The modern approach is not lacking all geometry, and the 19th century presentation is not lacking all algebra. Any respectable treatment will include both, the

The modern approach is not lacking all geometry, and the 19th century presentation is not lacking all algebra. Any respectable treatment will include both, the question is which makes the relationship clearer? The abstract algebraic approach talks of 'the rotation group' which maps on to geometric concepts as cleanly and directly as possible. Then we talk of different parameterizations or representations of this group, with the unit quaternions being the 'double cover' of the rotation group (which gives rise to the primary difficulty in understanding them - that a rotation by 360 degrees reverses orientation, and 720 degrees returns us to where we started). I think the modern approach is much clearer - the geometric ideas appear more directly, and the algebra is far less messy.