How did they do it?

I picked up an old desk who’s details I like and intend to make one similar to it. I am not a turner so this may be readily apparent to those who do this all the time however I’m wondering how they were able to turn the piece and have the flat for joinery. I took a scribe and drew the ark and the flat piece clearly falls outside of the ark. I do not see any glue lines indicating a piece has been added after the turning.

It appears that the builder used a large square blank and used a very large round-over cutter/bit to make the rounded edges. This looks like a unique design that I might want to use some day.
Others may chime in with their ideas.

I'm not much of a turner, but those sure look like marks from the lathe center on the top. I can only speculate that the square portion was let in after the leg was turned.

The piece (about 1/4) could have been sawn out of the blank, replace with a dummy piece to keep balance, and then replaced after turning to get a good grain match. But I'm just guessing.

Maybe mounted to a lathe with a box and then machined. Looks like something made in the 20's.

Looks likely that it was mounted between centers, however not "turned" between centers. I'm guessing that it was likely milled/machined for the curved sections. Possibly with some sort of indexing head as suggested by the regular spacing of the routed slots and round decorative features on the upper section. The cross drive imprint in the top surface indicates that it was held firmly by a drive center of some type.

The old Sears Router Crafter allowed hobbyists to build similar things.

I think Dueane is on the right track. The blank was lathe-mounted, indexed, and then carved with a secondary spinning cutter. The rose & flute patterns elsewhere on the leg would have been applied with the same technique.

In the hobby woodturning arena, I think that a fluting jig is what you're after.

Keith, I think it's more how would you do this now rather then how was this done. I would turn the entire leg except leave the top portion square...then carve the round part of the top and the mortise. The flutes and rosettes are done with a router while the leg is still on the lathe. Similarly you could use a router to create the mortise for the long attached square piece. Just create a shelf for the router that mounts on the lathe and run the router down the piece. Not too difficult.

I think that the round with flat is a separate piece that was added to the turned spindle. The "X" marks are there to get the orientation of the flutes to match the flat of the square piece that was likely cut with a pattern bit and router to get the round and leave the flat side. End grain is not visible on the face of the section and the gap tends to indicate hole fitting a tenon. That would be my process.