Jain Agamo Me Nihit Ganitiya Adhyayan Ke Vishay

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The book, "Jain Agamo me Nihit Ganitiya Adhyayan ke Vishay" (Mathematical Study Topics Contained in Jain Agamas), authored by Anupam Jain and Sureshchandra Agarwal, published by USA Federation of JAINA, explores the mathematical content embedded within Jain scriptures.

The text begins by defining the Agamas as the teachings of the Tirthankaras and their interpretation by their chief disciples (Ganadharas). It notes that the currently available Agamas were compiled between the 5th century BCE and the 5th century CE, based on the teachings of Lord Mahavira. The authors highlight the difference in acceptance of Agamas between the Digambara and Shvetambara traditions. The Digambara tradition recognizes texts like Shadkhandagama and Kashayaprahuti, along with the literature of Acharya Kundakunda, as Agamas. The Shvetambara tradition, following the Vallabhigani Vacana (453-456 CE), considers the Anga and Upanga literature, written in Ardhamagadhi Prakrit, as authoritative Agamas. The article focuses on these Anga and Upanga texts.

A significant text discussed is the Sthananga (Than), the third among the Anga literature. Originally created around 300 BCE and compiled in its current form by the 5th century CE, its tenth chapter, specifically the 100th gatha (verse), is of great importance to mathematicians. This verse lists various subjects of mathematical study, indirectly suggesting their presence within the Agamas, as Tirthankara Mahavira is considered an expert in numerical knowledge.

The gatha from Sthananga is presented in several variations, with the authors detailing its Sanskrit rendering as: "Parikarma, Vyavahara, Rajju, Rashi, Kalasavanne, Yavataavata, Varga, Ghana, Vargavarga, Kalpa."

The earliest known commentary on this gatha is by Abhayadeva Suri (10th century CE), who interpreted the terms as:

1. Parikarma: Compilation etc. (Fundamental operations)
2. Vyavahara: Series computation or table arithmetic.
3. Rajju: Plane geometry.
4. Rashi: Heap of grains.
5. Kalasavanne: Multiplication or addition of natural numbers.
6. Varga: Square.
7. Ghana: Cube.
8. Vargavarga: Fourth power.
9. Kalpa: Saw-like (or Krakachika) operation.

Later, B.B. Dutt (1929), approximately 900 years after Abhayadeva Suri, proposed a different interpretation, considering Abhayadeva's explanation incomplete. Dutt's interpretation, though made when Jain mathematics was less understood, is considered more logical. He identified the ten terms as:

1. Arithmetic's fundamental operations (Parikarma).
2. Arithmetic's subject of treatment (Vyavahara).
3. Geometry (Rajju).
4. Mensuration of solids (Rashi).
5. Fractions (Kalasavanne).
6. Simple equation (Yavataavata).
7. Quadratic equation (Varga).
8. Cubic equation (Ghana).
9. Biquadratic equation (Vargavarga).
10. Combination & Permutation (Kalpa).

The article then proceeds to analyze each term in detail, reviewing various scholarly interpretations:

• Parikarma (परिकर्म): Generally understood as fundamental operations in arithmetic. The text highlights its presence in Jain literature, including as a division of the Dristivada Anga and commentaries on Shadkhandagama. Indian mathematicians generally considered eight fundamental operations, with addition and subtraction being the most basic. The authors conclude that parikarma refers to the basic arithmetic operations.

• Vyavahara (ववहारो): Interpreted as applied arithmetic or practical mathematics. It is linked to various types of "Vyavahara" described by Brahmagupta and Mahaviracharya, such as mixed, series, field, pit, calculation of mounds, saw-like, and shadow calculations. The term is associated with Patiganita (arithmetic) and its practical applications.

• Rajju (रज्जु): This term is crucial in the subject list. Abhayadeva Suri linked it to plane geometry, involving calculations with ropes. Dutt expanded this to encompass all of geometry. Iyengar relates it to "Sulva" in Vedic literature. However, L.C. Jain emphasizes that in Jain literature, Rajju is derived from numerical principles and describes the dimensions of the cosmos, including concepts of infinite quantities. The authors suggest that Rajju in this context points to the mathematics related to the measurement and structure of the universe, possibly indicating a supra-mundane measurement. They believe both Abhayadeva and Dutt were unsuccessful in fully interpreting Rajju, with L.C. Jain providing a more accurate direction.

• Rashi (रासी): There is a significant divergence of opinion between Abhayadeva Suri and Dutt. Abhayadeva interpreted it as a heap of grains, while Dutt rejected this, arguing that measuring grain heaps lacks prominence in mathematical works. Dutt suggested Rashi broadly refers to geometry. L.C. Jain proposes that Rashi refers to sets, drawing parallels with modern set theory. The authors consider L.C. Jain's interpretation more appropriate, emphasizing its role in Jain cosmology and philosophy, particularly in relation to concepts of enumeration, aggregates, and the structure of the universe.

• Kalasavanne (कलासवण्णे): Unambiguously refers to the mathematics of fractions. The term signifies making fractions have a common denominator. The authors note its consistent usage from the Vakshali manuscript to Mahaviracharya.

• Yavataavata (जावंतावति): Also referred to as multiplication, Abhayadeva Suri interpreted it as the multiplication or addition of natural numbers. Dutt strongly suggests it relates to the "rule of false position" used to solve linear equations, a significant topic in early algebra. The authors find Dutt's interpretation more logical, linking the formula for the sum of natural numbers to simple equations. L.C. Jain sees it as representing limitations or an invitation to elaborate on results, and potentially related to indeterminate quantities or the science of algebra.

• Varga, Ghana, Vargavarga (वग्गो, घणो, वग्गवग्गो): These terms refer to squares, cubes, and fourth powers, respectively. While commonly understood in their current mathematical meanings, the authors question their significance as distinct topics if they are merely extensions of fundamental arithmetic operations. They note that later Jain texts and commentaries discuss higher powers. Dutt interprets these terms as quadratic, cubic, and biquadratic equations, which the authors find plausible, especially considering the absence of explicit mention in currently available Agamas but their presence in related mathematical works like Ganita Sara Sangraha.

• Vikapot or Kalpa (विकप्पोत या कप्पो): Abhayadeva Suri interpreted this as knowledge of cutting wood and building stones, relating it to "Krakachika Vyavahara" in Patiganita. However, Dutt and other scholars interpret it as combinations and permutations (Vikalpa or Bhanga in Jain literature). The authors strongly support the latter interpretation, highlighting the extensive and natural use of combinations and permutations in Jain philosophical explanations, indicating its deep integration into Jain thought. They consider Vikalpa Ganita as a significant mathematical subject, not a trivial one, where Jain scholars excelled.

The text also discusses another gatha from the Sthananga Sutra cited by Shilanka in the 9th century in his commentary on the Sutrakritanga. This gatha is similar but replaces Vikapot with Pudgala (matter), suggesting Pudgala as a subject of mathematical study. The authors acknowledge the complexity of this discrepancy. While Dutt dismisses Pudgala as a mathematical subject, the authors argue that in light of modern understanding of the mathematics of the soul (karma theory) and concepts like infinite quantities, Pudgala could be considered fundamental. They conclude that the interpretation of Pudgala versus Vikalpa requires further deliberation.

Finally, the article mentions another gatha from the fourth chapter of the Sthananga Sutra which lists four types of Sankhyana (numerical calculation): Parikarma, Vyavahara, Rajju, Rashi. The authors note that this enumeration differs from the ten topics listed earlier and requires further discussion.

In essence, the book delves into the mathematical knowledge present in Jain Agamas, analyzing specific terms and their interpretations by various scholars, highlighting the richness and unique contributions of Jain mathematics to the broader history of Indian mathematics, particularly in areas like geometry, fractions, algebra, and combinatorics, with a significant emphasis on their application in Jain cosmology and philosophy.