It is wrongly believed that science is purely European in origin. In fact, this belief grossly distorts the truth, since the scientific traditions of the ancient Eastern & African civilisations – including the Indian subcontinent, Arabia, Persia, Baghdad, China and Egypt – date back to several millennia before the Christian era, and thus vastly predate the European systems. Even the Greek traditions that European science claims descent from are not purely Greek but heavily dependent upon Egyptian traditions that predated the Greeks – however, that is not the subject of this essay, so we shall not dwell upon it here. The subject of this essay is to provide an outline of the main pre-occupations of Indian mathematicians – who concerned themselves with a wide range of disciplines, including medicine, commerce, grammar and prosody (rhythm and use of emphasis in prose and poetry), perfumery, astronomy, architecture, music and performing arts, an endless list.

It is wrongly believed that science is purely European in origin. In fact, this belief grossly distorts the truth, since the scientific traditions of the ancient Eastern & African civilisations – including the Indian subcontinent, Arabia, Persia, Baghdad, China and Egypt – date back to several millennia before the Christian era, and thus vastly predate the European systems. Even the Greek traditions that European science claims descent from are not purely Greek but heavily dependent upon Egyptian traditions that predated the Greeks – however, that is not the subject of this essay, so we shall not dwell upon it here. The subject of this essay is to provide an outline of the main pre-occupations of Indian mathematicians – who concerned themselves with a wide range of disciplines, including medicine, commerce, grammar and prosody (rhythm and use of emphasis in prose and poetry), perfumery, astronomy, architecture, music and performing arts, an endless list.

The reason to focus on topics such as this is to direct serious attention towards traditional knowledge systems. Non-Western knowledge systems are often regarded as mere intellectual curiosities and benign sources of intellectual entertainment, not relevant to real progress and development. This account of history needs to be challenged.

As a fairly stable and continuous tradition, Indian scholars developed uniquely Indian attributes in the study of nature and natural phenomena and in the application of knowledge and systematic procedure to solve practical problems. These attributes are evident in our literature, grammar, commentaries, astronomical observations, treatises on music and dance, etc. Thus, Panini’s Ashtadhyayi, Sharngadeva’s Sangita Ratnakara and the Ganita traditions come from the same stable of Indian scholarship – which includes the study of nature, application of procedure, reality check and problem-solving.

Let me now proceed to a discussion of the traditions of Indian mathematics – from the Vedic to the modern era. What is mathematics – in the Indian tradition? Ganite sankhyayate tad ganitam. (Calculation and numerical conversion constitutes the substance of Ganita). Thus Indian mathematics focuses on calculation and the use of numbers to determine certain other numbers.

Very early on, the Indian scholar Mahaviracharya (c. 800 CE) said that there is nothing in the three worlds that is not touched by Ganita. The all-pervasiveness of calculation was recognised and respected. Thus, worldly business and ritual – all used Ganita. The profusion of application of mathematics to worldly business however, is overwhelming compared to its explicit application to ritual – such as the construction of Vedic sacrificial places. It is interesting to note the qualities of the Ganitagnya (mathematician) – as stated in the Sulbasutras (approx. 500 yearsBCE). These are: sankhyagnya (arithmetician); parimanagnya (mensuration – a person capable of measurement); parashastrakutuhala (eager learner of other disciplines), etc! This shows that Ganita was not a stand-alone discipline but deeply integrated with practical problems in many walks of life.

Indian mathematical traditions from 500 BCE to the 19th century and even the 20th century CE (Srinivasa Ramanuja was a scholar in the Indian tradition) are a powerful expression of the range of thinking and problem solving that evolved in India. It is impossible in fact, to study Indian mathematics, without delving into the topics of language and grammar, music, commerce, etc. The following techniques were highly developed in India, to the point that it may even be claimed that they originated here: trigonometry, calculus, algebra, indeterminate equation solutions, combinatorics.

The Sulbasutras mentioned earlier clearly establish the so-called Pythagoras’ theorem – the square of the hypotenuse of a right triangle being the sum of the squares of the sides. The sutras arrive at a value of pi as 3.08, which is not so far from 3.14, the approximation that we currently use. The beauty of the Sulbasutras is that it studies geometry not by using a straight edge, but a flexible cord. Sulba itself means cord or rope, and Sulbasutras are thus the formulas, or brief conclusions, pertaining to the use of the cord. Thus Indian geometry easily measured curved surfaces something Descartes (whose eponymous plane geometry we learn in school) pronounced to be ‘impossible’. The reasons for Descartes’ view are complex and we will not go into those here: perhaps we can take up those issues in a future essay.

The Sulbasutras are dated prior to the 5th century BCE. The millennium from -500 to +500 CE is the classical period in Indian Mathematics – with the great scholars Panini, Pingala, Bauddhayana, Jaina Mahavira, Aryabhata – all hailing from this period.

The Vedas and Upanishads already refer to decimal numbers as well as to zero and infinity, both of which were assimilated into European mathematics only with great difficulty and many, many millennia later. The reference to arithmetic of infinity is available in the Shanti Shloka that refers to the formula that infinity minus infinity results in infinity, for example. The Vedas refer to many numbers that are powers and multiples of ten. These facts point to the sophistication achieved very early in the use of numbers.

A few highlights of the mathematical developments in India are listed below:

Varahamihira – used combinatorics (counting techniques, currently extensively employed in computer science) in perfumery, using four distinct perfumes (and mixing them).

Brahmagupta – performed calculations using zero and negative numbers (negative numbers were used very early in India, signifying loss or debt (rna), as opposed to gain or credit (dhana), in commercial arithmetic); he also derived the formula for the diagonals of a cyclic quadrilateral (four-sided figure inscribed inside a circle); he also used the technique of ‘bhavana’ – sensible estimations (bhavana here refers to production or generation of new solutions) – for going from one solution to another better solution.

Bhaskara – determined the concept of instantaneous velocity to study the changing planetary positions, thus creating the foundations of calculus. Madhava – developed infinite series for various trigonometric functions, including pi, sine, cosine. In the treatise Yuktibhasha, written by his disciple, techniques for accelerated convergence of series are described. Even today, this content is only taught at higher levels of mathematics.

Narayana Pandit – developed magic squares for recreational mathematics. These are trick squares with interesting properties. These continue to be widely used as puzzles for highly motivated mathematics students!

The fascinating story of Indian mathematics should include a reference to how ‘proof’ was treated in India. Ganesha (1540) – says that the scholar should be able to state his results in an audience of scholars, in order to rule out doubt. The Nyaya Vaisesika methods of Sabda (apta vakya) (this can be roughly translated as authoritative opinion), Pratyaksha (that which is observable), Anumana (estimation) and Yukti (reasoning) – are all applicable and mutually reinforcing methods. Indian scholarly tradition does not hold one method supreme or perfect, but allows multiple ways of presenting the opinion of the scholar, who then also remains open to challenge by a superior scholar. In fact, many scholarly propositions made by the earlier authorities are developed only via commentaries (called bhashya) written by their students or others and in the course, many of the propositions are also politely rejected or altered or improved, with a fresh set of evidence. As an example, the value of pi, earlier set at 3.08, was refined from time to time, and to an extremely high level of accuracy by the time of Madhava.

The problem-solving approach, validation of theories by gatherings of learned scholars, debate and discussion are a hallmark of Indian tradition of enquiry and scholarship. The application of procedure such as vidhi (procedure), kriya (action), prakriya (process), etc. and the employment of sutras – which are defined as extremely concise and clear statements – alpaksharam minimal use of words or syllables), asandigdham (unequivocal), etc. – are seen.

Indian scientific enquiry employed knowledge from one discipline to another. In Ayurveda, we see the employment of biochemistry, for example in the categories of doshas – kapha, pitta, vata. Further for example in the swasthavrtta (behaviours required for health), on cleaning teeth, the advice on using 4 types of rasas – bitter, astringent, pungent and sweet tastes – one sees a combinatorial bent of mind, listing, categorising and combining (as in perfumery).

Thus I would like to conclude by reiterating that problem solving – via observation, iteration, procedure, classification, learning, summarising, validating – is the hallmark of the Indian tradition of enquiry.

**Shailaja D Sharma hails from Palakkad in Kerala state, is a Ph.D. in Mathematics (IIT Bombay), works as a corporate executive, is currently based in Bangalore and has a strong and growing interest in Indian traditional knowledge.
Contact: shailajadsharma@gmail.com. **