Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. 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. Using the theorem of pappus and guldinuss, determine the volume of the storage tank shown in the figure. Theorems of pappus on surfaces of revolution wolfram. The theorems are attributed to pappus of alexandria and paul guldin. Me 2301 is a first semester, sophomore level class in statics. The pappusguldin theorem states the method of finding volumes and surface areas. 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. Answer to use the second pappusguldinus theorem to determine the volume generated by revolving the curve about the y axis.
Jan 01, 2001 pappus theorem submitted by plusadmin on january 1, 2001. The only proof of pappus s theorem ive ever seen is the one given here, using homogeneous coordinates. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft2. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal grid i. Suppose r is revolved about the line l which does not cut. Pappus of alexandria, flourished ad 320, 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.
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. In the projective environment, a modern version of pappus theorem is. 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. Suppose we have three points on one line represented by vectors a, b, and c and three points on a nonparallel line represented by vectors d, e, and f. I think that this needs a bit more explanation further down in the main article. 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. Euclidean version of pappuss theorem mathematics stack. Pappuss theorem appears in his text synagogue 17, a collection of classical greek geometry with insightful commentary. The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in. 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. If the pappus line u \displaystyle u and the lines g, h \displaystyle g,h have a point in common, one gets the socalled little version of pappus s theorem 2. Use the theorem of pappus to determine the surface area of this region as well.
To illustrate pappus s theorem, consider a circular. Theorem of pappus synonyms, theorem of pappus pronunciation, theorem of pappus translation, english dictionary definition of theorem of pappus. The conjecture was finally proven by hales 1999, 2001. May 24, 2014 if youd like to make a donation to support my efforts look for the tip the teacher button on my channels homepage. Nothing is known of his life, other than what can be found in his own writings.
Prove pappuss centroid theorems without calculus physics. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. 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. For gregory, the pappusguldin theorem and quite a few other results are easy consequences of a broader geometrical perspectivethat is, a perspective involving ratios between the. His great work a mathematical collection is an important source of information about ancient greek mathematics. The regular tessellation consisting of regular hexagons. 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. An analytic proof of the theorems of pappus and desargues. Pappus s 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.
Now the second pappusguldin theorem gives the volume when this region is rotated through. Any stretching of rin9 would provide a euclidean stretching of b, necessarily satisfying the premises of the main theorem. In this article w egiv an analytic proofpappus theorem and. To compute the volume of a solid formed by rotating a region. The following suggestions are leading to a relationship in plane geometry attributed to pappus.
How are these theorems proved without using calculus. Desarguess theorem is true for the real projective plane, for any projective space defined arithmetically from a field or division ring, for any projective space of dimension unequal to two, and for any projective space in which pappus s theorem holds. There are two results of pappus which relate the centroids to surfaces and solids of revolutions. James gregory and the pappusguldin theorem conclusion. The arc length of its right side is h h h and the distance traveled by its centroid is simply 2. Use the second pappusguldinus theorem to determine the volu. The only proof of pappuss theorem ive ever seen is the one given here, using homogeneous coordinates. Pappus s hexagon theorem theorem that, if the vertices of a hexagon lie alternately on two lines, then the three pairs of opposite sides meet in three collinear points upload media.
Convolution and parsevals theorem multiplication of signals multiplication example convolution theorem convolution example convolution properties parsevals theorem energy conservation energy spectrum summary e1. The history of mathematics cite on the link will give information about pappas and some of his work. Pappus also discusses the three and four lines theorem of apollonius. Theorem of pappus definition of theorem of pappus by the. The theorem of pappus and commutativity of multiplication. A centroid is easily visualized as the center of gravity or center of mass of a flat.
Contributor pappus alexandrinus, greek mathematician, approximately 3rd or 4th century ad. Pappus s theorem appears in his text synagogue 17, a collection of classical greek geometry with insightful commentary. Answer to use the second pappus guldinus theorem to determine the volume generated by revolving the curve about the y axis. We prove several theorems on orthopoles using the pappus theorem. Areas of surfaces of revolution and the theorems of pappus. Dec 25, 2011 areas of surfaces of revolution and the theorems of pappus. Construct external parallelograms on sides ab and ac.
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. Pdf on jun 1, 2002, elena anne marchisotto and others published the theorem of pappus. How to prove pappus theorem mathematics stack exchange. 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. Generalizations of the theorems of pappusguldin in the heisenberg. Z b a fx 2 dx, the familiar formula for volume of solid 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. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Aug 25, 2015 there are two theorems, both saying similar things.
Other than that he was born at alexandria in egypt and that his. The euclidean version of desargues theorem shows how projective geometry can provide an. Pappus centroid theorem provides a very simple way of computing the. From pappus theorem to the twisted cubic springerlink. Long before the invention of calculus, pappus of alexandria ca. The centroid of a region is essentially the one point on which the region should balance.
A similar calculation may be made using the y coordinate of the. However, there are many planes in which desarguess theorem is false. Pdf orthopoles and the pappus theorem researchgate. This proof, my current favourite, shows that the pappus con guration \closes if and only if two numbers a and b commute. Nothing is known of his life, except from his own writings that he had a son named hermodorus, and was a teacher in alexandria. Pappus theorem submitted by plusadmin on january 1, 2001. If one restricts the projective plane such that the pappus line is the line at infinity, one gets the affine version of pappus s theorem shown in the second diagram.
If youd like to make a donation to support my efforts look for the tip the teacher button on my channels homepage. 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. 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. Another exercise using pappus centroid theorem ucr math. The centroid of a rectangle with vertices 0,0, x,0, 0,y, and x,y. Pappuss area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Other than that he was born at alexandria in egypt and that his career coincided with the first three decades of the 4th. Desarguess theorem is true for the real projective plane, for any projective space defined arithmetically from a field or division ring, for any projective space of dimension unequal to two, and for any projective space in which pappuss theorem holds. The volume equals the product of the area of the region being rotated times the distance traveled by the centroid of the region in one rotation. Thanks for contributing an answer to mathematics stack exchange. Pappus involution theorem is useful for proving incidence rela. David hilbert observed that pappuss theorem is equivalent to the claim that the multiplication of lengths is commutative see, e. There are two theorems, both saying similar things. Pappus of alexandria greek mathematician britannica.
Jul 07, 2016 pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. Guldin 15771643 most of the remaining of the treatise is collections of lemmas that will assist the readers understanding of the original works. 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 three points are the points of intersection of the opposite sides of the hexagon. These quantities can be computed using the distance traveled by the centroids of the curve and region being revolved. 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. We discuss a classical result in planar projective geometry known as steiners theorem involving 12 interlocking applications of pappus theorem.
Pappuss hexagon theorem theorem that, if the vertices of a hexagon lie alternately on two lines, then the three pairs of opposite sides meet in three collinear points upload media. 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. Pappuss hexagon theorem disambiguation page providing links to topics that could be referred to by the same search term this disambiguation page lists. Pappuss centroid theorems are results from geometry about the surface area and volume of solids of revolution. Pappuss centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet.
Is there some height h of a rectangle of length 2 such that a semicircular cap will move its centroid exactly one unit to the right. We prove this result using three dimensional projective geometry then uncover the dynamics of this construction and relate them to the geometry of the twisted cubic. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Pdf if the vertices of a triangle are projected onto a given line, the per pendiculars. If the region does not cross the axis, then the volume of the resulting solid of revolution is v 2. Pappus s hexagon theorem disambiguation page providing links to topics that could be referred to by the same search term this disambiguation page lists mathematics articles associated with the same title. Use the second pappusguldinus theorem to determine the. A method for finding the volume of a solid of revolution.
1337 245 1412 1428 650 1484 1429 1600 521 316 286 219 100 1503 407 774 316 598 427 11 74 274 778 892 1310 1292 1433 339 932 458 661 1425 1068 57