Why Is Euclid Called the Father of Geometry? The Man Behind the Elements

Euclid is called the father of geometry because his book, the Elements, written around 300 BC, organised the geometry of his time into a single logical system: starting from a handful of definitions and simple assumptions, it proves hundreds of results one after another. The work has often been described as the most successful textbook ever written, and it is estimated to be second only to the Bible in the number of editions published, with well over a thousand. For more than two thousand years, learning geometry largely meant learning Euclid. Even today, the flat geometry you met at school is called "Euclidean geometry".

A famous name with almost no biography

Here is the surprising part: we know very little about Euclid the man. He is dated to around 300 BC, and he most likely worked in Alexandria, the great Greek city in Egypt. His birthplace is unknown. Most of what is said about him comes from writers who lived many centuries later, such as Proclus in the fifth century AD and Pappus of Alexandria in the fourth. Medieval Arabic writers supplied a detailed life story, but most scholars doubt it.

To make matters more confusing, later scholars often mixed him up with a different man, the philosopher Euclid of Megara. Some modern historians suspect that Euclid studied at Plato's Academy in Athens and that he was among the first scholars at the Musaeum, the research institution of Alexandria, but these remain educated guesses.

"There is no royal road to geometry"

The most famous story about Euclid comes from Proclus. King Ptolemy I of Egypt is said to have asked whether there was a quicker way to learn geometry than working through the Elements. Euclid replied that there is "no royal road to geometry".

It is a wonderful line, but treat it as a legend. A very similar conversation is told about another mathematician, Menaechmus, and Alexander the Great. Both versions were written down around the fifth century AD, more than seven hundred years later, and neither names its source. Whoever said it, the message remains true for anyone learning anything difficult: even kings have to do the exercises.

What is in the Elements?

The Elements is made up of 13 books, which we would call chapters today. Roughly, Books 1 to 6 deal with plane geometry, the geometry of flat shapes; Books 7 to 10 deal with numbers; and Books 11 to 13 deal with solid geometry. In a standard modern edition the whole work contains 465 propositions.

Euclid did not invent all of this himself. Much of the material came from earlier Greek mathematicians such as Eudoxus, Theaetetus and Hippocrates of Chios. His genius lay in the arrangement: choosing what to assume, putting the results in the right order, and proving each one from what came before.

Some highlights:

• The Pythagorean theorem. Book 1 ends with this famous result about right-angled triangles and its converse, including the earliest surviving proof of it.
• The Euclidean algorithm. Book 7 contains a step-by-step method for finding the largest number that divides two numbers exactly, more than two thousand years before anyone called such a recipe an "algorithm".
• Infinitely many primes. Book 9 proves that the list of prime numbers never ends, a result now called Euclid's theorem.
• The five Platonic solids. Book 13 builds the five regular solids, such as the cube and the twelve-sided dodecahedron, and shows that there are only five.

Five simple rules

The secret of the Elements is its method. Book 1 begins with definitions, five postulates and five "common notions". In plain words, the first four postulates say that you can draw a straight line between any two points, extend a line as far as you like, draw a circle with any centre and radius, and that all right angles are equal.

Everything else is proved from these. This approach, known as the axiomatic method, became the model for mathematics and later for science and even philosophy. When you learn a language, a similar idea applies: a small set of core rules can generate an enormous number of correct sentences.

The troublesome fifth postulate

The fifth postulate, about parallel lines, is longer and less obvious than the others. A common way to picture it: through a point beside a straight line, there is exactly one line that never meets it. For centuries, mathematicians tried to prove it from the other four and failed.

In the nineteenth century, they discovered why. You can replace the fifth postulate with a different assumption and still get a perfectly consistent geometry. In 1829 Nikolai Lobachevsky published a description of such a "hyperbolic" geometry. Euclid's choice to state the parallel postulate as an assumption, rather than pretending it was obvious, looks wiser with every century.

A book that travelled the world

The oldest physical fragments of the Elements come from Oxyrhynchus in Egypt. The book was translated into Arabic around 800, and into Latin from Arabic by Adelard of Bath around 1120. The first printed edition appeared in Venice in 1482, and in 1607 the Italian Jesuit Matteo Ricci and the Chinese scholar Xu Guangqi published the first Chinese edition. The first English edition followed in 1570, and in 1847 Oliver Byrne published a striking version that replaced many words with coloured diagrams.

The Elements was a standard university textbook through the Middle Ages and dominated mathematics teaching in both the Islamic world and Western Europe. It only fell out of favour as a school textbook in the nineteenth century.

Euclid's other books and his legacy

Euclid wrote more than the Elements. His Optics is the earliest surviving Greek work on perspective, and his Phaenomena deals with astronomy on a sphere. Several other books, including works on conic sections and on fallacies, are lost.

He is ranked with Archimedes and Apollonius among the greatest mathematicians of the ancient world. His name now flies in space too: the European Space Agency's Euclid space telescope is named after him, as are a lunar crater and an asteroid. Not bad for a man whose own life story is almost entirely lost.

Learn with Euclid: English phrases about learning step by step

Phrase	Meaning	Example
there's no royal road to...	there is no easy shortcut to...	There's no royal road to fluency.
step by step	gradually, one stage at a time	I learn new grammar step by step.
to start from scratch	to begin from the very beginning	I started Japanese from scratch last year.
to lay the foundations	to build the basic structure	Good listening habits lay the foundations.
it follows that...	this is the logical result	I practise daily; it follows that I improve.

Try this: Euclid built a whole system from five simple rules. Write three English sentences about the "basic rules" you rely on when learning a language, using "step by step", "lay the foundations" and "there's no royal road to".