First, set your compass to just over the radius of the circle.
Second, place the pin end of the compass on the and draw an arc in the middle of the circle. I have made a big black dot where I placed the compass pin and will continue to do this.
If you are needing to drill a hole in the new found center make sure to either start with a small drill bit or use a punch to keep the drill bit from "walking" when initially drilling the hole. If you are drilling through metal using a punch is necessary even when starting with a small drill bit. Lastly, no one uses drill bits properly but few properly maintain their bits by keeping them sharp. Buying a Drill Doctor will save you loads of time and money by keeping your bits nice and sharp. I tried hand sharpening drill bits but couldn't get it despite my machinist friend's help, so I spent the money on a great product. Best $80 I have spent thus far!
Quick note on variability with this method. Intersecting arcs need to be drawn with the same radius or compass length, ex: 1&2 or 3&4 but not all four. Tighten up your compass to it hold the length between placements if this is a problem. You don't need to make the second set of arcs perpendicular to the first. When making one set of intersecting arcs you don't have to place the compass in the exact location 180 degrees away but try to be close. I have attached a perspective free scan showing what you could get away with.
If you have any questions feel free to ask! If there are problems please include enough information to clearly outline what step/part, what I did and how it is an issue.