It’s all Greek to me

Famous Problems of Geometry and How To Solve Them by Benjamin Bold

When I studied for my mathematics degree, we were in the period analytic geometry and so we didn’t do a lot of study of the constructions using compass and straight edge for which the Greeks were famous. This book fills in some of the gaps in my knowledge, covering the ways the Greeks (and Gauss for some of the later material)  had for constructing lengths using a compass and a straight edge.

Chapter one covers the construction of ab, a/b, a^2, sqrt[a] and the roots of the quadratic equation x^2-ax+b=0, the first three using similar triangles, the sqrt using a semi-circle and Thales’ theorem and the roots using a circle whose diameter goes from (0,1) to (a,b).

Chapter two covers the analytic criteria for constructability, in particular the theorem: If a cubic equation with rational coefficients has no rational root, then none of its roots is constructible. This is used in later chapters on the Delian problem and the trisection of an angle, after a chapter on the complex numbers. There is then a chapter on squaring the circle.

This is followed by a chapter on the construction of regular polygons, covering the Greeks and their construction of regular polygons of  n sides where n = 2^m 3^r1 5^r2 where m is any non-negative integer and r1 and r2 are 0 or 1. This then goes on to cover Gauss’ work in this area.

The book has great explanations, and a set of exercises that take you through the proof of the constructions, and which also offer extra useful information. It gets a little dated in the end where it talks about the potential proofs of Fermat’s Last Theorem, but that’s the only bad thing I can say about it.

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