Elements

What It Is

Thirteen books of Greek mathematics, assembled in Alexandria around 300 BC. Very little in it is Euclid's own discovery. The achievement is the arrangement: he set two centuries of Greek mathematics in an order where nothing is assumed that has not first been established.

He begins with definitions, five postulates and five common notions, and from that small pile of assumptions builds 465 propositions, each one leaning on the ones before it. Nothing enters from outside. It is the first structure of its kind anyone built, and it still stands.

This is a book you work, not a book you read. Follow a proof with your eyes and you will understand nothing. Redraw the diagram, then work the argument until you see why each step could not have gone otherwise. Ten pages an hour is fast.

Why It Matters

Every proof written since is written in Euclid's form. State what you assume, argue only from that, and you may claim to have shown something. Outside mathematics the ambition proved irresistible: Newton cast the Principia as propositions and demonstrations, and Spinoza wrote his Ethics in the geometrical manner, with axioms and QEDs, as though morality could be forced the same way.

Lincoln, riding the circuit as a lawyer, worked through the first six books by lamplight because he could not stand not knowing what the word demonstrate meant. Einstein called it the holy little geometry book, and said its clarity never left him.

And Euclid's one visible flaw generated a new universe. His fifth postulate, on parallel lines, is longer and clumsier than the others, and for two thousand years mathematicians tried to prove it from the rest. They could not. In the nineteenth century a few of them tried assuming it was false, and non-Euclidean geometry fell out of the attempt. Einstein needed it. The most productive failure in the history of thought was a failure to fix one sentence of this book.

Before You Read

No arithmetic is required and no prior mathematics. What it demands is patience and a pencil.

Do not attempt the whole thing. Book I, which runs to 48 propositions and ends with the Pythagorean theorem, is the heart of it and the finest single stretch. Do a proposition or two an evening and you will be through in a month.

After that, take Books II and III, then the number theory of VII to IX, where you will find the proof that the primes never run out. Book V, on proportion, is beautiful and hard. Book X, on irrational magnitudes, is the longest and the most forbidding, and nearly every reader skims it. Book XIII, on the five regular solids, is the intended summit and worth reaching.

Which Translation to Read

• Thomas Heath (Dover, three volumes) — recommended. The standard English Euclid since 1908, revised in 1926 and in print from Dover ever since. Heath gives you the text and then a commentary that dwarfs it: the history of every proposition, what the Greek commentators said, where the reasoning is shaky. It is the only edition that teaches you the book rather than presenting it. The cost: three volumes, century-old typesetting, and notes so full they will hand you the point of a proposition before you have earned it. Do the proof first, then read Heath. View on Amazon →
• Green Lion Press (2002), using Heath's translation. The same words in one volume, stripped of the commentary, with a diagram on every spread so you never turn a page to find the figure. The copy to keep open on the desk. The cost: no commentary at all, so when you get stuck you are alone.
• Richard Fitzpatrick (2007), with Heiberg's Greek on the facing page. A clean modern rendering, free as a PDF and cheap in print. The cost: minimal apparatus, and the English is functional.

Heath is the Euclid to own, because the notes are the education. Green Lion is the one to work from. Fitzpatrick if you read Greek.

One edition is famous for other reasons. In 1847 Oliver Byrne printed the first six books using coloured figures in place of letters, and made one of the strangest and most beautiful books of the nineteenth century: Mondrian, eighty years early. Taschen has reissued it in facsimile. Own it; do not try to learn from it.

Getting the Most From This Book

Draw every diagram yourself, by hand, before you read the proof attached to it. That is the difference between reading Euclid and doing Euclid, and only one of them does anything to you.

Cover the proof and try to prove it yourself first. You will usually fail, and the failure is what makes Euclid's solution legible when you uncover it.

Do not treat it as a mathematics course. Modern proofs of the same results are shorter and easy to find. You are here for the architecture: to watch a mind decide what may be assumed, and then refuse itself everything else.