This section of the website includes a serialised guide to Euclid's first book of the elements in 
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Euclid's Mathematics

Euclid was a Greek mathematician who lived around 300 BC. He is believed to have learnt geometry from the students of Plato, in Athens, and to have then founded a school in Alexandria, Egypt. He is famous for having written thirteen books on the Elements of Geometry, which provide the most complete record that we have of the mathematics that was current in the world of Ancient Greece.

Up until the middle of the nineteenth century, Euclid's Elements ranked second only to the Bible as the most widely-printed and widely-read book in the world: they served as the standard mathematics textbook for over two thousand years.

Unfortunately they were used badly, and successive generations of schoolboys came to dislike Euclid with ever increasing intensity. Consequently, when secondary education became more widespread, in the late eighteenth century, Euclid's elements were abandoned in favour of a more 'modern' approach which taught those areas of Euclid's work that had practical applications in engineering and industry, but which did not try to explain the reasoning behind them.

This has led to an explosion in the use of mathematics, and to the development of ever more complex ideas, but it is possible that in the midst of all this activity, people have lost sight of the essential purpose and significance of mathematical thinking.

Western mathematics is still essentially derived from ancient Greece, and Euclid's work is still the best guide to the fundamental principles of the subject. The fault that was made in the past was to try to force it upon children  - public schoolboys, as young as twelve years old, were beaten until they could repeat Euclid's proofs by heart - while the work was originally  intended for adults who wanted to explore philosophy and the meaning of life. 

Book 1

Book 1 of the Elements starts from basic principles and leads, by a series of logical steps to a proof of Pythagoras's theorem.

Definitions

The book is traditionally considered to start with twenty-three 'definitions' but in fact the term definition is a mistranslation, the original word implies 'boundaries' or 'landmarks'. Thus the twenty-three 'definitions' are in fact twenty-three ideas or concepts that the student has to accept at face value before they can begin to study the Elements.
These 'definitions' are often glossed over, but, in some ways, the whole subject of mathematics is a game that plays with our willingness to accept these original concepts.

They are:

It is important to remember that, in pure geometry, things are only drawn to illustrate ideas. The subject itself is conducted in the realm of the imagination. Thus the fact that all the above things can be drawn, does not help to define them or to understand their essence. However, they are all things that should make sense to everyone from a 'common sense' point of view.

Over the years, many people have tried to provide watertight definitions for such things as the point, the line and the surface, straightness, flatness, and parallel, but it has always been found that they become meaningless under the scrutiny of logical examination.

For example:

The Point
It is easy to understand the need for the concept of a point in the study of geometry because lines have to start and finish in specific places; circles have to have a specific centre; lines have to meet at a specific place; etc.

The following are some of the ways that the point has been defined:
A point is that which has no parts.
A point is that which is indivisible into parts.
A point has position but no form.
A point is the extremity of a line.
A point is an indivisible line.

All of these definitions fall into the same difficulty: a point has to be something so small that it cannot be divided into smaller parts, but once we have imagined something to exist we can imagine it being cut into smaller pieces: it is therefore impossible for us to conceive of something that is small enough to be an exact point. Thus geometry is built upon a paradox: intuitively the idea of the point makes sense, but close examination shows that it is not something that can actually exist in the material universe.
The same applies to lines and surfaces.

Straightness
Everyone understands the concept of a straight line but it is has never been possible to define straightness without introducing some other words that themselves imply some prior understanding of what it means, i.e. the straight line is too simple and basic an idea to be defined. Attempted definitions include such things as:
A line lying evenly with itself.
A line stretched to the utmost.
The shortest distance between two points.
A line of which the middle covers the ends (when viewed from one particular position).

Ironically, straight lines do not exist in nature, and this fact was sometimes used by Greek Philosophers to infer the existence of the soul: everyone understands the idea of the straight line even though no one has ever seen one, therefore knowledge of the straight line must be an attribute of the soul and not of the mind.

Flatness is as difficult to define as straightness. The concept of parallel is perhaps even more fraught with difficulty. It suggests that it is possible to have two straight lines, in the same plane, that will never meet, no matter how far they are extended in either direction. This is clearly not something that could ever be proven and yet the idea of things being parallel is something that makes sense to people.

Mathematics, Science and Ancient Geometry
Discussion about the foundations of ancient geometry are not as irrelevant to modern life as may be imagined. It is  often stated that all of  modern science and all of modern mathematics has evolved directly from the the work of the ancient Greeks, but what is generally not appreciated is that it has evolved out of a misunderstanding of this work.

Scientists for the past five hundred years have assumed that the fact that the foundations of Greek geometry contained paradoxes was a problem that had to be solved (hence the whole quest for the atom, sub-atomic particles etc. - things that would perhaps provide the ultimate definition of a point), while clearly Greek philosophers were using mathematics specifically to highlight the existence of such paradoxes, and to make people question the assumptions that they make in every area of life. Thus whereas the refrain of modern science is 'We now understand almost everything, and have explained almost everything', the original purpose of geometry was to demonstrate to people that they knew nothing and could explain nothing.

The Real Purpose of Geometry
There is another side to geometry which has also been almost lost, it is a proposition that runs something like this: 'There are various concepts such as point, line, angle, circle, triangle, straight, parallel, etc. that do not make sense in terms of physical reality but which can exist in our imagination. Lets explore these ideas using only pure logic. And let's do this simply for fun.' 
This is what Euclid proceeds to do in his thirteen books on geometry. 

Next Instalment: 5 Postulates