Euclid's Elements

Euclid was a brainy Ancient Greek who wrote Elements, one of the definitive works of mathematics in about 300 BC.

Not much is known about Euclid's life. He lived around 300 BC, but we don't know when he was born or when he died. He founded and taught in a school in Alexandria, so he was probably from around there. Alexandria at the time was the foremost city in the Greek world, although it was situated in Egypt.

Euclid wrote a number of works which we know only by name: Data, Optics and Phaenomena (spherical geometry in astronomy). No copies of these have survived. There is also a work of his which survives only as an Arabic translation: On Divisions [of Figures]. However, it is for one work that Euclid is remembered, and considered one of the great names of mathematics and reasoning.

Elements

Euclid produced a huge work called Elements which was basically all the mathematics known by the Ancient Greeks at the time. It is based on plane geometry: Euclid considered a number to be the length of a line - this is more or less the same as the modern concept of a 'real number'.

Much of Elements was information which was freely available in the mathematical community at the time, but some of it was Euclid's own work. Euclid arranged the work starting with basic definitions, continuing with postulates (also known as axioms) and common notions, which are statements that are considered so obvious that they don't need to be proved. He then went on to a huge number of propositions. Every one of these was proved logically based only on what was already known from the postulates and previous propositions, so that the whole thing develops into a mathematical structure, all of which hangs together and is logically consistent.

The Structure of Elements

Elements is divided into 13 books. Each book starts with a number of definitions. The first has the postulates and common notions. Each book has a number of propositions. Some of these propositions state a fact about geometry or numbers and go on to prove it. Others explain a procedure for constructing a diagram, for example, a method to draw a perfect square using a compass and straightedge (a ruler with no markings).

Modern mathematics is not based as much on geometry as it was in Euclid's day, making much more use of algebra and symbols. We have no problem multiplying three or four numbers together. Euclid would only multiply two numbers together if they represented the lengths of two sides of a rectangle and the product was the area. When he talked about the square of the side of a triangle, he didn't just mean the length of the side multiplied by itself, he actually constructed a square and talked about its area.

Much of Elements, particularly the early propositions, may seem pointless to modern readers as it appears to fall into the category of 'stating the bleedin' obvious'. For example, Euclid proves that if you pick two points on a circle and join them with a straight line, then every point on that line is inside the circle. It's obvious, isn't it? But there is a great skill in being able to prove something as obvious as that. We can then be sure that it really is true.

Postulates and Common Notions

Euclid considered the following five points to be basic and obvious. They are known as postulates.

1. We can draw a straight line between any two points.

2. We can extend a straight line as far as we like at either end.

3. We can draw a circle with a given radius around any point.

4. All right angles are equal to each other.

5. If we have two straight lines and we draw a third one across them, and if the two interior angles formed on one side add up to less than two right angles, then the original two lines will meet if extended on that side.

He also included five 'common notions' which are also considered basic and obviously true.

1. Things which are equal to the same thing are also equal to one another.

2. If equals are added to equals, the sums are also equal.

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

4. Things that coincide with one another are equal to one another.

5. The whole is greater than the part.

Euclidean and Non-Euclidean Geometry

There are very few mistakes in Euclid's work - later commentators and editors often tried to 'correct' it based on their own misunderstandings of the text, but it has been possible to reconstruct the original text by examining different versions. We call the geometry that he outlines 'Euclidean Geometry' and it has been taught in schools for nearly two and a half millennia. Even if we have not studied geometry in school, we would recognise much of what it has to say as being reasonable.

Nobody has ever shown Euclid to be substantially wrong in his description of geometry, but about two centuries ago it was shown that two of his postulates, the second and the fifth, are in fact rather less obvious than the other three. Euclid assumed them to be true because he couldn't prove them. Assuming them to be false, however, leads to an alternative system, known as non-Euclidean geometry, which makes strange counter-intuitive predictions about the world. When this was discovered, it was considered to be just a mathematical oddity having no relevance to the real world, but Einstein's discovery of the 'bending' of space by gravity has shown that non-Euclidean geometry has its place in our universe alongside Euclidean geometry.

The Impossible Constructions

Euclid demonstrates how to draw a triangle, a square, a pentagon, a hexagon, all using only a compass and a straightedge (a ruler with no markings). He doesn't know how to draw a 7-sided figure or a 9-sided figure, or many bigger figures such as a 17-sided figure1. The 9-sided figure looks easy enough, as it should be easy to draw a triangle and then divide each angle into three. Here we come to the first of the impossible constructions - there is no way to divide an arbitrary angle into three equal parts using only a compass and straightedge. Many people have tried, but it was nearly 2,000 years later that it was finally proved impossible.

The other great impossible constructions are building a cube exactly twice the volume of a given cube, and building a square exactly equal in area to a given circle. All of these have been proved impossible in recent centuries, using methods far more elaborate than those available at the time of Euclid.

Number Theory

Not all of Elements was geometry. It also included a number of propositions relating to what is now known as 'number theory'; this is the study of whole positive numbers. The most famous of these is Euclid's proof that there are infinitely many prime numbers.

One of the simplest ways of classifying numbers is as composite or prime. A composite number, such as 6, can be composed by multiplying together two smaller numbers, in this case 2 and 3. A prime number, such as 13, can't be made in this way. There are no smaller numbers which divide evenly into 13, other than 1, which divides into every number.

Euclid proved there are an infinite number of primes by assuming there are a finite number, and then showing this led to a contradiction. A simplified version of the proof is as follows:

• If there is a finite number of primes, we can write them out in a list. We can multiply all the primes together and add 1 to get a new number. Let's call it N.
• N is not divisible by 2, because it is 1 greater than a multiple of 2.
• N is not divisible by 3, because it is 1 greater than a multiple of 3.
• In the same way, N is not divisible by any prime on the list, because it is 1 greater than a multiple of that prime.
• Since N is not divisible by any of the primes on the list, it must either be a prime itself, or a multiple of another prime we have not included in the list.
• So either way, we have found a prime which is not on the list. But the list includes every prime. So we have something that doesn't make sense. Therefore our original assumption, that the number of primes is finite, must be wrong; the number of primes must in fact be infinite.

Summary

Euclid's Elements is one of the recognised brilliant achievements of the Ancient Greek world. It is a classic demonstration of logical thought which anyone can appreciate at least some of, if they are willing to do a bit of work. Admittedly, very few people will have the stamina to trawl all the way through it. It is a glorious demonstration that in the field of pure thought, we haven't progressed much since the time of the Ancient Greeks.

1Carl Friedrich Gauss, an 18th/19th-Century mathematician, actually proved that it is possible to construct a 17-sided figure, and later mathematicians came up with a method for doing it.