In the 70's I had one of these http://www.victorialouise.co.uk/bottle-chopper-fun
In fact I think it is still in the garage in a box.
Basically an aluminium jig holds a bog-standard glasscutter in place so you can make a perfectly horizontal score on the bottle's outside surface.
Then you adjust a bent metal rod with effectively a washer on the end so that when it is placed inside the bottle it lines up horizontally exactly against the score line.
Then you tap the washer against the glass until you see a crack start, and continue tapping whilst rotating the bottle to propagate the crack all the way round.
It works well enough, but there are risks of imperfect results if the initial scoring isn't even - e.g. if there is grease/oil/fingerprints on the glass the cutter may slip rather than score.
Once chopped, you grind down the edge with wet and dry paper (coarse grades down to flour-grade) or use various grades of carborundum powder on a sheet of flat glass. A labour of love really but really quite satisfying when it works. I used to make pint (approx) beer glasses out of Spar orange-squash bottles (all I could afford to drink at uni!).. but boy was the glass quality dodgy?! - you could only tell once chopped - the thickness could vary from 1mm to 5mm.... another reason for failure in the initial chop.