Everyone has heard of the golden ratio, and that the golden rectangle made in this ratio is somehow the most aesthetically pleasing.
We know that knowledge of the ratio dates back to at least Euclid (300 BC), but he just noted it as interesting in mathematics, not aesthetics. Then, it doesn't seem to really turn up in writing until about 1500, when Italian mathematician Luca Pacioli wrote about it as "the divine proportion."
The story goes that the ancient Greeks used the ratio in art and architecture, and that the Renaissance masters did likewise.
But, on examination, all of the claimed uses only happen if you line things up just so, ignoring whatever inconvenient parts of the painting, building, or sculpture don't line up. And, no one from the time seems to write about it. All of the references that I can find to the golden ratio being used in such and such older work turn out to be modern in origin: someone (sometimes "expert", sometimes not) took measurements and declared them to match the golden ratio to some degree of precision. This appears to never be supported by direct evidence (construction lines, sketches, notes) from the time itself.
Now, one could perhaps buy the idea that, to the ancient Greeks, it was all about secret mysticism, and any texts have been lost. We certainly have lost more from that time than has carried over. But there's plenty of writing on art from the Renaissance — does any of it mention the golden ratio, by Pacioli's name "divine proportion", or by Euclid's term "extreme and mean ratio"?
From what I can tell, the idea that the number was important outside of being interesting in math comes from a guy named Adolph Zeising in 19th century Germany. There's pretty credible evidence that this is pretty much the start of the modern idea.
But, just because Zeising popularized the idea doesn't mean he's wrong. I'm just coming up empty coming up with anything credible before the 19th century.