We might divide Geometry into two parts:
- (Geometry of Euclid) The Geometry that can be done using a Compass and a straight edge to make drawings
- This earlier version of Geometry encompasses the basic Objects introduced in a High School Geometry class (and you probably learned most of them earlier).
- (Geometry of Descartes) The Geometry that makes use of graph paper and rulers.
Geometry calculates lengths, areas and volumes for triangles, squares rectangles, circles and spheres.
Geometry includes the study of parallel lines, bisectors, parallelograms, opposite angles, adjacent angles and corresponding angles. Be careful to note what changes when we go from two dimensions to three dimensions.
Trigonometric functions are introduced in Geometry sine cosine tangent. These functions are based on right triangles.
The Pythagorean theorem relates the lengths of the sides and hypotenuse of a Right Triangle.
Students get practice with Proofs as they study triangles and triangle congruency. We might argue that “Proofs” is not a topic in Geometry, but that Geometry “adopted” Proofs when it was found that Geometry ideas could be used to introduce Proofs.
Propositional Logic, another foreign subject, is introduced, along with Operators for Conjunction, Disjunction and Negation.
Appendix A
You may have a gut feeling that the distinction between Euclid and Descartes isn’t as simple as our two short sentences suggest.
Yeah. You can take your Compass and your Straight Edge, and you can draw a point that you call the origin and then you can go along the line making equidistant intervals corresponding to 1 2 3 4 and at that point you have a metric. You can then tell me, hey using Euclid I created a metric. But metric is a Descartes thing. I have to tell you that you crossed the line, oh, pardon the pun.
We have a problem once you know anything about what Descartes did, it makes so much sense, you can never take your brain back to that point where we didn’t have it.
Rob Sterling 