We explore the design of a space mission called Project Lyman that has the goal of quantifying the ionization history of the universe from the present epoch to a redshift of z ~ 3. Observations from WMAP and SDSS show that before a redshift of z (Symbol not available. See manuscript.) 6 the first collapsed objects, possibly dwarf galaxies, emitted Lyman continuum (LyC) radiation shortward of 912 Å that reionized most of the universe. Theoretical estimates of the LyC escape fraction ( fesc ) required from these objects to complete reionization is fesc ~10%. How LyC escapes from galactic environments, whether it induces positive or negative feedback on the local and global collapse of structures, and the role played by clumping, molecules, metallicity and dust are major unanswered theoretical questions, requiring observational constraint. Numerous intervening Lyman limit systems frustrate the detection of LyC from high z objects. They thin below z ~ 3 where there are reportedly a few cases of apparently very high fesc. At low z there are only controversial detections and a handful of upper limits. A wide-field multi-object spectroscopic survey with moderate spectral and spatial resolution can quantify fesc within diverse spatially resolved galactic environments over redshifts with significant evolution in galaxy assemblage and quasar activity. It can also calibrate LyC escape against Lyα escape, providing an essential tool to JWST for probing the beginnings of reionization. We present calculations showing the evolution of the characteristic apparent magnitude of star-forming galaxy luminosity functions at 900 Å, as a function of redshift and assumed escape fraction. These calculations allow us to determine the required aperture for detecting LyC and conduct trade studies to guide technology choices and balance science return against mission cost. Finally we review our efforts to build a pathfinding dual order multi-object spectro/telescope with a (0.5°)2 field-of-view, using a GSFC microshutter array, and crossed delay-line micro-channel plate detector.