Take a picture with your phone, with the hinge at an angle that shows a clearer detail and download it to google search, on your phone.
Send the picture to a restoration society.

It looks like a face mount 3/4" inset hinge.

Here is a link that might be useful: face mount 3/4

You are correct, Renovator8. I found a local hardware store that could identify it. After googling around, I found a company called Atlantic Pacific Industries, who don't sell to individuals, but who were able to put me in touch with a company called HardwareSource.com.

HardwareSource.com have a hinge identification service, where you send 3 photographs of your hinge (they specify how the photos must be taken). A helpful person called Dom confirmed what it was, and gave me the part number. The new hinges arrived after 3 days, and they are exactly right. I am very happy.

I hope this will help other people in similar situations.