1. Cisoide De Diocles Cisoide de diocles. Fecha de primera versión 0204-98 http://www.terra.es/personal/jftjft/Geometria/Diferencial/Curvas/Enelplano/Cisoi

Cisoide de Diocles Fecha de primera versión: 02-04-98 Fecha de última actualización: 04-12-00 La cisoide es el lugar geométrico de los puntos M, tal que OM = PQ. (ver dibujo) La ecuación genérica de la cisoide de Diocles en coordenadas cartesianas es: y = x /(a -x) La ecuación en coordenadas polares es: r = a sen q /cos q La ecuación genérica de la cisoide de Diocles en ecuaciones paramétricas es: x = a sen q y = a sen q /cos q La asíntota es: x = a El área entre la curva y la asíntota es: A = 3/4 p a Dibuja la curva. En la página http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html encontrarás todo sobre las curvas. Principal

2. No. 837: Diocles diocles' parabolic mirror in an old Arabic book that make our civilization run, and the people whose ingenuity created them. Who was diocles? We don't really know. http://www.uh.edu/engines/epi837.htm

No. 837: DIOCLES by John H. Lienhard Click here for audio of Episode 837. Today, we focus the rays of the sun. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them. W ho was Diocles? We don't really know. All we have is a text he wrote over 2000 years ago. It's not even in his own tongue. It was written in AD 1462 by a careless scribe who left only spaces where figures should've gone. But it's enough to tell us that it was Diocles who invented the parabolic mirror. Who was Diocles? Historian G.J. Toomer picks through this skimpy legacy this ancient text, titled On Burning Mirrors . Most of what we knew of Diocles came from reference to his work by a noted 6th-century mathematician. Now we finally read this copy of his book, penned 1600 years after the fact. Toomer does his historical detective work. He decides that Diocles flourished in Greece just after 200 BC. He was a mathematician a geometer. Toomer takes us through the text, recreating the figures. We read Diocles' opening: The burning-mirror surface submitted to you is the surface bounding the figure produced by a section of a ... cone ... revolved about [its axis].

3. Encyclopædia Britannica A resident of Athens, diocles was the first to write medical treatises in Attic Greek rather than in the Ionic Greek http://www.britannica.com/eb/article?eu=31017

4. Diocles diocles. Born about 240 BC in Carystus (now Káristos), Euboea (now Evvoia),Greece Died about 180 BC. diocles was a contemporary of Apollonius. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diocles.html

Diocles Born: Died: about 180 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Diocles was a contemporary of Apollonius . Practically all that was knew about him until recently was through fragments of his work preserved by Eutocius in his commentary on the famous text by Archimedes On the sphere and the cylinder. In this work we are told that Diocles studied the cissoid as part of an attempt to duplicate the cube . It is also recorded that he studied the problem of Archimedes to cut a sphere by a plane in such a way that the volumes of the segments shall have a given ratio. The extracts quoted by Eutocius from Diocles' On burning mirrors showed that he was the first to prove the focal property of a parabolic mirror. Although Diocles' text was largely ignored by later Greeks, it had considerable influence on the Arab mathematicians, in particular on al-Haytham . Latin translations from about 1200 of the writings of al-Haytham brought the properties of parabolic mirrors discovered by Diocles to European mathematicians. Recently, however, some more information about Diocles' life has come to us from the Arabic translation of Diocles'

5. Diocles Biography of diocles (240BC180BC) Next. Main index. diocles was a contemporary of Apollonius. http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Diocles.html

Diocles Born: Died: about 180 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Diocles was a contemporary of Apollonius . Practically all that was knew about him until recently was through fragments of his work preserved by Eutocius in his commentary on the famous text by Archimedes On the sphere and the cylinder. In this work we are told that Diocles studied the cissoid as part of an attempt to duplicate the cube . It is also recorded that he studied the problem of Archimedes to cut a sphere by a plane in such a way that the volumes of the segments shall have a given ratio. The extracts quoted by Eutocius from Diocles' On burning mirrors showed that he was the first to prove the focal property of a parabolic mirror. Although Diocles' text was largely ignored by later Greeks, it had considerable influence on the Arab mathematicians, in particular on al-Haytham . Latin translations from about 1200 of the writings of al-Haytham brought the properties of parabolic mirrors discovered by Diocles to European mathematicians. Recently, however, some more information about Diocles' life has come to us from the Arabic translation of Diocles'

6. Cissoid Cissoid of diocles. Cartesian equation y 2 = x 3 /(2a x). The Cissoid of dioclesis the roulette of the vertex of a parabola rolling on an equal parabola. http://www-gap.dcs.st-and.ac.uk/~history/Curves/Cissoid.html

Cissoid of Diocles Cartesian equation: y x a x Polar equation: r a tan( )sin( Click below to see one of the Associated curves. Definitions of the Associated curves Evolute Involute 1 Involute 2 ... Caustic curve wrt another point If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves. This curve (meaning 'ivy-shaped') was invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube by geometrical methods. The name first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found, in 1658, that the area between the curve and its asymptote was 3 a . From a given point there are either one or three tangents to the cissoid. The Cissoid of Diocles is the roulette of the vertex of a parabola rolling on an equal parabola. Newton gave a method of drawing the Cissoid of Diocles using two line segments of equal length at right angles. If they are moved so that one line always passes through a fixed point and the end of the other line segment slides along a straight line then the mid-point of the sliding line segment traces out a Cissoid of Diocles Diocles was a contemporary of Nicomedes . He studied the cissoid in his attempt to solve the problem of finding the length of the side of a cube having volume twice that of a given cube. He also studied the problem of Archimedes to cut a sphere by a plane in such a way that the volumes of the segments shall have a given ratio.

7. Books On Diocles Book Search for diocles. fair as the young palmtree that diocles saw beside the http://namesearchsix.tripod.com/diocles.html

Get Five DVDs for $.49 each. Join now. Tell me when this page is updated Diocles Search for books related to: Diocles Diocles Search Book Search Bestselling Books Childrens Books Search for other names All Products Books Popular Music Classical Music Video DVD Electronics Software var test=0; document.write("<");document.write("! "); document.write(" ");document.write(">"); Nordic Web Design About Diocles The once beautiful ruins carpeted with grass and wild flowers have been doubly desecrated by persons, academic persons, having authority and a plentiful lack of taste. The slim mountain-ashes, fair as the young palm-tree that Diocles saw beside the shrine of Apollo in Delos, have been cut down by the academic persons to whom power is given. Modified text originally written by Andrew Lang.

8. Cissoid Of Diocles Cissoid of diocles. diocles showed that if in addition you allow the useof the cissoid, then one can construct . Here is how it works. http://www.geocities.com/famancin/cissoid_diocles.html

Cissoid of Diocles Here is the definition of cissoid of two curves. Cissoid[ Let O be a fixed point and let L be a line through O intersecting the curves C and C at Q and Q . The locus of points P and P on L such that OP = OQ - OQ = Q Q is the cissoid of C and C with respect to O. The cissoid of Diocles is the cissoid of a circle and a tangent line, with respect to a fixed point O on the circumference opposite the point of tangency A. The screenshot below shows the cissoid drawn using Jeometry Let O be the origin and x = a be the line tangent to the circle. Let Ô be the angle BÔA in the picture above. Considering the right triangles OBA and CAO, we have OP OC OB a secÔ - a cosÔ a sinÔ tanÔ Hence the polar equation of the cissoid is r = a sinÔ tanÔ Then the Cartesian equation follows immediately by substitution, y (a - x) = x This is the same equation we found when considering the pedal of a parabola with respect to its vertex (let a = - We would like to find a parametric rapresentation of the curve. To do that, we note that the cissoid of Diocles is a cubic curve with a cusp in the origin, so we can find a rational parametrization by intersecting the cissoid with the line

9. Diocles Search diocles search by the name of diocles. Modified text originally written by Andrew Lang. http://www.geocities.com/namebookstwo/diocles.html

Book Search Diocles Toys Books Audio Music ... DVD's Begin searching for: Diocles Diocles Books Browse Books Search for other titles All Products Books Popular Music Classical Music Video DVD Electronics Software Design by NordicWebDesign.net var test=0; document.write("<");document.write("! "); document.write(" ");document.write(">"); About Diocles Thirty years have passed, like a watch in the night, since the earlier of the two sets of verses here reprinted, Ballades in Blue China, was published. At first there were but twenty-two Ballades; ten more were added later. They appeared in a little white vellum wrapper, with a little blue Chinese singer copied from a porcelain jar; and the frontispiece was a little design by an unknown artist by the name of Diocles. Modified text originally written by Andrew Lang.

10. I1069 DIOCLES ( - ) diocles. REFN 80438. Father HELENUS Family 1 + BASSANUS. _ PRIAM _ _ HELENUS_ _ diocles _ _ _ INDEX http://www.geocities.com/linniev2/eg/d0001/g0000014.html

11. Cissoid Of Diocles -- From MathWorld Cissoid of diocles, A cubic curve invented by diocles in about 180 BC in connectionwith his attempt to duplicate the cube by geometrical methods. http://mathworld.wolfram.com/CissoidofDiocles.html

Geometry Curves Plane Curves Algebraic Curves ... Geometric Construction Cissoid of Diocles A cubic curve invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube by geometrical methods. The name "cissoid" first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found, in 1658, that the area between the curve and its asymptote was MacTutor Archive ). From a given point there are either one or three tangents to the cissoid. Given an origin O and a point P on the curve, let S be the point where the extension of the line OP intersects the line and R be the intersection of the circle of radius a and center with the extension of OP . Then the cissoid of Diocles is the curve which satisfies OP = RS The cissoid of Diocles is the roulette of a parabola vertex of a parabola rolling on an equal parabola Newton gave a method of drawing the cissoid of Diocles using two line segments of equal length at right angles . If they are moved so that one line always passes through a fixed point and the end of the other line segment slides along a straight line, then the midpoint of the sliding line segment traces out a cissoid of Diocles.

12. Cissoid Cissoid of diocles. Cartesian equation y2 = x3/(2a x) http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cissoid.html

Cissoid of Diocles Cartesian equation: y x a x Polar equation: r a tan( )sin( Click below to see one of the Associated curves. Definitions of the Associated curves Evolute Involute 1 Involute 2 ... Caustic curve wrt another point If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves. This curve (meaning 'ivy-shaped') was invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube by geometrical methods. The name first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found, in 1658, that the area between the curve and its asymptote was 3 a . From a given point there are either one or three tangents to the cissoid. The Cissoid of Diocles is the roulette of the vertex of a parabola rolling on an equal parabola. Newton gave a method of drawing the Cissoid of Diocles using two line segments of equal length at right angles. If they are moved so that one line always passes through a fixed point and the end of the other line segment slides along a straight line then the mid-point of the sliding line segment traces out a Cissoid of Diocles Diocles was a contemporary of Nicomedes . He studied the cissoid in his attempt to solve the problem of finding the length of the side of a cube having volume twice that of a given cube. He also studied the problem of Archimedes to cut a sphere by a plane in such a way that the volumes of the segments shall have a given ratio.

13. Cissoid Of Diocles Inverse Curve -- From MathWorld Cissoid of diocles Inverse Curve, If the cusp of the cissoid of diocles istaken as the inversion center, then the cissoid inverts to a parabola. http://mathworld.wolfram.com/CissoidofDioclesInverseCurve.html

Geometry Curves Plane Curves Cissoids ... Inverse Curves Cissoid of Diocles Inverse Curve If the cusp of the cissoid of Diocles is taken as the inversion center , then the cissoid inverts to a parabola Author: Eric W. Weisstein Wolfram Research, Inc.

14. Diocles (2nd Century B.C.) -- From Eric Weisstein's World Of Scientific Biograph diocles (2nd century BC), This entry contributed by Margherita Barile.Greek mathematician who invented the cissoid of diocles. http://scienceworld.wolfram.com/biography/Diocles.html

Branch of Science Mathematicians Nationality Greek ... Barile Diocles (2nd century B.C.) This entry contributed by Margherita Barile Greek mathematician who invented the cissoid of Diocles This discovery appears in the collection of his writings that has reached us in an Arabic transcription entitled On Burning Mirrors ; the original has been lost. Diocles created the famous curve as an auxiliary tool for cube duplication For this construction problem, unrealizable with straightedge and compass he also presented a solution based on two intersecting parabolas Almost nothing is known about Diocles' life. His work contains a hint that he could have spent some time in Arcadia. A careful examination of the text, and the comparison with other sources allowed the historians to locate him as a contemporary of Apollonius , although Diocles' treatment of conic sections does not clearly indicate whether he ever had access to Apollonius' comprehensive work on the subject. On the other hand, he certainly knew some results by Archimedes: his construction of the mean proportional is explicitly intended as a completion of some of the latter's geometric proofs. References MacTutor History Topics. "How Do We Know About Greek Mathematicians?"

15. Cissoid Of Diocles Cissoid of diocles. Here is the definition of cissoid of two curves. http://www.geocities.com/CapeCanaveral/Hall/3131/cissoid_diocles.html

Cissoid of Diocles Here is the definition of cissoid of two curves. Cissoid[ Let O be a fixed point and let L be a line through O intersecting the curves C and C at Q and Q . The locus of points P and P on L such that OP = OQ - OQ = Q Q is the cissoid of C and C with respect to O. The cissoid of Diocles is the cissoid of a circle and a tangent line, with respect to a fixed point O on the circumference opposite the point of tangency A. The screenshot below shows the cissoid drawn using Jeometry Let O be the origin and x = a be the line tangent to the circle. Let Ô be the angle BÔA in the picture above. Considering the right triangles OBA and CAO, we have OP OC OB a secÔ - a cosÔ a sinÔ tanÔ Hence the polar equation of the cissoid is r = a sinÔ tanÔ Then the Cartesian equation follows immediately by substitution, y (a - x) = x This is the same equation we found when considering the pedal of a parabola with respect to its vertex (let a = - We would like to find a parametric rapresentation of the curve. To do that, we note that the cissoid of Diocles is a cubic curve with a cusp in the origin, so we can find a rational parametrization by intersecting the cissoid with the line

16. Xah: Special Plane Curves: Cissoid Of Diocles Table of Contents. Cissoid of diocles. Parallels of a cissoid of diocles diocles(~250~100 BC) invented this curve to solve the doubling of the cube problem. http://www.xahlee.org/SpecialPlaneCurves_dir/CissoidOfDiocles_dir/cissoidOfDiocl

Table of Contents Cissoid of Diocles Parallels of a cissoid of Diocles Mathematica Notebook for This Page History Description Formulas ... Related Web Sites History Diocles (~250-~100 BC) invented this curve to solve the doubling of the cube problem. (aka the Delian problem) The name cissoid (ivy-shaped) came from the shape of the curve. Later the method used to generate this curve is generalized, and we call curves generated this way as cissoids From Thomas L. Heath's Euclid's Elements translation (1925) (comments on definition 2, book one): This curve is assumed to be the same as that by means of which, according to Eutocius, Diocles in his book On burning-glasses solved the problem of doubling the cube. From Robert C. Yates' Curves and their properties (1952): As early as 1689, J. C. Sturm, in his Mathesis Enucleata, gave a mechanical device for the constructions of the cissoid of Diocles. From E.H.Lockwood A book of Curves (1961): The name cissoid ('Ivy-shaped') is mentioned by Geminus in the first century B.C., that is, about a century after the death of the inventor Diocles. In the commentaries on the work by Archimedes On the Sphere and the Cylinder , the curve is referred to as Diocles' contribution to the classic problem of doubling the cube. ... Fermat and Roberval constructed the tangent (1634); Huygens and Wallis found the area (1658); while Newton gives it as an example, in his

17. Xah: Special Plane Curves: Cissoid History. Cissoid is the generalization of Cissoid of diocles. Related Web Sites. seeGeneric Reference Page. MacTutor Famous Curve Index (on cissoid of diocles) http://www.xahlee.org/SpecialPlaneCurves_dir/Cissoid_dir/cissoid.html

Table of Contents Cissoid History Description Formulas Properties ... Related Web Sites History Cissoid is the generalization of Cissoid of Diocles . (*XahNote: Who generalized it? Around what era?*) Description Cissoid is a method of deriving a new curve based on two (or one) given curves C1, C2, and a fixed point O. A curve derived this way may be called the cissoid of C1 and C2 with the pole O. Step-by-step description: Given two curves C1 and C2, and given a fixed point O. Let P1 be a point on C1. Draw a line L passing O and P1. Let the intersection of L and C2 be P2. Mark a point Q on line L, such that distance[O,Q]==distance[P1,P2]. The locus of Q (as P1 moves on C1) is the cissoid of C1 and C2 with the pole O. Note: There are two points on line L such that distance[O,Q]==distance[P1,P2]. The two points are symmetric around point O, so either one will generate the same cissoid. Also, if L and C2 have more than one intersections, then we can label additional points P3, P4,... and the cissoid may have loops.) Formulas Properties Cissoids of two Lines or two Circles The cissoid of two concentric circles with pole on center is two concentric circles centered on pole. The cissoid of two circles in general is a combinations of various oval, figure-eight, or droplet-shaped curves.

18. Famous Curves Index Cayley's Sextic. Circle. Cissoid of diocles. Cochleoid. Conchoid. Conchoid of de Sluze http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html

Famous Curves Index Click on the name of a curve below to see its history and some of its associated curves. Astroid Bicorn Cardioid Cartesian Oval ... Witch of Agnesi Anyone with the Mathematical MacTutor system can investigate these curves and their associated curves in an interactive way. Similarly, anyone whose browser knows what to do with Java can experiment in the same way. Here is a list of those curves for which this facility is available Enter a word or phrase: Main index Definitions of Associated Curves History Topics Index Birthplace Maps ... Search Form JOC/EFR November 2000 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Curves/Curves.html

19. Cissoid Of Diocles The Cissoid of diocles diocles is one of many mathematicians who have attemptedto construct a cube whose volume is exactly twice that of a given cube. http://curvebank.calstatela.edu/diocles/diocles.htm

Click on the thumbnail images below to see experimental solar collectors near Barstow, California focus the sun's rays on a central tower where heat is converted to electricity. The famous Belvedere Apollo at the top of this column is a Roman copy of a much older Greek statue. This marble is now in the Pio Clementino Museum at the Vatican (Rome, Italy). The Burning Mirrows wall painting is from the Stanzino delle Matematiche in the Galleria degli Uffizi (Florence, Italy). Painted by Giulio Parigi (1571-1635) in the years 1599-1600. The Cissoid of Diocles Back to . . . Curve Bank Home This section . . . Another attempt to solve one of the three famous construction problems from Antiquity. Biographical Sketch Diocles is one of many mathematicians who have attempted to construct a cube whose volume is exactly twice that of a given cube. This is often called the "Delian" problem or "duplication of the cube". Legend: A number of legends surround this construction challenge. The good citizens of Athens were being devastated by a plague. History records that in 430 BC they sought advice from the oracle at Delos on how to rid their community of this pestilence. The oracle replied that the altar of Apollo, which was in the form of a cube, should be doubled. Thoughtless builders merely doubled the edges of the cube. Unfortunately the volume of the altar increased by a factor of 8. The oracle insisted the gods had been angered. As if to confirm this reprimand, the plague grew worse. Other delegations consulted Plato. When informed of the oracle's admonition, Plato told the citizens "the god has given this oracle, not because he wanted an altar of double the size, but because he wished in setting this task before them, to reproach the Greeks for their neglect of mathematics and their contempt of geometry."

20. References For Diocles References for the biography of diocles References for diocles. Biography in Dictionary of Scientific Biography (New York 19701990). http://www-gap.dcs.st-and.ac.uk/~history/References/Diocles.html

References for Diocles Biography in Dictionary of Scientific Biography (New York 1970-1990). Biography in Encyclopaedia Britannica. Books: T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921). G J Toomer, Diocles On Burning Mirrors, Sources in the History of Mathematics and the Physical Sciences (New York, 1976). Articles: J P Hogendijk, Diocles and the geometry of curved surfaces, Centaurus O Neugebauer, Note on Diocles' "burning mirror", in From ancient omens to statistical mechanics, Acta Hist. Sci. Nat. Med. (Copenhagen, 1987), 37-42. Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR April 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Diocles.html

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