Cut two 48-inch pieces of 1-by-2 boards, two 24-inch pieces of 2-by-2, and one 26-inch length of 2-by-3 wood with your saw. These pieces comprise the upright back of the deck chair. Sand the pieces, including the cut ends, with fine-grit sandpaper until they're smooth.
Lay out the two 48-inch pieces on their 1-inch edges so they're parallel and about 24 inches apart at the inside edge. Lay the 26-inch 2-by-3 across the two 48-inch pieces, so its ends are supported by the edges of the 1-by-2s, about 2 inches from the bottom of the pieces. Adjust the distance between the 48-inch pieces so the ends of the 26-inch pieces are flush with the edges of the 1-by-2s, as the actual measurements of dimensional lumber are rounded up, for instance a 1-by-2 is actually 13/16-by-1 1/2 inches, according to "Popular Mechanics" magazine.
Drill two holes, just smaller than the screws, through the face of each end of the 2-by-3 and into the edge of the 1-by-2 support. Apply wood glue at each point of contact and drive wood screws through the pre-drilled holes with the screwdriver.
Put two pieces of 2-by-2 between the two 48-inch pieces at the upper end of the two 48-inch supports. The first 2-by-2 should be flush with the ends of the 48-inch pieces, with a 1/2-inch gap between the 2-by-2s. Drill one hole through the 2-inch outer face of the 1-by-2 into the center of each end of each 2-by-2. Apply glue and wood screws as in Step 3. This completes the upright back and front legs of the sling chair frame.
Cut two 44-inch pieces of 1-by-2. Cut two pieces of 2-by-2 to 23 inches long and one piece of 2-by-2 to 25 inches long.
Clamp the two 1-by-2s together at their bottom ends along their 1-inch edge with the bar clamp. Install the 3/4-inch drill bit on your drill. Measure and mark points along the seam between the 1-by-2s every 2 inches, from 8 to 14 inches from the bottom end of the clamped pieces. Drill a 3/4-inch hole centered on each of the four marks. Unclamp the 1-by-2s. You should have four semi-circle grooves in the edge of each 1-by-2, perfectly aligned at the ends.
Lay the grooved 1-by-2s out, grooved ends up, so they're about 23 inches apart at their inside edges. Install the 25-inch 2-by-2 and the two 23-inch 2-by-2s and in the same manner as you did the 24-inch 2-by-2s in Steps 2, 3 and 4, making any width adjustments necessary. This completes the seat and back legs frame.
Lay the two frames one on top of the other with the seat-back frame face down and the front edge of the seat-bottom frame lined up with the bottom edge of the seat-back frame. The seat-bottom frame should nest inside the seat-back frame. Drill a 1/4-inch hole through both, 16 inches up from the lined-up edges on both sides. Insert the 2-inch bolt through the outer frame. Put a washer on from the other side, put the bolt through the inner frame and secure with a rounded, covered nut.
Cut two 1-by-2s 18 inches long. Cut a piece of 3/4-inch dowel 28 1/2 inches long. Drill a 1/4-inch hole 16 inches down from the top edge of the seat-back frame on both sides, and at one end of each 18-inch 1-by-2. Drill a 1/18-inch pilot hole all the way through the outside face of each of the 18-inch-long 1-by-2s, at the other end from where the 1/4-inch hole is drilled. Put the dowel between the two pieces so the cut ends of the dowel line up with the pilot holes. Apply glue and screw the drive a wood screw through the pilot holes on the face of the 1-by-2s and into the ends of the dowel to form a "U." This is the prop for the back of the slingback chair. Slide the open end of the U-shaped prop assembly over the frame for the seat back so the 1/4-inch holes at the top outside face of the "U" uprights line up with the 1/4-inch holes in the seat-back frame. Insert bolts with washers between the wood slats as in Step 8.
Cut a piece of canvas that's 52 inches long and 25 inches wide with scissors.
Fold the long edges of the fabric over twice, 1/2-inch toward the wrong side of the fabric -- the side that doesn't show -- and sew down along the folded fabric with the sewing machine so no raw edges are exposed.
Fold the short ends of the fabric over 1/2-inch toward the wrong side of the fabric and sew. Fold the short ends back 1-inch toward the wrong side of the fabric and sew again. This creates the channel for the 3/4-inch dowels.
Open the frame. Bend back the prop assembly and slide the dowel into one of the half-circle grooves.
Cut two 23 1/2-inch pieces of 3/4-inch dowel with a saw. Sand the ends smooth with fine-grit sandpaper.
Slide the rod-pocket ends of the canvas through the 1/2-inch slots created by the two 2-by-2s at the top of the seat back frame and at the front edge of the seat frame. Slide the 3/4-inch dowels through the rod pockets. The 2-by-2s hold the canvas in place as the dowel can't pass through the gap.