Has anyone any idea of how I can polish out (if that's the term) very fine scratches on a watch? The watch is one of those multi-function types that tell you the altitude, temperature, points of the compass etc and has a face of 40mm diameter.
If it scratches easily, it's probably plastic, not glass. You need a very fine abrasive on a slightly damp cloth - I have used a small amount of toothpaste with good results. Depends on depth of scratches, of course.
If the watch glass is indeed glass you will need emery or aluminium oxide abrasive. This will take quite a while, start off with 800 grade paper and keeping it wet polish ( small circular polishing movement)until most of the scratches appear to have gone, (don't try to get rid of all of them it will take you the rest of the year). finish off with 1200 grade or even 2400 grade. These grades of paper are only available from specialist shops so it would be easier and possibly cheaper to buy a new watch unless it is a rolex.
Some years ago when some televisions had a hard plastic dome over the front of the tube and this used to get scratched . It was possible to remove the scratches by rubbing it with a metal polish like Brasso. It was a slow process but it worked even on deep scratches.