In mathematics, pappuss centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. A classic example is the measurement of the surface area and volume of a torus. Pappus hat angedeutet, was guldin deutlich dargelegt. While most of the world refers to it as it is, in east asia, the theorem is usually referred to as pappus s theorem or midpoint theorem. A similar calculation may be made using the y coordinate of the. While most of the world refers to it as it is, in east asia, the theorem is usually referred to as pappuss theorem or midpoint theorem. Wikimedia commons has media related to pappusguldinus theorem. 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. David klein, sami hutchings, erin martin, ilam atmates of fall and winter 2007.
Proof of pappus theorem using affine geometry mathematics. Consider the curve c given by the graph of the function f. The theorems are attributed to pappus of alexandria and paul guldin. 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 traversed by. Nothing is known of his life, except from his own writings that he had a son named hermodorus, and was a teacher in alexandria. How are these theorems proved without using calculus. Jul 07, 2016 pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. James gregory and the pappusguldin theorem historical. German mathematician gerhard hessenberg proved that pappuss theorem implies desarguess theorem. Using the theorem of pappus and guldinuss, determi.
The proof above also shows that for pappuss theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a commutative field. Homework statement hey, im having issues with a problem, and my book doesnt seem to show me how to do it. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory. The first theorem of pappusguldinus says that the area of the sphere is given by a 2 rcl because we already know a 4 r2, we can solve this equation for rc in terms of r and l. Analytical expression of centroid, centre of mass and centre of gravity pappus guldinus. 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. 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 theorem from projective geometry. Now the second pappusguldin theorem gives the volume when this region is rotated through. Pappus chain of circles circles inscribed in an arbelos. An application of pappus involution theorem in euclidean.
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. 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. 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 published proof of the pappusguldin theorem appeared more than 20. First theorem is useful in calculating the surface of revolution of a given curve rotating around a given axis which does not intersect with the curve. James gregory and the pappusguldin theorem acknowledgements and references. Let s be the surface generated by revolving this curve about the xaxis. Areas of surfaces of revolution, pappuss theorems let f. Pappus was a greek geometer during the third century ad.
A video lecture that will explain both the theorems of pappus and guldinus with examples. A centroid is easily visualized as the center of gravity or center of mass of a flat. Long before the invention of calculus, pappus of alexandria ca. Areas of surfaces of revolution, pappuss theorems iitk. Pappus there gives a complete proof of the theorem that, if the distance of a point from a fixed point is in a given ratio to its distance from a fixed line, the locus of the point is a conic section which is an ellipse, a parabola, or a hyperbola according as the given ratio is less than, equal to, or greater than, unity. Original proof of pappus hexagon theorem mathoverflow. Pappus was a greek geometer during the third century ad his theorems about from eng 111 at rutgers university. The collection, his signature work, has been translated in its entirety in latin, french, german, and modern greek. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Me 2301 is a first semester, sophomore level class in statics. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. Pappus s 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 traversed by.
There are two theorems created by pappus and guldinus. Other than that he was born at alexandria in egypt and that his. A simple proof for the theorems of pascal and pappus. A torus may be specified in terms of its minor radius r and ma jor radius r by. 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. Pappus of alexandria greek mathematician britannica. Now the second pappus guldin theorem gives the volume when this region is rotated through. Use theorems of pappus and guldinus to calculate area created by revolving curve about an axis, or calculate the volume created by revolving area about an axis. Theorem of the day pappus theorem let a, b, c and a, b, c be two sets of collinear points.
It can be proved by pythagorean theorem from the cosine rule as well as by vectors. Now use theorem 1 to calculate the first contribution to surface area. A simplified proof of the pappusleisenring theorem. The collection, his signature work, has been translated in its entirety in latin, french. German mathematician gerhard hessenberg proved that pappus s theorem implies desarguess 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 ac and ca, ab and ba, and bc and cb lie on a common line called the pappus line of the configuration. James gregory and the pappusguldin theorem conclusion. Watch this short video on the first theorem, or read on below. In mathematics, pappus s centroid theorem also known as the guldinus theorem, pappus guldinus theorem or pappus s theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The proof above also shows that for pappus s theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a commutative field. There are two theorems, both saying similar things. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft2. An analytic proof of the theorems of pappus and desargues. 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. 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. If the region does not cross the axis, then the volume of the resulting solid of revolution is v 2. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Pappuss theorem also known as pappuss centroid theorem, pappusguldinus theorem or the guldinus theorem deals with the areas of surfaces of revolution. Using the theorem of pappus and guldinuss, determine the volume of the storage tank shown in the figure. Apolloniuss theorem is an elementary geometry theorem relating the length of a median of a triangle to the lengths of its sides. 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. Theorems of pappus and guldinus, centre of gravity and. Pappus theorem gives a proof and a java applet that lets you explore it. Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r. The first theorem states that the surface area a of a surface of revolution generated by rotating a plane curve c. 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.
Use the theorem of pappus to determine the surface area of this region as well. Pappuss centroid theorems are results from geometry about the surface area and volume of solids of revolution. Although pappus of alexandria is known mainly for his very informed commentaries on the work of earlier greek. The theorem of pappus and commutativity of multiplication. Using the theorem of pappus and guldinuss, determine the volume of the storage tank shown in. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. The main theorem of projective geometry that we will use is. Let a be a region in the upper half plane with boundary curve c, let e be the solid. James gregory and the pappusguldin theorem gregorys proof revealed. James gregory and the pappusguldin theorem selections from the gpu 1 james gregory and the pappusguldin theorem selections from. Pappuss centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Jul 18, 2015 use the theorem of pappus to determine the surface area of this region as well. 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. The centroid of a region is essentially the one point on which the region should balance.
To compute the volume of a solid formed by rotating a region. David eisenbud, mark green, and joe harris, cayleybacharach theorems and con. Prove pappuss centroid theorems without calculus physics. This is a partial version of desargues involution theorem see 3, p. These quantities can be computed using the distance traveled by the centroids of the curve and region being revolved.
Using motzkins theorem d3, further general analysis of possible simplices in the full matrix, including the last row, proves the main theorem. 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. Pappus was a greek geometer during the third century ad his. Theorems of pappus on surfaces of revolution wolfram. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Then the intersection points of the line pairs ab with ba, ac with ca and bc with cb are again collinear. Oct 25, 2017 a video lecture that will explain both the theorems of pappus and guldinus with examples. Pappusrelated things you might have been looking for when you found this page. Aug 01, 2017 use theorems of pappus and guldinus to calculate area created by revolving curve about an axis, or calculate the volume created by revolving area about an axis. An application of pappus involution theorem in euclidean and.
The centroid of a rectangle with vertices 0,0, x,0, 0,y, and x,y. 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. You have the full projective generality of the theorem, but are still asking for an affine proof. The first theorem of pappus states that the surface area s of a surface of revolution generated by the.
This is a generalization in a different direction from what the question asked for these references generalize in terms of finding volumes, but koundinya vajjha wanted a generalization in terms of finding the centroid. Pappus of alexandria was, and still is, a popular gure within the history of mathematics. The pappusguldin 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. Pappuss area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Determine the centroidal coordinate r c of a semicircular arc of radius r, given that the area of a sphere of radius r is known to be 4 r 2.
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 higher dimensional version by gray and miquel linked to below might yield this, but i havent read their paper yet. Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total. The theorem of pascal concerning a hexagon inscribed in a conic.
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