What exactly was William Herschel's Milky Way and how did he construct it?

William Herschel is famous for his "Construction of the Heavens," his pioneering quantitative view of the Milky Way based on extensive star counts ("gages") carried out while sweeping with his 20-ft telescope in the early 1780s. In 1785 he presented his main result in a single iconic diagram - a view of the Milky Way that was to dominate for over a century until "Kapteyn's Universe" finally overtook it in the early 20th century. In this paper we analyze the entire data sample of Herschel's published and unpublished star gages in order better to understand the nature of the data and how he used them. We examine such issues as his method of observing, his limiting magnitude, his consistency in star counting, his sky coverage, and his distance scaling (in units of "visual rays"). We also discuss how representative was his lone published 2-D slice through his model Milky Way, and the effects of his basic assumptions (such as a uniform density of stars). Where insight can be gained, we make comparisons with modern star counts (such as those of the HST Guide Star Catalog). To assist in this effort we have employed 3-D visualization software, as well as a plastic model (to be exhibited) constructed by a 3-D printer illustrating many aspects of the Herschelian Milky Way.