Bernhard Riemann — On the Hypotheses Which Lie at the Bases of Geometry

This book presents William Clifford’s English translation of Bernhard Riemann’s seminal text, accompanied by detailed mathematical, historical, and philosophical commentary. It explores Riemann’s revolutionary ideas on space, placing them within the broader framework developed by later thinkers such as Helmholtz, Ricci, Weyl, and Einstein. A historical introduction situates Riemann’s work in its 19th-century context, while subsequent chapters trace the evolution of the concept of space across philosophy, physics, and mathematics, and examine its enduring influence up to modern research.The second edition includes expanded mathematical commentary, a new section on metric geometry and machine learning, a systematic bibliography, and numerous updates throughout.Appealing to mathematicians, historians, and readers with an interest in physics or philosophy, this book provides a comprehensive perspective on Riemann’s groundbreaking contributions and their lasting impact.