Geometry: Euclid and Beyond

Rating:
Summary: Well beyond Euclid
Review: Hartshorne is a famous algebraist and one main contribution of this text is to show fascinating interrelations between classical geometries and modern algebra (of course the book contains lots of pure geometry as well). Example 1: Many texts show the impossibility of the classical problems of constructibility by straightedge and compass (by observing that the coordinates of any point so constructed lie in the smallest extension field of the rationals Q closed under taking square roots of positive numbers). Hartshorne's is the only text that goes further, solving the analogous problem when the straightedge is marked (real roots of cubic and quartic equations must also be allowed); Archimedes observed that any angle can be trisected with these tools. Example 2. Dehn's solution to Hilbert's Third Problem is given, whereby any two polyhedra equivalent under dissection must have equal Dehn invariants, and it shown that a tetrahedron has different invariant than a cube. Example 3. In hyperbolic geometry, Hilbert's arithmetic of ends is developed and applied. Example 4. Pejas' algebraic classification of Hilbert planes is discussed.

Hartshorne's text overlaps mine in correcting Euclid's errors, developing rigorous foundations for Euclidean and Non-Euclidean geometries, and covering much history, presented delightfully. He gives a thorough discussion of area and the open problems in that theory. He concludes with a nice chapter on polyhedra.

Rating:
Summary: A stunning book
Review: Hartshorne is a leading mathematician known for work in rather abstract geometry (see his book ALGEBRAIC GEOMETRY). He takes Euclid's ELEMENTS as great mathematics, no mere genial precursor, and collates it with Hilbert's FOUNDATIONS OF GEOMETRY.

Of course Harshorne proves that Euclid needed the parallel postulate, by exhibiting a non-Euclidean geometry. He gives a very pretty compass and straight-edge Euclidean theory of circles, which then turns into the Poincare plane model for hyperbolic geometry. He also proves that Euclid needed the method of exhaustion for volumes of solids: he gives the agreeably simple Dehn invariant proof that even a cube and a tetrahedron of equal volumes are not decomposable into congruent parts. It is a famous proof, rarely seen, and a beautiful use of the modern algebraic viewpoint in classical geometry. I had always supposed it must be hard but it is not.

Hartshorne also develops the contested "geometric algebra" of Euclid as a modern axiomatic algebra. Many commentators have shown it is wrong to think Euclid was doing "algebra" in the sense of a disguised theory of the roots of quadratic polynomials. But (unless and until Fowler's THE MATHEMATICS OF PLATO'S ACADEMY changes my mind) I think it is reasonable to say Euclid is doing algebra in this sense.

Rating:
Summary: Hartshorne's New Book, "Geometry: Euclid and Beyond", is a Masterpiece!
Review: I told my wife: "If I have to give up all my books but one, then this is the one I'd keep; no question about it." (More comments later.)

Rating:
Summary: Very good textbook but it repeats contemporary fallacies
Review: The book, like related literature, contends greater rigor in today's geometry or mathematics than in the time of Euclid. I challenge this and give some reasons why.

Examples occur in modern efforts to improve on Euclid's axioms. He, for instance, speaks in his proposition I.4 of applying one of equal figures to another so their parts coincide. This moving of figures is objected to (p.33 in Hartshorne) as not allowed by the axioms. However, equality, congruence, in figures is itself defined by their coinciding if placed upon each other. They are conceptualized accordingly, not requiring an axiom.

Most prominent to me are possibly fallacies connected with non-Euclidean geometries. One of their properties relied on is their understood consistency. This may be traced to past attempts at finding inconsistencies if the controversial parallel postulate is assumed false (see e.g. p.305 in Hartshorne). On finding no inconsistencies, it was inferred that the other geometries are logically valid. The inference commits the fallacy of "denying the antecedent". If inconsistency makes something invalid, it does not follow that consistency makes it valid. For example, inferring from "if A then B" that "if B then A" is not inconsistent, but it is invalid.

A further fallacy permeating non-Euclidean geometry is linguistic equivocation. Beside redefinitions (p.28) of terms like "line", now a straight line, or "curve", now including straight lines, or leaving such terms undefined (p.81), in perhaps confusing the following, concepts like straightness are reinterpreted to include various curvatures (e.g. pp.355-6). Moreover, these procedures are used to assert the parallel postulate unprovable, because they can contradict it. But one cannot make an inference about something, presently the postulate, by changing meanings of words in it. One is then speaking about something else.

Writing more elsewhere, I hope to have adequately touched on the subjects here.


Rating:
Summary: This book and course is not for the faint of heart!
Review: This is without exception the hardest math course I have ever taken. Your understanding of the concepts is pertinent. I had to read the 1st chapter over five times just to understand projective geometry. Hartshorne tries to simplify the material but only so much can be done. It is just a hard course, period. The book does contain many example and logical proofs but be ready to burn the midnight candle on this one.