This handy construction takes advantage proportional lengths and the properties of parallel lines to convert a square into a nicely formed regular octagon. Although regular octagons were not particularly uncommon traces for field fortifications, as the number of sides of the traced polygon increased so to was the fire of the garrison divided. Given the same size garrison and same length of interior crest fewer troops manning the parapet of an octagonal field work could strike the same target as troops manning the parapet of field works given a square, pentagonal, or hexagonal trace.

Given a right line AB construct a regular octagon with sides equal to the length of Line AB.

In fortification drawing the side of the octagon, that is, the length of Line AB, is determined by the number of troops and artillery armament composing the designed garrison and is  about one-eighth the overall length of the interior crest of the parapet.

Begin by constructing a square (ABCD) on Line AB.

Produce diagonal lines (AD and BC) through the square. Extend each diagonal beyond the limits of the square to points E and F so that line segments CE and DF are equal to the length of Line AB.

Construct two lines (AG and BH) from the lower extremities of the vertical sides of the square that are parallel to lines BE and AF that are equal to the length of Line AB.

Note that at this point in the constructions three sides of the regular octagon have been drawn: lines AB, AG, and BH.

Extend the vertical sides of the square (AC and BD) an indefinite distance.

Construct lines from points E and F that are parallel to lines AF and BE and extend them until they intersect the vertical extensions of the sides of the square at points K and M. Lines EK and FM will serve as two further sides of the regular octagon.

Complete the sides of the octagon by connecting points K and M, E and G, and F and H with right lines.

Carefully erase all of the construction lines and markers to complete the construction of the regular octagon.

~ An Instructive Animation Wherein the Foregoing Construction is Visually Explained ~