Each visible finger is shaped and sized differently and positioned differently and, depending on the type of work one engages in, soft or rough. But each, the mind perceives, participates in the one unchanging Form of the Finger. Mathematics thus can lead the philosopher warrior from practical application to pure formal entities and relationships. These will bring her closer to eternal truth. Geometry is added on, and solid geometry would be if more headway can be made in it.
Astronomy is the next study.
Here the position and movement of the planets is useful for navigation and also generalship. Socrates is reluctant to say that the study of astronomy has led souls to higher intellection, because stargazing without understanding cosmic laws of motions is as ignorant as shadow gazing. One can only guess what Plato thought of the mythology and astrology associated with the planets and the stars. True astronomy leads you to the Demiurge who rationally crafts the universe and sets it on a harmonious continuous direction. That there is a rational cosmic order inspires the philosopher to suspend her contemplation of it and work in public life to establish a civic order that reflects it.