Vartaman Sandarbh Me Jain Ganit Ki Upadeyata

Here is a comprehensive summary in English of the Jain text "Vartaman Sandarbh me Jain Ganit ki Upadeyata" by Parmeshwar Jha:

This article, "The Relevance of Jain Mathematics in the Present Context" by Dr. Parmeshwar Jha, argues for the profound and enduring significance of mathematics as developed and preserved within Jain tradition. The author begins by emphasizing the universal importance of mathematics, citing Vedic texts that place it at the pinnacle of knowledge. Jain mathematician Mahaviracharya is quoted, highlighting mathematics' utility in worldly, Vedic, and spiritual pursuits, stating that all existence in the three realms is inseparable from numbers.

The text asserts that the development of mathematical knowledge was a shared endeavor among ancient civilizations like Babylonia, Egypt, Greece, and India. However, it particularly emphasizes the immense contribution of Jain acharyas (religious scholars) to the enrichment and preservation of Indian culture, with mathematics being a subject of their deep contemplation. Jain acharyas used mathematical principles to explain the structure of the universe and the theory of karma, resulting in a wealth of mathematical content, both explicit and implicit, within Jain scriptural texts. The article contends that Jain acharyas played a crucial role in filling gaps in the history of Indian mathematics, especially during the so-called "dark age" of Indian mathematics (500 BCE - 500 CE). It claims that many principles attributed to Western mathematicians were documented centuries earlier by Jain scholars. The essay's primary aim is to analyze the relevance and utility of these mathematical principles in the current context.

The article then proceeds to provide an overview of the mathematical achievements of Jain acharyas, referencing key texts:

• Sthananga Sutra (324 BCE): This foundational text outlines ten branches of mathematics: Parikarma (fundamental operations), Vyavahara (applied mathematics), Raju (geometry), Rashi (sets, rule of three), Kalasa-varna (fractions), Yavat-tavat (simple equations), Varga (quadratic equations), Ghana (cubic equations), Varga-varga (biquadratic equations), and Vikalpa (permutations and combinations). This demonstrates that Jain scholars had a comprehensive understanding of these subjects well before the Common Era.

• The Decimal Place-Value System: The article highlights the invention of the decimal place-value system, including the concept of zero, as a revolutionary Indian contribution. It is supported by epigraphic evidence suggesting its invention in or before the first century BCE. Jain scriptures confirm its use, particularly for measuring vast quantities of space and time. Texts like Anuyogadvara Sutra (1st century BCE) and Tiloyapannatti by Yativrishabha (contemporary of Aryabhata I) define enormous units of time and space, showcasing the necessity and application of this system for Jain cosmological calculations. The article notes that this system was adopted in Europe much later, in the 12th and 13th centuries.

• Fractions: Jain acharyas are credited with significant contributions to the development of fractions. Their ancient texts contain methods for writing and performing operations on fractions, along with related formulas. Mahaviracharya is specifically mentioned for comprehensively detailing all aspects of fractions.

• Exponents and Logarithms: The article points to Uttaradhyayana Sutra (300 BCE) and Anuyogadvara Sutra as evidence that Jain acharyas understood the rules of exponents by the first century BCE. Anuyogadvara Sutra uses terms like "first square," "second square," etc., and demonstrates the application of powers in population calculations. Dhavala, a commentary on Panchkhandagama, extensively uses and discusses various rules of exponents (e.g., x^m * xⁿ = x^m+n, x^m / xⁿ = x^m-n, (x^m)ⁿ = x^mn), which are still in use today. The article also suggests that the principles of logarithms were used in Dhavala approximately seven centuries before their attributed discovery by Napier and Burgi.

• Set Theory (Rashi Siddhanta): The article states that the principles of set theory, now considered fundamental across scientific disciplines, were discussed in Jain texts like Shashkhandagama, Tiloyapannatti, Dhavala, and Trilokasara in ancient times. These texts describe concepts, classifications, examples, and operations related to sets, including finite, infinite, empty, and singleton sets, predating Georg Cantor.

• Series: Jain acharyas made unique contributions to the study of series, with detailed discussions found in Tiloyapannatti and Trilokasara. Nemichandra Siddhanta Chakravarti (10th century CE) extensively wrote about series, discussing 14 types and their formulas. The mention of a separate Jain text on series, Brihatdhara Parikarma, indicates the advanced stage of this field within Jain tradition.

• Geometry and Mensuration: Jain texts are rich in geometric discussions and mensuration. Surya Prajnapti discusses various geometric shapes. Bhagavati Sutra and Anuyogadvara Sutra contain formulas for solid shapes and plane figures. The commentary on Tattvarthadhigama Sutra by Umaswami (150 BCE) includes several mensuration formulas for circles, segments, and other figures. Tiloyapannatti and Trilokasara also contain numerous geometric formulas still in use today.

• The Value of Pi (π): The ratio of a circle's circumference to its diameter has been discussed in Jain scriptures since ancient times. Jain tradition accepted both an approximate value of 3 and a more precise value of √10. Surya Prajnapti discusses both. Dhavala provides the value of π as 355/113, which is very close to the modern value and is known as the Chinese value, but the article suggests Jain acharyas may have used it prior to its use in China.

• Probability: While Western mathematicians like Galileo, Fermat, and Pascal are credited with the discovery of probability, the article asserts that the foundational principles were laid by Jain acharyas Kundakunda (52 BCE – 44 CE) and Samantabhadra (2nd century CE) through the concept of Syadvada and its Sapta-bhangas (seven-valued logic).

• Key Jain Mathematicians: The article highlights two prominent Jain mathematicians:

  • Shridharacharya (8th century CE): He authored independent works like Patiganita and Trishatika, establishing a tradition of mathematical writing. He developed a scientific method for solving quadratic equations that is still in use today, as quoted by Bhaskaracharya.
  • Mahaviracharya (850 CE): His significant work, Ganita-Sara-Sangraha, presented mathematical rules with examples in a textbook style. He was the first to state that the square root of a negative number is not possible, thereby paving the way for the concept of imaginary numbers, a concept explored in Europe by Gauss in the 19th century. He also formulated general formulas for the sum of geometric series and permutations and combinations, which were later discovered in Europe. He also established formulas for the area of various geometric shapes and his formulas for cyclic quadrilaterals are modern in their form.

The article concludes by reiterating that Jain acharyas made significant contributions to the advancement of mathematics, and their fundamental principles remain widely applicable. Therefore, Jain mathematics holds the same relevance today as it did in ancient times. The author emphasizes the need to study and contemplate the neglected manuscripts found in various collections to accurately evaluate the contributions of Jain acharyas to mathematics.