Existence of parallels in axiomatic plane geometries

[lavinia]

Part of the axiom system is the ability to compare two segments and say whether they are equal and if not, which is larger. Euclid's axioms are full of conditions on the notion of "equal", which he seems to apply to lengths, areas, volumes and angles. His theorem statements also include statements about angles being equal, greater than or less than another. See his postulates 4,5 and all 5 common notions, as well as many propositions, e.g. all of the first 20 or so that I have quickly scanned, contain one of these comparison words. Hilbert has a set of axioms for congruence, and one can define less than and greater than in terms of congruence to a subset.

It would seem that one can lay multiples of one segment down on the other and get remainders. With the remainder one can repeat the process. If eventually one gets only repeats of congruent segments then the process ends in a finite number of steps and the two original segments are rational multiples of each other. If the process continues forever then they are not.

Examples.

The two segments are in a ratio of 5 to 3. Lay the smaller segment on the larger to get a remainder that is in a ratio of 5 to 2. Lay the smaller segment twice on the larger to get a remainder in a ratio of 5 to 1. All segments are multiples of this and the process stops.

If one introduces a unit it seems that one will end of with a segment whose length is the greatest common divisor to the original two.

 

[mathwonk]

yes! this is the famous euclidean algorithm for finding gcd's.