What is the history of the Roman ogee shape?
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The word ogee has its origin in the French ogive, the pointed arch, or each half of it, consisting of two opposing curves. A Dictionary of Greek and Roman Antiquities defines it nicely. The curve is visible in Vitruvius's description of Doric cornices:
The sigmoid compound of two opposing curves, is of course a Classical molding, also present on the taijitu ("yin-yang") symbol and in Moroccan designs.
Why we might call it "Roman" is less clear. The qualified term "Roman ogee" first appears in Google Books in 1813, and it's stayed much less common than plain "ogee" or "ogive". According to 1902 Britannica:
The mouldings used in Roman architectural works are the same as the Grecian in general form, but they vary materially from the in contour. The Roman cyma-recta is projected much more than the Greek, with a deeper flexure.
The compound curve appears often in Roman Woodworking.
In this set of bits, as in your image, the "Roman" one has a deep flexure and a small straight drop at top:
Was it used in Rome? Yes, such as in the bases of Roman Corinthian columns:
After all that, I don't know how the term "Roman ogee" came into favor, except that it supposedly represents Roman style in this regard. Perhaps another answer can do better.
For further reading, On the Terms Cyma Recta and Cyma Reversa has lots on how the ancients wrote about these curves. On the Cordovan ogive is a mathematical treatment of ogives.
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Source: stackoverflow.com
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