The division of the circle using only a ruler and a compass is an old and classical problem. For many years it had been an open and often discussed problem whether the regular 17-gon can be constructed with the use of these two tools only. The ancient Greeks knew how to construct an equilateral triangle (3-gon), a square (4-gon), and a regular pentagon (5-gon), and of course they could double the number of sides of any polygon simply by bisecting the angles, and they could construct the 15-gon by combining a triangle and a pentagon. For over 2000 years no other constructible n -gons were known. Then, on 30 March 1796, the 19 year old Gauss discovered that it was possible to construct the regular heptadecagon (17-gon). Subsequently Gauss presented this result at the end of Disquistiones Arithmeticae, in which he proves the constructibility of the n -gon for any n that is a prime of the form 
, also known as Fermat primes.
One of the nicest actual constructions of the 17-gon is Richmond’s (1893), as reproduced in Stewart’s Galois Theory. Draw a circle centered at O, and choose one vertex V on the circle. Then locate the point A on the circle such that OA is perpendicular to OV, and locate point B on OA such that OB is a quarter of OA. Then locate the point C on OV such that angle OBC is a quarter the angle OBV. Then find the point D on OV (extended) such that DBC is half of a right angle. Let E denote the point where the circle on DV cuts OA. Now draw a circle centered at C through the point E, and let F and G denote the two points where this circle strikes OV. Then, if perpendiculars to OV are drawn at F and G they strike the main circle (the one centered at O through V) at points V3 and V5, as shown in the figure.
The points V, V3, and V5 are the zeroth, third, and fifth verticies of a regular heptadecagon, from which the remaining verticies are easily found (i.e., bisect angle V3OV5 to locate V4, etc.). Gauss was clearly fond of this discovery, and there's a story that he asked to have a heptadecagon carved on his tombstone, like the sphere inscribed in a cylinder on Archimedes' tombstone.
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