Editor's Remarks

Mathematically a real challenge. You have 2000 years of mathematics to deal with here. We provide the algebraic framework for the 19th century solution in a mathematics course, 351, modern (abstract) algebra. This topic contains more than enough materi a l for a full one semester course.

The ancient Greek work on these problems is quite complex, and carried out without the aid of algebra. The best presentation I've seen of this material is van der Waerden's account in Science Awakening, which gives the clearest and most accurate account of Archytas' cube duplication, and the thinking behind it.

It would have been nice to incorporate some discussion of how Descartes' analytic geometry made was the link that made the algebraic reformulations of all these problems directly relevant. The point is, it is not enough to know what the problems mean al g ebraically, one must also understand what a "plane" solution means ( essentially, a solution gotten using only elementary arithmetic and the extraction of square roots. This is something that Vihte also understood, and wrote about, though as you say the natural analysis that follows from that was not actually carried out until the time of Wantzel (though really 1837 is very late ( by that point a number of people must have understood all of this).

I don't know why you quote a website for the treatment of the trisection of an angle when we did that as an exercise in class, which you did successfully. It seems a bit perverse to quote an authority for something you know perfectly well how to do your self.

Referring to Deepak's essay, this is the third point that Scott raises about algebraic geometry ( the conversion of one system into another, in this case the conversion of the system of Euclidean constructions into the arithmetic of square roots. References

Problematic. The underlying problem is that we settled on the specific focus of your paper rather late, and there are definitely problems coming up with good sources for this set of ideas. The best seems to be the text of van der Waerden, which I thought was going to make it into the second draft ... .

The Heath and Gow certainly have their uses here, and are helpful. The Knorr could be very interesting but is tough going and can't be used in a casual way. The web sites are not very helpful.

For the period after the ancient Greeks, we don't have much to go on. There is an account at a more advanced level in Hungerford's college algebra textbook, Chapter 15. That may be the best way to approach it, once one has the background.

The difficulty of course is why irreducible cubic equations correspond to nonconstructible numbers. Granted that, the rest all makes sense. (The transcendence of p however is a much deeper matter.)

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