
Re: A question about straight lines
Posted:
Jan 30, 2014 2:02 AM


It is a little too much to credit them with the outrageous demise of the course that brought many of us to the study of mathematics itself but the idea is right. SMSG had it about right. Do include (a little bit sloppily) the Ruler Postulate and Protractor Postulate (neither of which involve actual measurement) that would have been anathema to the Greeks but, per Pappus and much more recent analysis, a consequence thereof, don't be too formal but be formal enough for high school students  in fact, for everyone not working in the formality subtopic.
Wayne
At 10:27 AM 1/29/2014, Domenico Rosa wrote: >On 29 Jan 2014, Neighbor wrote: > > > that is to say, initially we have AB and finally > > (after the division being made) we have A(C) C (C)B, > > and A(C) and (C)B are said to be the two equal > > straightlines in which AB has been divided, for the > > point C cannot be divided between them, having no parts. > >One of the notions of Euclidean Geometry is that a point has no >length. Therefore all four lines [A,C], [B,C] [A,C) and [B,C) have >the same length. The fact that point C "has no parts" has no bearing >whatsoever on this. > >The gap in Euclid's construction has nothing to due with point C. As >pointed out in > >Edwin E. Moise and Floyd L. Downs, Jr. >Geometry, AddisonWesley (1982) 680p. > >an additional postulate is required in order to guarantee that the >arcs of equal radii, involved in the construction, will in fact >intersect. The following are some of the additional postulates >listed in the above book, a book that played a key role in the >demise of high school Euclidean Geometry and in the subsequent >promotion of junk geometry. > >CHAPTER Page > 2. Sets, real Numbers, and Lines 15 > The distance postulate > The ruler postulate > The ruler placement postulate > Betweenness. The pointplotting theorem > The line postulate > 3. Lines, Planes, and Separation 49 > The planespace postulate > The flat plane postulate > The plane postulate > Intersection of planes postulate > The plane separation postulate > The space separation postulate > 4. Angles and Triangles 77 > The angle measurement postulate > The angle construction postulate > The angle addition postulate > The supplement postulate

