Euclid index

An introduction to Euclid


Index --- introduction --- definitions --- axioms and postulates --- propositions --- other

This website is supposed to be an introduction to Euclid's geometry. It is neither mathematically rigorous or minutely faithful to Euclid, but it does its best. In making ths website, I made extensive use of Euclid's elements by David E. Joyce, Professor of Mathematics and Computer Science, Clark University. I suggest you consult that website for a more serious treatment. This website is trying to make it easier to understand the first book of Euclid which covers triangles and parallel lines.

Why learn Euclid? The obvious answer is that someone is making you! If that's true, then I hope this website makes it easier for you. However, there is a point to Euclid, or rather two points. He does describe a geometry (the study of lines, shapes and angles) which we can all recognise in the real world. There are other geometries, but if someone asked you to draw a triangle, you would draw a Euclidean triangle. However, Euclid does more than this. He (or perhaps his predecessors) wrote down definitions, axioms and postulates, and propositions which constructed a logical system without contradictions. Any assumptions were specified as such (this is what the definitions, axioms and postulates are), and then the body of propositions were made up, with each step only dependent on what had already been proved or assumed. This is real mathematics, and this is what mathematicians still do today. Euclid is the first known person to do this. What is more, since this is all about real things such as lines and angles, it is quite easy to see what is going on.

One problem about Euclid is that there are a lot of propositions, and I admit that they can get boring! At school, we used to study only a few propositions. This was less boring, but it did mean that we lost the sense of the gradual building of proof, with each proposition dependent on previous ones. I have listed what I think are the most interesting results on the index, with a complete list on the propositions page. If you look at a proposition, you will find that any previous propositions are given as links. So if you want, you can take them as read, or if you prefer, you can click on the link (which will open as a new page so you don't lose your place) and investigate the earlier proof. The first few propositions also tell you definitions, axioms and postulates used, but this got rather boring, so I drop it after that.

Euclid's propositions can be confusing to understand, as you need to identify various lines and angles. A diagram of the proposition can start getting messy. So I have set up each proposition so you can read it line by line. You get the first line, illustrated by a simple diagram. Click on 'Next line' and the next line will appear, with any line or angle mentioned marked on the diagram. So you can step through (or back) the diagram one line at a time. You can jump directly to the end of the proof if you wish, and all the lines will be seen. This will, I hope, stop you drowning in a sea of angles ABC, BAC, ACB and so on!

I have used conventional symbols on the diagrams, but used colour when I thought it would help. On the other hand, I have used the words 'triangle', 'line', 'parallel lines' and so on, during the proofs, rather than the more traditional symbols, as this makes it just ordinary text. I have used terms like 'congruent triangles' or '180°' just to simplify the explanations. I define these terms here.

Euclid described his propositions as sentences. These can be quite long, and it can be hard to find the proof that you are looking for. So while I have given Euclid's sentence at the start of each proof, I have used a summarised version in the index and headings and so on. I must confess that these may be rather vague! But they are there to help you find the correct proposition quickly.