## Development of Euclidean Geometry

Mathematics has existed for as long as civilisation itself, and the oldest type of mathematics is geometry. This is because of its practical uses in building and land surveying. The ancient Babylonians and Egyptians both had knowledge of advanced geometric techniques (used, among other things, to build the pyramids). These were passed down through generations of scribes working at palaces and temples, who slowly added to their knowledge base over time. The first people, however, to examine geometry in what we would call a scientific fashion where the ancient Greeks. The most important figures in the development of this rigorous, methodical approach were Thales and Pythagoras in the fifth century BC. They realised that the mish-mash of geometric rules the Greeks had inherited from previous civilisations could all be connected together. They came up with the idea that for something to be true it must be possible to prove it from other things, which are known to be true. Over the centuries that followed, various mathematicians constructed chains of geometric proofs, each starting with a few unproven assumptions (or axioms). About 300 BC, a mathematician living in Alexandria called Euclid, tied together these various chains of proofs into one inventory, a book called the elements. It started with five basic axioms and from these deduced 465 theorems, and it remained the fundamental book on geometry for over two centuries. The geometric system in the elements is called Euclidean geometry and most of the geometry taught in schools today is Euclidean.

## Description of Euclidean Geometry

Euclidean geometry consists basically of the geometric rules you learned at school. The angles of a triangle add up to 180°, the Pythagorean theorem, all the sine and cosine rules, and all the other two and three dimensional rules concerning shape areas, angles and so forth. The basis on which all these theorems are built is a collection of definitions and axioms. The definitions are all fairly obvious things such as "the extremities of a line are points" and "an acute angle is less than a right angle". The 5 axioms however are more problematic. The first four are all fairly straightforward and easy to accept, and indeed no one has ever seriously doubted them. They say that there is only one straight line between two points, and that this line may be produced (extended). That for a given centre and radius there is one circle, and that right angles are all identical. The fifth axiom however causes problems. It deals with parallel lines, and states that for a given line and point there is only one line parallel to the first line passing through the point. This may seem to be as obvious as the first four, but it isn't. Because of this doubt about axiom 5 Euclidean geometry came in two parts, the theorems which could be proved from the first 4 axioms alone (basic geometry), and the more complicated ones which relied on axiom 5 in their proof (Euclidean geometry).

## Problems with Euclidean Geometry

Many mathematicians after Euclid (and even Euclid himself) where not comfortable with axiom five, it is quite a complicated statement and axioms are meant to be small, simple and straightforward. Axiom five is more like a theorem than an axiom, and as such it should have to be proved to be true and not assumed. The problem that Euclid and every mathematician after him found for 200 years was that it could not be proven from the 4 axioms before it. However, all the theorems that can be proved from it worked and many mathematicians were happy just to leave it. It is something that seems obviously true and yet was impossible to prove mathematically in a satisfactory way.

## Development of Hyperbolic Geometry

Over the centuries many mathematicians tried to resolve the problem with axiom five, usually by rephrasing it and trying to express it as a theorem in terms of some new simple axioms. These attempts all failed. It was in the eighteenth century that the first attempt was made to prove axiom 5 by contradiction. That is to assume axiom 5 is false, and by making deductions from this which can be proved to be false to therefore prove it must be true. The first man to try was called Saccheri, and in the first half of the nineteenth century many others followed him. All these attempts produced alternate geometry's (non-Euclidean geometry's) which, although they seemed to be obviously wrong, could not be proved to be wrong. Eventually it was realised that, mathematically speaking, these geometry's where just as valid as traditional Euclidean geometry. The type of non-Euclidean geometry where axiom 5 is the opposite of Euclid's fifth axiom (there is more than one line passing through a given point, which is parallel to a given line) is called hyperbolic geometry. Hyperbolic geometry was developed separately (though in slightly different forms) and at roughly the same time by four different people. They where Gauss, Schweikart, Lobachevsky and János Bolyai. At first this met with ridicule and criticism, before being accepted.

## Description of Hyperbolic Geometry

Hyperbolic geometry is hard to describe. Its basic premise, that there can be multiple parallel lines through a point, is itself very hard to accept. In purely mathematical terms it is not so difficult. It consists of all Euclid's theorems that can be proved from the first four axioms (basic geometry), as well as many new theorems that can be proven using the new fifth axiom (hyperbolic geometry). In psychological terms, however, it is less easy to follow. To give some kind of a flavour, here are a few of the effects of hyperbolic geometry.

Straight lines will appear to be curves. Two lines may be inclined in such a way that if they are extended they will meet, and yet they will not meet, they will appear to curve and yet they will remain straight lines.

The Pythagorean theorem, a cornerstone of Euclidean geometry, is not true.

Scale models are impossible, as the size changes, so do the angles.

The larger the area of a triangle, the smaller the sum of its angles. And if two triangles have the same angle sum, they have the same area.

There is an upper limit on the area of a triangle.

The best way to understand hyperbolic geometry is to see it illustrated.

## The Current Situation

It took a while for hyperbolic geometry to be accepted. Many models were developed to show how hyperbolic geometry would work; the best was the one designed by Poincaré. As the legitimacy of non-Euclidean geometry was accepted it revolutionised the way scientists saw the world. The idea of the universe as a fixed and definite thing was banished forever. In the years that followed many other types of non-Euclidean geometry were developed, but hyperbolic remains the most important. As physics began to take in more and more accurate measurements of the vast distances between stars, it became apparent that traditional physics, built on Euclidean geometry was not fully correct. When Einstein developed his new physics theories it was hyperbolic geometry that formed the mathematical basis of relativity. Today it is generally accepted that when considered over the galactic scale, the universe is hyperbolic.