Here are the "open toe" hemmers. I have no idea if they have an official name, that is only what I called them.

http://i31.photobucket.com/albums/c3...ps23f4b882.jpg

The one on the right is from a metal box with "New Royal" on it. This set has the narrow gap like the Free has.

The second fits my Goodrich machine. it has the wider gap like what fits the Davis. This set came in a cardboard box with no name on it.

The New Royal has the offset the Davis needs, but is too narrow of a gap, the Goodrich had correct with gap, but the needle hole is centered. The Goodrich set came with slide plates to fit a round needle plate, the Royal came with flat end slide plates, like the Davis needs. So I figure I need to look for a wide gap set with the flat end slide plates . . .

For fun, here are some of the other type (non-open toe) hemmers. These all have the wide gap I'm not comparing them with the narrow gap ones.

http://i31.photobucket.com/albums/c3...psebdae877.jpg

These are an older set, and have the slice in them that indicates the center of the needle hole. Newer sets don't have this slice.

The are Standard (offset to the left) New Home (Offset to the right) and National and White (Centered).

Here is another difference.

http://i31.photobucket.com/albums/c3...ps2324cf68.jpg

The Standard is very tall, and the National is very short. Though with newer sets without the slot, they are all much closer in height.

The Standard, New Home and National all have about the same distance between gap and needle hole.

Here is the National lined up with the White - and you can see that the White has more distance between the gap and needle hole. On later White and National, the height is the same and the only difference is this distance.

http://i31.photobucket.com/albums/c3...ps694f289c.jpg

The feet that fit the New Home with the right offest will also fit the Davis well enough to be used.

Thanks Cathy, I subscribed to this thread, in hopes that I will be able to relocate it later. (fingers crossed...)
Very helpful, I just figured out how the square plates in some rotary sets fastens to the Damascus Grand Rotary, and assume likewise for the coming soon, Stag Electric Rotary.
http://www.sewnuts.com/auction/stag/IMG_1400.JPG

The "square" plate is different for White than for National, but so far all the National Rotary machines I've seen have the same attachment type.

I've since figured out that New Home also fits Singer. I was just trying to install it wrong, and could not get it to fit. You need to tip it completely vertical, and then the tab fits into the hole, and then you lay it down and install the bed screw to hold it in place. That tab won't go in unless you have it tipped all the way up.