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Did ancient mathematicians know Euler's characteristic for convex polyhedra?

there is no doubt the answer to your question is "no"; for a wonderful and scholarly recent book on the whole story, see Euler's Gem: The Polyhedron Formula and the Birth of Topology by David Richeson …
Carlo Beenakker's user avatar
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Source of a quote by Ferdinand Rudio

The quote is from a speech Rudio gave at the Town Hall in Zürich on the 6th December 1883; The German original is published in Felix Stähelin, Reden und Vorträge (1956, I have not found it online). …
Carlo Beenakker's user avatar