Pappus centroid theorem pdf the surface of revolution generated by a smooth curve. Jul 18, 2015 use the theorem of pappus to determine the surface area of this region as well. Full video on benchmark ktu mobile app download app in mathematics, pappuss centroid theorem also known as the guldinus theorem. Engineering mechanics pdf notes em pdf notes smartzworld. Pappus s hexagon theorem, often just called pappus s theorem, a theorem named for pappus of alexandria. Pappus theorem on volumes department of mathematics. Engineering mechanics notes pdf em notes pdf starts with topics covering introduction to engineering. Nothing is known of his life, other than what can be found in his own writings. Such a set of axioms was given by bachmann, and a proof of desargues theorem can be found in based on the following pappus theorem of euclidean geometry being considered as an axiom. 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.
The first theorem states that the surface area a of a surface of revolution generated by rotating a plane curve c. Original proof of pappus hexagon theorem mathoverflow. Jiwen he, university of houston math 1431 section 24076, lecture 23 december 4, 2008 16 16. The proposition that the volume of a solid of revolution. Carpus of antioch, conon of samos, demetrius of alexandria. Pappus of alexandria pdf pappus of alexandria pdf pappus of alexandria pdf download. A similar calculation may be made using the y coordinate of the. An application of pappus involution theorem in euclidean. If the region does not cross the axis, then the volume of the resulting solid of revolution is v 2. A simple proof for the theorems of pascal and pappus. Pappus centroid theorem pdf pappus centroid theorem pdf pappus centroid theorem pdf download. Theorems of pappus can also be used to determine centroid of plane curves if area.
The axiomatic destiny of the theorems of pappus and desargues. Full video on benchmark ktu mobile app download app in mathematics, pappus s centroid theorem also known as the guldinus theorem. In this paper, we report on the formalization of a synthetic proof of pappus theorem. Nov 17, 2014 homework statement the lampshade shown is constructed of 0. If the vertices of a triangle are projected onto a gi ven line, the per. Generalizations of pappus centroid theorem via stokes theorem adams, cole, lovett, stephen, and mcmillan, matthew, involve.
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. The theorem of pascal concerning a hexagon inscribed in a conic. Pappus botany, a structure within certain flowers pappus, a genus of insects in the tribe mirini. In mathematics, pappuss centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. Pappus definition of pappus by the free dictionary. Nowadays the theorem is known as pappus guldin theorem or pappus theorem. Pappus definition, a downy, bristly, or other tuftlike appendage of the achene of certain plants, as the dandelion and the thistle. Pappus of alexandria was a greek mathematician who lived around the end of the third century ad, although the exact date is uncertain.
To be honest most rmo geometry problems can be done using basic knowledge of similarity, congruence, collinearity and concurrency alone. Then the intersection points of the line pairs ab with ba, ac with ca and bc with cb are again collinear. Dec 25, 2011 areas of surfaces of revolution and the theorems of pappus. 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. Pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Pdf a synthetic proof of pappus theorem in tarskis. An application of pappus involution theorem in euclidean and noneuclidean geometry. 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.
Theorem of pappus and guldinus engineering mechanics. 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. A torus may be specified in terms of its minor radius r and ma jor radius r by. U dont need high level advanced theorems like pappus theorem or steiner leimus theorem. For the love of physics walter lewin may 16, 2011 duration. An analytic proof of the theorems of pappus and desargues. A centroid is easily visualized as the center of gravity or center of mass of a flat. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. Euclidean version of pappuss theorem mathematics stack. 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. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Center of gravity theorem of pappus free download as word doc.
A synthetic proof of pappus theorem in tarskis geometry. The theorems are attributed to pappus of alexandria and paul guldin. Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total. Which theorems in geometry are important and need to be. 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. Use the theorem of pappus to determine the surface area of this region as well.
Media in category pappus theorem the following 36 files are in this category, out of 36 total. Bayes theorem was rst proven in 1763 by thomas bayes. 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. Let s be the surface generated by revolving this curve about the xaxis. 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. His great work a mathematical collection is an important source of information about ancient greek mathematics. This is precisely what pappus centroid theorem gives. Long before the invention of calculus, pappus of alexandria ca. Pappus commentary on euclid smith, david eugene, bulletin of the american mathematical society, 1931. Pappus s centroid theorem, another 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. This file is licensed under the creative commons attributionshare alike 4. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. 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.
Theorems of pappus and goldinus mechanical engineering notes. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft2. 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. Centroid theorem of pappus guldinus volume and surface. Here you can download the free engineering mechanics pdf notes em pdf notes of latest materials with multiple file links to download. Media in category pappus s theorem the following 40 files are in this category, out of 40 total. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. Proving and generalizing desargues twotriangle theorem. Now the second pappus guldin theorem gives the volume when this region is rotated through. Let a be a region in the upper half plane with boundary curve c, let e be the solid of revolution formed by.
It is by some considered to the theory of probability what the pythagoras theorem is to geometry. Pappus theorem article about pappus theorem by the. Pdf orthopoles and the pappus theorem researchgate. 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.
Pdf a synthetic proof of pappus theorem in tarskis geometry. 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. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. Theorems of pappus on surfaces of revolution wolfram.
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. Although not so well known today, book 4 of pappus collection is one of the most. 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. 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. 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. Pappus theorem article about pappus theorem by the free.
For the sake of completeness, we give a proof of the pappus theorem for this case. The axiomatic destiny of the theorems of pappus and. Center of gravity theorem of pappus sphere area scribd. Of these pappus of alexandria was among the last, and from the point of view of. Now the second pappusguldin theorem gives the volume when this region is rotated through. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Theorem of the day pappus theorem let a, b, c and a, b, c be two sets of collinear points. An expression in the exterior algebra of a peano space yielding pappus theorem was originally given by doubilet, rota, and stein.
We refer to 3 for an excellent comprehensive historical survey of these two theorems. If one measures the ratio applicability over the di culty of proof, then this theorem even beats pythagoras, as no proof is required. An application of pappus involution theorem in euclidean and. There are two theorems, both saying similar things. Areas of surfaces of revolution and the theorems of pappus. Consider the curve c given by the graph of the function f. A classic example is the measurement of the surface area and volume of a torus. Areas of surfaces of revolution, pappuss theorems let f.
Let us find the area of the surface generated by revolving the curve y 1. A video lecture that will explain both the theorems of pappus and guldinus with examples. We do know that he recorded in one of his commentaries on the almagest2 that he observed a solar eclipse on october 18, 320. How are these theorems proved without using calculus. Mar 25, 2018 pappus and guldinus theorum explained. 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. These three points are the points of intersection of the opposite sides of the hexagon. Areas of surfaces of revolution, pappuss theorems iitk. Proving and generalizing desargues twotriangle theorem in 3. Moreover, very little is known of what his actual contributions were or even exactly when he lived. Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r. Theorems of pappus and goldinus mechanical engineering. An analytic proof of the theorems of pappus and desargues core.1305 319 1058 1045 1413 392 971 968 354 223 244 463 1062 486 1140 1129 1327 1036 1015 1179 167 1002 1404 1227 1465 845 273 1447 442 1213 548 1446 239 1182 649 772 843 229 597 846 1182 157 1051 608 376