The outside of the level still needs some final shaping and sanding. Despite this, I moved onto the next step because, well, I'm impatient.
Levels require a way to view the vial not only from the top but from the sides. Wooden levels (at least the antique ones I own) have sloped side holes which necessitate a bit of shaping.
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Two arcs create the slopes—a small arc that is closest to the vial, and a larger arc that terminates on the outside surface of the level.
I used a chisel, shallow gouge, and round file to shape the sloped holes—quick and easy.
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I bet if I had planned ahead, I could have drilled the inner arc with a Forstner bit before cutting the blank for the level and the channel for the vial.
But that's way too much forward-thinking for someone who's, well, impatient.
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Parts I and II for making this level are here and here.