Plane Geometry - Bruce E. Shapiro

30 SECTION 7. EUCLID’S ELEMENTSEuclid’s Axioms1 Let it have been postulated to draw a straight-line from any point to any point.2 And to produce a finite straight-line continuously in a straight-line.3 And to draw a circle with any center and radius.4 And that all right-angles are equal to one another.5 Euclid’s Parallel postulate. And that if a straight-line falling across two(other) straight-lines makes internal angles on the same side (of itself whosesum is) less than two right-angles, then the two (other) straight-lines, beingproduced to infinity, meet on that side (of the original straight-line) that the(sum of the internal angles) is less than two right-angles (and do not meet onthe other side) (see figure 7.1.)Historically the 5thpostulate has beenconsidered separatefrom the first fourand was followed byover 2000 years of attemptsto derive itfrom them. It wasnot until the 19thcentury (by EugenioBeltrami in 1868)that it was shownthat postulate 5 couldnot be derived fromthe other four. In the process, the existence of non-Euclidean geometrieswas proven, as well as neutral geometry, the study of the consequencesof the first four postulates.Euclid presumably stated his common notions to make clear what assumptionshe was making that he thought were obvious. Today we would probablystate them using algebra, but such expressions had not been inventedyet.0 Euclid’s line is what we call a plane curve.0 Euclid’s straight-line is what we would call a line segment. The modern concept of aline that extends infinitely in each direction was unknown to Euclid.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012