Here in the wikipedia article on quaternions, the following is written:
In general, let p and q be quaternions and write $${\displaystyle p=p_{\text{s}}+p_{\text{v}},}$$$$ {\displaystyle q=q_{\text{s}}+q_{\text{v}},}$$ where $p_\text{s}$ and $q_\text{s}$ are the scalar parts, and $p_\text{v}$ and $q_\text{v}$ are the vector parts of p and q. Then we have the formula $${\displaystyle pq=(pq)_{\text{s}}+(pq)_{\text{v}}=(p_{\text{s}}q_{\text{s}}-p_{\text{v}}\cdot q_{\text{v}})+(p_{\text{s}}q_{\text{v}}+q_{\text{s}}p_{\text{v}}+p_{\text{v}}\times q_{\text{v}}).}{\displaystyle pq=(pq)_{\text{s}}+(pq)_{\text{v}}=(p_{\text{s}}q_{\text{s}}-p_{\text{v}}\cdot q_{\text{v}})+(p_{\text{s}}q_{\text{v}}+q_{\text{s}}p_{\text{v}}+p_{\text{v}}\times q_{\text{v}}).}$$
Which I have no issue with. But then they go on to write:
Hamilton showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in Elliptic geometry.
For which I need more context to understand what it means. Are they saying that if $p_\text{v}$ and $q_\text{v}$ correspond to two vertices, then $(pq)_\text{v}$ is somewhat related to/corresponds to the third vertex? And how exactly are the arc-lengths included in this relation? The citation provided there is hard to comprehend for me, so I decided to ask here (I do not have any knowledge of quaternions prior to reading that much of the Wikipedia article). I am not asking for a proof, just want to know what the theorem is.
