The central and eastern United States has experienced only 5 historic earthquakes with Mw 7.0, four during the New Madrid sequence of 1811-1812: three principal mainshocks and the so-called «dawn aftershock» following the first mainshock. Much of the historic earthquake research done in the United States has focused on the New Madrid Seismic Zone (NMSZ), because the largest New Madrid earthquakes may represent the archetype for the most damaging earthquakes to be expected in intraplate regions. Published magnitude values ranging from 7.0 to 8.75 have generally been based on macroseismic effects, which provide the most direct constraint on source size for the events. Critical to the interpretation of these accounts is an understanding of their historic context. Early settlments clustered along waterways, where substantial amplification of seismic waves is expected. Analyzing the New Madrid intensity values with a consideration of these effects yields preferred values of Mw 7.2-7.3, 7.0, and 7.4-7.5 for the December, January, and February mainshocks, respectively, and of 7.0 for the «dawn aftershock». These values are consistent with other lines of evidence, including scaling relationships. Finally, I show that accounts from the New Madrid sequence reveal evidence for remotely triggered earthquakes well outside the NMSZ. Remotely triggered earthquakes represent a potentially important new wrinkle in historic earthquake research, as their ground motions can sometimes be confused with mainshock ground motions.