You can quickly and easily very finely dice an onion with only two cuts.  A horizontal cut is not necessary as long as your lengthwise cuts follow the shape of the onion and resemble the spokes of a cartwheel.  The important thing is to make all your cuts close together.

1.  Cut off the top of the onion, but not the root end with a chef’s knife, then cut the onion in half lengthwise and remove the skin.

2.  Place the onion cut side down and use the tip of your knife to make lengthwise cuts close together, from the outside of the onion to the center, making lots of narrow wedges.

3.  Thinly slice the onion crosswise.  You now have finely chopped onion.