jmc

From each vertex $V$, we can draw 4 diagonals: one to each vertex that is not $V$ and does not share an edge with $V$. There are 7 vertices in a heptagon, so we might be tempted to say the answer is $7\times 4=28$. However, note that this counts each diagonal twice, one time for each vertex. Hence there are $\frac{28}{2}=\boxed{14}$ distinct diagonals in a convex heptagon.