In this essay, I argue for the following theses. First, Kummer's concept of 'ideal prime factors of a complex number' drew inspiration from Poncelet's introduction of ideal elements in geometry as well as from the reconceptualization that Michel Chasles put forward for them in 1837. In other words, the idea of ideality in number theory derives from the introduction of ideal elements in the new geometry. This is where the term 'ideal' comes from. Secondly, the introduction of ideal elements in geometry as well as the subsequent reconceptualization of what was in play with these elements were linked to philosophical reflections on generality that practitioners of geometry in France developed in the first half of the 19th century to devise a new approach to geometry, which would eventually become projective geometry. These philosophical reflections circulated as such and played a key part in the advancement of other domains, including in Kummer's major innovation in the context of number theory.