You're standing in front of your picture window, staring out at a beautiful scene, when suddenly you spot a small scratch in the window glass. It's not a big scratch but it's an annoying one. It's too small to have the entire window pane replaced but large enough to irritate you. If your fingernail doesn't snag when you rub its edge across the scratch, you should be able to buff it out yourself. Some of the same materials used in lapidary to polish stones can be used to repair scratches in glass.
Wash the glass thoroughly on and around the scratch before starting work on it. If all grit isn't removed from the surface of the glass, the grit itself will end up scratching the window glass more.
Mix together a little 3000-grit diamond powder and olive oil, or a commercial lapidary lubricant, to make a paste. Diamond powder is a very fine grit abrasive. Using a small handheld, flexible shaft machine, insert a small lamb's wool buffing pad into the chuck and lock it in place. Pick up some of the paste on the buffing pad, set it against the scratch and turn on the power to low, about 1500 rpms. Buff over the scratch. Stop frequently to check the progress.
Clean off the paste from the surface of the glass with water and a soft cloth when all or most of the scratch is buffed away. Dry the glass. Give the glass a final cleaning with alcohol and a cotton ball.
Mix up a slurry of cerium oxide and water. The slurry should be the consistency of soft-serve ice cream. With a fresh buffing wheel in the power tool, load it with the slurry and begin to polish the glass over the location of the scratch. Cerium oxide is a very fine buffing compound used for final polishing of a surface.