Of course, with a ruler you can simplify it a bit:
1)Mark a point on the circumference of the circle.
2)Using a compass (or even a ruler) mark the circumference of the circle at a straight line distance of about 3/4 the diameter of the circle. (Using a compass is easier).
3) Draw a line between the two points you just made and use a ruler to mark the midpoint of the line.
4) Draw a line across the midpoint you just marked to the first mark you made (in step 1). Extend this line to the other side of the circle.
5) Use the ruler to find the midpoint on the line you drew in step 4; the midpoint will be the center of the circle.
Another method is to use a drafting triangle (the 90/45/45 one, not the 90/60/30 one). Use the edges adjacent the 90 degree angle to 'corner' the circle (put both edges tangent to the circumference and then draw the 90 degree corner). Now place the hypotenuse on one of the lines you just drew with a 45 degree angle at the corner and draw the line across the circle, which will bisect it. Either use a ruler to find the midpoint or draw another corner and 45 to bisect the circle again (where the second 45 crosses the first is, of course, the midpoint).
There was a center-finding tool that used that last method; it was a combination of a square and a rule at 45 degrees where you could just lay the squared edges adjacent to the circle and use the rule to mark the bisecting line, but I haven't seen one in a while. Can't remember if it was a drafting tool or something model-airplane related.