The next couple of sections lay, based on Einstein's suggestions, the measurable foundations needed for the discussion of relativistic effects coming afterwards. Although almost all of the concepts and results here seem intuitive (and even banal time to time), they will be of the utmost importance for clarity and understanding when proceeding further.
It's also demonstrated how cumbersome things can get when the classical ground is cut from under one's feet, i.e. when only those things can be taken for granted that either have been measured or were inferred from the symmetries of a given situation. Putting it another way, we're going to describe what one can say about space and time without blindly assuming anything that isn't supported by tangible evidence, while at the same time making no use of the constancy of the speed of light. The latter will be used only afterwards, in the parts on relativistic effects.
We do make one assumption though to start with: that the geometry in every inertial frame is Euclidean. Keep in mind that our final goal is an adjustment to classical mechanics, as minimal as possible, that eliminates all contradictions posed by the constancy of the speed of light. And the Euclidean-ness of geometry is not among the top suspects to doubt.
Lastly, the word "symmetry" appears in many of the proofs. In the context of an argument, it refers to a kind of reasonless-ness, the common sense that two things must be identical if there is no sensible reason for difference. It indicates a level where we still trust our intuition.
Note: the basis of all physical symmetries within a single inertial frame is the principle of relativity, as it ensures that the inertial frame can be considered on its own, while properties like its motion relative to other, maybe even "favored", inertial frames do not matter.