>>1950423People might have different opinions on this, and Im speaking solely from personal experience vs. being taught what Im what about to say, however I would assume a good number of people share my view. Using a circular motion to sharpen an edge is beneficial for a number of reasons. The surface area of the stone is much smaller than an axe, making dragging an along a stone a very inefficient process when compared to a knife. It also helps in maintaining a consistent bevel angle when sharpening free hand. Also keep in mind that when you're removing chips or damage from an edge, you're moving the entire edge back to make up for the material that no longer exists in a specific area. Using a stone along the bulk of the bit allows you to maintain your primary bevel angle as simply just moving both faces back until they form a point along the apex of the edge will cause the angle to become steeper. It (a stone in a circular motion) also allows material to be removed much more quickly. That being said, most work axes/users can be sharpened with a bastard file much more quickly than with a stone and can be put back to work without issue. As much as I enjoy having a "shaving sharp" edge, its not completely necessary. You can use the standard back and forth sweeping motion but in my opinion it only becomes efficient when using progressing grits of sandpaper as you brought up, which is also a perfectly fine way of sharpening an axe.
I use a circular motion regardless of the size or shape of the stone im using, because again, it allows me to maintain a more uniform surface and bevel angle across the entirety of the bit. A lot of my axes are convex grinds, and this method is great for that geometry. Here are my most used tools for sharpening and profiling axes. I use a piece of leather belt epoxied to a length of wood as a strop with the compound pictured. In the field I carry a 6" bastard file, a 250/1000 grit stone, and a small version of the strop i mentioned.
