And I guess as a tangential question, do we know when a majority of people really stopped believing that creatures like dragons or cyclops ever existed?

Someone with a background in late Middle Age Myths maybe can provide more information around the way the myths around dragons dissipated over time, but I’ll share information on Washington, his education, and how his contemporaries typically viewed ancient mythological monsters.

Mythical beings, whether dragons, sirens, or anything else were not commonly believed to be in existence by the late 18th century. Washington, who died in 1799, would not have likely believed in anything of this sort, nor would have others likely believed them around his time. I don't know when that belief died out, but to 18th century men, even less educated people, I've never seen a concern for mythical beings actually existing.

If you examine some of the writing of his contemporaries, evidence suggested it would be highly unlikely that the Founders believed in most mythological beings since they appear to regard ancient monsters as being fictions. For instance, this entry from 1787 by his soon to be Vice President John Adams includes a poem of sorts in with his entry, where he references mythological sirens. He, like his contemporaries studied the Odyssey while in college and had a habit of referencing these ancient myths in their personal writing. They gave no indication they ever believed they were more than myths. Washington’s personal library was made up of about 7% history books, many being ancient history, and 9% popular fiction, such as Don Quixote, so he likely knew of at least some of these mythological creature. However, if you were to search the US’s Library of Congress’ page on the founders, which has over 32,000 Washington letters transcribed, he never once mentions an actual dragon, which at least infers that he didn’t seem to believe they posed an actual risk. Part of why he likely did not believe in these mythical cheaters, is due to his own education.

While Washington may not have been formally educated, he was a very intelligent man who not only surrounded himself with intelligent men, but he himself was very-well read. He received an education by his mother during his childhood. He may have received a formal education before his father died when he was 11 in math, reading and writing at a day school, taught by one of his father's tenants.[1] There are other reports of him receiving an education in math at another school when he was older. While none of these claims have been fully confirmed, it seems likely that young George was educated early on in some formal fashion considering how strong of a skill he developed in math. He established himself early on in his professional life as a land surveyor, which is an occupation entirely rooted in math. As he grew into his roll as a planter and then as a field commander, Washington was known for taking his own measurements in surveying land and his math was reported to be impeccable by many sources. Washington continued to educate himself as he got older, but his lack of a formal liberal education did bother him personally as he grew older. As his well respected biographer, Ron Chernow stated:

One can't help but sums that Washington's life would have been vastly different had he attended college. He lacked the liberal education that then distinguished gentlemen, setting him apart from such illustrious peers as Jefferson, Hamilton, Adams, and Madison. He would always seem more provincial than other founders, his knowledge of European culture more second hand. A university education would have spared him a gnawing sense of intellectual inadequacy. We know that he regretted his lack of Latin, Greek, and French -- the major intellectual adornments of his day -- since he lecture wars in later years on their importance. The degree to which Washington dwelt upon the transcendent importance of education underscores the stigma that he felt about having missed college...

He seized every interval of leisure to improve himself and showed a steady capacity to acquire and retrain useful knowledge.

This last sentence is particularly important to the understanding of Washington's education. His personal library that he acquired over his life, shows a man who read fiction, history, philosophy, and geography. He was also obsessed with books that offered 'practical wisdom' and spent many years searching for as much information as possible to put him on an equal playing field to his contemporaries.

So in all, Washington lacked a formal education, but he still was well-read, and in 18th century America, that means something very significant. He was wise enough to dismiss myths and understand how to read and interpret facts from fictions.

1. Ron Chernow. Washington: A Life. Penguin Books. 2010.

Edited: to fix issues with speculation and add sources.