In mathematics , a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
82-418: One description of a parabola involves a point (the focus ) and a line (the directrix ). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section , created from the intersection of a right circular conical surface and a plane parallel to another plane that
164-405: A c = 0 , {\displaystyle b^{2}-4ac=0,} or, equivalently, such that a x 2 + b x y + c y 2 {\displaystyle ax^{2}+bxy+cy^{2}} is the square of a linear polynomial . The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of
246-539: A set of points; As an example, a line is an infinite set of points of the form L = { ( a 1 , a 2 , . . . a n ) ∣ a 1 c 1 + a 2 c 2 + . . . a n c n = d } , {\displaystyle L=\lbrace (a_{1},a_{2},...a_{n})\mid a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\rbrace ,} where c 1 through c n and d are constants and n
328-451: A connection with this curve, as Apollonius had proved. The focus–directrix property of the parabola and other conic sections was mentioned in the works of Pappus . Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope . Designs were proposed in
410-692: A definitive three-volume presentation of Commandino's translation with both the Greek and Latin versions (Berlin, 1875–1878). Using Hultsch's work, the Belgian mathematical historian Paul ver Eecke was the first to publish a translation of the Collection into a modern European language; his two-volume, French translation has the title Pappus d'Alexandrie. La Collection Mathématique. (Paris and Bruges, 1933). The characteristics of Pappus's Collection are that it contains an account, systematically arranged, of
492-405: A function f ( x ) = a x 2 with a ≠ 0. {\displaystyle f(x)=ax^{2}{\text{ with }}a\neq 0.} For a > 0 {\displaystyle a>0} the parabolas are opening to the top, and for a < 0 {\displaystyle a<0} are opening to the bottom (see picture). From
574-573: A graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape. An alternative proof can be done using Dandelin spheres . It works without calculation and uses elementary geometric considerations only (see the derivation below). Point (geometry) In geometry , a point is an abstract idealization of an exact position , without size, in physical space , or its generalization to other kinds of mathematical spaces . As zero- dimensional objects, points are usually taken to be
656-487: A parabola can then be transformed by the uniform scaling ( x , y ) → ( a x , a y ) {\displaystyle (x,y)\to (ax,ay)} into the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . Thus, any parabola can be mapped to the unit parabola by a similarity. A synthetic approach, using similar triangles, can also be used to establish this result. The general result
738-420: A parabola is the inverse of a cardioid . Remark 2: The second polar form is a special case of a pencil of conics with focus F = ( 0 , 0 ) {\displaystyle F=(0,0)} (see picture): r = p 1 − e cos φ {\displaystyle r={\frac {p}{1-e\cos \varphi }}} ( e {\displaystyle e}
820-399: A point is represented by an ordered pair ( x , y ) of numbers, where the first number conventionally represents the horizontal and is often denoted by x , and the second number conventionally represents the vertical and is often denoted by y . This idea is easily generalized to three-dimensional Euclidean space , where a point is represented by an ordered triplet ( x , y , z ) with
902-440: A quadrature of a curved surface. The rest of the book treats of the trisection of an angle , and the solution of more general problems of the same kind by means of the quadratrix and spiral. In one solution of the former problem is the first recorded use of the property of a conic (a hyperbola) with reference to the focus and directrix. In Book V, after an interesting preface concerning regular polygons, and containing remarks upon
SECTION 10
#1732766172634984-464: A remarkable exception. In many respects, his fate strikingly resembles that of Diophantus' , originally of limited importance but becoming very influential in the late Renaissance and Early Modern periods. In his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time (but see below) when he himself wrote. If no other date information were available, all that could be known would be that he
1066-645: A set of points ( locus of points ) in the Euclidean plane: The midpoint V {\displaystyle V} of the perpendicular from the focus F {\displaystyle F} onto the directrix l {\displaystyle l} is called the vertex , and the line F V {\displaystyle FV} is the axis of symmetry of the parabola. If one introduces Cartesian coordinates , such that F = ( 0 , f ) , f > 0 , {\displaystyle F=(0,f),\ f>0,} and
1148-542: A simple construction for the axes of an ellipse when a pair of conjugate diameters are given. Pappus's Collection was virtually unknown to the Arabs and medieval Europeans, but exerted great influence on 17th-century mathematics after being translated to Latin by Federico Commandino . Diophantus's Arithmetica and Pappus's Collection were the two major sources of Viète 's Isagoge in artem analyticam (1591). The Pappus's problem and its generalization led Descartes to
1230-418: A space of points is typically treated as a set , a point set . An isolated point is an element of some subset of points which has some neighborhood containing no other points of the subset. Points, considered within the framework of Euclidean geometry , are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In the two-dimensional Euclidean plane ,
1312-504: A topological space X {\displaystyle X} is defined to be the minimum value of n , such that every finite open cover A {\displaystyle {\mathcal {A}}} of X {\displaystyle X} admits a finite open cover B {\displaystyle {\mathcal {B}}} of X {\displaystyle X} which refines A {\displaystyle {\mathcal {A}}} in which no point
1394-524: A way that the operation "take a value at this point" may not be defined. A further tradition starts from some books of A. N. Whitehead in which the notion of region is assumed as a primitive together with the one of inclusion or connection . Often in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especially common in classical electromagnetism , where electrons are idealized as points with non-zero charge). The Dirac delta function , or δ function ,
1476-605: A way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction .) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola . The name "parabola" is due to Apollonius , who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has
1558-681: Is V = ( 0 , 0 ) {\displaystyle V=(0,0)} , and its focus is F = ( p 2 , 0 ) {\displaystyle F=\left({\tfrac {p}{2}},0\right)} . If one shifts the origin into the focus, that is, F = ( 0 , 0 ) {\displaystyle F=(0,0)} , one obtains the equation r = p 1 − cos φ , φ ≠ 2 π k . {\displaystyle r={\frac {p}{1-\cos \varphi }},\quad \varphi \neq 2\pi k.} Remark 1: Inverting this polar form shows that
1640-420: Is tangential to the conical surface. The graph of a quadratic function y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} (with a ≠ 0 {\displaystyle a\neq 0} ) is a parabola with its axis parallel to the y -axis. Conversely, every such parabola is the graph of a quadratic function. The line perpendicular to
1722-430: Is (informally) a generalized function on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line. The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized point mass or point charge . It was introduced by theoretical physicist Paul Dirac . In
SECTION 20
#17327661726341804-446: Is U-shaped ( opening to the top ). The horizontal chord through the focus (see picture in opening section) is called the latus rectum ; one half of it is the semi-latus rectum . The latus rectum is parallel to the directrix. The semi-latus rectum is designated by the letter p {\displaystyle p} . From the picture one obtains p = 2 f . {\displaystyle p=2f.} The latus rectum
1886-404: Is defined by dim H ( X ) := inf { d ≥ 0 : C H d ( X ) = 0 } . {\displaystyle \operatorname {dim} _{\operatorname {H} }(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.} A point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although
1968-420: Is defined similarly for the other two conics – the ellipse and the hyperbola. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, p {\displaystyle p} is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, p {\displaystyle p} ,
2050-456: Is exactly one straight line that passes through two distinct points" . As physical diagrams, geometric figures are made with tools such as a compass , scriber , or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve. Since the advent of analytic geometry , points are often defined or represented in terms of numerical coordinates . In modern mathematics,
2132-398: Is included in more than n +1 elements. If no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set. Let X be a metric space . If S ⊂ X and d ∈ [0, ∞) , the d -dimensional Hausdorff content of S
2214-1063: Is not mentioned above. It is defined and discussed below, in § Position of the focus . Let us call the length of DM and of EM x , and the length of PM y . The lengths of BM and CM are: Using the intersecting chords theorem on the chords BC and DE , we get B M ¯ ⋅ C M ¯ = D M ¯ ⋅ E M ¯ . {\displaystyle {\overline {\mathrm {BM} }}\cdot {\overline {\mathrm {CM} }}={\overline {\mathrm {DM} }}\cdot {\overline {\mathrm {EM} }}.} Substituting: 4 r y cos θ = x 2 . {\displaystyle 4ry\cos \theta =x^{2}.} Rearranging: y = x 2 4 r cos θ . {\displaystyle y={\frac {x^{2}}{4r\cos \theta }}.} For any given cone and parabola, r and θ are constants, but x and y are variables that depend on
2296-435: Is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. Therefore, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas do not. There are other simple affine transformations that map the parabola y = a x 2 {\displaystyle y=ax^{2}} onto
2378-435: Is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar . Parabolas have the property that, if they are made of material that reflects light , then light that travels parallel to
2460-652: Is the infimum of the set of numbers δ ≥ 0 such that there is some (indexed) collection of balls { B ( x i , r i ) : i ∈ I } {\displaystyle \{B(x_{i},r_{i}):i\in I\}} covering S with r i > 0 for each i ∈ I that satisfies ∑ i ∈ I r i d < δ . {\displaystyle \sum _{i\in I}r_{i}^{d}<;\delta .} The Hausdorff dimension of X
2542-401: Is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles . It is frequently used in physics , engineering , and many other areas. The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered
Parabola - Misplaced Pages Continue
2624-521: Is the dimension of the space. Similar constructions exist that define the plane , line segment , and other related concepts. A line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, that any two points can be connected by a straight line. This is easily confirmed under modern extensions of Euclidean geometry, and had lasting consequences at its introduction, allowing
2706-408: Is the distance of the focus from the directrix. Using the parameter p {\displaystyle p} , the equation of the parabola can be rewritten as x 2 = 2 p y . {\displaystyle x^{2}=2py.} More generally, if the vertex is V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} ,
2788-400: Is the eccentricity). The diagram represents a cone with its axis AV . The point A is its apex . An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle θ , as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola. A cross-section perpendicular to the axis of
2870-620: The Sphaerica of Theodosius , the Moving Sphere of Autolycus , Theodosius's book on Day and Night , the treatise of Aristarchus On the Size and Distances of the Sun and Moon , and Euclid's Optics and Phaenomena . Since Michel Chasles cited this book of Pappus in his history of geometric methods, it has become the object of considerable attention. The preface of Book VII explains
2952-871: The eccentricity . If p > 0 , the parabola with equation y 2 = 2 p x {\displaystyle y^{2}=2px} (opening to the right) has the polar representation r = 2 p cos φ sin 2 φ , φ ∈ [ − π 2 , π 2 ] ∖ { 0 } {\displaystyle r=2p{\frac {\cos \varphi }{\sin ^{2}\varphi }},\quad \varphi \in \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}} where r 2 = x 2 + y 2 , x = r cos φ {\displaystyle r^{2}=x^{2}+y^{2},\ x=r\cos \varphi } . Its vertex
3034-400: The hexagonal form of the cells of honeycombs , Pappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter (following Zenodorus 's treatise on this subject), and of the volumes of different solid figures which have all the same superficial area, and, lastly, a comparison of the five regular solids of Plato . Incidentally Pappus describes
3116-575: The Ἀνάλημμα ( Analemma ) of Diodorus of Alexandria . Pappus also wrote commentaries on Euclid 's Elements (of which fragments are preserved in Proclus and the Scholia , while that on the tenth Book has been found in an Arabic manuscript), and on Ptolemy's Ἁρμονικά ( Harmonika ). Federico Commandino translated the Collection of Pappus into Latin in 1588. The German classicist and mathematical historian Friedrich Hultsch (1833–1908) published
3198-415: The additional third number representing depth and often denoted by z . Further generalizations are represented by an ordered tuplet of n terms, ( a 1 , a 2 , … , a n ) where n is the dimension of the space in which the point is located. Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by
3280-407: The apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function in the equation. It is proved in a preceding section that if a parabola has its vertex at the origin, and if it opens in
3362-449: The arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as Cartesian coordinates of the points D and E, in a system in the pink plane with P as its origin. Since x is squared in the equation, the fact that D and E are on opposite sides of the y axis is unimportant. If the horizontal cross-section moves up or down, toward or away from
Parabola - Misplaced Pages Continue
3444-403: The axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel (" collimated ") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves . This reflective property
3526-527: The common definitions, a point is 0-dimensional. The dimension of a vector space is the maximum size of a linearly independent subset. In a vector space consisting of a single point (which must be the zero vector 0 ), there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non-trivial linear combination making it zero: 1 ⋅ 0 = 0 {\displaystyle 1\cdot \mathbf {0} =\mathbf {0} } . The topological dimension of
3608-399: The cone passes through the vertex P of the parabola. This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. Its centre is V, and PK is a diameter. We will call its radius r . Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described. It has a chord DE , which joins
3690-410: The construction of a curve of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time. The area of the surface included between this curve and its base is found – the first known instance of
3772-494: The construction of almost all the geometric concepts known at the time. However, Euclid's postulation of points was neither complete nor definitive, and he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics. In all of
3854-478: The context of signal processing it is often referred to as the unit impulse symbol (or function). Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1. Pappus of Alexandria Pappus of Alexandria ( / ˈ p æ p ə s / ; ‹See Tfd› Greek : Πάππος ὁ Ἀλεξανδρεύς ; c. 290 – c. 350 AD)
3936-415: The directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The " latus rectum "
4018-513: The directrix has the equation y = − 1 4 {\displaystyle y=-{\tfrac {1}{4}}} . The general function of degree 2 is f ( x ) = a x 2 + b x + c with a , b , c ∈ R , a ≠ 0. {\displaystyle f(x)=ax^{2}+bx+c~~{\text{ with }}~~a,b,c\in \mathbb {R} ,\ a\neq 0.} Completing
4100-752: The directrix has the equation y = − f {\displaystyle y=-f} , one obtains for a point P = ( x , y ) {\displaystyle P=(x,y)} from | P F | 2 = | P l | 2 {\displaystyle |PF|^{2}=|Pl|^{2}} the equation x 2 + ( y − f ) 2 = ( y + f ) 2 {\displaystyle x^{2}+(y-f)^{2}=(y+f)^{2}} . Solving for y {\displaystyle y} yields y = 1 4 f x 2 . {\displaystyle y={\frac {1}{4f}}x^{2}.} This parabola
4182-515: The early to mid-17th century by many mathematicians , including René Descartes , Marin Mersenne , and James Gregory . When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror . Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as
SECTION 50
#17327661726344264-665: The equation uses the Hesse normal form of a line to calculate the distance | P l | {\displaystyle |Pl|} ). For a parametric equation of a parabola in general position see § As the affine image of the unit parabola . The implicit equation of a parabola is defined by an irreducible polynomial of degree two: a x 2 + b x y + c y 2 + d x + e y + f = 0 , {\displaystyle ax^{2}+bxy+cy^{2}+dx+ey+f=0,} such that b 2 − 4
4346-509: The figure made of three semicircles and known as arbelos ("shoemakers knife") form the first division of the book; Pappus turns then to a consideration of certain properties of Archimedes's spiral , the conchoid of Nicomedes (already mentioned in Book I as supplying a method of doubling the cube), and the curve discovered most probably by Hippias of Elis about 420 BC, and known by the name, τετραγωνισμός, or quadratrix . Proposition 30 describes
4428-547: The first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus: Χωρογραφία οἰκουμενική ( Chorographia oikoumenike or Description of the Inhabited World ), commentary on the four books of Ptolemy 's Almagest , Ποταμοὺς τοὺς ἐν Λιβύῃ ( The Rivers in Libya ), and Ὀνειροκριτικά ( The Interpretation of Dreams ). Pappus himself mentions another commentary of his own on
4510-752: The focus F = ( v 1 , v 2 + f ) {\displaystyle F=(v_{1},v_{2}+f)} , and the directrix y = v 2 − f {\displaystyle y=v_{2}-f} , one obtains the equation y = 1 4 f ( x − v 1 ) 2 + v 2 = 1 4 f x 2 − v 1 2 f x + v 1 2 4 f + v 2 . {\displaystyle y={\frac {1}{4f}}(x-v_{1})^{2}+v_{2}={\frac {1}{4f}}x^{2}-{\frac {v_{1}}{2f}}x+{\frac {v_{1}^{2}}{4f}}+v_{2}.} Remarks : If
4592-638: The focus is F = ( f 1 , f 2 ) {\displaystyle F=(f_{1},f_{2})} , and the directrix a x + b y + c = 0 {\displaystyle ax+by+c=0} , then one obtains the equation ( a x + b y + c ) 2 a 2 + b 2 = ( x − f 1 ) 2 + ( y − f 2 ) 2 {\displaystyle {\frac {(ax+by+c)^{2}}{a^{2}+b^{2}}}=(x-f_{1})^{2}+(y-f_{2})^{2}} (the left side of
4674-574: The fundamental indivisible elements comprising the space, of which one-dimensional curves , two-dimensional surfaces , and higher-dimensional objects consist; conversely, a point can be determined by the intersection of two curves or three surfaces, called a vertex or corner . In classical Euclidean geometry , a point is a primitive notion , defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called axioms , that they must satisfy; for example, "there
4756-653: The lost Book I, like Book II, was concerned with arithmetic due to Book III being clearly introduced as beginning a new subject. The whole of Book II (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition) discusses a method of multiplication from an unnamed book by Apollonius of Perga . The final propositions deal with multiplying together the numerical values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2 × 10 and 2 × 10 . Book III contains geometrical problems, plane and solid. It may be divided into five sections: Of Book IV
4838-439: The most important results obtained by his predecessors, and, secondly, notes explanatory of, or extending, previous discoveries. These discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the various books as valuable, for they set forth clearly an outline of the contents and the general scope of the subjects to be treated. From these introductions one can judge of
4920-525: The notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e.g. noncommutative geometry and pointless topology . A "pointless" or "pointfree" space is defined not as a set , but via some structure ( algebraic or logical respectively) which looks like a well-known function space on the set: an algebra of continuous functions or an algebra of sets respectively. More precisely, such structures generalize well-known spaces of functions in
5002-398: The origin (0, 0) and the same semi-latus rectum p {\displaystyle p} can be represented by the equation y 2 = 2 p x + ( e 2 − 1 ) x 2 , e ≥ 0 , {\displaystyle y^{2}=2px+(e^{2}-1)x^{2},\quad e\geq 0,} with e {\displaystyle e}
SECTION 60
#17327661726345084-430: The origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the y axis as axis of symmetry. Hence the parabola P {\displaystyle {\mathcal {P}}} can be transformed by a rigid motion to a parabola with an equation y = a x 2 , a ≠ 0 {\displaystyle y=ax^{2},\ a\neq 0} . Such
5166-587: The other by a similarity , that is, an arbitrary composition of rigid motions ( translations and rotations ) and uniform scalings . A parabola P {\displaystyle {\mathcal {P}}} with vertex V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} can be transformed by the translation ( x , y ) → ( x − v 1 , y − v 2 ) {\displaystyle (x,y)\to (x-v_{1},y-v_{2})} to one with
5248-403: The parabola, which is the distance from the vertex to the focus. The focus and the point F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, the point F, defined above, is the focus of the parabola . This discussion started from the definition of a parabola as a conic section, but it has now led to a description as
5330-404: The perpendicular from the point V to the plane of the parabola. By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is complementary to θ , and angle PVF is complementary to angle VPF, therefore angle PVF is θ . Since the length of PV is r , the distance of F from the vertex of the parabola is r sin θ . It is shown above that this distance equals the focal length of
5412-399: The points where the parabola intersects the circle. Another chord BC is the perpendicular bisector of DE and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry PM all intersect at the point M. All the labelled points, except D and E, are coplanar . They are in the plane of symmetry of the whole figure. This includes the point F, which
5494-405: The positive y direction, then its equation is y = x / 4 f , where f is its focal length. Comparing this with the last equation above shows that the focal length of the parabola in the cone is r sin θ . In the diagram above, the point V is the foot of the perpendicular from the vertex of the parabola to the axis of the cone. The point F is the foot of
5576-551: The product of the remaining ones; (Pappus does not express it in this form but by means of composition of ratios, saying that if the ratio is given which is compounded of the ratios of pairs one of one set and one of another of the lines so drawn, and of the ratio of the odd one, if any, to a given straight line, the point will lie on a curve given in position); (b) the theorems which were rediscovered by and named after Paul Guldin , but appear to have been discovered by Pappus himself. Book VII also contains Chasles's citation of Pappus
5658-507: The relation of projective harmonic conjugates , and displayed an awareness of cross-ratios of points and lines. Furthermore, the concept of pole and polar is revealed as a lemma in Book VII. Book VIII principally treats mechanics, the properties of the center of gravity, and some mechanical powers. Interspersed are some propositions on pure geometry. Proposition 14 shows how to draw an ellipse through five given points, and Prop. 15 gives
5740-686: The same Theon), which states, next to an entry on Emperor Diocletian (reigned 284–305), that "at that time wrote Pappus". However, a verifiable date comes from the dating of a solar eclipse mentioned by Pappus himself. In his commentary on the Almagest he calculates "the place and time of conjunction which gave rise to the eclipse in Tybi in 1068 after Nabonassar ". This works out as 18 October 320, and so Pappus must have been active around 320. The great work of Pappus, in eight books and titled Synagoge or Collection , has not survived in complete form:
5822-411: The same preface is included (a) the famous problem known by Pappus's name, often enunciated thus: Having given a number of straight lines, to find the geometric locus of a point such that the lengths of the perpendiculars upon, or (more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to
5904-470: The section above one obtains: For a = 1 {\displaystyle a=1} the parabola is the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . Its focus is ( 0 , 1 4 ) {\displaystyle \left(0,{\tfrac {1}{4}}\right)} , the semi-latus rectum p = 1 2 {\displaystyle p={\tfrac {1}{2}}} , and
5986-425: The square yields f ( x ) = a ( x + b 2 a ) 2 + 4 a c − b 2 4 a , {\displaystyle f(x)=a\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {4ac-b^{2}}{4a}},} which is the equation of a parabola with Two objects in the Euclidean plane are similar if one can be transformed to
6068-484: The style of Pappus's writing, which is excellent and even elegant the moment he is free from the shackles of mathematical formulae and expressions. Heath also found his characteristic exactness made his Collection "a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us". The surviving portions of Collection can be summarized as follows. Book I has been completely lost. We can only conjecture that
6150-461: The terms analysis and synthesis, and the distinction between theorem and problem. Pappus then enumerates works of Euclid , Apollonius , Aristaeus and Eratosthenes , thirty-three books in all, the substance of which he intends to give, with the lemmas necessary for their elucidation. With the mention of the Porisms of Euclid we have an account of the relation of porism to theorem and problem. In
6232-431: The thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes , and finds, by a method recalling that of Archimedes, the surface and volume of a sphere. According to the preface, Book VI is intended to resolve difficulties occurring in the so-called "Lesser Astronomical Works" (Μικρὸς Ἀστρονομούμενος), i.e. works other than the Almagest . It accordingly comments on
6314-479: The title and preface have been lost, so that the program has to be gathered from the book itself. At the beginning is the well-known generalization of Euclid I.47 ( Pappus's area theorem ), then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two. This and several other propositions on contact, e.g. cases of circles touching one another and inscribed in
6396-435: The unit parabola, such as ( x , y ) → ( x , y a ) {\displaystyle (x,y)\to \left(x,{\tfrac {y}{a}}\right)} . But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see § As the affine image of the unit parabola ). The pencil of conic sections with the x axis as axis of symmetry, one vertex at
6478-476: Was a Greek mathematician of late antiquity known for his Synagoge (Συναγωγή) or Collection ( c. 340 ), and for Pappus's hexagon theorem in projective geometry . Almost nothing is known about his life except for what can be found in his own writings, many of which are lost. Pappus apparently lived in Alexandria , where he worked as a mathematics teacher to higher level students, one of whom
6560-404: Was later than Ptolemy (died c. 168 AD), whom he quotes, and earlier than Proclus (born c. 411 ), who quotes him. The 10th century Suda states that Pappus was of the same age as Theon of Alexandria , who was active in the reign of Emperor Theodosius I (372–395). A different date is given by a marginal note to a late 10th-century manuscript (a copy of a chronological table by
6642-399: Was named Hermodorus. The Collection , his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics that were part of the ancient mathematics curriculum, including geometry , astronomy , and mechanics . Pappus was active in a period generally considered one of stagnation in mathematical studies, where he stands out as
6724-406: Was repeated by Wilhelm Blaschke and Dirk Struik . In Cambridge, England, John J. Milne gave readers the benefit of his reading of Pappus. In 1985 Alexander Jones wrote his thesis at Brown University on the subject. A revised form of his translation and commentary was published by Springer-Verlag the following year. Jones succeeds in showing how Pappus manipulated the complete quadrangle , used
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