The main theorem of projective geometry that we will use is. A simple proof for the theorems of pascal and pappus. Prove pappuss centroid theorems without calculus physics. I would like to use only methods from those books, as proportions havent been introduced. There are two results of pappus which relate the centroids to surfaces and solids of revolutions.
I have seen the proof using thales theorem, and indeed it is much nicer. The book covers both statics and dynamics, and provides the students with a clear and thorough presentation of the theory as well as the applications. The pappus guldin theorems suppose that a plane curve is rotated about an axis external to the curve. 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. The first theorem of pappus states that the surface area s of a surface of revolution generated by the. 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.
To interpret the explanations on or computation meets knowledge you need to know what a centroid is. 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. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. Any stretching of rin9 would provide a euclidean stretching of b, necessarily satisfying the premises of the main theorem. 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. 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. Center of mass and centroids indian institute of technology. Vela murali, engineering mechanics, oxford university press 2010 engineering mechanics syllabus unit i. Statics and dynamics, 8th edition, tata mcgrawhill publishing company, new delhi 2004. An analytic proof of the theorems of pappus and desargues. The way in which mechanics has been taught in statics 9th edition by r. The students completing this course are expected to understand the concepts of forces and its resolution in different planes, resultant of force system, forces acting on a body, their free body diagrams using graphical methods.
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. These are lemmas xii, xiii, xv, and xvii in the part of book vii consisting of lemmas to the first of the three books of euclid s porisms. 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. Engineering mechanics pdf notes em pdf notes smartzworld. The theorems are attributed to pappus of alexandria and paul guldin. 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.
Jiwen he, university of houston math 1431 section 24076, lecture 23 december 4, 2008 16 16. An application of pappus involution theorem in euclidean and. Engineering mechanics is tailormade as per the syllabus offered in the first year of undergraduate students of engineering. Other than that he was born at alexandria in egypt and that his. Euclidean version of pappuss theorem mathematics stack.
How are these theorems proved without using calculus. Aug 25, 2015 there are two theorems, both saying similar things. 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. In mathematics, pappuss centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
The centroid of a region is essentially the one point on which the region should balance. Jul 18, 2015 use the theorem of pappus to determine the surface area of this region as well. Areas of surfaces of revolution, pappuss theorems iitk. Theorems of pappus guldinus 532 distributed loads on beams 539 forces on submerged surfaces 544 volumes center of gravity of a threedimensional body 552. Contributor pappus alexandrinus, greek mathematician, approximately 3rd or 4th century ad. I believe hartshorne refrains from using this as his book only assumes knowledge of the first four books of the elements thus far. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Pappus of alexandria greek mathematician britannica. 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. The theorem of pascal concerning a hexagon inscribed in a conic. Me 2301 is a first semester, sophomore level class in statics. In its earliest known form, pappuss theorem is propositions 8, 9, 141, and 143 of book vii of pappuss collection.
The centroid theorems of pappus, as they are often called, appear in many calculus textbooks today, but usually without the same prominence. 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. 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. Jul 07, 2016 pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Engineering mechanics pdf 1st year notes pdf download books. Hibbeler is very simple from very few easy example to high powerfull examples and problems has been discussed i really become very happy after reading 9th edition. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. 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. Jan 22, 20 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 plane region is revolved around the xaxis. In this article w egiv an analytic proofpappus theorem and. Summary of the formulas for plane laminas and curves 1.
Forder, the calculus of extension, cambridge university press. Now the second pappusguldin theorem gives the volume when this region is rotated through. Theorems of pappus and guldinus the theorems of pappus and guldinus are used to find the surfaces area and volume of any object of revolution provided the generating curves and areas do not cross the axis they are rotated surface area area of a surface of revolution product of length of the curve and distance traveled by the centroid in. Full video on benchmark ktu mobile app download app in mathematics, pappus s centroid theorem also known as the guldinus theorem.
The usual pappus theorem is just the situation whereby the conic degenerates into a pair of lines. There is also a french translation 1933 of the existing books see the. A fourth century theorem for twentyfirst century calculus. 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. Oct 08, 2008 homework statement hey, im having issues with a problem, and my book doesnt seem to show me how to do it. In the projective environment, a modern version of pappus theorem is. As an independent contribution pappus formulated the volume of a solid of revolution, the result we now call the the pappus guldin theorem. Engineering mechanics pdf notes em pdf notes unit v analysis of perfect frames analytical method types of frames assumptions for forces in members of a perfect frame, method of joints, method of sections, force table, cantilever trusses, structures with one end hinged and the other freely supported on rollers carrying. Media in category pappus theorem the following 36 files are in this category, out of 36 total.
The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in. Suppose r is revolved about the line l which does not cut. Alternatively, given a mystic hexagon, the pappus con. This is a partial version of desargues involution theorem see 3, p. When the six points are ordered as a, f, b, d, c, f the resulting polygon is just pascals mystic hexagon. 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. Guldin 15771643 most of the remaining of the treatise is collections of lem.
1504 901 497 38 645 558 519 416 1033 1018 1412 1471 593 540 1200 1112 457 266 1494 367 673 249 585 616 1493 562 705 868 238 387 79 1136 1370