Plane Geometry - Bruce E. Shapiro

Section 7Euclid’s ElementsHere and in the following sections we willpresent some of the axiomatic systems thathave been used to develop geometry in thewest since Euclid’s time. We will not be usingany of thse specific systems in our own developmentbut they are worth being familiarwith.Euclid’s Elements was written around 300 BC.It consists of 13 Books that are a compilationof geometric knowledge that had beed acquiredover the previous several centuries. Wedon’t know what (if any) parts of it are originalto Euclid. The great contribution of thisdocument is that it sets forth a set of basic assumptions(five postulates) from which all ofthe remainder of the treatise is logically derived.Euclid at the Oxford HistoryMuseum.Any logical system requires one to first define some basic concepts whichare accepted on faith; for geometry these are thinks like points and lines.In fact, Eculid states 23 definitions before his postulates, and follows themwith five additional common notions, which are statements that he expectsare so obvious they can be accepted by anyone with reason.It is worth looking at these basic statements to get a feeling for whereEuclid’s geometry starts, so they are extracted below[Euclid].29