Contribution Of Ancient Jaina Mathematicians - Book Summary

Here's a comprehensive summary of the provided Jain text, "Contribution of Ancient Jaina Mathematicians" by B.S. Jain:

The paper highlights the significant and often overlooked contributions of ancient Jain mathematicians to the field of mathematics, asserting that India led the world in mathematical advancements until the early 17th century.

1. Importance of Mathematics in Jainism:

Mathematics was deeply integrated into Jain religious life, considered one of the four "anuyogas" (auxiliary sciences) for spiritual liberation.
Knowledge of "Ganitanuyoga" (exposition of mathematical principles), also referred to as "Samkhyana" (science of numbers, including arithmetic and astronomy), was a key accomplishment for Jain priests.
This mathematical knowledge was essential for determining auspicious times and locations for religious ceremonies.
Jain tradition credits the first Tirthankara, Rishabhnath, with teaching writing and mathematics to his daughters.
Jain literature, known as Siddhanta or Agama, contains valuable scientific concepts, and Jain scholars made notable contributions to various scientific fields, including mathematics, medicine, physics, astronomy, cosmology, and atomic theory.

2. Key Jaina Mathematical Works and Figures:

Mahaviracārya (850 AD): Author of the renowned "Ganita Sara Samgraha" (Collection of Essence of Mathematics), a treatise on arithmetic and algebra. His appreciation of mathematics emphasizes its utility across various domains: worldly transactions, Vedic rituals, sciences of love, wealth, music, drama, cooking, medicine, architecture, prosody, poetry, logic, grammar, and celestial movements.
Surya Prajnapti and Chandra Prajnapti: Two astronomical treatises.
Umaswati (circa 150 BC or 135-219 AD): A reputed Jaina metaphysician and author of "Tattvartha-dhigama-Sutra-Bhashya," which is acknowledged by both Swetambara and Digambara sects. He also wrote "Ksetra Samasa" (Collection of Places), dealing with geography and mensuration, which summarized mathematical calculations in Jain canonical works. While not a mathematician himself, he cited mathematical results from contemporary treatises.
Bhadrabahu (4th Century BC): A great preceptor, astronomer, and mathematician, credited with memorizing the entire Jain canonical literature. He authored a commentary on Surya Prajnapti and an original work, "Bhadra bahavi Samihita."

3. Schools of Mathematics: The paper identifies three prominent schools of mathematics during the Sulva Sutra period (750 BC - 400 AD):

Kusumpura/Pataliputra School: A major learning center, associated with Nalanda University. Figures like Bhadrabahu and Umaswati belonged to this school. Aryabhata (476 AD), considered the father of Hindu algebra, was the Kulpati of Nalanda and influenced this school.
Ujjain School: Associated with Brahmagupta and Bhaskaracārya.
Mysore School: Associated with Mahaviracārya. These schools maintained close contact and exchanged knowledge.

4. Topics in Mathematics: According to the "Sthananga Sutra" (before 300 BC), the science of numbers ("Sankhyana") encompassed ten topics:

Parikarma: Fundamental arithmetic operations (addition, subtraction, multiplication, division).
Vyavahara: Applied arithmetic.
Rajju: Geometry (Sulva in the Vedic period).
Rasi: Mensuration of solid bodies.
Kala Savarnama: Fractions.
Yavat-tavat: Simple equations (symbol for an unknown quantity).
Varga: Quadratic equations (also refers to square and square-root).
Ghana: Cubic equations (also refers to cube and cube-root).
Varga-varga: Biquadratic equations.
Vikalpa or Bhong: Permutations and combinations.

5. Specific Mathematical Contributions:

Multiplication and Division by Factors: Umaswati's "Tattvartha dhigama-sutra-Bhashya" mentions methods of multiplication and division using factors, a simpler approach later found in Sridhara's work and transmitted to Italy.
Mensuration Formulas: Umaswati provided formulas for circle mensuration (circumference, area). Notably, formula (v) for the arrow (h) of a circular segment, $h = \sqrt{\frac{d^2 - c^2}{4}}$ (derived from $c^2 = 4h(d-h)$), demonstrates knowledge of solving quadratic equations. These formulas were known in India centuries earlier, appearing in texts like the Surya Prajnapti (500 BC) for calculating the dimensions of Jambudvipa. The paper highlights that the approximation for square roots mentioned in Jaina works predates Heron of Alexandria.
Value of Pi ($\pi$): The "Surya Prajnapti" (500 BC) provides values of $\pi$ as 3 and $\sqrt{10}$. The "Uttaradhyayana-sutra" (300 BC) implies $\pi \approx 3.16$. Medieval Jaina works consistently used $\pi = \sqrt{10}$.
Theory of Numbers: Mahaviracārya (850 AD) listed twenty-four notational places for numbers, exceeding the eighteen mentioned by other Indian mathematicians. This demonstrates the use of very large numbers in Jaina cosmology.
Infinity: The "Anuyogadwara-Sutra" (100 BC) classifies numbers into numerable, innumerable, and infinite. It defines a "highest numerable number" comparable to modern Aleph Zero. The "Sthanana ga-Sutra" (before 300 BC) classifies infinity into five types (direction, superficial, and all-encompassing expanse, and eternity).
Laws of Indices: The "Anuyogadwara-Sutra" and "Uttaradhyayana-sutra" show understanding of successive powers and roots, indicating knowledge of laws like $a^m \times a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$.
Permutations and Combinations: The "Sthananga-Sutra" (before 300 BC) identifies permutations and combinations as a mathematical topic. Mahaviracārya provided the general formulas for permutations ($P(n,r)$) and combinations ($C(n,r)$). The text also cites its use in Sushruta's medical work (600 BC) for calculating combinations of tastes.

6. Challenges in Studying Jaina Mathematics: The paper concludes by noting difficulties in studying ancient Indian mathematics, including the unavailability of original works, the poetic and concise nature of ancient Sanskrit texts, and the need for expertise in both mathematics and Sanskrit to fully interpret them. The author emphasizes the need for further research into Jaina manuscripts.

In essence, the paper argues that Jain mathematicians made profound and early contributions to various branches of mathematics, including arithmetic, algebra, geometry, mensuration, number theory, and combinatorics, laying crucial groundwork for India's global leadership in mathematics for centuries.