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Mathematical treatise by Euclid
The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid. The Elements is
Euclid's_Elements
Ancient Greek mathematician (fl. 300 BC)
conflated the two Euclids, as did printer Erhard Ratdolt's 1482 editio princeps of Campanus of Novara's Latin translation of the Elements. After the mathematician
Euclid
Geometric axiom
In geometry, the parallel postulate is the fifth postulate in Euclid's Elements and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional
Parallel_postulate
On prime factors of integer products
prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's lemma shows that in the integers irreducible elements are also
Euclid's_lemma
Mathematical model of the physical space
system attributed to Euclid, an ancient Greek mathematician, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming
Euclidean_geometry
Argument that leads to a logical absurdity
g_{k}} and derived a contradiction. Euclid's theorem states that there are infinitely many primes. In Euclid's Elements, the theorem is stated in Book IX
Reductio_ad_absurdum
5th-century Greek Neoplatonist philosopher
Commentary on the First Book of Euclid's "Elements" Proclus (1970). A Commentary on the First Book of Euclid's Elements. Princeton, N.J.: Princeton University
Proclus
Infinitely many prime numbers exist
proven by Euclid in his work Elements. There are at least 200 proofs of the theorem. Euclid offered a proof in his work Elements (Book IX, Proposition 20)
Euclid's_theorem
Formal and systematic written discourse on some subject
influential by scholars on the development of human civilization. Euclid's Elements has appeared in more editions than any other books except the Bible
Treatise
Relation between sides of a right triangle
mathematics." Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented, along with Euclid's formula for generating all
Pythagorean_theorem
Irish engineer & author (1810–1880)
geometry, and engineering. He is best known for his 'coloured' book of Euclid's Elements. He was also a large contributor to Spon's Dictionary of Engineering
Oliver_Byrne_(mathematician)
Topics referred to by the same term
an integral Euclid's Elements, a mathematical treatise on geometry and number theory An entry, or element, of a matrix Classical elements, ancient beliefs
Element
Field of knowledge
mathematical rigor began in ancient Greek mathematics, exemplified in Euclid's Elements. Mathematics was primarily divided into geometry and arithmetic until
Mathematics
Ancient Greek mathematician and astronomer (c. 190–120 BC)
authoring On Ascensions (Ἀναφορικός) and possibly the Book XIV of Euclid's Elements. Hypsicles lived in Alexandria. Although little is known about the
Hypsicles
Study of geometries as axiomatic systems
mathematician Euclid, which he described (although non-rigorously by modern standards) in his textbook on geometry: the Elements. Euclid's method consists
Foundations_of_geometry
Mathematics of Ancient Greece and the Mediterranean, 5th BC to 6th AD
mid-fifth century BC, but the earliest complete work on the subject is Euclid's Elements, written during the Hellenistic period. The works of renowned mathematicians
Ancient_Greek_mathematics
Line intersecting 2 coplanar lines at 2 points
of each of the other pairs are also congruent. Proposition 1.27 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and
Transversal_(geometry)
Algorithm for computing greatest common divisors
It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is an example of an algorithm, and is
Euclidean_algorithm
Mathematical book by Lewis Carroll
to or functionally identical to Euclid's Elements. In it, Dodgson supports using Euclid's geometry textbook The Elements as the geometry textbook in schools
Euclid_and_His_Modern_Rivals
Modern formulation of Euclid's parallel postulate
to L, since all interior angles are right angles, and there is in Euclid's Elements a proof, using the additional axiom, that L and the additional line
Playfair's_axiom
Shape that is the intersection of two circles with the same radius
dimensions is the lemon. This figure appears in the first proposition of Euclid's Elements, where it forms the first step in constructing an equilateral triangle
Vesica_piscis
Geometry treatise
The subject matter is closely related to the first four books of Euclid's Elements. The book contains 15 definitions and 94 propositions. Greek text
Euclid's_Data
Mathematical concept
London, Allen and Unwin. pp. 1–241. Retrieved 2020-01-09. Euclid (2008) [c. 300 BC]. Euclid's Elements of Geometry (PDF). Translated by Fitzpatrick, Richard
Infinity
Integers have unique prime factorizations
measure the product, it will also measure one of the original numbers. — Euclid, Elements Book VII, Proposition 30, (In modern terminology: if a prime p divides
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
On triangles inscribed in a circle with a diameter as an edge
mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes
Thales's_theorem
Generalization of Pythagorean theorem
explained by the side-side-angle congruence ambiguity. Book II of Euclid's Elements, compiled c. 300 BC from material up to a century or two older, contains
Law_of_cosines
British civil servant, mathematician and classicist (1861–1940)
Greek Algebra The Thirteen Books of Euclid's Elements: vol. 1, vol. 2, vol. 3 The Thirteen Books of Euclid's Elements - Second Edition Revised with Additions:
Thomas_Heath_(classicist)
American mathematician
quandles in knot theory, and for his online interactive edition of Euclid's Elements. He is a professor emeritus of mathematics at Clark University. Joyce
David_E._Joyce
Length of a line segment
the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's Elements, distances were not represented
Euclidean_distance
Expression which is not assigned an interpretation
which is not defined in terms of simpler concepts. For example, in Elements, Euclid defines a point merely as "that of which there is no part", and a line
Undefined_(mathematics)
English scholar and alderman (c. 1538–1606)
notes on Euclid's Elements, which he had with great pains drawn up and digested. Afterwards our author Billingsley translated the said Elements into English
Henry_Billingsley
Shape with three sides
defined in Book One of Euclid's Elements. The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin
Triangle
Basic framework of mathematics
under the name of Aristotle's logic and systematically applied in Euclid's Elements. A mathematical assertion is considered as truth only if it is a theorem
Foundations_of_mathematics
Straight figure with zero width and depth
which is a part of a line delimited by two points (its endpoints). Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly
Line_(geometry)
Geometric theorem about isosceles triangles
triangle theorem. The theorem appears as Proposition 5 of Book 1 in Euclid's Elements. Its converse is also true: if two angles of a triangle are equal
Pons_asinorum
Characterization of even perfect numbers
Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime. This is the final result on number theory in Euclid's Elements; the
Euclid–Euler_theorem
Mythological Greek character
mentioned a few times by Proclus in his Commentary on the First Book of Euclid's Elements. Antoninus Liberalis, 12 Apollodorus, E.7.27 Homer, Odyssey 18.395
Amphinomus
Exterior angle of a triangle is greater than either of the remote interior angles
The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than
Exterior_angle_theorem
Property of geometry, also used to generalize the notion of "distance" in metric spaces
inner product spaces. The triangle inequality theorem is stated in Euclid's Elements, Book I, Proposition 20: […] in the triangle ABC the sum of any two
Triangle_inequality
4th-century BC Greek mathematician
proceeding that of Euclid. He thinks it probable that the propositions in elementary geometry quoted by Aristotle were taken from this Elements of Theudius.
Theudius
Convex quadrilateral with at least one pair of parallel sides
a parallelogram; this definition is also exclusive and is used in Euclid's Elements. Professional mathematicians and post-secondary geometry textbooks
Trapezoid
Integer side lengths of a right triangle
Proclus, in his commentary to the 47th Proposition of the first book of Euclid's Elements, describes it as follows: Certain methods for the discovery of triangles
Pythagorean_triple
1st-century AD Hellenistic mathematician and engineer
the works of Ctesibius. In mathematics, he wrote a commentary on Euclid's Elements and a work on applied geometry known as the Metrica. He is mostly
Hero_of_Alexandria
Diagrams from Euclid". University of British Columbia. Retrieved 2008-09-26. (Boyer 1991, "Euclid of Alexandria" p.109) "Book II of the Elements is a short
History_of_algebra
Angle formed in the interior of a circle
arc. Its oldest appearance is in Propositions 20–21 in Book 3 of Euclid's Elements. The inscribed angle theorem states that an angle θ inscribed in a
Inscribed_angle
Maharaja of Amber (1688–1743)
at multiple places in India, including his capital Jaipur. He had Euclid's "Elements of Geometry" translated into Sanskrit. When Jai Singh acceded to the
Sawai_Jai_Singh
Sum of an (infinite) geometric progression
of positive numbers needing to add up to infinity was incorrect. Euclid's Elements has the distinction of being the world's oldest continuously used
Geometric_series
Greek scholar and mathematician (c. 335–405)
in Alexandria, Egypt. He edited and arranged Euclid's Elements and wrote commentaries on works by Euclid and Ptolemy. His daughter, Hypatia, also won
Theon_of_Alexandria
Number equal to the sum of its proper divisors
Euclid's Elements (Book VII, Definition 22) where it is called τέλειος ἀριθμός (téleios arithmós; 'perfect', 'ideal', or 'complete number'). Euclid also
Perfect_number
Geometry theorem relating the line segments created by intersecting chords in a circle
segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements. For example, for two chords AC and BD intersecting at point S, the
Intersecting_chords_theorem
Shape with four equal sides and angles
1007/s00591-016-0173-0. MR 3629442. Euclid's Elements, Book I, Proposition 47. Online English version by David E. Joyce. Euclid's Elements, Book VI, Proposition 31
Square
Primitive way of calculating area
"Euclid's Elements, Book XII, Proposition 2". aleph0.clarku.edu. "Euclid's Elements, Book XII, Proposition 5". aleph0.clarku.edu. "Euclid's Elements,
Method_of_exhaustion
5th-century Byzantine Greek architect and mathematician
Archimedes' works has been attributed to him. The spurious Book XV from Euclid's Elements has been partly attributed to Isidore of Miletus. Isidore of Miletus
Isidore_of_Miletus
Irish mathematician (1850–1922)
volumes of Euclid's Elements of Geometry, for the use of schools (Euclid's Elements of Geometry, books I and II (1897); Euclid's Elements of Geometry
Sophie_Bryant
Mathematical idealization of the trace left by a moving point
This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is […] the first species of quantity, which has
Curve
Relationship between two numbers of the same kind
until the 16th century. Book V of Euclid's Elements has 18 definitions, all of which relate to ratios. In addition, Euclid uses ideas that were in such common
Ratio
Solid with 2 parallel n-gonal bases connected by n parallelograms
(from Greek πρίσμα (prisma) 'something sawed') was first used in Euclid's Elements. Euclid defined the term in Book XI as "a solid figure contained by two
Prism_(geometry)
Japanese manga series
July 4, 2026 (2026-07-04) The Mongols steal the family's copy of Euclid's Elements, kill Sitara's mistress, sack the city, and take Sitara and the other
A_Witch's_Life_in_Mongol
5th-century BCE Greek mathematician and astronomer
Proclus. In Euclid's Elements. p. 66. Proclus. In Euclid's Elements. p. 80. Leonid Zhmud, The Menaechmi, March 24, 2023 Proclus. In Euclid's Elements. p. 283
Oenopides
Type of binary relation
relations are a class of binary relations that formalize "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other."
Euclidean_relation
12th-century English natural philosopher
introduced to Western Europe. The oldest surviving Latin translation of Euclid's Elements is a 12th-century translation by Adelard from an Arabic version. He
Adelard_of_Bath
Relation used in geometry
in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks, often as
Parallel_(geometry)
Topics referred to by the same term
theorem in Euclid's Elements, and in particular: Euclid's theorem that there are infinitely many prime numbers Euclid's lemma, also called Euclid's first theorem
Euclidean_theorem
Greek mathematician (c.417–c. 369 BCE)
contributions were on irrational lengths, which was included in Book X of Euclid's Elements and proving that there are precisely five regular convex polyhedra
Theaetetus_(mathematician)
Italian mathematician and astrologer (c. 1220–1296)
astronomer, astrologer, and physician who is best known for his work on Euclid's Elements. In his writings he refers to himself as Campanus Nouariensis; contemporary
Campanus_of_Novara
Danish philologist and historian
unknown texts in the Archimedes Palimpsest, and for his edition of Euclid's Elements that T. L. Heath translated into English. He also published an edition
Johan Ludvig Heiberg (historian)
Johan_Ludvig_Heiberg_(historian)
Geometric shape
semicircle Salinon Wigner semicircle distribution Euclid's Elements, Book VI, Proposition 13 Euclid's Elements, Book VI, Proposition 25 "Ford Circle". Weisstein
Semicircle
Fundamental space of geometry
of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in
Euclidean_space
Liberal arts of arithmetic, geometry, music and astronomy
Médiévales. pp. 18–19. Proclus. A Commentary on the First Book of Euclid's Elements, xii. trans. Glenn Raymond Morrow. Princeton: Princeton University
Quadrivium
Ancient Greek philosopher (c. 626 – c. 545 BC)
mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. Dante's Paradiso refers to Thales's theorem in the course of a speech
Thales_of_Miletus
5th-century BC Pythagorean philosopher
327. William Thompson (1930). The Commentary of Pappus on Book X of Euclid's Elements (PDF). Harvard University Press. p. 64. Couprie, Dirk L. (2011). "The
Hippasus
Abbreviation at completion of a proof
DEMONSTRANDUM". www.merriam-webster.com. Retrieved 2017-09-03. Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005 Valla, Giorgio. "Georgii
Q.E.D.
Geometry without using coordinates
approach to synthetic geometry is Euclid's Elements. However, it appeared at the end of the 19th century that Euclid's postulates were not sufficient for
Synthetic_geometry
Mathematical sequence of numbers
Babylonian mathematics beginning in 2000 BC. Books VIII and IX of Euclid's Elements analyze geometric progressions (such as the powers of two, see the
Geometric_progression
Bavarian printer (1442–1528)
the Historia Romana of Appianus (1477), and the first edition of Euclid's Elements (1482), where he solved the problem of printing geometric diagrams
Erhard_Ratdolt
Type of curvilinear angle
of Euclid's Elements. Vol. 2. The University Press. pp. 39–42. Weisstein, Eric W. "Horn Angle". MathWorld. David E. Joyce, "Definition 8" Euclid's Elements
Horn_angle
Medieval Arab mathematician
capital of the ʿAbbāsid Empire. He was the first author who translated Euclid's Elements from Greek into Arabic. His first translation was made for Yaḥyā ibn
Al-Ḥajjāj_ibn_Yūsuf_ibn_Maṭar
that Euclid (c. 300 BC) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time. The Elements introduced
History_of_mathematics
Number which when multiplied by x equals 1
proportion are described as reciprocall in a 1570 translation of Euclid's Elements. In the phrase multiplicative inverse, the qualifier multiplicative
Multiplicative_inverse
Line that intersects a curve at least twice
inside the circle this is Euclid III.35, but if the point is outside the circle the result is not contained in the Elements. However, Robert Simson following
Secant_line
Concept in geometry
then they are subtended by congruent sides (propositions I.5–6 in Euclid's Elements), forming an isosceles triangle. More generally, the law of sines
Subtended_angle
Property of objects which are scaled or mirrored versions of each other
proved in Euclid's Elements, Book VI, Proposition 4. For instance, Venema 2006, p. 122 and Henderson & Taimiņa 2005, p. 123. Euclid's Elements, Book VI
Similarity_(geometry)
4th-century Alexandrian astronomer and mathematician
unoriginal". His primary achievement was the production of a new edition of Euclid's Elements, in which he corrected scribal errors that had been made over the
Hypatia
Ancient Greek mathematician
of Eudoxus and Theaetetus, discovered many of the propositions in Euclid's Elements, and wrote about theorems on loci. Asper, Markus (May 2019). "Personae
Hermotimus_of_Colophon
Flat-sided three-dimensional shape
natures for each in his Timaeus, later soon treatment studied in Euclid's Elements. In Renaissance, toroidal polyhedra were used for sketching on polyhedral's
Polyhedron
Principle in compass and straightedge constructions
of Euclid's Elements. The proof of this theorem has had a chequered history. The following construction and proof of correctness are given by Euclid in
Compass_equivalence_theorem
compass, which in turn are based on the first three postulates of Euclid's Elements. The historical importance of rulers and compasses as instruments
Geometric_drawing
Type of non-Euclidean geometry
of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence
Hyperbolic_geometry
In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron
Etymology Dictionary. Strabo, Geography 2.1.36–37 Euclid, Elements book 11, proposition 28 Euclid, Elements book 11, proposition 38 Honsberger (1973). "A
Diagonal
Division of something into two equal or congruent parts
(collapsible) compass construction as per Proposition 10 in Book 1 of Euclid's Elements. This method served his purposes for writing deductive and logical
Bisection
Geometry theorem relating line segments created by a secant and tangent line
associated circle. This result is found as Proposition 36 in Book 3 of Euclid's Elements. Given a secant g intersecting the circle at points G1 and G2 and
Tangent–secant_theorem
Collection of minor Ancient Greek astronomical works
Euclid's Elements. The works contained in the collection are: Spherics by Theodosius of Bithynia: On spherical geometry, in the style of the Elements
Little_Astronomy
Geometrical theorem relating the lengths of two segments that divide a triangle
The angle bisector theorem appears as Proposition 3 of Book VI in Euclid's Elements. According to Heath (1956, p. 197 (vol. 2)), the corresponding statement
Angle_bisector_theorem
Ancient Chinese mathematics text
on the development of Eastern mathematical traditions to that of Euclid's Elements on the Western mathematical traditions. However, the influence of
The Nine Chapters on the Mathematical Art
The_Nine_Chapters_on_the_Mathematical_Art
That which is universally and eternally true
Spinoza sought to arrive at an ethical theory that is as precise as Euclid's Elements. In the history of philosophy, this way of proceeding may be contrasted
Sub_specie_aeternitatis
Relation between the side lengths and altitude of a right triangle
This theorem should not be confused with proposition 48 in book 1 of Euclid's Elements, the converse of the Pythagorean theorem, which states that if the
Inverse_Pythagorean_theorem
Chinese intellectual (1562–1633)
of several classic Western texts into Chinese, including part of Euclid's Elements. He was also the author of the Nong Zheng Quan Shu, a treatise on
Xu_Guangqi
Property of magnitude or multitude
respect of size between two magnitudes of the same kind. — Euclid, Elements For Aristotle and Euclid, relations were conceived as whole numbers (Michell, 1993)
Quantity
Theorem concerning ratios of line segments
Babylonians and Egyptians, although its first known proof appears in Euclid's Elements. A mechanical device which produces geometrically-similar shapes is
Intercept_theorem
Scottish mathematician (1687–1768)
reading Sinclair's Tuyrocinia Mathematica in Novem Tractatus and then Euclid's Elements Simson soon became deeply interested in mathematics and especially
Robert_Simson
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