**Lived c. 325 – c. 270 BC**

Euclid authored the *Elements*, the most famous and most published mathematical work in history. The *Elements* is concerned mainly with geometry, proportion, and number theory. Enormously influential in mathematics teaching for over two thousand years, the *Elements* provided the spark that inspired many of the world’s greatest mathematicians and scientists to embark on their remarkable intellectual journeys.

Euclid is also famous for another enormously influential book, *Optics*, in which he explained light’s behavior using geometrical principles he had developed in the *Elements*. His theory of light was the basis of artistic perspective, astronomical methods, and navigation methods for more than two thousand years.

### Historical Introduction:

Little is known about Euclid personally and we do not know what he looked like. He was born in around 325 BC, was probably educated in Plato’s school in Athens, and he taught mathematics in Alexandria, the great new city of commerce and academia constructed in Egypt on the orders of Alexander the Great during Euclid’s lifetime.

Alexander built his city in a strategic location where the river Nile meets the Mediterranean sea.

#### Lifetimes of Selected Ancient Greek Scientists and Philosophers

Proclus, a 5th-century AD mathematician and philosopher tells us Euclid lived in the time of Ptolemy I (323 to 285 BC) and wrote the *Elements*, employing many of Eudoxus’ theorems and perfecting many of Theaetetus’s concepts. Proclus also stated that the *Elements* proved concepts which had been only loosely established by Euclid’s predecessors.

Proclus tells us that when Ptolemy I, who was presumably finding geometry hard work, asked if there was a shorter path to learning geometry than Euclid’s Elements, Euclid told him:

“There is no royal road to geometry.”

Serenus of Antinouplis, via Joannes Stobaeus, tells us that when a student asked Euclid what he could gain from learning geometry, Euclid said to a servant:

“Give him threepence, and then he will have gained something.”

### Euclid’s Elements

Euclid’s Elements is a masterpiece, a work of genius whose importance to the intellectual development of our species is difficult to exaggerate. It inspired ancient Greeks, such as Archimedes; Persians, such as Omar Khayyam; and, following the Renaissance, thousands of individual scientists such as Nicolaus Copernicus, Galileo Galilei, Isaac Newton, James Clerk Maxwell, Albert Einstein, and Thomas Gold.

Working in Alexandria, Euclid compiled mathematical proofs from the Pythagoreans, Eudoxus, and other earlier Greek mathematicians, strengthened the logical rigor anywhere it was weak, added his own proofs, and produced a work of stunning intellectual power.

Euclid was not concerned with solving mundane problems in mathematics such as how many tiles you need to cover a roof. His goal was to discover eternal, universal truths. The only tools he allowed himself were a straight edge and compass.

Starting with a few self-evident principles, such as that all right-angles are equal, Euclid deduced and proved a large number of ever more sophisticated mathematical theorems placing them in the *Elements’* 13 books.

The *Elements* deals with three fields: geometry in two dimensions; number theory; and geometry in three dimensions.

It includes an extraordinarily beautiful proof that there are infinitely many prime numbers.

It also includes the first ever nontrivial mathematical algorithm, perhaps devised by followers of Pythagoras, which Euclid uses to calculate the greatest common divisor of two numbers.

Following Johannes Gutenberg’s introduction of movable type printing in 1450, Euclid’s *Elements* – first printed in 1482 – is second only to the Bible in the number of editions released.

### The Elements

The *Elements* is Euclid’s most famous work. The book is logically set out into thirteen books so that it can be used easily as a reference.

In Book 1, Euclid lists twenty-three definitions, five postulates (or rules) and five common notions (assumptions) and uses them as building blocks; from these all other proofs and theorems are derived. For example, the first postulate states that it is possible to draw a straight line between any two points.

- Book 1 proves elementary theorems about plane geometry.
- Book 2 deals with geometric algebra.
- Book 3 investigates the properties of circles and this book is believed to be the work of Pythagoras and his followers.
- Book 4 concerns the construction of regular polygons, in particular the pentagon.
- Book 5 establishes the arithmetic theory of proportion and ratio and is the work of Eudoxus.
- Book 6 applies the theory of ratios in Book 5 to plane geometry.
- Book 7 deals with elementary number theory including prime numbers and contains the Euclidean algorithm for finding the greatest common divisor of two numbers.
- Book 8 looks at geometric series.
- Book 9 concerns the application of results from Books 7 and Book 8.
- Book 10 deals with the theory of irrational numbers and is mainly the work of Theaetetus and contains his “method of exhaustion”.
- Book 11 examines three-dimensional geometry giving basic definitions.
- Book 12 continues with three-dimensional geometry, calculating the relative volumes of cones, pyramids, cylinders, and spheres using “the method of exhaustion” as invented by Eudoxus.
- Book 13 investigates the five Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere, based on a work by Theaetetus.

**A Quick, Easy Proof that √2 is Irrational**

### Euclid’s Optics

Euclid’s *Optics* was an immensely influential book on light and vision. Euclid explained light’s behavior using geometrical principles he had developed in the *Elements*. His theory of light was the basis of artistic perspective, astronomical methods, and navigation methods for more than two thousand years.

Euclid considered the geometrical behavior of light rays. He got one major point wrong – he adopted the Greek consensus view of the time that we see things because our eyes emit rays rather than receive rays. Nevertheless, Euclid’s theory of light works perfectly well, because as can be seen in the image below, it is the geometry that is important, not whether a ray is travelling into or out of an eye.

### Other Contributions and Accomplishments:

Four other works of Euclid have survived:

*The Data*, a work on geometrical problems.*On Divisions of Figures*, which concerns the division of geometrical figures into two or more equal parts or into various ratios.*Catoptrics*, which examines the mathematical theory of mirrors, especially images formed by plane and spherical concave mirrors.*Phaenomena*, a treatise on spherical astronomy.

A Latin translation of the *Elements* was made around 1120 AD by English monk Adelard of Bath, who had acquired a copy of an Arabic version in Spain and the first complete English translation of the *Elements* was made in 1570 by merchant Sir Henry Billingsley.

The increasing development of sciences and mathematics in the 18th and 19th centuries earned Euclid a crucial place in the curriculum of schools and universities throughout the Western world.

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