Euclid writes the Elements

The story

An author without a biography

Euclid is one of the most widely read authors in history, but we know almost nothing about him. We do not know his date of birth, or his birthplace, or his appearance, or his family. Scholars place him between the end of the 4th and the beginning of the 3rd century BC. The philosopher Proclus, who wrote many centuries later, tells that he was younger than Plato's disciples and older than Eratosthenes and Archimedes, and that he lived in the time of Ptolemy I, the king of Egypt. He probably taught at the Mouseion of Alexandria (this is a reconstruction, not a certain fact), the great centre of study set up by the Ptolemies. That is why our site marks the date as “about 300 BC”: it is an estimate, not a precise year.

We also know an anecdote, which may be a legend. King Ptolemy is said to have asked him whether there was a shorter way to learn geometry. Euclid is said to have answered that there is no “royal road” to geometry. As if to say: study is the same for everyone, even the powerful.

The book that put mathematics in order

The work that made him famous is called the “Elements”. It is divided into 13 books, which gather and put in order what Greek mathematicians knew in his time. Books one to six deal with plane geometry: triangles, quadrilaterals, circles, proportions. Seven to nine treat numbers and their properties. The tenth studies magnitudes that cannot be measured with whole numbers or fractions, called incommensurable. The last three books, the eleventh to the thirteenth, concern geometry in space, that is, solids.

The “Elements” are not a collection of random results. Euclid's merit is the method. Each book opens with definitions, that is, the explanation of what a point, a line, a circle is. Then come the postulates and the common notions, the few simple statements that are accepted without proof: for example, that a straight line can be drawn between two points. From these foundations Euclid proves, step by step, hundreds of propositions and theorems: each result rests on the previous ones, with no jumps. It is a building made of bricks, in which every brick is checked.

The fifth postulate

Among the postulates there is one, the fifth, that has always seemed different from the others. It concerns parallel lines, and in its simplest formulation, that of the scholar Playfair in 1795, it says that, given a line and a point outside it, only one line can be drawn parallel to the given one. If you deny this postulate you do not fall into contradiction, but you obtain other geometries: the elliptic one, in which no parallel passes through a point outside the line, and the hyperbolic one, in which at least two pass. These are the “non-Euclidean geometries”.

A book that has lasted two thousand years

The “Elements” were copied by hand for centuries and translated many times. One of the Latin versions, that of Campanus of Novara, revised starting from the translation of Adelard of Bath, was printed in Venice in 1482. For more than twenty centuries it was regarded as the model of how to write a scientific text, clear and rigorous, and until the early 20th century it was one of the main texts for teaching geometry in schools. According to a scholar cited by Wikipedia, the number of its editions has been exceeded only by that of the Bible.

Euclid also wrote other works. The “Elements” is the most famous, but among those that have come down to us, some attributed to him, are the “Optics” and the “Catoptrics”, on light and mirrors, the “Phenomena”, on astronomy, the “Data”, the “Division of the Canon”, on music, and “On Divisions of Figures”. Others, like the “Surface Loci” and the “Conics”, have been lost.