Pappus theorem pdf writer

Use the theorem of pappus to determine the surface area of this region as well. Pappus article about pappus by the free dictionary. The first two books were devoted to arithmetic, and the third through fifth books deal primarily with geometry. A theorem for cubica generalization of carnot theorem. 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. Spin b around the x axis, creating a shape of revolution. Let s be the surface generated by revolving this curve about the xaxis.

How are these theorems proved without using calculus. Pdf orthopoles and the pappus theorem researchgate. 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. As a writer, pappus must have been quite versatile if the following list of. Other than that he was born at alexandria in egypt and that his. Long before the invention of calculus, pappus of alexandria ca. The proposition that the volume of a solid of revolution. Pappus was the author of mathematical collections in eight books, only the last six of which are extant.

The centroid of a region is essentially the one point on which the region should balance. Z b a fx 2 dx, the familiar formula for volume of solid of revolution. The inscribed circles are tangent to one another, and to the boundaries of the arbelos. James gregory and the pappusguldin theorem conclusion. 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. Then three pairwise intersections 1 bc bc, 2 ac ac, and 3 ab ab are incident to a third straight line. 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 admits an interesting geometrical interpretation. 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 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. Compute the volume of the shape using cylindrical coordinates. Euclidean version of pappuss theorem mathematics stack.

We have already mentioned desargues theorem as an example of a result which. Theon made a marginal note in one of his manuscripts stating that pappus wrote during the reign of roman emperor diocletian, which places him in the period from 284 to 305 ad, but it also seems. 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. 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 theorem of pappus and commutativity of multiplication. 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. The first theorem states that the surface area a of a surface of revolution generated by rotating a plane curve c. Its power is illustrated by proving with it some theorems about euclidean and noneuclidean polygons of di erent types. The pappuss theorem can also be used in reverse to find the centroid of a curve or figure. Pappuss centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. 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 trunk, the cylinder, and the solid of revolution. Pappus of alexandria was a greek mathematician who lived around the end of the third century ad, although the exact date is uncertain. To satisfy 2, we would require at a minimum that each switch connects the same number of users. Consider the curve c given by the graph of the function f.

Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r. Here are some definitions, which culminate in a formal definition of a convex. Pappuss centroid theorems are results from geometry about the surface area and volume of solids of revolution. The theorems are attributed to pappus of alexandria and paul guldin wikimedia commons has media related to pappus guldinus theorem. Every f2cx of degree ncan be factored into nlinear factors. 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. From pascals theorem to d constructible curves will traves abstract.

If p is a projective point 1dimensional subspace then p. 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. Pappuss area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. 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. A bridge between algebra and geometry article pdf available in the american mathematical monthly 1096 june 2002 with 2,548 reads how we measure reads.

Theorem of pappus to find volume of revolution calculus 2. Areas of surfaces of revolution, pappuss theorems iitk. Section 6 and is the key to our proof of pappustheorem. The rst writer to assert that any nth degree polynomial has a root is peter roth in 1600 334, proven rst by carl friedrich gauss and nalized in 1920 by alexander ostrowski who xed a topological mistake in gauss proof. Let us find the area of the surface generated by revolving the curve y 1. We prove a generalization of both pascals theorem and its converse, the braikenridge maclaurin theorem. Now the second pappusguldin theorem gives the volume when this region is rotated through. In the situation with zero slope both lines are parallel and the intersection point vanishes. The first example shows the picture most often drawn in textbooks with the final.

The history of mathematics cite on the link will give information about pappas and some of his work. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. 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. To compute the volume of a solid formed by rotating a region. A communications network what are the desirable properties of the switching box. For instance, if b is a circle the result is a torus. Then pappus theorem implies that the volume of this torus is the same as that of a cylinder with base sand height 2.

There are two theorems, both saying similar things. Contributor pappus alexandrinus, greek mathematician, approximately 3rd or 4th century ad. Any stretching of rin9 would provide a euclidean stretching of b, necessarily satisfying the premises of the main theorem. All structured data from the file and property namespaces is available under the creative commons cc0 license.

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. Pappus involution theorem is a powerful tool for proving theorems about noneuclidean triangles and generalized triangles in cayleyklein models. These quantities can be computed using the distance traveled by the centroids of the curve and region being revolved. This proof, my current favourite, shows that the pappus con guration \closes if and only if two numbers a and b commute.

Greek mathematician of the second half of the third century. Theorem of the day pappus theorem let a, b, c and a, b, c be two sets of collinear points. A centroid is easily visualized as the center of gravity or center of mass of a flat. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. 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. This means that p is a point on the surface of uv if. Pappus theorem article about pappus theorem by the. Theorems of pappus on surfaces of revolution wolfram. Nothing is known of his life, other than what can be found in his own writings. An analytic proof of the theorems of pappus and desargues. 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. The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in. The centroid of a rectangle with vertices 0,0, x,0, 0,y, and x,y. Center of mass, pappus theorem pappus theorem let b be a blob in the xy plane, floating entirely above the x axis.

These three points are the points of intersection of the opposite sides of the hexagon. When the region sis rotated around e, a torus with section sis obtained. Files are available under licenses specified on their description page. 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. 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. If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irreducible curve c of degree d, then the remaining k. If the region does not cross the axis, then the volume of the resulting solid of revolution is v 2. The fewest number but bigger than 1 of switches should be used. Areas of surfaces of revolution, pappuss theorems let f.

An application of pappus involution theorem in euclidean. A similar calculation may be made using the y coordinate of the. The chain of inscribed circles is sometimes called a pappus chain, for pappus of alexandria, who studied and wrote about it in the 4th century a. The theorem of pascal concerning a hexagon inscribed in a conic. A simple proof for the theorems of pascal and pappus. The only objects involved in the statement of pappuss theorem are points and lines and the only re. The following suggestions are leading to a relationship in plane geometry attributed to pappus. Then the intersection points of the line pairs ab with ba, ac with ca and bc with cb are again collinear. We may write y wejj, where w is a word in the box operations t.