Bijganit Purvarddh - Book Summary

Here's a comprehensive summary of the Jain text "Bijganit Purvarddh" by Bapudev Shastri, based on the provided pages:

Title: Bijganit Purvarddh (Algebra - First Part) Author: Pandita Bapu Deva Shastri (Professor of Mathematics and Astronomy at the Sanskrit College, Benares) Publisher: Medical Hall Press, Benares Edition: Second Edition

Overview:

This book, "Bijganit Purvarddh," is the first part of a treatise on Algebra written in Hindi by the esteemed scholar Pandita Bapu Deva Shastri. The preface highlights Shastri's extensive knowledge and academic affiliations, including honorary memberships in prestigious societies and a fellowship at Calcutta University. The book aims to introduce European Algebra to a Hindi-speaking audience, compiled from various European and Native authors, occasionally quoting Bhaskaracharya.

Key Themes and Content:

The book begins with a preface that meticulously outlines the author's perspective on the history and classification of mathematics, emphasizing India's significant contributions.

1. Classification of Mathematics: Shastri categorizes the science of computation into three branches:

Vyakta-Ganita (Arithmetic): Deals with numbers and provides specific, case-dependent results. He notes its Indian origin and its transmission to other cultures.
Rekha-Ganita (Geometry/Geometry of lines): Deals with lines and provides general conclusions. He refers to his treatise on Geometry (Kshettramiti) for further details and credits Pandita Jagannatha for coining the term Rekha-Ganita, though he prefers Kshettramiti.
Avyakta-Ganita (Algebra): Deals with the relations of abstract quantities using letters and symbols. He argues for its Indian origin, citing its presence in ancient texts like the Surya Siddhanta. He provides a detailed algebraic derivation of a rule from the Surya Siddhanta concerning the sine of the altitude of the sun.

2. Historical Perspective on Algebra: Shastri asserts that Algebra originated in India, not with the Arabs or Greeks. He argues that Arab mathematicians borrowed from other nations and that their algebra differs significantly from Diophantus, suggesting they, too, learned from India. He traces the spread of Algebra from India to Europe via the Arabs, noting its development in Europe with key figures like Lucas de Burgo and Stifel.

3. The Need for this Treatise: Shastri explains that the current European methods of Algebra are superior and can solve problems that Hindú methods cannot easily address. He undertook this work at the request of Mr. D. F. M'Leod, then Magistrate of Benares, to make European Algebra accessible in Hindi.

4. Structure of the First Part (Purvarddh): The first part of the book is divided into five chapters:

Chapter I: Definitions: Introduces fundamental algebraic terms and symbols, including the notation for known and unknown quantities, operations (addition, subtraction, multiplication, division), powers, roots, and the meaning of various symbols. It also explains the concepts of algebraic terms (monomials, binomials, etc.), like and unlike terms, and the idea of substitution. Numerous examples are provided to illustrate these definitions.
Chapter II: Simple Rules and Miscellaneous Topics: Covers basic operations like involution (powers) and evolution (roots), properties of prime quantities, and various miscellaneous useful techniques (prakirnaka) such as equating expressions, transposition, substitution, and multiplication of algebraic entities.
Chapter III: Greatest Common Measure and Least Common Multiple: Explains the methods for finding the G.C.M. and L.C.M. of algebraic terms.
Chapter IV: Algebraic Fractions: Discusses algebraic fractions, their types, operations (addition, subtraction, multiplication, division), and miscellaneous topics like solving for variables, negative and fractional exponents, and decimal fractions.
Chapter V: Equations: Covers the nature and classification of equations, simple equations with one or more unknown quantities, problems leading to simple equations, and methods like single and double position.

5. Key Concepts Introduced:

Symbols and Notation: The book systematically introduces the use of letters to represent unknown quantities (e.g., 'y', 'r', 'l' for unknowns, 'a', 'k', 'g' for knowns) and the symbols for operations (+, -, ×, ÷).
Basic Operations: Detailed explanations and numerous examples are provided for addition, subtraction, multiplication, and division of algebraic expressions.
Involution and Evolution: Concepts of powers (e.g., a², a³) and roots (square root, cube root) are explained.
Brackets/Parentheses: The use of different types of brackets to group terms and clarify the order of operations is illustrated.
Like and Unlike Terms: The distinction between terms with the same variables and powers and those with different ones is clarified, as it's crucial for addition and subtraction.
Equations: The concept of an equation as a statement of equality is introduced, along with the process of solving for unknown variables through algebraic manipulation (transposition, cancellation, etc.).
Fractions: Algebraic fractions and operations involving them are covered.
Properties of Numbers: Several fundamental properties of numbers and algebraic quantities are stated as axioms or proved, such as the commutative and associative properties of addition and multiplication, and distributive properties.
Problems: The book is rich with examples and exercises at the end of each section and chapter, designed to help students grasp the concepts.

Significance:

"Bijganit Purvarddh" represents a significant effort to introduce the sophisticated system of European Algebra into the Indian educational framework through the medium of Hindi. It demonstrates the author's commitment to bridging the gap between traditional Indian mathematics and the advancements in the West. The book's comprehensive nature, covering foundational concepts and providing numerous solved examples, makes it a valuable pedagogical tool for its time. The preface itself serves as an important historical document on the appreciation and integration of mathematical knowledge across cultures.