Introduction

Mathematics has always been an integral part of Indian culture. From the cities of Indus Valley Civilization like Mohenjidaro to the Vedic Sulbasutras to the Golden Age of Indian Mathematics (500 – 1200 CE) to the Kerala School of Mathematics, Indian culture had always given huge importance fields of mathematics- whether arithmetic, geometry, algebra and even trigonometry. In this blog we will discuss India’s various contributions to mathematics particularly in the Vedic, the Classical Age & also the later periods.The blog has intentionally been  made free from  equations and formulae  in order to make it more of an easy read.

Now let’s address the title of this blog. Who or What was Lilatvati? Well, Lilavati is a treatise on mathematics by Indian mathematician Bhaskara II around early 12th Century CE. The legend behind the name is that Lilavati was the name of the daughter of Bhaskara II. He studied his daughter’s horoscope that she would remain both childless and unmarried. Despite his efforts he failed to get his daughter married and so dedicated his book to her so that her name remains immortal through time. Even in the book, many mathematical problems are addressed to her, also claiming her to be an intelligent young woman. For example, “Oh Lilavati, intelligent girl, if you understand addition and subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10, and 100, as well as those when subtracted from 10000”. So, I am dedicating this blog to all the intelligent women who contributed in the field of mathematics even if in a very minuscule manner.

Chapter 1: Ancient Indian Mathematics- The Harrappan & The Vedic Periods

The earliest concept of mathematics in India can be traced to the Harappan or the Indus Valley Culture around 3000 BCE. Weights and Scales of various measures are found in the ruins of this Pre-Vedic Civilization. If we consider 27.584 grams of weight as a standard unit of weight, plumb bob weighing around 0.05, 0.1, 0.2, 0.5, 2, 5, 10, 20, 50, 100, 200 & 500 units were discovered from the ruins. Scales of different lengths were discovered in sites like Harappa, Mohenjodaro and Lothal. The Mohenjodaro Scale has a length of 66.2 mm with nine carefully sawn equally spaced parallel lines of 6.7056 mm each. One of the lines is marked by a hollow circle, and the sixth line from the circle is indicated by a large circular dot. The distance between the two markers is 1.32 inches(335mm) also known as the “Indus Inch”. Another important discovery is the Harappan Bricks which had a ratio of 4:2:1 in terms of length, breadth & thickness.

Next came the vedic age which is suggested to be roughly around 1800-800 BCE as accepted by most scholars, gets its name from the four vedas– the Rig Veda (contains hymns & prayers during rituals), the Yajur Veda (contains rules & guidance for sacrifices), the Sama Veda (have melodies in praises to the gods) & the Atharva Veda (is a collection of magical spells). Each veda has four parts- Samhitas, Brahmanas, Aranyakas & Upanishads. A Brahmana named Satapatha Brahmana (Brahmana with hundred paths) has one of the earliest references to mathematics in the world. The Satapatha Brahmana (~1200BCE) contains technical details of altar constructions. Details of isosceles trapezoidal altars; a circular, semi-circular and a square altar -all three of equal areas are mentioned. Another text called Vedanga Jyotisha gave procedures for calculating time & positions of the sun & the moon in accordance with various nakshatras (zodiac signs).

Some of the important scriptures depicting Indian Vedic Mathematics are the Sulbasutras. The Sulbasutras used instructions for two types of rituals- personal & communal. Three of the most mathematically significant Sulbasutras are those recorded by Baudhayana, Apastamba & Katyayana. The earliest of them is considered to be the Baudhayana Sulbasutra (~800 BCE), followed by the Apastamba Sulbasutra (~600 BCE) and lastly the Katyayana Sulbasutra (~200 BCE). The Baudhayana Sulbasutra states Pythagoras Theorem with an approximation procedure for obtaining the square root of 2 correct to 5 decimal places. The actual statement of the Pythagorean theorem, in terms of the sides and diagonals of squares and rectangles, is found in both the Baudhayana and the Apastamba Sulbasutras. The Sulbasutras primarily uses geometric instructions for altar construction of various shapes- squares,rectangles, circles & trapeziums. Some examples include- how to turn a pair of equal or unequal squares into a third square, how to turn a rectangle into a square without changing the area, process of squaring a circle or circling a square. An extraordinary achievement of vedic mathematics was the procedure of finding square roots with a high degree of approximation like that of 2.

The earliest types of numerals found in India are Kharosthi numerals of around 400 BCE and  Brahmi numerals of around 300 BCE and later Gwalior numerals around 850 CE. The earliest form of symbol for zero is found in the Gwalior script of around 876 CE. In fact the word zero comes from the Arabic al-sifr. Sifr in turn is a transliteration from Sanskrit Shunya which means void or empty. The symbol for Shunya first began as a dot or a Bindu found in India, Cambodia & Sumatra which later became a circle.

The rise of Jainism & Buddhism in India in around 700 BCE also impacted mathematics as the use of it slowly shifted from religious to personal purposes of philosophy & inquisition. Both the Jaina and the Bauddha traditions also developed their own schools.

The mathematics was soon integrated for astronomical works called Siddhantas. These Siddhantas contained not only an explanation of the methods involved but also a discussion of the technical instruments available then for measuring time and angles which soon led to the Golden Age of Indian Mathematics.

Chapter 2: Classical Indian Mathematics- The Golden Age

The Golden Age of India which is generally considered between 500 to 1200 CE saw the birth of a number of “Great Indian Mathematician-Astronomers”. Some of them and their contributions are mentioned below:-

1.  Aryabhata I (b.476 CE)– Best known for his work Aryabhatiya, Aryabhata I was the pioneer of the Golden Age of Indian Mathematics. His work contains details of an alphabet-numeral system of notation, rules for arithmetical operations, and methods of solving simple and quadratic equations and indeterminate equations of the first degree. The book pays some attention to trigonometry and introduces the sine and versine (i.e., 1 – cosine) functions—a notable innovation on earlier work both in and outside India. He also calculated 3.1416 as a close approximation to the ratio of the circumference of a circle to its diameter. India’s first satellite launched in 1975 was also named after him.
2.  Brahmagupta (b.598 CE)– Brahmagupta was from Ujjain which is in the modern state of  Madhya Pradesh. He is known for his work Brahma Sphuta-Siddhanta. It is an astronomical text with many chapters in mathematics. Brahmagupta called the twelfth chapter Ganita (Arithmetical Calculation). although it includes a discussion of mathematical series and a few geometric topics . The eighteenth chapter, Kuttaka (literally Pulverizer, but also translated as Algebra), contains solutions of indeterminate equations of the first and second degree which later directly influenced the evolution of algebra in the Islamic World. His other important work is Khanda Khadyaka which gives further developments in trigonometry including a method of obtaining the sines of intermediate angles from a given table of sines.
3. Sridhara (fl.800 CE)– Sridhara was from Bengal. His most important work is the Trisatika which is one of the greatest works on arithmetic before Bhaskaracharya’s Lilavati. In it he deals with elementary operations, including extracting square and cube roots, and fractions. Eight rules are given for operations involving zero. His methods of summation of different arithmetic and geometric series were to become standard references in later works.
4. Mahavira (fl.850 CE)– He was a Jain and thus was familiar with Jaina Mathematics. His works include Ganita-sara-sangraha which deals with arithmetic operations and a number of examples to illustrate the rules.
5. Bhaskara II (b.1114 CE)– Bhaskaracharya( or Bhaskara the teacher) was from Sahyadri region of modern Maharashtra. His most famous work is Lilavati which contains a profound understanding of arithmetic, permutations & combinations, rules to work with zero. His other work Bijaganita contains problems on determining unknown quantities and solving simple & quadratic equations.

Classical India was greatly  affected by both import & export of foreign cultures. Indian mathematics got influenced by Greece & China and in turn influenced cultures like Persia, Arabia and even China & Greece. India was a power house when it came to arithmetic,geometry, and especially algebra and trigonometry. 

Chapter 3: Legacy & Influences

During the later medieval period, after the Islamic Conquests in India, much of the mathematical tradition declined. But it managed to survive through regional schools. The most famous of those schools was the Kerala School of Astronomy and Mathematics between 14th to 16th century CE. Their most important members were the Madhava-its founder, Paramesvara, Damodara, Nilakantha, Jyesthadeva, Achyuta Pisaroti, Citrabhanu & Sankara Variyar & their most important contribution was series expansion for trigonometric functions of sine,cosine & arctangent including an infinite series for pi by Madhava. Their works were completed two centuries before the invention of calculus in Europe, which is now considered the first example of a power series other than the geometric series.

The school later inspired S.Ramanujan(1887-1920) who made great contributions to mathematical analysis,number theory, infinite series & continued fractions. There is a book and a movie after him named The Man Who Knew Infinity.

Other important legacies  of Indian Mathematics include the inspiration to Al-Khwarizmi who studied Brahmagupta’s Brahma Sphuta-Siddhanta and wrote Al-Jabr from which  the word algebra comes, and the fibonacci sequence which originally is credited to an Indian  poet & mathematician Pingala in 200 BCE. There are also other contributions which are beyond the scope of this blog.

Conclusion

India has a huge contribution in the field of mathematics which many modern scholars highly neglect. On the other hand the extra ordinary claim of India being the sole ancient hub of knowledge is also false. In truth India was one of the major contributors in mathematics & science, which shared its knowledge with Greece, China, Persia, Egypt & others in a bidirectional manner. Hope you liked this blog. If you find this interesting please share and subscribe. Also comment for any suggestions, queries or criticism. I will be happy to answer them.

Suggested Readings-

1. Indian Mathematics- Internet Encyclopedia of Philosophy
2. MacTutor History of Mathematics- Online Archive
3. The Crest of the Peacock: Non-European Roots of Mathematics- George Gheverghese Joseph

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