# How did Aristotle influence Euclid?

In other words, how is Aristotle's logic represented in Euclid's Elements?

I have read many articles where Euclid's Elements is linked to Aristotle's logic, but I do not understand, and I can't find any examples explaining how deductive logic (i.e. syllogism) or laws of thought or any other work from Aristotle could have influenced the development of geometry and the concepts Euclid came up with when writing his book.

• Can you give an example of such an article? – Mikhail Katz Dec 10 '17 at 11:29
• @MikhailKatz The article Logic and Mathematics, by Stephen G. Simpson, contains the sentence “The methodological ideas of Aristotle decisively influenced the structure and organization of Euclid's monumental treatise on geometry, the Elements.” – José Carlos Santos Dec 11 '17 at 13:15

As Simpson clearly states in the section "Aristotelian logic" of his article, principle 3 is the law of excluded middle: $P \vee \neg P$ is true. Euclid's Elements indeed rely on classical logic rather than intuitionistic logic. There have been attempts recently to rewrite Euclid in an intuitionistic framework; see e.g., the work of Michael J. Beeson.