Engineering Science and Technology in Vedic Period and Post Vedic Period

Vedic age marked a new era of development in the field of science and technology. During Vedic times science developed out of religious plane. Vedic people influenced the growth of Indian science. Understanding real-world phenomenon in rational and scientific way started with study of natural phenomenon in the context of rainfall, appearance of the sun, the moon, changes in season and agriculture. This naturally led to theories about physical processes and the forces of nature that are today studied as specific topics within different branches of physical science. Later advancements in the field of mathematics, astronomy, astrology, medicine, surgery etc. during ancient India were significant by all standards. It can be understood by the contributions made by scholars in different field /subjects.

Sushruta

Rishi Sushruta lived in circa 6th century BC and belonged to clan of Rishi Vishwamitra. He wrote Sushruta Samhita, which throws light on several achievements of ancient Indians in field of medical science. This treatise has one of the earliest mentions of leprosy, cataract surgery, rhinoplasty etc. Sushruta is sometimes called the father of plastic surgery and general surgery.

Charaka

Charaka was a major contributor to Ayurveda system of Medicine and author of Charaka Samhita.

Panini

During the 5th century BC, the scholar Panini made several discoveries in the field of phonetics, phonology and morphology.

Pingala

Pingala was a BC era mathematician who used binary numbers in the form of short and long syllables, very much similar to Morse code in current times. This indicates his deep understanding of arithmetic. Binary representation has now become the basis of information storage as sequences of 0s and 1s in modern-day computers. Pingala’s work also contains the basic ideas of what we know now as Fibonacci number and a presentation of the Pascal’s triangle.

Aryabhatta-I

Aryabhatta-I, the legendary mathematician was a resident of Patliputra, (5th century AD, Gupta Era). He collected existing concepts and developed the algebraic theories and other mathematical concepts. He wrote a mathematical treatise named Aryabhattiya (circa AD 499) and referred to Algebra as Bijaganitam. He successfully calculated the value of pi and this was much more accurate than the value calculated by the Greeks and is very dose to the present value accepted by mathematicians. In trigonometry, he concluded for a triangle, the result of a perpendicular with the half-side is the area. He also worked on the motions of the solar system and calculated the length of the solar year to 365.8586805 days.

His work Aryabhatiyam sketches his mathematical, planetary, and cosmic theories. This book is divided into four chapters:

1. The astronomical constants and the sine table
2. Mathematics required for computations,
3. Division of time and rules for computing the longitudes of planets using eccentrics and epicycles,
4. The armillary sphere, rules relating to problems of trigonometry and the computation of eclipses.
5. Aryabhata took the earth to spin on its axis; this idea appears to have been his innovation.

He also considered the heavenly motions to go through a cycle of 4.32 billion years; here he went with an older tradition, but he introduced a new scheme of subdivisions within this great cycle. According to the historian Hugh Thurston, Not only did Aryabhata believe that the earth rotates, but there are glimmerings in his system (and other similar systems) of a possible underlying theory in which the earth (and the planets) orbits the sun, rather than the sun orbiting the earth. The evidence is that the basic planetary periods are relative to the sun.

That Aryabhata was aware of the relativity of motion is clear from this passage in his book “Just as a man in a boat sees the trees on the bank move in the opposite direction, so an observer on the equator sees the stationary stars as moving precisely toward the west.”

In his book named ‘Aryabhattiyam’, Aryabhatta has given lot of references of Suryasidhanta. He had developed instruments like chakra yantra (disk instrument), Gola yantra (type of armillery sphere) and shadow instruments. Aryabhatta deduced that earth is a rotating sphere: the stars do not move, it is the earth that rotates. Its diameter is 1,050 yojanas. Its circumference is therefore 1050 x 13.6 x π = 44,860 km.

Aryabhatta also deduced that: “The moon eclipses the sun, and the great shadow of the earth eclipses the moon.”

Varahamihira (500 AD)

Varahmihira has done a valuable job of compilation of five astronomical theories, which were in use before Christ, and suryasidhanta is one of them. This compiled book is known as ‘Panchasidhanta’. He had developed some ring and string instruments.

Brahmagupta (598–668 AD)

Brahmagupta (598–668 AD) wrote important treatise on mathematics and astronomy in Brahamasphutasiddhanta in 628 AD. He gave solutions for the general linear equation, two equivalent solutions to the general quadratic equation, explained how to find cube and cube root of an integer, rules for facilitating the computation of squares and square roots and gave rules for dealing with five types of combinations of fractions. He was able to find (integral) solutions of Pell’s equation. Brahmagupta’s most famous result in geometry is his formula for cyclic quadrilaterals, a theorem on rational triangles and values of (pi).

In chapter seven of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun, an idea maintained in scriptures. He does this by explaining the illumination of the Moon by the Sun.

Bhaskara-I

Bhaskara-I was a 7th-century Indian mathematician, who was apparently the first to write numbers in the Hindu decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata’s work. This commentary, Aryabhatiyabhasya, written in 629 CE, is the oldest known prose work in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata’s school, the Mahabhaskariya and Laghubhaskariya.

Lalla (700 AD)

Lalla was an astronomer and mathematician who followed the tradition of Aryabhata-I and wrote Shishyadhividdhidatantra. He was well known because of twelve instruments which he brought into practice.

Vachaspati Misra (circa. AD 840)

Vachaspati anticipated solid (co-ordinate) geometry eight centuries before Descartes (AD 1644).

Brahmadeva

Brahmadeva (1060–1130) was an Indian mathematician. He was the author of Karanaprakasa, which is a commentary on Aryabhata’s Aryabhatiya. Its contents deal partly with trigonometry and its applications to astronomy.He was well known and intelligent scientist.

Halayudha (10 Century AD)

Halayudha was a 10th-century Indian mathematician who wrote the Mritsanjivani, a commentary on Pingala’s Chandashastra, containing a clear description of Pascal’s triangle, which was expressed as Meru Prastara (Staircase of Meru)

Bhaskara-II (Born 1114)

Bhaskara-II was a 12th century astronomer mathematician. He authored several mathematical treatises. The most important being Siddhanta Siromani. This has four parts :

• Leelavati - Named after his daughter deals with arithmetic and geometry.
• Bijaganita - It has contents related to algebra and has a chapter that deals with fundamental operation with positive and negative quantities with zero, solving many types of equations including quadratic equation.
• Grahaganita - Ideals with astronomy.
• Goladhyaya - It deals with astronomy.

He was the first to conceive the concept of differential calculus. He was also well versed in geometry and has given many theories concerning topics like the calculation of altitude, area of triangle and quadrilaterals. He gave formula for calculating area of circle in terms of the chord and vice-versa. The method of graduated calculation was documented in the book entitled the Five Principles (PanchSiddhantika). Bhaskara-II has been called the greatest mathematician of medieval India. He contributed to the fields of mathematics, arithmetic, algebra, trigonometry, calculus, astronomy and engineering. Conceptual design for a perpetual motion machine by Bhaskara II dates to 1150 AD. He described a wheel that he claimed would run forever. He used a measuring device known as Yasti-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.

Madhava

Madhava (c. 1340-1425) developed a procedure to determine the positions of the moon every 36 minutes. He also provided methods to estimate the motions of the planets. He gave power series expansions for trigonometric functions, and for pi correct to eleven decimal places.

Nilakantha Somayaji

Nilakantha (c. 1444-1545) was a very prolific scholar who wrote several works on astronomy. It appears that Nilakantha found the correct formulation for the equation of the center of the planets and his model must be considered a true heliocentric model of the solar system. He also improved upon the power series techniques of Madhava. The methods developed by the Kerala mathematicians were far ahead of the European mathematics of the day.

History of Mathematics

Science and Mathematics were highly developed during the ancient period in India. Ancient Indians contributed immensely to the knowledge in Mathematics as well as various branches of Science.

In this section, we will read about the developments in Mathematics and the scholars who contributed to it. You will be surprised to know that many theories of modern day mathematics were actually known to ancient Indians.

However, since ancient Indian mathematicians were not as good in documentation and dissemination as their counterparts in the modern western world, their contributions did not find the place they deserved. Moreover, the western world ruled over most of the world for a long time, which empowered them to claim superiority in every way, including in the field of knowledge.

Baudhayan

Baudhayan was the first one ever to arrive at several concepts in Mathematics, which were later rediscovered by the western world. The value of pi was first calculated by him. As you know, pi is useful in calculating the area and circumference of a circle. What is known as Pythagoras theorem today is already found in Baudhayan‟s Sulva Sutra, which was written several years before the age of Pythagoras.

Aryabhatta

Aryabhatta was a fifth century mathematician, astronomer, astrologer and physicist. He was a pioneer in the field of mathematics. At the age of 23, he wrote Aryabhattiya, which is a summary of mathematics of his time.

There are four sections in this scholarly work. In the first section he describes the method of denoting big decimal numbers by alphabets. In the second section, we find difficult questions from topics of modern day Mathematics such as number theory, geometry, trigonometry and Beejganita (algebra). The remaining two sections are on astronomy.

Aryabhatta showed that zero was not a numeral only but also a symbol and a concept. Discovery of zero enabled Aryabhatta to find out the exact distance between the earth and the moon. The discovery of zero also opened up a new dimension of negative numerals. As we have seen, the last two sections of Aryabhattiya were on Astronomy.

Evidently, Aryabhatta contributed greatly to the field of science, too, particularly Astronomy. In ancient India, the science of astronomy was well advanced. It was called Khagolshastra. Khagol was the famous astronomical observatory at Nalanda, where Aryabhatta studied.

In fact science of astronomy was highly advanced and our ancestors were proud of it. The aim behind the development of the science of astronomy was the need to have accurate calendars, a better understanding of climate and rainfall patterns for timely sowing and choice of crops, fixing the dates of seasons and festivals, navigation, calculation of time and casting of horoscopes for use in astrology. Knowledge of astronomy, particularly knowledge of the tides and the stars, was of great importance in trade, because of the requirement of crossing the oceans and deserts during night time.

Disregarding the popular view that our planet earth is „Achala‟ (immovable), Aryabhatta stated his theory that „earth is round and rotates on its own axis‟ He explained that the appearance of the sun moving from east to west is false by giving examples. One such example was: When a person travels in a boat, the trees on the shore appear to move in the opposite direction.

He also correctly stated that the moon and the planets shined by reflected sunlight. He also gave a scientific explanation for solar and lunar eclipse clarifying that the eclipse were not because of Rahhu and/or Ketu or some other rakshasa (demon,). Do you realize now, why the first satellite sent into orbit by India has been named after Aryabhatta?

Brahmgupta

In 7th century, Brahmgupta took mathematics to heights far beyond others. In his methods of multiplication, he used place value in almost the same way as it is used today. He introduced negative numbers and operations on zero into mathematics. He wrote Brahm Sputa Siddantika through which the Arabs came to know our mathematical system.

Bhaskaracharya

Bhaskaracharya was the leading light of 12th Century. He was born at Bijapur, Karnataka. He is famous for his book Siddanta Shiromani. It is divided into four sections: Lilavati (Arithmetic), Beejaganit (Algebra), Goladhyaya (Sphere) and Grahaganit (mathematics of planets). Bhaskara introduced Chakrawat Method or the Cyclic Method to solve algebraic equations. This method was rediscovered six centuries later by European mathematicians, who called it inverse cycle. In the nineteenth century, an English man, James Taylor, translated Lilavati and made this great work known to the world.

Mahaviracharya

There is an elaborate description of mathematics in Jain literature (500 B.C -100 B.C). Jain gurus knew how to solve quadratic equations. They have also described fractions, algebraic equations, series, set theory, logarithms and exponents in a very interesting manner. Jain Guru Mahaviracharya wrote Ganit Sara Sangraha in 850A.D., which is the first textbook on arithmetic in present day form. The current method of solving Least common Multiple (LCM) of given numbers was also described by him.

Concept of Zero

The concept of zero is not only a mathematical innovation but also a profound philosophical and cultural development. In the context of Vedic mathematics and the history of zero, its evolution in ancient India played a critical role in shaping both mathematics and the worldview of ancient scholars. Vedic mathematics, which refers to ancient Indian mathematical principles embedded in the Vedas and other classical Indian texts, includes methods that were later developed into a formal concept of zero.

Historical Development of Zero in India

The journey of zero as a concept is deeply rooted in the development of Indian mathematical thinking. While zero itself wasn’t fully conceptualized as a number in the earliest texts of the Vedas, India was one of the first places where a formal concept of zero, both symbolically and functionally, was developed.

Early Use of Zero-like Concepts

Vedic Period (around 1500 BCE–500 BCE): In the Vedic texts, there were references to shunya (meaning "void" or "nothingness") in philosophical and cosmological contexts, but zero was not yet treated as a number. The philosophical idea of "shunya" symbolized nothingness or the absence of something. Ancient Indian thinkers considered the notion of void or emptiness, as seen in the Upanishads, where the concept of Brahman (the ultimate reality) is sometimes described in terms that resemble the concept of zero—endless, without form, and yet the source of everything.

Shunya as a Number

The realization of zero as a number came much later, particularly in the context of Indian mathematics. This development took place during the classical period of Indian mathematics (around 5th to 6th century CE), when mathematicians began developing more structured methods for computation. The term "Shunya" became more closely associated with the concept of zero around the 5th century CE, especially in the works of mathematicians like Brahmagupta, who formalized zero as both a concept and a number.

Brahmagupta’s Contribution (598–668 CE)

Brahmagupta, one of the greatest mathematicians and astronomers of ancient India, is credited with providing the first clear mathematical definition and rules for zero. In his seminal work "Brahmasphutasiddhanta" (628 CE), Brahmagupta wrote:

Zero is the result of subtracting a number from itself (e.g., a−a=0).

He also discussed the rules for addition, subtraction, and multiplication involving zero:

Addition and Subtraction: The sum or difference of a number and zero is the number itself (e.g., a+0=a).

Multiplication: Any number multiplied by zero equals zero (e.g.,a×0=0).

Division

Brahmagupta mentioned that dividing a number by zero was undefined, but it was still considered a challenging concept. This formal treatment of zero as a mathematical entity marked a milestone in the history of mathematics and was a significant leap toward understanding zero's potential.

The Role of Zero in Indian Numeral System

Alongside Brahmagupta’s work, the Hindu-Arabic numeral system (which included zero) began to take shape in India. By the time of Aryabhata (476–550 CE), the full integration of zero into mathematical notation and calculation was being used in astronomical and mathematical texts. Aryabhata used zero and place-value notation in his works, such as the Aryabhatiya, which dealt with mathematical concepts related to astronomy and trigonometry.

Symbolization of Zero

The symbol for zero (a dot or circle) is believed to have originated in India in the 6th century CE. This symbol was used in the Bakhshali manuscript, a text that dates back to the 7th or 8th century CE and is considered one of the earliest Indian mathematical texts to clearly illustrate the use of zero as a place-holder in numerals. Zero as a digit played a critical role in the development of the place-value system (also known as the positional numeral system), which is the basis of the decimal system used today in mathematics globally. The place-value system was essential in simplifying arithmetic operations and expressing large numbers efficiently.

Transmission to the Arab World

The concept of zero and the Hindu-Arabic numeral system was transmitted to the Arab world during the 8th century CE through translations of Indian mathematical texts. Al-Khwarizmi, the Persian mathematician, played a key role in bringing these concepts to the Arab scholars, which later found their way into Europe. The Arab mathematicians further refined the use of zero and expanded its application in their own works, which contributed to the spread of zero in medieval Europe.

Zero in Europe

The Hindu-Arabic numeral system spread to Europe through translations of Arabic texts, especially during the Renaissance (14th to 16th century). Fibonacci, an Italian mathematician, was instrumental in introducing the system to Europe with his work "Liber Abaci" (1202), which included the use of zero. However, the full acceptance of zero as a number and its implications for computation took some time in Europe, as the Roman numeral system, which had no symbol for zero, was dominant.

Zero in Vedic Mathematics

While the explicit use of zero as a number is not present in the earliest Vedic texts, Vedic mathematics—a collection of ancient Indian mathematical techniques—contains methods that are intrinsically linked to the development of place-value systems and the concept of zero.

Vedic Mathematics is a system that employs mental calculation techniques and principles that were likely influenced by the development of the Hindu-Arabic numeral system, including zero. Some Vedic techniques allow rapid multiplication, division, and other operations that rely on an understanding of place value, which would not be possible without the concept of zero. The ancient Indians’ ability to perform calculations with large numbers, understand square roots, and solve problems in geometry and trigonometry implicitly relied on the place-value system, which includes zero.

Key Features of Zero in Indian Mathematics:

• Zero as a Place Value - In the Indian numeral system, zero was essential for representing large numbers by holding the place of empty positions. For example, 100 is understood as 1 in the hundreds place, 0 in the tens place, and 0 in the ones place. Zero enables the distinction between numbers like 100 and 10.

• Zero in Calculations - In addition to its role as a place-holder, zero became a functional element in performing arithmetic operations like addition, subtraction, and multiplication. For example, the process of carrying over and borrowing in addition and subtraction would not be feasible without the use of zero.

• Zero and Infinity - In Indian philosophy and mathematics zero was often viewed not just as the absence of quantity but also as a symbol of the infinite, representing both the void and the boundless potential of the universe.

History and Culture of Astronomy in India

The history of astronomy in India is a fascinating and rich narrative that spans thousands of years, deeply intertwined with the country’s cultural, philosophical, and religious heritage. Ancient Indian astronomy has contributed significantly to the development of the field, particularly through its mathematical and observational advancements. Indian astronomers have developed early cosmological models, made astronomical observations, and devised detailed systems for timekeeping, navigation, and calendar-making.

Early Beginnings of Astronomy in India The origins of Indian astronomy can be traced to the Vedic period (circa 1500 BCE to 500 BCE), when celestial phenomena were often linked to spiritual and religious practices. The early astronomical knowledge of the ancient Indians was not only functional (used for timekeeping, agricultural activities, and rituals) but also deeply philosophical.

Vedic Astronomy (circa 1500 BCE to 500 BCE)

The Vedas, particularly the Rigveda, contain references to celestial bodies and their movements. The Rigveda mentions the Nakshatras (lunar constellations), which were used for timekeeping and astrological purposes. In the Vedic worldview, the universe was cyclical, and celestial phenomena were often linked to divine actions. The solar year, the lunar month, and the sidereal year were all calculated in Vedic astronomy. The Yajurveda and the Atharvaveda also contain references to astronomical observations. These texts also mention the solstices and the equinoxes, which were important for ritualistic purposes.

Pre-Classical and Classical Indian Astronomy (500 BCE to 500 CE)

During this period, Indian astronomy moved from its purely ritualistic and philosophical basis to a more scientific and observational approach. Aryabhata (476–550 CE), one of the most prominent ancient Indian astronomers and mathematicians, is often credited with laying the foundation for modern astronomy in India. In his famous work Aryabhatiya (499 CE), Aryabhata introduced key concepts such as the rotation of the Earth, the sidereal day, and the heliocentric model of the solar system (where the Earth rotates on its axis and the Moon orbits the Earth). Aryabhata also made significant contributions to trigonometry and algebra, both of which are essential for astronomical calculations. Varahamihira (circa 505–587 CE) was another important figure during this time. His work Brihat Samhita includes important contributions on the movements of celestial bodies and the impact of planetary positions on human affairs (astrology). Varahamihira's Pancha-siddhantika deals with the five major astronomical systems of that era. Brahmagupta (598–668 CE) further advanced astronomy by formalizing the concept of zero and providing rules for astronomical calculations, including the positions of planets and lunar eclipses.

Medieval Indian Astronomy (500 CE to 1500 CE)

In the medieval period, Indian astronomers built on the foundational works of Aryabhata, Brahmagupta, and Varahamihira, while also engaging with other cultures and their astronomical ideas, particularly those from the Arab world. Madhava of Sangamagrama (c. 1350 CE) is often regarded as the founder of the Kerala School of Astronomy and Mathematics. His work on infinite series for trigonometric functions anticipated later developments in calculus by several centuries. The Kerala School developed the sine function and the cotangent function and applied them to astronomical calculations, making significant advancements in trigonometry. Raghunatha Siromani (15th century) and Parameshvara made significant contributions in refining the methods for calculating the positions of planets and the calculation of eclipses.

Key Contributions and Concepts in Indian Astronomy:

Calendar Systems

One of the most important aspects of Indian astronomy was the development of accurate calendar systems. Ancient Indian astronomers observed the cycles of the moon and sun, and based on these observations, they created complex lunar, solar, and sidereal calendars. The Hindu calendar is a lunisolar calendar, which combines the cycles of the moon with the position of the sun. The Tithi (lunar day), Paksha (fortnight), and Masa (month) were defined based on lunar phases and the position of the sun. Indian astronomy also developed detailed calculations for the timing of eclipses and the positions of planets. These were often used for religious and astrological purposes, such as determining the right time for sacrifices or rituals.

Sidereal and Tropical Year

Ancient Indian astronomers distinguished between the sidereal year (the time it takes for the Earth to complete one orbit around the Sun relative to the fixed stars) and the tropical year (the time it takes for the Earth to return to the same position in its seasonal cycle). Aryabhata was one of the first to correctly calculate the length of the sidereal year, which was measured as 365.2588 days. This was remarkably close to the modern value of 365.2564 days.

Heliocentric Modern

Aryabhata proposed a heliocentric model of the solar system, suggesting that the Earth rotates on its axis and that the apparent movement of the Sun is due to the Earth's rotation. Though his model was not universally accepted at the time, it anticipated the later work of Copernicus and Galileo by over a thousand years.

Observational Instrumentry

Indian astronomers developed a variety of observational instruments, including astrolabes, gnomons, and armillary spheres, to track the movements of celestial bodies. These instruments were used to measure the altitude and azimuth of stars and planets, as well as to determine the times of eclipses and other celestial phenomena. The Jantar Mantar observatories, built by Maharaja Jai Singh II in the 18th century (in cities like Jaipur and Delhi), are famous examples of large astronomical instruments that were used to measure celestial positions and time. Astronomy and Culture in India

Astronomy and Religious

Indian astronomy was deeply intertwined with religion. Celestial phenomena such as eclipses, planetary alignments, and the movement of stars were often interpreted as signs from the gods, with religious texts, rituals, and festivals being tied to astronomical events. The zodiac signs (Rashis), which are an important part of Hindu astrology, are based on the ecliptic plane of the Sun’s path across the sky. The Navagraha (nine celestial bodies in astrology) are also significant in religious contexts, with the planets and their movements influencing people's fortunes and destinies.

Cosmology

The Indian cosmological view was deeply influenced by philosophical concepts found in the Upanishads, which describe the universe as cyclical and infinite. The concept of time in Indian cosmology is vast, with the universe going through endless cycles of creation, preservation, and destruction (the triloka—the three realms of heaven, earth, and the underworld). Ancient Indian cosmology, as seen in texts like the Vishnu Purana, presents a universe composed of countless worlds and realms, with a constant cycle of creation and destruction governed by cosmic laws. Modern Era and the Revival of Indian Astronomy

Colonial and Post-Independence Development

During the British colonial period, Indian astronomy was largely influenced by Western ideas, but there was also an interest in preserving traditional knowledge. After independence in 1947, India made significant progress in modern astronomy. Jawaharlal Nehru, India’s first Prime Minister, was a great supporter of scientific development, and under his leadership, India established institutions like the Indian Space Research Organisation (ISRO), which has made significant strides in space exploration. India has launched numerous satellites and space missions, including the Chandrayaan missions to the Moon and the Mangalyaan (Mars Orbiter Mission), placing India among the leading space-faring nations.

Kerela School of Astronomy and Mathematics

The Kerala School of Astronomy and Mathematics is one of the most fascinating yet lesser-known chapters in the history of Indian science. Flourishing between the 14th and 16th centuries CE, this school produced groundbreaking mathematical concepts, many of which anticipated modern calculus by several centuries.
The Kerala School was founded by Madhava of Sangamagrama (c. 1350–1425 CE), a brilliant mathematician and astronomer. His successors, including Parameshvara, Nilakantha Somayaji, and Jyesthadeva, expanded on his ideas, refining mathematical and astronomical theories that were far ahead of their time.

Madhava of Sangamagrama (c. 1350–1425 CE)

• Often regarded as the father of mathematical analysis in India.
• Developed infinite series expansions for trigonometric functions (sine, cosine, and arctangent).
• Estimated π (pi) with high accuracy using an infinite series, now known as the Madhava-Leibniz series (discovered in Europe centuries later).
• Introduced concepts resembling Taylor series and calculus long before Newton and Leibniz.

Nilakantha Somayaji (1444–1544 CE)

• Authored Tantrasangraha, a detailed astronomical treatise that refined planetary models.
• Proposed an improved version of the geocentric model, which closely resembles aspects of Tycho Brahe’s later work in Europe.
• Corrected earlier astronomical errors and provided a more precise understanding of planetary motion.

Jyesthadeva and the Yuktibhasa (c. 16th century CE)

• Yuktibhasa, written by Jyesthadeva, is considered the first known text on calculus.
• Explained the derivation of infinite series and their practical applications.
• Covered topics like integration, differentiation, and convergence of series—concepts essential to modern calculus.

Mathematical Contributions

The Kerala School’s achievements in mathematics were revolutionary:

• Infinite Series for Trigonometric Functions:

  • Derived series expansions for sine, cosine, and arctangent, centuries before European mathematicians.
  • Example: The Madhava-Leibniz series for π: $$\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \dots$$
  • This method was later rediscovered by James Gregory and Gottfried Wilhelm Leibniz.

• Early Concepts of Differentiation & Integration:

  • Developed techniques similar to calculus for calculating areas under curves.
  • Used integration to compute the volume of curved surfaces, a principle fundamental to modern mathematical physics.

• Improvements in Astronomical Calculations:

  • Made significant advancements in planetary motion models and eclipse predictions.
  • Defined precise sine and cosine tables, essential for navigation and astronomy.