Jain Ganit Ane Teni Mahatta - Book Summary

Here is a comprehensive summary in English of the provided Jain text, "Jain Ganit ane teni Mahatta" (Jain Mathematics and its Importance) by Narsinh M. Shah:

This work, "Jain Ganit ane teni Mahatta," authored by Narsinh M. Shah, delves into the significant contributions of Jainism to the field of mathematics. The author asserts that Jain philosophy, structured into four anuyogas (Prathamanuyoga, Charnanuyoga, Karananuyoga, and Dravyanuyoga), explicitly includes Karananuyoga as the branch concerned with both transcendental and worldly mathematics, thus also being known as Ganitanuyoga. This highlights the high regard for mathematics within Jainism.

The text emphasizes the originality and importance of Jain worldly mathematics, citing scholars like Professor Hiralal Kapadia who stated that Indians, and particularly Jains, were not behind any other nation in their attention to mathematics. The author draws a parallel with the South Indian mathematician Mahaviracharya (c. 850 CE) and his work "Ganitasarasangraha," which demonstrated the applicability of mathematics in diverse fields such as music, logic, dramaturgy, architecture, medicine, grammar, poetry (Pingala), economics, and even love science. Mahaviracharya's analysis of triangle and quadrilateral mathematics, with unique insights, further illustrates the practical application of Jain mathematics beyond purely religious contexts.

A significant point made is that Jain Acharyas played a principal role in the development of Indian mathematics. During periods when mathematics was in its nascent stages, Jain mathematicians made valuable contributions to solving problems in algebra and mensuration. Dr. G. Thibaut is quoted praising the Jain contribution, suggesting that the Suryaprajnapthi text predates Greek influence in India due to its lack of any such impact.

The book refutes a common misconception that only Brahmins studied mathematics in ancient India, while other religious groups like Buddhists and Jains were not interested. The author argues that this notion arises from the limited availability and relative obscurity of ancient mathematical texts written by Jain mathematicians, possibly due to their sectarian nature. However, a study of Jain Agamas and other religious scriptures reveals that Jains actively engaged with and contributed to mathematics. The knowledge of mathematics and astronomy was considered a distinct achievement within the Jain monastic institution.

The text identifies several prominent centers of mathematical study in ancient India, including Pataliputra, Ujjain, Khambhat, Mysore, Malabar, Valabhi, Varanasi, and Taxila. While direct evidence of the connections between these centers is scarce, the similarity in mathematical works found from different locations suggests interaction and exchange of ideas among scholars. The spread of Jainism and Buddhism fostered the study of various sciences and arts, and Jain religious literature frequently features large numbers.

A key contribution highlighted is the invention of the decimal system and the place-value system of notation. The text explains how large numbers were cumbersome to write in earlier systems like those of the Egyptians (requiring repetition of symbols) and Romans (using letters). The Jain mathematicians are credited with developing the decimal system, which uses ten digits, enabling the representation of any large number by simply appending zeros. This system significantly simplified arithmetic operations like addition and subtraction. The development of the decimal system is placed in the early centuries of the Common Era, coinciding with the zenith of Jain and Buddhist influence, and tracing its roots from Vedic times through the works of mathematicians like Aryabhata and Varahamihira.

The text addresses the apparent gaps in the continuity of mathematical literature from the Vedic, Buddhist, and Jain periods, noting the scarcity of manuscripts predating Aryabhata's Aryabhatiya (499 CE). An exception is the Vakshali manuscript, believed to be from the 2nd or 3rd century CE, which indicates knowledge of numerical arrangement systems, though it lacks the detailed mathematical explanations found in later works.

The discovery of Ganitasarasangraha in 1912 and its subsequent research by Rangacharya led scholars to recognize the existence of a Jain mathematical tradition. Professor B.D. Bose, in his article "Jain School of Mathematics," shed light on Jain mathematical formulas and references, highlighting that many works by Jain mathematicians are still undiscovered. The author calls for the exploration of Jain manuscript collections to bring these important works to light. The Western notion that all sciences originate from Greece or Rome is challenged.

Notable Jain mathematicians mentioned include Bhadrabahu (author of commentary on Suryaprajnapthi and Bhadrabahavi Samhita, d. 278 BCE) and Siddhasena (known for his astronomical work). References to their mathematical contributions are found in Ardhamagadhi and Prakrit literature. The text also discusses Virasena's commentary on the Digambara text Khandagama, known as Dhavala, written in the early 9th century. While Virasena was a philosopher, the mathematical content in Dhavala is attributed to earlier mathematicians and is dated around 200-600 CE, providing insights into the "dark age" of Indian mathematics. The mathematical methods described in Dhavala are unique and less sophisticated than those in later works like Aryabhatiya.

The text elaborates on the Jain understanding and application of the place-value system, providing three examples of how large numbers were represented, showcasing variations in how numbers were grouped (e.g., from right to left, or by units of 100 in Prakrit and Pali literature).

The prevalence of large numbers in Jain texts, discussing concepts like jivaraashi (aggregate of souls) and dravyapramana (quantity of substances), is noted. Texts like Karmagrantha and Dhavala contain astronomical numbers such as "crore-crore-crore" souls. Basic arithmetic operations, square roots, and cube roots are also mentioned.

A significant discussion revolves around the Jain understanding of logarithms and related concepts, which are presented with surprising modernity. The text defines terms like ardhachchheda (related to base-2 logarithm), trikarchhed (related to base-3 logarithm), and chaturthachhed (related to base-4 logarithm). It also details several logarithmic identities derived in Dhavala, including:

Log (m/n) = log m - log n
Log (m.n) = log m + log n
Log₂ m = m
Log (x²) = 2 x log x
log log (x²) = log (2 x log x) = log x + log 2 + log log x (with further simplification)
log (x^x) = x^x log x^x

These identities suggest that Jain mathematicians were familiar with modern exponential laws and logarithmic principles centuries before their widespread recognition elsewhere.

The text also briefly touches upon fractions and proportions, with the use of specific terminology like phala (fruit), ichchha (desire), and pramana (measure).

The concept of infinity is extensively discussed, with Jain philosophers credited with classifying its eleven types, including namanananta, sthananananta, dravyananta, gananananta (numerical infinity), apradheshikananta, ekananta, ubhayanananta, vistarananta, sarvananta, bhavanananta, and shashwatananta (eternal). This classification is considered comprehensive, covering all uses of the word "infinity" in Jain literature.

The text contrasts the Indian approach to large numbers and infinity with that of European mathematicians like Archimedes. While Greek philosophers struggled with the concept of infinity, Indian thinkers, especially Jains, developed appropriate frameworks. The Jain classification of infinity and their use of terms like sankhyata (numerable), asankhyata (innumerable), and ananta (infinite) are highlighted.

The three methods used by Jains for representing large numbers are reiterated:

Place-value system: Using powers of ten (e.g., 10⁴⁰).
Laws of exponents (varga-samvarga): For concise representation of large numbers, exemplified by (2) raised to the power of itself repeatedly, producing enormous numbers.
Logarithms (ardhachchheda or logeritham): For simplifying calculations with very large numbers.

These methods are shown to be foundational to modern mathematical practices in fields like physics. The text suggests that these principles were known in India before the 7th century CE, with a significant contribution from Jain mathematicians.

The text acknowledges George Cantor's work on transfinite numbers in the 19th century, which provided a rigorous foundation for infinity. It notes that Jain concepts like utkrushtha-asankhyata (excellent-innumerable) approach the idea of infinity, and their early attempts to develop principles for infinite cardinal numbers are considered remarkable.

Finally, the text touches upon Jain geometric knowledge, mentioning discussions of minimal regions for creating geometric shapes in the Bhagavati Sutra and detailed descriptions of mountain levels in the Budhiprajnapthi. The text also discusses the calculation of Pi (π), citing three values found in Jain scriptures: √10, "a little more than three" (3 < π), and 316 (related to the ratio of circumference to diameter). The approximation of π as 19/6 is also mentioned from Digambara texts.

The author concludes by emphasizing that the Jain contribution to the development of mathematics requires extensive research, urging the systematic exploration of undiscovered ancient manuscripts. While acknowledging efforts made so far, more comprehensive research is deemed essential to fully appreciate the depth of Jain mathematical achievements.