I'm interested in learning the origin of compass/straight-edge constructions. In particular, I am interested in the historical interplay between Euclid's axioms for plane geometry, and compass/straightedge constructions: Were the axioms designed to formalize the process of using a compass and straightedge? Or were the compass and straightedge created as tools with which to practice the operations allowed within Euclid's geometry? Neither, perhaps?
Any resources or information is much appreciated.
Euclid never uses the words straightedge and compass; his axioms include the idea that we can draw a circle of any known radius at any known point, and that one can extend any line indefinitely.
These axioms are purely mathematical in nature, and can be given non physical interpretations (as in hyperbolic geometry or finite planes). However, they can also be interpreted physically by saying that the geometer has access to a compass and a straightedge. So in this sense, they are a formalization of the compass and straightedge.
Euclid did not invent the compass or the straightedge; compasses and straightedges were used before Euclid for many purposes. In fact, they had better tools. A labelled straightedge (i.e. a ruler) allows one to do much more, including trisect an angle, and this was known to the Greeks. Euclid chose his axioms not for their mathematical power, but for clarity of development. The Greeks valued the ideal and abstract more than the practical.
Here is an interesting link describing how Egyptians and others handled compasses and rulers: http://php.math.unifi.it/users/archimede/archimede_NEW_inglese/curve/curve_giusti/prima.php?id=1