Survey of Indian Mathematics over the Ages

In this post, I have done a Survey of Indian Mathematics over the Ages starting from Yajur Veda, through Sulba Sutras, Aryabhata, Brahmagupta, Kerala School upto Shrinivasa Ramanujam. 

Indian Mathematics has had a glorious development over the ages since the most ancient times. Hindu Mathematciains gave the world 0 and the place value system - the basis of modern science and technology. Yet they never asked for or received any credit for their work.

Why did Indian Mathematics decline after 11th Century CE?
This is a common question that many people ask - why did Indian Mathematics decline with the advent of Islam in India? Let us see if we can answer the question.

Mathematics in India developed in parallel to Greek and Babylonian tradition, although there were interactions and collaborations between these civilizations from time to time. However, Hindu Mathematics suffered a devastating set-back from 10th century AD with the advent of Islamic rule. This happened because of:

1. large-scale destruction of all major universities like Nalanda and Odantapuri and others by barbaric Islamic general Bakhtiyar Khilji^[1], and destruction of numerous other centers of learning by the Mughal hordes. This destroyed the back-bone of collaborative education which was common prior to that
2. burning of major libraries where millions of manuscripts were burnt
3. killing and slaughter of Hindu, Buddhist and Jain scholars in Universities by invading Islamic warlords
4. destruction of Hindu temples, Buddhist monasteries and Jain viharas - These religious centers were also the seat of secular science and learning - however they were razed and their inhabitants slaughtered by Mughals and other invaders like Mahmud Ghazni, Allah-uddin Khilji and the likes

The Arab geographer and scholar, Alberuni, who wrote an account of India and spent much time at Mahmud of Ghazni's^[2] court, wrote of his raids that "the Hindus became like the atoms of dust scattered in all directions and like a tale of old in the mouths of people. This is the reason, too, why Hindu sciences have retired far away from those parts of the country conquered by us, and have fled to places which our hands cannot yet reach, to Kashmir, Benaras, and other places." ^[3],[4]

Despite such brutality, the spirit of Jain and Hindu scientists never wavered - they kept on producing one great result after another, often pre-dating their Western counterparts by centuries!

Survey of Indian Mathematics
This survey of Indian Mathematics is based on NPTEL course Mathematics in India - From Vedic Period to Modern Times by Prof. M.D.Srinivas, Prof.M.S.Sriram & Prof. K. Ramasubramanian, Department of mathematics, IIT Bombay.

Please do watch this lecture.

1500 BCE - Yajurveda
Yajurveda-Samhita talks about powers of 10 upto 10¹² (parardha)

1000 BCE to 800 BCE - Baudhayana Sulvasutra

• Methods of Construction and transformation of geometrical figures and alters using rope (rajju) and gnomon (shariku)
• Units of Measurement (bhumiparimana)
• Construction of a square of a given side (sama chaturashra karana)
• Construction of √2, √3 and 1/√3 times a given length
• Oldest Theorem in geometry (or the so-called Pythagoras Theorem) - The square of diagonal of a rectangle is the sum of the square of its sides
• Construction of Squares which are the sum and differences of different squares
• Transforming a square into a rectangle, isosceles trapezium, isosceles triangle and a rhombus of equal area and vice versa
• Approximate conversion of a square of side a into a circle of radius: r = a/3 * (2 + √2)

Gnomon - A gnomon is the part of a sundial that casts the shadow.

750 BCE - Upanishads

• Talks of zero (shunya, kha) and infinity (Purna)
• Ishavasya Upanishad of Shukla Yajur Vead - पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते [When infinity is removed from infinity, infinity remains]

500 BCE - Astronomical Siddhantas

500 BCE - Panani's Ashtadhyayi
Uses idea of zero-morpheme (lopa)

300 BCE - Pingala and Combinatorial Methods (Varna Meru)

250 BCE - Katyayana Sulvasutra
How to construct a square which is n-times a given square

1 CE - Complete formalization of decimal place-value system

50 CE - Works of Vasumitra uses place-value system

100 CE - Vyasya Bhashya on Yogasutra refers to place-value system

270 CE - Vriddha-yavana-jataka of Sphujidhvaja uses a full-fledged place-value system

499 CE - Aryabhata
Ganitapada of Ariyabhatiya | Most of the standard procedures in arithmetic, algebra, geometry and trigonometry are perfected by this time.

• Place value, square root, cube root, squaring, cubing
• Area of triangle, circle, trapezium
• Volume of sphere, equilateral tetrahedron
• Approximate value of π 3.1416
• Computing table of rsines
• Obtaining shadows of gnomons
• Square of the hypotenuse is the square of the sides
• Summing an Arithmetic progression, repeated summations
• Obtaining the sum of squares and cubes of natural numbers
• Interest and Principal
• Rule of three
• Arithmetic of fractions
• Inverse Processes
• Linear Equations with one unknown
• Meeting time of two bodies
• Solution of linear indeterminate equation

550 CE - Varahamihira

• In Brihat Samhita, Varahamihira talks about combinatorics and how to create a prastara
• Gandhayukti / Panchasiddhantika

628 CE - Brahmagupta

• Brahma-sphuta-siddhanta / Khanda-khadyaka
• Mathematics of zero and negative numbers
• Development of algebra
• Chapter 12 - Ganitadhyaya deals with fractions, cube root, interest, areas of various figures, shadow problems
• Chapter 18 - Kuttakadhaya (Algebra) of Brahmasphutasiddhanta deals with solution of linear indeterminate equations, surds, operations with unknown, equations with single or several unknowns, second order indeterminate equation

629 CE - Bhaskara I 

• Efficient squaring of number
• Aryabhatiya Bhashya, Mahabhaskariya and Laghubhaskariya

750 CE - Sridhara and Lalla

800 CE - Govindasvamin

850 CE - Mahaviracharya's Ganita-sara-sangraha

860 CE - Prithudaka-svamin

932 CE - Munjala

950 CE - Important Commentaries

• Halayudha's commentary on Pingala-sutra
• Bhattotpala's commentary on Varahamihira's brihat samhita

950 CE - Aryabhata II

1039 CE - Sripati

1050 CE - Jayadeva

1150 CE - Bhaskara II

• Lilavati, Bijaganita and Siddhanta Siromani become canonical texts
• Bhaskara II solved X² - 61Y² = 1 for integral solutions. The results are X= 1766319049 and Y = 226153980, which are huge numbers.
• Fermat posed this problem 500 years later in 1657 as a challenge to European Mathematicians, unaware that Hindu Mathematicians had already solved it half a millennia earlier!

1250 CE - Regional Works

• Vyavahara ganita in Kannada of Rajaditya
• Pavuluri ganitamu in Telugu of Pavuluri Mallana

1350 CE - Narayana Pandita

• Varasankalita of Narayana Pandita / Construction of Magic Squares
• Ganitakaumudi / Bija-ganitavatamsa

1350 CE - Madhava

• Infinite series for π. After this it was Newton after 200 years later who came up with an infinite series approximation of π in 1665!
• Infinite series for sine and cosine
• Fast convergent variants

1400 CE - Works of Paramershvara

1500 CE - Nilakantha

• Formula for instantaneous velocity
• Revised planetary model (sun-centric)

1500 CE - Jnana Raja

1530 CE - Yukti Bhasha of Jyeshtha Deva
A Malayalam Text with the most detailed exposition of upapattis (proof/ logic/ rationale) in Indian Mathematics

1540 CE - Ganesha Daivajna, commentary Buddhivilasini on Lilavati

1541 CE - Suryadasa

1580 CE - Achyuta Pisarati of Later Kerala School

1600 CE - Krishna Daivajna

1603 CE - Munishvara

1616 CE - Kamalakara

1700 CE - Putumana Somayaji of Later Kereala School

1830 CE - Sankara Varman

1869 CE - Chandrasekhara Samanta of Odisha
- All major lunar inequalities

1900 CE - Srinivasa Ramanujam, Mathematical Genius

• Modular equation for π upto 17 million digits accuracy
• 3000 results in his "Lost Notebook" most of which are coming true now almost 100 years after his death

This completes the brief survey of Indian mathematics through the ages. Hope you enjoyed it and appreciate the fact that what we have been made to believe regarding our heritage through English media and NCERT text-books is but a fraction of the true worth of our civilizational achievements in Mathematics and Technology. Courses like these help us appreciate the true genius of our ancestors and of Jain and Hindu science. We Indians need to recognize our heritage and stop believing that science and mathematics came to India from the west.

Additional References

1. Bakhtiyar Khilji's Conquest of Bengal and Bihar 
2. Shocking Facts about Mahmud Ghazni
3. https://www.sscnet.ucla.edu/southasia/History/Mughals/mahmud_mughals.html
4. Quotes on Ghaznavids Conquest of India
5. http://economictimes.indiatimes.com/slideshows/people/indian-maths-wizards-and-their-ideas/bhaskara-ii-1114-1185/slideshow/40354408.cms
6. http://www.lotuswise.co.uk/communities/6/004/011/906/426//images/4604203443.jpg 

You may also like the following.

• Interesting Facts about Rig Veda
• Interesting Facts about Atharva Veda
• An Introduction to Tantra 
• The Hindu World View 
• Introduction to Trika

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