Pdf orthopoles and the pappus theorem researchgate. Let us find the area of the surface generated by revolving the curve y 1. 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 theorem from projective geometry. Nothing is known of his life, other than what can be found in his own writings. An application of pappus involution theorem in euclidean. Engineering mechanics pdf notes em pdf notes smartzworld. Pappus of alexandria pdf from his descriptions, we may surmise that either the classical works. A classic example is the measurement of the surface area and volume of a torus. Centroid theorem of pappus guldinus volume and surface. This is precisely what pappus centroid theorem gives. Center of gravity theorem of pappus sphere area scribd. Now the second pappusguldin theorem gives the volume when this region is rotated through.
Pappus theorem for a conic and mystic hexagons ross moore macquarie university sydney, australia pappus theorem is a wellknown result for triples of points on two lines in the. How are these theorems proved without using calculus. Media in category pappus s theorem the following 40 files are in this category, out of 40 total. Areas of surfaces of revolution and the theorems of pappus. This file is licensed under the creative commons attributionshare alike 4. The axiomatic destiny of the theorems of pappus and. Pappus of alexandria pdf pappus of alexandria pdf pappus of alexandria pdf download. The first theorem states that the surface area a of a surface of revolution generated by rotating a plane curve c.
The theorem of pascal concerning a hexagon inscribed in a conic. For the sake of completeness, we give a proof of the pappus theorem for this case. His great work a mathematical collection is an important source of information about ancient greek mathematics. Center of gravity theorem of pappus free download as word doc. Pappus of alexandria was a greek mathematician who lived around the end of the third century ad, although the exact date is uncertain. A simple proof for the theorems of pascal and pappus. Pappus theorem on volumes department of mathematics. Let s be the surface generated by revolving this curve about the xaxis. Let a be a region in the upper half plane with boundary curve c, let e be the solid of revolution formed by. An application of pappus involution theorem in euclidean and noneuclidean geometry. 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. Pappus theorem article about pappus theorem by the free.
Full video on benchmark ktu mobile app download app in mathematics, pappus s centroid theorem also known as the guldinus theorem. There are two theorems, both saying similar things. Theon made a marginal note in one of his manuscripts stating that pappus wrote during the reign of roman emperor diocletian, which places him in the period from 284 to 305 ad, but it also seems. The notes contain the usual topics that are taught in those courses as well as a few extra topics that i decided to include just because i wanted to. In mathematics, pappus s hexagon theorem attributed to pappus of alexandria states that given one set of collinear points,, and another set of collinear points,, then the intersection points, of line pairs and, and, and are collinear, lying on the pappus line. In mathematics, pappus s centroid theorem also known as the guldinus theorem, pappus guldinus theorem or pappus s theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Which theorems in geometry are important and need to be. Pappus centroid theorem pdf the surface of revolution generated by a smooth curve. Pappus definition, a downy, bristly, or other tuftlike appendage of the achene of certain plants, as the dandelion and the thistle. Pappus botany, a structure within certain flowers pappus, a genus of insects in the tribe mirini.
Pappus commentary on euclid smith, david eugene, bulletin of the american mathematical society, 1931. Of these pappus of alexandria was among the last, and from the point of view of. An analytic proof of the theorems of pappus and desargues. Theorems of pappus can also be used to determine centroid of plane curves if area. Century ad proposed two theorems for determining the area and volume of surfaces of revolution.
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. It is by some considered to the theory of probability what the pythagoras theorem is to geometry. Nowadays the theorem is known as pappus guldin theorem or pappus theorem. Use the theorem of pappus to determine the surface area of this region as well. Mar 25, 2018 pappus and guldinus theorum explained. 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 definition of pappus by the free dictionary. When r is rotated about the xaxis, it generates a cone of volume use the theorem of pappus to determine the ycoordinate of the centroid of r. U dont need high level advanced theorems like pappus theorem or steiner leimus theorem. Pappus theorem article about pappus theorem by the. Full video on benchmark ktu mobile app download app in mathematics, pappuss centroid theorem also known as the guldinus theorem. Long before the invention of calculus, pappus of alexandria ca. Generalizations of pappus centroid theorem via stokes theorem adams, cole, lovett, stephen, and mcmillan, matthew, involve.
Original proof of pappus hexagon theorem mathoverflow. Theorems of pappus and goldinus mechanical engineering notes. Here you can download the free engineering mechanics pdf notes em pdf notes of latest materials with multiple file links to download. Pdf a synthetic proof of pappus theorem in tarskis. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. We do know that he recorded in one of his commentaries on the almagest2 that he observed a solar eclipse on october 18, 320. Pappus s hexagon theorem, often just called pappus s theorem, a theorem named for pappus of alexandria. In mathematics, pappuss centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. Nov 17, 2014 homework statement the lampshade shown is constructed of 0. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft2. Jiwen he, university of houston math 1431 section 24076, lecture 23 december 4, 2008 16 16.
Euclidean version of pappuss theorem mathematics stack. 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 ac and ca, ab and ba, and bc and cb lie on a common line called the pappus line of the configuration. Of course, this does not make the computation trivial in general, since computing the centroid of a region or curve is not easy, even for relatively simple shapes. Moreover, very little is known of what his actual contributions were or even exactly when he lived. Proving and generalizing desargues twotriangle theorem. 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. Contained in this site are the notes free and downloadable that i use to teach algebra, calculus i, ii and iii as well as differential equations at lamar university. Proving and generalizing desargues twotriangle theorem in 3. Pappus s centroid theorem, another theorem named for pappus of alexandria. 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 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. Media in category pappus theorem the following 36 files are in this category, out of 36 total. For the love of physics walter lewin may 16, 2011 duration. Consider the curve c given by the graph of the function f.
Jul 18, 2015 use the theorem of pappus to determine the surface area of this region as well. These three points are the points of intersection of the opposite sides of the hexagon. If the vertices of a triangle are projected onto a gi ven line, the per. Pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. 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. A similar calculation may be made using the y coordinate of the. Im not sure what hartshorne has in mind, but pappus theorem is a simple consequence of similarity of euclidean triangles in guise of the intercept theorem and theres no need of introducing the circle.
To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Theorem of pappus and guldinus engineering mechanics. We refer to 3 for an excellent comprehensive historical survey of these two theorems. Carpus of antioch, conon of samos, demetrius of alexandria. Theorems of pappus and goldinus mechanical engineering. Engineering mechanics notes pdf em notes pdf starts with topics covering introduction to engineering. A synthetic proof of pappus theorem in tarskis geometry. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. If one measures the ratio applicability over the di culty of proof, then this theorem even beats pythagoras, as no proof is required. The theorems are attributed to pappus of alexandria and paul guldin.
If the region does not cross the axis, then the volume of the resulting solid of revolution is v 2. 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. Pdf a synthetic proof of pappus theorem in tarskis geometry. An expression in the exterior algebra of a peano space yielding pappus theorem was originally given by doubilet, rota, and stein. An analytic proof of the theorems of pappus and desargues core. Bayes theorem was rst proven in 1763 by thomas bayes.
A torus may be specified in terms of its minor radius r and ma jor radius r by. Theorem of the day pappus theorem let a, b, c and a, b, c be two sets of collinear points. Theorems of pappus on surfaces of revolution wolfram. Areas of surfaces of revolution, pappuss theorems let f. Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. The axiomatic destiny of the theorems of pappus and desargues. An application of pappus involution theorem in euclidean and. A centroid is easily visualized as the center of gravity or center of mass of a flat.
Dec 25, 2011 areas of surfaces of revolution and the theorems of pappus. A video lecture that will explain both the theorems of pappus and guldinus with examples. Areas of surfaces of revolution, pappuss theorems iitk. To be honest most rmo geometry problems can be done using basic knowledge of similarity, congruence, collinearity and concurrency alone. The proposition that the volume of a solid of revolution. Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total. This lead to an alternative statement and proof of pappus s theorem retrieving pappus s and hesse configurations of lines as special points in complex projective grassmannian. Then the intersection points of the line pairs ab with ba, ac with ca and bc with cb are again collinear. Now the second pappus guldin theorem gives the volume when this region is rotated through.
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