Morasses and finite support iterations

We introduce a method of constructing a forcing along a simplified $(\kappa,1)$-morass such that the forcing satisfies the $\kappa$-chain condition. Alternatively, this may be seen as a method to thin out a larger forcing to get a chain condition. As an application, we construct a ccc forcing that adds an $\omega_2$-Suslin tree. Related methods are Shelah's historic forcing and Todorcevic's $\rho$-functions.