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.
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.”
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.
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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