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. I also undertake a discussion of dutch book theorems in general and remark on. James gregory and the pappus guldin theorem conclusion. The second theorem of pappus is very similar to the first. Tbtf and tbtj in economics, the term usually refers to a sequence of trades that would leave one party strictly worse off and another strictly better off. Learn how to use the theorem of pappus to find the volume of a solid, in this particular case, a right circular cone. The author would like to thank dennis schneider for his help in using mathematica to prepare the 3d images used in this article. Watch this short video on the first theorem, or read on below. 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. It is associated with probabilities implied by the odds not being coherent. This proof appears in the book iv of mathematical collection by pappus of.
Centroid theorem of pappus guldinus volume and surface. Although, the last part of the question describe a dutch book. We say that a quadrilateral in euclidean, hyperbolic or ellip. 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. T l heath, the thirteen books of euclids elements 3 volumes new york, 1956. A geometric impossibility theorem can be conceived as the negative counterpart of a the. The dutch book theorem shows that, if your credences are not probabilistic, then theres a series of decision problems and a dominated series of options from them that those credences require you to choose. Translations into english are somewhat more piecemeal. There are two theorems, both saying similar things. They are named after pappus of alexandria, who worked on them. 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. This book was originally published in dutch, and this will be the first english. Contributor pappus alexandrinus, greek mathematician, approximately 3rd or 4th century ad.
Jul 18, 2015 use the theorem of pappus to determine the surface area of this region as well. Ramsey 1926 and finetti 1937, offers prudential grounds for action in conformity with personal probability. Dutch book is the very means las vegas, racetracks, and brokerage houses make their dough. Quite bluntly, the logical structure of newtons book is slipshod, its level of verbal. The theorems are attributed to pappus of alexandria and paul guldin. James gregory and the pappusguldin theorem historical. The six and a half extant books as mentioned book i and book ii are partly lost were first edited and translated into latin by commandinus in 1588 pappus of alexandria mathematicae collectiones a federico commandino urbinate in latinum conversae, et commentariis illustratae. 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 published proof of the pappus guldin theorem appeared more than 20 years before. A bridge between algebra and geometry article pdf available in the american mathematical monthly 1096 june 2002 with 2,579 reads how we measure reads.
Mar 03, 2009 the theorem of pappus says that you can take the area of a figure and multiply it times the distance traveled by the centroid when it is rotated about some axis. Dutch book theorem is a type of probability theory that postulates that profit opportunities will arise when inconsistent probabilities are assumed in a. It tells us that the volume of a solid of revolution which is generated by rotating a plane figure about an external axis equals the area of the plane figure call this a times the distance d which is traveled by the geometric centroid of f. Areas of surfaces of revolution and the theorems of pappus. To compute the volume of a solid formed by rotating a region. I will also suggest that any successful dutch book defense of bayesianism cannot be disentangled from decision theory. Theorem of pappus tells us that volume is equal to area of the plane region. Finding surface area and volume of a sphere using only pappus centroid theorem. Consider the curve c given by the graph of the function f. Is there a way the bookie can find a dutch book against the gamblers. Of course, students might prefer to use pappus dual theorem. 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.
Pappus of alexandria 290 350, was a hellenized egyptian born in alexandria. The present study argues that a number of theorems on spirals in pappus collectio are based on early. Pascals theorem is the polar reciprocal and projective dual of brianchons theorem. It is by some considered to the theory of probability what the pythagoras theorem is to geometry. Pappus theorem article about pappus theorem by the free. A type of probability theory that postulates that profit opportunities will arise when inconsistent probabilities are assumed in a given context and are in violation of the.
The dutch book argument, tracing back to independent work by f. The dutch book arguments attempt to justify the bayesian approach to science and belief. In addition, this book contains recent, geometric theorems which have been. Dutch book cannot be made against a bayesian bookie. Here we will exhibit its power by using it for giving a simple proof of a little theorem about certain euclidean and noneuclidean quadrilaterals. Pascals theorem is a special case of the cayleybacharach theorem. But mostly this post is to introduce people to the argument and to get people thinking about a solution.
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. Pappus, on the pappusguldin theorem figures generated by acomplete revolution ofa plane figure about an axis are in a ratio compounded 1 of the ratio of the areas of the figures, and 2 of the ratio of the straight lines similarly drawn to i. His main contribution to mathematics was primarily as an encyclopedist. An introduction to a long conversation the diachronic dutch book argument the willtowager assumption or just what does this prove, anyway. A w grootendorst, geometrical algebra in euclid dutch, in summer course. 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. Theorem of pappus this section ends with a discussion of the theorem of pappus for volume, which allows us to find the volume of particular kinds of solids by using the centroid. In economics, the term usually refers to a sequence of trades that would leave one party strictly worse off. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft2. 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 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.
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. Perhaps the best apologist for gregorys work is gregory himself. The exact formulation of the theorem already has some subtleties, which we want to mention here. There is also a theorem of pappus for surface area, but it is much less useful than the theorem for volume. Greek mathematician of the second half of the third century. Other than that he was born at alexandria in egypt and that his. In gambling, a dutch book or lock is a set of odds and bets which guarantees a profit, regardless of the outcome of the gamble. It was formulated by blaise pascal in a note written in 1639 when he was 16 years old and published the following year as a broadside titled essay povr les coniqves. 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. These quantities can be computed using the distance traveled by the centroids of the curve and region being revolved.
Moments and centers of mass calculus volume 2 openstax. My aim in this post is to present a particularly powerful way of thinking about the mathematics of these theorems. Pappus theorem article about pappus theorem by the. The first two books were devoted to arithmetic, and the third through fifth books deal primarily with geometry. An application of pappus involution theorem in euclidean and. Dutch book arguments stanford encyclopedia of philosophy. Optimization of the courses in geometry by the usage of dynamic geometry software sam mature plants can produce more than flower heads with the potential to yield 25000 seeds with pappus, which allows them major wind dispersal vigna and lopez, 2008. He also gives his name to the pappus chain, and to the pappus configuration and pappus graph arising from his hexagon theorem.
Pappuss area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Pappus s centroid theorems are results from geometry about the surface area and volume of solids of revolution. Let v be the set of all realvalued functions on,sov is a linear space of dimension card. Given a set of betting quotients that fails to satisfy the probability. This can be achieved using the second theorem of pappus, that states. References for euclid mactutor history of mathematics. The conclusion of the dba is that the degrees of belief, or credences, that an agent attaches to the members of a set \x\ of sentences, statements, or propositions, should satisfy the axioms of probability.
Use the theorem of pappus to determine the surface area of this region as well. The dutch book argument, tracing back to independent work by. A theory stating that when an assumption is made that is not accurate with regard to the likelihood of an event occurring, and then. James gregory and the pappusguldin theorem conclusion. Although pappuss theorem usually refers to pappuss hexagon theorem, it may also refer to pappuss centroid theorem. This book was originally published in dutch, and this will be the first english translation. The mathematical principles underlying newtons principia.
The volume v of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area a of the lamina and the distance d traveled by the laminas geometric centroid bar. Pappus article about pappus by the free dictionary. I understand that a dutch book is a gambling term wherein everyone wins. Pappus theorem pronunciation sign in to disable all ads. 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 of alexandria greek mathematician britannica. Finding surface area and volume of a sphere using only. May 24, 2014 theorem of pappus to find volume of revolution calculus 2. Any stretching of rin9 would provide a euclidean stretching of b, necessarily satisfying the premises of the main theorem.
Dutch book arguments typically take the bookie to be the clever person who is assured of winning money off some irrational agent who has posted vulnerable odds, whereas at the racetrack it is the bookie who posts the odds in the first place. A centroid is easily visualized as the center of gravity or center of mass of a flat. Under several structural assumptions about combinations of stakes that is, assumptions about the combination of wagers your betting policy is undominated. 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. The converse dutch book theorem shows that, if your credences are instead probabilistic, then there is no such series of decision problems. The book is a collection of 367 proofs of the pythagorean theorem and has been. The first theorem of pappus states that the surface area s of a surface of revolution generated by the. In gambling, a dutch book or lock is a set of odds and bets which guarantees a profit. Let s be the surface generated by revolving this curve about the xaxis. A dutch book theorem for quantificational credences. Now turn everything around and look at it from the bookies perspective. This small book has for a long time been a unique place to find classical results from geometry, such as pythagoras theorem, the ninepoint circle, morleys triangle, poncelets. If one measures the ratio applicability over the di culty of proof, then this theorem even beats pythagoras, as no proof is required.
Let r be the triangular region bounded by the line y x, the xaxis, and the vertical line x r. Theorem of pappus definition of theorem of pappus by the. Pappus definition of pappus by the free dictionary. 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. This book has the same format as both the dutch and. In order to prove the result, i introduce the novel concept of a quantificational bet.
Finding surface area and volume of a sphere using only pappus. Very little is known about his life, but the written records suggest he was probably a teacher. The euclidean pseudoline arrangement b is derived from a by taking line 0 as the line at in. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. A bridge between algebra and geometry elena anne marchisotto 1. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Right after the introduction pappus gives a definition of analysis and of. It appeared in pythagoras a dutch math magazine for schoolkids in the.
However, there are many planes in which desarguess theorem. It is associated with probabilities implied by the odds not being coherent, namely are being skewed e. A video lecture that will explain both the theorems of pappus and guldinus with examples. The collection, his signature work, has been translated in its entirety in latin, french, german, and modern greek. The second is always a mathematical theorem sometimes known as the conjunction of the dutch book theorem and the converse dutch book theorem. Dutch book theorem is a type of probability theory that postulates profit opportunities will arise when inconsistent probabilities are assumed in. Theorem of pappus to find volume of revolution calculus 2. Theorem of pappus synonyms, theorem of pappus pronunciation, theorem of pappus translation, english dictionary definition of theorem of pappus. Pappus s theorem admits that any pair of lines that do not have a point of the con. The theorem of pascal concerning a hexagon inscribed in a conic. The proposition that the volume of a solid of revolution. Listen to the audio pronunciation of pappus theorem on pronouncekiwi how to pronounce pappus theorem. Pappus was the author of mathematical collections in eight books, only the last six of which are extant. Bayes theorem was rst proven in 1763 by thomas bayes.
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. Objectivists believe in frequency theory definitions of probability, which refer to objective outcomes of events like coin flips. 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. An analytic proof of the theorems of pappus and desargues. If we use as criteria for greatness evidence of usefulness, importance, or beauty, then there is a simple theorem proposed by pappus of alexandria that qualifies. Areas of surfaces of revolution, pappuss theorems let f. Fluxional reworkings of theorems in its first book are indeed still in. Pappus of alexandria was, and still is, a popular gure within the history of mathematics. Pythagorean theorem and its many proofs cut the knot. A simple proof for the theorems of pascal and pappus. 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.
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