Cut out the old caning and clean the holes with an awl to remove dust. Use sandpaper to smooth the hole rims and inner seat frame.
Divide the holes into four curved lines at the top, bottom and both sides, with a free hole between each. Dampen a cane and insert one end down through the farthest left of the holes in the top set. Draw down about 6 inches and peg the hole.
Draw the cane toward you and feed it down through the hole farthest left of the bottom set of holes, smooth side uppermost. Pull tight and secure with a peg. Feed the cane up through the hole to the right, pull tight and move the peg over to keep it taut. Cross to the opposite side, and repeat the pattern across all top and bottom holes, pegging the final hole.
Dampen a new length of cane and repeat the same pattern horizontally. Start at the top hole in the left hand set, peg it, and take the cane across to the top hole in the right hand set.
Take the peg out of the first hole bottom left, push another cane down through the hole, and repeg. Ease both canes to lie side by side in the hole. Pull the cane tight and repeat the vertical section across the top of the first two layers.
Repeat for the second horizontal caning. This time, feed it under the first and over the second strand in each vertical set of two. When reversing, match the weave by feeding it over the first and under the second of each pair.
Change to the wider diameter cane. Dampen the cane, feed it down through the empty top left hole, and peg.
Aim for the bottom right hand hole, and pass the cane over the first set of horizontal canes and under the first set of verticals. Repeat the pattern across the settings.
Peg the cane, then feed it up through the hole directly to the left, pull it taut, and peg. Repeat the over-and-under pattern across the left semicircle of the seat.
Feed a second length of cane through the top left hole and repeg. Repeat the diagonal pattern, working across the opposite semicircle of the seat toward the top right-hand section.
Work the third diagonal crossing in the opposite direction, beginning at the bottom left-hand hole and working across to the top right, covering the semicircle up toward the top left. Repeat for the lower half of the seat down to the bottom right-hand section.
Insert a dowel plug into each of the holes along the seat frame, knocking it in with the hammer and punch until it is just below the top of the hole. Turn the chair over and cut away the cane ends.