A square as large as a triangle

“Der do machenn wil ein firung vnd ein driangel daz die firung vnd der driangel itlichs als vil in im helt als daz ander”: so the workshop booklet Geometria deutsch, attributed to Mathes Roriczer, sets out to make a square that holds as much as a triangle. Divide one side of the triangle into three equal parts and make the square on two of them: “so helt die fierung gleich als vil in als der driangel”. That is true only for a triangle whose height is eight ninths of the divided side. For the triangle the compass gives first, the equilateral one drawn here, the square holds 2.6 percent more. The booklet does not say how to divide in three. Here two sides are run on past the far corner and cut there by the compass at the side’s own opening; the lines from the vesica’s other point to those two cuts divide the side in three, exactly.