The ruler has a fixed distance marked on it and one mark is kept on a given line while the other traces the conchoid curve. The construction is explained in more detail in Nicomedes ' biography in this archive. Now this is exactly the curve needed to solve both versions of trisection of an angle given above and Nicomedes solved the problem with his curve. However, in practice the method of moving a ruler until the required configuration was found was on the whole much easier than drawing the conchoid so Nicomedes method was more of theoretical rather than practical interest. In the same work Pappus writes about how the problem of trisecting an angle was solved by Apollonius using conics.
Pappus gives two solutions which both involve the drawing of a hyperbola.
PBA is a hyperbola. The hyperbola has eccentricity 2, focus B and directrix the perpendicular bisector of AB . The hyperbola is given in the left of the two diagrams. The right of the two diagrams shows how this hyperbola can be used to trisect the angle AOB . With O as centre draw a circle through A and B .