The next morning, the narrator and Chris bid farewell to the DeWeeses, who wanted them to stay longer. The narrator, however, is anxious to get going and to continue with his thoughts. He wants to begin a new Chautauqua by talking about a person he has studied extensively for this talk, an “eminent scientific man” named Jules Henri Poincaré. Poincaré was a philosopher, mathematician, astronomer and physicist. The narrator mentions that as Phaedrus was a poor scholar, he never ventured to see if others had traced the same inroads he had into Quality. Poincaré seems to come closest. He was also interested in testing the limits scientific reasoning. His basis was with Euclidian geometry, and as various mathematical axioms were discovered during his lifetime, the revelations completely changed the purportedly rational system of mathematics. Poincaré dealt with the apparent predicament, as well as what the predicament indicated, that—like Phaedrus had discovered—there can actually be an infinite number of hypotheses, by proposing that some facts were better than others based on attributes. This was in fact similar to Phaedrus’s ideas concerning Quality.

The narrator and Chris stop in Missoula, and then continue westward.