṇatti — Yativṛṣabha, Śaurasenī Prakrit

Geometry in Ancient and Medieval India c8: Chapter III. Early Jaina Geometry

Contents

Chapter III. Early Jaina Geometry

3.5. Solid figures

3.6. The trapezium and the trapezoidal solid

Linked texts (11)