Hippocrates, Part 1 hippocrates of chios, the lunesquarer, was a contemporary of Plato. hippocrates of chios was the first known mathematician to begin to assemble the http://doe.ncia.net/~bobmead/hippoc1.htm
Extractions: Home Math History Articles Graduate Projects ... Other GSP Sketchpads Introduction The story of mathematics, and the achievements and biographies of its practitioners, is intriguing in every way. This series of articles will trace one concept, that of the area of planar figures, through four eras in history. We will see those particular problems and applications of area measurement that faced mathematicians in each era. We will examine the strengths and weaknesses of various approaches. We will see evidence of startling creativity in the solutions, and what is more, we will see the "look" and substance of mathematics change forever. In Part One we visit ancient Greece and witness their best minds struggle with making geometry a logical system. We will see their attempts to create a basis of comparison for all planar areas in a topic known as quadrature. Each achievement gave rise to many new questions. An important one to keep in mind as you read is: could Euclidian geometry ever reach the degree of comprehensiveness and efficiency needed to solve all the quantitative problems of our universe? In Part Two we will see the achievements of ancient cultures, both Eastern and Western, congregate and synthesize in the Arabian Empire after the fall of Rome. A system to be known as algebra will unify much of Greek geometry, Hindu number theory, and application problems from the earliest civilizations. Area will be a key link in the theory of quadratics.
Lecture 3 Hippocrates Quadrature Of Lunes hippocrates of chios, c. 410 ( Plato -400, Euclid -360, Archimedes -250) was afailed merchant of For more about hippocrates of chios, click here. http://www.maths.uwa.edu.au/~schultz/3M3/L3Hippocrates.html
Extractions: The distinguishing feature of Greek mathematics is that it is concerned with logical development, not problem solving. We use the term Greek Mathematics to denote mathematics written in the Greek language between about -600 (Thales) and about 250 (Diophantos). The mathematicians were not necessarily ethnically Greek nor living in the region we now call Greece. In fact the major developments occurred in the Greek colonies now known as Turkey, Egypt and Italy. The Greeks did not have a sophisticated number system. The integers were expressed by concatenating the letters a-k for 19, and l-u for1090 etc. Special letters were invented for larger numbers. Later, Archimedes in the "Sand Reckoner", (in which he calculated the number of grains of sand needed to fill the Universe) developed an exponential system for arbitrarily large numbers. The Greeks used a decimal system for common purposes and a sexigesimal system for scientific purposes, for example astronomy. Concatenations of unit fractions were used for rationals, although later Diophantos developed special symbols for rationals. In Greek mathematics the numbers were 2,3,4,.. The unity 1 was not a number, but the unit in which the numbers were measured. There were no negative numbers or zero. Geometrical quantities such as line segments, angles, areas and volumes were called
Hippoarea.html hippocrates of chios. Introduction the Area Problem. The Babylonians, theEgyptians, and indeed every ancient civilization had knowledge of basic geometric http://cerebro.xu.edu/math/math147/02f/hippocrates/hippoarea.html
Extractions: The Babylonians, the Egyptians, and indeed every ancient civilization had knowledge of basic geometric concepts like how to calculate the areas of simple plane figures (triangles, squares, rectangles, parallelograms, trapezoids, and the like) and the volumes of simple solid bodies (parallelopipeds and pyramids). The Greeks however turned geometry into a real science by applying to it the deductive methods they were systematizing through philosophy. For the first time, epistemological questions were being studied about mathematical ideas: how do we know that the results we have discovered are true? Are these ideas interrelated? Dialectical reasoning strove to find the first principles of mathematical knowledge as a foundation for understanding the real world. This created an architecture of logical structure for mathematical ideas based on cause and effect relationships: if a certain theorem was a consequence of another, then the second was given an a priori precedence over the first. It became standard for geometers to communicate in a very spare language, consisting of statements of theorems followed by their proofs followed the next theorem in the logical development, with little in the way of discussion or explanation. It may not have had the same emotive force as the epic poetry of Homer, but it was beautiful in its own abstract way, like music to the listener. Moreover it was seen as uncovering the secrets of the physical universe, since physical objects and phenomena like light and sound behaved according to geometric principles.
Hippotext.html hippocrates of chios the quadrature of a lune 1 hippocrates of chios was amerchant who fell in with a pirate ship and lost all his possessions. http://cerebro.xu.edu/math/math147/02f/hippocrates/hippotext.html
Extractions: From Philoponus Commentary on Aristotle's Physics Hippocrates of Chios was a merchant who fell in with a pirate ship and lost all his possessions. He came to Athens to prosecute the pirates and, staying a long time in Athens by reason of the indictment, consorted with philosophers, and reached such proficiency in geometry that he tried to affect the quadrature of the circle. He did not discover this, but having squared the lune he falsely thought from this that he could square the circle also. For he thought that from the quadrature of the lune the quadrature of the circle could also be calculated. From Simplicius Commentary on Aristotle's Physics Eudemus , however, in his History of Geometry says that Hippocrates did not demonstrate the quadrature of the lune on the side of a square but generally, as one might say. For every lune has an outer circumference equal to a semicircle or greater or less, and if Hippocrates squared the lune having an outer circumference equal to a semicircle and greater and less, the quadrature would appear to be proved generally. I shall set out what Eudemus wrote word for word, adding only for the sake of clearness a few things taken from Euclid's Elements on account of the summary style of Eudemus, who set out his proofs in abridged form in conformity with the ancient practice. He writes thus in the second book of the History of Geometry.
Extractions: He was born in the year 470 BC on the island of Chios and died in 410 BC. Hippocrates, began his career as a merchant but Aristotle has said that Hippocrates was not a very smart merchant. Hippocrates was cheated of his money by corrupted tax officials in Byzantium other say pirates robbed him. Hippocrates was a brilliant mathematician and One of the greatest geometers of antient Greece. He moved to Athens around 430 BC. He attended lectures in his spare time and eventually became a teacher of geometry. One of Hippocrates achievements is squaring of the circle or the quadrature. In Hippocrates attempt to square the circle, he was able to find the areas of lunes, certain crescent-shaped figures, using his theory that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. Hippocrates proofed that if between a number and its double, two mean proportionals can be found that the cube can be doubled. This finding had a major influence and changed the attempts on duplicating the cube. More of Hippocrates achievements can be learned here at the university of St Andrews: Hippocrates of Chios Oenopides Theopompus Chios home The Fragrant Island Hippocrates Hippocrates In a beautiful surrounding with a breathtaking view.
Philosophers: H hippocrates of chios (c.450c.480 BCE). Articles Web pages + Hippocratesof Chios Article by JJ O Connor and EF Robertson, for the MacTutor History http://users.ox.ac.uk/~worc0337/phil-H.html
Extractions: Philosophers: Haack to Hypatia Han Fei-zi (280-233 BCE) Han Feizi Short Wikipedia article. Legalism, Qin Empire and Han Dynasty From Sanderson Beck's Ancient Wisdom and Folly Han Fei Tzu Page of links maintained by by Kyle M. (14) at Iolani School, Honolulu. Han Fei John Knoblock's "intellectual biography of the 3rd century 'Legalist' philosopher Han Fei and excerpts from two of his most important works." Selection from Han Fei Zi From W.K. Liao [trans.]
Grecia Heroica hippocrates of chios (Athens), squaring the circle or how to draw a square hippocrates of chios (430 BC). He spent his life studying geometry after http://descartes.cnice.mecd.es/ingles/maths_workshop/A_history_of_Mathematics/Gr
Extractions: THE GREEK HEROIC AGE History THE HEROIC AGE (Vth century B.C.) One of the most important personalities of this century is Pericles Athens attracted intellectuals from all parts of the Greek world wanting to satisfy their thirst for knowledge. Rather than coming up with necessary solutions to practical problems at that time, the scholars were more interested in developing their own personal intellect. This desire for wisdom lead them to focus their learning on theoretical issues. During this period the three famous (or classical) problems were dealt with and two methods of reasoning were put into use The table below lists the mathematicians who lived during this period and the problems that formed the focus of their study. Anaxagoras of Clazomenae (Athens) Hippocrates of Chios (Athens) squaring the circle or how to draw a square whose area is the same as that of a circle using a ruler and compass. Hippias de Elis (Attic peninsular) the trisection of the angle or how to construct an angle equal to a third of another given angle Philolaus of Tarentum (Southern Italy) Archytas of Tarentum the duplication of the cube or how to construct another cube whose volume is double that of the given cube Hippasus of Metapontum (Southern Italy) Incommensurability or line segments which are not in rational proportion to one another (THE GOLDEN SECTION)
Chios Chios was the birthplace of the great mathematician, hippocrates of chios (notthe same as the physician Hippocrates of Cos). http://library.thinkquest.org/26264/tools/map/site003.htm
More Word Origins 5 hippocrates of chios, the mathematician, is often confused with Hippocrates ofCos, who is considered to be the father of medicine, and for whom the http://www.pballew.net/arithme5.html
Extractions: Math Words, pg 5 Back to Math Words Alphabetical Index Barycenter The word barycenter is another term for the center of gravity or centroid. The Greek root is barus which generally refers to weighty or heavy. The more ancient Indo-European root seems to have come from a word like "gwerus" and has relatives in our words for gravity and grave. Another word derived from the same root is baryon , the name for a family of particles that are heavier (more massive) than mesons. The word barometer also comes from the same root and is so named because, in a sense, it measures how heavy the air is. Another related word still in current use is baritone, which literally means heavy voiced. The science names for the chemical barium and the ore from which we obtain it, barite, also called "heavy spar", are both from the same root. The History of Math web site at St. Andrews University in Scotland credits the creation of barycenters to August Möbius (1790-1868): In 1827 Möbius published Der barycentrische Calcul, a geometrical book which studies transformations of lines and conics. The novel feature of this work is the introduction of barycentric coordinates. Given any triangle ABC then if weights a, b and c are placed at A, B and C respectively then a point P, the center of gravity, is determined. Möbius showed that every point P in the plane is determined by the homogeneous coordinates [a,b,c], the weights required to be placed at A, B and C to give the center of gravity at P. The importance here is that Möbius was considering directed quantities, an early appearance of vectors.
Extractions: This paper was written by Andrew Wiesner and submitted to the classics department of the Colorado College in May of 1994. The majority of attempts to search out these origins have been determined by two presuppositions: The first of these is that "proof" in this context means deduction from axioms, and that the origins of the mathematics of the Elements are at one and the same time the origins of this conception of proof; the second is that the field of texts which might testify to these origins includes only those texts which exemplify what we recognize today as "Greek science" (including philosophy, mathematics, logic, medicine, geography, natural history, and so on). This paper proceeds by modifying the first of these presuppositions, and by rejecting the second. Visible equality... in things of the same form is manifestly the ground of the entire proof. For there are two axioms here that comprise the whole procedure of this theorem. One is that things which coincide are equal to one another. This is true without qualification... The other is that things that are equal coincide with one another. This is not true in all cases, but only of things that are similar in form.
Richard Delaware Talks hippocrates of chios Squares a Lune, But Can t Square a Circle! Apr. 29,1998, KCATM (Kansas City Area Teachers of Mathematics) Awards Banquet Speaker. http://d.faculty.umkc.edu/delawarer/RDtalks.htm
Angle Trisection By Hippocrates Hippocrates (470410 BC) of Chios, famous for his work on quadrature of circular hippocrates of chios, a mathematician and astronomer, should not be http://www.cut-the-knot.com/Curriculum/Geometry/Hippocrates.shtml
Extractions: Sites for parents Hippocrates (470-410 B.C.) of Chios, famous for his work on quadrature of circular lunes and the arrangement of theorems in a logical manner, later used by Euclid in his Elements , also left the first known construction for the trisection of an angle. As the trisection of Archimedes , this one, too, is not done by straightedge and compass, which we know is impossible . (Such constructions that employ tools beyond straightedge and compass are known as neusis constructions .) Hippocrates of Chios, a mathematician and astronomer, should not be confused with the physician Hippocrates of Cos (460-ca. 370 B.C.) , whom we owe the Oath of Hippocrates For a given angle BAC, let D be the foot of the perpendicular from C to AB. Form a rectangle ADCF and extend CF beyond C. On the extension we shall place an additional point E. If H is the intersection of AE and CD, then E is chosen so as to satisfy When this is done, the angle BAE is one third the angle BAC.
Extractions: (circa 287 - 212 B.C.) Archimedes of Syracuse is popularily known for the law he discovered on occasion of taking his bath . "Eurika" he exclaimed and made it into the history. (Along with Newton and Gauss he is counted among the greatest mathematicians of all times. As an engineer he frustrated numerous attempts by the Romans to capture the city of Syracuse.) The problem of constructing an angle equal to the one third of the given one has been pondered since the times of antiquity. Probably to make the notion of 'geometric construction' more exciting the Ancient Greeks have restricted the allowed operations to using a straightedge and a compass. It's thus specifically forbidden to use a ruler for the sake of measurement. Three famous construction problems lingered until early 19th century when it was shown that it's impossible to solve them with the help of only a straightedge and a compass. The three problems are: to trisect a given angle, to double a cube, and to square a circle . However, one illicit solution that has been found in the works of Archimedes is demonstrated below. (This is Proposition 8 from his
History Of Mathematics: Greece 480411); Oenopides of Chios (c. 450?) Leucippus (c. 450); hippocrates of chios (c.450); Meton (c. 430) *SB; Hippias of Elis (c. http://aleph0.clarku.edu/~djoyce/mathhist/greece.html
History Of Mathematics: Chronology Of Mathematicians 480411) *SB *MT; Oenopides of Chios (c. 450?) *SB; Leucippus (c. 450) *SB *MT;hippocrates of chios (fl. c. 440) *SB; Meton (c. http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html
Extractions: Note: there are also a chronological lists of mathematical works and mathematics for China , and chronological lists of mathematicians for the Arabic sphere Europe Greece India , and Japan 1700 B.C.E. 100 B.C.E. 1 C.E. To return to this table of contents from below, just click on the years that appear in the headers. Footnotes (*MT, *MT, *RB, *W, *SB) are explained below Ahmes (c. 1650 B.C.E.) *MT Baudhayana (c. 700) Thales of Miletus (c. 630-c 550) *MT Apastamba (c. 600) Anaximander of Miletus (c. 610-c. 547) *SB Pythagoras of Samos (c. 570-c. 490) *SB *MT Anaximenes of Miletus (fl. 546) *SB Cleostratus of Tenedos (c. 520) Katyayana (c. 500) Nabu-rimanni (c. 490) Kidinu (c. 480) Anaxagoras of Clazomenae (c. 500-c. 428) *SB *MT Zeno of Elea (c. 490-c. 430) *MT Antiphon of Rhamnos (the Sophist) (c. 480-411) *SB *MT Oenopides of Chios (c. 450?) *SB Leucippus (c. 450) *SB *MT Hippocrates of Chios (fl. c. 440) *SB Meton (c. 430) *SB
The Dark Side Of The Moon Both Cos and Chios are islands in the Dodecanese group; but hippocrates of chios in This conclusion seems to have encouraged Hippocrates, as well as his http://members.fortunecity.com/templarser/moon.html
Extractions: Duncan Graham-Rowe Astronomers are taking the search for somewhere quiet to work to new extremes with a plan to put a radio telescope on the far side of the Moon. The advantage of this unusual location is that the Moon would act as a massive shield, protecting the telescope against radio emissions from Earth. Astronomers could also listen to low radio frequencies that don't penetrate the Earth's atmosphere. Claudio Maccone, an astronomer at the Centre for Astrodynamics in Turin, Italy, is assessing the concept for the International Academy of Astronautics. He even has his eye on a plot of lunar real estate. A 100-kilometre-wide crater called Daedalus the Moon . Maccone is due to present the results of his study to the International Astronautical Congress next October. If the plans are approved, the first step will be to design a satellite probe to orbit the Moon and check there really is a quiet zone. Jupiter's giant light show SOMETHING strange is happening on Jupiter. Its magnetic field extends hundreds of times further out into space than previously thought, creating auroras that make the Earth's northern lights seem feeble in comparison. Jupiter is the giant of the Solar System, more than a thousand times as massive as Earth. In January 2001, the combined power of the Cassini and Galileo space probes, the Chandra X-ray telescope and the Hubble Space Telescope were all trained on the Jovian magnetosphere - the region controlled by the planet's magnetic field. Magnetic field lines fan out from a planet like the lines of iron filings from the poles of a bar magnet. Auroras are caused by ions zipping along these lines, so researchers can use the location of auroras to track how far out into space the planet's magnetic field lines can trap ions from the solar wind.
Extractions: Crockett Johnson Homepage Bibliography of Crockett Johnson's Works Cartoons Magazines Pamphlet Books ... About Crockett Johnson Cartoons Editorial cartoons The New Masses , April 1934 - May 1940. For a more complete bibliographic listing, please click here . Two of these cartoons appear in Robert Forsythe's Redder Than the Rose (listed under " Illustrated By ," below) and one in Joseph North's New Masses: An Anthology of the Rebel Thirties (listed under " About...
Timeline Of Greek And Roman Philosophers hippocrates of chios (c. 470c. 410 BC) Greek geometer, Hippocrates. Socrates (c.469-399 BC) Greek philosopher, Socrates Links. Plato (c. 427-347 BC) http://ancienthistory.about.com/library/bl/bl_time_philosophers.htm
Extractions: zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Homework Help Ancient / Classical History Ancient Greece ... Presocratic Philosophy Timeline of Greek and Roman Philosophers Homework Help Ancient History Essentials Ancient World Maps ... Help zau(256,140,140,'el','http://z.about.com/0/ip/417/C.htm','');w(xb+xb+' ');zau(256,140,140,'von','http://z.about.com/0/ip/496/6.htm','');w(xb+xb); Sign Up Now for the Ancient / Classical History newsletter!
