Euclid's Elements

The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid.

The Elements is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus, and Theaetetus, the Elements is a collection in 13 books of definitions, postulates, geometric constructions, and theorems with their proofs that covers plane and solid Euclidean geometry, elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the construction of regular polygons and polyhedra.

Often referred to as the most successful textbook ever written, the Elements has continued to be used for introductory geometry. It was translated into Arabic and Latin in the medieval period, where it exerted a great deal of influence on mathematics in the medieval Islamic world and in Western Europe, and has proven instrumental in the development of logic and modern science, where its logical rigor was not surpassed until the 19th century.

Euclid's Elements is the oldest extant large-scale deductive treatment of mathematics. Proclus, a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians, including Eudoxus, Hippocrates of Chios, Thales, and Theaetetus, while other theorems are mentioned by Plato and Aristotle. It is difficult to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics. The Elements version available today also includes "post-Euclidean" mathematics, probably added later by later editors such as the mathematician Theon of Alexandria in the 4th century. The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps" and the historian Serafina Cuomo described it as a "reservoir of results". Despite this, historian Michalis Sialaros opines that its "remarkably tight structure" suggests that Euclid wrote the Elements himself rather than merely editing together the works of others.

The detailed attribution of parts of the Elements to specific mathematicians is still the subject of scholarly debate. According to W. W. Rouse Ball, Pythagoras was probably the source for most of books I and II, Hippocrates of Chios for book III, and Eudoxus of Cnidus for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. Wilbur Knorr ascribes the origin of the material in Books I, III, and VI of the Elements to the time of Hippocrates of Chios, and of the material in books II, IV, X, and XIII to the later period of Theodorus of Cyrene, Theaetetus, and Eudoxos. However, this suggested history has been criticized by van der Waerden, who believed that books I through IV were largely due to the much earlier Pythagorean school.

Other similar works are also reported to have been written by Hippocrates of Chios, Theudius of Magnesia, and Leon, but are now lost.

The Elements does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry (books I-VI), basic number theory (books VII-X) and solid geometry (books XI-XIII), though book V (on proportions) and X (on incommensurability) do not exactly fit this scheme. The heart of the text is the theorems scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια). The postulates (that is, axioms) and common notions occur only in book I. Close study of Proclus suggests that older versions of the Elements may have followed the same distinctions but with different terminology, instead calling each definition a "hypothesis" (ὑπόθεσις) and each common notion an "axiom" (ἀξίωμα). The second group consists of propositions, presented alongside mathematical proofs and diagrams. It is unknown whether Euclid intended the Elements as a textbook, despite its wide subsequent use as one. As a whole, the authorial voice remains general and impersonal.

Number theory, the theory of the arithmetic of natural numbers, is covered by books VII to X. Book VII begins with a set of 22 definitions for parity (whether a number is even or odd), prime numbers, and other arithmetic-related concepts. The first of these definitions is for the unit (in modern terms, the number one), while the second states that "a number is a multitude composed of units"; this is generally interpreted to mean that, for Euclid, one is not a number, and the natural numbers begin at two.

The final three books primarily discuss solid geometry. By introducing a list of 37 definitions, Book XI contextualizes the next two. Although its foundational character resembles Book I, unlike Book I it features no axiomatic system or postulates.

Two additional books, that were not written by Euclid, Books XIV and XV, have been transmitted in the manuscripts of the Elements:

• Book XIV was likely written by Hypsicles, following a treatise by Apollonius of Perga. It continues the study in Book XIII of the Platonic solids and their circumscribed spheres. It concludes that, for a dodecahedron and icosahedron inscribed in a common sphere, the ratio of their surface areas and the ratio of their volumes are equal, both being\[\sqrt{\frac{10}{3(5-\sqrt{5})}} = \sqrt{\frac{5+\sqrt{5}}{6}}.\]
• Book XV may have been written by a student of Isidore of Miletus. It also studies the Platonic solids; it inscribes some of them within each other, counts their edges and vertices (without however finding Euler's formula \(V-E+F=2\) relating these counts to each other), and computes the dihedral angles between their faces.

The practice of adding to the works of famous authors, exemplified by these books, was not unusual in ancient Greek mathematics.

Euclid's axiomatic approach and constructive methods were widely influential.

Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass (circle-drawing tool) and straightedge (unmarked ruler). His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier.

The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then the 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation.

No indication is given of the method of reasoning that led to the result, although a different book by Euclid, Data, does provide instruction about how to approach the types of problems encountered in the first four books of the Elements. For proofs involving case analysis, the Elements often includes details only of the most difficult case; some of these case analyses have been filled out by later editors such as Theon.

Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles, the number 1 was sometimes treated separately from other positive integers, and, as multiplication was treated geometrically, as the area of a rectangle with given side lengths, he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals, an alphabetic numeral system in which each Greek letter represented a single-digit multiple of a power of ten.

Euclid's Elements has been referred to as the most successful textbook ever written. The Elements is often considered after the Bible as the most frequently translated, published, and studied book in history. With Aristotle's Metaphysics, the Elements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Islamic world and Western Europe. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482, the number reaching well over one thousand.

The oldest extant evidence for Euclid's Elements are a set of six ostraca (clay fragments with writing scratched onto them) found among the Elephantine papyri and ostraca, from the 3rd century BC, that deal with propositions XIII.10 and XIII.16, on the construction of a dodecahedron. A papyrus recovered from Herculaneum contains an essay by the Epicurean philosopher Demetrius Lacon on Euclid's Elements. The earliest extant papyrus containing the actual text of the Elements is Papyrus Oxyrhynchus 29, a fragment containing the text of Book II, Proposition 5 and an accompanying diagram, dated to c. 75-125 AD.

Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian Library in Oxford. The manuscripts available are of variable quality, and often incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text. Also of importance are the scholia, or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study.

In the 4th century AD, Theon of Alexandria produced an edition of Euclid which was so widely used that it became the only surviving Greek-language source (in multiple manuscripts) until François Peyrard's 1808 discovery at the Vatican of a manuscript not derived from Theon's. This manuscript, MS. Vat.gr.190, was transcribed in the 10th century. It does not include text identifying itself as edited by Theon, and is missing a corollary to Book VI Proposition 33 claimed by Theon to be his own addition. Both Greek versions include many explanations beyond the propositions and their proofs that are missing from the Arabic translations of the Elements. This sparked a 19th-century academic debate between M. Klamroth and J. L. Heiberg over whether the differences between the various versions reflected abridgements or additions to Euclid's text. Revisiting this issue, Wilbur Knorr sides with Klamroth in suggesting that the Arabic sources were closer to the original, but concludes that "We have never had a 'genuine' text of Euclid, and we never will have one."

Although Euclid was known to Cicero, for instance, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century.

From classical antiquity until the western invention of printing, texts such as the Elements were preserved and duplicated through the process of copying manuscripts. This was laborious and expensive so manuscripts were often confined to the collections of the wealthy or to institutions such as the House of Wisdom in the medieval Islamic world or the monasteries and early universities of medieval Europe.

The Islamic world received the Elements from the Byzantine Empire around 760. According to sources from that milieu, this version was translated into Arabic under Harun al-Rashid (c. 800), in two versions by Al-Ḥajjāj ibn Yūsuf ibn Maṭar. Another Arabic translation was made later in the 9th century by Ishaq ibn Hunayn and revised by Thābit ibn Qurra. Although most Arabic manuscripts have been attributed to one or another of these translations, some mix material from both, and their attributions are not always in accord with the evidence from textual similarities in surviving manuscripts. This mixture was also passed down into medieval translations into Hebrew from the Arabic.

The Byzantine scholar Arethas commissioned the copying of one of the Greek manuscripts of Euclid in the late ninth century; it and another Byzantine manuscript are the two oldest surviving copies of the Greek text. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, except through fragments of a translation into Latin by Boethius (c. 500), quoted in other works. In about 1120, the English monk Adelard of Bath translated the Elements into Latin from an Arabic translation. A relatively recent discovery was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the Almagest to Latin. The Euclid manuscript is extant and quite complete.

After Adelard's translation (which became known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period include Herman of Carinthia who wrote an edition around 1140, Robert of Chester (his manuscripts are referred to collectively as Adelard II, written on or before 1251), John of Tynemouth (late 12th century; his manuscripts are referred to collectively as Adelard III), and Gerard of Cremona (sometime after 1120 but before 1187). The detailed transmission history of these translations is still an active area of research. Campanus of Novara relied heavily on these Arabic translations to create his edition (sometime before 1260) which ultimately came to dominate Latin editions until the availability of Greek manuscripts in the 16th century. There are more than 100 pre-1482 Campanus manuscripts still available today. After its availability in Europe, the first books of the Elements became standard in medieval universities as part of the quadrivium, the second stage of instruction after the trivium of grammar, logic, and rhetoric.

The first printed edition of the Elements was published by Erhard Ratdolt in 1482, based on Campanus's version, and since then it has been translated into many languages and published in over a thousand different editions. A manuscript descended from Theon's Greek version was recovered and a Latin translation was published in Venice in 1505 by Bartolomeo Zamberti. The Greek text itself was published in 1533. The first to translate the Elements into a modern European language was Nicolo Tartaglia, who published an Italian edition in 1543.

In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley. In 1607, The Italian Jesuit Matteo Ricci and the Chinese mathematician Xu Guangqi published the first Chinese edition of Euclid's Elements.

The Renaissance also saw the creation of new works about polyhedra, illustrated with perspective drawings, including Piero della Francesca's De quinque corporibus regularibus (late 1400s), its plagiarism by Luca Pacioli as Divina proportione (1498, illustrated by Leonardo da Vinci), and Wenzel Jamnitzer's Perspectiva corporum regularium (1568). The Elements were the main inspiration behind della Francesca's initial work in this direction. Pacioli lectured in Venice on Euclid, and his commentary was included in a 1509 edition of the Elements. Jamnitzer, likewise, credits the Elements in the subtitles of his book.

Although this period also saw an explosion in newly published textbooks, teachers often stuck to the classics: a list of recommended readings by 16th century Dutch humanist Joachim Sterck van Ringelbergh, for instance, lists the Elements as its only mathematics book. Even after printed versions existed, a university might expect its students to copy by hand material from the university's copy of the Elements.

In the 19th century the Elements fell out of favor as a geometry textbook, in part supplanted by newer textbooks such as one by Adrien-Marie Legendre, in part because of the rise of other forms of geometry including non-Euclidean geometry, analytic geometry, and descriptive geometry, and in part out of pressure for an approach to mathematics education with more emphasis on intuition and less on memorization. Charles Dodgson (better known as Lewis Carroll), in particular, railed against this replacement of Euclid in his book Euclid and His Modern Rivals (1879). Another defender of the Elements, mathematician and historian W. W. Rouse Ball, remarked that "the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose." Despite falling out of wide use in education, the Elements is still occasionally used as a textbook in experimental education projects.

The Elements remains an object of scholarly study for the history of mathematics, and it has had significant influence on two areas of modern mathematics, the development of non-Euclidean geometry and of the axiomatic method.

Over one thousand editions of Euclid's Elements have been published, in Greek, Latin, English, and other languages. Some of the more significant of these include:

Condensed: the full section is in Wikipedia.