Cut the 1 ¾-inch dowel to 14 inches with a handsaw. Sand the cut edge to smooth the surface.
Cut an eight-inch circle out of the ¾-inch wood board with a band saw or scroll saw to make the base of the paper towel holder. Cut an 8-inch x8-inch square and trim the edges with the band saw to make rounded corners instead of making a circle base, if desired.
Use a drill press to drill a 1 ¾-inch thick hole through the center of the base and top. Use a smaller drill bit if using a hand drill since it will be harder to drill a straight hole, then widen and straighten the hole with a round file.
Round the top edge of the paper towel base with a belt sander. Lightly sand the inside hole to remove jagged or splintering wood.
Apply wood glue to the bottom on the dowel and insert into the hole. Slice off chucks of overflowing glue with a razor after the glue has set for an hour, then allow the glue to dry for 24 hours.
Brush on a wood stain or wood finish of choice. Apply two or three layers to seal the wood.