Definition of a Tangent Plane We all have and intuitive idea of what the tangent plane to a point on a given surface should be. It is the plane that goes through that point and that it is tangent to the surface at that point. Suppose. z = f(x,y) is a smooth surface above the xy-plane (see the picture below).
The two prokaryotic domains (or empires),  Bacteria  (or  Eubacteria ) and  Archaea  (or  Archaeobacteria ), are subdivided into 35 phyla (or divisions): 30 phyla in the domain  Bacteria , and 5 phyla in the domain  Archaea .
Garrity et al. (2001) separate the Eubacteria (a group that they call Bacteria) into 23 groups. Also, the problems of lateral gene transfer further blur the distinctions of the groups. We present a tentative system for the Eubacteria with 9 kingdoms (see below). This system is based largely on Margulis and Schwartz (1998), with modifications from Garrity et al. (2001, 2003, and 2005), Tudge (2000), and Black (2002). The All-Species Living Tree Project (Figure 2; Yarza et al.