TITLE: History of Compass/Straight Edge Construction
QUESTION [11 upvotes]: 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.

REPLY [2 votes]: First of all, just as one distinguishes the Straightedge from the Ruler, one must distinguish the Collapsible Compass from a Compass in general; the idea here is that one cannot "carry" a fixed Distance, or Segment, by lifting the Compass completely off the drawing surface. This is just as important as not having "marked" Ruler. In fact, this distinction should remind one of Descartes' revolutionary move in which he introduced the Unit (=!) and thereby laid the foundation (first step) of Analytic Geometry. In short, one can Construct more Figure with the "regular" Compass which permits effectively to move a "unit" Segment from one place to another. In Euclidean terms, it means being Given (a "Datum"/"Data" in Latin) a Finite Line equal to another. But I'm not prepared to show here exactly how this is a significant restriction, since with a Collapsible Compass one can Multiples and Fractions of any Diameter.
Second. In fact, your question involves what one may call simply another (parallel [no pun intended]) Language of Geometry. The Greeks had a sense of Perfection. So one should imagine that these two tools were absolutely exact. Curiously, the ancient Greeks did not postulate the a special Pen (or Pencil) which would permit one to draw a Point. One cannot write a point with a Straightedge, or a Ruler for that matter, and the instrument used to draw actually the line is implicitly assumed to be a part of the Straightedge. Furthermore, one can argue that any kind of Compass permits one to draw a Center alone, which is the single point. So perhaps the hidden premiss here is that every Compass gives rise to a Center and a Circumference Point. But one can also argue that the Compass always produces a Center and at least an Arch of a Circle determined by the previous Center - in other words, a compass doesn't just produce two points; it produces a Given Pair, a Center and the whole Circle, or a Part of a Circle.
A great book on your question is a small 20th century anthology, "Squaring the Circle," in which the lead author is E. W. Hobson: [http://www.worldcat.org/title/squaring-the-circle-hobson-ruler-and-compass-hudson-the-theory-and-construction-of-non-differentiable-functions-singh-how-to-draw-a-straight-line-a-lecture-on-linkages-kempe/oclc/258683688].