What is the axiomatic approach in Euclidean Geometry?

Answer 1

In Euclid's version, all geometric theorems are deduced from just ten assumptions divided among five axioms and five postulates. Euclid's axioms (also called "common notions") are algebraic statements such as " equals added to equals are equal". The postulates are geometrical in nature and thus embody the essence of Euclidean geometry.

Euclid's postulates are:

1. Any two points can be joined by a straight line.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. Parallel postulate: If two lines intersect a third line in such a way that the sum of the inner angles on one side of the third line is less than two right angles, then the two lines, extended indefinitely if necessary, will meet on that side of the third line.

Euclid's axioms are:

1. Things which are equal to the same thing are also equal to one other.

2. If equals are added to equals, the sums are also equal.

3. If equals are subtracted from equals, the remainders are also equal.

4.Things that coincide with one another are equal to one another.

5. The whole is greater than the part.

However, Euclid's axioms turned out not to be enough. For example, the first proof in Euclid (Book I, Proposition 1) shows how to construct an equilateral triangle on a given base, by drawing two circular arcs and locating their point of intersection. But Euclid doesn't have any axiom allowing him to conclude that there is a point of intersection.

This isn't to knock Euclid. He did an extraordinary job, and it took well over two thousand years before substantially better treatments were given.

As far as I know, the definitive modern treatment was given by David Hilbert. The first edition of his book (published in German in 1899) is available for download in English translation from Project Gutenberg at

http://www.gutenberg.org/files/17384/17384-pdf.pdf

It has twenty axioms. However, this version intentionally doesn't cover everything that can be done by ruler and compass. Hilbert revised his work, and his version of 1930 does cover everything that can be done by ruler and compass. The axiom that Euclid needed for his first Proposition is that called by Hilbert the Axiom of Linear Completeness.

To be really complete, one would need to add another layer underneath the geometrical axioms, a layer of axioms for symbolic logic. Hilbert undoubtedly knew this, since he worked in logic, but it would have made the book much harder to read.

My information for Hilbert's 1930 work comes from Morris Kline Mathematical Thought from Ancient to Modern Times, Oxford University Press New York 1972, Chapter 42 "The Foundations of Geometry".