A letter signed L. Cassegrain described a concave primary and a convex secondary that sends the beam back through a hole in the primary. The tube is short for its focal length. Newton published a sharp reply about spherical aberration; the name still stuck.
Who Cassegrain was is still a bit thin. The layout is not: Hubble’s main optics are a Cassegrain cousin. On this bench the return beam comes to a focus on a screen behind the primary, having crossed the hole bored through it.