James gregory and the pappusguldin theorem latin originals. Both subjects were discussed with pickert in the last year of his life. Suppose r is revolved about the line l which does not cut. Pappus of alexandria, mathematical collection iv 2125, pp. James gregory and the pappusguldin theorem conclusion. Pappus was a greek geometer during the third century ad his theorems about from eng 111 at rutgers university. Then, using the second theorem of pappusguldinus, compute the volume of the solid generated by revolving the shaded area about the aa axis. In this article w egiv an analytic proofpappus theorem and. Chiltern announces agreement to acquire theorem clinical research. His great work a mathematical collection is an important source of information about ancient greek mathematics. In the situation with zero slope both lines are parallel and the intersection point vanishes.
The pappusguldin theorems suppose that a plane curve is rotated about an axis external to the curve. He noted14 that 6th degree curves have 27 parameters, whilst the corresponding pappus problem would involve 11 or 12 lines. 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. How are these theorems proved without using calculus. Remarks on orthocenters, pappus theorem and butterfly.
But the 12 line problem requires that d 1d 2d 3d 4d 5d. Jiwen he, university of houston math 1431 section 24076, lecture 23 december 4, 2008 16 16. As its a bit long, the full source and resulting pdf can be found on my webpage. 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. Introduction a century ago, mathematics history began with the greeks, then skipped a thousand years and continued with developments in the european renaissance. Modernization is a vague concept that tends to dissolve when one focuses on the concrete historical process, but its comprehensiveness makes it a useful starting point. Prioritybeing the first to publish a proof of the resultcertainly isnt an issue here. They show that the second theorem you stated has a generalization when the circle that the centroid of f travels on is replaced with any sufficiently smooth simple closed space curve whose curvature never. Compute the volume of the shape using cylindrical coordinates. The theorems are attributed to pappus of alexandria and paul guldin. Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r.
Treasury of analysis ancient geometry books britannica. Pappuss centroid theorem volume by george kotzabassis on prezi. However, planning for post merger integration typically focuses on operational issues such as harmonizing product lines. Using integration, compute both the area and the centroidal distance xc of the shaded region. 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.
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. After this the point comes back from a very far position on. Business wire chiltern, a leading global contract research organization cro, announced it has signed a purchase agreement to acquire theorem clinical research. 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. In his principia 1687 newton also found inspiration in pappus. Prove pappuss centroid theorems without calculus physics. Create marketing content that resonates with prezi video. Filename, size file type python version upload date hashes. Indeed, descartes claimed that every algebraic curve is the solution of a pappus problem of nlines, which newton shows to be false. The pappus consists of one to many dry scales, awns small pointed processes, or capillary hairlike bristles. You may combine the document with other documents released under this. Areas of surfaces of revolution, pappuss theorems let f. Jul 18, 2015 use the theorem of pappus to determine the surface area of this region as well. Center of mass, pappus theorem pappus theorem let b be a blob in the xy plane, floating entirely above the x axis.
There was sometimes a brief mention that the arabs preserved greek knowledge during the dark ages so that it was avail. Pappus s centroid theorem may refer to one of two theorems. Dec 11, 2015 we present a generalization of the notion of the orthocenter of a triangle and of pappus theorem. The theorem of pappus can be either one of two related theorems that can help us derive formulas for the volumes and surface areas of solids or surfaces of revolution they are named after pappus of alexandria, who worked on them. An analytic proof of the theorems of pappus and desargues. The reader might be left wondering why we should be interested in james gregorys proof of this result. Pappuss theorempappuss theorem proves that the volume of the solid torus obtained by rotating the disk of radius a around line l that is b units away is. Adding the zero vector given by 5 to this representation of q, we obtain the following representation of q. Idea the theorem of pappus allows the centroid of a region to be used as a from cal at ecole normale superieure.
Idea the theorem of pappus allows the centroid of a region to. There are two results of pappus which relate the centroids to surfaces and solids of revolutions. 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. The analytic proof involved demonstrating a relationship between the sought object and the given ones such. Other articles where treasury of analysis is discussed. Pappus definition of pappus by the free dictionary. Consider the curve c given by the graph of the function f. 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. Construction of archimedes spiral and basic symptom. Then three pairwise intersections 1 bc bc, 2 ac ac, and 3 ab ab are incident to a third straight line. Solved using integration, compute both the area and the. Watch this short video on the first theorem, or read on below.
Let s be the surface generated by revolving this curve about the xaxis. Dtd, and standardconforming simple html, postscript or pdf designed for human modification. Pappus problem thus inspired the new method of analytical geometry that has proved such a powerful tool in subsequent centuries. Pappus was a greek geometer during the third century ad his. Pappuss theorem 1 3 2 4 5 6 9 8 7 the collinearity of 123, 456, 159, 168, 249, 267, 348, 357 imples the collineartity of 7,8,9. Spin b around the x axis, creating a shape of revolution. The section contains the latin originals of propositions 23,27, 29, 31, 33, and 35. Pappusguldin rule the area of the lateral surface of a solid of revolution is equal. Jul 07, 2016 pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. This means that p is a point on the surface of uv if and only if there is a point so, to.