From the Science Museum:
Example of how a sector is used
Suppose you want to divide a line that is six inches long into five equal parts. Measure off six inches with dividers, and open the sector until 5 and 5 on the two linear scales is six inches. By the principle of similar triangles the distance between 1 and 1 on the linear scales is then one fifth of the distance between 5 and 5. Measure the distance 1 to 1 with the dividers and the five equal lengths can then be marked off on the original six inch line.
I wrote an Instructable for the sector. By using the principle of similar triangles, the Pythagorean Theorem, or trigonometric tables, you could play with these babies for hours. The only problem is that dividers get a little stabby. I’m wondering if a bit of graduated wire might be a more useful partner for the sector.