Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft2. By corrected axiom 3, there is a line not containing x. 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. Pdf a geometric identity for pappus theorem michael. Pappus s centroid theorem may refer to one of two theorems.
Now the second pappusguldin theorem gives the volume when this region is rotated through. 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. The axiomatic destiny of the theorems of pappus and desargues. The theorems are attributed to pappus of alexandria and paul guldin.
Pappus, working in alexandria about 600 years after euclid, made valuable compilations of greek mathematics, as well as. The concept of projectivity lies at the very heart of projective geometry and provides. 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 surface area of a solid of revolution is the arc length of the generating curve multiplied by.
Theorem of pappus tells us that volume is equal to area of the plane region, times the distance traveled by the centroid of the same plane region, if the. Areas of surfaces of revolution, pappuss theorems let f. If one measures the ratio applicability over the di culty of proof, then this theorem even beats pythagoras, as no proof is required. Generalizations of the theorems of pappusguldin in the heisenberg. 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.
Consider the curve c given by the graph of the function f. Pappus theorem kelly mckinnie history pappus theorem geometries picturing the projective plane lines in projective geometry back to pappus theorem proof of pappus theorem pappus of alexandria pappus theorem as you can see from the java applet, pappus theorem should really read. In this note we prove some theorems on orthopoles by using a well known result from projective geometry, the pappus theorem. An analytic proof of the theorems of pappus and desargues. His great work a mathematical collection is an important source of information about ancient greek mathematics. The theorem of pappus concerning the volume of a solid of revolution can be found in any book on the calculus and runs as follows. A similar calculation may be made using the y coordinate of the. 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. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. It is by some considered to the theory of probability what the pythagoras theorem is to geometry. Consider two straight lines emanating from point o and containing the points p 1 through p 6 as shown in the figure below. Areas of surfaces of revolution, pappuss theorems iitk.
Contributor pappus alexandrinus, greek mathematician, approximately 3rd or 4th century ad. An analogue to pappus chain theorem with division by zero. Theorem of pappus to find volume of revolution calculus 2. An expression in the exterior algebra of a peano space yielding pappus theorem was originally given by doubilet, rota, and stein doubilet, p. The pappusguldin theorems suppose that a plane curve is rotated about an axis external to the curve. Pappus theorem on volumes department of mathematics. This is precisely what pappus centroid theorem gives. 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. 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. A simple proof for the theorems of pascal and pappus.
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. 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. This is the theorem of pappus or the pappus guldin theorem. Pdf orthopoles and the pappus theorem semantic scholar. There are two results of pappus which relate the centroids to surfaces and solids of revolutions. It is well known that pappus theorem implies the commutativity of the multiplication in the field k of segment arithmetic see the discussion in 3 and a proof of this fact in 4, pp. Dec 22, 2019 applications of the theorems of pappus. Letting l ij denote the line through points p i and p j, and letting a,b,c denote the points of intersection between the pairs of lines l 15,l 24, l 16,l 34, and l 35,l 26 respectively, pappus theorem asserts that the points a,b,c lie on a straight line. This is a partial version of desargues involution theorem see 3, p. 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.
Euclidean version of pappuss theorem mathematics stack. Theorem of pappus and guldinus engineering mechanics. His great work a mathematical collection is an important source of information about ancient greek. The theorem of pascal concerning a hexagon inscribed in a conic. Moreover, very little is known of what his actual contributions were or even exactly when he lived.
May 24, 2014 theorem of pappus to find volume of revolution calculus 2. Mar 25, 2018 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. 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 chain, we give a theorem analogue to pappus chain theorem.
Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Pappus theorem the cross ratio of four lines of a pencil of lines equals the cross ratio of the four points at which an arbitrary line cuts the lines. 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. It is best to position a beyond cin order to construct. Pappus centroid theorem pdf pappus centroid theorem pdf pappus centroid theorem pdf download. 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. We denote the intersection of two lines g and g by g. 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. From collectiones mathematicae, pappus of alexandria, fourth century a. Pappus centroid theorem pdf the surface of revolution generated by a smooth curve. Pappus of alexandria greek mathematician britannica. Other than that he was born at alexandria in egypt and that his.
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. Pappus theorem article about pappus theorem by the free. 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. 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. The pappusguldin theorem states the method of finding volumes and surface areas respectively for any solid of revolution into two parts. Summary of the formulas for plane laminas and curves 1. It remains to prove pappus theorem and theorems 3 and 4. Suppose r is revolved about the line l which does not cut. Bayes theorem was rst proven in 1763 by thomas bayes. James gregory and the pappusguldin theorem mathematical. Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total. In englishspeaking countries, these two theorems are known as pappuss theorems, after the ancient greek geometer pappus of alexandria. An application of pappus involution theorem in euclidean and.
In continental europe, these theorems are more commonly associated with the name of paul guldin who rediscovered them. The first theorem of pappus states that the surface area s of a surface of revolution generated by the. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. What we need is a simple affine theorem which is a special case of the pappus theorem. Section 6 and is the key to our proof of pappustheorem. The pappus guldin theorems suppose that a plane curve is rotated about an axis external to the curve.
Notably, we need not even use it in the general case. Theorems of pappus and goldinus mechanical engineering notes. 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. The second theorem of pappus guildinus gives the volume as v 2 40360r c a 1 where r c is the distance to the centroid of the generating area, a is the magnitude of the area, and the factor 40360 accounts for the fact that the dam corresponds to 40 rather than to a complete circle. The axiomatic destiny of the theorems of pappus and. The main theorem of projective geometry that we will use is.835 731 1254 64 378 1273 1544 1534 568 1362 519 990 754 631 752 1630 1305 1472 507 1110 692 1168 244 263 532 181 227 820 1135 1023 345 244 1223 409 728