I decided to continue with my ‘Forgotten Mathematicians’ series with Indian mathematics.
Mathematics owes a huge debt to the extraordinary contributions given by Indian mathematicians over many hundreds of years, however there has been a reluctance to recognise this.
Vedic Period (between 1500 BC and 800 BC)
The earliest expression of mathematical understanding is linked with the origin of Hinduism as mathematics forms an important part of the Sulbasutras (appendices of the Vedas – the original Hindu scriptures). They contained geometrical knowledge showing a development in mathematics, although it was purely for practical religious purposes. Additionally, there is evidence of the use of arithmetic operations including square, cubes and roots.
The Sulbasutras were composed by Baudhayana (around 800 BC), Manava (about 750 BC). Apastamba (about 600 BC) and Katyayana (about 200 BC).
Before the end of this period – around the middle of the 3rd century BC – the Brahmi numerals began to appear. Indian mathematicians refined and perfected the numeral system, particularly with the representation of numerals and, thanks to its dissemination by medieval Arabic mathematicians, they developed into the numerals we use today.
Jainism was a religion and philosophy which was founded in India around the 6th century BC. The main topics of Jaina mathematics in around 150 BC were the theory of numbers, arithmetical operations, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.
Furthermore, Jaina mathematicians, such as Yativrsabha, recognised five different types of infinities: infinity in one direction, two directions, in area, infinite everywhere and perpetually infinite.
Mathematical advances were often driven by the study of astronomy as it was the science at that time that required accurate information about the planets and other heavenly bodies.
Yavanesvara (2nd century AD) is credited with translating a Greek astrology text dating from 120 BC. In doing so, he adapted the text to make it work into Indian culture using Hindu images with the Indian caste system integrated into his text, thus popularising astrology in India.
Aryabhata was also an important mathematician. His work was a summary of Jaina mathematics as well as the beginning of the new era for astronomy and mathematics. He headed a research centre for mathematics and astronomy where he set the agenda for research in these areas for many centuries to come.
Brahmagupta (beginning of 7th century AD)
Brahmagupta made major contributions to the development of the numbers systems with his remarkable contributions on negative numbers and zero. The use of zero as a number which could be used in calculations and mathematical investigations, would revolutionise mathematics. He established the mathematical rules for using the number zero (except for the division by zero) as well as establishing negative numbers and the rules for dealing with them, another huge conceptual leap which had profound consequences for future mathematics.
As well as this, he established the formula for the sum of the squares of the first n natural numbers as and the sum of the cubes of the first n natural number as .
He even wrote down his concepts using the initials of the names of colours to represent unknowns in his equations. This one of the earliest intimations of what we now know as algebra.
Additionally, he worked on solutions to general linear equations, quadratic equations and even considered systems of simultaneous equations and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657. Furthermore, he dedicated a substantial portion of his work to geometry. His biggest achievements in this area was the formula for the area of a cyclic quadrilateral, now known as Brahmagupta’s Formula, as well as a celebrated theorem on the diagonals of a cyclic quadrilateral, usually referred to as Brahmagupta’s Theorem.
‘Golden Age’ (from 5th to 12th centuries)
In this period, the fundamental advances were made in the theory of trigonometry. They utilised sine, cosine and tangent functions to survey the land around them, navigate seas and chart the skies. For example, Indian astronomers used trigonometry to calculate the relative distances between the Earth and the Moon and the Earth and the Sun. They realised that when the Moon is half full and directly opposite the Sun, then the Sun, Moon and Earth form a right angled triangle. By accurately measuring the angle as 1⁄7°, using their sine tables that gave a ratio of the sides of such triangle as 400:1, it shows that the Sun is 400 times further away from the Earth than the Moon.
Bhaskara II lived in the 12th century and is considered one of the most accomplished of India’s mathematicians. He is credited with explaining that the division by zero – a perviously misunderstood calculation – yielded infinity.
He also made important contributions to many different areas of mathematics including solutions of quadratic, cubic and quartic equations, solutions of Diophantine equations of the second order, mathematical analysis and spherical trigonometry. Some of his discoveries predate similar ones made in Europe by several centuries, and he made important contributions in terms of the systemisation of knowledge and improved methods for known solutions.
This Kerala School of Astronomy and Mathematics school was founded late 14th century by Madhava of Sangamagrama. Madhava also developed an infinite series approximation for π. He did this by realising that by successively adding and subtracting different odd number fractions to infinity, he could establish an exact formula for π, a conclusion that was made by Leibniz in Europe two centuries later. Applying this series, Madhava obtained a value for π correct to 13 decimal places! Using this mathematics he went on to obtain infinite series expressions for sine, cosine, tangent and arctangent. Arguably more remarkable though was the fact that he gave estimates of the correction term, implying that he had an understanding of the limit nature of the infinite series.
In addition, he made contributions to geometry and algebra and laid the foundations for later development of calculus and analysis, such as the differentiation and integration for simple functions. It is argued that these may have been transmitted to Europe via Jesuit missionaries, making it possible that the later European development of calculus was influenced by his work to some extent.
In astronomy, Madhava discovered a procedure to determine the positions of the Moon every 36 minutes and methods to estimate the motions of the planets.
I have only included some of the earlier Indian mathematicians, missing out magnificent mathematicians such as Ramanujan, as I feel that these are the most forgotten. To find out more about other Indian mathematicians, this may be a good starting point.
Hope you enjoyed this post; I was thinking of doing Chinese mathematicians next. Let me know what you think! M x