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At 33.4mm (wears more like a 36mm Datejust or my 37mm 6205 or 5508) I believe the 3660 with its center second was Rolex's counterpart to the Omega Medicus. (Thank you Bloke for identifying that connection.)
Lovely borrowed photo of the Omega Medicus:
The center second was all the rage in thr mid 30's. Here is a Rolex ad from the mid 30's noting the center second.and it's intended application.
Here are some additional pics from the service:
This watch sat in the drawer of my watchmaker's mentor and teacher since the 1960's. My watchmaker acquired it a couple of years ago and offered it to me.
I believe the configuration and size of the watch to be designed for a physician. To that end, dare I say this was an early tool watch?
Thanks for looking.