Welcome to an America you've never seen. Where anyone can drop by the White House and visit the President between 10 a.m. and noon; where cowcatchers are bloodied daily on train tracks between New York and Boston; where spent bullets are strewn across Civil War battlefields, and Indians still roam Yosemite Valley; where pigs rut in the sand-and-clay streets of Washington, DC., and the weather-bleached skeletons of oxen and horses line the old mail roads across the West.

For three hot summer months in 1869, Ernst Mendelssohn-Barthody, the nephew of famed composer Felix Mendelssohn,...