Introduction

In the early decades of the 14th century, a mathematical renaissance took place in Kerala, with the emergence of the Kerala School of Mathematics and Astronomy. Central to the work of the Kerala School were explorations into trigonometry and circular properties, including circumference, diameter, arcs, and corresponding chords. Remarkably, many of the ideas and techniques used in the Kerala School were later recognised as the precursors to infinitesimal calculus. Focus on these topics stemmed from necessity as Indian astronomers relied heavily on trigonometric tables and values to compute the positions of stars to predict astronomical events such as eclipses. These events were important in scheduling Indian religious ceremonies, and so more accurate approximations were necessary. The historical timeline of Kerala Mathematics is anchored by Sangamagrama Madhavan (1340–1425), who achieved the remarkable feat of calculating Pi (π) correctly to eleven decimal places. Throughout the 14th to 16th centuries, astronomers and mathematicians from Kerala made pioneering contributions, predating European mathematicians by a staggering 300 years.

Europe during this period witnessed little mathematical activity until the late 1600s. Notable figures like Fibonacci in the early 1200s marked the last major influences before the emergence of great mathematicians such as Fermat, Newton, and Leibniz. However, by the time these European luminaries rose to prominence, the Kerala School had already completed much of its major work and was entering a period of decline. However, a historical paradox emerged as many of these groundbreaking advances made by Kerala’s scholars and teachers were later attributed to European mathematicians, who benefitted from the ability to publish and circulate these ideas to larger audiences.

Background affecting development in Kerala

The medieval period in Kerala, spanning from the Perumals of Mahodayapuram to the establishment of Swarupams, witnessed significant historical developments such as the rise of agricultural communities, expansion of trade, and complex power struggles. The establishment of Swarupams brought about a unique socio-political landscape characterised by incessant competition among them. The Kerala School’s foundations were laid amidst these historical currents, through the meticulous study and translation of earlier Indian traditions along with external influences permeating through trade. Additionally, the Nambuthiris, Jains, and Buddhists played crucial roles in introducing ideas from earlier Indian traditions.

Under the Kulasekhara dynasty, particularly during the reign of Sthanu Ravi Varma, there was significant progress in education, astronomy, astrology, mathematics, and healthcare. The establishment of observatories, such as the one in Mahodayapuram, reflected a keen interest in astronomy during this period. The socio-economic background of political changes, cultural evolution, and external influences collectively shaped Kerala’s development during this period. The regionalisation of culture and the formation of a distinct Kerala identity were significant outcomes of this transformative period. Medieval scholars, including those from the Kerala School, were instrumental in shaping the intellectual landscape.

Lives and works of its members

MADHAVA OF SANGAMAGRAMA

Madhava, often referred to as Sangamagrama Madhavan (1340–1425), played a pivotal role in the evolution of mathematical thought. Born in Bakulavihara, modern-day Irinjalakuda, he pioneered the exploration of infinite power series, a remarkable feat predating Newton’s rediscovery by nearly two and a half centuries. His groundbreaking series, π/4 =1 -1/3 + 1/5 – 1/7 + …, predates the Leibniz series by centuries. Madhava’s focus predominantly dwelt on trigonometric functions. Beyond his mathematical innovations, Madhava applied his keen intellect to astronomy, calculating the exact positions of the moon with remarkable precision—correct to the seconds for every 36 minutes of the day. He also delved into discussions on the computation of the longitudes of planets and the ascendant. Most of Madhava’s works are known through his disciples such as Nilakantha, Jyesthadeva, Narayana, and Sankara Variyar.

VATASSERI PARAMESHVARA

Vatasseri Parameshvara Namboothiri (1360-1460) was a prominent Indian mathematician and astronomer who belonged to the Vatasseri illam in Malappuram. Born into a Namboothiri family specialising in the study of astronomy and astrology, Paramesvara’s contributions to the field have left a lasting impact. One of Paramesvara’s major works was the development of the Drk system for astronomical calculations, which aimed at achieving greater accuracy in predicting celestial events. He made significant strides in mathematics; he formulated a formula for the radius of the circle. This mathematical insight was presented in his commentary on Leelavati.

Paramesvara’s teacher, Madhava, played a crucial role in shaping his intellectual pursuits. Paramesvara placed great emphasis on the importance of verifying theoretical concepts against empirical observations. The scope of Paramesvara’s astronomical work extended to various facets of spherical astronomy, including the apparent and true motions of planets.

NILAKANTHA SOMAYAJI

Nilakantha Somayaji, born in 1444 into a Namboothiri Brahmin family in Trikkantiyur, Malappuram district, was a distinguished Indian mathematician and astronomer. He was from the Kelallur illam, a renowned centre of learning. Tantrasangraha, authored by him around 1500 AD, covered topics related to the computation of sine, movement of planets, fixing of gnomon, calculation of meridian, latitude, declensions, and the prediction of eclipses.

One of Nilakantha’s achievements is the major revision he made to the Aryabhatan model for the interior planets—Mercury and Venus. His modifications resulted in a more accurate specification of the equation of the centre for these planets than existing ones in Islamic or European astronomy before Johannes Kepler, who came 130 years later. In the Aryabhatiyabhasya, Nilakantha developed a computational scheme for planetary motion that surpassed later models, including that of Tycho Brahe. His scheme implied a heliocentric model of planetary motion where Mercury, Venus, Mars, Jupiter, and Saturn moved in eccentric orbits around the Sun, which, in turn, revolved around the Earth.

His interactions with contemporaries extended beyond Kerala, as evident in his text Sundararajaprasanottara, a dialogue with the Tamil astronomer Sundararaja. This interaction is particularly valuable as it provides rare evidence of interest in Kerala Mathematics and astronomy in other areas of South India.

JYESTHADEVAN

Jyesthadevan Namboothiri, born in the Parannottu family in Alathur around 1500–1610, received his education under the guidance of Damodaran and emerged as a notable disciple of Nilakanda. He is best known as the author of Yukti Bhasha. This critically important text is divided into two parts. The first part composed in Malayalam focuses on the logical demonstration of various mathematical results and includes discussions on Madhava’s contributions to infinite series, especially regarding Pi and sine values, along with detailed rationale and computations. The second part of Yukti Bhasha transitions into an astronomical treatise, providing the rationale behind various astronomical results.

Jyesthadevan’s contributions to the Kerala School and the broader field of mathematics were recognised by C.M. Whish, the first Western scholar to acknowledge the Kerala School’s work. In 1834, Whish highlighted Yukti Bhasha as one of the key astronomical and mathematical texts, alongside works by other eminent scholars such as Nilakantha, Putumana Somayaji, and Sankara Varman. The legacy of Jyestha Devan extends to his impact on subsequent generations, as evidenced by his disciple Achutha Pisharady, who, in 1592, authored Uparaga Kriyakarmam. Melpathur Narayanabhattathiri, the renowned author of Narayaneeyam, was a disciple of Achutha Pisharady, underscoring the intellectual lineage and influence of Jyesthadevan in shaping the mathematical landscape of the Kerala School.

Recognition

In 1825, John Warren published the Kala Sankalita, a memoir focused on the division of time in southern India. Warren briefly mentioned the discovery of infinite series by Kerala astronomers, providing a glimpse into the mathematical sophistication present in the region. The true introduction of the Kerala School’s works to the Western world occurred in 1835 when Englishman C.M. Whish highlighted Kerala mathematicians’ achievements, including the Gregory series for the inverse tangent, the Leibniz power series for π, and the Newton power series for sine and cosine functions.

Despite Whish’s efforts, Kerala Mathematics faced a delay in gaining widespread attention. It was not until the 1940s that C.T. Rajagopal and his collaborators revisited and emphasised the significant contributions of the Kerala School, including a detailed analysis of Yukti Bhasa’s proof of sine and cosine series, and the presentation of Sanskrit verses from Tantrasangrahavakhya for arctan, sin, and cosine series with English translation and commentary. This re-examination shed light on the depth and sophistication of Kerala’s mathematical and astronomical achievements.

In the broader context of India’s mathematical heritage, historical narratives often focus on iconic figures like Aryabhata or the invention of zero. However, as highlighted by Kim Plofker in Mathematics in India (2008), the richness of India’s mathematical tradition extends far beyond these well-known aspects. Kerala’s contributions to mathematics also include key elements such as decimal place value numerals, the use of negative numbers, solutions to indeterminate equations, and innovative power series.

In recent times, extensive research, notably in works like The Golden Crest of Peacock and The Passage to Infinity by Dr George Geevarghese Joseph, has played a crucial role in establishing the validity of a non-Eurocentric history of mathematics.

Conclusion

The emergence of mathematical analysis in the form of infinite series and their finite approximations marked a breakthrough in Kerala Mathematics. Despite its extensive explorations and documentation of ideas, the dissemination of its discoveries faced formidable challenges. Around 400 palm leaf manuscripts dedicated to astronomy and approximately 350 manuscripts on astrology have been discovered in Kerala. However, many of these manuscripts remain hidden from public view, primarily due to the conservative attitudes of their custodians. The climatic conditions of Kerala have posed a threat to the preservation of these manuscripts, with a significant number either perishing or deteriorating to such an extent that deciphering them becomes a formidable challenge.

The physical separation of the school, along with cultural and linguistic barriers, constrained the spread of knowledge beyond the local region. Operating in a society where educational involvement and information exchange were predominantly limited to Brahmins, this cultural division hindered the sharing of publications and impeded broader access to knowledge among scholars. Additionally, the publications from the Kerala School were primarily in Sanskrit or Malayalam, creating a significant language barrier. Even if Madhava’s publications had reached Europe during Gregory and Leibniz’s era, understanding the content would have posed a challenge for the majority of the mathematical community.

Despite these obstacles, the mathematical and astronomical contributions of the Kerala School should not be underestimated. While it may not have received due credit in its time, the Kerala School stands as a remarkable representation of academic excellence beyond the confines of a Eurocentric perspective.