Euclid

Chapter 1 · A Life That Cannot Be Recovered

What we actually know, which is almost nothing

fl. c. 300 BC · Alexandria

It is worth being blunt at the start. We do not know when Euclid was born or died, where he came from, what he looked like, whether he had a family, or whether he taught. We are reasonably confident he worked at Alexandria around 300 BC, because Proclus says so — writing roughly seven hundred and fifty years later.

The two anecdotes everybody repeats are equally late. One has King Ptolemy asking for a shorter way through the subject and Euclid replying that there is no royal road to geometry; that is told of others too. The other has a student asking what he will gain from learning geometry, whereupon Euclid tells a slave to give the boy a coin, since he must profit from what he learns. Both are good stories and neither is evidence.

Some scholars have gone further and suggested that Euclid may not have been one person at all, but a name attached to a school or a team, in the manner of Bourbaki in the twentieth century. That is a minority position, but it cannot be ruled out on the evidence available.

This is not a gap to be filled with plausible invention. It is a fact about the ancient record: a work of overwhelming importance survived, and the person behind it did not.

Why this matters

Euclid is the clearest case in the history of science of a work surviving completely while its author's life vanishes entirely.

You have a name with almost nothing attached to it. What would you ask?

Ask Euclid

• “Is there really no royal road to geometry?”
• “Did you teach, and if so what was a lesson like?”
• “Would you mind if you turned out to be several people?”
• “What was Alexandria like to work in?”
• “Does it matter that nothing of your life survives?”

Chapter 2 · The Short List at the Front

Five postulates, five common notions, and everything else

c. 300 BC · Alexandria

The *Elements* opens with twenty-three definitions, five postulates and five common notions. That is the whole of what is assumed. Everything in the thirteen books that follow is derived from it.

The postulates are startlingly modest. That a straight line can be drawn between any two points. That a finite line can be extended. That a circle can be drawn with any centre and radius. That all right angles are equal. And the fifth — the awkward one — which amounts to saying that through a point not on a line, exactly one parallel can be drawn.

The common notions are more general still: things equal to the same thing are equal to each other; if equals are added to equals the wholes are equal; the whole is greater than the part.

The design decision is the achievement. Everything you are being asked to grant is printed at the front, in ten short sentences, where any reader can examine it and refuse. Nothing is smuggled in later. If you accept the list, you must accept the consequences, and if you reject a consequence you have to say which of the ten you are rejecting.

That structure — stated assumptions, then derived consequences, with nothing else permitted — is now so ordinary that its invention is invisible. It was not ordinary. It was made.

Why this matters

Putting every assumption at the front where a reader can inspect and reject it is the foundational move of the axiomatic method, and it was a deliberate design choice.

You have ten sentences holding up thirteen books. What would you ask?

Ask Euclid

• “Why put every assumption at the front rather than argue as you go?”
• “Why is the fifth postulate so much longer than the others?”
• “Did you suspect the fifth postulate could be proved from the rest?”
• “What is the difference between a postulate and a common notion?”
• “Could you have managed with fewer assumptions?”

Chapter 3 · Proved, Not Checked

Proposition 47, Proposition 48, and the primes

c. 300 BC · Alexandria

Book I, Proposition 47 is the theorem about the squares on the sides of a right-angled triangle, attributed to Pythagoras and proved here from the postulates. Book I, Proposition 48 is its converse, and it is the more useful of the two in practice: if the square on one side equals the sum of the squares on the other two, then the angle *is* a right angle. That gives you a test. It is how a builder checks a corner.

Book IX, Proposition 20 shows that the prime numbers do not run out. The argument is short. Take any finite list of primes; multiply them together and add one. The number you get leaves a remainder of one when divided by any prime on your list, so either it is itself prime or it has a prime factor not on the list. Either way, your list was incomplete. No finite list of primes can be complete.

That proof is the model of what Euclid means by knowing something. It does not check a thousand cases. It settles all cases, including the ones nobody will ever write down, in five lines.

Book V preserves Eudoxus's theory of proportion, which handles magnitudes with no common measure — the diagonal of a square against its side — and does so rigorously enough that nineteenth-century mathematicians building the real numbers found it already essentially correct.

Why this matters

The distinction between checking a result in many cases and proving it for all cases is the single most transferable idea in the Elements.

You have the primes settled for ever in five lines. What would you ask?

Ask Euclid

• “Why is Proposition 48 more useful than Proposition 47?”
• “How can five lines settle infinitely many cases?”
• “What is wrong with establishing a truth by examples?”
• “Why does Book V need a whole theory just for proportion?”
• “Which proof in the Elements do you consider the best made?”

Chapter 4 · What Euclid Did Not Do

The gaps, and the diagram that argues

c. 300 BC and after · Alexandria

The *Elements* is not perfect, and pretending otherwise makes it less interesting.

Several proofs quietly use facts read off the diagram that the postulates do not supply. The very first proposition of Book I constructs an equilateral triangle by drawing two circles and taking their intersection — and nothing in the postulates guarantees that the two circles actually meet. It is obvious from the picture. It is not derivable from the list. David Hilbert had to add a whole apparatus of axioms of order and continuity in 1899 to close such gaps.

Euclid also discovered comparatively little of the content. The theorem of Book I.47 is Pythagoras's school; Book V is Eudoxus; the classification of irrationals and the regular solids in Books X and XIII come substantially from Theaetetus. He does not claim otherwise, and he does not credit them either — attribution in the modern sense was not the convention.

And the fifth postulate turned out to be the most interesting thing in the book. For two thousand years mathematicians tried to derive it from the other four and failed. In the nineteenth century Lobachevsky, Bolyai and Riemann tried assuming it false and found perfectly consistent geometries. Those geometries are what general relativity is written in.

Why this matters

The one assumption in the Elements that looked least secure turned out, when dropped, to produce the geometry that describes the universe.

You have the gaps and the postulate that would not behave. What is your question?

Ask Euclid

• “How do you know those two circles in Proposition 1 actually meet?”
• “When is a diagram an illustration and when is it an argument?”
• “How much of the Elements is your own discovery?”
• “What would you say to somebody who assumes your fifth postulate is false?”
• “Does finding a gap in a proof destroy it or improve it?”

Chapter 5 · The Shape of an Argument

Newton, Spinoza, Jefferson, and every proof since

c. 300 BC – present · Alexandria · Cambridge · Amsterdam · Philadelphia

The *Elements* was a working textbook for over two thousand years. Editions in Greek, Arabic and Latin ran continuously; the first printed edition appeared in 1482, and it was still used in British schools in the early twentieth century. Only the Bible has been printed more often.

But the deeper influence is the shape. Newton wrote the *Principia* in Euclidean form — definitions, axioms, then propositions with proofs — because that was what a serious demonstration looked like. Spinoza wrote the *Ethics* *more geometrico*, in the geometrical manner, with definitions, axioms and propositions about God and the emotions. The American Declaration of Independence opens by holding certain truths to be self-evident and deriving consequences: that is a Euclidean move, made by men who had learned geometry.

The pattern is always the same. State what you are assuming, in public, at the start. Derive what follows, step by step. Allow the reader to reject the assumptions or accept the conclusions, and nothing in between.

That is the standard every student meets when they are asked to *prove* something rather than check it. It has a source, it was constructed by somebody, and it is possibly the most successful piece of intellectual architecture in history.

Why this matters

The Euclidean shape of argument — public assumptions, derived consequences — spread far outside mathematics into physics, philosophy and political founding documents.

You have the shape that outlived everything else. What would you ask?

Ask Euclid

• “Why did Newton write the Principia in your form?”
• “Can the geometrical manner really be used for ethics?”
• “What does 'self-evident' actually mean in an argument?”
• “Should geometry still be taught the way you arranged it?”
• “Is the arrangement of results really an achievement?”

What Euclid changed

The Elements is the most successful textbook ever written and the source of the standard every student meets when asked to prove rather than check a result. Its shape — assumptions stated in public, consequences derived from them — was borrowed by Newton for the Principia, by Spinoza for the Ethics, and by the drafters of the American Declaration of Independence. Dropping its least secure postulate in the nineteenth century produced the geometries that general relativity is written in.

A debate that continues

Nothing reliable is known about Euclid's life, and a minority of scholars have argued the name may cover a school rather than an individual. Several proofs also contain gaps that were only closed by Hilbert in 1899.

Keep exploring — ask Euclid

• “What would you add to the Elements if you wrote it again?”
• “Is a proof still a proof if nobody can follow it?”
• “Which of your postulates would you most like to test?”

Related lives

• Archimedes — Give Me a Place to Stand
• Ibn al-Haytham — Who Tested What the Ancients Wrote
• Isaac Newton — Lucasian Professor · President of the Royal Society
• René Descartes — Who Conserved The Quantity Of Motion

Related themes

Proof and reasoning · Geometry · Prime numbers