I found an acceptable-quality copy of the public-domain painting I need to illustrate a point in my new book. The same image is available from a museum, but the museum would charge me a permissions fee. Can I just use the copy I found for my book?
Yes, you can use the copy. You do not need to pay a permissions fee, which is not the same as a copyright charge; there is no copyright, courts have found, in a simple photographic reproduction of a flat item. And the work you want to use for illustration has no copyright anyway, since it’s in the public domain. If you didn’t already have access to the image, then you might have had to pay the fees the museum is asking. But since you do, it sounds like you’re ready to publish. If you have any other concerns, you can consult the Code of Best Practices in Fair Use for the Visual Arts to see what else your peers have to say about permissions and fair use.