Firstly, you should be using a very sharp knife. A dull knife will crush the onion cells that release the noxious fumes into the air. Since the chemicals in the onion that irritate the glands in our eyes are concentrated by the root, you should avoid cutting this part.
Some other techniques include holding a piece of bread in your mouth. It will not completely prevent your eyes from stinging, but it'll help to delay the tears. A method I highly suggest is putting white vinegar on the cutting board to neutralize the chemicals. When I do this, my eyes rarely sting. Unfortunately, the smell was not so nice. Also, the vinegar on the onion interferes with frying.
The best way to avoid crying is to place the cutting board near your stove and turn on a burner or two. Then, turn on the hood exhaust fan. The heat from the flames will burn the noxious fumes from the onion, and the exhaust fan will get rid of it along with the flame exhaust. While this is the most effective solution, it clearly only works for those of us who are cooking with gas. Apparently, an electric range on high doesn't really work. However, instead, you can light several candles. The candle flame pulls air from around it and sends it up the smoke plume.
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