 The rules for synthetic demonstrations lead to a list of criticisms of “geometers”. Chapter 4 notes some equivocations in Euclid’s use of “angle”, “ratio”, and “proportion.” Later mathematicians are criticizes for not respecting the distinction between nominal and real definitions. The former should be stipulations about how words are associated with ideas. The latter should characterize the nature of the object being defined. Examples are given about disputes among prominent mathematicians over nominal definitions as if they were real (IV.5). Another mistake the authors find geometers making is “proving things that do not need to be proved” (IV.9, 254). This mistake is attributed to inattention to the connection between truth and clear and distinct perception. Euclid proves that that any two sides of a triangle are greater than the third side. Because this is immediately evident, the proof violates method by mis-ordering what is known. Another mistake Euclid makes in ordering is tacking back and forth from more general to more specific cases; a number of examples are provided. Proper order requires always moving from more simple items to more complex ones. A more significant disagreement with geometrical practice is about indirect proofs by reductio ad absurdum. Because the denial of a true proposition is always confused, the derivation of a contradiction will usually offer no insight into why the conclusion is true (IV.9, 255–56). It is conceded, however, that indirect proofs might be necessary when no direct proof can be found, and they can be useful when a result is already obtained directly.