Science & Technology
Śulbasūtra (शुल्बसूत्र): Tradition of Mathematics in Ancient India
Śulbasūtras are part of the Kalpa Vedāṅga (कल्प वेदांग), particularly of the Śrautasūtra (श्रौतसूत्र) texts which deal with the rites, rituals and ceremonies. Apparently, each Śrautasūtra had its own Śulba (शुल्ब) section. These sections constitute the text of appendices to the main Śrauta (श्रौत) ritual texts and are known as Śulbasūtras.
Yajñya (यज्ञ) was considered to be an act of noblest order. It was necessary to perform the yajñyas properly and exactly in accordance with the systems and methods laid down by the corresponding treatises. The two prerequisites were: 1. To determine and fix the appropriate and auspicious time for performing the yajñya and 2. To construct the altars – Citis (चिति) or Vedis (वेदि), of exact shape and dimensions. These Vedis were of 33 shapes from simple shapes like triangle, square to complex shapes like hawk and tortoise. The altars erected in different shapes are meant to achieve different results or objectives.
To meet the first need, Jyotiṣaśāstra (ज्योतिषशास्त्र) or astrology and astronomy came into being. Mathematics emerged as a consequence of calculations required for Jyotiṣaśāstra. Science of algebraic geometry emerged to meet requirements of measuring the dimensions
and constructing Vedis. Around 800 BCE, Baudhāyana (बौधायन) codified these geometric principles and equations in sūtra (सूत्र) format into manuals called Śulbasūtras. They reveal
an important aspect of Indian genius. Subsequently, other Śulbasūtras were composed.
Unlike many works of Vedic and Sanskrit literature, it is not too difficult to ascertain the period of Śulbasūtras. Period of Sūtra literature could be determined around 800 BCE to 200 CE. So far eight Śulbasūtras are available.
Śulbasūtras can be divided into two categories:
1. Manuals of Baudhāyana (बौधायन), Āpastamba (आपस्तम्ब) and Kātyāyana (कात्यायन) – they focus on the early state of Indian geometry.
2. Later Śulbas of Mānava (मानव), Varāha (वराह), Maitrāyaṇa (मैत्रायण), Vādhūla (वाधूल), Hiraṇyakeśi (हिरण्यकेशि) are considered to be of lesser significance. These texts are of paramount significance in the history of Indian mathematics. Not only do they contain concepts of algebraic geometry but pioneering discoveries such as the theorem of the square on the hypotenuse of a right-angled triangle enunciated long before Pythagoras in the West. Historians trace the origin of Greek mathematics to Thales of Miletus – 624-548 BCE. He is acknowledged as the first natural philosopher in Greek
tradition. Śulbasūtras date to 800 BCE. The knowledge of seminal geometry and astronomy definitely predates the mathematics of the Greeks.
History of Indian Mathematics
In the evolution of man’s intellect numeration must have preceded mensuration (measurement of area, length and volume) by centuries. The Ṛgvedic Puruṣasūkta (ऋग्वेद पुरुषसूक्त) begins with – सहस्रशीर्षा पुरुषः सहस्राक्षः सहस्रपात् (sahasraśīrṣā puruṣaḥ sahasrākṣaḥ sahasrapāt). Although here ‘sahasra (सहस्र)’ is not definitive but descriptive. The Ṛgvedic people knew what sahasra (thousand) was. For cowherds and shepherds counting of livestock was important and for the delineation of their grazing pastures, mensuration became a necessity. This was the seed of the geometry measuring of ground. The Vedic people had devised their own ways of appeasing, celebrating, and extolling forces of nature and making offerings to them – the elaborate performance of yajñyas. Yajñya has a wide range of connotation from Ṛgveda through out the entire expanse of Sanskrit literature. The noun Yajñya is derived from the verbal root yaj (यज्) which means to worship, adore, honour, consecrate, offer, invoke deities to accept offerings. Sacrifice is a very insufficient and misleading translation. From the Saṅgrāma- adhvara (सङ्ग्राम-अध्वर) (adhvara- yajñya in the form of war of the Kauravas (कौरव) and Pāṇḍavas (पाण्डव)) to Vinoba Bhave’s Bhūdāna (भूदान) yajñya , the concept is clear and it rises much above the ritual called ‘sacrifice’.
Śatapatha Brāhmaṇa (शतपथ ब्राह्मण) shows perfect knowledge of construction of a vedi of rhombus shape with parallel sides 24 and 30 vikramas (विक्रम) and the two equal sides of 36 vikramas. It names 6 citis – droṇa (द्रोण), rathacakra (रथचक्र), kaṅka (कङ्क), prauga (प्रउग), ubhayataḥ prauga (उभयत: प्रउग), samuhyapurīṣa (समुह्यपुरीष). Kātyāyana Śulbasūtras
enumerates these citis and the commentators elaborate them with descriptive details. The Mahīdhara Vṛtti (महीधर वृत्ति) on Kātyāyana Śulbasūtras mentions about taking extra care while preparing bricks for the vedic altars. The brick decreases in size (shrinks) when bedecked and on drying by 1/32 units, so wise people should take the wet mould 1/32 units more than required so as to finally get a brick of accurate measurement. This accuracy is of measurement to 1/32 nd part of a small brick.
Although the primary purpose of construction of altars was for yajñyas, they could possibly have had other uses. Dr N. G. Dongre, a scientist and accomplished scholar, hypothesized that these altars in different shapes and sizes were possibly meant to function as furnaces for extracting metal from its ores by a process involving heating and melting. In the light of evidence of the development made in the field of metallurgy in ancient India, this process of smelting at that time does not seem far from reality. Due to the different shapes and sizes of altars it was possible to conserve heat and maintain temperatures to different degrees required for smelting particular ores. This is however a hypothesis and requires scientific testing.
With the discovery of the Sindhu-Harappan civilization with its excellent town planning, it was established without an iota of doubt that ancient Indians were utilizing mathematics in their civil life, road alignments, residential quarters, drainage systems way back in 3000 BCE. Far earlier than the rise of natural sciences-mathematics-philosophy of the ancient Greeks.
Śaulbīya (शौल्बीय) right angle triangles:
The Sūtras do not use the word Śulba for measuring cord. They use the term Rajju (रज्जु). Śulbasūtras engage themselves in the construction and not in the theory of plane figures, traingles, squares, trapezia. Theories in the form of propositions and proofs were taken for granted. Indian altar builders realised theoretically the relationship between the sides of
any right-angled triangle. Baudhāyana, the earliest writer of Śulbasūtras had arrived at this theory around 800 BCE, almost 250-300 years prior to Pythagoras.
The Śulbasūtras had known and named the three sides of the right-angled triangle through rectangle as akṣṇayā (अक्ष्णया), tiryaṅmānī (तिर्यङ्मानी) and pārśvamānī (पार्श्वमानी).
Post Śulbasūtra Mathematics:
Trigonometry:
The origin of trigonometry is not to be found in pure mathematical works but in the astro- mathematical treatises (siddhānta, सिद्धान्त) summarised by Varāhamihira (वराहमिहिर) in his Pañcasiddhāntikā (पंचसिद्धांतिका). However, the classical era of mathematics was heralded by Āryabhaṭṭa (आर्यभट्ट, 476-550 CE), a mathematician-astronomer, who wrote many treatises on mathematics and astronomy. His main work is Āryabhaṭīya (आर्यभटीय). The mathematical section of his work contains arithmetic, algebra, plane geometry, trigonometry and spherical trigonometry. His work Āryasiddhānta (आर्यसिद्धान्त) is lost but is referred to by his contemporary Varāhamihira (505-587 CE) and later mathematicians and commentators like Brahmagupta (ब्रह्मगुप्त, 598-670 CE) and Bhāskara I (भास्कर१, 600-680 CE). Āryabhaṭṭa describes several astronomical instruments like gnomon (Śaṅku yantra, शङ्कु यन्त्र), shadow- instrument (Chāyā yantra, छाया यन्त्र), possibly angle measuring instruments and other devices like semi-circular (Dhanur yantra, धनुर् यन्त्र), circular (Chakra yantra, चक्र यन्त्र) and water clocks of at least two types- bow-shaped and cylindrical. In his work Golapāda (गोलपाद) are discussed geometric and trigonometric aspects of the celestial sphere, ecliptic, celestial equator, cause of day and night, rising of zodiacal signs on the horizon. His work was in verses and the cryptic ideas were elaborately explained by his disciples Bhāskara I and Nīlakaṇṭha Somayājī (नीलकण्ठ सोमयाजी, 1444 – 1550 CE). Āryabhaṭṭa gave definitions of sine, cosine, versine, inverse sine and devised complete sine and versine tables in the intervals of 3.75 0 from 0 0 to 90 0 to an accuracy of four decimal places.
In the 14 th century, one of the greatest mathematician-astronomers from Kerala, Mādhava (माधव, 1340- 1425 CE) developed the infinite series approximation for many trigonometric functions, sines and cosines including π. He laid the foundation of calculus – differential and integral, which were elaborated by the successive generations of mathematicians from Kerala. Mādhava also derived the approximate value of π to upto 11 decimal places.
Value of π:
Śulbasūtras do not mention circle or the quadrature of a circle. They were satisfied with the proportion circumference/diameter = 3/1 for the purpose of altars. Reaching nearer the real value of π (as the Greeks called it), which can never be attained, was not their primary aim. This academic exercise was undertaken by later scholars the Jainas (जैन). Vīrasena’s (वीरसेन) Dhavalā Ṭīkā (धवला टीका) on the Ṣaṭkhaṇḍāgama (षट्खण्डागम), calculates the value of π to up to six decimal places. Both Āryabhaṭṭa and Bhāskara give the value of π as 3.1416. Āryabhaṭṭa mentions that diameter (Viṣkambha, विष्कम्भ) multipled by the value 3.1416 gives an approximate (āsanna, आसन्न) or near value of the circumference of the circle (vṛttapariṇāha, वृत्तपरिणाह) i.e., 2πr, as known in modern mathematics.
Nīlakaṇṭha, commentator of Āryabhaṭīya explains the word approximate (āsanna). The circumference cannot be calculated by the value of the diameter without a remainder. Similarly, the diameter cannot be calculated by the value of the circumference without a remainder. Thus, the measure of the two will never be without a remainder. With greater calculations what can be achieved is a smaller and even smaller remainder, but never will it be remainderless. This transcendental number (vaktum aśakya, वक्तुम् अशक्य) is today expressed as π saying that in decimal notation it is a non-recurring, non-termination decimal.
Conclusion:
The Śulbasūtras could not keep their identity independent of the yajñyas. It appears the Śulbasūtras were probably considered relevant only with regards to performance of Yajñyas. Once the performing of yajñyas stopped, the Śulbasūtras were lost to oblivion. Had they been made part of the principle texts of Śrautasūtras and not put as appendices, mathematics would have advanced centuries before Āryabhaṭṭa and Bhāskara I and Bhāskara II. The studies in the form of commentaries which exist today, however scant, show that they are immensely useful in understanding the texts and carrying forward discussions on diverse mathematical topics. Research is or should be a never-ending process. Vast treasure of manuscript literature is waiting to be discovered and interpreted. What has come to light so far may be interpreted and re-interpreted. Every sincere probe can give new insights and new inputs for further quests.
(For detailed studies please refer “Critical Edition of KātyāyanaŚulbasūtra with Commentary of Ramachandra Vajpeyi Somayajin” by Shankar Gopal Nene, National Manuscript Mission, Delhi, 2016.)