Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume, by Hermann Schwarz

TL;DR

This paper provides an English translation of Schwarz's 1884 proof demonstrating that among bodies of equal volume, the sphere has the smallest surface area, a fundamental result in geometric analysis.

Contribution

It offers the first accessible English translation of Schwarz's classical proof, preserving original notations and grammatical structure for historical and educational purposes.

Findings

Confirmed the minimal surface area property of the sphere for given volume

Preserved original mathematical notation and language for historical accuracy

Facilitated understanding of Schwarz's original proof in modern context

Abstract

We present English translation of the classical article of Hermann Amadeus Schwarz (1843--1921) "Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume" which was published in 1884, in Proceedings of the K"onigliche Gesellschaft der Wissenschaften and the Georg-Augusts-Universit"at, G"ottingen. We preserved the author notations throughout the text and tried to follow his grammar construction of the sentences.