1. Diocles --  Encyclopædia Britannica MLA style " diocles." Encyclopædia Britannica. 2004. Encyclopædia Britannica Premium Service. APA style diocles. Encyclopædia Britannica. Retrieved April 25, 2004, from http://www.britannica.com/eb/article?eu=31017

2. No. 837: Diocles diocles' parabolic mirror in an old Arabic book make our civilization run, and the people whose ingenuity created them. Who was diocles? We don't really know to tell us that it was diocles 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. ComPilots.com - Aviation Portal Last 10 Forum Topics by diocles Find all posts by diocles Last 10 News Submissions sent by diocles Last 10 Aerodrome Modifications sent by diocles http://www.compilots.com/userinfo-diocles.html

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 the cylinder. In this work we are told that diocles studied the cissoid as part of an attempt to 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. Diocles Search diocles search Book Search. diocles. Toys. Books. Audio 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 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.

8. 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

9. 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

10. Cissoid CISSOID OF diocles. include graphics.h include math.h main() {int gd=DETECT,gm,x,y,i,j; float a; initgraph( gd, gm, c\\tc\\bgi http://www.geocities.com/ashwinkrec/cissoid.htm

CISSOID OF DIOCLES int gd=DETECT,gm,x,y,i,j; float a; setcolor(RED); line(x,0,x,480); line(0,y,640,y); setcolor(LIGHTMAGENTA); outtextxy(20,20,"CISSOID OF DIOCLES"); for(a=0.0001;a <4;a+=0.001) putpixel(x+50*sin(a)*sin(a),y+20*pow(sin(a),3)/cos(a),LIGHTBLUE); delay(100); Go to the programs page click here!!

11. Perseus Lookup Tool Collections Classics ·. Papyri ·. Renaissance ·. London ·. California ·. Upper Midwest ·. Chesapeake ·. Boyle ·. Tufts History. Configure display ·. Help ·. Tools ·. Copyright ·. FAQ ·. Publications http://www.perseus.tufts.edu/cgi-bin/vor?type=phrase&alts=0&group=typeca

12. 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

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Curves Plane 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

13. Cissoid Of Diocles From MathWorld Cissoid of diocles from MathWorld 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 http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/CissoidofDiocles.h

14. Cissoid Of Diocles Catacaustic -- From MathWorld Cissoid of diocles Catacaustic. (6). is the equation of a cardioid, the catacausticof the cissoid of diocles for radiant point at is a cardioid with a = 2. http://mathworld.wolfram.com/CissoidofDioclesCatacaustic.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Curves Plane Curves ... Cissoids Cissoid of Diocles Catacaustic For the parametric representation the catacaustic of this curve from the radiant point is given by Eliminating t gives the Cartesian equation Therefore, since is the equation of a cardioid , the catacaustic of the cissoid of Diocles for radiant point at is a cardioid with a Cardioid Catacaustic Cissoid of Diocles search Eric W. Weisstein. "Cissoid of Diocles Catacaustic." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/CissoidofDioclesCatacaustic.html Wolfram Research, Inc.

15. Cissoid Of Diocles Cissoid of diocles. Here is the definition of cissoid of two curves. The cissoid of diocles is the cissoid of a circle and a tangent line, with respect to 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. 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."

17. Xah: Special Plane Curves: Cissoid Of Diocles diocles (~250~100 BC) invented this curve to solve the doubling of the cube problem a century after the death of the inventor diocles. In the commentaries on the work http://xahlee.org/SpecialPlaneCurves_dir/CissoidOfDiocles_dir/cissoidOfDiocles.h

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

18. Cissoid Of Diocles Inverse Curve From MathWorld Cissoid of diocles Inverse Curve from MathWorld If the cusp of the cissoid of diocles is taken as the inversion center, then the cissoid inverts to a parabola. http//mathworld.wolfram.com/ http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/CissoidofDioclesIn

19. Diocles diocles. In Greek mythology, diocles was one of the first priests of Demeterand one of the first to learn the secrets of the Eleusinian mysteries. http://www.fact-index.com/d/di/diocles.html

Main Page See live article Alphabetical index Diocles In Greek mythology Diocles was one of the first priests of Demeter and one of the first to learn the secrets of the Eleusinian mysteries This article is from Wikipedia . All text is available under the terms of the GNU Free Documentation License

20. Cisoide De Diocles Translate this page Cisoide de diocles. (ver dibujo). La ecuación genérica de la cisoidede diocles en coordenadas cartesianas es y 2 = x 3 /(a -x). 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

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