TheInfoList

Euclidean geometry is a mathematical system attributed to
Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; : Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the in after and , in , and a major economic centre. With a total population of 5,200,000, Alexandria is the ...

n Greek mathematician
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης ; 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematics, Greek mathematician, often referred to as the "founder of ge ...

, which he described in his textbook on
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

: the '' Elements''. Euclid's method consists in assuming a small set of intuitively appealing
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

s, and deducing many other
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same mea ...
s (
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s) from these. Although many of Euclid's results had been stated by earlier mathematicians,. Euclid was the first to show how these propositions could fit into a comprehensive
deductive Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...
and
logical system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
. The ''Elements'' begins with plane geometry, still taught in
secondary school A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both lower secondary education (ages 11 to 14) and upper secondary educat ...
(high school) as the first
axiomatic system In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and the first examples of
mathematical proof A mathematical proof is an inferential argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek ...
s. It goes on to the
solid geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of
three dimensions Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...
. Much of the ''Elements'' states results of what are now called
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other
self-consistent In classical deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditiona ...
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
are known, the first ones having been discovered in the early 19th century. An implication of
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

's theory of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
is that physical space itself is not Euclidean, and
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
is a good approximation for it only over short distances (relative to the strength of the
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

). Euclidean geometry is an example of
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of
coordinates In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

to specify those objects. This is in contrast to
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
, which uses coordinates to translate geometric propositions into algebraic formulas.

# The ''Elements''

The ''Elements'' is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. There are 13 books in the ''Elements'': Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

"In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47) Books V and VII–X deal with
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as
prime numbers A prime number (or a prime) is a natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the coun ...

and
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
and
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s are introduced. It is proved that there are infinitely many prime numbers. Books XI–XIII concern
solid geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The
platonic solid In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...
s are constructed.

## Axioms

Euclidean geometry is an
axiomatic system In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, in which all
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s ("true statements") are derived from a small number of simple axioms. Until the advent of
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...
, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.The assumptions of Euclid are discussed from a modern perspective in Near the beginning of the first book of the ''Elements'', Euclid gives five
postulate An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
s (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): :Let the following be postulated: # To draw a
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...

from any
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
to any point. # To produce (extend) a continuously in a straight line. # To describe a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

with any centre and distance (radius). # That all
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s are equal to one another. # [The
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

]: 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 the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique. The ''Elements'' also include the following five "common notions": # Things that are equal to the same thing are also equal to one another (the transitive property of a
Euclidean relationIn mathematics, Euclidean relations are a class of binary relations that formalize ":wikisource:Page:First_six_books_of_the_elements_of_Euclid_1847_Byrne.djvu/26, Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to eac ...
). # If equals are added to equals, then the wholes are equal (Addition property of equality). # If equals are subtracted from equals, then the differences are equal (subtraction property of equality). # Things that coincide with one another are equal to one another (reflexive property). # The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more extensive and complete sets of axioms.

## Parallel postulate

To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the ''Elements'': his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated which are
logically equivalent Logic (from Greek: grc, λογική, label=none, lit=possessed of reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=n ...
to the parallel postulate (in the context of the other axioms). For example,
Playfair's axiom In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
states: :In a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
, through a point not on a given straight line, at most one line can be drawn that never meets the given line. The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists.

# Methods of proof

Euclidean Geometry is ''
constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has a ...
''. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.Within Euclid's assumptions, it is quite easy to give a formula for area of triangles and squares. However, in a more general context like set theory, it is not as easy to prove that the area of a square is the sum of areas of its pieces, for example. See
Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...
and
Banach–Tarski paradox The Banach–Tarski paradox is a theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, ...
.
Strictly speaking, the lines on paper are '' models'' of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.g., some of the Pythagoreans' proofs that involved irrational numbers, which usually required a statement such as "Find the greatest common measure of ..." Euclid often used
proof by contradiction In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stateme ...
. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I.4, side-angle-side congruence of triangles, is proved by moving one of the two triangles so that one of its sides coincides with the other triangle's equal side, and then proving that the other sides coincide as well. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition.

# System of measurement and arithmetic

Euclidean geometry has two fundamental types of measurements:
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

and
distance Distance is a numerical measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and ...
. The angle scale is absolute, and Euclid uses the
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

as his basic unit, so that, for example, a 45-
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. Measurements of
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...
and
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

are derived from distances. For example, a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20. Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term "
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
" refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similar.
Corresponding angles Correspondence may refer to: *In general usage, non-concurrent, remote communication Communication (from Latin ''communicare'', meaning "to share") is the act of developing Semantics, meaning among Subject (philosophy), entities or Organization ...
in a pair of similar shapes are congruent and
corresponding sides In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
are in proportion to each other.

# Notation and terminology

## Naming of points and figures

Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C.

## Complementary and supplementary angles

Angles whose sum is a right angle are called
complementary A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...

. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. Angles whose sum is a straight angle are . Supplementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite.

## Modern versions of Euclid's notation

In modern terminology, angles would normally be measured in
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
s or
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s. Modern school textbooks often define separate figures called
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

s (infinite), (semi-infinite), and
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s (of finite length). Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" in Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary.

# Some important or well known results

File:pons_asinorum_dzmanto.png, The ''
pons asinorum In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
'' or ''bridge of asses theorem'' states that in an isosceles triangle, α = β and γ = δ. File:Sum_of_angles_of_triangle_dzmanto.png, The ''triangle angle sum theorem'' states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees. File:Pythagorean.svg, The ''
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

'' states that the sum of the areas of the two squares on the legs (''a'' and ''b'') of a right triangle equals the area of the square on the hypotenuse (''c''). File:Thales' Theorem Simple.svg, ''
Thales' theorem In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

'' states that if AC is a diameter, then the angle at B is a right angle.

## Pons asinorum

The
pons asinorum In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
(''bridge of asses'') states that ''in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another''. Its name may be attributed to its frequent role as the first real test in the ''Elements'' of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.

## Congruence of triangles

Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent.

## Triangle angle sum

The sum of the angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one or
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

.

## Pythagorean theorem

The celebrated
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

(book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

## Thales' theorem

Thales' theorem In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

, named after
Thales of Miletus Thales of Miletus ( ; el, Θαλῆς Thales of Miletus ( ; el, Θαλῆς (ὁ Μιλήσιος), ''Thalēs''; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (fr ...

states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after the manner of Euclid Book III, Prop. 31.

## Scaling of area and volume

In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, $A \propto L^2$, and the volume of a solid to the cube, $V \propto L^3$. Euclid proved these results in various special cases such as the area of a circle and the volume of a parallelepipedal solid. Euclid determined some, but not all, of the relevant constants of proportionality. E.g., it was his successor
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

who proved that a sphere has 2/3 the volume of the circumscribing cylinder.

# Applications

Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here. File:us land survey officer.jpg, A surveyor uses a
level Level or levels may refer to: Engineering * Level (instrument), a device used to measure true horizontal or relative heights *Canal pound or level *Regrading or levelling, the process of raising and/or lowering the levels of land *Storey or level ...
File:Ambersweet oranges.jpg,
Sphere packing In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

applies to a stack of
orange Orange most often refers to: *Orange (colour), occurs between red and yellow in the visible spectrum *Orange (fruit), the fruit of the tree species '' Citrus'' × ''sinensis'' ** Orange blossom, its fragrant flower *Some other citrus or citrus-li ...
s. File:Parabola with focus and arbitrary line.svg, A parabolic mirror brings parallel rays of light to a focus.
As suggested by the etymology of the word, one of the earliest reasons for interest in geometry was
surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them. A land surveying professional is called a land survey ...

, and certain practical results from Euclidean geometry, such as the right-angle property of the 3-4-5 triangle, were used long before they were proved formally. The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor. Historically, distances were often measured by chains, such as
Gunter's chain Gunter's chain (also known as Gunter’s measurement) is a distance measuring device used for surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial or three-dimensional positions ...
, and angles using graduated circles and, later, the
theodolite A theodolite is a precision optical instrument for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), ...

. An application of Euclidean solid geometry is the determination of packing arrangements, such as the problem of finding the most efficient in n dimensions. This problem has applications in
error detection and correction In information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to ...
.
Geometric optics Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...
uses Euclidean geometry to analyze the focusing of light by lenses and mirrors. File:Damascus Khan asad Pacha cropped.jpg, Geometry is used in art and architecture. File:Water tower cropped.jpg, The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry. File:Origami crane cropped.jpg, Geometry can be used to design origami. Geometry is used extensively in
architecture upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted), Plan of the second floor (attic storey) of the Hôtel de Brionne in Paris – 1734. Architecture (Latin ''archi ...

. Geometry can be used to design
origami ) is the paper art, art of paper folding, which is often associated with Japanese culture. In modern usage, the word "origami" is used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to tra ...

. Some classical construction problems of geometry are impossible using
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexan ...
, but can be solved using origami. Quite a lot of CAD (computer-aided design) and CAM (computer-aided manufacturing) is based on Euclidean geometry. Design geometry typically consists of shapes bounded by planes, cylinders, cones, tori, etc. In the present day, CAD/CAM is essential in the design of almost everything, including cars, airplanes, ships, and smartphones. A few decades ago, sophisticated draftsmen learned some fairly advanced Euclidean geometry, including things like
Pascal's theorem Image:Pascal'sTheoremLetteredColored.PNG, 250px, Self-crossing hexagon , inscribed in a circle. Its sides are extended so that pairs of opposite sides intersect on Pascal's line. Each pair of extended opposite sides has its own color: one red, one ...

and Brianchon's theorem. But now they don't have to, because the geometric constructions are all done by CAD programs. File:Motor partsolutions.gif

# As a description of the structure of space

Euclid believed that his
axioms An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms, in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called ''Euclidean motions'', which include translations, reflections and rotations of figures. See, for example: and The ''group of motions'' underlie the metric notions of geometry. See Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries; postulate 4 (equality of right angles) says that space is
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by ...
and figures may be moved to any location while maintaining
congruence Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
; and postulate 5 (the
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

) that space is flat (has no
intrinsic curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a Surface (mathematics), surface deviates from being a plane (ge ...
). As discussed in more detail below,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

's
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...
significantly modifies this view. The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infiniteHeath, p. 200. (see below) and what its
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

is. Modern, more rigorous reformulations of the system typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in
elliptic geometry Elliptic geometry is an example of a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conc ...
), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a
torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ...

for two-dimensional Euclidean geometry).

# Later work

## Archimedes and Apollonius

Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

(c. 287 BCE – c. 212 BCE), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original. He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the
Archimedean property In abstract algebra and analysis Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics Mathematics ( ...

of finite numbers.
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος ''Apollonios o Pergeos''; la, Apollonius Pergaeus; ) was an Ancient Greece, Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the ...
(c. 262 BCE – c. 190 BCE) is mainly known for his investigation of conic sections.

## 17th century: Descartes

René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

(1596–1650) developed
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
, an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its Cartesian (''x'', ''y'') coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. The equation :$, PQ, =\sqrt \,$ defining the distance between two points ''P'' = (''px'', ''py'') and ''Q'' = (''qx'', ''qy'') is then known as the ''Euclidean
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
'', and other metrics define
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., ''y'' = 2''x'' + 1 (a line), or ''x''2 + ''y''2 = 7 (a circle). Also in the 17th century,
Girard Desargues Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( ...
, motivated by the theory of perspective, introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry,
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.

## 18th century

Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect. Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until
Pierre Wantzel Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...
published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include
doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume Volume is the quantity of three-dimensional space ...

and
squaring the circle Squaring the circle is a problem proposed by ancient Ancient history is the aggregate of past events
. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, while doubling a cube requires the solution of a third-order equation.
Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithm ...

discussed a generalization of Euclidean geometry called
affine geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).

## 19th century and non-Euclidean geometry

In the early 19th century, and systematically developed the use of signed angles and line segments as a way of simplifying and unifying results. The century's most significant development in geometry occurred when, around 1830,
János Bolyai János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the ...
and
Nikolai Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russia Russia (russian: link=no, ...
separately published work on
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...
, in which the parallel postulate is not valid. Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates. In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the ''Elements''. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the ''Elements,'' shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third
vertex Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers. Starting with
Moritz Pasch Moritz Pasch (8 November 1843, Breslau, Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a historically prominent Germans, German state that originated in 1525 with Duchy of Prussia, a duchy centered on the Prussia (region), reg ...

in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
,
George Birkhoff George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during ...
, and Tarski.Tarski (1951).

## 20th century and relativity

theory of
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
involves a four-dimensional
space-time In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ...
, the
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclid ...
, which is non-Euclidean. This shows that non-Euclidean geometries, which had been introduced a few years earlier for showing that the
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

cannot be proved, are also useful for describing the physical world. However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, for which the geometry of the space part of space-time is not Euclidean geometry. For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the Sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting these deviations in rays of light from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the
GPS The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government The federal government of the United States (U.S. federal government) is the national ...
system.

# Treatment of infinity

## Infinite objects

Euclid sometimes distinguished explicitly between "finite lines" (e.g., Postulate 2) and "
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

lines" (book I, proposition 12). However, he typically did not make such distinctions unless they were necessary. The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, as implying that space is infinite. The notion of infinitesimal quantities had previously been discussed extensively by the
Eleatic School The Eleatics were a pre-Socratic school of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of min ...
, but nobody had been able to put them on a firm logical basis, with paradoxes such as
Zeno's paradox Zeno's paradoxes are a set of philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge Knowledge is a familiarity, awareness, or understanding of someo ...
occurring that had not been resolved to universal satisfaction. Euclid used the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

rather than infinitesimals. Later ancient commentators, such as
Proclus Proclus Lycius (; 410/411/ 7 Feb. or 8 Feb. 412 –17 April 485 AD), called Proclus the Successor, Proclus the Platonic Successor, or Proclus of Athens (Greek: Προκλου Διαδοχου ''Próklos Diádochos'', ''"''in some Manuscript ...
(410–485 CE), treated many questions about infinity as issues demanding proof and, e.g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it. At the turn of the 20th century, Otto Stolz,
Paul du Bois-Reymond Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
,
Giuseppe Veronese Giuseppe Veronese (7 May 1854 – 17 July 1917) was an Italy, Italian mathematician. He was born in Chioggia, near Venice. Education Veronese earned his laurea in mathematics from the Istituto Tecnico di Venezia in 1872. Work Although Veronese's w ...

, and others produced controversial work on models of Euclidean geometry, in which the distance between two points may be infinite or infinitesimal, in the
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...
sense. Fifty years later,
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
provided a rigorous logical foundation for Veronese's work.

## Infinite processes

One reason that the ancients treated the parallel postulate as less certain than the others is that verifying it physically would require us to inspect two lines to check that they never intersected, even at some very distant point, and this inspection could potentially take an infinite amount of time. The modern formulation of
proof by induction Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Mathematical induction is a mathematical proof A mathematical proof is an Inference, inferential Argument-deduction-proof distinct ...
was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g., the proof of the infinitude of primes. Supposed paradoxes involving infinite series, such as
Zeno's paradox Zeno's paradoxes are a set of philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge Knowledge is a familiarity, awareness, or understanding of someo ...
, predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of the
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
in IX.35 without commenting on the possibility of letting the number of terms become infinite.

# Logical basis

## Classical logic

Euclid frequently used the method of
proof by contradiction In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stateme ...
, and therefore the traditional presentation of Euclidean geometry assumes
classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world. Char ...
, in which every proposition is either true or false, i.e., for any proposition P, the proposition "P or not P" is automatically true.

## Modern standards of rigor

Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries.A detailed discussion can be found in The role of primitive notions, or undefined concepts, was clearly put forward by
Alessandro Padoa Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian language, Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new ...

of the
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

delegation at the 1900 Paris conference: That is, mathematics is context-independent knowledge within a hierarchical framework. As said by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
: Such foundational approaches range between
foundationalism Foundationalism concerns philosophical theories of knowledge resting upon justified belief, or some secure foundation of certainty such as a conclusion inferred from a basis of sound premises.Simon Blackburn, ''The Oxford Dictionary of Philosoph ...
and
formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scienti ...
.

## Axiomatic formulations

*Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time. It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that the
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or non-Euclidean. *
Hilbert's axioms Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an exten ...
: Hilbert's axioms had the goal of identifying a ''simple'' and ''complete'' set of ''independent'' axioms from which the most important geometric theorems could be deduced. The outstanding objectives were to make Euclidean geometry rigorous (avoiding hidden assumptions) and to make clear the ramifications of the parallel postulate. * Birkhoff's axioms: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. This system relies heavily on the properties of the
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

. The notions of ''angle'' and ''distance'' become primitive concepts. * Tarski's axioms:
Alfred Tarski Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician ...
(1902–1983) and his students defined ''elementary'' Euclidean geometry as the geometry that can be expressed in
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
and does not depend on
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
for its logical basis, in contrast to Hilbert's axioms, which involve point sets. Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain
sense A sense is a biological system used by an organism for sensation, the process of gathering information about the world and responding to Stimulus (physiology), stimuli. (For example, in the human body, the brain receives signals from the senses ...
: there is an algorithm that, for every proposition, can be shown either true or false. (This doesn't violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
for the theorem to apply.Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. AK Peters. . Pp. 25–26.) This is equivalent to the decidability of real closed fields, of which elementary Euclidean geometry is a model.

*
Absolute geometry Absolute geometry is a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with pro ...
*
Analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
* Birkhoff's axioms *
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
*
Hilbert's axioms Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an exten ...
*
Incidence geometry Incidence may refer to: Economics * Benefit incidence, the availability of a benefit * Expenditure incidence, the effect of government expenditure upon the distribution of private incomes * Fiscal incidence, the economic impact of government tax ...
*
List of interactive geometry software Interactive geometry software (IGS) or dynamic geometry environments (DGEs) are computer programs which allow one to create and then manipulate geometry, geometric constructions, primarily in Euclidean plane geometry, plane geometry. In most IGS, ...
*
Metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...
*
Ordered geometry Ordered geometry is a form of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, ...
*
Parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

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Type theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

## Classical theorems

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Angle bisector theorem In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that Bisection, bisects the opposite angle. It equates their relative lengths to the relative lengt ...

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Butterfly theorem The butterfly theorem is a classical result in Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكن ...

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Ceva's theorem Ceva's theorem is a theorem about triangle A triangle is a polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one o ...
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Heron's formula In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

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Menelaus' theorem Menelaus's theorem, named for Menelaus of Alexandria Menelaus of Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; el, Αλεξά ...
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Nine-point circle In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

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Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

# References

* * * * In 3 vols.: vol. 1 , vol. 2 , vol. 3 . Heath's authoritative translation of Euclid's Elements, plus his extensive historical research and detailed commentary throughout the text. * * * *