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Differential cross sections for elastic ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{-}p$ scattering have been measured at lab momenta of 8 and 12 GeV/c in a momentum-transfer region corresponding to $1.2\ensuremath{\le}\ensuremath{-}t\ensuremath{\le}6$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$. Also, differential cross sections near 180\ifmmode^\circ\else\textdegree\fi{} were measured for 4 and 8 GeV/c pions. At momentum transfers greater than $\ensuremath{-}t=2$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$, the $\ensuremath{\pi}\ensuremath{-}p$ cross sections drop much faster with increasing angle than the corresponding $p\ensuremath{-}p$ cross sections. Also, in the region $\ensuremath{-}t\ensuremath{\simeq}1.3$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$, there is structure in the $\ensuremath{\pi}\ensuremath{-}p$ angular distribution but not in the $p\ensuremath{-}p$ angular distribution. At $\ensuremath{-}t\ensuremath{\simeq}3$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$, the drop in cross section appears to stop and from then on the angular distribution is consistent with isotropy. But in the angular region 170\ifmmode^\circ\else\textdegree\fi{} to 180\ifmmode^\circ\else\textdegree\fi{}, the cross sections have become much larger, and sharp backward peaks are observed. Information is given on the energy and charge dependences and widths of these backward peaks.