Indian Mathematical Lore
The forgotten contributions of primeval Indian academics
Ancient India is undoubtedly one of the oldest and most cultured civilizations in the world as evidenced by its prehistoric roots and philosophical heritage. The tenets espoused in the native belief systems of the Indic people, including the Hindu, Buddhist, Jain, and Sikh faiths of have governed the way millions throughout the Indian subcontinent lead their lives and commit themselves towards reaching their full potential. This has enabled the region to become the bedrock of art, dance, science, and other diverse fields. Its influence quickly spread from the Middle East to even the far reaches of China and allowed various civilizations to reap the benefits of this work in terms of education and wellbeing. However, one of the greatest gifts that ancient India could bestow upon contemporary society was its profound knowledge of how numbers work. Math has a rich history that dates back over 3,000 years in the Indian subcontinent, as it has flourished for many centuries before Europeans could make their own discoveries. The work of Indian mathematicians and academics was prized for its groundbreaking contributions to trigonometry, calculus, as well as the creation of negative numerals and zero.
During the Golden Age of Indian Mathematics that occurred between the 5th and 12th centuries, mathematicians made crucial progress in their understanding of trigonometry, which is a method of linking both numbers and geometry. As early as 400 CE, Indian scholars published a text known as the “Surya Siddhanta’’, which contains the roots of arguably one of the most important branches of math, trigonometry.
It was in this Sanskrit treatise that the use of sines, cosines, and several other trigonometric functions was first recorded. Almost 200 years later, the renowned mathematician and astronomer Aryabhata produced the categorical definitions of these functions and specified the complete tables of both sine and versine. Aryabhata also determined the solutions to simultaneous quadratic equations and created an approximation for the value of π. Ancient Indians employed these concepts for a variety of purposes, including ocean navigation, land surveying, and even the mapping of stars and planets. For example, astronomers were able to determine the relative distance between the Earth and its Moon by using trigonometric tables. Aryabhata himself managed to calculate the circumference of the Earth by using his approximation of π, and in doing so he revealed that he knew it was an irrational number, which was undiscovered in Europe until the 1700s. Apart from Aryabhata, many other stalwarts made brilliant contributions to this branch of mathematics. Among those great men was Madhava of Sangamagrama, a scholar so learned that he was celebrated as the greatest mathematician-astronomer of medieval India. A title that was extremely difficult to possess, and one which he rightfully deserved with his extraordinary achievements.
Madhava is said to have been responsible for the development of the infinite series approximations for a range of trigonometric functions. His outstanding intellect and humble nature made him one of the most sought after teachers or gurus in the country, and by the 1300s, he had founded the Kerala school of astronomy and mathematics to cater to thousands of eager students or shishyas. Under his fine tutelage, his disciples would carry on his legacy for generations. This would include his work in trigonometry as well as that of his and his predecessors in another fundamental branch of math, calculus, which is the study of the continuous change in that takes place in the universe.
The importance of calculus cannot be stressed enough as it is used to measure all rates of change and is crucial in almost every branch of science, underpinning countless breakthroughs in modern physics. Many believe that Gottfried Wilhelm Leibniz is the father of calculus, as he was one of the first Europeans to attain success in his development of calculus in the late 1600s, along with the reputable Sir Isaac Newton.
However, the Indian mathematician Bhaskaracharya had already discovered his theories almost 500 years in advance. Bhaskara was a highly accomplished mathematician, who made major contributions to algebra, arithmetic, geometry, and trigonometry. He provided the solutions of “Diophantine’’ equations and established the preliminary concepts of infinitesimal calculus. Ultimately, his ideas would become the rudimentary basics that successive legions of mathematicians would go on to strengthen and develop, including Madhava of Sangamagrama. The latter’s compositions on the subject of calculus were heavily influenced by Bhaskara’s work and by the 1300s, Madhava and his shishyas were responsible for many mathematical firsts. These include several early calculus related results as well as the use of mathematical induction. The proponents of this Kerala school conceived systems like the Taylor series expansions and differentiation, many of which were not repeated in Europe until several hundred years later. As a matter of fact, Jesuit missionaries who visited Madhava’s homeland were actually the first foreigners to be imbued with this knowledge. This demonstrates how far back the Europeans were when it came to mathematics as they chose to reject two key Indian principles, which were zero and negative numerals.
Perhaps the most endearing and prominent achievement of Indian scholars was their development of both zero and negative digits. This was because it not only transformed how the world viewed mathematics but also enabled future advances in a plethora of modern subjects such as computer programming and binary code. The first recorded zeros appeared in the Bakshali manuscript but were merely placeholders that were found in other ancient cultures as well. But what made India different was that zero eventually progressed to become a value of its own. In fact, one must note that the earliest recorded usage of a circular symbol for zero was in a Gwalior Hindu temple. The man who was responsible for this amazing leap was none other than Brahmagupta, a 7th-century mathematician and astronomer who revolutionized mathematics with his findings.
He established rules for working with zero and his theories led to the development of the decimal system. Before his advent, much of the mathematical knowledge of ancient India was organized as part of the Vedas. In these texts, numbers were expressed as combinations of powers of 10. This representation is believed to have played an important role in the introduction of the decimal-place value system in India. After Brahmagupta’s addition of the zero value, the Indian numeral system that much of the world uses today would be complete. He also helped formulate the edicts of how negative numbers should be dealt with and pointed out that quadratic equations can have two possible solutions, one of which could be negative. Many European mathematicians were hesitant to accept the idea of negatives as they could not see how they could be useful. It was Brahmagupta that recognized and demonstrated that one thing that could be counted with negatives were debts. He essentially revolutionized mathematics with all of his discoveries, yet why exactly were he and other intellectuals so devoted to mere numbers?
As mentioned before, ancient Indians did not just script complex formulas and equations as a pastime. Mathematics was born out of the practical needs of society, which was why several cultures of yore regarded it as a fundamental subject. With its unrivaled proficiency over this key discipline, ancient India was able to usher in a wave of prosperity for itself and its neighbors. Temples, universities, and fascinating ideologies were introduced by Ancient Indians to the rest of the Asia, including Cambodia, Indonesia, and Myanmar.
In the same period when ancient India was experiencing its cultural renaissance under the reign of the Guptas, its universities received students from all over the world. One of these pupils was the reputed Chinese scholar, Hsuan Tsang, who studied and taught at the Nalanda University for five years. Apart from being the first known university, Nalanda possessed over 10,000 students like Tsang and remained the center of scholarship and studies from the 5th to 12th centuries. Various other institutions such as Takshashila also harbored foreigners and sought to instill them with an understanding of mathematics. This resulted in many nations giving affording rare respect to ancient India maintaining friendly relations with its people. This effectively contributed to keeping peace and stability in the region, whereas other empires of the time were engaged in warmongering.
One thing that mathematics significantly developed was architecture. Hindu temples in the ancient India such as the Chennakeshava or Elllora temples displayed many common mathematical principles in their pillars and engravings. The same applies to the several structures built in other nations, including Angkor Wat, Borobudur, and Prambanan. These bear evidence of how deeply the Hindu and Buddhist culture of ancient India had penetrated its neighboring nations. This influence was a product of India’s maritime expertise, which mathematicians extensively contributed to by discovering new methods of navigation and introducing new shipbuilding techniques. With its vast geopolitical influence, economic prosperity, and boundless intelligence, ancient India should have become a modern superpower. Sadly, it was not to be as this nation was not only stripped of its wealth and power, but also the credit and dignity that its mathematicians rightfully deserved.
Brahmagupta, Aryabhata, Bhaskaracharya, and countless others didn’t dedicate all their lives to mathematics just so that they could win riches or laurels. Their work was for the advancement of all of humanity, not just for their state, kingdom, or country. This explains why ancient India was so willing to share its findings with numerous civilizations and eager to teach students from China, Arabia, Europe, and Africa. Unfortunately, foreign invasions and conquests lead to the downfall of Indic education as groups ranging from the Mongols to the British exploited this sacred philosophy of Vasudaiva Kutumbam, the world is one, to their advantage and pillaged Indian kingdoms of their wealth and learning. Some individuals from these groups even took credit for many of the seminal advancements that ancient India was responsible for and sought to erase any evidence of India’s wisdom. Even today, the effects of this Eurocentric bias are visible as a majority of Indians themselves refuse to accept that they were once part of such a mathematically enlightened culture. This indicates how important it is for contemporary Indians to educate themselves about their wise ancestors so that they can continue to work towards restoring modern day India to its former glory.
All in all, one can rightfully proclaim that the Indian subcontinent is the cradle of modern-day mathematics as its achievements remain unparalleled in the history of mankind. Although there will always be detractors, it is irrefutable that ancient India has a strong mathematical heritage that has prevailed in even the 21st century, where people of Indian-origin continue to be at the forefront of math, science, engineering, and much more.
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