  The discussion of how to apply geometric algebra to euclidean $n$-space has
been clouded by a number of conceptual misunderstandings which we first
identify and resolve, based on a thorough review of crucial but largely
forgotten themes from $19^{th}$ century mathematics. We then introduce the dual
projectivized Clifford algebra $\mathbf{P}(\mathbb{R}^*_{n,0,1})$ (euclidean
PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry.
We compare euclidean PGA and the popular 2-up model CGA (conformal geometric
algebra), restricting attention to flat geometric primitives, and show that on
this domain they exhibit the same formal feature set. We thereby establish that
euclidean PGA is the smallest structure-preserving euclidean GA. We compare the
two algebras in more detail, with respect to a number of practical criteria,
including implementation of kinematics and rigid body mechanics. We then extend
the comparison to include euclidean sphere primitives. We conclude that
euclidean PGA provides a natural transition, both scientifically and
pedagogically, between vector space models and the more complex and powerful
CGA.
