[ High School and Self-Education Board ]
Posted by Myrtle on 15:18 Jan 11
In Reply to: This is probably a stupid question, showing how little I remember my geometry, but... posted by Nan in Mass
The basic idea behind axiomatic geometry is that you start with a few axioms use those as premises in syllogisms that result more and more sophisticated conclusions. You create a domino effect in logic: You use the result of one theorem as the premise for the following and that last domino (theorem) that you get to is supposed to be some interesting result.
If the author has you do "proofs" in which he simply gives you new premises (postulates) to assume rather than requiring you to base your premises on prior results then you are really losing a lot.
There is some discussion about the ideal set of axioms (premises) to begin with. The short list of geometry programs out there that are actually written by mathematicians are:
1. Birkhoff and Beatly. You can look up Birkhoff in Wiki.
Sample pages:
http://books.google.com/books?hl=en&id=fMULJZsTTWkC&dq=birkhoff+beatley+geometry&printsec=frontcover&source=web&ots=o6bzk-wkhM&sig=xpSPpIBVIZb9QTDl-gXE6XTozmU
We bought our copy here:
http://www.ams.org/bookstore?co1=AND&co2=AND&co3=AND&d=BOOK&f=G&fn=105&l=100&op1=AND&op2=AND&op3=AND&p=1&pg1=&pg2=&pg3=ALLF&r=3&s1=&s2=&s3=Geometry%2BBirkhoff%20Beatley&subject=genint&u=
The teacher's manual is very useful and also explains scheduling.
2. Kiselev. Recently translated to English this book was used for many decades in the Soviet Union:
We got our copy here:
http://www.sumizdat.org/
Sadly, there is not answer key.
3. Pre-1970s Moise and Downs based on the SMSG axioms. Good luck finding one! I talked an old math teacher out of my copy.
4. Solomonovich. - order from author's website.
Solomonovich has an awesome article about what to look for in a Geometry program that includes reviews of Jacob's (interesting to see what a PhD research mathematician opines about geometry books) A few years ago I read through the whole article and it raised many questions for me.
http://www.solomonovich.com/geometry/textbook.html#1a
PS I have a copy of Moise and Downs and it's recommended by Barry Garelick from NYC Hold as one of the best geometry books that he's come across. Solomonovich quotes Kline as saying that the axioms are "sloppy" and I suppose I'll have to figure out what that is supposed to be all about. (Morris Kline was a crackpot in so many ways though.)