Nov 29, 20 pappus and desargues finite geometries 1. Wedderburn we show that every finite desarguesian affine plane is pappian. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. An application of pappus involution theorem in euclidean and noneuclidean geometry. But avoid asking for help, clarification, or responding to other answers. Other than that he was born at alexandria in egypt and that his. The usual pappus theorem is just the situation whereby the conic degenerates into a pair of lines. The first two books were devoted to arithmetic, and the third through fifth books deal primarily with geometry. The theorem of pappus and commutativity of multiplication leroy j dickey 20120518 abstract the purpose of this note is to present a proof of the theorem of pappus that reveals the role of commutativity of multiplication. From pascals theorem to d constructible curves will traves abstract. If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irreducible curve c of degree d, then the remaining k. Areas of surfaces of revolution, pappus s theorems let f. A method for finding the volume of a solid of revolution. Long before the invention of calculus, pappus of alexandria ca.
At this time higher geometry was in complete abeyance until. Descartes first studied pappus problem during late 1631 and early 1632, on the instigation of golius. Moreover, very little is known of what his actual contributions were or even exactly when he lived. This is crucial to obtain a coordinatefree version of the proof. Pappus was the author of mathematical collections in eight books, only the last six of which are extant. We do know that he recorded in one of his commentaries on the almagest2 that he observed a solar eclipse on october 18, 320. This provides a free source of useful theorems, courtesy of reynolds. Pappus s hexagon theorem, often just called pappus s theorem, a theorem named for pappus of alexandria. Pappus theorem if points a,b and c are on one line and a, b and c are on another line then the points of intersection of the lines ab and ba, ac and ca, and bc and cb lie on a common line called the pappus line of the configuration. Pappus theorem definition of pappus theorem by the free. Motivated by an identity of rota, we give an identity in a grassmanncayley algebra of step 3, involving joins and meets alone, which expresses the theorem of pappus. Spin b around the x axis, creating a shape of revolution. Alternatively, given a mystic hexagon, the pappus con. Every function of the same type satisfies the same theorem.
The centroid of a region is essentially the one point on which the region should balance. Pappuss area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Pappus s first theorem states that the area of a surface generated by rotating a figure about an external axis a distance from its centroid equals the product of the arc length of the generating figure and the distance traversed by. Jiwen he, university of houston math 1431 section 24076, lecture 23 december 4, 2008 16 16. When the six points are ordered as a, f, b, d, c, f the resulting polygon is just pascals mystic hexagon. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Pappus article about pappus by the free dictionary.
Media in category pappus theorem the following 36 files are in this category, out of 36 total. The theorem of pappus and commutativity of multiplication. A simple proof for the theorems of pascal and pappus. Throughout this course you will learn to do an analyses of particles, rigid bodies, trusses, frames, and machines in static equilibrium with applied forces and couples.
In mathematics, pappuss centroid theorem also known as the guldinus theorem, pappusguldinus theorem or pappuss theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. Compute the volume of the shape using cylindrical coordinates. A bridge between algebra and geometry article pdf available in the american mathematical monthly 1096 june 2002 with 2,653 reads how we measure reads. Theorems of pappus on surfaces of revolution wolfram.
Aditionally to several features for defining the layout of theorem like environments which can be regarded to be standard requirements for a theorem package, it provides solutions for two related problems. Pi is a command line utility able to check the dimensional validity of an equation. The theorem, which can also be thought of as a generalization of the pythagorean theorem, is named after the greek mathematician pappus of alexandria 4th century ad, who discovered it. The theorem of pappus states that when a region r is rotated about a line l, the volume of the solid generated is equal to the product of the area of r and the distance the centroid of the region has traveled in one full rotation. Euclidean construction of the principal axes of an ellipse, in magnitude as well as in position, given any pair of its conjugate.
In chapter 19 i argued that the confrontation with the problem was decisive for the final stage of the development of his programmatic ideas on geometry. Without using the representation theorem and a theorem of j. The theorems are attributed to pappus of alexandria and paul guldin. The wonder of it all is that the plan of salvation is set before us in the night sky. Invision employees share their remote work secrets. The volume equals the product of the area of the region being rotated times the distance traveled by the centroid of the region in one rotation. A classic example is the measurement of the surface area and volume of a torus. Pappus theorem submitted by plusadmin on january 1, 2001. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft2. Pappuss first theorem states that the area of a surface generated by rotating a figure about an external axis a distance from its centroid equals the product of the arc length of the generating figure and the distance traversed by. Theorems of pappus and goldinus mechanical engineering. Pappus s theorem, in mathematics, theorem named for the 4thcentury greek geometer pappus of alexandria that describes the volume of a solid, obtained by revolving a plane region d about a line l not intersecting d, as the product of the area of d and the length of the circular path traversed by the centroid of d during the revolution. Greek mathematician of the second half of the third century. Mar 25, 2018 pappus and guldinus theorum explained.
Pappus theorem on volumes department of mathematics. Pappus s theorem is very important theorem in ge ometry as pappus s theorem holds for some projective plane if and only if it is a projective plane over a commutativ e. Pappuss hexagon theorem, often just called pappuss theorem, a theorem named for pappus of alexandria. Euclidean version of pappuss theorem mathematics stack. A proof of the theorem of pappus in finite desarguesian. The theorem of pascal concerning a hexagon inscribed in a conic. A bridge between algebra and geometry article pdf available in the american mathematical monthly 1096 june 2002 with 2,653 reads. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Theorem of pappus tells us that volume is equal to area of the plane region, times the distance traveled by the centroid of the same plane region, if the. Pappus definition of pappus by the free dictionary. For instance, if b is a circle the result is a torus.
An application of pappus involution theorem in euclidean and. Me 2301 is a first semester, sophomore level class in statics. Aditionally to several features for defining the layout of theoremlike environments which can be regarded to be standard requirements for a theorempackage, it provides solutions for two related problems. Does anyone know where i can find an english translation, preferably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of pappus hexagon. We prove a generalization of both pascals theorem and its converse, the braikenridge maclaurin theorem. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. Pappus of alexandria greek mathematician britannica. This proof, my current favourite, shows that the pappus con guration \closes if and only if two numbers a and b commute. Pappus of alexandria was one of the last great greek mathematicians of antiquity, known for. An expression in the exterior algebra of a peano space yielding pappus theorem was originally given by doubilet, rota, and stein. A matrixvector analytic construction for the pappuseuler. Pappuss theorem, in mathematics, theorem named for the 4thcentury greek geometer pappus of alexandria that describes the volume of a solid, obtained by revolving a plane region d about a line l not intersecting d, as the product of the area of d and the length of the circular path traversed by. Jan 01, 2001 pappus theorem submitted by plusadmin on january 1, 2001. Pappus botany, a structure within certain flowers pappus, a genus of insects in the tribe mirini.
Adding the zero vector given by 5 to this representation of q, we obtain the following representation of q. Full video on benchmark ktu mobile app download app in mathematics, pappuss centroid theorem also known as the guldinus theorem. Euclidean construction, pappuseuler problem 1 introduction at the conclusion of chapter 17 of book viii of his collection 7, pp. Pappus himself probably intended collection 4 to be an introductory survey of the classical geometrical tradition from the point of view of mathematical methods and strategies for readers that had a basic training in elementary geometry elements i vi. There are two theorems, both saying similar things. Original proof of pappus hexagon theorem mathoverflow. Thanks for contributing an answer to mathematics stack exchange. The pappusguldin theorems suppose that a plane curve is rotated about an axis external to the curve. To illustrate pappus s theorem, consider a circular. A geometric identity for pappus theorem internet archive. Pdf a synthetic proof of pappus theorem in tarskis. A synthetic proof of pappus theorem in tarskis geometry halinria. Pappuss theorem is a very important theorem in geometry, since pappuss theorem holds for.
Solid of rotation, pappus centroid theorem a solid of rotation is the figure that results from rotating a plane figure about an external axis an axis on the same plane as the figure such that no two points of the figure are on opposite sides of the axis. Let s be the surface generated by revolving this curve about the xaxis. Consider the curve c given by the graph of the function f. Areas of surfaces of revolution, pappuss theorems iitk. Pappus theorem article about pappus theorem by the free. These quantities can be computed using the distance traveled by the centroids of the curve and region being revolved. Since considering the kolob theorem my mind has not rested.
Pappuss centroid theorems are results from geometry about the surface area and volume of solids of revolution. Likewise, this edition can be used as a textbook in advanced undergraduate and. From these introductions one can judge of the style of pappuss writing, which is excellent and even elegant the moment he is free. Pappus and desargues finite geometries linkedin slideshare. Pappus s centroid theorems are results from geometry about the surface area and volume of solids of revolution.
A torus may be specified in terms of its minor radius r and ma jor radius r by. The first theorem of pappus states that the surface area sof a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance d 1 traveled by the curves geometric centroid kern and bland 1948, pp. Pappus of alexandria, the most important mathematical author writing in greek during the later roman empire, known for his synagoge collection, a voluminous account of the most important work done in ancient greek mathematics. A centroid is easily visualized as the center of gravity or center of mass of a flat. We do not present here the first formal proof of pappus theorem.
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