Newton’s System or World and the non-euclidean geometries in the XIX century: possible relationships?

In the present work we aimed to investigate how Isaac Newton’s theory of gravitation, present in Book III of Principia – System of World – was approached during the 19th century, when non-Euclidean geometries were being used in the study of mechanics on surfaces with curvature other than zero. For this, we carried out a study of the main aspects of Book III and the geometry used by Newton; we present the principles of non-euclidean geometry under discussion in the 19th century and we carried out a bibliographic search in the Royal Society’s database in search of works that related these two themes. As a result, we find only one work during the 19th century in which some Newtonian concepts of motion are discussed under Riemannian ideas of metric, in which the curvature of the adopted space does not influence the equations of motion. We therefore conclude that there was no differentiation, in terms of the movement of celestial bodies, in Newton’s absolute space even when non-Euclidean geometry brought new interpretations and possibilities.

Keywords:
19th century; World System; Non-Euclidean geometries