Very little is known of Euclid's life, and most information comes from the scholarsProclus andPappus of Alexandria many centuries later.Medieval Islamic mathematicians invented a fanciful biography, and medievalByzantine and earlyRenaissance scholars mistook him for the earlier philosopherEuclid of Megara. It is now generally accepted that he spent his career inAlexandria and lived around 300 BC, afterPlato's students and before Archimedes. There is some speculation that Euclid studied at thePlatonic Academy and later taught at theMusaeum; he is regarded as bridging the earlier Platonic tradition inAthens with the later tradition of Alexandria.
The English name 'Euclid' is the anglicized version of theAncient Greek nameEukleídes (Εὐκλείδης).[4][a] It is derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious".[6] In English, bymetonymy, 'Euclid' can mean his most well-known work,Euclid'sElements, or a copy thereof,[5] and is sometimes synonymous with 'geometry'.[2]
As with manyancient Greek mathematicians, the details of Euclid's life are mostly unknown.[7] He is accepted as the author of four mostly extant treatises—theElements,Optics,Data,Phaenomena—but besides this, there is nothing known for certain of him.[8][b] The traditional narrative mainly follows the 5th century AD account byProclus in hisCommentary on the First Book of Euclid's Elements, as well as a few anecdotes fromPappus of Alexandria in the early 4th century.[4][c]
According to Proclus, Euclid lived shortly after several ofPlato's (d. 347 BC) followers and before the mathematicianArchimedes (c. 287 – c. 212 BC);[d] specifically, Proclus placed Euclid during the rule ofPtolemy I (r. 305/304–282 BC).[7][8][e] Euclid's birthdate is unknown; some scholars estimate around 330[11][12] or 325 BC,[2][13] but others refrain from speculating.[14] It is presumed that he was of Greek descent,[11] but his birthplace is unknown.[15][f] Proclus held that Euclid followed thePlatonic tradition, but there is no definitive confirmation for this.[17] It is unlikely he was a contemporary of Plato, so it is often presumed that he was educated by Plato's disciples at thePlatonic Academy in Athens.[18] HistorianThomas Heath supported this theory, noting that most capable geometers lived in Athens, including many of those whose work Euclid built on;[19] historian Michalis Sialaros considers this a mere conjecture.[4][20] In any event, the contents of Euclid's work demonstrate familiarity with the Platonic geometry tradition.[11]
In hisCollection, Pappus mentions thatApollonius studied with Euclid's students inAlexandria, and this has been taken to imply that Euclid worked and founded amathematical tradition there.[8][21][19] The city was founded byAlexander the Great in 331 BC,[22] and the rule of Ptolemy I from 306 BC onwards gave it a stability which was relatively unique amid the chaoticwars over dividing Alexander's empire.[23] Ptolemy began a process ofhellenization and commissioned numerous constructions, building the massiveMusaeum institution, which was a leading center of education.[15][g] Euclid is speculated to have been among the Musaeum's first scholars.[22] Euclid's date of death is unknown; it has been speculated that he diedc. 270 BC.[22]
Identity and historicity
Domenico Maroli's 1650s paintingEuclide di Megara si traveste da donna per recarsi ad Atene a seguire le lezioni di Socrate [Euclid of Megara Dressing as a Woman to Hear Socrates Teach in Athens]. At the time, Euclid the philosopher and Euclid the mathematician were wrongly considered the same person, so this painting includes mathematical objects on the table.[25]
Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopherEuclid of Megara, a pupil ofSocrates included indialogues of Plato with whom he was historically conflated.[4][14]Valerius Maximus, the 1st century AD Roman compiler of anecdotes, mistakenly substituted Euclid's name forEudoxus (4th century BC) as the mathematician to whom Plato sent those asking how todouble the cube.[26] Perhaps on the basis of this mention of a mathematical Euclid roughly a century early, Euclid became mixed up with Euclid of Megara in medievalByzantine sources (now lost),[27] eventually leading Euclid the mathematician to be ascribed details of both men's biographies and described asMegarensis (lit.'of Megara').[4][28] The Byzantine scholarTheodore Metochites (c. 1300) explicitly conflated the two Euclids, as did printerErhard Ratdolt's 1482editio princeps ofCampanus of Novara's Latin translation of theElements.[27] After the mathematicianBartolomeo Zamberti [fr;de] appended most of the extant biographical fragments about either Euclid to the preface of his 1505 translation of theElements, subsequent publications passed on this identification.[27] Later Renaissance scholars, particularlyPeter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.[27]
Medieval Arabic sources give vast amounts of information concerning Euclid's life, but are completely unverifiable.[4] Most scholars consider them of dubious authenticity;[8] Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world.[17] There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man".[29] The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading hisElements, which Euclid replied with "there is no royal road to geometry".[29] This anecdote is questionable since a very similar interaction betweenMenaechmus and Alexander the Great is recorded fromStobaeus.[30] Both accounts were written in the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature.[31]
Any firm dating of Euclid's activityc. 300 BC is called into question by a lack of contemporary references.[4] The earliest original reference to Euclid is in Apollonius'prefatory letter to theConics (early 2nd century BC): "The third book of theConics contains many astonishing theorems that are useful for both the syntheses and the determinations of number of solutions of solidloci. Most of these, and the finest of them, are novel. And when we discovered them we realized that Euclid had not made the synthesis of the locus on three and four lines but only an accidental fragment of it, and even that was not felicitously done."[26] TheElements is speculated to have been at least partly in circulation by the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted;[4] however, Archimedes employs an older variant of thetheory of proportions than the one found in theElements.[8] The oldest physical copies of material included in theElements, dating from roughly 100 AD, can be found onpapyrus fragments unearthed in an ancient rubbish heap fromOxyrhynchus,Roman Egypt. The oldest extant direct citations to theElements in works whose dates are firmly known are not until the 2nd century AD, byGalen andAlexander of Aphrodisias; by this time it was a standard school text.[26] Some ancient Greek mathematicians mention Euclid by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author ofElements").[32] In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from acorruption of Greek mathematical terms.[33]
Euclid is best known for his thirteen-book treatise, theElements (Ancient Greek:Στοιχεῖα;Stoicheia), considered hismagnum opus.[3][35] Much of its content originates from earlier mathematicians, includingEudoxus,Hippocrates of Chios,Thales andTheaetetus, while other theorems are mentioned by Plato and Aristotle.[36] It is difficult to differentiate the work of Euclid from that of his predecessors, especially because theElements essentially superseded much earlier and now-lost Greek mathematics.[37][h] 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 historianSerafina Cuomo described it as a "reservoir of results".[38][36] Despite this, Sialaros furthers that "the remarkably tight structure of theElements reveals authorial control beyond the limits of a mere editor".[9]
TheElements does not exclusively discuss geometry as is sometimes believed.[37] It is traditionally divided into three topics:plane geometry (books 1–6), basicnumber theory (books 7–10) andsolid geometry (books 11–13)—though book 5 (on proportions) and 10 (onirrational lines) do not exactly fit this scheme.[39][40] The heart of the text is thetheorems scattered throughout.[35] Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles".[41] The first group includes statements labeled as a "definition" (Ancient Greek:ὅρος orὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια);[41][42] only the first book includes postulates—later known asaxioms—and common notions.[37][i] The second group consists of propositions, presented alongsidemathematical proofs and diagrams.[41] It is unknown if Euclid intended theElements as a textbook, but its method of presentation makes it a natural fit.[9] As a whole, theauthorial voice remains general and impersonal.[36]
To draw a straight line from any point to any point[j]
2
To produce a finite straight line continuously in a straight line
3
To describe a circle with any centre and distance
4
That all right angles are equal to one another
5
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles
Book 1 of theElements is foundational for the entire text.[37] It begins with a series of 20 definitions for basic geometric concepts such aslines,angles and variousregular polygons.[44] Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions.[45][k] These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as anaxiomatic system.[46][l] The common notions exclusively concern the comparison ofmagnitudes.[48] While postulates 1 through 4 are relatively straightforward,[m] the 5th is known as theparallel postulate and particularly famous.[48][n] Book 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions of plane geometry andtriangle congruence (1–26);parallel lines (27–34); thearea oftriangles andparallelograms (35–45); and thePythagorean theorem (46–48).[48] The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".[41]
Book 2 is traditionally understood as concerning "geometric algebra", though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later.[41] The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes.[37][48] It focuses on the area ofrectangles andsquares (seeQuadrature), and leads up to a geometric precursor of thelaw of cosines.[50] Book 3 focuses on circles, while the 4th discussesregular polygons, especially thepentagon.[37][51] Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion".[52][o] Book 6 utilizes the "theory ofratios" in the context of plane geometry.[37] It is built almost entirely of its first proposition:[53] "Triangles and parallelograms which are under the same height are to one another as their bases".[54]
From Book 7 onwards, the mathematicianBenno Artmann [de] notes that "Euclid starts afresh. Nothing from the preceding books is used".[55]Number theory is covered by books 7 to 10, the former beginning with a set of 22 definitions forparity,prime numbers and other arithmetic-related concepts.[37] Book 7 includes theEuclidean algorithm, a method for finding thegreatest common divisor of two numbers.[55] The 8th book discussesgeometric progressions, while book 9 includes the proposition, now calledEuclid's theorem, that there are infinitely manyprime numbers.[37] Of theElements, book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes.[41]
The final three books (11–13) primarily discusssolid geometry.[39] By introducing a list of 37 definitions, Book 11 contextualizes the next two.[56] Although its foundational character resembles Book 1, unlike the latter it features no axiomatic system or postulates.[56] The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) andparallelepipedal solids (24–37).[56]
In addition to theElements, at least five works of Euclid have survived to the present day. They follow the same logical structure asElements, with definitions and proved propositions.
Catoptrics concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors, though the attribution is sometimes questioned.[57]
TheData (Ancient Greek:Δεδομένα), is a somewhat short text which deals with the nature and implications of "given" information in geometrical problems.[57]
On Divisions (Ancient Greek:Περὶ Διαιρέσεων) survives only partially inArabic translation, and concerns the division of geometrical figures into two or more equal parts or into parts in givenratios. It includes thirty-six propositions and is similar to Apollonius'Conics.[57]
Four other works are credibly attributed to Euclid, but have been lost.[9]
TheConics (Ancient Greek:Κωνικά) was a four-book survey onconic sections, which was later superseded by Apollonius' more comprehensive treatment of the same name.[58][57] The work's existence is known primarily from Pappus, who asserts that the first four books of Apollonius'Conics are largely based on Euclid's earlier work.[59] Doubt has been cast on this assertion by the historianAlexander Jones [de], owing to sparse evidence and no other corroboration of Pappus' account.[59]
TheSurface Loci (Ancient Greek:Τόποι πρὸς ἐπιφανείᾳ) is of virtually unknown contents, aside from speculation based on the work's title.[58] Conjecture based on later accounts has suggested it discussed cones and cylinders, among other subjects.[57]
The cover page ofOliver Byrne's 1847 colored edition of theElements
Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity.[11] Many commentators cite him as one of the most influential figures in thehistory of mathematics.[2] The geometrical system established by theElements long dominated the field; however, today that system is often referred to as 'Euclidean geometry' to distinguish it from othernon-Euclidean geometries discovered in the early 19th century.[61] Among Euclid'smany namesakes are theEuropean Space Agency's (ESA)Euclid spacecraft,[62] the lunar craterEuclides,[63] and the minor planet4354 Euclides.[64]
TheElements is often considered after theBible as the most frequently translated, published, and studied book in theWestern World's history.[61] With Aristotle'sMetaphysics, theElements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds.[61]
The first English edition of theElements was published in 1570 byHenry Billingsley andJohn Dee.[27] The mathematicianOliver Byrne published a well-known version of theElements in 1847 entitledThe First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners, which included colored diagrams intended to increase its pedagogical effect.[65]David Hilbert authored amodern axiomatization of theElements.[66]Edna St. Vincent Millay wrote that "Euclid alone has looked on Beauty bare."[67]
References
Notes
^In modern English, 'Euclid' is pronounced as/ˈjuːklɪd/.[5]
^Euclid'soeuvre also includes the treatiseOn Divisions, which survives fragmented in a later Arabic source.[9] He authored numerouslost works as well.[9]
^Some of the information fromPappus of Alexandria on Euclid is now lost and was preserved inProclus'sCommentary on the First Book of Euclid's Elements.[10]
^SeeHeath 1981, p. 354 for an English translation on Proclus's account of Euclid's life.
^Later Arab sources state he was a Greek born in modern-dayTyre, Lebanon, though these accounts are considered dubious and speculative.[8][4] SeeHeath 1981, p. 355 for an English translation of the Arab account. He was long held to have been born in Megara, but by theRenaissance it was concluded that he had been confused with the philosopherEuclid of Megara,[16] see§Identity and historicity
^TheElements version available today also includes "post-Euclidean" mathematics, probably added later by later editors such as the mathematicianTheon of Alexandria in the 4th century.[36]
^The use of the term "axiom" instead of "postulate" derives from the choice ofProclus to do so in his highly influential commentary on theElements. Proclus also substituted the term "hypothesis" instead of "common notion", though preserved "postulate".[42]
^The distinction between these categories is not immediately clear; postulates may simply refer to geometry specifically, while common notions are more general in scope.[45]
^The mathematician Gerard Venema notes that thisaxiomatic system is not complete: "Euclid assumed more than just what he stated in the postulates".[47]
^SeeHeath 1908, pp. 195–201 for a detailed overview of postulates 1 through 4
^Since antiquity, enormous amounts of scholarship have been written about the 5th postulate, usually from mathematicians attempting toprove the postulate—which would make it different from the other, unprovable, four postulates.[49]
^Much of Book 5 was probably ascertained from earlier mathematicians, perhaps Eudoxus.[41]
^SeeJones 1986, pp. 547–572 for further information on thePorisms
Goulding, Robert (2010).Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History. Dordrecht: Springer Netherlands.ISBN978-90-481-3542-4.
Sialaros, Michalis (2020). "Euclid of Alexandria: A Child of the Academy?". In Kalligas, Paul; Balla, Vassilis; Baziotopoulou-Valavani, Chloe; Karasmanis, Effie (eds.).Plato's Academy. Cambridge:Cambridge University Press. pp. 141–152.ISBN978-1-108-42644-2.
