History of Compass/Straight Edge Construction

Solution 1:

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

Solution 2:

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].