Chasles and the Projective Geometry The Birth of a Global Foundational Programme for Mathematics, Mechanics and Philosophy

This monograph meticulously examines the contributions of French mathematician Michel Chasles to 19^th-century geometry. Through an in-depth analysis of Chasles' extensive body of work, the author examines six pivotal arguments which collectively reshape the foundations of geometry. Chasles introduces a novel form of polarity, termed "parabolic," to the graphic context, so expressing the metric properties by means of this specific polarity—a foundational argument. Beyond the celebrated "Chasles theorem," he extends his analysis to the movement of a rigid body, employing concepts derived from projective geometry. This approach is consistently applied across diverse domains. Chasles employs the same methodology to analyze systems of forces. The fourth argument examined by the author concerns the principle of virtual velocities, which can also be addressed through a geometric analysis. In the fifth chapter, Chasles' philosophy of duality is explained. It is grounded on theduality principles of projective geometry. Finally, the author presents Chasles’ synthetic solution for the intricate problem of ellipsoid attraction—the sixth and concluding chapter. Throughout these explorations, Chasles engages in a dynamic scientific dialogue with leading physicists and mathematicians of his era, revealing diverse perspectives and nuances inherent in these discussions.

Introduction.- Chasles’ foundational programme for geometry.- Displacement of a rigid body.- Chasles and the systems of forces.- The principle of virtual velocities.- Chasles’ philosophy of duality.- Chasles and the ellipsoid attraction.- Conclusion.

Paolo Bussotti is Associate Professor in History of Science and Techniques at the University of Udine (Italy). His research areas are history of science and mathematics, in particular history of geometry and number theory between the 17th and the 19th centuries, and history of physics and astronomy in the 17th century. He is the author of more than 150 scientific publications, among which a monograph on the history of the method of infinite descent (number theory), From Fermat to Gauss. Indefinite descent and methods of reduction in Number Theory (2006), one on Leibniz’s planetary theory, The complex itinerary of Leibniz’s planetary theory (Birkhäuser, 2015) and one, written jointly with Prof. Brunello Lotti, titled Cosmology in the Early Modern Age. A Web of Ideas (Springer, 2022). He is the co-author (jointly with prof. R. Pisano) of many papers on the Geneva Edition of Newton’s Principia published in important journals dedicated to the history of science. Furthermore, he is reviewer for leading scientific journals and well-known reviewing services such as Zentralblatt für Mathematik.