Ancient Indian Mathematics

Right from the Vedic period, various sciences took birth and flourished in India, leading to well-developed branches of knowledge. Mathematics i.e गणित ,with measurement at its root(गण्) was one of them. Known as संख्यान originally, it was then included in Jain scriptures with the name गणितानुयोग while the Baudhdhas named it गुणन and divided it into 3 types -counting by fingers, solving the problem in head, and advanced mathematical operations. The Indians had realized the importance of mathematics as the most perfect language to impart knowledge. Over the period of time that stretched over several millenniums, mathematics grew into 3 separate branches as follows-

Arithmetic (पाटि गणित), Algebra (बीजगणित) and Geometry (क्षेत्रगणित).

Ø Arithmetic (पाटिगणित)-

There are 3 major contributions of India to world’s mathematical heritage –

· Indian notational system- In the period of 5th cent BC to 3rd cent BC, the numbers were denoted by words, such as-1 =पृथ्वी , 2 = हस्त, 0 = अवकाश, fractions= कला. Later the ‘ कटपयादि system’ came into vogue, which assigned a particular letter in the वर्णमाला to denote a particular number. This system was adopted by ancient Indian astronomers to form formulae in their treatises and also by कर्नाटक संगीत to name their रागs in order to simplify the complicated sequences of स्वरs. Such numerical data was read from left to right thus the numbers were known to be वामतोगति:. It was after this, that the current notational system evolved, which has now been accepted universally. This important invention accelerated the progress of Indian numerals up to the 18th power of 10 known by names such as अयुत, कोटी, महाक्षोभ while the Greeks and the Romans had reached only the 4th power of 10.

· Zero- Philosophical formulations concerning शुन्य – i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (बिंदु) as an empty place holder in the place-value numeral system appears much earlier like in पिंगलाचार्य‘s treatise छंदशास्त्र, algebraic definitions of the zero and it’s relationship to mathematical functions appear in the mathematical treatises of ब्रह्मगुप्त in the 7th C. AD. Ancient Indians were also familiar with the concept ‘Infinite’ i.e. अनंत.

· Decimal system – Harappan scales with decimal divisions have been found, which were used in implementing town planning rules that required roads of fixed widths to run at right angles to each other, for drains to be constructed of precise measurements. The treatise- ललितविस्तर has also mentioned शतमानपद्धति.

Ø Algebra ( बीजगणित/कुट्टक/कोडे)-

Addition (संकलित), Substraction (व्युत्कलन), Multiplication (गुणाकार), Division (भाजन), Square (वर्ग) – Squareroot (वर्गमूळ), Cube (घन)- Cuberoot (घनमूळ) were considered to be the 8 fundamental mathematical operations. All these operations could be done in various ways such as उत्क्रम was one of the types of addition where the gratest place values were added first. Or a type of multiplication कपाटसन्धि that was taken to Europe by Arabs. Apart from these more complicated operations such as L.C.M(निरुद्ध), Simple interest, Compound interest, Ratio and proportion(गुणोत्तरप्रमाण), Calculus(शुन्यलब्धि), त्रैराशिक were also known to Indians. Even the theorem of Pythagoras was invented by शुल्बसूत्रकार at a much earlier time. Apart from this the method of इष्टकर्म – i.e. to solve the problem in such a way that the desired solution will be obtained and विलोमगति: where a problem was solved in reverse order were also known. ब्रह्मगुप्त’s description of negative numbers as debts and positive numbers as fortunes points to a link between trade and mathematical study.

On the other hand, there are 16 सूत्रs like एकाधिकेन पूर्वेण , – compiled by आद्य शंकराचार्य from अथर्ववेद, now known as vedic maths which are used for quick mathematical operations with large numbers. Even today, these सूत्रs have high importance as they are taught to students to crack IIT and MBA entrance exams.

Geometry – It was further divided into 3 sub-branches-i.e.शुल्बसूत्रcalculations with the chord, खातव्यवहार i.e. calculations of volumes and राशिगणित i.e. calculations connected with heaps. शुल्बसूत्र describe techniques for the construction of ritual altars in use during the Vedic era. While बौधायन सूत्र displays an understanding of basic geometric shapes and techniques of converting one geometric shape (such as a rectangle) to another of equivalent area (such as a square). आपस्तंब also looked at the problems of dividing a segment of a circle into seven equal parts. Jain texts from the 6th C BC such as सूर्यप्रग्यप्ति describe ellipses. Developments also took place in applied mathematics such as in creation of trigonometric tables and measurement units.
4 types of measurements were practiced – मान – volume ,तुलामान –weight,अवमान – area,कालप्रमाण – time. It was prevalent in India since Harappan period as shown by an analysis of Harappan weights and measures , appreciated for their remarkable accuracy. Weights corresponding to ratios ranging from 0.05 to 500 have been identified . The treatises of much later period like Aqa-Saas~, ललितविस्तरetc. have given micro-measurements such as- 7 परमाणु = 1 रेणु, , 7रेणु =1त्रुटि , 7 त्रुटि= 1वातायन रज , 7वातायन रज = 1शशराज . The distance was used in योजन. 1000 धनु =1 क्रोश, 4 क्रोश= 1 योजन.

No doubt, astronomy was at the nucleus of this mathematical progress which can be deduced from the importance of jyaaoitYa which is one of the वेदांग.

Astronomy – The science studying the motion of planets and celestial objects was spurred by the need to have accurate calendars for determining मुहुर्त- for sacrifices and for crossing oceans and deserts by adjusting the schedules according to high and low tides and also for a better understanding of climate and rainfall patterns for timely sowing and choice of crops. भास्कराचार्य in his astronomical text सिद्धांतशिरोमणि postulated that the earth had a gravitational force. He also wrote several chapters relating to the study of the sphere and it’s properties and applications to planetary mean motion, eccentric epicyclical model of the planets, first visibilities of the planets, the seasons, the lunar crescent etc. He also discussed astronomical instruments and spherical trigonometry.

In this way, the ancient Indian गणित revolutionized the spheres of religion, trade and commerce. The famous Indian astronomers such as भास्कराचार्य ,आर्यभट्ट ,लल्ल,ब्रह्मगुप्त etc. not only made significant contributions to world’s intellectual heritage, but helped it spread among the ignorant masses through simple and interesting teaching methods as seen from their milestone treatises.

१ प्रतिक्रिया (+add yours?)

  1. Satya
    जुलै 25, 2009 @ 08:50:30

    Hi Archana.
    This is a nice article.


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