Euclidean geometry
plane 

Geometers 
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,^{[1]} Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.^{[2]}
The Elements begins with plane geometry, still taught in
For more than two thousand years, the adjective "Euclidean" was unnecessary because Euclid's axioms seemed so intuitively obvious (with the possible exception of the
Euclidean geometry is an example of
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 rightangled 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
Books XI–XIII concern solid geometry. 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 solids are constructed.
Axioms
Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of simple axioms. Until the advent of nonEuclidean geometry, 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 independently from the physical reality.^{[4]}
Near the beginning of the first book of the Elements, Euclid gives five
 Let the following be postulated:
 To draw a straight line from any pointto any point.
 To produce (extend) a finite straight line continuously in a straight line.
 To describe a circle with any centre and distance (radius).
 That all right angles are equal to one another.
 [The parallel postulate]: 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 explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them 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 relation).
 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.^{}[6] 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.^{[7]} 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 to the parallel postulate (in the context of the other axioms). For example, Playfair's axiom states:
 In a plane, 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
Euclid often used proof by contradiction.^{[citation needed]}
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
Angles whose sum is a straight angle are
Modern versions of Euclid's notation
In modern terminology, angles would normally be measured in degrees or radians.
Modern school textbooks often define separate figures called
Some important or well known results

The pons asinorum or bridge of asses theorem states that in an isosceles triangle, α = β and γ = δ.

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.

The Pythagorean theorem 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).

Thales' theoremstates that if AC is a diameter, then the angle at B is a right angle.
Pons asinorum
The pons asinorum (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.^{[11]} 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 surefooted donkey could cross.^{[12]}
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).^{}
Pythagorean theorem
The celebrated Pythagorean theorem (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
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, , and the volume of a solid to the cube, . Euclid proved these results in various special cases such as the area of a circle^{}[16] and the volume of a parallelepipedal solid.^{[17]} Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder.^{[18]}
System of measurement and arithmetic
Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, for example, a 45degree 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
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 "
In engineering
Design and Analysis
 Stress Analysis: structural integrity and durability.
 Gear Design: mechanical systems, relies heavily on Euclidean geometry to ensure proper tooth shape and engagement for efficient power transmission.
 Heat Exchanger Design: heat exchangers, where the geometric configuration greatly influences thermal efficiency. See shellandtube heat exchangers and plate heat exchangersfor more details.
 Lens Design: mirrors.
Dynamics
 Vibration Analysis: vibrationseffectively.
 Wing Design: Aircraft Wing Design  The application of Euclidean geometry in aerodynamics is evident in aircraft wing design, airfoils, and hydrofoils where geometric shape directly impacts lift and drag characteristics.
 Satellite Orbits: astrodynamics, celestial mechanics, and elliptic orbit.
CAD Systems
 3D Modeling: In CAD (computeraided design) systems, Euclidean geometry is fundamental for creating accurate 3D models of mechanical parts. These models are crucial for visualizing and testing designs before manufacturing.
 Design and Manufacturing: Much of planes, cylinders, cones, tori, and other similar Euclidean forms. Today, CAD/CAM is essential in the design of a wide range of products, from cars and airplanes to ships and smartphones.
 Evolution of Drafting Practices: Historically, advanced Euclidean geometry, including theorems like Pascal's theorem and Brianchon's theorem, was integral to drafting practices. However, with the advent of modern CAD systems, such indepth knowledge of these theorems is less necessary in contemporary design and manufacturing processes.
Circuit Design
 PCB Layouts: circuit performance.
Electromagnetic and Fluid Flow Fields
 Antenna Design: electromagnetic waves.
 Field Theory: solenoidal field or a conservative vector field.
Controls
 Calculation Tools: diagnostics.
Other general applications
Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here.

A surveyor uses alevel

Sphere packing applies to a stack of oranges.

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 and also one of the most common current uses of geometry is surveying.^{[19]} In addition it has been used in classical mechanics and the cognitive and computational approaches to visual perception of objects. Certain practical results from Euclidean geometry (such as the rightangle property of the 345 triangle) were used long before they were proved formally.^{[20]} 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, and angles using graduated circles and, later, the theodolite.
An application of Euclidean solid geometry is the

Geometry is used in art and architecture.

The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry.

Geometry can be used to design origami.
Geometry is used extensively in architecture.
Geometry can be used to design
Later history
Archimedes and Apollonius
Archimedes (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.^{[22]} He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers.
Apollonius of Perga (c. 240 BCE – c. 190 BCE) is mainly known for his investigation of conic sections.
17th century: Descartes
René Descartes (1596–1650) developed analytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra.^{[23]}
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 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
defining the distance between two points P = (p_{x}, p_{y}) and Q = (q_{x}, q_{y}) is then known as the Euclidean metric, and other metrics define nonEuclidean geometries.
In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first and secondorder equations, e.g., y = 2x + 1 (a line), or x^{2} + y^{2} = 7 (a circle).
Also in the 17th century, Girard Desargues, 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, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.^{[24]}
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.^{[25]}
Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of
Euler discussed a generalization of Euclidean geometry called affine geometry, 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 between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).
19th century
In the early 19th century, Carnot and Möbius systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.^{[27]}
Higher dimensions
In the 1840s
At midcentury Ludwig Schläfli developed the general concept of Euclidean space, extending Euclidean geometry to higher dimensions. He defined polyschemes, later called polytopes, which are the higherdimensional analogues of polygons and polyhedra. He developed their theory and discovered all the regular polytopes, i.e. the dimensional analogues of regular polygons and
Regular convex 4polytopes
 

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors


Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube

4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 squares x 3  4 rectangles x 4  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon x 2  1 octagon x 3  2 octagons x 4  2 dodecagons x 4  4 30gons x 6

20 30gons x 4
 
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
Schläfli performed this work in relative obscurity and it was published in full only posthumously in 1901. It had little influence until it was rediscovered and
In 1878 William Kingdon Clifford introduced what is now termed geometric algebra, unifying Hamilton's quaternions with Hermann Grassmann's algebra and revealing the geometric nature of these systems, especially in four dimensions. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions. The Clifford torus on the surface of the 3sphere is the simplest and most symmetric flat embedding of the Cartesian product of two circles (in the same sense that the surface of a cylinder is "flat").
NonEuclidean geometry
The century's most influential development in geometry occurred when, around 1830,
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. 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 in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of Hilbert,^{[31]} George Birkhoff,^{[32]} and Tarski.^{[33]}
20th century and relativity
However, the threedimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with general relativity, for which the geometry of the space part of spacetime is not Euclidean geometry.^{[34]} 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 system.^{[35]}
As a description of the structure of space
Euclid believed that his
As discussed above, Albert Einstein's theory of relativity 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 infinite^{[39]} (see below) and what its topology is. Modern, more rigorous reformulations of the system^{[40]} 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), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a torus for twodimensional Euclidean geometry).
Treatment of infinity
Infinite objects
Euclid sometimes distinguished explicitly between "finite lines" (e.g., Postulate 2) and "infinite 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.^{[39]}
The notion of
Later ancient commentators, such as Proclus (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.^{[42]}
At the turn of the 20th century,
Infinite processes
Ancient geometers may have considered the parallel postulate – that two parallel lines do not ever intersect – less certain than the others because it makes a statement about infinitely remote regions of space, and so cannot be physically verified.^{}[45]
The modern formulation of
Supposed paradoxes involving infinite series, such as
Logical basis
This section needs expansion. You can help by adding to it. (June 2010) 
Classical logic
Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, 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.^{[47]} The role of primitive notions, or undefined concepts, was clearly put forward by Alessandro Padoa of the Peano delegation at the 1900 Paris conference:^{[47]}^{[48]}
...when we begin to formulate the theory, we can imagine that the undefined symbols are completely devoid of meaning and that the unproved propositions are simply conditions imposed upon the undefined symbols.
Then, the system of ideas that we have initially chosen is simply one interpretation of the undefined symbols; but..this interpretation can be ignored by the reader, who is free to replace it in his mind by another interpretation.. that satisfies the conditions...
Logical questions thus become completely independent of empirical or psychological questions...
The system of undefined symbols can then be regarded as the abstraction obtained from the specialized theories that result when...the system of undefined symbols is successively replaced by each of the interpretations...
— Padoa, Essai d'une théorie algébrique des nombre entiers, avec une Introduction logique à une théorie déductive quelconque
That is, mathematics is contextindependent knowledge within a hierarchical framework. As said by Bertrand Russell:^{[49]}
If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
— Bertrand Russell, Mathematics and the metaphysicians
Such foundational approaches range between
Axiomatic formulations
Geometry is the science of correct reasoning on incorrect figures.
— George Pólya, How to Solve It, p. 208
 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.^{[50]} 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 was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or nonEuclidean.
 Hilbert's axioms: 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.
 real numbers.^{[51]}^{[52]}^{[53]} The notions of angle and distance become primitive concepts.^{[54]}
 real closed fields, of which elementary Euclidean geometry is a model.
See also
 Absolute geometry
 Analytic geometry
 Birkhoff's axioms
 Cartesian coordinate system
 Hilbert's axioms
 Incidence geometry
 List of interactive geometry software
 Metric space
 NonEuclidean geometry
 Ordered geometry
 Parallel postulate
 Type theory
Classical theorems
 Angle bisector theorem
 Butterfly theorem
 Ceva's theorem
 Heron's formula
 Menelaus' theorem
 Ninepoint circle
 Pythagorean theorem
Notes
 ^ ^{a} ^{b} Eves 1963, p. 19.
 ^ Eves 1963, p. 10.
 ^ Misner, Thorne, and Wheeler (1973), p. 47.
 ^ The assumptions of Euclid are discussed from a modern perspective in
Harold E. Wolfe (2007). Introduction to NonEuclidean Geometry. Mill Press. p. 9. ISBN 9781406718522.
 ^ tr. Heath, pp. 195–202.
 ISBN 9780131437005.
 .
 ^ Ball, p. 56.
 ^ 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 and Banach–Tarski paradox.
 ^ Daniel Shanks (2002). Solved and Unsolved Problems in Number Theory. American Mathematical Society.
 ^ Euclid, book I, proposition 5, tr. Heath, p. 251.
 Sir Thomas L. Heathmentions another interpretation. This rests on the resemblance of the figure's lower straight lines to a steeply inclined bridge that could be crossed by an ass but not by a horse: "But there is another view (as I have learnt lately) which is more complimentary to the ass. It is that, the figure of the proposition being like that of a trestle bridge, with a ramp at each end which is more practicable the flatter the figure is drawn, the bridge is such that, while a horse could not surmount the ramp, an ass could; in other words, the term is meant to refer to the surefootedness of the ass rather than to any want of intelligence on his part." (in "Excursis II", volume 1 of Heath's translation of The Thirteen Books of the Elements).
 ^ Euclid, book I, proposition 32.
 ^ Heath, p. 135. Extract of page 135.
 ^ Heath, p. 318.
 ^ Euclid, book XII, proposition 2.
 ^ Euclid, book XI, proposition 33.
 ^ Ball, p. 66.
 ^ Ball, p. 5.
 ^ Eves, vol. 1, p. 5; Mlodinow, p. 7.
 ^ Tom Hull. "Origami and Geometric Constructions". Archived from the original on 20190618. Retrieved 20131229.
 ^ Eves, p. 27.
 ^ Ball, pp. 268ff.
 ^ Eves (1963).
 ^ Hofstadter 1979, p. 91.
 ISBN 0486647250.
 ^ Eves (1963), p. 64.
 polyschemesin 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in threedimensional space. Although he described a quaternion as an ordered fourelement multiple of real numbers, the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.
 ^ PerezGracia & Thomas 2017; "It is actually Cayley whom we must thank for the correct development of quaternions as a representation of rotations."
 ^ Ball, p. 485.
 ^ * Howard Eves, 1997 (1958). Foundations and Fundamental Concepts of Mathematics. Dover.
 ^ Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)", Annals of Mathematics 33.
 ^ ^{a} ^{b} Tarski (1951).
 ^ Misner, Thorne, and Wheeler (1973), p. 191.
 ^ Rizos, Chris. University of New South Wales. GPS Satellite Signals Archived 20100612 at the Wayback Machine. 1999.
 ISBN 9780817647827.
 ^
See, for example: Luciano da Fontoura Costa; Roberto Marcondes Cesar (2001). Shape analysis and classification: theory and practice. CRC Press. p. 314. ISBN 0486434818.
 .
 ^ ^{a} ^{b} Heath, p. 200.
 ^ e.g., Tarski (1951).
 ^ Ball, p. 31.
 ^ Heath, p. 268.
 ^ Giuseppe Veronese, On NonArchimedean Geometry, 1908. English translation in Real Numbers, Generalizations of the Reals, and Theories of Continua, ed. Philip Ehrlich, Kluwer, 1994.
 ^ Robinson, Abraham (1966). Nonstandard analysis.
 ^ Nagel and Newman, 1958, p. 9.
 ^ Cajori (1918), p. 197.
 ^ ISBN 0471251836.
 ^ Société française de philosophie (1900). Revue de métaphysique et de morale, Volume 8. Hachette. p. 592.
 ISBN 0486411516.
 ^ Bertrand Russell (1897). "Introduction". An essay on the foundations of geometry. Cambridge University Press.
 ISBN 0821821016.
 .
 .
 .
 .
We regard as elementary that part of Euclidean geometry which can be formulated and established without the help of any settheoretical devices
 .
 . Pp. 25–26.
References
 .
 Coxeter, H. S. M.(1961). Introduction to Geometry. New York: Wiley.
 Eves, Howard (1963). A Survey of Geometry (Volume One). Allyn and Bacon.
 . Heath's authoritative translation of Euclid's Elements, plus his extensive historical research and detailed commentary throughout the text.
 . W. H. Freeman.
 Mlodinow (2001). Euclid's Window. The Free Press. ISBN 9780684865232.
 Nagel, E.; Newman, J. R. (1958). Gödel's Proof. New York University Press.
 Tarski, Alfred (1951). A Decision Method for Elementary Algebra and Geometry. Univ. of California Press.
 Stillwell, John (January 2001). "The Story of the 120Cell" (PDF). Notices of the AMS. 48 (1): 17–25.
 PerezGracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. S2CID 12350382.
External links
 "Euclidean geometry", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 "Plane trigonometry", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Kiran Kedlaya, Geometry Unbound Archived 20111026 at the Wayback Machine (a treatment using analytic geometry; PDF format, GFDL licensed)