12 1. Trisecting Angles Archimedes’s willingness to introduce the spiral in order to trisect an- gles underscores a major difference between him and the ancient Greeks. The School of Euclid was a school of the discrete and finite. The axioms of geometry were set and all proofs had to follow from these axioms in a finite number of steps. The famous Paradox of Zeno1 arises precisely because the Greeks were unwilling to allow infinite, continuous processes. Archimedes broke away from this long established tradition — the mark of a truly cre- ative mind. Though staying within the framework of acceptable logic, he forged ahead into the realm of infinite processes. Much of what Archimedes did was subsumed by the work of later mathematicians, especially after the invention of calculus by Newton and Leibniz. This might make it more dif- ficult to appreciate his contribution. After he invented calculus, Newton is quoted as saying, “If I have been able to see further than others, it is because I have been able to stand on the shoulders of giants.” Surely one of the giants he had in mind was Archimedes. For more on Archimedes see Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?, Mathematical Association of America (1999) and http://en.wikipedia.org/wiki/Archimedes. 1.5. Constructible points and constructible numbers In this section we begin the proof that some angles cannot be trisected. To do this we must first make mathematically precise what it is we are given. Of course we have our straightedge and compass, but we are not allowed to randomly draw lines and circles. We are only allowed to draw lines through points we have been given or that have been constructed. Therefore we have to start this process with two given points in the plane. We need at least two points to use our straightedge, and, though we need only one point to use as the center of a circle, we will need two points to determine a distance to use as a radius. We will take as the two given points the origin, O = (0, 0), and the point I = (1, 0). This is all we are given. From here we are allowed to construct any lines and circles we wish, as long as: (a) we only draw a straight line connecting two points that are either the two given points or points that have been previously constructed (b) we only draw circles with a center that is a given point, O or I, or some point we have previously constructed and the radius used is the distance between two of the given or constructed points. Let us start. First we can draw the line connecting O and I — the x-axis. This is an example of a constructible line. Now draw a circle using O as the center and radius |OI| = 1. This is an example of a constructible 1See http://en.wikipedia.org/wiki/Zeno’s paradoxes for the paradox and its history as well as refutations.

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