  Traves and Wehlau recently gave a straightedge construction that checks
whether 10 points lie on a plane cubic curve. They also highlighted several
open problems in the synthetic geometry of cubics. Hermann Grassmann
investigated incidence relations among points on cubic curves in three papers
appearing in Crelle's Journal from 1846 to 1856. Grassmann's methods give an
alternative way to check whether 10 points lie on a cubic. Using Grassmann's
techniques, we solve the synthetic geometry problems introduced by Traves and
Wehlau. In particular, we give straightedge constructions that find the
intersection of a line with a cubic, find the tangent line to a cubic at a
given point, and find the third point of intersection of this tangent line with
the cubic. As well, given 5 points on a conic and a cubic and 4 additional
points on the cubic, a straightedge construction is given that finds the sixth
intersection point of the conic and the cubic. The paper ends with two open
problems.
