Euclidean geometry is a mathematical system attributed toEuclid, anancient Greek mathematician, which he described in his textbook ongeometry,Elements. Euclid's approach consists in assuming a small set of intuitively appealingaxioms (postulates) and deducing many otherpropositions (theorems) from these. One of those is theparallel postulate which relates toparallel lines on aEuclidean plane. Although many of Euclid's results had been stated earlier,[1] Euclid was the first to organize these propositions into alogical system in which each result isproved from axioms and previously proved theorems.[2]
For more than two thousand years, the adjective "Euclidean" was unnecessary becauseEuclid's axioms seemed so intuitively obvious (with the possible exception of theparallel postulate) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many otherself-consistentnon-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication ofAlbert Einstein's theory ofgeneral relativity is that physical space itself is not Euclidean, andEuclidean space is a good approximation for it only over short distances (relative to the strength of thegravitational field).[3]
Euclidean geometry is an example ofsynthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This is in contrast toanalytic geometry, introduced almost 2,000 years later byRené Descartes, which usescoordinates to express geometric properties by means ofalgebraic formulas.
TheElements 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 theElements:
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 thePythagorean theorem "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 withnumber theory, with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such asprime numbers andrational andirrational numbers are introduced. It is proved that there are infinitely many prime numbers.
Books XI–XIII concernsolid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. Theplatonic solids are constructed.
The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Euclidean geometry is anaxiomatic system, in which alltheorems ("true statements") are derived from a small number of simple axioms. Until the advent ofnon-Euclidean 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 theElements, Euclid gives fivepostulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):[5]
[Theparallel postulate]: That, if a straight line falling on two straight lines makes 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.
TheElements also include the following five "common notions":
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] Moderntreatments use more extensive and complete sets of axioms.
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 theElements: his first 28 propositions are those that can be proved without it.
Many alternative axioms can be formulated which arelogically equivalent to the parallel postulate (in the context of the other axioms). For example,Playfair's axiom states:
In aplane, 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.
A proof from Euclid'sElements that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.
Euclidean Geometry isconstructive. 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 acompass and an unmarked straightedge.[8] In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such asset theory, 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.[9] Strictly speaking, the lines on paper aremodels 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 have. Though nearly all modern mathematicians considernonconstructive proofs just as sound as constructive ones, they are often considered lesselegant, intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring a statement such as "Find the greatest common measure of ..."[10]
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.
Angles whose sum is a right angle are calledcomplementary. 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 aresupplementary. 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.
In modern terminology, angles would normally be measured indegrees orradians.
Modern school textbooks often define separate figures calledlines (infinite),rays (semi-infinite), andline segments (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" for Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary.
Thepons asinorum orbridge of asses theorem states that in an isosceles triangle, α = β and γ = δ.
Thetriangle 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.
ThePythagorean theorem states that the sum of the areas of the two squares on the legs (a andb) of a right triangle equals the area of the square on the hypotenuse (c).
Thales' theorem states that if AC is a diameter, then the angle at B is a right angle.
Thepons asinorum (bridge of asses) states thatin 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.[12] Its name may be attributed to its frequent role as the first real test in theElements 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.[13]
Congruence of triangles is determined by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless the angle specified is a right angle.
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.
The sum of the angles of a triangle is equal to a straight angle (180 degrees).[14] 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 oneobtuse orright angle.
The celebratedPythagorean 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).
An example of congruence. The two figures on the left are congruent, while the third issimilar to them. The last figure is neither. Congruences alter some properties, such as location and orientation, but leave others unchanged, likedistance andangles. The latter sort of properties are calledinvariants and studying them is the essence of geometry.
Thales' theorem, named afterThales of Miletus 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.[15][16]
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[17] and the volume of a parallelepipedal solid.[18] Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successorArchimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder.[19]
Euclidean geometry has two fundamental types of measurements:angle anddistance. The angle scale is absolute, and Euclid uses theright angle as his basic unit, so that, for example, a 45-degree 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 ofarea andvolume are derived from distances. For example, arectangle 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" 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 assimilar.Corresponding angles in a pair of similar shapes are equal andcorresponding sides are in proportion to each other.
Lens Design:Lens - In optical engineering, Euclidean geometry is critical in the design of lenses, where precise geometric shapes determine thefocusing properties.Geometric optics analyzes the focusing oflight bylenses andmirrors.
Vibration Analysis:Vibration - Euclidean geometry is essential in analyzing and understanding thevibrations inmechanical systems, aiding in the design of systems that can withstand or utilize thesevibrations effectively.
3D Modeling: InCAD (computer-aided design) systems, Euclidean geometry is fundamental for creating accurate 3D models of mechanical parts. These models are crucial for visualizing and testing designs beforemanufacturing.
Design and Manufacturing: Much ofCAM (computer-aided manufacturing) relies on Euclidean geometry. The design geometry in CAD/CAM typically consists of shapes bounded byplanes,cylinders,cones,tori, and other similar Euclidean forms. Today, CAD/CAM is essential in the design of a wide range of products, fromcars andairplanes toships andsmartphones.
Evolution of Drafting Practices: Historically, advanced Euclidean geometry, including theorems likePascal's theorem andBrianchon's theorem, was integral to drafting practices. However, with the advent of modern CAD systems, such in-depth knowledge of these theorems is less necessary in contemporary design and manufacturing processes.
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 issurveying.[20] In addition it has been used inclassical mechanics and thecognitive and computational approaches to visual perception of objects. 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.[21] 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 asGunter's chain, and angles using graduated circles and, later, thetheodolite.
A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.
René Descartes (1596–1650) developedanalytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra.[24]
In this approach, a point on a plane is represented by itsCartesian (x,y) coordinates, a line is represented by its equation, and so on.
In Euclid's original approach, thePythagorean 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 pointsP = (px,py) andQ = (qx,qy) is then known as theEuclideanmetric, and other metrics definenon-Euclidean geometries.
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 = 2x + 1 (a line), orx2 +y2 = 7 (a circle).
Also in the 17th century,Girard Desargues, motivated by the theory ofperspective, 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.[25]
Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealizedcompass and straightedge.
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.[26]
Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem oftrisecting 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, untilPierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible includedoubling the cube andsquaring the circle. 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,[27] while doubling a cube requires the solution of a third-order equation.
Euler discussed a generalization of Euclidean geometry calledaffine 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 anequivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).
Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions
In the early 19th century,Carnot andMöbius systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.[28]
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 andfully documented in 1948 byH.S.M. Coxeter.
In 1878William Kingdon Clifford introduced what is now termedgeometric algebra, unifying Hamilton's quaternions withHermann 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. TheClifford torus on the surface of the3-sphere 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").
The century's most influential development in geometry occurred when, around 1830,János Bolyai andNikolai Ivanovich Lobachevsky separately published work onnon-Euclidean geometry, in which the parallel postulate is not valid.[31] 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 theElements. 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 theElements, 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 thirdvertex. 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 thecompleteness property of the real numbers. Starting withMoritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those ofHilbert,[32]George Birkhoff,[33] andTarski.[34]
A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solareclipse. The rays of starlight were bent by the Sun's gravity on their way to Earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry.
However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case withgeneral relativity, for which the geometry of the space part of space-time is not Euclidean geometry.[35] 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 theGPS system.[36]
Euclid believed that hisaxioms were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms,[37] in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-calledEuclidean motions, which include translations, reflections and rotations of figures.[38] 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 isisotropic and figures may be moved to any location while maintainingcongruence; and postulate 5 (theparallel postulate) that space is flat (has nointrinsic curvature).[39]
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[40] (see below) and what itstopology is. Modern, more rigorous reformulations of the system[41] 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 inelliptic geometry), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or atorus for two-dimensional Euclidean geometry).
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.[40]
The notion ofinfinitesimal quantities had previously been discussed extensively by theEleatic School, but nobody had been able to put them on a firm logical basis, with paradoxes such asZeno's paradox occurring that had not been resolved to universal satisfaction. Euclid used themethod of exhaustion rather than infinitesimals.[42]
Later ancient commentators, such asProclus (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.[43]
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.[46]
The modern formulation ofproof by induction 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.[47]
Supposed paradoxes involving infinite series, such asZeno's paradox, predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of thegeometric series in IX.35 without commenting on the possibility of letting the number of terms become infinite.
Euclid frequently used the method ofproof by contradiction, and therefore the traditional presentation of Euclidean geometry assumesclassical 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.[48] The proof by contradiction (orreductio ad absurdum method) rests on two cardinal principles of classical logic: thelaw of contradiction and thelaw of the excluded middle. In simple terms, the law of contradiction saysthat if S is any statement, then S and a contradiction (that is, the denial)of S cannot both hold. And the law of the excluded middle states, thateither S or the denial of S must hold (that is, there is no third, or middle, possibility). This method therefore consists of assuming (by way of hypothesis) that a proposition that is to be established is false; if an absurdity follows, one concludes that the hypothesis is untenable and that the original proposition must then be true.[49]
Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries.[50] The role ofprimitive notions, or undefined concepts, was clearly put forward byAlessandro Padoa of thePeano delegation at the 1900 Paris conference:[50][51]
...when we begin to formulate the theory, we can imagine that the undefined symbols arecompletely devoid of meaning and that the unproved propositions are simplyconditions imposed upon the undefined symbols.
Then, thesystem of ideas that we have initially chosen is simplyone interpretation of the undefined symbols; but..this interpretation can be ignored by the reader, who is free to replace it in his mind byanother interpretation.. that satisfies the conditions...
Logical questions thus become completely independent ofempirical orpsychological questions...
The system of undefined symbols can then be regarded as theabstraction obtained from thespecialized 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 context-independent knowledge within a hierarchical framework. As said byBertrand Russell:[52]
If our hypothesis is aboutanything, 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
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.[53] It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that theparallel postulate was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean ornon-Euclidean.
Hilbert's axioms: Hilbert's axioms had the goal of identifying asimple andcomplete set ofindependent 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 thereal numbers.[54][55][56] The notions ofangle anddistance become primitive concepts.[57]
Tarski's axioms:Alfred Tarski (1902–1983) and his students definedelementary Euclidean geometry as the geometry that can be expressed infirst-order logic and does not depend onset theory for its logical basis,[58] in contrast to Hilbert's axioms, which involve point sets.[59] Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certainsense: there is an algorithm that, for every proposition, can be shown either true or false.[34] (This does not violateGödel's theorem, because Euclidean geometry cannot describe a sufficient amount ofarithmetic for the theorem to apply.[60]) This is equivalent to the decidability ofreal closed fields, of which elementary Euclidean geometry is a model.
^Venema, Gerard A. (2006),Foundations of Geometry, Prentice-Hall, p. 8,ISBN978-0-13-143700-5.
^Florence P. Lewis (Jan 1920), "History of the Parallel Postulate",The American Mathematical Monthly,27 (1), The American Mathematical Monthly, Vol. 27, No. 1:16–23,doi:10.2307/2973238,JSTOR2973238.
^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. SeeLebesgue measure andBanach–Tarski paradox.
^Daniel Shanks (2002).Solved and Unsolved Problems in Number Theory. American Mathematical Society.
^Mancosu, Paolo (1991). "On the Status of Proofs by Contradiction in the Seventeenth Century".Synthese.88 (1):15–41.doi:10.1007/BF00540091.JSTOR20116923.
^Euclid, book I, proposition 5, tr. Heath, p. 251.
^Ignoring the alleged difficulty of Book I, Proposition 5,Sir Thomas L. Heath mentions 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 sure-footedness of the ass rather than to any want of intelligence on his part." (in "Excursis II", volume 1 of Heath's translation ofThe Thirteen Books of the Elements).
^Stillwell 2001, p. 18–21; In four-dimensional Euclidean geometry, aquaternion is simply a (w, x, y, z) Cartesian coordinate.Hamilton did not see them as such when hediscovered the quaternions.Schläfli would be the first to considerfour-dimensional Euclidean space, publishing his discovery of the regularpolyschemes in 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 three-dimensional space. Although he described a quaternion as anordered four-element multiple of real numbers, the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.
^Perez-Gracia & Thomas 2017; "It is actually Cayley whom we must thank for the correct development of quaternions as a representation of rotations."
^Giuseppe Veronese, On Non-Archimedean Geometry, 1908. English translation in Real Numbers, Generalizations of the Reals, and Theories of Continua, ed. Philip Ehrlich, Kluwer, 1994.
^Matthews, Bennie (2019).Statics and Analytical Geometry (1st ed.). Edtech (published June 21, 2019). p. 27.ISBN9781839473333.
^Eves, Howard Whitley (1997).Foundations and fundamental concepts of mathematics. Dover books on mathematics (3rd ed.). Mineola, NY: Dover Publications. p. 55.ISBN978-0-486-69609-6.
^Bertrand Russell (2000)."Mathematics and the metaphysicians". In James Roy Newman (ed.).The world of mathematics. Vol. 3 (Reprint of Simon and Schuster 1956 ed.). Courier Dover Publications. p. 1577.ISBN0-486-41151-6.
^Alfred Tarski (2007)."What is elementary geometry". In Leon Henkin; Patrick Suppes; Alfred Tarski (eds.).Studies in Logic and the Foundations of Mathematics – The Axiomatic Method with Special Reference to Geometry and Physics (Proceedings of International Symposium at Berkeley 1957–8; Reprint ed.). Brouwer Press. p. 16.ISBN978-1-4067-5355-4.We regard as elementary that part of Euclidean geometry which can be formulated and established without the help of any set-theoretical devices
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