The Elements

by Euclid of Alexandria


Euclid, also called "Euclid of Alexandria" (to distinguish him from the Socratic philosopher Euclid of Megara) is considered the founder of Geometry, often referred to as the "elementer" (στοιχειωτής). He was active in Alexandria during the reign of Ptolemy 323–283 BCE. The Elements is one of the most influential works in Mathematics, and was used for teaching Mathematics from its appearance until the beginning of the 20th century; and although it was written so early, it is an exemplar of Mathematical rigour.

His name (Euclid) means "renowned", "glorious". We don't know much about his life. According to Arabic sources, he was born in Tyre around 325 BCE. Proclus briefly refers to Euclid in his commentary work on the Elements. According to Proclus, Euclid belonged to Plato's circle, and he composed the Elements from the works of various students of Plato such as Eudoxus of Cnidus, Theaetetus and Philip Opundius. Euclid died around 270 BCE in Alexandria.

Although the Elements are composed from various earlier works, Euclid's great contribution is that he integrated them very well in one work with an absolute commitment to mathematical rigor, a rigor that has remained the basis of Mathematics to this day.

In todays Greece, several books on Euclidean "Geometry" are available, on which the teaching of geometry in secondary education is based. In our view, although several of these are very worthy efforts, none has the mathematical beauty, charm, and rigor that the Elements themselves have. Unfortunately the Elements are not available to Greek Mathematicians and we hope that this edition will fill-in this gap. The text does not target the Historians of Mathematics, but the Mathematician colleagues who have to teach geometry in today's modern school classroom.

We firmly believe that Euclid's text is unsurpassed, and it is in our opinion unfortunate that it is not taught verbatim in secondary education. Typical is the example of the Pythagorean Theorem (Book 1, Proposition 47) which is taught with various proofs, such as that of similar triangles, preferring a fast way to prove it, over the deeper understanding one gets from Euclid's beautiful proof.

The following pdf contains a release of the Elements in modern Greek using modern mathematical notation but following the essence of Euclid's proofs. It also contains the ancient text at the end of the file, while each Proposition in the modern Greek text is connected with a hyperlink to the corresponding Proposition in the ancient Greek from the original.

The comments added are minimal, so as not to affect the flow of the text, but to provide necessary the clarifications so that the text can be understood by modern Mathematicians.

The source code of the book is available under restrictions.

Download the Elements in pdf

We also find it helpful to take a detour when studying Book 12 Proposition 2. Because in this proposition it is proved that the area of the circle of radius r is equal to the area of ​​the circle of radius 1 times r2. In order for this part to be complete, we recommend the study of Archimedes' Measurement of a Circle, which estimates the area of a circle of radius 1, i.e. π, which he finds to be 3.14.

Download Measurement of a Circle in pdf

First release of the Elements in Braille: download the text in Modern Greek and the figures here.
First release of the original ancient Greek of Elements in full polytonic Braille: download here.

The conversions were performed by latex2nemeth.

Samos, Greece, 2022

N. Rokopanos
S. Sakellari
A. Tsolomitis