How do you find the center of a circle with a pencil and a book?

Let the corner of the book just touch the circle's edge. You can draw two perpendicular lines along book's edge which intersect with the circle at two different points. Connect the two points and it's a diameter.

Repeat this and get another diameter. The intersect of these two diameters is the center of the circle.

Not as simple as the accepted answer, but you have another solution:

Use the book to draw two parallel lines that cross the circle at different distances from the center to obatain 4 intersection points in the circle. Draw 4 lines connecting these points. The two points of intersection from these 4 lines define a diameter. Repeat the process with two parallels with different direction to obtain another diameter. Finally the intersection of two diameters returns the circle center.

If the circle is a foldable material (like paper), you could fold it in half twice, the intersection of the folds will indicate the center.