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Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds.
This phrase was a translation by Prof. George Herbert Palmer, Harvard University, from an ancient Greek work of Herodotus describing the Persian system of mounted postal carriers c. 500 B.C.E. The inscription was added to the building by William Mitchell Kendall of the architectural firm of McKim, Mead & White, the building's architects. It derives from a quote from Herodotus' Histories, referring to the courier service of the ancient Persian Empire:
It is said that as many days as there are in the whole journey, so many are the men and horses that stand along the road, each horse and man at the interval of a day’s journey; and these are stayed neither by snow nor rain nor heat nor darkness from accomplishing their appointed course with all speed.—Herodotus, Histories (8.98) (trans. A.D. Godley, 1924)
In 2001, the USPS created a television commercial edited to Carly Simon's song "Let the River Run". The commercial, which ran after the September 11, 2001, attacks and the anthrax mailings, featured no voice over, only the following text interspersed on title cards. A portion of this variation also appeared without citation in the USPS 2001 Comprehensive Statement on Postal Operations (1.A-1):
We are mothers and fathers. And sons and daughters. Who every day go about our lives with duty, honor and pride. And neither snow, nor rain, nor heat, nor gloom of night, nor the winds of change, nor a nation challenged, will stay us from the swift completion of our appointed rounds. Ever.
In Adventures in Odyssey, the character Wooton Bassett said the mailman's motto is:
Rain or shine, snow or sleet, we deliver your mail! (But sunny days are optional...)
In Terry Pratchett's Discworld novel Going Postal, the motto for the Ankh-Morpork Post Office is very similar, reading "Neither rain nor snow nor gloom of night can stay these messengers about their duty." 
The creed is said by a mail carrier in the Arthur episode "What's Cooking? / Buster's Special Delivery".