It seems like math could only have been developed to serve entirely practical purposes, like building, measuring, trading, and managing labor (like we talked about for the megamachine that was building the pyramids). It is amazing that a simple compass could hold such an abstract meaning to ancient Greek geometers.

The theory connected with this thing created the level of abstraction to inspire millenia of mathematical development. The abstract algebra required to completely understand what simple circles and lines can do is only a modern tool. If certain open problems had not motivated its development, who knows when it would have come around.

The ability of an artifact to change the path of mathematical fact is based on its theoretical restrictions, described below.

The Rules, according to Euclid

Postulate I. Let it be granted, that a straight line may be drawn from any one point to any other point.

Postulate II. That a terminated straight line may be produced to any length in a straight line.

Postulate III. And that a circle may be described from any center, at any distance from that center.

Of course "any" means "any that exists already" or "any that can be constructed."

Though Euclid stuck to these rules in his problems and theorems, such tight restrictions in studying constructions were probably not emphasized. According to the late Stanford professor Wilbur Richard Knorr, a scholar of the history of Greek mathematics,

"It is imperative, I maintain, to raise these doubts about the formal nature of the work of Oenopides and his contemporaries. While we might consider it obvious and natural to classify constructions according to the means employed and to assign privileged status to those demanding only compass and straightedge for their execution, our intuitions in such matters are thoroughly conditioned through knowledge and adoption of the objectives of formal geometric tradition, advanced primarily through the works of Euclid and Appollonius. But in fact this formal restriction on the treatment of problems in itself betokens attainment of a sophisticated theoretical level."

True, geometers used other instruments such as set squares, forms of cocmpasses and sectors, angle-measuring devices, plummets, and sliding marked rulers long after Euclid. However, these are the rules he followed, and the ones we connect with a theoretical compass and straightedge. It is unclear when they developed or when mathematicians decided certain restrictions were "ideal."

However, "Plato viewed as heresy the use in geometry of any instruments other than the compass and the straight-edge, or--what is the same thing--the use of loci other than the circle and the straight line...Plato maintained that all of that 'spoils and destroys the good of geometry, for geometry thus strays away from incorporeal and mentally perceivable things and moves toward sthe sensorial, making use of bodies that are needed in the application of instruments of vulgar handicrafts'" (Smilga, 54).

The Theory of the Straightedge

In theory, all a straightedge can do is draw a line or line segment which connects two given points. A straightedge IS one-dimensional and infinite in length. A straightedge CANNOT measure lengths, construct right angles, have any width or shape.

In 1833, Steiner proved that everything that can be constructed with compass and straightedge can be constructed with a straightedge only, as long as you are given a single circle and its center to begin.

The Theory of the Compass

The only thing a compass can do is mark a circle about a point and through another point. A theoretical compass is automatically COLLAPSIBLE, meaning that after it constructs a circle of some length, it collapses, and cannot mark off that length or circle somewhere else. No matter how far apart two points are, the compass can construct a circle with their distance as the radius. This means the theoretical compass is infinitely big. It also has to make arbitrarily small circles, so the pivot point must touch the pencil when the compass is closed.

Except for connecting vertices, a compass alone can construct everything that a compass and straightedge can construct together. It was proven by Mascheroni in 1797.

Note that the items we use as compass and straightedge do very little of what they theoretically should be able to do when it comes to size. However, our instruments can do a lot more than they should be able to, like mark off and measure lengths. This shows that the purpose of construction is not just to draw perfect pictures, it is also to prove theorems. For all practical purposes, one could square the circle using (16/9)^2 as an approximation for pi. In fact, that is what early engineers did. The introduction of compass and straightedge in the study of geometry meant a change from the practical uses of math to abstract explorations. This artifact is the beginning of pure math!

Back to my project---> Compass and Straightedge