A physician, Sloper has made a career of calculation: “He had passed his life in estimating people (it was part of the medical trade), and in nineteen cases out of twenty he was right” (p. 124). From his quantifying perspective, the lavish “red satin gown trimmed with gold fringe” (p. 75) that Catherine wears to her cousin’s engagement party makes her look as though she “had eighty thousand a year” (p. 83). When his embarrassed daughter “illogically” denies the charge, Sloper’s rejoinder gives brutal testimony to his own relentless logic of equivalency: “So long as you haven’t you shouldn’t look as if you had” (p. 83). Like other such ironic barbs, Sloper’s comments are “perfectly calculated” to dumbfound the deferential child. As a student boggled by the complexities of life’s long division, Catherine must make sense of the world’s messy remainders. To extract the least scrap of happiness from her father’s mordant remarks, she must perform a kind of linguistic surgery, cutting “her pleasure out of the piece, as it were” and ignoring the “portions left over, light remnants and snippets of irony, which she never knew what to do with, which seemed too delicate for her own use” (p. 83).

For his part, Sloper is fascinated by a different mathematical concept : that of a limit. According to the Oxford English Dictionary, “limit” is a “finite quantity to which the sum of a converging series progressively approximates, but to which it cannot become equal.” Although Catherine acknowledges that “her deepest desire was to please” her father, she has in fact “never succeeded beyond a certain point,” and “to go beyond the point in question seemed to her really something to live for” (p. 72). When Catherine’s Aunt Almond accuses the doctor of toying with his daughter’s filial devotion, Sloper coolly responds that he is merely interested in locating the limit of Catherine’s affection: “It is the point where the adoration stops that I find it interesting to fix” (p. 164).

Sloper meets his mathematical match in Morris Townsend—a good-looking scoundrel whose name itself postulates a limit—a town’s end. Like his would-be father-in-law, Morris reckons his relationship with Catherine in mathematical terms. “Dr. Sloper’s opposition,” he observes, “was the unknown quantity in the problem he had to work out. The natural way to work it out was by marrying Catherine; but in mathematics there are many short cuts, and Morris was not without a hope that he should yet discover one” (p. 169). Nonetheless, when Catherine returns from Europe, the young man’s confidence begins to flag. “He will never give us a penny; I regard that as mathematically proved,” he impatiently tells Catherine’s Aunt Penniman (p. 197). Like Sloper, Townsend subscribes to a rigid logic of quid pro quo. When Catherine pleads with him to stay—“Think of what I have done! ... Morris, I have given up everything!”—Townsend replies with “a sort of calculated brutality,” “You shall have everything back!” (p. 206). Despite the mathematical tidiness of this response, and the geometric precision of Townsend’s farewell note—a letter of some “five large square pages”—Catherine again confronts the shabby remainders of a broken heart.

As its title would suggest, geometry is Washington Square’s mathematical discipline of choice. The story itself takes place on a square, itself the geographic border between Manhattan’s irregular network of roads snaking across the island’s southern tip, and its neat grid of avenues and streets that was then beginning to extend northward (see Bell, “‘This Exchange of Epigrams,”’ p. 53).