The Pacific seafloor is littered with small fragments of lithosphere captured from adjacent plates by past plate boundary reorganizations. One of the clearest examples of such a reorganization is documented in the Mathematician Seamounts region, where a distinctive geomorphology and well-developed magnetic anomalies are present. This reorganization involved a short-lived microplate between the failing Mathematician Ridge and a new propagating spreading center: the East Pacific Rise. It produced a transfer of a fragment of lithosphere from the Farallon to the Pacific plate, and also created a number of landforms and magnetic patterns, within and on the margins of the captured fragment; these make up the Mathematician paleoplate. In many cases, two sides of a microplate are active spreading ridges. A microplate evolves into a paleoplate when dual spreading ceases and full spreading resumes at the prevailing spreading ridge. We define a paleoplate as the area of the seafloor, from the axis of a failed rift to the boundary of resumed, full spreading. It includes a fragment of captured lithosphere and the lithosphere slowly accreted to it during the period of dual spreading, prior to complete abandonment of the failed rift. The Mathematician paleoplate has the following boundaries and components from west to east: the axis of the Mathematician failed rift, the fragment of captured Farallon plate, a complex rift initiation site at the Moctezuma Trough, a zone of slow spreading, and an as yet ill-defined eastern boundary where dual spreading stopped and full spreading resumed. The northern boundary of the paleoplate is the Rivera fracture zone; its southeastern boundary a now-inactive transform fault, the West O'Gorman fracture zone. In this case, as well as in other more poorly documented ones, relict landforms, and magnetic patterns are carried on the aging lithosphere, away from the spreading ridge, recording a former geometry. ¿ American Geophysical Union 1988