Bharatiya Ganit Ke Andh Yuga Me Jainacharyo Ki Upalabdhiya

Here's a comprehensive summary of the provided Jain text, focusing on the achievements of Jainacharya in the "dark age" of Indian mathematics:

The text, titled "Jainacharyo ki Upalabdhiya" (Achievements of Jain Acharyas) in the Dark Age of Indian Mathematics by Dr. Parmeshwar Jha, published as part of the Z_Kusumvati_Sadhvi_Abhinandan_Granth, highlights the significant, often overlooked, contributions of Jain scholars to the field of mathematics during a period when other Indian mathematical traditions may have been less active.

Introduction to Indian Mathematics and the Jain Context:

The article begins by establishing the ancient roots of Indian culture and its rich mathematical traditions, referencing early texts like the Vedas, Brahmanas, Upanishads, and epics. It notes that early scientific facts were intertwined with spiritual principles. The development of Jyotirvidya (astronomy) and mathematics was initially linked to religious activities. The period from the 5th to the 12th century CE is described as the "Golden Age of Indian Mathematics," which saw the emergence of renowned mathematicians like Aryabhata, Brahmagupta, Bhaskara I, Sridharacharya, Mahaviracharya, and Bhaskara II. These scholars made foundational contributions that were later rediscovered in other parts of the world.

The author raises a pertinent question: Was there no original work in mathematics during the period between the 5th century BCE and the 5th century CE? Was this the "dark age" for Indian mathematics, or did it simply experience a different trajectory? The text asserts that to answer these questions, it's crucial to examine the works from this period, particularly Jain Agama texts, which serve as a crucial link in the chain of Indian mathematical history.

Jain Acharyas and Mathematics:

The article emphasizes that mathematics has consistently been a subject of contemplation for Jain monks. "Sankhyana" (numbers and astrology) is an integral part of their spiritual pursuit, holding a prominent place among the 14 essential limbs of knowledge and being considered the most important among the 72 sciences and arts. The entire Jain literary corpus is divided into four "Anuyogas" (categories), with Karananuyoga, also known as Ganitanuyoga, being dedicated to mathematics. Jain Acharyas utilized mathematical tools to explain the structure of the universe and the theory of karma. Therefore, understanding even the non-mathematical texts (like Dravyanuyoga) requires a mature understanding of mathematics.

The text quotes Mahaviracharya, a Jain mathematician, who stated the universal applicability of mathematics in all aspects of life, be it worldly, Vedic, or religious. This explains the pervasiveness of mathematical elements, both explicit and implicit, in Jain Agama texts. Jain Acharyas and scholars also composed independent mathematical and astronomical treatises for the convenience of their students, some of which are still extant, while others have been lost to time.

Key Jain Texts and their Mathematical Contributions:

The article details several significant Jain texts and their mathematical content:

• Surya-prajnapti, Chandra-prajnapti, and Jambu-dweepa Prajnapti (circa 500 BCE): These are considered authoritative texts of ancient Jain astronomy, written in Prakrit. Surya-prajnapti even discusses two values for Pi (π): 3 and 10.
• Bhadrabahu (318 BCE): The last Shruta-kevali (omniscient being) of Jainism, Bhadrabahu, is credited with two astronomical works: a commentary on Surya-prajnapti and the Bhadrabahu Samhita.
• Dwadashanga Literature (400-1000 BCE): Texts like Sthānāṅga, Praśnavyākaraṇaṅga, Samavāyaṅga, and Sūyagaḍaṅga discuss concepts like the nine planets, constellations, zodiac signs, solstices, eclipses, etc.
• Sthānāṅga Sūtra (325 BCE): This text enumerates ten branches of mathematics in a verse:
  • Parikarma (fundamental operations)
  • Vyavahāra (various applications)
  • Rajjū (unit of measurement, geometry)
  • Rāśi (sets, rule of three)
  • Kalā-savarna (calculations with fractions)
  • Yāvat-tāvat (simple equations)
  • Varga (quadratic equations)
  • Ghana (cubic equations)
  • Varga-varga (biquadratic equations)
  • Vikalpa (combinations and permutations) Mahaviracharya's Ganita-sara-sangraha elaborates on these ten subjects.
• Uttarādhyayana and Bhagavatī Sūtra (300 BCE): These texts discuss topics like squares, cubes, combinations, and permutations. Bhagavatī Sūtra uses terms like "samyoga" (combination) for one, two, and three elements, indicating knowledge of combinatorial principles. It also describes the minimum number of points required to form geometric shapes like lines, squares, cubes, rectangles, triangles, circles, and spheres.
• Umasvati (150 BCE): The author of the comprehensive Tattvārthādhigama-Sūtrabhāṣya, Umasvati's work contains mathematical principles like place value, fraction reduction, multiplication and division methods, and formulas related to the area, circumference, chord, and height of circles. It suggests the availability of mathematical texts during his time. He is also credited with the astronomical work Jambudweepa Samāsa.
• Anuyoga Dwāra Sūtra (1st Century BCE): This text mentions "prathama varga" (first square), "dvitiya varga" (second square), etc., indicating knowledge of exponents by the 1st century BCE. It also discusses a unique system of naming places with "koti-koti," demonstrating understanding of the decimal system for large numbers.
• Kundakunda (52 BCE - 44 CE): A prolific Jain scholar, Kundakunda's works like Samaya-sara, Pravachana-sara, and Panchastikaya Sara use concepts like "sankhya" (number), "asankhya" (innumerable), and "ananta" (infinite). Panchastikaya Sara defines time and "nimisha" (a fleeting moment) and uses the term "ananta" with various classifications. Pravachana-sara contains the fundamental principles of probability, a concept often attributed to later Western mathematicians.
• Yativrisha (circa 176 CE): This Jain philosopher and mathematician composed the significant work Tiloyapanṇatti (Triloka-Prajnapti) in Prakrit, which describes the structure of the universe and spiritual principles. It also contains numerous mathematical principles, defining vast numbers and units for time and space. The smallest unit of time is "samaya," and the largest unit of "sankyata" (countable) is "achalātma." It establishes relationships between different types of countable, uncountable, and infinite quantities.
• Kashaya Pahuda (406 CE) and Shatkhandagama (1st or 2nd Century CE): These works are rich in mathematical content. Virasenacharya's commentary Dhavala on Kashaya Pahuda contains references and quotations from many ancient Prakrit mathematical texts, including Aggayanīya, Ditthivāda, Parikamma, Loyavinicchaya, Loka Vibhāga, and Logāiṇi. The text mentions Pariyamma-Sūtra, a prose text with mathematical verses. Yativrisha also mentions a text called Karana-Sūtra, and there is mention of Karana Bhāvanā.
• Virasenacharya's Siddha Handa Paddhati: His commentary indicates the existence of a text on geometry.
• Mahaviracharya (850 CE): His Ganita-sara-sangraha is a comprehensive work that elaborates on the ten mathematical subjects mentioned in the Sthānāṅga Sūtra.
• Nemichandra Siddhanta Chakravarti (10th Century CE): His Triloka-sara mentions Vrihattara, an ancient Jain text on series, which was available at the time but is now lost.
• Bhaskara I (7th Century CE): His commentary on Aryabhata's work contains five Prakrit verses related to arithmetic, possibly extracted from a Jain work. Bhaskara I's writings also suggest that mathematicians like Maskari, Purna, Mridgala, and Putana, who each wrote separate works on the eight "vyavaharas," might have been Jain scholars.

Unavailability of Texts and the Need for Research:

The article notes that many ancient Jain mathematical texts, such as Loka-vibhāga, Ganita Tilaka, Maha-dhawala, and others, contain references to lost works. The author laments that while information about these texts exists, they are not currently available.

Conclusion:

The text concludes by stating that from the 5th century BCE to the 5th century CE, major Jain Agama texts presented important principles from various branches of mathematics, utilizing numerous formulas. This indicates that the knowledge of these mathematical principles was prevalent in India much earlier. Furthermore, it suggests that independent mathematical texts were being written in India during this period, many of which are now lost. The "dark age" of Indian mathematics still saw the development of mathematics, with Jain Acharyas making invaluable contributions. The author stresses the need for researchers to focus on this area, explore manuscripts in various libraries, and conduct scientific study of these texts to accurately evaluate the mathematical achievements of Jain Acharyas and present India's true contribution to the world.

In essence, the article argues that the so-called "dark age" of Indian mathematics was not a period of decline but rather one where Jain scholars actively preserved and advanced mathematical knowledge, laying crucial groundwork for future developments and contributing significantly to India's rich mathematical heritage.