Oh dear indeed. In suggesting the use of slopes instead of degrees, I
deliberately left out the gory details because it isn't necessary to deal with
gory details in order to make furniture. Yes, I did use trigonometry to
calculate the degree equivalents in order to illustrate that the results are
essentially the same, but it's much simpler not to deal with degrees at all. You
don't suppose that 18th-century galoots used degrees in marking out their
furniture, do you? It is much easier to set your adjustable bevel gauge using
slopes rather than degrees, and much more repeatable.

Another practical use of slopes is in evaluating bevel angles of chisels and
plane irons. If the width of the bevel is twice the thickness of the blade/iron,
then you have a 30-degree bevel. If it's more than 2:1, then you have a
shallower angle. If it's less, then you have a steeper angle.

For the curious, I used an HP48 calculator emulator on my smartphone (set to
calculate in degrees) to make that table, and I did indeed use the arctangent
function.

Cheers,
Chuck Taylor
north of Seattle USA