Theorem of pappus pdf download

The main theorem of projective geometry that we will use is. Request pdf the axiomatic destiny of the theorems of pappus and desargues we present the largely twentieth century history of the discovery of the. Pdf pappusguldin theorems for weighted motions researchgate. An application of pappus involution theorem in euclidean. Theorem of pappus synonyms, theorem of pappus pronunciation, theorem of pappus translation, english dictionary definition of theorem of pappus. Latin, old man, down on certain seeds, from greek pappos. A centroid is easily visualized as the center of gravity or center of mass of a flat. Expert answer 100% 1 rating previous question next question get more help from chegg.

Consider the curve c given by the graph of the function f. Summary of the formulas for plane laminas and curves 1. The first two books were devoted to arithmetic, and the third through fifth books deal primarily with geometry. 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. How to prove pappus theorem mathematics stack exchange.

Any stretching of rin9 would provide a euclidean stretching of b, necessarily satisfying the premises of the main theorem. Much of it is overwhelming, and some of it is wrong. The pappusguldin theorems suppose that a plane curve is rotated about an axis external to the curve. Summarythe centroid theorems of pappus or the pappusguldin. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. In mathematics, pappuss 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. An expression in the exterior algebra of a peano space yielding pappus theorem was originally given by doubilet, rota, and stein. 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 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.

Theorem of pappus definition of theorem of pappus by the. Let cp be a sufficiently smooth simple closed curve that bounds a region 5d. Long before the invention of calculus, pappus of alexandria ca. 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. Pappuss area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Pappus centroid theorem pdf pappus centroid theorem pdf pappus centroid theorem pdf download. Now the second pappusguldin theorem gives the volume when this region is rotated through.

Pappus centroid theorem pdf the surface of revolution generated by a smooth curve. 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. Original proof of pappus hexagon theorem mathoverflow. Let the plane of d move on es in such a way that a the point ro is always on. A similar calculation may be made using the y coordinate of the. Let ro be a fixed point in the plane of d but not necessarily contained in sd. The axiomatic destiny of the theorems of pappus and. 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. Pappuss 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. Engineering mechanics pdf material is available for all the students who are visited this page at free of cost. Pappus definition of pappus by the free dictionary.

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. It is well known that pappus theorem implies the commutativity of the multiplication in. Pdf on mar 1, 2006, ximo gualarnau and others published pappusguldin theorems for weighted motions find, read and cite all the. 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 the centroid of d during the revolution. Pappus, working in alexandria about 600 years after euclid, made valuable compilations of greek mathematics, as well as. Theorems of pappus on surfaces of revolution wolfram.

Let es be a sufficiently smooth space curve on which the curvature ks is never zero. Let a be a region in the upper half plane with boundary curve c, let e be the solid of revolution formed by rotating a about the. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. Areas of surfaces of revolution, pappuss theorems let f. 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 is either of two related theorems dealing with the. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Some fundamental theorems in mathematics oliver knill abstract. The proposition that the volume of a solid of revolution. How are these theorems proved without using calculus. Theorems of pappus and goldinus mechanical engineering. A fourth century theorem for twentyfirst century calculus.

We have provided all the pdf links to download study material for all the 1st year engineering b. There are two theorems, both saying similar things. Lesson 45 centroid theorem of pappus guldinus volume and surface area duration. Criteria for the current list of 172 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide 6 without leading to panic. Pappus theorem article about pappus theorem by the. Pappuss centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Areas of surfaces of revolution, pappuss theorems iitk. Pappuss collection smith, david eugene, bulletin of the american mathematical society, 1934. Z b a fx 2 dx, the familiar formula for volume of solid of revolution.

Pappus was the author of mathematical collections in eight books, only the last six of which are extant. Use the theorem of pappus to find the volume of the given solid. These three points are the points of intersection of the opposite sides of the 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. Of these pappus of alexandria was among the last, and from the point of view of. The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in. Pappus and desargues finite geometries linkedin slideshare. Greek mathematician of the second half of the third century. The theorems are attributed to pappus of alexandria and paul guldin. An expository hitchhikers guide to some theorems in mathematics. 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.

Let s be the surface generated by revolving this curve about the xaxis. Euclidean version of pappuss theorem mathematics stack. The theorem of pascal concerning a hexagon inscribed in a conic. Pappus article about pappus by the free dictionary. Research monograph 20 pdf download the companion programs octapet and bonepet. Pappus definition, a downy, bristly, or other tuftlike appendage of the achene of certain plants, as the dandelion and the thistle.

Changing the angle of the lines ac or ac, or the position of the points b or b on these lines, will change the length or slope of xz but keep it collinear with y. This lead to an alternative statement and proof of pappuss theorem retrieving pappuss and hesse configurations of lines as special points in complex projective grassmannian. This is precisely what pappus centroid theorem gives. A simple proof for the theorems of pascal and pappus. 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. For the love of physics walter lewin may 16, 2011 duration. Generalizations of the theorems of pappusguldin in the heisenberg groups. Nothing is known of his life, except from his own writings that he had a son named hermodorus, and was a teacher in alexandria. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. The solid obtained by rotating the triangle with vertices 5, 2, 5, 4, and 8, 3 about the xaxis.