In Second Replies, Descartes draws a distinction between two methods of demonstration: analysis and synthesis. He nowhere offers a formal definition or account of the two methods, but he does make claims throughout his corpus, but especially in the Second Replies, that provide important clues as to the details of their nature. For example, he identifies analysis as a method of instruction, and he says that indeed it “is the best and truest method of instruction” and is the method that he employs in the Meditations. He says that synthesis “is very suitable to deploy in geometry” (AT VII 156, CSM II 111) and that it characteristically involves the presentation of a series of definitions, postulates, axioms, and theorems that together form a deductive chain of reasoning that forces even the most stubborn of minds to affirm its conclusion (AT VII 156, CSM II 110–11) (see deduction). He makes additional claims as well: that analysis is a version of a method that was highly regarded in ancient geometry; that it helps us to have clear and distinct perceptions of the primary notions of metaphysics; and that it is a method of discovery (AT VII 155–57, CSM II 110–12). He says that synthesis and analysis are complementary methods but one difference is that a successful analytic demonstration does not compel our assent (AT VII 156, CSM II 110–11).

Descartes draws a further distinction between the method of demonstration and the order of demonstration. Both analysis and synthesis must employ the proper order: claims that are put forward initially cannot depend for their support on claims that come later, and claims that are derived thereafter must depend solely on claims that have already been established (Gueroult 1984, 1:8–11). Descartes emphasizes that in the Meditations he tried to adhere to this order: in the First Meditation he refrains from affirming claims that are dubitable, and when he does finally stand behind metaphysical principles, they are either primary notions that are known through themselves or the conclusions of arguments whose premises comprised such notions (AT VII 155, CSM II 110). Any method must adhere to the proper order, and so it is in other respects that analysis and synthesis diverge.