As Coleridge aged, his prose no longer resonates with the animated descriptions of his rambling but, instead, dwells even more on working out his thoughts concerning social and political problems and the dynamics of human relations, as well as the intricacies of religious and philosophical thought. Throughout his indebtedness to the geometric idiom remains evident. However, Euclid’s hold on England’s spatial imagination was shifting. To many, such as Charles Dodgson, he was still important, but this orientation was not without its challenges, especially on the Continent, where theoretical mathematicians were exploring alternate ways of regarding one’s surroundings. The book closes by suggesting that Coleridge, though still basically indebted to Euclidean thought, did unwittingly and occasionally anticipate the challenges to Euclidean thought and the ensuing paradigm shifts.

This chapter introduces some of the basic tools of a cosmologist, including scale factor, redshift, and comoving distance. We start with the Hubble law, which is a key consequence of the expanding universe. Next, we cover the possible geometries of space (positively and negatively curved, and flat), and the associated Friedmann--Lemaître--Robertson--Walker metric that describes them. This leads us to define distance measures in cosmology, and introduce the Friedmann equation that describes the evolution of the universe given its contents. We end by discussing the role of critical density and curvature.

Roberto Bonola, in his celebrated history of non-Euclidean geometry, imported a distinction between elementary and advanced mathematics commonly drawn around 1900 back to the discoveries of the 1820s and 1830s. In so doing he fell into anachronism and misrepresented the subject, most obviously in his treatment of the work of Lobachevskii.