Mahaviracharya Krut Ganitasar Sangraha

This document is a detailed academic article, likely a chapter or essay, analyzing Mahaviracharya's Ganitasara Sangraha. The author, Dr. Alexander Volodraski, provides a comprehensive overview and evaluation of this significant medieval Indian mathematical text.

Here's a summary of the key points covered in the provided pages:

Introduction and Mahaviracharya's Context:

• Significance: Ganitasara Sangraha holds a special place in the development of medieval Indian mathematics, yet it hasn't received adequate attention in historical texts on science.
• Biographical Information: Very little is known about Mahaviracharya's life. He doesn't mention his birth, place, parents, or teachers.
• Dating: The text addresses an Indian ruler who reigned between 814-815 and 877-878 CE. Since Mahaviracharya wishes for the ruler's future success, the book is estimated to have been composed in the mid-9th century.
• Location: While difficult to pinpoint, most scholars believe Mahaviracharya resided in South India. This is supported by the existence of manuscripts where questions and answers are provided in Kannada (popular in South India during the medieval period) in addition to Sanskrit, and because Jainism, Mahaviracharya's faith, was more prevalent in South India.

Content and Structure of Ganitasara Sangraha:

• Scope: The text covers arithmetic and geometry thoroughly, along with many topics in algebra and number theory.
• Distinction: Unlike earlier scholars who integrated mathematics into astronomical works, Mahaviracharya dedicated this book solely to mathematics.
• Completeness: While earlier works offered rules, examples, and problems were scarce. Mahaviracharya provides rules, examples, and problems, though proofs are absent, which is common for medieval Indian, Arabic, and Western texts.
• Typical Structure of Indian Mathematical Texts: Indian mathematical treatises generally consist of three parts:
  1. Main Body: Contains rules and problem statements (often in verse).
  2. Special Section: Presents problem conditions and examples in tables or charts for ease of calculation (extensive use of symbols, diagrams for geometry).
  3. Commentary: Detailed solutions and examples by later scholars, with references.
• Mahaviracharya's Text: The Ganitasara Sangraha has nine chapters and 1131 verses, with 452 for rules and 679 for examples and problems. It is considered the largest among medieval Indian mathematical texts, due to the detailed rules and a significant portion dedicated to examples (3/5ths of the main text).
• Word-Symbols for Numbers: A unique system of word-symbols is presented for numbers (e.g., 0-space, 1-moon, 2-eyes, 3-fire). This system was primarily for expressing numbers and not for solving entire problems. Understanding it requires knowledge of ancient Indian literature, religion, and mythology.

Praise for Mathematics:

• Mahaviracharya begins his work by praising mathematics, stating its utility in all aspects of life, from worldly and religious matters to arts, sciences, and astronomy.

Key Mathematical Concepts and Contributions:

• Algebra (Avyakta Ganita):
  • Discusses rules for operations with negative numbers, similar to those found in Brahmagupta's work. These rules cover multiplication, division, addition, subtraction, and squaring of positive and negative numbers.
  • Notes that the square of a negative number is positive, but a negative number cannot have a real square root.
  • Compares the earliest mention of negative numbers to the Chinese text "Nine Chapters on the Mathematical Art."
• Linear Equations:
  • Highlights the use of linear equations in solving problems related to percentages, speed, and payments.
  • Explains the principle of "imaginary rule" (कल्पित नियम) for solving equations like $ax = b$, where the unknown is assumed to be 1.
  • Provides the solution formula for $ax=b$ as $x = \frac{b}{a}$.
  • Mentions similar rules found in later Arabic and European literature.
  • Discusses the solution of equations of the form $ax + b = px$ and $ax + c = bx + d$, referencing contributions from Aryabhata I, Brahmagupta, Sripathi, Bhaskara II, and Narayana.
  • Includes examples of solving problems using linear equations, such as problems involving fractions of a total quantity and problems with two unknowns (like finding the price of lemons and apples).
  • Presents the general form of simultaneous linear equations: $$ax + by = c$$ $$bx + ay = d$$ and their solutions: $$x = \frac{ac-bd}{a^2-b^2}$$ $$y = \frac{ad-bc}{a^2-b^2}$$
• Quadratic Equations:
  • While there isn't a dedicated chapter, several problems require solving quadratic equations.
  • An example involves finding the total number of camels based on fractions and the square root of the total.
  • Presents a rule for solving quadratic equations of the form $x^2 + cx + p = x$ or $(1-c)x - \sqrt{x} = p$. The rule involves manipulating coefficients and finding the square root of a sum.
  • Notes that Mahaviracharya was aware of the dual nature of square roots, which was also known to Brahmagupta (as indicated by commentator Prthudaka Swami).
  • An example demonstrates solving a problem involving the square root of a quantity and the quantity itself, leading to a quadratic equation.
• Higher-Order Equations:
  • Some problems require solving equations of higher order than quadratic.
  • An example involves finding the common ratio of a geometric progression, leading to a fifth-degree equation.
  • Discusses methods for finding roots of $n$-th degree equations, though explicit rules are not provided, suggesting a curated list of solutions.
  • Examples include problems related to geometric progressions that lead to equations solvable by specific rules.
• Series:
  • Arithmetic Progression (AP): Discusses rules for finding sums and terms of APs, referencing earlier mathematicians.
  • Geometric Progression (GP): States that rules for GPs are not found in the works of Aryabhata and Brahmagupta. Mahaviracharya is credited with the first rules for sums and terms of GPs, later presented by Shridhara and Bhaskara II.
• Permutations and Combinations:
  • Provides a formula for calculating combinations of mixed numbers and gives examples, such as forming garlands from different types of gems and pearls. Similar formulas are attributed to Shridhara and Narayana.
• Sum of Series:
  • Provides rules for the sum of series, including the sum of squares and cubes of natural numbers and the sum of squares and cubes of terms in an arithmetic progression. These methods were known in Babylonian, Egyptian, Chinese, and later Arabic and Western European mathematics.
• Number Theory (Diophantine Equations):
  • Discusses the algorithmic method for solving indeterminate equations of the first and second degree, building upon the work of Aryabhata I, Brahmagupta, and Bhaskara II.
  • Presents methods for solving specific problems, such as determining the gold content in two rods, which translates to an indeterminate linear equation.
  • Explains how to find rational solutions for equations like $x^2 + a^2 = z^2$ and $x^2 + y^2 = c^2$, mentioning Pythagorean triples.
  • Discusses solving systems of equations with three unknowns, including those with terms like $pxy$.
  • Provides methods for solving problems related to the area of a rectangle being equal to its perimeter, or its area being equal to its diagonal.
  • Mentions indeterminate equations of the third degree, such as finding the first term, number of terms, and common difference of an arithmetic progression.

Geometry:

• Measurement of Area: Mahaviracharya differentiates between approximate (practical) and exact measurements of area.
• Classification of Figures: Introduces a classification of shapes: three types of triangles, five types of quadrilaterals, and eight types of figures formed by curved lines (circle, semicircle, ellipse, etc.).
• Area Formulas:
  • Provides approximate formulas for triangles and quadrilaterals, similar to those known by Egyptian scholars.
  • Presents the exact area formula for a quadrilateral (Heron's formula).
  • Offers formulas for calculating diagonals of quadrilaterals, applicable to cyclic quadrilaterals.
  • Discusses formulas for circles, ellipses, and other curved shapes (like a shell, or hollow and raised circles), including methods for calculating perimeter and area.
  • Explains how to calculate the area of shapes resembling barley, drums, and vajra (diamond), often by approximating them as rectangles with average widths.
  • Details methods for finding the area of segments of a circle and related quantities like the chord and sagitta.
  • Provides formulas for the area of wheel rims and the space between four touching circles.

Volumes:

• Prisms and Spheres: The text describes methods for calculating the volume of prisms (base area multiplied by depth) and spheres (approximated and exact). The formulas are compared to those of Bhaskara and Shridhara.

Conclusion:

• Mahaviracharya's Place: Mahaviracharya's Ganitasara Sangraha is larger than earlier mathematical sections within astronomical works. Many of his rules and examples were foundational.
• Firsts: Mahaviracharya is credited with being the first to:
  • Provide rules for division.
  • Address specific cases of converting numbers into squares and cubes, and methods for cube roots of fractions.
  • Develop rules for percentage and gold purity.
  • Formulate solutions for systems of two linear equations with two unknowns, indeterminate linear equations, and indeterminate quadratic equations.
  • Develop fundamental methods for solving linear equations with multiple unknowns.
  • Provide methods for finding roots of fourth and eighth-degree equations.
  • Determine the first term and common difference of an arithmetic progression.
  • Find any term and the sum of a geometric progression.
  • Define the Least Common Multiple (LCM).
  • Explain the dual meaning of square roots for positive numbers and state that negative numbers do not have square roots.
• Influence: Many rules and problems from Ganitasara Sangraha were adopted by later mathematicians like Shridhara and Narayana, indicating a significant influence.

The article is a scholarly examination of Mahaviracharya's contributions, highlighting his originality and the comprehensive nature of his mathematical work.