This is a partial version of desargues involution theorem. This arrangement has a complex constraint involving the sines of its angles. The first theorem states that the surface area a of a surface of revolution generated by rotating a plane curve c. Pdf a new proof of pappuss theorem semantic scholar. The surface area of a solid of revolution is the arc length of the generating curve multiplied by. 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. Nothing is known of his life, except from his own writings that he had a son named hermodorus, and was a teacher in alexandria. An application of pappus involution theorem in euclidean and. 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. Pappus s centroid theorem may refer to one of two theorems.
Profrobbob i introduce the theorem of pappus and then work. Here are some definitions, which culminate in a formal definition of a convex marked box, the. Theorem of pappus synonyms, theorem of pappus pronunciation, theorem of pappus translation, english dictionary definition of theorem of pappus. Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r. Use the theorem of pappus to determine the surface area of this region as well. 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. In englishspeaking countries, these two theorems are known as pappus s theorems, after the ancient greek geometer pappus of alexandria. Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total. Any stretching of ringels non pappus pseudoline arrangement when projected into the euclidean plane, implicitly contains a particular arrangement of nine triangles. Now the second pappusguldin theorem gives the volume when this region is rotated through. Now the second pappus guldin theorem gives the volume when this region is rotated through. This is a partial version of desargues involution theorem see 3, p. Dec 22, 2019 applications of the theorems of pappus.
Pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. In addition, pappus gave some apparently original results, such as the proposition that is commonly called pappus theorem involving a hexagon inscribed between two lines. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. Theorems of pappus on surfaces of revolution wolfram. Moreover, very little is known of what his actual contributions were or even exactly when he lived. Consider two straight lines emanating from point o and containing the points p 1 through p 6 as shown in the figure below. This is precisely what pappus centroid theorem gives. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. 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 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. Furthermore we add a projective butterfly theorem which covers all known affine cases. Pappus of alexandria greek mathematician britannica. Pdf a synthetic proof of pappus theorem in tarskis geometry. Theorems of pappus and goldinus mechanical engineering. A simple proof for the theorems of pascal and pappus. Section 6 and is the key to our proof of pappustheorem. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. We denote the intersection of two lines g and g by g. The theorems are attributed to pappus of alexandria and paul guldin. An analytic proof of the theorems of pappus and desargues. Theorem 2 pappus involution theorem the three pairs of opposite sides of a complete quadrangle meet any line not through a vertex in three pairs of an involution. The axiomatic destiny of the theorems of pappus and desargues.
A bridge between algebra and geometry article pdf available in the american mathematical monthly 1096 june 2002 with 2,579 reads how we measure reads. Pappus generalisation of pythagoras theorem volume 89 issue 516 john h. The volume v of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area a of the lamina and the distance d traveled by the laminas geometric centroid bar. There are two theorems, both saying similar things. If a plane area is rotated about an axis in its plane, but which does not cross the area, the volume swept out equals the area times the distance moved by the centroid. An analogue to pappus chain theorem with division by zero. These three points are the points of intersection of the opposite sides of the hexagon. Then the intersection points of the line pairs ab with ba, ac with ca and bc with cb are again collinear.
Euclidean version of pappuss theorem mathematics stack. Theorem of the day pappus theorem let a, b, c and a, b, c be two sets of collinear points. The equivalence of the pappus s theorem and the theorem of pappus bria nchon 5. This file is licensed under the creative commons attributionshare alike 4. 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. Any stretching of rin9 would provide a euclidean stretching of b, necessarily satisfying the premises of the main theorem. The theorem of pascal concerning a hexagon inscribed in a conic. Triangles d abc and d au bu cu are perspective from a point o if lines aau, bbu and ccu. The surface area of a solid of revolution is the arc length of the generating curve multiplied by the distance traveled by the centroid of the curve. 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. These constraints cannot be satisfied by any projection of the initial arrangement. Media in category pappus s theorem the following 40 files are in this category, out of 40 total.
We do know that he recorded in one of his commentaries on the almagest2 that he observed a. 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. Pappus was the author of mathematical collections in eight books, only the last six of which are extant. In this note we prove some theorems on orthopoles by using a well known result from projective geometry, the pappus theorem. A centroid is easily visualized as the center of gravity or center of mass of a flat. What we need is a simple affine theorem which is a special case of the pappus theorem. If a plane area is rotated about an axis in its plane, but which does not cross the area, the volume swept out equals the area times the distance moved by the centroid if a plane curve is rotated about an axis in its plane, but which does not cross the curve. If the region does not cross the axis, then the volume of the resulting solid of revolution is v 2. Other than that he was born at alexandria in egypt and that his. Centroids and centers of gravity theorem of pappus and guldinus centroids and centers of gravity. 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. A similar calculation may be made using the y coordinate of the.
Pappus s hexagon theorem theorem that, if the vertices of a hexagon lie alternately on two lines, then the three pairs of opposite sides meet in three collinear points. Both subjects were discussed with pickert in the last year of his life. This video is highly rated by mechanical engineering students and has been viewed 267 times. Areas of surfaces of revolution, pappus theorem 4 1. Two triangles that are perspective from a point are perspective from a line, and converseley, two triangles that are perspective from a line are perspective from a point. Pappus definition of pappus by the free dictionary. Full video on benchmark ktu mobile app download app in mathematics, pappus s centroid theorem also known as the guldinus theorem. Contributor pappus alexandrinus, greek mathematician, approximately 3rd or 4th century ad. Areas of surfaces of revolution, pappuss theorems let f. 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.
Theorem of pappus and guldinus engineering mechanics. Theorem of pappus to find volume of revolution calculus 2. 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. Theorem of pappus and guldinus centroids and centers of. The first example shows the picture most often drawn in textbooks with the. How are these theorems proved without using calculus. Lesson 55 centroid theorem of pappus guldinus volume and surface area jeff hanson. The galluccis axiom is equi v alent to the pappus s theorem in a space of incide nce. Mar 25, 2018 pappus and guldinus theorum explained. Pappus definition, a downy, bristly, or other tuftlike appendage of the achene of certain plants, as the dandelion and the thistle.
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. Notably, we need not even use it in the general case. Jiwen he, university of houston math 1431 section 24076, lecture 23 december 4, 2008 16 16. If the vertices of a triangle are projected onto a given line, the per pendiculars from the projections to the corresponding sidelines of the triangle intersect at one point, the orthopole of the line with respect to the triangle. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. We present a generalization of the notion of the orthocenter of a triangle and of pappus theorem. We prove several theorems on orthopoles using the pappus theorem, a fundamental result of projective geometry. Consider the curve c given by the graph of the function f. If points a,b and c are on one line and a, b and c are on another line then the.
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. The first example is finding the volume of a tarus at 2. The proposition that the volume of a solid of revolution. A synthetic proof of pappus theorem in tarskis geometry. The first two books were devoted to arithmetic, and the third through fifth books deal primarily with geometry. We do know that he recorded in one of his commentaries on the almagest2 that he observed a solar eclipse on october 18, 320. Greek mathematician of the second half of the third century. An application of pappus involution theorem in euclidean. In continental europe, these theorems are more commonly associated with the name of paul guldin who rediscovered them. Let s be the surface generated by revolving this curve about the xaxis. Theorems of pappus and goldinus mechanical engineering notes.
Pappus theorem article about pappus theorem by the free. Pappus chain, we give a theorem analogue to pappus chain theorem. The first theorem of pappus states that the surface area s of a surface of revolution generated by the. His great work a mathematical collection is an important source of information about ancient greek mathematics. Theorems of pappus and guldinus mechanical engineering video edurev is made by best teachers of mechanical engineering. The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in.
Pappus article about pappus by the free dictionary. 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. We have already mentioned desargues theorem as an example of a result which is. Media in category pappus s theorem the following 40 files are in this category, out of.
The fewest number but bigger than 1 of switches should. 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. This can be achieved using the second theorem of pappus, that states. 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.
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