21. Encyclopædia Britannica Encyclopædia Britannica, heron of alexandria Encyclopædia Britannica Article. MLAstyle heron of alexandria. 2003 Encyclopædia Britannica Premium Service. http://www.britannica.com/eb/article?eu=41048

22. Encyclopædia Britannica heron of alexandria University of St Andrews Biography of this geometer, best knownfor his formula relating the area and the semiperimeter of a triangle by http://www.britannica.com/search?query=heron&ct=

23. Heron Of Alexandria heron of alexandria Heron of Alexendria, also known as Hero, was ageometer and a worker with mechanics. There was a mixup on when http://www.geocities.com/type3kids/adamheron.html

Heron of Alexandria Heron of Alexendria, also known as Hero, was a geometer and a worker with mechanics. There was a mix-up on when he lived, so information about his family, childhood, and education is unknown. But now we know that he was born about 10 AD and died about 75 AD. Some of his aquaintences were Ptolemy, Archimedes, Euclid, Pappus, Eutocius, Neuburger, and Heath. Heron wrote books and made formulas about area, exponents, and more. Some thought Heron didn't understand them himself. Famous Mathematicians Return to the Meet the Mathematicians Homepage

24. The Area Of A Triangle Numerically unstable and numerically stable solution The classical formula due toheron of alexandria first calculates a semiperimeter, S, as half the sum of http://www.geocities.com/SiliconValley/Garage/3323/aat/a_atria.html

The area of a triangle PROBLEM Given the lengths of three sides of a triangle X Y Z . How to calculate the area of this triangle? Numerically unstable and numerically stable solution The classical formula due to Heron of Alexandria first calculates a semiperimeter, S , as half the sum of the sides, and then evaluates the area as in the following internal function HERON IMPLEMENTATION Unit: internal function Interface: the functions SQRT Parameters: positive real numbers X Y Z - lengths of three sides of a triangle a positive integer P - number of significant digits of result, default is Returns: the area of a triangle HERON: procedure parse arg X, Y, Z, P if P = "" then P = 9; numeric digits P S = (X + Y + Z) / 2 return SQRT (S * (S - X) * (S - Y) * (S - Z), P) This function is numerically unstable for needle-shape triangle. W. Kahan describes a good, numerically stable, function. It follows: ATRIAN: procedure parse arg X, Y, Z, P if P = "" then P = 9; numeric digits P then do; W = X; X = Y; Y = W; end then do; W = X; X = Z; Z = W; end then do; W = Y; Y = Z; Z = W; end

25. Wundersames, Heron Translate this page Technical works by heron of alexandria, Aristides Quintilianus andJohannes Pediasimos, with diagrams, later 16th cent. http//image http://www.wundersamessammelsurium.de/Heron/

Heron von Alexandrien Hero von Alexandrien, Heron aus Alexandrien, Hero aus Alexandrien Heron of Alexandria, Hero of Alexandria, Heron from Alexandria, Hero from Alexandria Unter allen "alten" Findern und Erfindern ist der Grieche Heron von Alexandrien als außerordentlich bemerkenswert hervorzuheben. Er war ein gelehrter Wissenschaftler, Mathematiker, Mechaniker, Physiker, Naturforscher, Techniker, Ingenieur der Antike und lebte in Alexandrien, Ägypten. Von wann bis wann er genau lebte ist unklar, ein Gelehrtenstreit, aber es war wohl in der Zeitspanne 150 v. Chr. - 250 n. Chr. De Automatis (Automat = griech.: Selbstbeweger). Eine Sammlung von Konstruktionen von 'thaumata', "Wunder"-erzeugenden Geräten speziell für Tempel. Heron beschreibt u. a.: sich selbst entzündende Opferfeuer, drehende und bewegende und tanzende Figuren, Bühnenblitze, Bühnendonner, selbsttätig öffnende und schließende Türen, automatische Musik auf Zimbeln und Trommeln, den Ausfluss von Wein oder Milch aus dem Becher einer Figur. Automaten gibt es mindestens seit dem 3. Jh. v. Chr. Der griechische Mechaniker Philon von Byzanz beschreibt in seiner "Mechanike syntaxis" pneumatische Apparate. Damals gab es sogar automatische astronomische Anzeigegeräte.

26. Final Project Contents. I. A brief history of heron of alexandria. II. V. Resources. Heronof Alexandria. Not much is known about the man named heron of alexandria. http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Umberger/MATH7200/HeronFormulaPr

MATH 7200 : Foundations of Geometry I University of Georgia, Fall 2000 Dr. McCrory, Instructor Final Project: Heron's Formula by Shannon Umberger Contents I. A brief history of Heron of Alexandria II. Heron's Formula , including a GSP sketch to test III. Three proofs of Heron's Formula: one algebraic, one geometric, and one trigonometric IV. Related topics: A. Brahmagupta's Generalization , including a GSP sketch to test and a proof B. An extension of Brahmagupta's Generalization, including a GSP sketch to test C. The Pythagorean Theorem , including a proof using Heron's Formula V. Resources Heron of Alexandria Not much is known about the man named Heron of Alexandria. Even his name is not definite; he has been called Heron and Hero. No one knows exactly when he lived, though it is commonly believed that he lived sometime between 150 B.C. and 250 A.D. Heron did live in the great scholarly city of Alexandria, Egypt, where many Greek mathematicians and scientists studied. Yet it is not known whether he was a Greek or actually an Egyptian with Greek training. What is sure, though, is that Heron of Alexandria was a brilliant man who gave the modern world much insight into the mathematical and physical sciences. Heron wrote so many works on mathematical and physical subjects that "it is customary to described him as an encyclopedic writer in these fields" (Eves, p. 178). Most of these works can be divided into two categories: geometric and mechanical. While approximately fourteen of his treatises have been uncovered, there are references to other lost works.

27. Math Forum - Ask Dr. Math heron of alexandria Born about 65 in (possibly) Alexandria, Egypt Died about125 Sometimes called Hero, Heron was an important geometer and worker in http://mathforum.org/dr.math/problems/kilgore11.12.97.html

Associated Topics Dr. Math Home Search Dr. Math Who was Hero (or Heron)? Date: 11/12/97 at 12:50:48 From: Jeff Kilgore Subject: Who was Hero? I have been trying to find information on the Greek mathmatician Hero. The encyclopedias that I have referenced either do not give any information, or they just state that he was a Greek who found a formula for the area of triangles. Any references would be helpful. http://mathforum.org/dr.math/ Associated Topics High School History/Biography Middle School History/Biography Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

28. About "Heron's Calculation Of Triangle Area (MSTE)" The area of a triangle can be calculated using a formulaattributed to heron of alexandria. This calculation requires...... http://mathforum.org/library/view/7888.html

Heron's Calculation of Triangle Area (MSTE) Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.mste.uiuc.edu/dildine/heron/triarea.html Author: James P. Dildine; University of Illinois at Urbana-Champaign (UIUC) Description: The area of a triangle can be calculated using a formula attributed to Heron of Alexandria. This calculation requires that the lengths of all three sides be known. This activity includes an Excel file to calculate the area of a triangle, and Java and Sketchpad files to experiment with interactive triangles. Explore the Triangle Inequality Theorem while exmining the acceptable ranges for the third side of a triangle. Levels: Middle School (6-8) High School (9-12) Languages: English Resource Types: Lesson Plans and Activities Problems/Puzzles Documents/Sketches/Galleries General Software Miscellaneous ... Web Interactive/Java Math Topics: Triangles and Other Polygons Suggestion Box Home The Math Library ... Search http://mathforum.org/ webmaster@mathforum.org

29. Heron1 ? Now there is a formula, attributed to heron of alexandria,that enables us to calculate the area of a triangle directly in terms of http://www.mathwright.com/library2/heron1.htm

MPBodyInit('heron1_files') Maximizing Areas and a Formula of Heron The Story… Go to Table of Contents Press F11 for Full Screen             This Mathwright Microworld develops Heron's formula for calculating the areas of triangles in a surprising way.  It shows that Heron's classical formula is actually a special case of a more general construction on quadrilaterals.  Hero of Alexandria (ad 20?-after 62), was a Greek mathematician and scientist. His name is also spelled Heron. He appears to have been of Egyptian birth, to have done his work in Alexandria, Egypt, and to have written at least 13 works on mechanics, mathematics, and physics. He developed various mechanical devices, including the aelopile, a rotary steam engine; Hero's fountain, a pneumatic apparatus in which a vertical jet of water is produced and sustained by air pressure; and the dioptra, a primitive theodolite, a surveying instrument. He is best known, however, as a mathematician. In geometry and geodesy he handled problems of mensuration more successfully than anyone of his time. He also devised a method of approximating the square roots and cube roots of numbers that are not perfect squares or cubes. The formula attributed to him, however, for finding the area of a triangle in terms of its sides, was devised before his time.                                     - From "Hero of Alexandria," Microsoft (R) Encarta, 1993.

30. Mathematician Biographies explain the theory of the planets. Link to more information Back toTop heron of alexandria. Born about 10 (possibly in Alexandria http://carrie.soffietti.students.noctrl.edu/mathbios.htm

Biographies of Mathematicians The table below is alphabetical order; however, the biographies that follow are in chronological order, according to birth dates. Agnesi deLagny Newton Thales ... References Return to: Carrie Soffietti's Home Page Thales of Miletus Born: about 624 BC in Miletus, Asia Minor (modern day Turkey) Died: about 547 BC in Miletus, Asia Minor (modern day Turkey) He was a pre-Socratic philosopher, who specialized in geometry. After being a merchant toward the beginning of his life, his interests shifted to astronomy, then to philosophy, and then to mathematics. Thales calculated the height of the Great Pyramid in Egypt with the sun and a stick. He is also known for his five geometric theorems that state (1) a closed angle circumscribed in a semicircle is a right angle, (2) a diameter bisects a circle, (3) if two sides of a triangle are equal then their bases are equal and vice versa (the definition of an isosceles triangle), (4) the vertical angle theorem, and (5) the theorem for similar triangles: if two triangles have equal angles, then any ratio of corresponding sides has the same value as any other (which is the basis for trigonometry). No direct writings are attributed to him, for they may have been lost or his findings were only recorded by others. Link to more information Back to Top Pythagoras of Samos Born: about 569 BC in Samos, Ionia

31. History Of Mathematics: Greece c. 77 BCE); Cleomedes (c. 40? BCE); heron of alexandria (fl. c. 62 CE)(Hero); Theodosius of Tripoli (c. 50? CE?); Menelaus of Alexandria http://aleph0.clarku.edu/~djoyce/mathhist/greece.html

Greece Cities Abdera: Democritus Alexandria : Apollonius, Aristarchus, Diophantus, Eratosthenes, Euclid , Hypatia, Hypsicles, Heron, Menelaus, Pappus, Ptolemy, Theon Amisus: Dionysodorus Antinopolis: Serenus Apameia: Posidonius Athens: Aristotle, Plato, Ptolemy, Socrates, Theaetetus Byzantium (Constantinople): Philon, Proclus Chalcedon: Proclus, Xenocrates Chalcis: Iamblichus Chios: Hippocrates, Oenopides Clazomenae: Anaxagoras Cnidus: Eudoxus Croton: Philolaus, Pythagoras Cyrene: Eratosthenes, Nicoteles, Synesius, Theodorus Cyzicus: Callippus Elea: Parmenides, Zeno Elis: Hippias Gerasa: Nichmachus Larissa: Dominus Miletus: Anaximander, Anaximenes, Isidorus, Thales Nicaea: Hipparchus, Sporus, Theodosius Paros: Thymaridas Perga: Apollonius Pergamum: Apollonius Rhodes: Eudemus, Geminus, Posidonius Rome: Boethius Samos: Aristarchus, Conon, Pythagoras Smyrna: Theon Stagira: Aristotle Syene: Eratosthenes Syracuse: Archimedes Tarentum: Archytas, Pythagoras Thasos: Leodamas Tyre: Marinus, Porphyrius Mathematicians Thales of Miletus (c. 630-c 550)

32. History Of Mathematics: Chronology Of Mathematicians A list of all of the important mathematicians working in a given century.Category Science Math Mathematicians Directories...... BCE?) *SB 1 CE. Theodosius of Tripoli (c. 50? CE?); Pamphila (c. 60 CE);heron of alexandria (fl. 62 CE) (Hero) *SB *MT 100 CE. Balbus (fl. http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html

Chronological List of Mathematicians 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 Table of Contents 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 List of Mathematicians 1700 B.C.E. Ahmes (c. 1650 B.C.E.) *MT 700 B.C.E. Baudhayana (c. 700) 600 B.C.E. 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) 500 B.C.E. 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

33. "Natural Magick" - "Glossary/Index - H" heron of alexandria Greek mathematician heron of alexandria, who livedin 1 AC. Heron teaches, that in burning of walls, after http://members.tscnet.com/pages/omard1/h.htm

Home Page "Natural Magick" The Author And His Work Editor/Producer ... Book Store "H" A B C D ... Z note : herbs/plants ref. w/" The English Physitian ", Culpeper, Nicholas, 1616-1654. where possible. or ( B otanical.com A Modern Herbal , Mrs. M. Grieve) Habergeon Habergeon (Hauberk) - Properly, a short hauberk, but often used loosely for the hauberk. A coat of mail; especially, the long coat of mail of the European Middle Ages, as contrasted with the habergeon, which is shorter and sometimes sleeveless. "...How an Habergeon or Coat of Arms is to be tempered..." Hackle Hackle - One of the peculiar, long, narrow feathers on the neck of fowls, most noticeable on the cock. "...And if a women with a child meets a Serpent , her fruit becomes abortive. Hence it is, that when a woman is in very Fore Travel , if she does but smell the fume of an Adder's Hackle , it will presently either drive out, or destroy her child. .." Hack Hack - A frame or grating of various kinds; as, a frame for drying bricks, fish, or cheese; a rack for feeding cattle; a grating in a mill race, etc. And it is fit for to make wicks for candles. Yet that is stuck in with

34. Historical Background Reflection. 2.4 Reflected Light. Historical Background. Euclid and heron of alexandria. Heron(Hero) of Alexandria (c.100 AD) also occupies a place in history. http://www.qesn.meq.gouv.qc.ca/mst/sapco/opticks/Chapter2/4_historique.html

Reflection 2.4 Reflected Light Historical Background Euclid and Heron of Alexandria The first to formulate the second law of reflection was Euclid of Alexandria (330-260 BC). Euclid was without doubt the most important teacher and mathematician of his time. His book, The Elements , is a compilation of knowledge that remained the standard in teaching of geometry for 2000 years. The Elements consists of 13 books on plane geometry, the theory of numbers, irrational numbers, and the geometry of solids. The discovery of these ideas are not all attributed to him. Yet The Elements, to his credit, remains a masterpiece in its clarity of explanation and in its simplicity in the demonstration of theorems. Since 1482, more than 1000 editions of The Elements have been printed. Euclid's famous fifth postulate, that one and only one line can be drawn through a point parallel to a given line, gave birth to Euclidean geometry. It was only in the 19th century that this axiom was abandoned (non-Euclidean geometry). The portrait of Euclid comes to us from an 1800 edition of The Elements . It is unlikely that it is a faithful representation of the great mathematician. Heron (Hero) of Alexandria (c.100 AD) also occupies a place in history. He explained that angles of incidence and reflection which are equal would provide light with the shortest path (in distance). He proved that this was also the case with curved mirrors. Fermat then modified this law, noting that light follows a trajectory which demands the least time.

35. The History Of Mathematics - Library Center For E-courses ?. heron of alexandria The Mac Tutor History of MathematicsArchive, University of St. Andrews . heron of alexandria http://www-lib.haifa.ac.il/www/mesila/math/sites.htm

The History of Mathematics Trinity College, Dublin:á åôñàðù íåçúá íéøúà David R. Wilkins éãé ìò The History of Mathematics David R. Wilkins : é"ò êøòð History of mathematics resources Indexes of Biographies MacTutor History of Mathematics archive:êåúî Mathematicians of the Seventeenth and EigHteenth Centuries Mathematics Genealogy Project Mathematical Journey through Time The Mactutor History of Mathematics archive University of st Andrews Scotland,School of Mathematics and Statistics:êåúî Philosophy and History of Science Kyoto University World of Scientific Biography Erics Treasure Trove of Scientific Biography Arabic mathematics : forgotten brilliance? Doubling the cube History Topics: Babylonian mathematics History Topics: Ancient Egyptian mathematics ... udoxus of Cnidus The Mac Tutor History of Mathematics Archive, University of St. Andrews êåúî Eudoxus of Cnidus An Introduction to the works of Euklid with an Emphasis on the Elements Euclid of Alexandria The Mac Tutor History of Mathematics Archive University of St. Andrews:êåúî

36. First Steam Ball Framework, Projet  Europe Des Découvertes BC and the 3rd cent. AD). Données biographiques / Biographical data,heron of alexandria or Hero, mathematician and inventor. He http://www.inrp.fr/lamap/activites/projet/europe/grece/form2.htm

Accueil Activités The project Le projet Projet l'Europe des découvertes Framework for teachers Activités : Document de travail Katerina Garga garga@cti.gr Computer Technology Institute, Educational Technology Sector Athènes Grèce Publication : august 2001 Mise en ligne : august 2001 Titre / Title First Steam Ball Date 1st cent. A.D. Domaine scientifique / Scientific field Physics (mechanics-pneumatics) Nom du scientifique / Name of the scientist Heron (The dates of his birth and death are unknown; conjecture places them between the 2nd cent. B.C. and the 3rd cent. A.D.) Données biographiques / Biographical data Heron of Alexandria or Hero, mathematician and inventor. He is believed to have lived in Alexandria; although he wrote in Greek, his origin is uncertain. Several of his works survive either in Greek or in Latin translation. He wrote on the measurement of geometric figures, and a formula for finding the area of a triangle has been ascribed to him. Known for his study of mechanics and pneumatics, he invented many contrivances operated by water, steam, or compressed air; these include a fountain, a fire engine, siphons, and an engine in which the recoil of steam revolves a ball or a wheel. Description de la découverte ou de l'invention/ description of the discovery or invention .

37. Szolovits Sections Of 6.001, Fall 2002 one might build a computer algorithm for computing the square root of a nonnegativereal number (1) due to heron of alexandria, successively approximating y http://medg.lcs.mit.edu/people/psz/6.001/

Notes for Sections 5 and 6 of 6.001, Fall 2002 (equal? Recitation-Instructor ' Szolovits This Web page contains notes and program fragments that I have used in presenting some of the 6.001 recitations this term. A number of students have asked for these to be available, therefore I have collected them here. Higher-order Procedures "Alternative" methods of computing square-root Printing Pairs vs. Printing Lists Reader, Evaluator and Printer ... Maintaining a priority queue for simulation Higher-order Procedures In the recitation following the introduction of higher-order procedures, I argued that there is a necessary sense of design elegance for building complex systems that remain comprehensible. One aspect of that elegance is related to regularity and the ability to compose operations. For example, we looked at two alternative implementations of how one might take a derivative of an arbitrary function of one variable: The typical approach in ordinary programming languages is to define a procedure, say ddx , that takes as inputs the procedure that computes the function of one argument and the value, x , at which the derivative is to be evaluated.

38. Heron's Calculation Of Triangle Area This formula is attributed to heron of alexandria but can be traced back toArchimedes. This formula is represented by. Area=SQRT(s(sa)(sb)(sc)),. http://www.mste.uiuc.edu/dildine/heron/triarea.html

James P. Dildine Quickies: Triangles are fascinating mathematical topics. Many other mathematical topics typically encounter or utilize elemements discovered within triangles. Let us explore some aspects of triangles in an interactive fashion. This site will take you through some historical explorations, some interactive activities, as well as some intriguiging connections in mathematics. A triangle in typical plane geometry is descibed as a three-sided figure or polygon whose interior angle sum is equal to 180 degrees. The perimeter of a triangle is the sum of all three of the sides 1. Experiment with this file The area of a triangle is quite interesting. Often it will be represented by Area=(1/2) x Base x Height . Where the height is an altitude drawn from the base to the opposite angle. This formula makes for a relatively easy calculation of the area of a triangle but it is rather difficult to naturally find a triangle that is given in terms of at least one side (the base) and a height. We typically can determine or are given the sides of a triangle when a triangle is present. A formula does exist that can calculate the area of a triangle when all three sides are known. This formula is attributed to Heron of Alexandria but can be traced back to Archimedes.

39. Heron Triangle Area A formula for calcualting the area of a triangle when all sides are known is attirbutedto heron of alexandria but it is thought to be the work of Archimedes. http://www.mste.uiuc.edu/dildine/js/heron.html

James P. Dildine, 1999 A formula for calcualting the area of a triangle when all sides are known is attirbuted to Heron of Alexandria but it is thought to be the work of Archimedes. At any rate, the formula is as follows: A triangle has sides a, b, and c. After Calculating S, where S = (a+b+c)/2 The Area of a Triangle = SQRT(s*(s-a)(s-b)(s-c)) Use the Fields Below to calculate the Area of a Triangle with side lengths A,B, and C. Heron's Formula for Calculating the Area of a Triangle Sides Lengths A = B = C = or Area of Triangle = If you get a message that details the lengths do not fulfill the triangle inequality theorem. Recall that the triangle inequality theorem states that the SUM of TWO SIDES MUST ADD UP TO BE GREATER THAN THE LENGTH OF THE REMAINING THIRD SIDE. OR: If these are not fulfilled then you do not have a triangle and thus cannot calculate the area of an unbounded figure.

40. Untitled Document is about the begining of scientific research of propulsive motion in ancienttimes, including the first reactive engine created by heron of alexandria. http://www.informatics.org/museum/origins.html

The Origins Of Ideas of Space Flight. This exposition is about humanity's dreams about wings, flight, and space flight. From the earliest folk tales to fantasy and science fiction. The exhibition includes some information about Icarus and Kai-Kaus, Russian folk tales, and Lucian of Samosata, among others. - Lucian of Samosata (II Cent. A.D.). He was a Greek sophist and satirist. He wrote the very first science fiction space travel novels "True History" and "Icaro-Menippus". He described an accidental trip to the Moon by means of a sailing vessel. - Kai-Kaus. The hero king from epic poem "Shah-Nama", was published by the Persian poet Firdausi (1010 B.C.). Kai-Kaus acheived flight with the help of four eagles. This part of the exposition is about the begining of scientific research of propulsive motion in ancient times, including the first reactive engine created by Heron of Alexandria. Visitors can see here a model of the Ho-Tsyan Arrow (Fiery Arrow), and to learn about Van-Ghu's tragic flight. Additional information about Middle Century rocketry and Mongolphier's new hot air baloon flight technology are here as well. - Heron of Alexandria. Two thousand years ago, in his "Pneumatick", he described the principles of reactive motion. "The Sphere of Heron" shows the possibility of reactive motion.

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