Final answer:
The rhetorical device in the phrase is an allusion to Spartan culture to emphasize living a life focused on essentials.
Step-by-step explanation:
The rhetorical device used in the phrase "...to live so sturdily and Spartan-like as to put to rout all that was not life" is an allusion. This device references Spartan culture, known for its austere and rigorous lifestyle, to reinforce the ideal of living a life stripped of nonessentials and focused on the fundamental aspects. It implies a sturdy, disciplined approach to life, consistent with the Spartan ethos.