A new theoretical approach to sample problems and deductive reasoning in Sanskrit mathematical texts

Abstract

Geometry is the earliest recorded branch of Indian mathematics. Mathematics of the Vedic period consists of those geometric techniques needed for the construction of the altars and fire-places described by the priestly hereditary class for the performance of their rites. The link between geometry and ritual suggests that mathematical accuracy was considered of the utmost importance in this context. The study of rational figures in the Sanskrit work Gaṇita Sāra Saṃgraha, to which Mahāvīrācārya, a ninth-century ce Jaina mathematician, dedicates a special treatment, reveals striking parallelism with the earlier geometry developed in connection with the Vedic sacrifice. Mahāvīra makes extensive use of the uddeśaka or ‘sample problem’, and I suggest a new way of interpreting the uddeśaka as a significant device for constructing an ‘actual proof’ which validates and links a mathematical rule to its unmentioned premises and provides a system of knowledge based on deductive syllogism.

Notes

¹ This paper is an elaborated extract of my 2012 Cambridge University MPhil thesis ‘The Gaṇita Sāra Saṃgraha: a linguistic approach to a Sanskrit mathematical treatise’. I would like to thank my supervisor, Dr Vincenzo Vergiani, whose guidance and wisdom have been a precious help. I also would like to express my gratitude to Howard Emmens, of the BSHM Bulletin’s editorial team, for all his valuable suggestions.

² Jamison and Witzel (1992) define the Vedic period as ‘the earliest period of Indian history for which we have direct textual evidence’, suggesting 1500–500 bce as ‘convenient limiting dates of the period’.

³ It is not possible to date these works. The oldest layers of Baudhāyana’s text, the earliest among the four Śulbasūtras, seem to go back approximately to sometime between the seventh and fifth centuries bce.

⁴ The Vedas are a collection of religious hymns and rituals and are the earliest known texts in any Indian language. They are considered sacred texts and the historical foundation of what has been later referred to as ‘Hinduism’.

⁵ The best-known authors of the ŚS, approximately in chronological order on the basis of the style and grammar of the language of their texts, are Baudhāyana, Mānava, Āpastamba, and Kātyāyana. Nothing is known about the lives of these authors or the circumstances in which their works were written.

⁶ Seidenberg (1978) claims that the geometries of Greece and Vedic India were basically similar and similarly related to ritual. He underlines that the ideas about the religious importance of exact geometrical altar constructions were consistent in both countries.

⁷ Staal (1999, 105–127) proposes the idea of a common Indo-European ancestor for Indian and Greek geometry, mainly based on the ritual associations of the Śulbasūtras techniques and the Greek altar of Delos legend.

⁸ The Vedic altar used in the ceremonies of the sacrificial ritual of the soma, the sacred beverage offered to the gods. It had to be set up on the east of three fires in the shape of an isosceles trapezium with its base facing west, using prescribed dimensions.

⁹ The Brāhmaṇas are glosses on the mythology, philosophy, and rituals of the Vedas and are seminal works in the development of later Indian thought.

¹⁰ Some of the texts mentioned in this section belong to the Jaina canonical literature. They can roughly be placed between the end of the first millennium bce and the first centuries ce. See Dundas 2002 for a more in-depth study on Jainism.

¹¹ See Mahāvīra’s description in GSS 1.17–23.

¹² Rāśi can mean ‘quantity’ or ‘heap’ and hence came to be applied to finding the volume of solids such as stacks of bricks or of sawn timber.

¹³ The medieval siddhānta genre was developed by astronomers starting in about the fifth century ce. The evolution of siddhāntas and astronomical schools are fully described in Plofker 2009, 66–72. Plofker suggests that even before the middle of the first millennium ce, mathematical texts developed separately from astronomy but failed to survive.

¹⁴ All the Sanskrit translations found in this paper are my own, unless otherwise stated.

¹⁵ The Bakhshālī Manuscript (eighth–twelfth century ce), the Pātīgaṇita and Triśatikā of Śrīdhara (eighth–ninth century ce),and the Gaṇita Sāra Saṃgraha of Mahāvīra (ninth century ce) are the only early works dealing exclusively with mathematics.

¹⁶ See for example Plofker 2009, 162.

¹⁷ Nīlakaṇṭha (fifteenth century ce) was one of the most influential mathematician-astronomers of the Kerala school.

¹⁸ According to Bag (1979, 15) ‘the Āryabhaṭīya is essentially a systematisation of the results contained in the siddhāntas and it is of particular value because of the picture it gives of the state of mathematical knowledge of the period, no less than for the impulse it gives to the study of the subject’.

¹⁹ The word śloka denotes a verse consisting of two lines of sixteen syllables each. This is by far the commonest metre in classical Sanskrit poetry. Single line half-verses also occur throughout the GSS, giving rise to verse numbers ending ‘’.

²⁰ It is not unusual for Sanskrit words to be used as technical terms having specific meanings within a given field of expertise. Brahmagupta uses the technical term jātya (literally ‘noble, legitimate’) for a rational right-angled triangle. Sarasvati Amma (2007, 136) underlines that the use of this term seems to suggest the idea that ‘the rational right triangle alone belonged to the highest or original species. It is quite likely that all rectilinear figures were in one sense viewed as formed by the juxtaposition of right triangles’.

²¹ For example, Brahmagupta gives solutions for isosceles and scalene triangles, the rectangle, and the isosceles trapezium (Brāhmasphuṭasiddhānta 12.38).

²² Mahāvīra was not in fact a Hindu but a Jaina.

²³ The Sanskrit term śāstra is used to refer to a ‘treatise’ or ‘scripture’ written to explain an idea. It also denotes a branch of Sanskrit learning that involves specialized or technical knowledge in a defined area of practice. For instance, vāstuśāstra is the ‘science of architecture’ and jyotiḥśāstra is the ‘science of astral sciences’. Mathematics (gaṇita) pertains to the tradition of jyotiḥśāstra.

²⁴ In the sample problems given by Mahāvīra, bāhu (or bhuja) is longer than koṭi. I use the terms ‘base’ and ‘height’ respectively to denote the lengths of these two dimensions.

²⁵ Datta and Singh (1962, 207) point out that Brahmagupta (BrSpSi 12.33) was the first to give this solution.

²⁶ A dvisamatribhuja is ‘that trilateral which has two equal sides’.

²⁷ That is, the area of the generated rectangle is the same as that of the isosceles triangle obtained from it.

²⁸ The method suggested for the scalene triangle follows Bramagupta (Brāhmasphuṭasiddhānta 12.34), but Mahāvīra gives more details and explains how to find the bījas for the two rational right triangles.

²⁹ Read chittvā for chitvā, which is clearly a misprint.

³⁰ Read lambaka for lambakā, which is probably a misprint.

³¹ Mahāvīra says ‘any’ number probably in order to fit the metre of the verse, but it would have been obvious to him that further conditions were needed for the rule to work out.

³² Such a condition is hypothetical in this case; however, it is not always necessary. It depends on the data of the first rectangle and on the bījas chosen. Nevertheless, one rectangle should be bigger than the other.

³³ The proportion could be also reversed, with the second generated rectangle smaller than the first one. In such case, all the conditions regarding the relation of the sides of the two figures are also inverted. Also, the first generated rectangle can be placed at the right side of the second figure.

³⁴ In ślokas 154 and 156 Mahāvīra gives the rule for finding the numerical values of the sides, the base, and the height of all triangles having the same given area and the rule to find the base and the height of an isosceles triangle whose area is given.

³⁵ Ābādhā is the segment of the base of a triangle.

³⁶ The mahāvedi altar is in the shape of an isosceles trapezium.

³⁷ It is the longest chapter of the GSS (337 and a half ślokas) and deals with problems involving the solution of quadratic equations and linear equations in one unknown.

³⁸ Read bhāge for bhāgo, which is probably a misprint. See Raṅgācārya in Mahāvīrācārya 1912, 31.

³⁹ Read saptahṛtaḥ for saptahṛte, which is probably a misprint. See Raṅgācārya in Mahāvīrācārya 1912, 31.

⁴⁰ My theoretical formulation of the uddeśaka as a constructive proof and a specific way to carry out mathematical activity thus also applies to texts such as the Pāṭīgaṇita, the Gaṇitatilaka, and the Līlāvatī.

⁴¹ See for instance the Gaṇitatilaka by Śrīpati or the Līlāvatī by Bhāskarācārya.

⁴² The commentaries usually repeat the verse rule in a more articulated and comprehensible prose form, or illustrate it with sample problems, explaining it in more detail. See Plofker 2009.

⁴³ The fire-place in the shape of the isosceles triangle.

⁴⁴ In Vedic India, mathematical accuracy and the perfect pronunciation of Sanskrit hymns were means for the efficacy of ritualistic performances.