Euclid

How to use this course

Euclid wrote the most successful textbook in history and left almost no trace of himself. No contemporary mentions him. The earliest surviving reference comes about a century after his time, and the fullest account was written some seven hundred years later. Yet his Elements has been copied, translated, printed and taught without a break since antiquity, and it is often called the most published book after the Bible.

The biographical lessons are therefore short and careful. Each claim is tied to its source, and later legend is marked as legend. The technical lessons follow the Elements book by book, then turn to the other works.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Euclid’s statements are paraphrased in modern English. Citations such as I.47 mean Book I, Proposition 47, in the numbering of Heiberg’s edition and Heath’s translation. Euclid never multiplies lengths or writes formulas; where the course uses algebra, it is a translation.

Timeline

Dates BC are approximate. Euclid’s own dates are estimates resting on Proclus’ statement that he lived under Ptolemy I.

Alexandria under the first Ptolemies

Alexander the Great founded Alexandria on the Egyptian coast in 331 BC. After his death in 323 BC his general Ptolemy, son of Lagus, took Egypt, and in 305 or 304 BC he declared himself king. Ptolemy I and his son Ptolemy II built a Greek capital on Egyptian soil and drew scholars to it with royal money. The Museum, a community of scholars attached to the palace, and the Library grew up in these decades. Whether Euclid belonged to the Museum is not recorded, though he is often placed among its first members.

What the sources say

The fullest ancient statement is by the Neoplatonist philosopher Proclus, writing in Athens in the fifth century AD in his commentary on Book I. Proclus says that Euclid was younger than the circle of Plato and older than Eratosthenes and Archimedes, that he lived in the time of the first Ptolemy, and that he was a Platonist in outlook. Proclus probably drew his history of earlier geometry from the lost history by Eudemus of Rhodes, a pupil of Aristotle, though Eudemus wrote too early to mention Euclid himself.

Proclus’ dating partly rests on a reference to the Elements in Archimedes’ On the Sphere and Cylinder. The Danish mathematician Johannes Hjelmslev argued that this reference is a later insertion, which would remove one support. The earliest secure mention is in the preface to Book I of the Conics of Apollonius of Perga, around 200 BC. Apollonius remarks that Euclid had not fully worked out the locus with respect to three and four lines.

Pappus of Alexandria, writing in the early fourth century AD, adds two details. Apollonius spent a long time with the pupils of Euclid in Alexandria, which suggests that Euclid founded a school there. And Euclid was fair and kindly toward anyone who could advance mathematics, careful, and not boastful. Some scholars suspect this character sketch is a later addition, since it appears in a passage that criticizes Apollonius.

Who was Euclid?

Two anecdotes

Two stories about Euclid survive, both from late sources. Proclus says that Ptolemy asked whether there was a shorter way to geometry than the Elements, and Euclid replied that there is no royal road to geometry. The fifth-century anthologist Stobaeus tells the same story about Menaechmus and Alexander the Great, which makes both versions doubtful. Stobaeus also reports that a beginner, having learned the first proposition, asked what he would gain from it; Euclid told his slave to give the young man a coin, since he had to profit from what he learned. Neither story appears in earlier Greek literature.

Confusions and legends

Confusion began early. The Roman writer Valerius Maximus (1st century AD) names Euclid where the story requires Eudoxus, as the geometer to whom Plato sent those asking about doubling the cube. Medieval writers merged him with Euclid of Megara, a philosopher and pupil of Socrates who lived a century earlier. The Byzantine scholar Theodore Metochites made the merger explicit around 1300, and the first printed edition of 1482 carried it to Renaissance readers. Peter Ramus and others later pulled the two apart. Arabic authors of the Middle Ages give Euclid a father named Naucrates and a birthplace in Tyre. These details appear in no Greek source and are regarded as inventions.

Three hypotheses

The French historian Jean Itard set out three possibilities, reported by MacTutor:

1. Euclid was a single historical person who wrote the Elements and the other works attributed to him.
2. Euclid led a group of mathematicians in Alexandria who wrote under his name, some of them after his death.
3. There was no such person; an Alexandrian group took the name from Euclid of Megara.

Itard favoured the second. Most historians accept the first as the simplest reading of the evidence: Apollonius and Pappus speak of Euclid as an individual with pupils, and no ancient source doubts his existence. The modern collective “Nicolas Bourbaki” shows that the third is not absurd, but there is no positive evidence for it. This course uses the name for the author of the Elements, whoever that was.

Before the Elements

Euclid’s book was not the first of its kind. Proclus, following Eudemus, says that Hippocrates of Chios, in the later fifth century BC, was the first to compile elements of geometry. Leon, a younger contemporary of Plato, wrote a better collection, and Theudius of Magnesia wrote another in the Academy. None of these survives. Proclus then says Euclid put together the Elements, arranged many of Eudoxus’ theorems, perfected many of Theaetetus’, and gave rigorous proofs of results his predecessors had argued loosely.

Layers in the text

Historians use this report, and the internal evidence of the text, to sort the Elements into layers:

• Books I–IV: plane geometry of lines, areas and circles, much of it older material. Book IV, on regular polygons in circles, is often linked with the Pythagoreans.
• Books V–VI: the general theory of proportion, attributed to Eudoxus of Cnidus by an ancient scholium, and its application to similar figures.
• Books VII–IX: the arithmetic of whole numbers, in a separate, older style that does not use Book V.
• Book X: the classification of incommensurable lines, based on the work of Theaetetus.
• Books XI–XIII: solid geometry; Book XII uses Eudoxus’ method of exhaustion, and Book XIII builds the five regular solids, again drawing on Theaetetus.

The text also contains signs of an editor at work on inherited material. Book I defines the oblong, the rhombus and the rhomboid, which are never used. Book VII proves results about numbers that Book V had proved for magnitudes in general. Euclid’s own contribution is best seen in the architecture: the order of propositions, the choice of what to assume, and the insistence that every step rest on what came before. Probably few individual theorems were new.

The earliest physical traces

Six ostraca (inscribed potsherds) found on Elephantine Island in 1906–1908 and dated to the third century BC carry notes on figures that appear in Book XIII. They do not follow Euclid’s text, so they may reflect a parallel tradition. The oldest papyrus with words matching the Elements comes from Oxyrhynchus and dates from around AD 100.

Book I: definitions, postulates, first proofs

For the general reader. Euclid begins by saying what he will assume. There are a few definitions, five postulates about what may be constructed with straightedge and compass, and five common notions about equality. Every later result must follow from these. The idea that a whole subject can be built this way is Euclid’s most lasting gift.

The starting points

Book I opens with 23 definitions (a point is that which has no part; a line is breadthless length; and so on), then five postulates and five common notions.

The first propositions

I.1 constructs an equilateral triangle on a given segment \(AB\): draw the circle with centre \(A\) through \(B\) and the circle with centre \(B\) through \(A\); join a point \(C\) where they meet to \(A\) and \(B\). Then \(CA=AB\) and \(CB=BA\), so \(CA=CB\) by Common Notion 1.

I.2 copies a given segment to a given point, and I.3 cuts off from the greater of two segments a part equal to the less. These look fussy. They show that the compass of Postulate 3 may “collapse” when lifted, yet still transfers lengths.

Euclid proves I.4 by placing one triangle on the other and appealing to Common Notion 4. This “superposition” assumes that figures can be moved without changing shape, which no postulate states. I.5 then proves that the base angles of an isosceles triangle are equal, the pons asinorum. Proclus reports a shorter proof attributed to Pappus: apply I.4 to the triangle \(ABC\) and the same triangle read as \(ACB\).

The gaps

• In I.1, nothing in the postulates guarantees that the two circles meet. That needs a continuity principle.
• I.4 needs a principle of rigid motion, or an axiom in its place.
• Many proofs rely on the diagram to decide which of several points lies between the others. Moritz Pasch showed in 1882 that order needs its own axioms. Pasch’s axiom says that a line entering a triangle through one side, and not passing through a vertex, leaves through one of the other two sides.

Hilbert’s Grundlagen der Geometrie (1899) gave axioms of incidence, order, congruence, parallels and continuity, and took a weakened form of I.4 as an axiom of congruence. With these, Euclid’s proofs go through with small repairs.

Parallels and the theorem of Pythagoras

For the general reader. Euclid’s fifth postulate, about lines that eventually meet, is longer and less obvious than the others. He avoided it for 28 propositions and then used it to prove that the angles of a triangle add to two right angles and that the square on the hypotenuse equals the squares on the other two sides. For two thousand years mathematicians tried to prove it from the other postulates. They could not, because it is independent of them.

The structure of Book I

Propositions I.1–I.28 never use Postulate 5. Among them, I.27 shows that if a transversal makes equal alternate angles, the two lines are parallel (they never meet). I.29 proves the converse, and it is the first use of Postulate 5: a transversal to parallel lines makes equal alternate angles. From I.29 come I.32 (the angles of a triangle sum to two right angles), I.35–I.41 on areas of parallelograms and triangles, and I.47.

Proof sketch. Let the right angle be at \(A\). Erect squares \(BCED\) on \(BC\), \(ABFG\) on \(AB\), and \(ACKH\) on \(AC\), all outward. Draw \(AL\) through \(A\) parallel to \(BD\), meeting \(DE\) at \(L\). Since \(\angle BAC\) and \(\angle BAG\) are right, \(G,A,C\) are collinear (I.14). Triangles \(ABD\) and \(FBC\) have \(AB=FB\), \(BD=BC\), and \(\angle ABD=\angle FBC\) (each is a right angle plus \(\angle ABC\)), so they are equal by I.4. The parallelogram \(BL\) is double triangle \(ABD\), on the same base and between the same parallels (I.41). The square \(GB\) is double triangle \(FBC\) for the same reason. So the rectangle \(BL\) equals the square on \(AB\). Likewise the rectangle \(CL\) equals the square on \(AC\), and adding gives the result. The hero figure at the top of this page shows the construction.

Equivalents and independence

Assuming the other axioms (in Hilbert’s form, including continuity), each of the following is equivalent to Postulate 5:

• Playfair’s axiom (John Playfair, 1795): through a point not on a line there is at most one parallel to the line.
• The angle sum of every triangle is two right angles; equivalently, of some one triangle.
• There exist two similar triangles that are not congruent. John Wallis based an attempted proof on this assumption in 1663.

Proclus records an attempted proof by Ptolemy and gives his own. Arabic mathematicians, among them Ibn al-Haytham and ʿUmar al-Khayyām, tried again, as did Girolamo Saccheri in 1733, who derived many theorems of what is now hyperbolic geometry while trying to reach a contradiction. Carl Friedrich Gauss came to believe the postulate unprovable but did not publish. Lobachevsky (1829) and Bolyai (1832) published a consistent geometry in which it fails. Eugenio Beltrami’s models of 1868 showed that this geometry is consistent if Euclidean geometry is, so Postulate 5 cannot be proved from the others.

Geometric algebra, the golden cut, the pentagon

For the general reader. Book II states, with rectangles and squares, facts that we would write as algebraic identities. Book IV uses them to inscribe a regular pentagon in a circle. The key is a special division of a line, later called the golden section, which reappears in Book XIII in the construction of the dodecahedron and icosahedron.

Book II

Historians of the late nineteenth century, notably Hieronymus Georg Zeuthen and Paul Tannery, read Book II as “geometric algebra”: algebra disguised as geometry, perhaps after the Pythagorean discovery of incommensurables made numerical algebra unsafe. In 1975 Sabetai Unguru argued that this reading projects modern thought onto Greek texts, and a long debate followed. A neutral position: Book II gives area relations among rectangles and squares that can be used to solve problems we would solve with quadratic equations.

I.45 had already made a rectangle equal to any polygon. II.14 turns a rectangle with sides \(a,b\) into a square, using II.5: on the line of length \(a+b\) draw a semicircle; the perpendicular at the join meets it at height \(h\) with \(h^2=ab\). Every polygon can therefore be “squared”. Whether a circle can be squared with straightedge and compass remained open until Ferdinand von Lindemann proved \(\pi\) transcendental in 1882.

Extreme and mean ratio

If the line has length \(a\) and the greater segment \(x\), the condition is \(a(a-x)=x^2\), so \(x=\tfrac{\sqrt5-1}{2}\,a\) and \(a/x=\tfrac{1+\sqrt5}{2}=\varphi\approx1.618\). The name “golden section” is modern; Euclid simply says extreme and mean ratio.

The regular pentagon (IV.10–IV.11)

Cut \(AB\) at \(C\) in extreme and mean ratio, draw the circle with centre \(A\) and radius \(AB\), and fit into it a chord \(BD\) equal to the greater segment \(AC\). Triangle \(ABD\) has angles \(36^\circ,72^\circ,72^\circ\) in modern units. IV.11 inscribes a triangle with these angles in a given circle and bisects the two base angles; the five points cut off on the circle are the vertices of a regular pentagon. IV.16 combines a triangle and a pentagon at a common vertex to inscribe a regular fifteen-sided polygon. Carl Friedrich Gauss added the 17-gon in 1796, the first new constructible regular polygon with a prime number of sides since antiquity.

Ratio, similarity and incommensurables

For the general reader. The Greeks knew that the side and diagonal of a square have no common unit of measure. That made it hard to say what it means for two such pairs of lengths to be “in the same ratio”. Book V, based on the work of Eudoxus, gives a definition that works for every kind of magnitude. Book VI applies it to similar figures, and Book X uses it to classify the incommensurable lines that arise in geometry.

Book V

Definition V.4 excludes infinitely small or infinitely large magnitudes relative to one another; it is a form of what is now called the Archimedean property. Definition V.5 compares two ratios by comparing them with every rational number \(n/m\). Thomas Heath and many later writers compare it with the definition of real numbers by cuts that Richard Dedekind gave in 1872, though some historians stress how far the two differ in aim. The course on Eudoxus treats Book V in full.

Book VI

From VI.1 follow VI.2 (a line parallel to one side of a triangle cuts the other two proportionally), the similarity criteria VI.4–VI.7, and VI.31: in a right triangle, a figure on the hypotenuse equals the similar and similarly drawn figures on the legs, a generalization of I.47.

Book X

Book X is the longest book of the Elements, with 115 propositions in Heath’s numbering. Two magnitudes are commensurable if some magnitude measures both. Euclid calls a line rational (Greek rhētos) if it is commensurable with a fixed line, or if its square is commensurable with the square on that line; so for him the side of a square of area 2 is rational, which differs from modern usage.

This procedure of alternate subtraction is called anthyphairesis; for numbers it is the Euclidean algorithm of Book VII, and for magnitudes it produces the continued fraction of their ratio. X.9, attributed to Theaetetus, shows that two squares have sides commensurable in length exactly when the squares have the ratio of a square number to a square number. Euclid then classifies thirteen kinds of irrational lines built from rational ones by taking means, sums and differences, such as the binomial, which in modern terms is of the form \(\sqrt{p}+\sqrt{q}\) with \(\sqrt p,\sqrt q\) commensurable only in square.

The arithmetic books

For the general reader. Books VII–IX are about whole numbers, which Euclid pictures as line segments made of units. They contain the oldest surviving algorithm still in daily use, the proof that there are infinitely many primes, and a recipe for perfect numbers that took two thousand years to complete.

Book VII defines a unit as that by which each existing thing is called one, a number as a multitude composed of units (so 1 is not a number for Euclid), and a prime number as one measured by a unit alone.

In modern form: \(a=q_1b+r_1\), \(b=q_2r_1+r_2\), …, until \(r_{k}=0\). Each common divisor of \(a,b\) divides every remainder, and the last non-zero remainder divides \(a\) and \(b\). The algorithm terminates because the remainders decrease, a step Euclid justifies by the finiteness of numbers.

VII.31 shows that every composite number is measured by some prime, and IX.14 that a least number measured by given primes is measured by no other prime. Together these contain the core of unique factorization, though Euclid never states it as such; the first complete statement and proof is in Gauss’s Disquisitiones arithmeticae (1801).

Proof. Given primes \(p_1,\dots,p_k\), let \(N=p_1p_2\cdots p_k+1\). If \(N\) is prime, it is a new prime. If not, some prime \(q\) measures it (VII.31). If \(q\) were one of the \(p_i\), it would measure \(N\) and \(N-1\), hence the unit, which is absurd. Euclid writes the proof for three primes, as a representative case.

Leonhard Euler proved the converse for even numbers in a work published after his death: every even perfect number has Euclid’s form. Whether an odd perfect number exists is still unknown.

Exhaustion and the five regular solids

For the general reader. How do you compare the areas of two circles without a formula for either? Book XII squeezes each circle between polygons until any difference is smaller than any amount you care to name. Book XIII ends the Elements by building the five regular solids, the cube and its four companions, and proving that there are no others.

The proof uses Definition V.4: some multiple of the lesser exceeds the greater, and halving the greater that many times leaves less than the lesser.

Proof sketch. Let circles \(C_1, C_2\) have diameters \(d_1,d_2\). XII.1 shows that similar polygons inscribed in them are as \(d_1^2:d_2^2\). Suppose \(d_1^2:d_2^2=C_1:S\) with \(S<C_2\). Inscribe a square in \(C_2\): it is half the circumscribed square, so it takes more than half the circle. Doubling the number of sides, each new triangle on a side is half the rectangle around its segment, so it takes more than half that segment. By X.1, after finitely many doublings the polygon \(P_2\) leaves less than \(C_2-S\), so \(P_2>S\). Inscribe the similar polygon \(P_1\) in \(C_1\). Then \(P_1:P_2=d_1^2:d_2^2=C_1:S\). Since \(P_1<C_1\), we get \(P_2<S\), a contradiction. The case \(S>C_2\) reduces to the first by exchanging the circles. Hence \(S=C_2\).

The same technique gives XII.10 (a cone is one third of the cylinder with the same base and height) and XII.18 (spheres are as the cubes on their diameters). Archimedes credits Eudoxus with the first proofs of the cone and pyramid results. The method is called “exhaustion”, a name from the seventeenth century.

Book XIII

XIII.13–XIII.17 construct the tetrahedron, octahedron, cube, icosahedron and dodecahedron inscribed in a given sphere and compare each edge with the diameter. The icosahedron and dodecahedron use the extreme and mean ratio of Lesson 06.

Proof. At a vertex at least three faces meet, and the plane angles there sum to less than four right angles (XI.21). Equilateral triangles (\(60^\circ\)) allow 3, 4 or 5 at a vertex; squares (\(90^\circ\)) only 3; regular pentagons (\(108^\circ\)) only 3; regular hexagons already give \(3\cdot120^\circ=360^\circ\). So at most five types exist, and XIII.13–17 construct all five.

Optics, Data and the lost works

For the general reader. Euclid wrote more than the Elements. His Optics is the first surviving geometrical theory of vision; the Data teaches how to analyse a problem; the Phaenomena applies spherical geometry to the rising and setting of stars. Several further books, on conic sections and on “porisms”, are lost and known only from later descriptions.

The Optics

The Optics assumes that straight visual rays go out from the eye and form a cone with its vertex at the eye; what the rays fall on is seen, and objects seen under a larger angle appear larger. From these postulates it derives propositions on apparent size and shape, which later served writers on perspective. Ptolemy built on the visual cone in his own Optics. Because the rays are separated by gaps, the theory also explains why small objects can escape notice.

The inequality, with its companion \(\frac{\sin\beta}{\sin\alpha}<\frac{\beta}{\alpha}\), was a standard tool of Greek mathematical astronomy; Aristarchus of Samos and Ptolemy use such bounds.

The Data and On Divisions

The Data (94 propositions) shows that if certain parts of a figure are “given” in magnitude, position or form, then others are given too. It supports the method of analysis: assume the problem solved and trace what is then determined. On Divisions (of figures) gives constructions that divide a figure by a line into parts in a given ratio. It survives in Arabic; Franz Woepcke published an Arabic version in 1851, and Raymond Clare Archibald reconstructed the work in 1915 with 36 propositions.

The Phaenomena and doubtful works

The Phaenomena is an introduction to the geometry of the rotating celestial sphere, close in content to Autolycus of Pitane’s On the Moving Sphere. A Catoptrics on mirrors is attributed to Euclid, but the surviving text may be a later compilation, possibly by Theon. Proclus mentions Elements of Music; the surviving Sectio canonis (Division of the Canon) is sometimes ascribed to Euclid, but the attribution is doubtful.

Lost works

• Conics, four books, later superseded by Apollonius. Pappus says Euclid built on the solid loci of Aristaeus.
• Porisms, three books, described by Pappus as containing 171 theorems and 38 lemmas. A porism was a proposition asserting that something can be found. Robert Simson in the eighteenth century and Michel Chasles in 1860 attempted reconstructions.
• Surface Loci, two books, contents uncertain.
• Pseudaria (Fallacies), described by Proclus as training in detecting false reasoning in geometry.

Two thousand years of copies

Greek antiquity and Byzantium

Archimedes and Apollonius already take some propositions of the Elements for granted. Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries, of which Proclus’ on Book I survives in full. In the fourth century AD Theon of Alexandria, father of Hypatia, issued an edition that smoothed the language and filled small gaps. Nearly all Greek manuscripts descend from it. Theon himself mentions, in his commentary on Ptolemy, an addition he made to VI.33, which allows scholars to recognize his edition.

In 1808 François Peyrard recognized that a tenth-century Vatican manuscript, Vat. gr. 190, then in Paris after Napoleon’s armies had removed it from Rome, preserves a text that does not derive from Theon. Heiberg used it as the main witness for his critical edition (1883–1888). The oldest complete dated copy is the Bodleian’s MS D’Orville 301, copied by Stephen the Clerk in Constantinople in 888 for Arethas of Patras, later archbishop of Caesarea.

Arabic, Latin and print

Around 800 al-Ḥajjāj ibn Yūsuf ibn Maṭar translated the Elements into Arabic in Baghdad, reportedly twice, under Hārūn al-Rashīd and again under al-Maʾmūn. In the ninth century Isḥāq ibn Ḥunayn made a new translation, revised by Thābit ibn Qurra. Arabic commentators, among them al-Nayrīzī, Ibn al-Haytham and ʿUmar al-Khayyām, discussed the parallel postulate and the theory of ratio.

Latin Europe recovered the work from Arabic. Adelard of Bath made a translation around 1120, Gerard of Cremona another in the twelfth century, and Campanus of Novara’s version of about 1260 became the standard. Erhard Ratdolt printed Campanus in Venice in 1482, with diagrams in the margins, one of the first mathematical books ever printed. Bartolomeo Zamberti published a Latin translation from the Greek in 1505, and Simon Grynaeus printed the Greek text in Basel in 1533. Federico Commandino’s Latin edition (1572) became the scholarly reference, and Christopher Clavius’ annotated edition (1574) shaped Jesuit teaching. Henry Billingsley’s English translation of 1570 carried a long preface by John Dee on the uses of mathematics. In 1607 Matteo Ricci and Xu Guangqi published Books I–VI in Chinese.

Euclid and modern mathematics

The model of a deductive science

For more than two thousand years the Elements was the standard of rigour. Baruch Spinoza set out his Ethics (1677) “in geometrical order”, with definitions, axioms and propositions, and Isaac Newton’s Principia (1687) presents its mechanics in a similar style. Students in Europe, the Islamic world and later the Americas and China learned to prove things by learning Euclid.

Non-Euclidean geometry

The failure to prove the fifth postulate, described in Lesson 05, became one of the great discoveries of the nineteenth century. Once Lobachevsky, Bolyai and Beltrami had shown that a consistent geometry could deny it, Bernhard Riemann’s 1854 lecture opened the study of curved spaces in any dimension, and Felix Klein’s Erlangen programme of 1872 classified geometries by their groups of motions. Einstein’s general relativity uses Riemann’s geometry to describe gravity. Whether physical space is Euclidean became a question for measurement.

Axioms made complete

Pasch (1882) and Hilbert (1899) supplied what Euclid left implicit: order, continuity and congruence. Hilbert also showed which theorems depend on which axioms, for instance that the theory of area for polygons can be built without the Archimedean axiom. Alfred Tarski later gave a first-order axiomatization of elementary Euclidean geometry and proved it complete and decidable.

Euclid in the classroom

In nineteenth-century Britain Euclid’s text was taught almost verbatim. Critics urged change, and Lewis Carroll (Charles Dodgson) defended Euclid in Euclid and His Modern Rivals (1879). Oliver Byrne’s edition of 1847 replaced letters with coloured diagrams. In 1959, at a seminar at Royaumont, Jean Dieudonné called for an end to Euclid in schools with the slogan “Down with Euclid!”, and the “new math” reforms followed. School geometry today is usually taught with coordinates and transformations, though Euclid’s proofs remain the first encounter with proof for many students.

Legacy

• Method: the axiomatic method, from Euclid to Hilbert and modern set theory.
• Geometry: “Euclidean geometry” names the flat geometry of \(\mathbb R^n\); the parallel postulate led to non-Euclidean and Riemannian geometry.
• Number theory: the Euclidean algorithm, the infinitude of primes and the form of even perfect numbers; “Euclidean domain” names rings with a division algorithm.
• Names: the lunar crater Euclides, the asteroid 4354 Euclides, and the European Space Agency’s Euclid space telescope, launched in 2023 to map the geometry of the dark universe.

Connections

People linked to Euclid across this collection. A name without a link has no page here yet. See Euclid in the network of all 337 lives →

Influenced by

• Archytas of Tarentum
• Eudoxus of Cnidus
• Plato
• Autolycus of Pitane
• Hippocrates of Chios
• Theaetetus

Influenced

• Apollonius of Perga
• Archimedes
• Aristarchus of Samos
• Claudius Ptolemy
• David Hilbert
• Diophantus of Alexandria
• Eratosthenes
• Fibonacci
• Giovanni Girolamo Saccheri
• Hypatia of Alexandria
• Ibn al-Haytham
• Ibn Sina
• János Bolyai
• Leonhard Euler
• Muhammad ibn Musa al-Khwarizmi
• Nasir al-Din al-Tusi
• Niccolò Tartaglia
• Nikolai Lobachevsky
• Omar Khayyam
• Pappus of Alexandria
• Proclus
• Theon of Alexandria

Also mentioned in this life

• Aristotle
• Eugenio Beltrami
• Michel Chasles
• Richard Dedekind
• Albert Einstein
• Carl Friedrich Gauss
• Felix Klein
• Isaac Newton
• Bernhard Riemann
• Alfred Tarski
• John Wallis

Fields

• geometry 32 lives
• number theory 45 lives
• optics 38 lives

On CoursesHub

• Pythagoras & Euclid · History of Mathematics
• Optics & Photonics

Sources and further reading

Biography and transmission

1. J. J. O’Connor and E. F. Robertson, Euclid of Alexandria, MacTutor History of Mathematics.
2. Euclid and Euclid’s Elements, Wikipedia (with references to the ancient sources).
3. Proclus, A Commentary on the First Book of Euclid’s Elements, transl. G. R. Morrow, Princeton University Press, 1970.
4. Pappus of Alexandria, Collection, Book VII, ed. and transl. A. Jones, Springer, 1986.
5. Clay Mathematics Institute, Euclid’s Elements: the D’Orville manuscript (digitized MS D’Orville 301).
6. S. Unguru, On the need to rewrite the history of Greek mathematics, Archive for History of Exact Sciences 15 (1975).

Texts and mathematics

1. J. L. Heiberg and H. Menge (eds.), Euclidis Opera omnia, Teubner, 1883–1916.
2. T. L. Heath, The Thirteen Books of Euclid’s Elements, 3 vols., Cambridge, 1908; 2nd ed. 1926; Dover reprint.
3. D. E. Joyce, Euclid’s Elements, online edition with interactive diagrams, Clark University.
4. O. Byrne, The First Six Books of the Elements of Euclid, Pickering, London, 1847.
5. D. Hilbert, Grundlagen der Geometrie, Teubner, 1899.
6. R. Hartshorne, Geometry: Euclid and Beyond, Springer, 2000.
7. H. E. Burton, The Optics of Euclid, Journal of the Optical Society of America 35 (1945).
8. R. C. Archibald, Euclid’s Book on Divisions of Figures, Cambridge University Press, 1915.
9. Parallel postulate, Wikipedia.