37 Science in Ancient India- an overview

Michel Danino

Introduction:

As in all ancient cultures, astronomy was born in India before mathematics, beginning with observations of the periodicity of the moon’s phases, a few identifiable planets, the northward or southward journey of the sunrise on the eastern horizon through the year, or to trace imaginary lines between the stars.

1. Harappan Beginnings

In India, those beginnings are not adequately documented. It has been suggested that patterns of rock art found in Kashmir, such as a double sun or concentric circles, may have been depictions of a supernova and meteor showers respectively, perhaps witnessed some 7,000 years ago. Ring-stones found at Mohenjo Daro, the largest city of the Indus civilization (2600-1900 BCE), have been interpreted as calendrical devices keeping track of the sunrise at different times of the year. The alignment of main streets along the cardinal directions has been attributed to the sighting of the star cluster Pleiades (Krittikā), which rose due east at the time. These interpretations are sound but remain conjectural.

Harappan town planning, including the use of simple ratios in the dimensions of major structures (see module on The Aryan Issue), implies some knowledge of basic geometric principles and an ability to measure angles, as evidenced by a few cylindrical compasses made of shell, with slits cut every 45°. Besides, for trading purposes the Harappans developed a standardized system of weights (see Module on The Aryan Issue), in which, initially, each weight was double the preceding one, then, 10, 100 or 1,000 times the value of a smaller weight. This shows that the Harappans could not only multiply a quantity by such factors, but also had an inclination for a decimal system of multiples. However, there is no agreement among scholars regarding the numeral system used by Harappans.

2. Science from the Vedic to the Early Historical Era

A few thousand years ago, the Rig-Veda, the oldest of the four Vedas, spoke of a year of 360 days divided into twelve equal parts. It clearly recorded a solar eclipse, although in metaphorical language. A few centuries later, the Yajur-Veda considered a lunar year of 354 days and a solar year of 365 days, and divided the year into six ritus or seasons of two months each; it also gave the first list of 27 nakshatras or lunar mansions, that is, constellations along the path of the moon on the celestial sphere.

The Vedas mention multiples of 10 up to a million millions in the Yajur Veda — a number called parārdha. (By comparison, much later, the Greeks named numbers only up to 10,000, which was a “myriad”; and only in the 13th century CE was the concept of a “million” adopted in Europe.) The Brāhmanas, commentaries on the Vedas, knew the four arithmetical operations as well as basic fractions.

Because of the need to keep time for the proper conduct of rituals, calendrical astronomy grew more sophisticated in the late Vedic period, with the Vedānga Jyotisha of Lagadha as its representative text. On the basis of its own astronomical data, it has been dated between the 12th and the 14th centuries BCE by most scholars. The length of the sidereal day (i.e. the time taken by the earth to complete one revolution with respect to any given star) it uses is 23 h 56 min 4.6 s, while the correct value is 23 h 56 min 4.091 s; the tiny difference is an indication of the precision reached in that early age. The Vedānga Jyotisha also discusses solstices (ayanānta) and equinoxes (vishuva ) and uses an intercalary lunar month (adhikamāsa) to catch up with the solar calendar over a five-year yuga (era): the solar year is about 365.24 solar days, while the lunar year is, at most, 360 days; after a few years, the difference between the two will be such that a month (or two, depending on the system) needs to be added to the lunar year to catch up with the solar cycle.

In that era, a simple water clock was used to measure time; the gnomon, a vertical stick whose shadow is measured, must have come into use a little later.

The first Indian texts dealing explicitly with mathematics are the Shulbasūtras, dated between the 8th and 6th centuries BCE. They were written in Sanskrit in the highly concise sūtra style and were, in effect, manuals for the construction of fire altars (citis or vedis ) intended for specific rituals and constructed with bricks. The altars often had five layers of 200 bricks each, the lowest layer symbolizing the earth, and the highest, heaven; they were thus symbolic representations of the universe, to which the sacrifice was in effect offered. Because their total area needed to be carefully defined and constructed from bricks of specified shapes and size, complex geometrical calculations followed, some of which led to interesting corollaries, such as the so-called Pythagoras theorem, or a rational approximation √2 (correct to the fifth decimal).

The Shulbasūtras also introduced a system of linear units based on dimensions of the human body; they were later slightly modified and became the traditional units used across India.

A couple of centuries later, Jain astronomy also developed in this period, based on a peculiar model of two sets of 27 nakshatras, two suns and two moons; it nevertheless resulted in precise calendrical calculations. Jaina texts indulged in cosmological speculations involving colossal numbers and dealt with geometry, combinations and permutations, fractions, square and cube powers; they were the first in India to come up with the algebraic notion of an unknown (yāvat-tāvat), and introduced a value of π equal to √10, which remained popular in India for centuries.

This is also the period when huge scales of time were conceived of such as a “day of Brahmā” (or kalpa ) of 4.32 billion years, which happens to be close to the age of the earth (4.5 billion years). There are much longer time scales to be found in Jain texts and in the Purānas, but also infinitesimal dimensions of time, space, weight or angle. Indian scholars were keen to explore the two extremities of the infinite.

With the appearance of the Brāhmī script a few centuries BCE, India’s first numerals come into use, on Ashoka’s edicts in particular, but as yet without any decimal positional value. These numerals will evolve in shape; eventually borrowed by Arabs scholars, they will be transmitted, with further alterations, to Europe and become our modern “Arabic” numerals.

Babylonian and Greek influences are clear in the introduction of the 24-hour day, the seven-day week and of the solar zodiac of 12 signs (rāshi), first recorded in the Yavanajātaka (c.269 CE).

3. The Siddhāntic or Classical Era

The full-fledged place-value system of numeral notation worked out in India in the first centuries CE was a crucial advance. While positional systems existed earlier (e.g., in Babylonia and China), they had failed to integrate zero with the nine numerals. The new system was gradually adopted across India, and later taken to Europe through the Arabs.

The Siddh āntic era opened in the 5th century CE, when texts called siddhāntas were composed — a Sanskrit word meaning ‘principle’ or ‘conclusion’, but which applies here to a collection of conclusions or a treatise. Āryabhata I (born 476 CE), working near what is today Patna, ushered in this era with his Āryabhatīya, which dealt concisely but systematically with developments in mathematics and astronomy. Brilliant scholars followed, such as Varāhamihira, Bhāskara I, Brahmagupta, Lalla, Srīdhara, Mahāvīra, Āryabhata II and many more. This was the time of great advances in algebra, geometry and the first steps in calculus, and in astronomy effective algorithms to track the paths of celestial bodies and predict eclipses.

Among those advances, let us mention solutions of indeterminate, quadratic and cubic equations, finite and infinite series, negative numbers, trigonometry, expansions of fractions, permutations and combinations, and in astronomy issues of coordinate systems, time measurement and division, mean and true positions of celestial bodies, epicyclic models for the computations of planetary positions, and calculations of solar and lunar eclipses. The contributions of Bhāskara II (born 1114 CE), better known as Bhāskarāchārya, were the culmination of the Siddhāntic era.

During those centuries, astronomy’s interface with the society was mostly through calendars and pañchāngas (almanacs), and the prediction of eclipses, which had great religious and social significance. Indeed, an astronomer’s fame was guaranteed if he could accurately predict the occurrence, nature and duration of eclipses, and many inscriptions record a king’s reward to such an astronomer. Another interface was architecture, and many monuments, including temples, show clear astronomical alignments with events such as the sunrise at solstices and equinoxes.

4. The Kerala School

The common belief that there was no progress in Indian astronomy and mathematics after Bhāskara II ignores the developments that took place in the southern state of Kerala. The so-called Kerala School of Astronomy and Mathematics flourished there from the 14th to the 17th century, when networks of knowledge transmission in north India were severely disrupted in the wake of repeated invasions.

Mādhava (c. 1340–1425 CE) laid some of the foundations of calculus by working out power series expansions for the sine and cosine functions (the so-called Newton and Gregory– Leibniz series). Parameshvara (c. 1360-1455 CE) was extremely productive in astronomical works and observations. Nīlakantha Somayājī (1445-1545 CE) carried out a major revision of the older Indian planetary model, achieving a quasi-heliocentric model. Results and methods produced by the Kerala School, both in mathematics and astronomy, remained in use well into the 19th century.

5.Cross-cultural Developments

About the same time, a complex interface with Islamic astronomy took place, which, among other benefits, brought instruments such as the astrolabe to India. The famous and massive yantramantra or Jantar Mantar observatories built in the early 18th century by the Maharaja of Jaipur, Sawai Jai Singh (1688-1743 CE), represent a convergence between Indian, Arabic and European astronomy. Indian astronomy interacted not only with Islamic (or Zīj) and European astronomies, but also with Chinese astronomy, in complex interplays that invariably enriched both players.

In a general way, Indian scientists were more interested in efficient methods of computation than in theoretical models. Nevertheless, they did often provide logically rigorous justifications for their results, especially in the longer texts. Bhā skarācārya states that presenting proofs (upapattis) is part of the teaching tradition, while Jyeshthadeva devotes considerable space to them in his Yukti Bhāshā. Some of the algorithmic methods used to calculate planetary positions and eclipses yielded remarkably precise results and impressed by their speed European astronomers such as Le Gentil, a French savant who stayed in Puducherry for two years to observe a solar transit of Venus in June 1769.

Whether those specificities of Indian science limited the further growth of Indian mathematics is open to debate. Other factors have been discussed by historians of science, such as historical disruptions of centres and networks of learning (especially in north India,), limited royal patronage, or the absence of a conquering impulse (which, in Europe, did fuel the growth of science and technology in the colonial age). Be that as it may, India’s contribution in the field was enormous by any standard. Through the Arabs, many Indian inputs, from the decimal place-value system of numeral notation to some of the foundations of algebra and analysis, travelled on to Europe and provided important components to the development of modern mathematics and astronomy.

Web links

• Jantar Mantar Observatories
• Resource on Indian mathematicians by J.J. O’Connor & E.F. Robertson An overview of Indian mathematics by J.J. O’Connor & E.F. Robertson
• Resource of primary texts on the history of science
• Mathematics in India: From Vedic Period to Modern Times by M.D. Srinivas, K. Ramasubramanian and M.S. Sriram
• Mathematics in India: From Vedic Period to Modern Times by M.D. Srinivas, K. Ramasubramanian and M.S. Sriram (another website)
•  Papers on Indian Science and Technology, resource prepared by S and HI, IIT Bombay

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