Polygon
In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".
It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.
The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.
This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1
Parallel (geometry)
In geometry, parallel lines are lines in a plane which do not meet. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same three-dimensional space. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism; the parallel symbol is ∥. For example, A B ∥ C D indicates that line AB is parallel to line CD. In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation "equal and parallel to". Given parallel straight lines l and m in Euclidean space, the following properties are equivalent: Every point on line m is located at the same distance from line l.
Line m is in the same plane as line l but does not intersect l. When lines m and l are both intersected by a third straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, so, are "more complicated" than the second. Thus, the second property is the one chosen as the defining property of parallel lines in Euclidean geometry; the other properties are consequences of Euclid's Parallel Postulate. Another property that involves measurement is that lines parallel to each other have the same gradient; the definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein.
Simplicius mentions Posidonius' definition as well as its modification by the philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools; the traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines; these reform texts were not without their critics and one of them, Charles Dodgson, wrote a play and His Modern Rivals, in which these texts are lambasted. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing the idea may be traced back to Leibniz. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, the difference of their directions is the angle between them."
Wilson In definition 15 he introduces parallel lines in this way. Wilson Augustus De Morgan reviewed this text and declared it a failure on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson devotes a large section of his play to denouncing Wilson's treatment of parallels. Wilson edited this concept out of the third and higher editions of his text. Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better; the main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line; this must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles all transversals must do so.
Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of parallel lines; because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines, y = m x + b 1 y = m x + b 2
Angle
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane. Angles are formed by the intersection of two planes in Euclidean and other spaces; these are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is used to designate the measure of an angle or of a rotation; this measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation; the word angle comes from the Latin word angulus, meaning "corner".
Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow". Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, do not lie straight with respect to each other. According to Proclus an angle must be a relationship; the first concept was used by Eudemus. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower case Roman letters are used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted ∠BAC or B A C ^. Sometimes, where there is no risk of confusion, the angle may be referred to by its vertex. An angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign.
However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise angle from B to C, ∠CAB to the anticlockwise angle from C to B. An angle equal to 0° or not turned is called a zero angle. Angles smaller than a right angle are called acute angles. An angle equal to 1/4 turn is called a right angle. Two lines that form a right angle are said to be orthogonal, or perpendicular. Angles larger than a right angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle. Angles larger than a straight angle but less than 1 turn are called reflex angles. An angle equal to 1 turn is called complete angle, round angle or a perigon. Angles that are not right angles or a multiple of a right angle are called oblique angles; the names and measured units are shown in a table below: Angles that have the same measure are said to be equal or congruent.
An angle is not dependent upon the lengths of the sides of the angle. Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. A reference angle is the acute version of any angle determined by subtracting or adding straight angle, to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, an angle of 150 degrees has a reference angle of 30 degrees. An angle of 750 degrees has a reference angle of 30 degrees; when two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other. A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles, they are abbreviated as vert. opp. ∠s. The equality of vertically opposite angles is called the vertical angle theorem.
Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note, w
Straightedge and compass construction
Straightedge and compass construction known as ruler-and-compass construction or classical construction, is the construction of lengths and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, no markings on it; the compass is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates, it turns out to be the case that every point constructible using straightedge and compass may be constructed using compass alone. The ancient Greek mathematicians first conceived straightedge and compass constructions, a number of ancient problems in plane geometry impose this restriction; the ancient Greeks developed many constructions. Gauss showed that most are not; some of the most famous straightedge and compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields.
In spite of existing proofs of impossibility, some persist in trying to solve these problems. Many of these problems are solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In terms of algebra, a length is constructible if and only if it represents a constructible number, an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots; the "straightedge" and "compass" of straightedge and compass constructions are idealizations of rulers and compasses in the real world: The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to extend an existing segment; the compass can be opened arbitrarily wide.
Circles can only be drawn starting from two given points: a point on the circle. The compass may not collapse when it is not drawing a circle. Actual compasses do not collapse and modern geometric constructions use this feature. A'collapsing compass' would appear to be a less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a checkered history; each construction must be exact. "Eyeballing" it and getting close does not count as a solution. Each construction must terminate; that is, it must have a finite number of steps, not be the limit of closer approximations. Stated this way and compass constructions appear to be a parlour game, rather than a serious practical problem; the ancient Greek mathematicians first attempted straightedge and compass constructions, they discovered how to construct sums, products and square roots of given lengths.
They could construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, a regular polygon with 3, 4, or 5 sides. But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides. Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed.
In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He showed that Gauss's sufficient constructibility condition for regular polygons is necessary. In 1882 Lindemann showed that π is a transcendental number, thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle. All straightedge and compass constructions consist of repeated application of five basic constructions using the points and circles that have been constructed; these are: Creating the line through two existing points Creating the circle through one point with centre another point Creating the point, th
Plane (geometry)
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry; when working in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, trigonometry, graph theory, graphing are performed in a two-dimensional space, or, in other words, in the plane. Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry, he selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.
Euclid never used numbers to measure angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane. A plane is a ruled surface; this section is concerned with planes embedded in three dimensions: in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines; the following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues: Two distinct planes are either parallel or they intersect in a line. A line intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other. Two distinct planes perpendicular to the same line must be parallel to each other. In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it to indicate its "inclination".
Let r0 be the position vector of some point P0 =, let n = be a nonzero vector. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that n ⋅ = 0. Expanded this becomes a + b + c = 0, the point-normal form of the equation of a plane; this is just a linear equation a x + b y + c z + d = 0, where d = −. Conversely, it is shown that if a, b, c and d are constants and a, b, c are not all zero the graph of the equation a x + b y + c z + d = 0, is a plane having the vector n = as a normal; this familiar equation for a plane is called the general form of the equation of the plane. Thus for example a regression equation of the form y = d + ax + cz establishes a best-fit plane in three-dimensional space when there are two explanatory variables.
Alternatively, a plane may be described parametrically as the set of all points of the form r = r 0 + s v + t w, where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, r0 is the vector representing the position of an arbitrary point on the plane. The vectors v and w can be visualized as vectors starting at r0 and pointing in different directions along the plane. Note that v and w can be perpendicular, but cannot be parallel. Let p1=, p2=, p3= be non-collinear points; the plane passing through p1, p2, p3 can be described as the set of all points that satisfy the following determinant equations: | x − x 1 y − y 1 z − z 1 x 2 − x 1 y 2 − y
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More one can be obtained from the other by uniformly scaling with additional translation and reflection; this means that either object can be rescaled and reflected, so as to coincide with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. For example, all circles are similar to each other, all squares are similar to each other, all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, isosceles triangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle the triangles are similar.
Corresponding sides of similar polygons are in proportion, corresponding angles of similar polygons have the same measure. This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are similar, but some school textbooks exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar. In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional, it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of, necessary and sufficient for two triangles to be similar: The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:If ∠BAC is equal in measure to ∠B′A′C′, ∠ABC is equal in measure to ∠A′B′C′ this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar. All the corresponding sides have lengths in the same ratio:AB/A′B′ = BC/B′C′ = AC/A′C′; this is equivalent to saying. Two sides have lengths in the same ratio, the angles included between these sides have the same measure. For instance:AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′; this is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; when two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′. There are several elementary results concerning similar triangles in Euclidean geometry: Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other.
Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if one other side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, find a point F such that △ABC ∼ △DEF; the statement that the point F satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by G. D. Birkhoff the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms. Similar triangles provide the basis for many synthetic proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles provide the foundations for right triangle trigonometry.
The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles. Equality of all angles in sequence is not sufficient to guarantee similarity. A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given n, all regular n-gons are similar. Several types of curves have the property; these include: Circles Parabolas Hyperbolas of a specific eccentricity Ellipses of a specific eccentricity Catenaries Graphs of the logarithm function for different bases Graphs of the exponential function for different bases Logarithmic spirals are self-similar A similarity of a Euclidean space is a bijection f from the space onto itself that multiplies all distances
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp