Indian mathematics

• 1. Indus Valley Foundations

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  Excavations at Harappa and Mohenjo-daro have uncovered evidence of practical mathematics used by the Indus Valley Civilization. The people of this ancient society manufactured bricks with dimensions in the proportion 4:2:1, a ratio considered favorable for structural stability. They employed a standardized system of weights based on specific ratios including 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Each unit weight equaled approximately 28 grams, which is roughly equal to the English ounce or Greek uncia. These weights were mass-produced in regular geometrical shapes such as hexahedra, barrels, cones, and cylinders. This production demonstrated knowledge of basic geometry long before written records existed. Inhabitants also designed rulers like the Mohenjo-daro ruler to standardize length measurement with high accuracy. That ruler divided its unit of length into ten equal parts, where each part measured approximately 1.32 inches or 3.4 centimeters. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this specific unit of length. Hollow cylindrical objects made of shell found at Lothal and Dholavira from around 2200 BCE could measure angles in a plane. These same hollow objects helped determine the position of stars for navigation purposes.

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• 2. Vedic Ritual Geometry

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  The Śulba Sūtras list rules for constructing sacrificial fire altars during the Vedic period between 700 and 400 BCE. Most mathematical problems considered in these texts spring from a single theological requirement regarding altar construction. Altars required five layers of burnt brick with each layer consisting of exactly 200 bricks. No two adjacent layers could have congruent arrangements of bricks. The diagonal rope of an oblong rectangle produces both which the flank and horizontal ropes produce separately. This statement implies square areas constructed on their lengths as explained by teachers to students. Lists of Pythagorean triples appear within the text including examples like 3, 4, 5 and 5, 12, 13. Baudhayana composed the best-known Sulba Sutra containing simple Pythagorean triples such as 3, 4, 5 and 5, 12, 13. He also provided a general statement of the Pythagorean theorem for rectangle sides. The expression he gave for the square root of two is accurate up to five decimal places. True value being 1.41421356... The value of this approximation 577/408 is the seventh in a sequence of increasingly accurate approximations known as Pell numbers. Three Sulba Sutras were composed including the Manava Sulba Sutra and Apastamba Sulba Sutra. These contained results similar to the Baudhayana Sulba Sutra. Kātyāyana wrote the Katyayana Sulba Sutra presenting much geometry including the general Pythagorean theorem. His work included a computation of the square root of 2 correct to five decimal places.

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• 3. Classical Golden Age

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  Aryabhata I lived from 476 to 550 CE and wrote the Aryabhatiya treatise containing 332 shlokas. This work described fundamental principles of mathematics including quadratic equations and trigonometry. He calculated the value of π correct to four decimal places. Aryabhata defined sine as the modern relationship between half an angle and half a chord. He also defined cosine, versine, and inverse sine functions. His tables listed values at 3.75° intervals from 0° to 90° with four decimal places accuracy. Varahamihira produced the Pancha Siddhanta in 575 CE compiling five earlier astronomical works. He made important contributions to trigonometry including sine and cosine tables accurate to four decimal places. Brahmagupta published his astronomical work in 628 CE containing two chapters devoted to arithmetic and algebra. Chapter 18 contained rules for arithmetical operations involving zero and negative numbers. These rules were all correct except one exception regarding division by zero. He gave the first explicit solution of the quadratic equation using methods equivalent to modern approaches. Brahmagupta discovered an identity that generalized an earlier identity of Diophantus. He used this lemma to generate infinitely many integral solutions of Pell's equation given one solution. Bhāskara II lived from 1114 to 1185 CE and wrote several treatises including Siddhanta Shiromani and Lilavati. He computed π correct to five decimal places and calculated Earth's revolution around the Sun to nine decimal places. His work included preliminary concepts of differentiation and discovery of differential coefficients. He stated early forms of Rolle's theorem and derived the differential of the sine function.

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• 4. Kerala School Series

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  The Kerala school of astronomy and mathematics was founded by Madhava of Sangamagrama in South India. It flourished between the 14th and 16th centuries with original discoveries ending around 1559-1632. Members included Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri, and Achyuta Panikkar. They developed infinite series expansions for trigonometric functions several centuries before calculus appeared in Europe. The Tantrasangraha-vakhya gave these series in verse form translated into mathematical notation as standard power series. Jyesthadeva provided proofs for the series for sine, cosine, and inverse tangent in his Yuktibhāśā written in Malayalam. Their results include applications of ideas from what became differential and integral calculus to obtain Taylor-Maclaurin infinite series. They used rectification computation of arc length to prove these results instead of quadrature methods. A rational approximation error term allowed derivation of faster converging series for π. Using improved series they derived a rational expression 104348/33215 for π correct up to nine decimal places. These mathematicians employed intuitive notions of limits to compute results without formal function theory. They did not formulate exponential or logarithmic functions despite their advanced work. Works were first written up for Western world by Englishman C.M. Whish in 1835. His claims about fluxional forms remained neglected until over a century later when investigated again by C. Rajagopal.

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• 5. Transmission To West

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  The decimal place-value system in use today was first recorded in India then transmitted to Islamic world and eventually to Europe. Syrian bishop Severus Sebokht wrote mid-7th century CE about nine signs Indians used for expressing numbers. Earliest surviving evidence includes copper plate from Gujarat mentioning date 595 CE written in decimal notation though authenticity remains doubtful. Decimal numerals recording years 683 CE found in stone inscriptions in Indonesia and Cambodia where Indian cultural influence substantial. Buddhist philosopher Vasumitra dated likely 1st century CE discussed counting pits of merchants showing knowledge of decimal representation. Number four could be represented word Veda since four religious texts existed while number thirty-two called teeth representing full set. Moon symbolized one as only moon exists in sky. Yavanajātaka versification of earlier Greek horoscopy adaptation from c. 150 CE used object numbers convention by mid-3rd century CE. Mathematical concepts spread to Asia Middle East and eventually Europe through various transmission routes. Some scholars suggest knowledge might have reached Europe via trade routes from Kerala by traders and Jesuit missionaries around 1500. No evidence of transmission has been found despite existing communication routes making possibility plausible. David Bressoud states no evidence that Indian work series known beyond India until nineteenth century. Both Arab and Indian scholars made discoveries before 17th century now considered part of calculus history. They did not combine differing ideas under derivative and integral themes like Newton and Leibniz later did.

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• 6. Oral Tradition Mechanics

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  Mathematicians ancient early medieval India were Sanskrit pandits trained in language literature grammar exegesis logic. Memorization recitation played major role transmitting sacred texts mathematical treatises philosophical works. Modern scholars noted truly remarkable achievements preserving enormously bulky texts orally for millennia. Prodigious energy expended ensuring texts transmitted generation to generation with inordinate fidelity. Eleven forms recitation included mesh recitation where every two adjacent words first recited original order then repeated reverse order finally repeated original order again. Flag recitation paired first two last two words proceeding sequence word one word two word N minus one word N. Dense recitation took form combining multiple word groupings demonstrating effectiveness preservation most ancient religious text Rigveda c. 1500 BCE single text without variant readings. Mathematical texts remained exclusively oral until end Vedic period c. 500 BCE. Sūtras expressed highly compressed mnemonic form achieving brevity through ellipsis technical names abridging lists markers variables. Communication through text only part whole instruction rest transmitted guru-shishya parampara uninterrupted succession teacher student. Extreme brevity achieved using ellipsis beyond tolerance natural language using technical names instead longer descriptive names. Abiding lists by mentioning first and last entries using markers variables created impression communication text only part whole instruction.

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