On this page
Most of school maths asks you to calculate. Euclid's Elements asks something harder: why must this be true? Written around 300 BC, it set the pattern for proof that every later mathematician learned from. This post explains how the Elements is built, works through two of its arguments with real numbers, lists the mistakes students make when they try proofs, and ends with what you can take from Euclid's way of working.
Who was Euclid?
Very little is known about his life. The MacTutor biography from the University of St Andrews says only that he taught at Alexandria in Egypt, and gives his dates as roughly 325 to 265 BC. Most of what we have comes from Proclus, a philosopher writing about 750 years later, who said Euclid collected and arranged earlier results and brought "to irrefutable demonstration" things his predecessors had proved only loosely. MacTutor also notes that some historians have asked whether "Euclid" was one person or a team at Alexandria.
What is certain is the book. MacTutor says the Elements is divided into 13 books, that books one to six deal with plane geometry, and that more than one thousand editions have been published since it was first printed in 1482.
How is the Elements built?
Book I, in the edition edited by David Joyce at Clark University, opens with 23 definitions, 5 postulates and 5 common notions. Then come 48 propositions, each proved only from what came before.
Definitions say what words mean: "A point is that which has no part."
Postulates are assumptions about geometry, such as "To draw a straight line from any point to any point."
Common notions are general rules of reasoning, such as "Things which equal the same thing also equal one another" and "The whole is greater than the part."
Everything else must be proved. Proposition 1 constructs an equilateral triangle on a given line. Proposition 47 is the theorem you know as Pythagoras: "In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle."
Worked example 1: why do primes never run out?
Book IX, Proposition 20, in Joyce's edition, states: "Prime numbers are more than any assigned multitude of prime numbers." Joyce's commentary outlines the argument. Take any finite list of primes, multiply them, and add 1. The result is either prime, or divisible by some prime. That prime cannot be on your list, because every prime on the list leaves remainder 1. Either way, there is a prime you did not list.
Try it with numbers:
List 2, 3, 5, 7. Product = 210. Add 1: 211. No prime up to 13 divides 211, and 17 x 17 = 289 is bigger than 211, so 211 is prime, a new one.
List 2, 3, 5, 7, 11, 13. Product = 30,030. Add 1: 30,031 = 59 x 509. It is not prime, but 59 and 509 are both new primes.
The second case shows a common misreading: the proof does not claim that "product plus one" is always prime. It claims only that it has a prime factor missing from your list.
Worked example 2: Euclid's method for the gcd
Book VII, Proposition 2 shows how "to find the greatest common measure of two given numbers": keep taking the smaller from the larger until one measures the one before it. In modern form, divide and keep the remainder. For 1071 and 462:
1071 = 2 x 462 + 147
462 = 3 x 147 + 21
147 = 7 x 21 + 0
The last non-zero remainder is 21, so gcd(1071, 462) = 21. Check: 1071 = 21 x 51 and 462 = 21 x 22, and 51 and 22 share no common factor.
Common mistakes when you write a proof
Using what you are trying to prove. Every step must rest on earlier results, not on the conclusion.
Checking examples and calling it a proof. Finding that 211 is prime proves nothing about all lists of primes. The argument must work for any list.
Skipping the reason. Write why each step follows, as Euclid cites earlier propositions. Our post on George Pólya's How to Solve It has a useful checklist for this.
When you get stuck on a proof or a number-theory problem, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
What can a student learn from how Euclid worked?
State your assumptions. Euclid listed his up front. Write down what is given before you start a problem.
Order matters. He proved results in a sequence where each one is available for the next. Building your notes the same way makes revision faster.
Prefer an argument to a pile of examples. Examples suggest; proofs settle.
Practice routine: find gcd(252, 198) and gcd(1001, 143) by Euclid's method, then rewrite the primes proof in your own words for the list 2, 3, 5.
Next step: read Book I, Proposition 1 in Joyce's online edition and redraw the construction with a compass.
