Pdf orthopoles and the pappus theorem semantic scholar. 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. The equivalence of the pappus s theorem and the theorem of pappus bria nchon 5. The first two books were devoted to arithmetic, and the third through fifth books deal primarily with geometry. Pappus generalisation of pythagoras theorem volume 89 issue 516 john h. Theorems of pappus and goldinus mechanical engineering notes.
If the region does not cross the axis, then the volume of the resulting solid of revolution is v 2. Nothing is known of his life, except from his own writings that he had a son named hermodorus, and was a teacher in alexandria. The first example is finding the volume of a tarus at 2. What we need is a simple affine theorem which is a special case of the pappus theorem. 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. An application of pappus involution theorem in euclidean. Lesson 55 centroid theorem of pappus guldinus volume and surface area jeff hanson. These three points are the points of intersection of the opposite sides of the hexagon. Media in category pappus s theorem the following 40 files are in this category, out of. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul.
Contributor pappus alexandrinus, greek mathematician, approximately 3rd or 4th century ad. There are two theorems, both saying similar things. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. Pappus definition of pappus by the free dictionary. Now the second pappus guldin theorem gives the volume when this region is rotated through. If points a,b and c are on one line and a, b and c are on another line then the.
Then the intersection points of the line pairs ab with ba, ac with ca and bc with cb are again collinear. This file is licensed under the creative commons attributionshare alike 4. The first theorem states that the surface area a of a surface of revolution generated by rotating a plane curve c. How are these theorems proved without using calculus. 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. Oct 25, 2017 a video lecture that will explain both the theorems of pappus and guldinus with examples. 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. This arrangement has a complex constraint involving the sines of its angles. Let s be the surface generated by revolving this curve about the xaxis. 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. We do know that he recorded in one of his commentaries on the almagest2 that he observed a.
The galluccis axiom is equi v alent to the pappus s theorem in a space of incide nce. Pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Pappus of alexandria greek mathematician britannica. The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in. Theorem of the day pappus theorem let a, b, c and a, b, c be two sets of collinear points. Areas of surfaces of revolution, pappuss theorems let f. Pappus s centroid theorem may refer to one of two theorems. Theorem of pappus and guldinus centroids and centers of. In continental europe, these theorems are more commonly associated with the name of paul guldin who rediscovered them.
Consider the curve c given by the graph of the function f. Profrobbob i introduce the theorem of pappus and then work. Pappus was the author of mathematical collections in eight books, only the last six of which are extant. Long before the invention of calculus, pappus of alexandria ca. 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 similar calculation may be made using the y coordinate of the. These constraints cannot be satisfied by any projection of the initial arrangement.
Triangles d abc and d au bu cu are perspective from a point o if lines aau, bbu and ccu. In englishspeaking countries, these two theorems are known as pappus s theorems, after the ancient greek geometer pappus of alexandria. This video is highly rated by mechanical engineering students and has been viewed 267 times. 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 theorems are attributed to pappus of alexandria and paul guldin. Media in category pappus s theorem the following 40 files are in this category, out of 40 total. 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. Here are some definitions, which culminate in a formal definition of a convex marked box, the. The fewest number but bigger than 1 of switches should. 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. The theorem of pascal concerning a hexagon inscribed in a conic.
Greek mathematician of the second half of the third century. Pdf a synthetic proof of pappus theorem in tarskis geometry. The first theorem of pappus states that the surface area s of a surface of revolution generated by the. Remarks on orthocenters, pappus theorem and butterfly. Both subjects were discussed with pickert in the last year of his life. 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. 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. 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. 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. 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. Section 6 and is the key to our proof of pappustheorem. Dec 22, 2019 applications of the theorems of pappus. 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. Century ad proposed two theorems for determining the area and volume of surfaces of revolution.
Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total. This is a partial version of desargues involution theorem. Jiwen he, university of houston math 1431 section 24076, lecture 23 december 4, 2008 16 16. This can be achieved using the second theorem of pappus, that states. Now the second pappusguldin theorem gives the volume when this region is rotated through. Mar 25, 2018 pappus and guldinus theorum explained. We denote the intersection of two lines g and g by g. 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. Pappus chain, we give a theorem analogue to pappus chain theorem. We have already mentioned desargues theorem as an example of a result which is. In this note we prove some theorems on orthopoles by using a well known result from projective geometry, the pappus theorem. A synthetic proof of pappus theorem in tarskis geometry.
Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r. Any stretching of rin9 would provide a euclidean stretching of b, necessarily satisfying the premises of the main theorem. The first example shows the picture most often drawn in textbooks with the. We prove several theorems on orthopoles using the pappus theorem, a fundamental result of projective geometry. An analytic proof of the theorems of pappus and desargues. Theorem of pappus to find volume of revolution calculus 2. Theorem of pappus and guldinus engineering mechanics.
A communications network what are the desirable properties of the switching box. 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. 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. 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. The axiomatic destiny of the theorems of pappus and desargues. Other than that he was born at alexandria in egypt and that his.
To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Pdf a new proof of pappuss theorem semantic scholar. A simple proof for the theorems of pascal and pappus. Theorems of pappus and goldinus mechanical engineering. Pappus theorem article about pappus theorem by the free. It is not necessary that the plane or the curve be rotated through a full 360o. 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. Furthermore we add a projective butterfly theorem which covers all known affine cases. 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. 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. Full video on benchmark ktu mobile app download app in mathematics, pappus s centroid theorem also known as the guldinus theorem.
The main theorem of projective geometry that we will use is. This is precisely what pappus centroid theorem gives. Consider two straight lines emanating from point o and containing the points p 1 through p 6 as shown in the figure below. The surface area of a solid of revolution is the arc length of the generating curve multiplied by. We do know that he recorded in one of his commentaries on the almagest2 that he observed a solar eclipse on october 18, 320. Any stretching of ringels non pappus pseudoline arrangement when projected into the euclidean plane, implicitly contains a particular arrangement of nine triangles. Centroids and centers of gravity theorem of pappus and guldinus centroids and centers of gravity. The proposition that the volume of a solid of revolution. 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. 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.
An application of pappus involution theorem in euclidean and. Theorems of pappus on surfaces of revolution wolfram. We present a generalization of the notion of the orthocenter of a triangle and of 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. His great work a mathematical collection is an important source of information about ancient greek mathematics. Theorem of pappus synonyms, theorem of pappus pronunciation, theorem of pappus translation, english dictionary definition of theorem of pappus. Pappus theorem article about pappus theorem by the. Euclidean version of pappuss theorem mathematics stack. Original proof of pappus hexagon theorem mathoverflow. Use the theorem of pappus to determine the surface area of this region as well. Notably, we need not even use it in the general case. A centroid is easily visualized as the center of gravity or center of mass of a flat. Moreover, very little is known of what his actual contributions were or even exactly when he lived.
1248 1598 328 1570 1038 1112 13 1556 340 1039 13 1001 774 1631 1594 1486 251 1538 1354 210 231 1663 504 1585 759 433 818 786 910 1520 189 248 1238 199 1399 1036 537 693 1362 568 1141 781