the hierarchy of geometries

Klein's program does more than describe each geometry separately; it stacks them. Some geometries are more permissive than others — they let you do more to a figure and therefore notice less about it. Lining the geometries up by how permissive their transformation groups are gives a tidy tower: at the top sits projective geometry, the most permissive, and beneath it, each a special case of the one above, come affine, then similarity, then Euclidean geometry at the bottom, the strictest.

The organizing principle is plain once you see it: if the transformation group of geometry A is contained inside the transformation group of geometry B, then B is more permissive and A is finer. For the plane, the Euclidean group of rigid motions sits inside the similarity group (which adds uniform scalings), which sits inside the affine group (which adds shears and non-uniform scalings), which sits inside the projective group (which adds perspective maps sending parallel lines to meeting lines). Each step up enlarges the group and so discards invariants: from Euclidean to similarity you lose absolute length but keep shape and angle; from similarity to affine you lose shape and angle but keep parallelism and ratios along a line; from affine to projective you lose parallelism but keep incidence and the cross-ratio. The smaller the group, the richer the geometry — Euclidean geometry sees the most because it allows the least.

This nesting is what lets a single theorem live in several geometries at once and explains why some results 'belong' to one level. A statement about incidence and collinearity is really projective, true in every geometry below as well, while a statement about angles is genuinely Euclidean and meaningless higher up. A frequent confusion is to picture the hierarchy upside down, thinking the 'biggest' geometry is the most powerful; it is the reverse — a bigger group means a coarser, weaker geometry with fewer invariants to work with.