Euclid's Elements (c. 300 BC): The Most Influential Maths Book

Euclid’s Elements, compiled in Alexandria around 300 BC, is a Greek treatise in 13 books that builds geometry and arithmetic from a handful of definitions and assumptions, proving every result in logical order. It became the standard way to learn mathematics in the Greek, Arabic and European worlds for more than 2,000 years. Its method of deriving theorems from axioms still defines what mathematicians mean by a proof.

Background: Greek mathematics before Euclid

Greek geometry began, according to later tradition, with Thales of Miletus in the 6th century BC and the followers of Pythagoras. By the 4th century BC mathematicians connected with Plato’s Academy, such as Theaetetus and Eudoxus of Cnidus, had made major advances in the theory of irrational lengths, proportion and solid geometry.

Earlier collections called “Elements” existed. Proclus, writing in the 5th century AD, names Hippocrates of Chios as the first to compose one, in the 5th century BC. None of these survive, because Euclid’s version replaced them.

Euclid and Alexandria

Very little is known about Euclid himself. Proclus says he lived in the time of Ptolemy I Soter, the Macedonian general who ruled Egypt from 323 BC and became king in 305 BC. Pappus of Alexandria, writing around AD 320, says Euclid taught in Alexandria and had students there.

Alexandria was a new city, founded by Alexander the Great in 331 BC. The Ptolemies made it a centre of scholarship, with the Museum and the Library of Alexandria attracting scholars from across the Greek world. Euclid’s work belongs to the first generation of this scientific culture.

A famous anecdote, told by Proclus, has Ptolemy asking if there was a shorter way to learn geometry, and Euclid replying that there is no royal road to geometry. A similar story is told of other mathematicians and kings, so it is probably not authentic. Medieval writers sometimes confused him with Euclid of Megara, a philosopher and pupil of Socrates.

What the Elements contains

The Elements opens with 23 definitions, five postulates and five “common notions” (general axioms, such as “things equal to the same thing are equal to each other”). Everything else is proved from these, with each proposition relying only on what came before.

The proof that there are infinitely many prime numbers (Book IX, Proposition 20) and the method for finding the greatest common divisor of two numbers (Book VII, Propositions 1–2) are still taught in essentially Euclid’s form. Book XIII ends by showing that exactly five regular solids exist.

Each proposition follows a fixed pattern: a statement, a diagram with lettered points, a construction, a step-by-step proof and a conclusion. Proofs often end with a Greek phrase later translated into Latin as “quod erat demonstrandum” (QED), “which was to be demonstrated”. The very first proposition shows how to construct an equilateral triangle on a given line using two circles.

The work contains about 465 propositions. Much of it reworks earlier discoveries: Book V is generally credited to Eudoxus, Books X and XIII draw on Theaetetus, and Book XII uses Eudoxus’ “method of exhaustion”, an early approach to limits.

Euclid’s other works

Euclid wrote several other books, some of which survive. The Data deals with what can be deduced when certain parts of a figure are known. The Optics, the earliest surviving Greek treatise on perspective, treats vision as straight rays travelling from the eye. The Phaenomena applies the geometry of the sphere to astronomy, and On Divisions of Figures survives mainly through an Arabic version.

Other works are lost, including the Conics, a four-book study of curves later surpassed by Apollonius of Perga, and the Porisms, known only through summaries by Pappus. Books XIV and XV, sometimes printed with the Elements, were added later by other authors; Book XIV is usually attributed to Hypsicles in the 2nd century BC.

The parallel postulate

The fifth postulate is longer and less obvious than the others. It says that if a line crossing two other lines makes interior angles on one side adding to less than two right angles, those two lines will meet on that side. In practice, it guarantees that parallel lines behave as we expect.

Many mathematicians, from Ptolemy in antiquity to Arabic scholars such as Omar Khayyam and later Europeans, tried to prove it from the other postulates and failed. In the 1820s and 1830s Nikolai Lobachevsky and János Bolyai, and privately Carl Friedrich Gauss, showed that consistent geometries exist in which the postulate is false. These non-Euclidean geometries later became essential to Einstein’s general theory of relativity.

How the text survived

Euclid’s original manuscript is lost. Small fragments survive on papyri from Roman Egypt, including a piece from Oxyrhynchus showing a diagram from Book II. Most medieval Greek manuscripts descend from an edition by Theon of Alexandria in the 4th century AD. In 1808 the French scholar François Peyrard identified a manuscript in the Vatican that preserves an earlier, pre-Theonine version.

The Elements reached the Islamic world in the late 8th and 9th centuries, when al-Hajjaj ibn Yusuf and later Ishaq ibn Hunayn, revised by Thabit ibn Qurra, translated it into Arabic. Adelard of Bath made a Latin translation from Arabic around 1120, and Campanus of Novara produced a widely used Latin version around 1260.

The first printed edition appeared in Venice in 1482, from the press of Erhard Ratdolt. The first full English translation, by Henry Billingsley, followed in 1570 with a preface by John Dee.

Legacy

The Elements is often said to be second only to the Bible in the number of editions, with estimates of more than 1,000. That claim cannot be checked precisely, but no other mathematical book has been used for so long. Until the 20th century, studying Euclid was a normal part of education in Europe and North America. In Britain the text was still taught almost word for word in the 19th century, and in 1871 reformers founded the Association for the Improvement of Geometrical Teaching to argue for more modern textbooks. In the United States, Abraham Lincoln wrote in 1860 that he had studied and nearly mastered the first six books of Euclid after serving in Congress.

Its influence went beyond mathematics. Philosophers such as Spinoza wrote works “in geometrical order”, and Isaac Newton set out his Principia in the style of definitions, axioms and propositions. In 1899 David Hilbert published a modern set of axioms for geometry that filled logical gaps in Euclid’s system, but kept its overall plan.

Timeline

• c. 440 BC: Hippocrates of Chios writes an early “Elements”, now lost.
• c. 390–340 BC: Theaetetus and Eudoxus develop theories later used in the Elements.
• 331 BC: Alexandria is founded.
• c. 300 BC: Euclid compiles the Elements in Alexandria.
• 4th century AD: Theon of Alexandria produces his edition of the text.
• c. 800–900: The Elements is translated into Arabic.
• c. 1120: Adelard of Bath translates it into Latin.
• 1482: First printed edition appears in Venice.

Why it matters

The Elements showed that a large body of knowledge could be built from a few clear assumptions by strict logic. That idea shaped mathematics, science and philosophy, and Euclidean geometry still describes the world at everyday scales.