Prarambhik Jain Grantho Me Bijganit

Here is a comprehensive summary in English of the provided Jain text, "Prarambhik Jain Grantho me Bijganit" by Mukutbiharilal Agarwal:

This text, "Algebra in Early Jain Texts" by Mukutbiharilal Agarwal, explores the significant contributions of Jain scriptures to the field of mathematics, specifically algebra. The author highlights how early Jain texts, dating back to around 300 BCE, already contained sophisticated concepts and terminology related to algebra.

Key Discoveries and Concepts:

• Unknown Variables: The "Sthananga Sutra" (circa 300 BCE) is cited for its use of the term "Yavat-tavat" to represent unknown quantities, which is analogous to modern algebraic variables.
• Powers and Exponents: The "Uttaradhyayana Sutra" (circa 300 BCE) is noted for providing ancient Hindu terms for powers of a variable. These include:
  • "Varga" for the second power (a²).
  • "Dhan" for the third power (a³).
  • "Varga-varga" for the fourth power (a⁴, meaning "square of a square").
  • "Ghana-varga" for the sixth power (a⁶, meaning "cube of a square").
  • "Ghana-varga-varga" for the twelfth power (a¹², meaning "cube of a square of a square"). The text also indicates the understanding of exponent rules like (aᵐ)ⁿ = aᵐⁿ and aᵐ × aⁿ = aᵐ⁺ⁿ. Later Jain texts expanded on this, introducing terms for higher odd powers like "Varga Ghana Ghati" for the fifth power (a⁵) and "Varga-varga Ghana Ghati" for the seventh power (a⁷).
• Roots: The "Anuyogadvara Sutra" (pre-Christian era) introduced specific terms for higher powers, including integer and fractional powers. It defined the "nth square" of a variable 'a' as aⁿ and the "nth square root" of 'a' as a¹/ⁿ.
• Rules of Signs: The "Ganitasarasangraha" elaborates on the rules of signs for addition, subtraction, and multiplication. It clearly states that:
  • Adding a positive and negative number results in their difference.
  • Adding two negative numbers results in a negative sum.
  • Adding two positive numbers results in a positive sum.
  • Subtracting a positive number is equivalent to adding a negative number, and vice versa.
  • Multiplying two positive or two negative numbers results in a positive number.
  • Multiplying a positive and a negative number results in a negative number.
  • Division of two positive or two negative numbers yields a positive result, while division of a positive and a negative number results in a negative number.
  • The square of both a positive and a negative number is positive. The square root of a positive number can be positive or negative, but a negative number is considered "avarga" (non-square) and thus has no real square root.
• Types of Equations: Jain texts categorize equations into four types:
  1. Ekavarna Samikarana (Single-Variable Equations): These are linear equations, also called "Yavat-tavat" equations. The text explains the "Rule of False Position" and provides examples of solving them, as demonstrated by Aryabhata I and Acharya Mahavira.
  2. Dwighatiya Samikarana (Quadratic Equations): These are referred to as "Varga Samikarana" (square equations).
  3. Anekavarna Samikarana (Multi-Variable Equations): Equations involving multiple variables.
  4. Bhavita Samikarana: Equations involving the product of two variables.
• Solving Linear Equations: The text details methods for solving linear equations, including those with multiple variables. An example is given of finding the individual prices of two fruits based on their combined costs in different scenarios.
• Solving Systems of Linear Equations: Acharya Mahavira's "Ganitasarasangraha" presents methods for solving systems of linear equations. An example involves determining the amount of money each owner of fighting cocks had, based on the profits gained in different betting situations.
• Problems with Multiple Unknowns: The text discusses problems involving several unknown quantities, such as those faced by four merchants calculating their shares of a joint venture. It outlines a method to determine the total value of goods and then the individual shares.
• "Vichitra Kutti Kar Vidhi" (Peculiar Indeterminate Equation Method): Acharya Mahavira introduced this method for solving problems where individuals request sums of money from each other, resulting in their wealth becoming multiples of others' remaining amounts. The text provides a detailed explanation of how to solve these complex problems.
• Interest-Related Problems: Jain texts also addressed problems involving simple interest, requiring the calculation of different interest amounts based on principal, time periods, and interest rates, often leading to systems of linear equations.
• Quadratic Equations (Varga Samikarana): The text traces the origin of quadratic equations to Vedic compositions and their geometrical solutions in early Jain texts (500-300 BCE) and Umaswati's "Tattvartha Adhigama Sutra" (150 CE). The "Bakshali Manuscript" (200 CE) also mentions quadratic equations. "Ganitasarasangraha" provides examples, including those involving square roots. The text explicitly states that Mahavira was aware of the two roots of quadratic equations and provided methods for solving them, though some examples in his work focus on finding only one (typically the positive) root.
• Higher-Degree Equations: Mahavira discussed simple higher-degree equations related to geometric progressions, presenting formulas for finding the common ratio when the first term, number of terms, and sum are known.
• "Vishama Sankraman Niyam" (Rule of Odd Transition): This refers to the Hindu method of solving specific types of simultaneous quadratic equations, which Mahavira termed "Vishama Sankraman." The text provides the formulas for solving equations like x² - y² = m and x + y = p.
• Indeterminate Equations (Ekaghat Anirnit Samikaran): The study of indeterminate equations began with Aryabhata and was further developed by subsequent Indian mathematicians like Brahmagupta and Mahavira. These equations were known by names such as "Kuttak," "Kuttakar," and "Kuttikar." The text details Aryabhata's method for solving linear indeterminate equations of the form ax + c = by, which involves a systematic process of division and calculation to find the values of x and y. Mahavira's method for solving a related form (ax = by + c) is also explained, utilizing a process similar to finding the Highest Common Factor (HCF).

Conclusion:

The author concludes that Jain literature is rich with algebraic concepts, demonstrating the depth and breadth of Jain scholars' contributions to mathematics. Their efforts not only highlight the ancient roots of algebra but also its modern relevance. The text emphasizes that Jain scholars provided a comprehensive understanding of algebra, encompassing its form, theoretical principles, and practical applications. The summary also includes a quote from the Encyclopedia Britannica acknowledging the Indian origin of the numeral system and its transmission through Arab scholars.

In essence, the book "Prarambhik Jain Grantho me Bijganit" by Mukutbiharilal Agarwal is a testament to the advanced mathematical knowledge present in early Jain scriptures, showcasing their pioneering work in algebra, including the treatment of variables, exponents, roots, equations, and complex problem-solving techniques.