… he immediately followed it up, leapt out of the vessel in joy, and, returning home naked, cried out with a loud voice that he had found that of which he was in search, for he continued exclaiming, in Greek, εὑρηκα, (I have found it out).
Book IX, preface
Look closely at what is inside the parentheses. “I have found it out” is Gwilt’s translation of the Greek word immediately before it — a footnote for readers who don’t read Greek, not a second sentence Archimedes shouted. Morgan’s 1914 version, as reproduced on Chris Rorres’s Archimedes pages, renders the same moment as the Greek doubled: “Εὕρηκα, εὕρηκα.”
✓Public domain
Vitruvius wrote in the first century BC, so the underlying text is long out of copyright, and rights here turn on the translation you quote. Both standard English versions that carry this passage predate 1931 and are public domain in the US: Joseph Gwilt’s (London: Priestley and Weale, 1826), reproduced at LacusCurtius ↗, and Morris Hicky Morgan’s (Harvard University Press, 1914), which Project Gutenberg ↗ hosts in full. A post-1931 translation of the same passage would be a different matter.
⚠ This reflects the original work, which is public domain. If your exact English wording comes from a specific modern translation, that translation may itself still be under copyright — check the edition you quote.