Geometrical Exposition

In the Second Set of Objections to the Meditations, Descartes is asked to “set out the entire argument in geometrical fashion, starting from a number of definitions, postulates and axioms” (see Objections and Replies). The objectors then add, “You are highly experienced in employing this method, and it would enable you to fill the mind of each reader so that he could see everything as it were at a single glance, and be permeated with awareness of the divine power” (AT VII 128, CSM II 92). Mersenne and his fellow objectors, in effect, are asking Descartes to do for his metaphysics what Euclid had done for geometry. Euclid in The Elements systematizes geometry by using what came to be known as the geometrical method. That is, by using a small set of stipulated definitions, axioms, which are taken to be self-evident, along with postulates, which are sometimes understood to be assumptions that are not self-evident (Heath 1926, 1:123–24), together with the rules of deduction, various theorems can then be demonstrated.Descartes replies to the second set of objectors’ request by making a distinction between the order and the method of demonstration when writing in a “geometrical manner.” The geometrical order, as he describes it in both the Second Replies and the synopsis to the Meditations, is one by which claims or items that come first must be entirely known without the aid of those that come later in the demonstration (AT VII 154, CSM II 110; AT VII 4, CSM II 5). In turn, what comes later in the demonstration must rely solely on what came before. Some have wondered whether Descartes means these items to be ontologically or epistemically prior – that is, first in the order of being or first in the order of discovery (CSM II 110 n. 2 and 112 n. 1). However, given that Descartes proves God's existence in the Third Meditation and his own existence, which depends on God, in the Second Meditation, it seems fair to assume that he means first in the order of discovery and not in the order of being.