Create a mixture of 1/2 tablespoon liquid dish soap and 3 cups warm water. Stir the ingredients to create a sudsy mixture.
Dampen a soft cloth with the mixture, and wipe down the affected section of mahogany to remove any dirt, debris or grime. Wipe down the mahogany with a separate soft cloth to remove the excess moisture.
Remove a small sliver of wood from an extra piece of mahogany with a plane, a specialized woodworking tool also used to shave warped doors. Set the plane on the mahogany. Adjust the height to remove a piece large enough to fill the crack, and slide the plane over the wood. Remove the shavings, and set them into the crack until a suitable piece is located.
Cover the crack’s inside with a thin layer of wood glue. Insert the wood shaving into the crack, and press it down with a toothpick or your finger. Place pressure on the wood shaving for one to two minutes to allow the glue to set.
Allow the glue to dry for at least two hours before applying a thin layer of wood filler over the glue. Use a plastic scraper to spread the wood filler over the crack, wiping away the excess from the surrounding wood with a damp cloth. Allow the wood filler to cure according to the package directions.
Sand down the filler with 120-grit sandpaper until it is flush with the surrounding wood. Wipe down the mahogany with a tack cloth, and cover the repaired area with a fresh coating of shellac, varnish or whatever was used to originally finish the piece.