Euclid's Mathematics - 3

Previous articles on Euclid’s elements of geometry described his twenty-three definitions – in which he defined things such as points, lines, and shapes – and five postulates in which he asks the student to accept that certain things, such as drawing a straight line between two points, are possible.

Euclid’s next step is to put forward five common notions (or axioms) which he suggests should be readily accepted by everyone. These are:

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

2. If equals be added to equals, the wholes are equal.

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

4. Things which are in exactly the same place as one another are equal to one
   another.

5. The whole is greater than the part.

Over the years, many people have tried to use logic to prove that these common notions are true, but without success – every attempted proof calls for more assumptions to be made than are actually contained within the common notions themselves.

Euclid appears to have succeeded in building his system of geometry upon the simplest possible foundations.

The reason that his definitions, postulates and common notions are accepted is firstly because they make sense and secondly because they are simple. Simplicity has always been regarded as a guiding principle in determining the validity of mathematical ideas: the simpler they are, the more highly they are regarded.

The ancient Greek mathematicians held this idea of simplicity very close to their hearts and their mathematics rigorously adheres to the simplest possible solution to every problem.

Modern mathematics, although paying lip service to the idea of simplicity, has in fact become extremely complicated and this is perhaps why it is so unappealing to the majority of people.

Having established the ground rules, Euclid proceeds to put forward a series of propositions. His object is to prove that they are true, simply with the aid of the definitions, postulates and common notions that have already been agreed.

It is important to remember that Euclid’s geometry is meant to be conducted in the mind, not on paper. The object is to try to imagine the things that he describes and to decide for oneself whether or not one agrees with his sequence of thought.

Diagrams are used to illustrate the concepts that are expressed but the fact that something can be drawn does not prove that a proposition is correct.

Proposition 1 The first proposition is that it is possible to construct an equilateral triangle on any given, finite straight line.
Let AB be the given straight line. (Thus it is required to construct an equilateral triangle on the straight line AB)
With centre A and distance AB, draw a circle. [Postulate 3: A circle can be drawn at any distance around any point.]
With centre B and distance BA draw another circle. [Postulate 3: A circle can be drawn at any distance around any point.]
Join point C, where the circles join to points A and B. [Postulate 1: It is possible to draw a straight line between any two points.]
AC is equal to AB because they are both radii of the same circle. (a radius is a line going from the centre to the outside of a circle) [Definition 15: A circle is a shape such that all the straight lines falling upon it from one point within it are equal to each other.] BC is equal to BA because they are both radii of the same circle. [Definition 15] This means that both AC and BC are equal to AB. Things that are equal to the same thing are also equal to each other. [Common Notion 1] Therefore CA is equal to CB. Therefore the three straight lines AC, AB, BC are equal to each other. Therefore the triangle is equilateral; and it has been constructed on the given, finite straight line AB – which is what it was required to do.