James alludes to the poet Henry Wadsworth Longfellow. This allusion helps convey the superficiality of Arthur Townsend's attempts to chat with Catherine:
I always try and keep up with the new things of every kind. Don’t you think that’s a good motto for a young couple—to keep “going higher”? That’s the name of that piece of poetry—what do they call it?—Excelsior!’
In this scene, Catherine is barely listening as Arthur drones on about his preferences and his wife. He is trying very hard to seem interesting, which of course has quite the opposite effect. The mention of Longfellow's 1842 poem "Excelsior" is both funny and revealing, given Henry James's own relationship with the poem. James was known to have hated Longfellow's poetry. He publicly said on many occasions that he found it insipid and commonplace. By having the dull Arthur Townsend clumsily reference the poem, James uses it as a vehicle to portray his dullness and desperation to appear cultured. Townsend’s uncertainty about the poem's title, despite using it as a proposed "motto," also displays his lack of real engagement with literature.
His ignorance is amplified by the fact that "Excelsior" is already a motto: indeed, it’s the state motto of New York, where this conversation is taking place. The fact that he thinks “Excelsior!” is an original “motto” is a testament to his own lack of originality. His ineffectual attempts to engage with Catherine, who remains unimpressed with him, build on each other to create a picture of an insufferably boring and unoriginal person.
When discussing a questionable choice by Dr. Sloper, James incorporates situational irony through an allusion to an infamous insane asylum of late-1800s New York. This both provides local color for the audience and indirectly emphasizes the Doctor’s weighty professional reputation:
About a year after this, the accident that the Doctor had spoken of occurred; he took a violent cold. Driving out to Bloomingdale one April day to see a patient of unsound mind, who was confined in a private asylum for the insane, and whose family greatly desired a medical opinion from an eminent source, he was caught in a spring shower [...]
By referencing "Bloomingdale," James is alluding to a notorious private hospital, Bloomingdale Insane Asylum, which opened its doors in 1821. This location has nothing to do with the department store Bloomingdales, but it does remain an important New York landmark. The former facility is actually now the site of one of the buildings of Columbia University.
Bloomingdale Asylum—as it was in the Gilded Age—would have been used to confine or imprison people believed to be dangerously mentally ill. The mere fact that Dr. Sloper is visiting such a place underscores his professional distinction and “eminent” reputation. Conditions in the asylum were famously terrible. Very little legislation existed to protect people diagnosed with mental illnesses in this period, and the treatment of "patients" in private hospitals could be horrifyingly bad. The violence and lack of sanitation within Bloomingdale was the subject of a popular undercover report in the New York Tribune by the journalist Julius Chambers in 1871, titled “Among the Maniacs.” When writing Washington Square—which was published in 1880—James would have been familiar with this infamous article, as would his contemporary readers.
There are two situationally ironic twists in this passage. The first is in Dr. Sloper's relationship with the facility itself: it is, much like the Doctor himself, an institution that seems to do good while really only operating for its own self-interest and profit. The second lies in the “distinguished physician” falling ill and dying because of this very visit. Having escaped unscathed from his dangerous appointment, he quickly expires. Hence, by detailing Dr. Sloper's visit to provide an "eminent" medical opinion, James establishes the doctor's professional prominence, gives the reader another insight into his character, and provides Dr. Sloper with some final, ironic justice in his final moments.