Geometry.Net - Basic_E: Euclid Geometry

High School Euclid Paper
can be proven. 7 Lobachevsky's geometry grew out of his unsuccessfulattempts to prove euclid's parallel postulate. 8 Zeno of
http://www.obkb.com/dcljr/euclidhs.html

Extractions: bottom of page One of the most influential mathematicians of ancient Greece, Euclid flourished around 300 B.C. Not much is known about the life of Euclid. One story which reveals something about Euclid's character concerns a pupil who had just completed his first lesson in geometry. The pupil asked what he would get from learning geometry. So Euclid told his slave to give the pupil a coin so he would be gaining something from his studies. Included in the many works of Euclid is Data , concerning the solution of problems through geometric analysis, On Divisions (of Figures) , the Optics , the Phenomena , a treatise on spherical geometry for astronomers, several lost works on higher geometry, and the Elements , a thirteen volume textbook on geometry. The Elements , which surely became a classic soon after its publication, eventually became the most influential textbook in the history of civilization. In fact, it has been said that apart from the Bible , the Elements is the most widely read and studied book in the world.
Edu2000 On-Line Resources
euclid's geometry History and Practice A series of interdisciplinary lessons oneuclid's Elements, researched and written by Alex Pearson, a Classicist at The
http://www.education2000.com/web_resources/web_resources.htm

Extractions: Visual Mathematics Series Online Demo Ask Dr. Math The Geometry Center, University of Minnesota The Geometry Junkyard by Eppstein Euclid's Geometry: History and Practice PBS Mathline The Public Broadcasting System's Math Service. Combining computing and telecommunications technologies, public television offers interactive data services in addition to interactive video and voice services for education based on the mathematics standards set by the the National Council of Teachers of Mathematics (NCTM). Middle School Math Project (MSMP); Elementary School Math Project (ESMP); Math Electronic Resource Center (MERC).
Home Subjects Geometry
Noneuclid (Rice U. Math Dept) - Java Software for Interactively Creating Ruler andCompass Disk and the Upper Half-Plane Models of Hyperbolic geometry for use
http://www.jbmconsult.com/math/subjects_geometry.htm

Extractions: Geometry Resources on the Web Updated: July 14, 2002 Euclid's Elements All About Geometry (AAA Math Site) - These pages teach geometry facts covered in K8 math courses. Each page has an explanation, interactive practice and challenge games about geometry Ancient Geometry, by Ralph H. Abraham Cut the Knot - This is a great site for imaginative, interactive, geometric, and mathematical game problems and illustrations. You can spend many hours here. The site is available on CD-ROM. Author: Alexander Bogomolny. CIGS (Corner for Interactive Geometry Software at the Math Forum Computational Geometry at the City University of Hong Kong Computational Geometry Code - This page lists "small" pieces of geometric software available on the Internet. (Dr. Jeff Erickson, Assistant Professor of Computer Science University of Illinois at Urbana-Champaign)
Foundations Of Geometry
axioms. The axioms of euclidean geometry were not correctly writtendown by euclid, though no doubt, he did his best. There are
http://www.imsc.ernet.in/~kapil/geometry/euclid/node1.html

Extractions: Next: The Axioms of Incidence Up: euclid Previous: euclid Euclidean Geometry is the attempt to build geometry out of the rules of logic combined with some ``evident truths'' or axioms. The axioms of Euclidean Geometry were not correctly written down by Euclid, though no doubt, he did his best. There are now a number of different ways of giving the formal basis for the same geometry. These are The ``High School Geometry'' text book approach. Hilbert's ``Foundations of Geometry'' approach. Through Projective Geometry as in Coxeters' ``Non-Euclidean Geometry''. Trough the study of the Euclidean group as done by Sophus Lie. We shall examine the middle two approaches in the following text. The first method which was learned in school should now be forgotten since we are looking at (to paraphrase Klein) ``elementary mathematics from an advanced standpoint''. The method that (to my mind) comes closest to the original approach is that of Hilbert's Foundations of Geometry. Unlike the ``High School Geometry'' text books, this makes no reference to the ``Ruler Placement Postulate'' or a ``Protractor Placement Postulate'', both of which are anti-thetical to a purely geometric approach. The arithmetic aspects of geometry should grow out of it rather than be imposed from outside. Another difference is that the notion of a line is not as a set of points in Euclid's approach; points, lines and planes are distinct notions in Hilbert's approach too.
Preface Of
In addition to euclid's geometry, there is another geometry, fractal, which seemscloser to nature, is more complex, and has beauties of its own that challenge
http://nsr.bioeng.washington.edu/Documents/fracphys/preface.html

Extractions: Fractal Physiology I know that most men, including those at ease with the problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. Joseph Ford quoting Tolstoy ( Gleick, 1987) We are used to thinking that natural objects have a certain form and that this form is determined by a characteristic scale. If we magnify the object beyond this scale, no new features are revealed. To correctly measure the properties of the object, such as length, area, or volume, we measure it at a resolution finer than the characteristic scale of the object. We expect that the value we measure has a unique value for the object. This simple idea is the basis of the calculus, Euclidean geometry, and the theory of measurement. However, Mandelbrot (
Clark University
community. euclid demonstrating principles of geometry. Detail from TheSchool of Athens by Raphael. In the Vatican. Click to enlarge.
http://www.clarku.edu/research/access/math_cs/joyce/joyceD.shtml

Extractions: Angles, lines and logic Star Trek's science officer Mr. Spock is greatly admired by fans throughout the world for his ability to solve problems using logic. But according to Clark mathematics professor David Joyce, U.S. high school students are at a disadvantage when it comes to learning deductive reasoning, an important branch of logic. He maintains that geometry, one of the most important subjects for teaching deductive reasoning, is often poorly taught in U.S. high schools.
Euclid's Postulates -- From MathWorld
euclid himself used only the first four postulates ( absolute geometry ) for thefirst 28 propositions of the Elements, but was forced to invoke the parallel
http://mathworld.wolfram.com/EuclidsPostulates.html

Extractions: Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates (" absolute geometry ") for the first 28 propositions of the Elements , but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent " non-Euclidean geometries " could be created in which the parallel postulate
Science Timetable
describe Newton's law of gravity by using infinitesimal calculus made by Newton andLeibniz, while Newton himself proved the law in terms of euclid's geometry.
http://www.scienceall.com/menu/time/m02.html

Extractions: Euclid Far more famous and important than any other mathematical textbook is 'the Elements' written by B. C. Euclid(active around 259 B.C.) and ranked classical geometry for more than one thousand years. Now we describe Newton's law of gravity by using infinitesimal calculus made by Newton and Leibniz, while Newton himself proved the law in terms of Euclid's geometry. 'The Elements' composed of 13 volumes is a complete compilation of the ancient geometry in which he explained the theorems of his predecessors, especially of Pythagoras and Eudoxus. Despite of the well-known episode that once the Egyptian Ptolemy asked him if there were a shorted way to study geometry, to which he replied that there was no royal road to geometry, little is known of Euclid's life except that he, Archimedes' contemporary, lived during the Hellenistic period, and taught mathematics at Alexandria in Egypt. He left several works besides 'the Elements'; 'Data' and 'On Division' remain to the present while 'Book of Fallacies' on geometrical fallacies and 'Optics' to which Apollonius added his share remain only in terms of the posterity's commentaries.
Are All Triangles Isosceles?
In retrospect it's clear that euclid's geometry, rather than giving rigorous proofsof abstract concepts suggested by roughly drawn figures, actually gave
http://www.mathpages.com/home/kmath392.htm
HallPhysic.com :: Euclid\'s Window: The Story Of Geometry From Parallel Lines To
HallPhysic.com euclid's Window The Story of geometry from Parallel Linesto Hyperspace. HallPhysic.com. the most comprehensive Physics portal.
http://hallphysic.com/index.php/Mode/product/AsinSearch/0684865238/name/Euclid%2
Euclid
In addition to the Elements, euclid wrote other works on geometry, including thetheory of conics, and on astronomy, optics, and music; many of these works
http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib

Extractions: Euclid (ca. 365 ca. 300 B.C Euclid lived in Alexandria, Egypt and was the most talented and influential mathematician of his time. He was younger than Plato and Aristotle , but older than Archimedes While he was probably educated in Athens, he taught at the Museum in Alexandria, a research institute stressing science and literature. Euclid recorded, collated, and extended the mathematics of the ancient world. He was one of the most influential mathematicians of all time and a prolific author. Euclid is best known for his foundational work in geometry, which was presented in the classic work entitled The Elements This work laid the foundation for the subject of geometry and in general for axiomatic mathematics. All of the facts of the subject (e.g., geometry) must be proven deductively as statements of theorems and propositions. The reasoning can depend only on the assumptions made at the start (i.e., the definitions and axioms) and on relevant previously established theorems and propositions. The Elements contains 13 books and begins with definitions and axioms, including the famous parallel postulate, which states that one and only one line can be drawn through a given point parallel to a given line.
Non-Euclidean Geometry Seminar
Seminar notes by Greg Schreiber.Category Science Math geometry Noneuclidean We began with an exposition of euclidean geometry, first from euclid's perspective(as given in his Elements) and then from a modern perspective due to Hilbert
http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html

Extractions: Seminar on the History of Hyperbolic Geometry Greg Schreiber In this course we traced the development of hyperbolic (non-Euclidean) geometry from ancient Greece up to the turn of the century. This was accomplished by focusing chronologically on those mathematicians who made the most significant contributions to the subject. We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible. References: Four general references were used throughout this course: Bonola's Non-Euclidean Geometry, Jeremy Gray's Ideas of Space, Greenberg's Euclidean and Non-Euclidean Geometries, and McCleary's Geometry from a Differential Viewpoint. In addition, original works of these mathematicians were used whenever possible, as well as biographies of them. These books included Euclid's Elements, Hilbert's Foundations of Geometry, Proclus's A Commentary on the First Book of Euclid's Elements, Saccheri's Euclid Vindicated, Bolyai's Science of Absolute Space, Lobachevskii's Geometrical Researches in the Theory of Parallels, and Riemann's "On the Hypotheses Which Lie at the Foundations of Geometry," among others.
NonEuclid - Hyperbolic Geometry Article + Software Applet
Features software that simulates hyperbolic straightedge and compass constructions. Provides basic information about nonEuclidean geometry. of Hyperbolic geometry. for use in High School and Undergraduate Education. Hyperbolic geometry is a geometry of
http://math.rice.edu/~joel/NonEuclid

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