## Table of contents

**Euclid of Alexandria** (Ancient Greek: Εὐκλείδης – *Eukleídēs*, lived c. 300 BCE, Alexandria, Egypt) systematized ancient Greek and Near Eastern mathematics and geometry. He wrote *The Elements*, the most widely used mathematics and geometry textbook in history. Older books sometimes confuse him with Euclid of Megara.

Modern economics has been called “a series of footnotes to Adam Smith,” who was the author of *The Wealth of Nations* (1776 CE). Likewise, much of Western mathematics has been a series of footnotes to Euclid, either developing his ideas or challenging them.

## Biography

Euclid is mentioned by Pappus, but the most important testimony on which the historiography about him is based comes from Proclus, who places him among the followers of Plato, but younger than the disciples of the latter.

At the end of the fourth century BC, Ptolemy I, then Pharaoh, an enlightened ruler, punctilious and proactive in his governmental efforts, established a school in Alexandria, called the Museum. Those who taught in this school were a group of scholars, including Euclid.

Euclid was one of the initiators of the axiomatic set-up of mathematical theories, an endeavor that was undertaken beginning in his century and that involves axioms and theorems, which are consequences of the former. This model is applied to all deductive scientific disciplines, such as logic and mathematics, and has allowed them to appropriate the methodicality that we now attribute to them, thanks to the articulation of first principles and results derived from them. Despite the very few historical precedents of axiomatic theory in mathematical and other fields, it must be said that the axiom itself is nevertheless the basis of mathematics. Provided that the initiation to this kind of approach is a huge credit to the mathematician of Alexandria, he proposed a kind of geometry strongly based on axiomatic theory, while, in an antithetic way, many of his contemporary colleagues clearly rejected a kind of geometry that started from axioms.

With regard to the teachings conducted by Euclid in the Museum, he was remembered by his students above all for his wide knowledge in various fields and for expository skills that made him one of the most appreciated and prepared teachers in the Alexandrian school. These unique qualities also helped him in the writing of his great work, the Elements.

Controversial is the news that he was a convinced Platonic. Today there is a tendency to consider this judgment as unfounded and probably dictated by Proclus’ desire to annex the greatest mathematician of antiquity to the ranks of Neoplatonists to which Proclus himself belonged.

## Euclid’s Elements

Euclid, who was given the epithet of στοιχειωτής (composer of the Elements), formulated the first organic and complete representation of geometry in his fundamental work: the Elements, divided into 13 books. Of these, six deal with elementary plane geometry, three with number theory, one (book X) with incommensurables, and the last three with solid geometry. Each book begins with a page containing statements that can be considered as a kind of definitions that serve to clarify the subsequent concepts; they are followed by other propositions that are instead real problems or theorems: these differ from each other for the way they are stated and for the ritual phrase with which they close.

To give an idea of the complexity of writing Euclid’s Elements, one need only think of the statement that, in the incipit of the first part of one of his essays on Euclid, Pietro Riccardi, a 19th century scholar, makes about the disproportionate number of editions of Euclid’s work: “The number of editions of the mentioned work of Euclid, and of the translations and reductions that were published under his name, is certainly higher than can be commonly conjectured; and indeed I maintain that there is no book of considerable importance, except for the Bible, which can boast a greater number of editions and illustrations”.

The work does not review all the geometrical knowledge of the time, as was mistakenly assumed, but deals with all the so-called elementary arithmetic, i.e. related to the theory of numbers, in addition to “synthetic geometry” (ie an axiomatic approach to the subject), and algebra (not in the modern sense of the word, but as an application of the discipline to the geometric field).

### Tradition of the text

This text has been handed down through the first reconstruction Theon of Alexandria made of it, which was translated into Latin by Adelard of Bath.

In 1270, Adelard’s translation was revised, partly in light of other Arabic sources (themselves derived from other Greek versions of Theon’s manuscript) by Campano da Novara. This version (or a copy of a copy) was printed in Venice in 1482.

Subsequently, other Greek versions of Theon’s manuscript and a Greek copy that probably predates Theon’s were found. The current reconstruction is based on the version of the Danish philologist J. L. Heiberg dating from 1880 and on that of the English historian T. L. Heath from 1908. The first translation in Chinese language from Latin was the work of the Jesuit Matteo Ricci, in 1607.

Regarding further translations in Latin, the most ancient ones are all attested between the XV and XVI centuries. The most accredited Latin translations, however, date back to the seventeenth and eighteenth centuries and, in chronological order, the most corroborated are those of Barrow (1639), Borelli (1658), Keill (1701), Gregory (1703), and Simson, considered one of, if not the most prestigious, so as to be still today the first reference text for Scottish surveyors (1756).

According to some sources, the Elements are not the work of Euclid alone: he collected together, reworking and arranging axiomatically, the mathematical knowledge available in his time. His work has been considered for over 20 centuries an exemplary text for clarity and rigor of exposition, and can be considered the most successful text for the teaching of mathematics and argumentative precision in history.

The Elements is not a compendium of the mathematics of the time, but an introductory manual that encompasses all “elementary” mathematics, that is, arithmetic (the theory of numbers), synthetic geometry (of points, lines, planes, circles and spheres) and algebra (not in the modern sense of symbolic algebra, but of an equivalent in geometric terms).

No direct copies of this work have come down to us; in the version that has come down to us, the Euclidean treatise merely presents a sober and logical exposition of the basic elements of elementary mathematics.

Many ancient editions contain two other books that more recent critics attribute to Ipsicles (2nd century B.C.) and Isidore of Miletus (5th-6th century A.D.), respectively.

### Modern vision

In 1899 David Hilbert posed the problem of giving a rigorous axiomatic foundation to geometry, that is to describe Euclidean geometry without leaving any axiom unexpressed. He thus comes to define 28 axioms, expressed in his work Grundlagen der Geometrie (foundations of geometry). Many of these axioms are assumed implicitly by Euclid in the Elements: for example, Euclid never expressly says “there is at least one point outside the line”, or “given three non-aligned points, there is only one plane containing them”, yet he uses them implicitly in many demonstrations.

Taking a cue from Hilbert, and inspired by the spirit of Euclid, the collaboration of some of the best mathematicians active from 1935 to 1975 gathered under the pseudonym Nicolas Bourbaki composed the monumental work, Elements of Mathematics, in 11 volumes and tens of thousands of pages, giving an axiomatic treatment to the various branches of mathematics. However, due to Gödel’s incompleteness theorem, no axiomatization of mathematics that contains at least arithmetic can be complete.

Not without interest is the unique edition of the first six books of Euclid’s Elements proposed by the Irish engineer and mathematician Oliver Byrne in 1847. In the author’s intentions, the use of colors for the diagrams and the search for new symbolic languages should have facilitated the understanding and consolidation of arithmetic knowledge, that is, it was not intended to be purely illustrative but didactic. The result, rather eccentric, is an authentic work of art that anticipates the artistic avant-garde of the twentieth century. “None of those who hold this book in their hands can escape the fascination that emanates from these pages, precisely because by this means the understanding of the most complex and abstract mathematical regularities is proposed in the simplest manner, as it appears for now, and demonstrated in a completely concrete way ad oculos.”

Euclid had an enormous influence on culture; primarily, of course, in the areas of mathematics and geometry. Reducing to the bone some of the important theories he expounded in the “Elements” and still studied today, Euclid defined all geometric and arithmetic entities, starting from the point and ending with the theory of parallel lines. This is not a construction of concepts, but a description of the entities, so that they can be easily recognized through a satisfactory nomenclature. The geometric entities, therefore, already exist; to define them implies only to recognize them.

Geometry, originally, was not supposed to have anything to do with ontology. Actually, the documentation about Greek geometers is quite scarce, so we have no certainties of any kind. What transpires in the following centuries, however, is the common understanding that Euclidean geometry is primarily aimed at describing space. Immanuel Kant, the last of the rationalist theorists, confirms this hypothesis, asserting that Euclidean geometry is the a priori form of our knowledge of space.

## Other works

Euclid was the author of other works: the Data, closely related to the first 6 books of the Elements; the Porisms, in 3 books, which have come down to us thanks to the summary made by Pappo of Alexandria; the Surface Places, which has been lost; the Conics, which has been lost; the Optics and the Catoptrics, the first of which is a valuable work, since it is one of the first treatises on perspective, understood as the geometry of direct vision. In the Optics Euclid proposes an original theory on the vision of reality, of effusive or emissive type, according to which rays depart from the eye and spread in space, until they meet the objects.

This kind of definition is in sharp contrast with the previous perspective theory of Aristotle, who, instead, assumed that there was a straight line that ideally joined the eye with the object, allowing the action of the eye on the object itself. Euclid’s Optics had, among its many objectives, that of fighting the Epicurean concept according to which the dimensions of an object were the same as those perceived by the eye, without taking into account the shrinking caused by the perspective from which the object was seen.

He wrote the Phenomena, description of the celestial sphere; Section of the Canon and the Harmonic Introduction, treatises on music.

Another consideration deserves the Division of Figures, a work that has come to us thanks to a salvific maneuver of translation by some Arab scientists. The original work in Greek, in fact, was lost, but before its disappearance was used a translation into Arabic which was in turn translated into Latin and then again in major modern languages.