Lecture 16 We first ended the discussion of Book I of Euclid's Elements. Namely, we proved Proposition 47 (Pythagoras' Theorem), following Euclid's original argument ("the windmill proof"). We then gave an alternative proof using similar triangles (read the commentaries). After that we finished Book I by proving Proposition 48, that is the converse of Pythagoras' Theorem: Proposition 47 Proposition 48. We then passed to (the much shorther) Book II (2 Definitions and 14 Propositions). In this book (together with the end of Book I and Book VI) Euclid describes what is commonly called Geometric Algebra. After some inspection we noticed that this approach reminds us of the methods that Babylonians used in solving algebraic problems. Here, though, proofs are supplied! We gave two samples of geometric algebraic statements by commenting (without proofs) the following propositions: Proposition 1 Proposition 4. Finally, we talked about Book III (11 Definitions and 37 Propositions), which deals with the properties of circles. We proved the following well known propositions from our high-school days: Proposition 1 Proposition 20 Proposition 21 Proposition 22. Click on the above links to go directly to the proofs and commentaries of the results.