If $G$ is a group acting on a set $S$, then the "orbit" of a point $x$ in $S$ is defined as the set of all elements of the form $gx$ where $g \in G$. My question: why was the word "orbit" chosen for this concept? It is not obviously related to previous uses of this word, such as the path of a planet around the Sun.
It seems the concept goes back further than D.W. Hall (a student of Whyburn) - at least to Kuratowski, cf. the following excerpt from Kuczma et al. Iterative functional equations, p. 14. $\quad $ One can probably find further historical details by Googling "Kuratowski-Whyburn orbit" etc.
Note: for a simple yet enlightening example of the key role that orbits play in the solution of functional equation see this post.