History of the construction of regular polygons using compass and ruler

We find in Euclid’s Elements (ca −300) the geometric construction of regular polygons with 3, 4, 5, 15 sides, from which may be inferred those obtained therefrom by successive doubling. As usual in Euclid’s geometry, only compass and ruler are admitted for these constructions. With this restriction, the problem did not witness any progress until Gauss 1796 discovered a few other constructible cases, notably the heptadecagon. This put also, as it might seem, an end to this question.