Lesson 55 centroid theorem of pappus guldinus volume and surface area. Alternatively, given a mystic hexagon, the pappus con. How to prove pappus theorem mathematics stack exchange. Then the intersection points of the line pairs ab with ba, ac with ca and bc with cb are again collinear. Theorem of pappus definition of theorem of pappus by the. 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.
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. Areas of surfaces of revolution, pappuss theorems let f. Guldin 15771643 most of the remaining of the treatise is collections of lemmas that will assist the readers understanding of the original works. Jan 01, 2001 pappus theorem submitted by plusadmin on january 1, 2001. This rephrasing of gregorys proposition 35 may be familiar to those who have seen second semester calculus. There are two theorems, both saying similar things. The following table summarizes the surface areas calculated using pappuss centroid theorem for various surfaces of revolution. Theorems of pappus and guldinus, centre of gravity and moment. Theorem of pappus synonyms, theorem of pappus pronunciation, theorem of pappus translation, english dictionary definition of theorem of pappus. Jul 07, 2016 pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Pappusguldin theorems for weighted motions gualarnau, ximo and miquel, vicente, bulletin of the belgian mathematical society simon stevin, 2006.
Pappus theorem synonyms, pappus theorem pronunciation, pappus theorem translation, english dictionary definition of pappus theorem. The pappus guldin theorem is simultaneously one of the last great results in greek mathematics and one of the first novel results in the 16th and 17th century renaissance in. If the pappus line u \displaystyle u and the lines g, h \displaystyle g,h have a point in common, one gets the socalled little version of pappus s theorem 2. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Theorems of pappus and guldinus, centre of gravity and. There are two results of pappus which relate the centroids to surfaces and solids of revolutions. The theorem of pascal concerning a hexagon inscribed in a conic is very useful in many geometrical constructions and ought to be included in a normal course on descriptive geometry. A similar calculation may be made using the y coordinate of the. Now the second pappusguldin theorem gives the volume when this region is rotated through. It states that the volume of each solid of revolution is equal to the area of its base multiplied by the circumference of the circle in which the center of gravity of that figure is revolved.
James gregory and the pappus guldin theorem introduction. An application of pappus involution theorem in euclidean and. 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. Pappuss centroid theorems are results from geometry about the surface area and volume of solids of revolution. The main theorem of projective geometry that we will use is. Pappus theorem let a, b, c and a, b, c be two sets of collinear points. 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. This is the theorem of pappus or the pappusguldin theorem. Prove pappuss centroid theorems without calculus physics. Theorems of pappus and guldinus assignment help, theorems of pappus and guldinus homework help, centre of gravity and moment of inertia tutors. Pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Profrobbob i introduce the theorem of pappus and then work. A copy of the license is included in the section entitled gnu free documentation license.
This theorem is also known as the pappusguldinus theorem and pappuss centroid theorem, attributed to pappus of alexandria. James gregory and the pappusguldin theorem conclusion. Me 2301 is a first semester, sophomore level class in statics. Area of surface of revolution the area of a surface of revolution is equal to the length of the generating curve multiplied by the distance traveled by the centroid of the curve while the surface is being genera. Suppose c is revolved about the line l which does not cut through the interior of c, then the area of the surface generated is s 2l where is the distance from the axis of revolution to the centroid and l is the length of the curve c see figure 3. These quantities can be computed using the distance traveled by the centroids of the curve and region being revolved. 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. Summary of the formulas for plane laminas and curves 1. Area of surface of revolution the area of a surface of revolution is equal to the length of the generating curve multiplied by the distance traveled by the centroid of the curve while the surface is being. An application of pappus involution theorem in euclidean and noneuclidean geometry. Original proof of pappus hexagon theorem mathoverflow. The theorem of pascal concerning a hexagon inscribed in a conic. 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 of alexandria greek mathematician britannica. 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. Prpsanchez 1 of 2 centroids and centers of gravity theorem of pappus and guldinus theorem 1. The first theorem states that the surface area a of a surface of revolution generated by rotating a plane curve c about an axis external to c and on the same plane is equal to the product of the. For gregory, the pappusguldin theorem and quite a few other results are easy consequences of a broader geometrical perspectivethat is, a perspective involving ratios between the. 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. Pappus related things you might have been looking for when you found this page. Any two distinct points are incident with exactly one line. In this article w egiv an analytic proofpappus theorem and. Using the theorem of pappus and guldinuss, determine the volume of the storage tank shown in the figure. Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total.
Nothing is known of his life, other than what can be found in his own writings. To compute the volume of a solid formed by rotating a region. The centroid of a region is essentially the one point on which the region should balance. Any two lines are incident with at least one point. After this the point comes back from a very far position on. Pappus theorem definition of pappus theorem by the free. James gregory and the pappusguldin theorem historical. In the situation with zero slope both lines are parallel and the intersection point vanishes. The centroid of a rectangle with vertices 0,0, x,0, 0,y, and x,y. Nowadays the theorem is known as pappusguldin theorem or pappus theorem. The pappusguldin theorems suppose that a plane curve is rotated about an axis external to the curve.
Homework statement hey, im having issues with a problem, and my book doesnt seem to show me how to do it. Suppose r is revolved about the line l which does not cut. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. A simple proof for the theorems of pascal and pappus. Answer to use the second pappusguldinus theorem to determine the volume generated by revolving the curve about the y axis. Theorem of pappus to find volume of revolution calculus 2. The arc length of its right side is h h h and the distance traveled by its centroid is simply 2.
When the six points are ordered as a, f, b, d, c, f the resulting polygon is just pascals mystic hexagon. 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 on mar 1, 2006, ximo gualarnau and others published pappusguldin theorems for weighted motions find, read and cite all the. He discovered the guldinus theorem to determine the surface and the volume of a solid of revolution. Gregorys geometrical approach toward proving this result and just why this result ended up in gregorys text in the first place are the subjects of this article. Pappus guldin theorems for weighted motions gualarnau, ximo and miquel, vicente, bulletin of the belgian mathematical society simon stevin, 2006. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. Use the pappusguldin theorem to find the volume of the surface of revolution obtained by rotating the triangular region in the plane with vertices 0, 0, 1, 0, and 0, 1 about the line x 2. Z b a f x 2 dx, the familiar formula for volume of solid of revolution. Aug 25, 2015 there are two theorems, both saying similar things. The centroids are at a distance a in red from the axis of rotation. 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 pappus guldin theorem is simultaneously one of the last great results in greek mathematics and one of the first novel results in the 16th and 17th century renaissance in european mathematics. This proof, my current favourite, shows that the pappus con guration \closes if and only if two numbers a and b commute. A centroid is easily visualized as the center of gravity or center of mass of a flat. This is a partial version of desargues involution theorem see 3, p. This is the theorem of pappus or the pappus guldin theorem. An analytic proof of the theorems of pappus and desargues. 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. Pappus also discusses the three and four lines theorem of apollonius. Oct 08, 2008 homework statement hey, im having issues with a problem, and my book doesnt seem to show me how to do it. Pappus of alexandria, flourished ad 320, 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. Get answer use the pappusguldin theorem to find the.
Pappus theorem, which is more about points and lines, and is a special case of pascals theorem, in which a hexagon is inscribed in a conic. Every line is incident with at least three distinct points. 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. A simplified proof of the pappus leisenring theorem. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. If one restricts the projective plane such that the pappus line is the line at infinity, one gets the affine version of pappus s theorem shown in the second diagram. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Pappus commentary on euclid smith, david eugene, bulletin of the american mathematical society, 1931. The theorem of pappus and commutativity of multiplication. Pappus theorem definition of pappus theorem by the. The surface area of the cylinder, not including the top and bottom, can be computed from pappuss theorem since the surface is obtained by revolving its right side around its left side. Summarythe centroid theorems of pappus or the pappusguldin theorems, or the guldin theorems show deep connections between areas.
Full video on benchmark ktu mobile app download app in mathematics, pappus s centroid theorem also known as the guldinus theorem. Areas of surfaces of revolution, pappuss theorems iitk. Let s be the surface generated by revolving this curve about the xaxis. Euclidean version of pappuss theorem mathematics stack. Other than that he was born at alexandria in egypt and that his. Nowadays the theorem is known as pappus guldin theorem or pappus theorem. Let point, line, and incidence be primitive concepts. The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in. Other than that he was born at alexandria in egypt and that his career coincided with the first three decades of the 4th. Pappuss centroid theorem volume by george kotzabassis.
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 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. An application of pappus involution theorem in euclidean. A fourth century theorem for twentyfirst century calculus.
Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. James gregory and the pappus guldin theorem historical. Theorem of pappus and guldinus engineering mechanics. A simple proof for the theorems of pascal and pappus marian palej. Consider the curve c given by the graph of the function f. A simple proof for the theorems of pascal and pappus 3. I link to it here, because you may have been looking for it when you found this page. How are these theorems proved without using calculus. Create marketing content that resonates with prezi video.
James gregory and the pappusguldin theorem mathematical. In mathematics, pappuss centroid theorem also known as the guldinus theorem, pappus guldinus theorem or pappuss theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to pappus of alexandria and paul guldin. Pdf pappusguldin theorems for weighted motions researchgate. The usual pappus theorem is just the situation whereby the conic degenerates into a pair of lines.