\documentclass{article}
\usepackage{amsmath, amssymb, geometry, xcolor}
\geometry{a4paper, margin=1in}
\title{Understanding \textbf{Varga} and \textbf{Nija} in Sanskrit Mathematics}
\author{Perplexity AI}
\date{\today}
\begin{document}
\maketitle
\section*{Introduction}
The terms \textbf{varga} (वर्ग) and \textbf{nija} (निज) are fundamental to interpreting classical Indian mathematical texts like Aryabhata’s \textit{Aryabhatiya}. Their meanings and contextual usage reveal critical insights into ancient mathematical methodologies.
\section{\textbf{Varga} (वर्ग): The Concept of "Square"}
\subsection{Literal Meaning}
\textbf{Varga} translates directly to "square" or "group" in Sanskrit. In mathematics, it specifically denotes:
\begin{itemize}
\item The \textbf{square} of a number (e.g., \textit{pañcavarga} = \(5^2 = 25\))
\item A \textbf{class} or \textbf{category} of numbers (e.g., odd/even \textit{varga})
\end{itemize}
\subsection{Mathematical Applications}
Aryabhata uses \textit{varga} extensively:
\begin{itemize}
\item \textbf{Square of a number}:
\textit{Yavad vargād vargaśodhanaṃ} ("Subtract the square from the square as much as possible") refers to algebraic operations involving squares.
\item \textbf{Area of a square}:
\textit{Vargaṃ caturasraṃ} ("A square is quadrilateral") implies \textit{varga} as a geometric square.
\item \textbf{Astronomical cycles}:
\textit{Varga} also denotes divisions of planetary orbital periods.
\end{itemize}
\subsection{Example from \textit{Aryabhatiya} (Verse 2.3)}
\begin{quote}
\textit{Vargādvargaṃ śuddhiḥ} \\
("The purification [result] from the square of squares")
\end{quote}
This likely refers to iterative squaring in astronomical calculations.
\section{\textbf{Nija} (निज): The Nuanced Meaning of "Own"}
\subsection{Literal Meaning}
\textbf{Nija} means "own," "inherent," or "intrinsic." It emphasizes a \textbf{self-contained property} of an object.
\subsection{Mathematical Context in \textit{Aryabhatiya}}
In the sphere volume formula:
\begin{quote}
\textit{तत्र निजमूले हतं घनगोलः फलं त्रिघ्नविशेषम्} \\
(\textit{tatra nijamūle hataṃ ghanagolaḥ phalaṃ trighnaviśeṣam})
\end{quote}
\subsection{Interpretation Challenges}
\begin{itemize}
\item Traditional translation: "multiplied by its own square root" \\
\( V = \pi r^2 \times \sqrt{\pi r^2} \approx 1.77\pi r^3 \)
\item Problem: Overestimates true volume (\( \frac{4}{3}\pi r^3 \)) by 33\%.
\end{itemize}
\subsection{Reinterpreting \textbf{Nija} as a Geometric Ratio}
Scholars argue \textit{nijamūle} may instead mean \textbf{"inherent base ratio"}:
\begin{align*}
\text{If } \textit{nijamūle} &= \frac{4}{3} \times r \text{ (radius):} \\
V &= \pi r^2 \times \frac{4}{3}r = \frac{4}{3}\pi r^3
\end{align*}
\section*{Conclusion}
The term \textbf{nija} exemplifies how Sanskrit mathematical texts encode complex ideas through compact phrasing. Aryabhata’s formula, when decoded as \( \frac{4}{3}\pi r^3 \), reveals a sophisticated understanding of solid geometry that parallels Archimedes’ work.
\end{document}