The Posterior Analytics is a rather dull and uninspired work even by Aristotelian standards. Aristotle's trademarked method of pointless classification is here running at high gear at its pointless best. Although the relation between Aristotle and mathematicians such as Euclid is never made explicit in the historical record, it seems clear to me that the mathematicians owe nothing to Aristotle, and that the Posterior Analytics is an awkward attempt at saying something about the geometrical method by an outsider who is not really attuned to it. To understand the essence of the geometrical method one will be better off reading philosophers with a natural affinity with mathematics, such as Plato and Descartes. Aristotle puts it well when he says that "you should not argue about geometry among non-geometers---for those who argue poorly will escape detection" (77b). Unfortunately, generations of geometrically ignorant readers have ignored this sound advice and ended up greatly overestimating this rather trifling treatise.
Be that as it may, the essence Aristotle's view of the axiomatic-deductive method is summed up in the following sentence: "Demonstrative understanding ... must proceed from items which are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusions." (71b)
Three notable consequences of this thesis are:
The axiomatic-deductive method is much more than mere logic. "There can be a deduction even if these conditions are not met, but there cannot be a demonstration---for it will not bring about understanding." (71b)
There is a fundamental distinction between "demonstrations which are said to demonstrate and those which lead to the impossible" (85a), i.e., proofs by contradiction, which must be seen as intrinsically inferior (87a).
Axioms stem from perception. "I call prior and more familiar in relation to us items which are nearer perception" (72a), so immediate perception must be the ultimate foundations of "demonstrative understanding." "We must get to know the primitives [i.e., axioms] by induction; for this is the way in which perception instills universals." (100b) However, "for the principles [i.e., axioms] a geometer as geometer should not supply arguments" (77b). Note the two coextensive words for "axiom"---indeed, "I call the same things principles and primitives" (72a), since immediately given truths and logical starting points of a deductive system should be the same thing.