Often referred to as the most successfultextbook ever written, theElements 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 onmathematics in the medieval Islamic world and in Western Europe, and has proven instrumental in the development oflogic and modernscience, where its logical rigor was notsurpassed until the 19th century.
Euclid'sElements is the oldest extant large-scale deductive treatment of mathematics.[1]Proclus, a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on theElements: "Euclid, who put together theElements, collecting many ofEudoxus' theorems, perfecting many ofTheaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".[a] Scholars believe that theElements is largely a compilation of propositions based on books by earlier Greek mathematicians,[2] includingEudoxus,Hippocrates of Chios,[b]Thales, andTheaetetus, while other theorems are mentioned by Plato and Aristotle.[3] 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difficult to differentiate the work of Euclid from that of his predecessors, especially because theElements essentially superseded much earlier and now-lost Greek mathematics.[4] TheElements version available today also includes "post-Euclidean" mathematics, probably added later by later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.[3] 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".[5][3] Despite this, historian Michalis Sialaros opines that its "remarkably tight structure" suggests that Euclid wrote theElements himself rather than merely editing together the works of others.[6]
The detailed attribution of parts of theElements to specific mathematicians is still the subject of scholarly debate. According toW. W. Rouse Ball,Pythagoras was probably the source for most of books I and II, Hippocrates of Chios for book III, andEudoxus of Cnidus for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.[7] TheElements 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.[8]Wilbur Knorr ascribes the origin of the material in Books I, III, and VI of theElements to the time of Hippocrates of Chios, and of the material in books II, IV, X, and XIII to the later period ofTheodorus of Cyrene, Theaetetus, and Eudoxos. However, this suggested history has been criticized byvan der Waerden, who believed that books I through IV were largely due to the much earlierPythagorean school.[9]
Other similar works are also reported to have been written by Hippocrates of Chios,Theudius of Magnesia, andLeon, but are now lost.[10][11]
Summary Contents of Euclid'sElements (Heath edition)
Book
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
Totals
Definitions
23
2
11
7
18
4
22
–
–
16
28
–
–
131
Postulates
5
–
–
–
–
–
–
–
–
–
–
–
–
5
Common Notions
5
–
–
–
–
–
–
–
–
–
–
–
–
5
Propositions
48
14
37
16
25
33
39
27
36
115
39
18
18
465
TheElements does not exclusively discuss geometry as is sometimes believed.[4][12] It is traditionally divided into three topics:plane geometry (books I–VI), basicnumber theory (books VII–X) andsolid geometry (books XI–XIII)—though book V (on proportions) and X (onincommensurability) do not exactly fit this scheme.[13][14] The heart of the text is the theorems scattered throughout.[15] Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles".[16] The first group includes statements labeled as a "definition" (Ancient Greek:ὅρος orὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια).[16][17] The postulates (that is,axioms) and common notions occur only in book I.[4] Close study ofProclus suggests that older versions of theElements may have followed the same distinctions but with different terminology, instead calling each definition a "hypothesis" (ύπόΘεςιζ) and the common notions "axioms" (άξιώμα).[17] The second group consists of propositions, presented alongsidemathematical proofs and diagrams.[16] It is unknown whether Euclid intended theElements as a textbook,[6] despite its wide subsequent use as one.[18] As a whole, theauthorial voice remains general and impersonal.[3]
To draw a straight line from any point to any point.
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
Euclid's 5th postulate: Lineg falls on the two linesh andk, making interior anglesα andβ that sum to less than 180°, so linesh andk must meet at some pointS on the same side ofg as the angles.TheBride's Chair from the proof of thePythagorean theorem, in the colored version used by Byrne's 1847 edition. The proof shows that the black and yellow areas are equal, as are the red and blue areas.
Book I of theElements is foundational for the entire text.[4] It begins with a series of 20 definitions for basic geometric concepts such aspoints,lines,angles and variousregular polygons.[20] Euclid then presents 10 assumptions (see table, right), grouped into five postulates and five common notions.[21] These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as anaxiomatic system.[22] The common notions exclusively concern the comparison ofmagnitudes, the sizes of geometric objects.[23] In modern mathematics these magnitudes would be treated asreal numbers measuringarc length,angle, orarea, and compared numerically, but Euclid instead found ways of comparing the magnitude of shapes using geometric operations, without interpreting these magnitudes as numbers.[24] While the first four postulates are relatively straightforward, the fifth is not. It is known as theparallel postulate, and the question of its independence from the other four postulates became the focus of a long line of research leading to the development ofnon-Euclidean geometry.[23]
Proposition 5, that the base angles of anisosceles triangle are equal, became known in theMiddle Ages as thepons asinorum, or bridge of asses, separating the mathematicians who could prove it from the fools who could not.[25]Papyrus Oxyrhynchus 29, a 3rd-century CE papyrus, contains fragments of propositions 8–11 and 14–25.[c] The last two propositions of Book I comprise the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".[16] The figure for the Pythagorean theorem has itself become well known under multiple names: theBride's Chair, the windmill, or the peacock's tail.[26]
Euclid's subdivision of line segmentAB into thegolden ratio from Book II Proposition 11, with an arc added to the traditional diagram. The construction finds the midpointE of sideAC of squareABCD, intersects lineAC atF with a circle of radiusEB, and constructs a second squareAFGH, whose vertexH is the subdivision point.
The second book focuses onarea, measured throughquadrature, meaning the construction of asquare of equal area to a given figure. It includes a geometric precursor of thelaw of cosines, and culminates in the quadrature of arbitraryrectangles.[23] In the late 19th and 20th centuries, Book II was interpreted by some mathematical historians to establish a "geometric algebra", an expression of algebraic manipulation of linear and quadratic equations in terms of geometric concepts of length and area,[27][28] centered on the quadratic case of thebinomial theorem.[23] This interpretation has been heavily debated since the 1970s;[28] critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later.[16] Nevertheless, taken as statements about geometry, many of the propositions in this book are superfluous to modern mathematics, as they can be subsumed by the use of algebra.[29]
Proposition 11 of Book II subdivides a given line segment into extreme and mean proportions, now called thegolden ratio. It is the first of several propositions involving this ratio: It is later used in Book IV to construct agolden triangle andregular pentagon and in Book XIII to construct theregular dodecahedron andregular icosahedron, and studied as a ratio in Book VI Proposition 30.[30][31]
Book III begins with a list of 11 definitions, and follows with 37 propositions that deal withcircles and their properties. Proposition 1 is on finding the center of a circle. Propositions 2 through 15 concernchords, and intersecting andtangent circles.Tangent lines to circles are the subjects of propositions 16 through 19. Next are propositions oninscribed angles (20 through 22), and on chords, arcs, and angles (23 through 30), including theinscribed angle theorem relating inscribed to central angles as proposition 20. Propositions 31 through 34 concern angles in circles, includingThales's theorem that an angle inscribed in asemicircle is aright angle (part of proposition 31). The remaining propositions, 35 through 37, concern intersecting chords and tangents; proposition 35 is theintersecting chords theorem, and proposition 36 is thetangent–secant theorem.[32]
Book V, which is independent of the previous four books, concernsratios ofmagnitudes (intuitively, how much bigger or smaller one shape is relative to another) and the comparison of ratios.[34] Heath and other translators have formulated its first six propositions in symbolic algebra, as forms of thedistributive law of multiplication over division and theassociative law for multiplication. However,Leo Corry argues that this is anachronistic and misleading, because Euclid did not treat magnitudes as numbers, nor taking a ratio as a binary operation from numbers to numbers.[35]
Much of Book V was probably ascertained from earlier mathematicians, perhaps Eudoxus,[16] although certain propositions, such as V.16, dealing with "alternation" (ifa :b ::c :d, thena :c ::b :d) likely predate Eudoxus.[36]
Christopher Zeeman has argued that Book V's focus on the behavior of ratios under the addition of magnitudes, and its consequent failure to define ratios of ratios, was a flaw that prevented the Greeks from finding certain important concepts such as thecross ratio (central toprojective geometry).[37]
Book VI uses the theory of ratios from Book V in the context of plane geometry,[4] especially the construction and recognition ofsimilar figures. It is built almost entirely of its first proposition:[38] "Triangles and parallelograms which are under the same height are to one another as their bases". That is, if two triangles have the same height, the ratio of their areas is the same as the ratio of lengths of their two base segments (and analogously for two parallelograms of the same height). This proposition provides a connection between ratios of lengths and ratios of areas.[39] Proposition 25 constructs, from any twopolygons, a third polygon similar to the first and with the same area as the second.Plutarch attributes this construction to Pythagoras, calling it "more subtle and more scientific" than the Pythagorean theorem. The famous ancient Greek problem ofdoubling the cube, now known impossible with compass and straightedge, is a special case of the analogous 3d problem of constructing a figure with a specified shape and volume.[40] The book ends as it begins, by connecting two types of ratios: ratios of angles, and ratios of circular arc lengths, in proposition 33.[41]
Number theory, the theory of the arithmetic ofnatural numbers, is covered by books VII to X. Book VII begins with a set of 22 definitions forparity (whether a number is even or odd),prime numbers, and other arithmetic-related concepts.[4] 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";[42] this is generally interpreted to mean that, for Euclid, one is not a number, and the natural numbers begin at two.[43]
Book VII deals with elementary number theory, and includes 39 propositions, which can be loosely divided into: theEuclidean algorithm, a method for determining whether numbers arerelatively prime and for finding thegreatest common divisor (1–4), fractions (5–10), the theory of proportions for numbers (11–19), prime and relatively prime numbers and the theory of greatest common divisors, (20–32), andleast common multiples (33–39).[44]
The topic of Book VIII isgeometric progressions.[44] For Euclid, these were defined by the property of being in continued proportion (each two consecutive magnitudes have the same ratio) rather than, as in modern treatments, byexponentiation (theth term of the progression has the form for constants and). This allowed Euclid to avoid multiplication of more than two values, but led to some awkward proofs of facts that exponential notation would make obvious.[45]
The first part of Book VIII (propositions 1 through 10) deals with the construction and existence of geometric progressions of integers in general, and thedivisibility of members of a geometric progression by each other.Propositions 11 to 27 deal withsquare numbers andcube numbers in geometric progressions, and the relation between these special progressions and the elements two or three steps apart in an arbitrary geometric progression.[44]
After continuing the investigations of Book VIII on squares and cubes in geometric progressions,[44] Book IX applies the results of the preceding two books and gives theinfinitude of prime numbers (Euclid's theorem, proposition 20), the formula for the sum of afinite geometric series (proposition 35) and a construction using this sum for evenperfect numbers (proposition 36). Here, a number is perfect if it equals the sum of itsproper divisors, as for instance 28 = 1 + 2 + 4 + 7 + 14.[4][46]Alhazen conjectured c. 1000, and in the 18th centuryLeonhard Euler proved, that this construction generatesall even perfect numbers. This result is theEuclid–Euler theorem.[47]
Of theElements, book X is by far the largest and most complex, dealing with (in modern terms)irrational numbers in the context of magnitudes.[16][48] Proposition 9 (as restated in modern terms) proves the irrationality of the square roots of all non-square integers such as, thesquare root of 2.[49] A lemma to Proposition 29 givesEuclid's formula for producing all fundamentalPythagorean triples.[50] Additionally, this book classifies irrational lengths into thirteen disjoint categories, related to their construction by various combinations of other lengths that are integers and their square roots.[51] However,Wilbur Knorr warns that "The student who approaches Euclid's Book X in the hope that its length and obscurity conceal mathematical treasures is likely to be disappointed. ... the mathematical ideas are few."[52]
Rather than treating magnitudes asreal numbers and asking whether these arerational numbers, Euclid handles this material in terms of thecommensurability of lengths or areas: whether two line segments or two rectangles can both be measured by an integer number of copies of a common subunit.[48] His classification of lengths as rational or irrational differs from the modern meaning: for Euclid, a line segment is rational when the square on its side has a rational area. That is, for Euclid, a length such as that is the square root of a rational area is itself rational.[53]
This book is connected to a short passage inPlato's dialogueTheaetetus amongSocrates,Theodorus of Cyrene, andTheaetetus, a younger mathematician. This passage discusses a proof by Theodorus that the non-square integers from 3 to 17 have irrational square roots (after the much earlier discovery of the irrationality of), the generalization of this result to all non-square integers by Theaetetus, and a partial classification of the irrational numbers (with fewer than 13 classes).[54][55]
The final three books primarily discusssolid geometry.[13] By introducing a list of 37 definitions, Book XI contextualizes the next two.[56] Although its foundational character resembles Book I, unlike Book I it features no axiomatic system or postulates.[56]
Book XI generalizes the results of book VI to solid figures: perpendicularity, parallelism, volumes, and similarity ofparallelepipeds (polyhedra with three pairs of parallel faces). The three sections of Book XI include content on: solid geometry (1–19), solid angles (20–23), and parallelepipeds (24–37).[56]
Book XII studies the volumes ofcones,pyramids, andcylinders in detail by using themethod of exhaustion, a precursor tointegration,[56] and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder.[57] It concludes by showing that the volume of asphere is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids.[58]
Book XIII constructs the fivePlatonic solids (regular polyhedra) inscribed in a sphere, compares the ratios of their edges to the radius of the sphere,[59] and concludes theElements by proving that these are the only regular polyhedra.[60]
Two additional books, that were not written by Euclid, Books XIV and XV, have been transmitted in the manuscripts of theElements:[61]
Book XIV was likely written byHypsicles, following a treatise byApollonius 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[61]
Book XV may have been written by a student ofIsidore of Miletus. It also studies the Platonic solids; it inscribes some of them within each other, counts their edges and vertices (without however findingEuler's formula relating these counts to each other), and computes thedihedral angles between their faces.[61]
The practice of adding to the works of famous authors, exemplified by these books, was not unusual in ancient Greek mathematics.[61]
An animation showing how Euclid constructed a hexagon (Book IV, Proposition 15). Every two-dimensional figure in theElements can be constructed using only a compass and straightedge.[62]
Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used toconstruct the object using acompass (circle-drawing tool) andstraightedge (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.[65]
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.[66]
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 theElements.[67] For proofs involvingcase analysis, theElements often includes details only of the most difficult case; some of these case analyses have been filled out by later editors such as Theon.[68]
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,[69] 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 awkwardAlexandrian system of numerals,[70] analphabetic numeral system in which each Greek letter represented a single-digit multiple of a power of ten.[71]
Euclid'sElements has been referred to as the most successfultextbook ever written.[18][72] TheElements is often considered after theBible as the most frequently translated, published, and studied book in history.[73] With Aristotle'sMetaphysics, theElements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Islamic world and Western Europe.[73][72] It was one of the very earliest mathematical works to be printed after theinvention of the printing press and has been estimated to be second only to theBible in the number of editions published since the first printing in 1482,[74][75] the number reaching well over one thousand.[75]
The oldest extant evidence for Euclid's Elements are a set of sixostraca (clay fragments with writing scratched onto them) found among theElephantine papyri and ostraca, from the 3rd century BC, that deal with propositions XIII.10 and XIII.16, on the construction of a dodecahedron.[76] A papyrus recovered fromHerculaneum[77] contains an essay by the Epicurean philosopherDemetrius Lacon on Euclid'sElements.[76] The earliest extant papyrus containing the actual text of theElements isPapyrus Oxyrhynchus 29, a fragment containing the text of Book II, Proposition 5 and an accompanying diagram, dated toc. 75–125 AD.[78]
Copies of the Greek text still exist, some of which can be found in theVatican Library and theBodleian Library in Oxford.[d][e] The manuscripts available are of variable quality, and often incomplete.[79] By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text.[80] Also of importance are thescholia, 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.[81]
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) untilFrançois Peyrard's 1808 discovery at theVatican of a manuscript not derived from Theon's.[82] This manuscript, MS. Vat.gr.190,[d] 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 theElements. This sparked a 19th-century academic debate between M. Klamroth andJ. 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."[82]
Although Euclid was known toCicero, for instance, no record exists of the text having been translated into Latin prior toBoethius in the fifth or sixth century.[83]
A woman teaches geometry, from a manuscript (c. 1309–1316) ofAdelard of Bath's 12th century translation of theElements from Arabic into Latin.[83]
From classical antiquity until the western invention of printing, texts such as theElements were preserved and duplicated through the process of copyingmanuscripts. This was laborious and expensive so manuscripts were often confined to the collections of the wealthy or to institutions such as theHouse of Wisdom in themedieval Islamic world or themonasteries and early universities of medieval Europe.[84]
The Islamic world received theElements from theByzantine Empire around 760. According to sources from that milieu, this version was translated intoArabic underHarun al-Rashid (c. 800),[83] in two versions byAl-Ḥajjāj ibn Yūsuf ibn Maṭar. Another Arabic translation was made later in the 9th century byIshaq ibn Hunayn and revised byThābit ibn Qurra.[85][86] Although most Arabic manuscripts have been attributed to one or another of these translations, some mix material from both,[85] and their attributions are not always in accord with the evidence from textual similarities in surviving manuscripts.[86] This mixture was also passed down into medieval translations intoHebrew from the Arabic.[85]
The Byzantine scholarArethas commissioned the copying of one of the Greek manuscripts of Euclid in the late ninth century;[87] it and another Byzantine manuscript are the two oldest surviving copies of the Greek text.[88] Although known in Byzantium, theElements was lost to Western Europe until about 1120,[89] except through fragments of a translation into Latin byBoethius (circa 500), quoted in other works.[90][f] In about 1120, the English monkAdelard of Bath translated theElements into Latin from an Arabic translation.[91] 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 theAlmagest to Latin. The Euclid manuscript is extant and quite complete.[89]
After Adelard's translation (which became known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period includeHerman 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[92] (late 12th century; his manuscripts are referred to collectively as Adelard III), andGerard of Cremona (sometime after 1120 but before 1187). The detailed transmission history of these translations is still an active area of research.[93]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.[90][g] After its availability in Europe, the first books of theElements became standard in medieval universities as part of thequadrivium, the second stage of instruction after thetrivium of grammar, logic, and rhetoric.[84]
The ItalianJesuitMatteo Ricci (left) and the Chinese mathematicianXu Guangqi (right) published the firstChinese edition ofEuclid's Elements (Jīhé yuánběn幾何原本) in 1607.
The first printed edition of theElements was published by Erhard Ratdolt in 1482, based on Campanus's version,[94] and since then it has been translated into many languages and published in over a thousand different editions.[75] A manuscript descended from Theon's Greek version was recovered and a Latin translation was published in Venice in 1505 byBartolomeo Zamberti [de].[95] The Greek text itself waspublished in 1533.[96] The first to translate theElements into a modern European language wasNicolo Tartaglia, who published an Italian edition in 1543.[97]
In 1570,John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition byHenry Billingsley.[98][99][100] In 1607, The Italian JesuitMatteo Ricci and the Chinese mathematicianXu Guangqi published the first Chinese edition of Euclid's Elements.[101]
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 humanistJoachim Sterck van Ringelbergh, for instance, lists theElements as its only mathematics book.[107] Even after printed versions existed, a university might expect its students to copy by hand material from the university's copy of theElements.[108]
In the 19th century theElements fell out of favor as a geometry textbook, in part supplanted by newer textbooks such as one byAdrien-Marie Legendre,[109][110][111] in part because of the rise of other forms of geometry includingnon-Euclidean geometry,analytic geometry, anddescriptive geometry,[112][113] and in part out of pressure for an approach to mathematics education with more emphasis on intuition and less on memorization.[112][114]Charles Dodgson (better known as Lewis Carroll), in particular, railed against this replacement of Euclid in his bookEuclid and His Modern Rivals (1879).[109] Another defender of theElements, mathematician and historianW. W. Rouse Ball, remarked that "the fact that for two thousand years [theElements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."[69] Despite falling out of wide use in education, theElements is still occasionally used as a textbook in experimental education projects.[115]
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.[73]
One of the most notable influences of Euclid on modern mathematics and, beyond mathematics, modern physics and the discovery ofgeneral relativity, is the discussion of theparallel postulate.[116][117] In Book I, Euclid lists five postulates, the fifth of which stipulates
If aline segment intersects two straightlines forming two interior angles on the same side that sum to less than tworight angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematicianNikolai Lobachevsky published a description of acute geometry (orhyperbolic geometry), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate (elliptic geometry). If one takes the fifth postulate as a given, the result isEuclidean geometry.[120]
The axiomatic reasoning of Euclid'sElements was long considered to set the standard for mathematical rigor,[121] but the issues of the soundness and completeness of Euclid's axioms came to the foreground in the late 19th century, when gaps were found in his reasoning[122] and whenDavid Hilbert began seeking "to revive Euclid's axiomatic point of view", to develop improved axiom systems through which all mathematical and physical questions could be answered by simple calculations.[118] Hilbert's hopes were dashed in thefoundational crisis of the early 20th century, in whichKurt Gödel and others discovered that any sound axiom system forset theory must necessarily be incomplete.[123] In the 21st century, a new standard for rigor arose,computer-assisted proofs, and the propositions of theElements (with some updates to their proofs) have withstood computer checking.[119][124]
Two circles sharing a radius cross each other, forming avesica piscis. This is the first step of Book I Proposition 1, the construction of anequilateral triangle from the radius and one of the crossing points.
Some of the foundational proofs of theElements use assumptions that Euclid did not state explicitly as axioms. For example, in the first construction of Book 1, of anequilateral triangle, Euclid used a premise that was neither postulated nor proved: that two circles sharing the same line segment as a radius will cross each other in two points, rather than somehow not crossing.[125][126] This example depends only on topological properties of its diagram, which remain evident even if the diagram is drawn inaccurately.[127] However, in other cases, Euclid did not prove that certain objects were distinct or separated from each other, and the possibility that they might coincide (a type ofdegeneracy) might not be evident from a single diagram. An example occurs in Euclid'sbisection of an angle, by constructing an isosceles triangle on the given angle and an equilateral triangle with the same base, and connecting by a line the apexes of the two triangles. This breaks down when the initial angle is 60° and the two apexes coincide.[124]
Later editors of theElements have included these implicit axiomatic assumptions, such asPasch's axiom,[128][129] in their editions' lists of formal axioms.[130] Early attempts to construct a more complete set of axioms includeHilbert's geometry axioms[131][132] andTarski's.[128][133] In 2017, Michael Beeson et al. used computerproof assistants to create and check a set of axioms similar to Euclid's. Beeson et al. chose Tarski's system as their starting point, instead of Hilbert's, because it is closer to Euclid's, and uses only points as the variables in its formulas. They provided computer-verified proofs of all propositions in Book I, using these axioms, and they also proved (using a separate logical formalization of thereal numbers) that all of their axioms are valid for the points of theCartesian coordinate system.[124]
Over one thousand editions of Euclid'sElements have been published,[75] in Greek, Latin, English, and other languages. Some of the more significant of these include:
Zamberti, Bartolomeo, ed. (1505).Euclidis megarẽsis philosophi platonici. Venice. First full-text Latin translation directly from the Greek.[95] The confusion in the title between Euclid of Alexandria, the author of theElements, andEuclid of Megara, a philosopher from almost a century earlier, was a common mistake in the Middle Ages and Renaissance.[135]
Grynaeus, Simon, ed. (1533).Ευκλείδου Στοιχεῖον. Basel: Johann Herwagen.Editio princeps of the Greek text. Based on two Greek manuscripts, Paris gr. 2343 and Venetus Marcianus 301,[137] both "very inferior ones".[95]
Magnien, Jean, ed. (1557).Euclidis Elementorum libri XV grœce et latine. Paris: Cavellat. Greek with Latin translation. Revised posthumously by Stephanus Gracilis; incorporates a translation of Book X by Pierre de Montdoré. Most proofs omitted.[138]
Commandino, Federico, ed. (1572).Euclidis Elementorum Libri XV. Pisauri [Pesaro, Italy]: Apud Camillum Francischinum. In Latin. "The reference edition for the scholarly community up until the early nineteenth century"; Italian translation published in 1575 by Commandino's son-in-law, Valerio Spaccioli.[139]
Clavius, Christopher, ed. (1574).Euclidis elementorum libri XV. Rome: Apud Vincentium Accoltum. Latin, with added commentary by Clavius. Published in an expanded 2nd edition in 1584. Based on multiple sources, but most closely related to the version of Magnien and Gracilis.[136]
Ricci, Matteo;Xu, Guangqi, eds. (1607).Jī hé yuán běn幾何原本 [Source of quantity] (in Chinese). Beijing. Translated from the Latin edition of Clavius, but including only books I-VI.[101] This translation, together with a later translation of the remaining books, can be found ons:zh:幾何原本.
Briggs, Henry, ed. (1620).Eukleidou Stoicheiōn biblia 13 / Elementorum Euclidis libri tredecim. London: William Jones. Despite the title this includes only the first six books, with parallel columns of Greek from Grynaeus 1533 and Latin corrected from Commandino 1572. The first edition in either language published in England.[100]
Heiberg, Johan Ludvig, ed. (1883–1888)Euclidis Opera omnia [Euclid's complete works, in Greek]. Leibzig: Teubner. Volumes 1–5 comprise theElements.Vol.I,Vol.II,Vol.III,Vol.IV,Vol.V. Heiberg consulted multiple Greek manuscripts for his work, taking the position that the single version not edited by Theon, MS. Vat.gr.190, was the most authentic, but following the others at points where he suspected his primary text to be faulty.[82]
Heath, Thomas, ed. (1908).The Thirteen Books of Euclid's Elements. Cambridge University Press. 2nd ed., 1926. In three volumes:Vol. I,Vol. II,Vol. III. Reprints include Dover, 1956; Green Lion Press, 2002,ISBN1-888009-18-7 (single volume, without Heath's commentary);[148] Barnes & Noble, 2006,ISBN0-7607-6312-7 (single volume).
^"MS. D'Orville 301".Bodleian Library. University of Oxford. Retrieved2025-09-11.
^It was once thought that a complete Latin translation by Boethius remained available in England before Adelard, and was annotated byAlfred the Great. However, there is no evidence of any full translation by Boethius surviving (if it ever existed) and a manuscript found in theBiblioteca Marciana in Venice, holding a marginal note that it was annotated by Alfred, is one of many post-Adelard versions. SeeBusard 2005, p. 3 andClagett 1954.
Ackerberg-Hastings, Amy (2023). "Analysis and synthesis in Robert Simson'sThe Elements of Euclid". In Zack, Maria; Waszek, David (eds.).Research in history and philosophy of mathematics—the CSHPM 2021 volume. Annals of the Canadian Society for History and Philosophy of Mathematics. Cham: Birkhäuser/Springer. pp. 133–147.doi:10.1007/978-3-031-21494-3_8.ISBN978-3-031-21493-6.MR4633160.
Baldasso, Renzo (2013). "Printing for the Doge: On the first quire of the first edition of theLiber elementorum Euclidis".La Bibliofilía.115 (3):525–552.JSTOR26202240.
Brentjes, Sonja (2018). "Who translated Euclid'sElements into Arabic?". In Hämeen-Anttila, Jaakko; Lindstedt, Ilkka (eds.).Translation and Transmission: Collection of articles. The Intellectual Heritage of the Ancient and Mediaeval Near East. Vol. 3. Münster: Ugarit-Verlag. pp. 21–54.
Brock, W. H. (January 1975). "Geometry and the universities: Euclid and his modern rivals 1860–1901".History of Education.4 (2):21–35.doi:10.1080/0046760750040203.
Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1988).The Historical Roots of Elementary Mathematics. Dover.
Clagett, Marshall (September 1954). "King Alfred and theElements of Euclid".Isis.45 (3). University of Chicago Press:269–277.doi:10.1086/348338.JSTOR226713.
Corry, Leo (July 2013). "Geometry and arithmetic in the medieval traditions of Euclid'sElements: a view from Book II".Archive for History of Exact Sciences.67 (6):637–705.doi:10.1007/s00407-013-0121-5. See especially p. 637.
Corry, Leo (2021). "Distributivity-like results in Euclid'sElements".Distributivity-like Results in the Medieval Traditions of Euclid's Elements. SpringerBriefs in History of Science and Technology. Springer International Publishing. pp. 5–25.doi:10.1007/978-3-030-79679-2_2.ISBN9783030796792.
Damiani, Giacomo (June 2024). "Form and matter of regular geometrical bodies in Luca Pacioli'sSumma (1494) andCompendium de divina proportione (1498)".Early Science and Medicine.29 (3):230–270.doi:10.1163/15733823-20240106.
Davis, Margaret Daly (1977).Piero Della Francesca's Mathematical Treatises: The Trattato D'abaco and Libellus de Quinque Corporibus Regularibus. Longo Editore.
De Young, Gregg (April 2012). "Further adventures of the Rome 1594 Arabic redaction of Euclid's Elements".Archive for History of Exact Sciences.66 (3). Springer Science and Business Media LLC:265–294.doi:10.1007/s00407-012-0094-9.JSTOR41472233.
Dorandi, Tiziano (2018). "L'Euclide di P.Oxy. 5299".Zeitschrift für Papyrologie und Epigraphik (in Italian).205:92–95.JSTOR26603972.
Dyde, Samuel Walters, ed. (1899).The Theaetetus of Plato. J. Maclehose.
Elior, Ofer (2021). "Niccolò Tartaglia's 1543 edition of Euclid'sElements and the sources of an early Modern Hebrew version of theElements".Aleph: Historical Studies in Science and Judaism.21 (1). Indiana University Press:123–148.doi:10.2979/aleph.21.1.0123.JSTOR10.2979/aleph.21.1.0123.
Elior, Ofer (2024). "The Hebrew translation of Euclid'sElements ascribed to Rabbi Jacob: a new analysis following the 'discovery' of the Arabic version in MS Paris, BULAC ARA. 606".Historia Mathematica.68:1–21.doi:10.1016/j.hm.2024.02.005.MR4799290.
Folkerts, Menso (1989).Euclid in Medieval Europe(PDF). The Benjamin Catalogue for History of Science. Collected inFolkerts, Menso (2006).The Development of Mathematics in Medieval Europe. Variorum Collected Studies Series. Vol. 811. Aldershot: Ashgate Publishing Limited.ISBN0-86078-957-8.MR2229235.
Goulding, Robert (2009). "The puzzling lives of Euclid".Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History. Archimedes. Vol. 25. Springer Netherlands. pp. 117–142.doi:10.1007/978-90-481-3542-4_5.ISBN978-90-481-3542-4.
Grafton, Anthony T. (2008). "Textbooks and the disciplines". In Campi, Emidio; Angelis, Simone De; Goeing, Anja-Silvia; Grafton, Anthony T. (eds.).Scholarly Knowledge: Textbooks in Early Modern Europe. Librairie Droz. pp. 11–38.
Jameson, Tom (2019).Euclid in the Modern Classroom (Thesis). Harvard University.ProQuest2511164716.
Knorr, Wilbur R. (1975).The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry. Synthese Historical Library. Dordrecht, Netherlands: D. Reidel Publishing Co.
Knorr, Wilbur (1983). "'La croix des mathématiciens': the Euclidean theory of irrational lines".American Mathematical Society. New Series.9 (1):41–69.doi:10.1090/S0273-0979-1983-15155-8.MR0699316.
Laubenbacher, Reinhard; Pengelley, David (1999). "Chapter 1: Geometry: The parallel postulate".Mathematical Expeditions. Undergraduate Texts in Mathematics. Springer New York. pp. 1–53.doi:10.1007/978-1-4612-0523-4.ISBN9781461205234.
Majer, Ulrich (2006). "Hilbert's axiomatic approach to the foundations of science—A failed research program?". In Hendricks, Vincent F.; Jørgensen, Klaus Frovin; Lützen, Jesper; Pedersen, Stig Andur (eds.).Interactions: Mathematics, Physics and Philosophy, 1860–1930. Boston Studies in the Philosophy of Science. Vol. 251. Springer Netherlands. pp. 155–184.doi:10.1007/978-1-4020-5195-1_5.ISBN9781402051944.
Mueller, Ian (1970). "Preface". In Glenn R. Morrow (ed.).Proclus: A Commentary on the First Book of Euclid's Elements. Princeton University Press.
Mueller, Ian (December 1969). "Euclid's Elements and the axiomatic method".The British Journal for the Philosophy of Science.20 (4). University of Chicago Press:289–309.doi:10.1093/bjps/20.4.289.JSTOR686258.
Rothstein, Bret (2013). "Making Trouble: Strange Wooden Objects and the Early Modern Pursuit of Difficulty".Journal for Early Modern Cultural Studies.13 (1). Project MUSE:96–129.doi:10.1353/jem.2013.0000.
Schubring, Gert (2022). "Textbooks before the invention of the printing press: orality and teaching".Analysing Historical Mathematics Textbooks. International Studies in the History of Mathematics and its Teaching. Springer International Publishing. pp. 15–53.doi:10.1007/978-3-031-17670-8_2.ISBN9783031176708.
Szmielew, Wanda (1974)."The role of the Pasch axiom in the foundations of Euclidean geometry". In Henkin, Leon; Addison, John; Chang, Chen Chung; Craig, William; Scott, Dana; Vaught, Robert (eds.).Proceedings of the Tarski Symposium: An international symposium held at the University of California, Berkeley, June 23–30, 1971, to honor Alfred Tarski on the occasion of his seventieth birthday. Proceedings of Symposia in Pure Mathematics. Vol. 25. Providence, Rhode Island: American Mathematical Society. pp. 123–132.doi:10.1090/pspum/025/0373872.ISBN978-0-8218-1425-3.MR0373872.
Tarski, Alfred (1959). "What is elementary geometry? The axiomatic method, With special reference to geometry and physics". In Henkin, L.; Suppes, P. (eds.).Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam. pp. 16–29.MR0106185.
Toumasis, Charalampos (December 1990). "The epos of Euclidean Geometry in Greek secondary education (1836-1985): Pressure for change and resistance".Educational Studies in Mathematics.21 (6):491–508.doi:10.1007/BF00315941.JSTOR3482388.
Unguru, Sabetai (2018). "Counter-revolutions in mathematics". In Sialaros, Michalis (ed.).Revolutions and continuity in Greek mathematics. Science, Technology, and Medicine in Ancient Cultures. Vol. 8. De Gruyter, Berlin. pp. 17–34.ISBN978-3-11-056365-8.MR3822179.
Wardhaugh, Benjamin (2021). "Defacing Euclid: Reading and Annotating the Elements of Geometry in Early Modern Britain". In Goeing, Anja-Silvia; Parry, Glyn; Feingold, Mordechai (eds.).Early Modern Universities: Networks of Higher Learning. BRILL.doi:10.1163/9789004444058_015.ISBN9789004442412.
Wardhaugh, Benjamin (2023).Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World. Princeton University Press.ISBN9780691235769.
