The Clifford, Grace, and de Longchamps chains: historical developments and inversive properties

This thesis investigates the historical origins and developments of specific configurations of points, lines, and circles within the framework of elementary geometry. The demonstrations originally presented by the authors examined in this thesis have been carefully reconstructed and analysed in detail, with particular attention to their methodological aspects. The aim of this historical reconstruction is to clarify how different approaches to geometric reasoning evolved over time and to highlight the methodological choices that shaped these developments. The primary focus is on the chain of theorems introduced by W. K. Clifford in 1871, along with its generalisations to higher-dimensional spaces. A significant advancement in this field was made by J. H. Grace in 1898, who not only extended Clifford’s chain but also applied circular inversion to it, giving the entire construction a more symmetric form. The thesis also explores the foundational ideas of recursive geometry introduced in 1877 by G. de Longchamps, who constructed a number of chains of theorems that depended on each other in an iterative manner, starting from objects related to geometry of the triangle. These series of theorems extend beyond the traditional boundaries of elementary geometry, as many have found significant applications in mathematical research during the second half of the 20th century. For instance, in 1972, M. S. Longuet-Higgins showed a correspondence between Clifford’s and Grace’s chains of configurations and n-dimensional polytopes. In 1988, H. L. Dorwart highlighted a strong connection between configurations of points and circles and block designs. Once again, in 1976, Longuet-Higgins further analysed the inversive properties of both Clifford’s chain and de Longchamps’ chain related to the circumcentre, highlighting their analogies and structural connections. Throughout the topic discussed, the thesis sought to underscore the surprising and unexpected connections between various theories, including elementary geometry, synthetic geometry, the theory of polytopes, and blockdesign theory. In the concluding reflections, the educational potential of the addressed geometric themes is also considered.