The Anatomy of Infinitesimals: A Unified Theory for Modern Analysis

This paper explores the historical development and modern rigorous formalization of infinitesimals, proposing a unified theoretical framework for their application in modern analysis. Historically a cornerstone of early calculus, infinitesimals faced rigorous scrutiny and eventual displacement by the epsilon-delta definitions of continuity and limits due to a lack of formal foundations. The advent of Abraham Robinson's Non-Standard Analysis (NSA) in the 20th century provided a consistent and robust mathematical structure for these intuitively powerful quantities. This work details how NSA, through the construction of hyperreal numbers and the Transfer Principle, offers a coherent framework to re-integrate infinitesimals into foundational analysis. We demonstrate how key concepts such as continuity, differentiation, and integration can be elegantly reformulated using infinitesimals, often providing a more intuitive understanding without sacrificing rigor. The paper systematically reviews the transition from heuristic use to modern formalization, outlines the core principles of NSA, and presents a series of analytical re-interpretations. The discussion critically assesses the pedagogical and theoretical advantages of this unified approach, its challenges, and its implications for mathematical research and education. We elaborate on how the equivalence to standard analysis is established for each reformulated concept, reinforcing the rigor of the non-standard approach. Finally, a comprehensive conclusion summarizes the findings and reiterates the significance of this unified theory.