Like the square in which he dwells, Sloper adheres to the systematic principles of Euclidean geometry. When Mrs. Almond, testing his resolve, asks whether he plans to yield to Catherine’s importunities, Sloper icily replies, “Shall a geometrical proposition relent? I am not so superficial.” Hearing his sister’s clever rejoinder, “Doesn’t geometry treat of surfaces?” Sloper only reaffirms the tenets of his mathematical faith: “Yes; but it treats of them profoundly. Catherine and her young man are my surfaces; I have taken their measure” (p. 164).

Sloper’s confidence in the incontrovertibility of Euclid’s time-honored axioms reveals and fortifies his low opinion of women. Like the majority of nineteenth-century educators, Sloper believes that “geometry [is] impossible for women to understand” due to the latter’s “insufficient abstract reasoning power” (Cohen, A Calculating People, p. 8). Indeed, the doctor finds his “idea of the beauty of reason ... on the whole meagrely gratified by what he observed in his female patients” (p. 70). To this axiom, Catherine is no exception. From her father’s pitiless perspective, “she is about as intelligent as the bundle of shawls” (p. 178). In Sloper’s binary world, boys are “sent to college or placed in counting-rooms,” and girls are “married very punctually” (p. 78). In such a world, the gendered discourse of geometry admits no exceptions.

Yet when Sloper asks, “Shall a geometrical proposition relent?” (p. 164), James invites us to take a critical, rather than merely rhetorical, view of the matter. Indeed, by the late nineteenth century, geometrical propositions were relenting. In the mid-1870s, mathematicians in England, where James lived while writing Washington Square, were learning of the first fundamental challenge to the gold standard of mathematical thought in nearly two thousand years. Based on the work of Carl Friedrich Gauss and Georg Rie mann, both German geometers, the Russian mathematician Nikolai Ivanovich Lobachevsky, a Hungarian named Janós Bolyai, and their English-language spokesmen—Hermann von Helmholtz, William Clifford and, ultimately, Albert Einstein himself—Euclidean geometry underwent a major revolution. At the base of this transformation was a startling mathematical premise: that Euclid had not exhausted the possibilities of spatial knowledge, and that other spaces were equally possible—including curved spaces. These so-called “hyperbolic spaces,” though difficult to imagine, yielded a radically new insight: that, in the words of Leonard Mlodinow, “Euclidean form is approachable but not attainable, like the speed of light, or your ideal weight” (Euclid’s Window, p. 121 ). In these new, non-Euclidean spaces, brave new worlds were possible. The sum of a triangle’s angles, for example, was always less or more than 180 degrees; similar triangles did not exist, and parallel lines intersected (Mlodinow, p. 121). The “curved space revolution” challenged the most fundamental tenets of nineteenth-century thought, among them, the concepts of absolute space, absolute time, and Immanuel Kant’s moral imperative (Mlodinow, p. 95). As the British geometer William Clifford declared in 1875, “the idea of the Universe, the Macrocosm, the All, as subject of human knowledge, and therefore of human interest, has fallen to pieces” (quoted in Richards, Mathematical Visions, p. 112). A new spirit of empiricism had raised doubts about Euclid’s formerly absolute truths.