We propose a new global and quadratically convergent algorithm for the linear $l_{\infty}$ problem. This method works on the piecewise $l_{\infty}$ problem directly by generating descent directions - via a sequence of weighted least squares problems - and using piecewise linear linesearches to ensure a decrease in the $l_{\infty}$ function at every step. We prove that ultimately full Newton-like steps are taken where the Newton step is based on the complementary slackness condition holding at the solution. Numerical results suggest a very promising method; the number of iterations required to achieve high accuracy is relatively insensitive to problem size.