41. Beauty In Euclid\'s Geometry House Of Essays Category science. Essay Title Beauty in euclid s geometry. Click here for professional written papers on Beauty in euclid s geometry. http://www.houseofessays.com/viewpaper/16030.html

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42. Biography Of Euclid Another story says that Ptomlemy asked the mathematician if there was an easier way to learn geometry, euclid replied, there is no royal road to geometry http://www.andrews.edu/~calkins/math/biograph/199899/bioeucli.htm

Biography of Euclid (330?-275? B.C. Back to the Table of Contents His Books The Elements Five Postulates ... Bibliography One of the most influential mathematicians of ancient Greece, Euclid, lived around 300 B.C. For his work in the field of geometry he is known as the father of geometry . He created the geometry called Euclidean Geometry. Very little is known about his life. It is believed that he was educated at Plato's academy in Athens, Greece. Most sources believe that he lived somewhere around 300 B.C. His 13 books, the Elements , are some of the most famous books in the world. He wrote them at about 300 B.C. According to Proclus (410-485 A.D. ) he said that Euclid came after the first pupils of Plato and lived during the reign of Ptomlemy I (306-283 B.C. ). It is said that Euclid established a mathematical school in Alexandria. Most history states that he was a kind, fair, patient man. One story that reveals something of his character, concerns a pupil that has just finished his first geometry lesson. The pupil asked what he would gain from learning geometry. So Euclid told his slave to get the pupil a coin so he would be gaining from his studies. Another story says that Ptomlemy asked the mathematician if there was an easier way to learn geometry, Euclid replied, "there is no royal road to geometry", and sent the king to study. His Books Euclid wrote many books such as: Data, On Divisions of Figures, Phaenomena, Optics

43. Euclid Proofs And The BOOK says that two lines are parallel if they never met, even if extended indefinitely in both directions which defines the fifth postulate of euclid s geometry. http://www.mlahanas.de/Greeks/Euclid.htm

Euclid's proofs in THE BOOK and the parallel postulate Godfrey Harold Hardy Euclid with Pythagoras Ptolemy and Aristotle (not shown here) in the famous Cathedral at Chartres in France built around 1260 AD http://people.cornellcollege.edu/S-Wilson/ THE BOOK THE BOOK of the BOOKS The BOOK will be included in the library of Babel of J. L. Borges that contains all books that can be formed by all combinations of all alphabetical symbols. The constraint that each book has 410 pages is not a problem. The library contains books that describe the life of each of us in all details and at all moments, even our future. What is the most important book in this library (The BOOK of the BOOKS)? What information can the library contain? Is something that cannot be expressed in words? The Elements and The BOOK Euclid's Book Stoicheion Elements , written around 300 B.C., has gone through more than 1000 editions since the invention of printing. The Romans were not very much interested in Science if it is not related to Military or Engineering purposes and until 480 AD no copy of the Elements was available in Latin. Euclid teaching geometry to his students Euclids book seems to be a series of axioms, postulates, proofs very strict arranged without any examples and this some have considered was the reason that generations of students had problems in learning mathematics and geometry. It is difficult to see how important this book is and even Euclid was asked by a student if he will have some benefit from learning geometry. Euclid is said to have given the student some money so that at least the student could not say that he has no benefit.

44. What Is Non-Euclidean Geometry? The philosopher Immanuel Kant (17241804) called euclid s geometry, the inevitable necessity of thought. Such philosophical opinions impeded mathematical http://njnj.essortment.com/noneuclideange_risc.htm

What is non-Euclidean geometry? An introduction to the history and mathematics of non-Euclidean geometries. Euclid's geometrical thesis, "The Elements" (c. 300 B.C.E), proposed five basic postulates of geometry. Of these postulates, all were considered self-evident except for the fifth postulate. The fifth postulate asserted that two lines are parallel (i.e. non-intersecting) if a third line can intersect both lines perpendicularly. Consequently, in a Euclidean geometry every point has one and only one line parallel to any given line. For centuries people questioned Euclid's fifth postulate. Even Euclid seemed suspicious of the fifth postulate because he avoided solving problems with it until his 29th example. Mathematicians stumbled with ways to prove the validity of the fifth postulate from the first four postulates, which we now call the postulates of absolute geometry. Those mathematicians who didn't fail were soon seen to have fallacious errors in their reasoning. These errors usually occurred because a mathematician had made self-fulfilling assumptions pertaining to parallel lines, rather than working with the other postulates. Essentially, they were forcing a result through the application of faulty logic. bodyOffer(29808) Though many mathematicians questioned Euclidean geometry, Euclidean thought prevailed through school mathematical programs. "The Elements" became the most widely purchased non-religious work in the world, and it still remains the most widely received of mathematical texts. Furthermore, mathematical inquiries into the nature of non-Euclidean geometries were often devalued as frivolous. The philosopher Immanuel Kant (1724-1804) called Euclid's geometry, "the inevitable necessity of thought." Such philosophical opinions impeded mathematical progress in the field of geometry. Karl Friedrich Gauss (1777-1855), who began studying non-Euclidean geometries at the age of 15, never published any of his non-Euclidean works because he knew the mathematical precedent was against him.

45. Euclid Although, the formal system advocated by euclid is incomplete, for the majority of us euclid s geometry is good enough, because it is simple yet concise. http://members.fortunecity.com/kokhuitan/euclid.html

The Mathematics of Euclid (325-265 BC)(Greek) Euclid of Alexandria is well-known for his great work, The Elements , which is a compilation of all the Geometry of his time. However, Euclid's effort is more than just a simple compilation of previous works. It is the first systematic approach to Geometry; the first axiomatic approach of its kind. Although usually associated with Geometry, The Elements also contained some works on Number Theory. Although, the formal system advocated by Euclid is incomplete, for the majority of us Euclid's Geometry is good enough, because it is simple yet concise. In the 20th century, David Hilbert completed the system by increasing the number of axioms from 5 to 20, and proved that the resulting system is complete and consistent. However, Euclid's effort cannot be underestimated. In fact, The Elements remained the Geometry text for 2000 years and is still used because of its simplicity. An it is the fifth postulate of The Elements that sprout the various non-Euclidean Geometry known to us now. The Elements This great work contains 13 books.

46. Euclid Mathematician Picture Math Gift Shirt Mathematicians Famous picture math gift shirt mathematicians, Famous mathematician pictures, mathematicians pictures, mathematician gifts, euclid, geometry, Eudlid s Elements http://www.mathematicianspictures.com/Mathematicians/Euclid.htm

Home about The Mathematicians Gift Shop about us ... Cantor Euclid Gift items available include: Euclid Poster Euclid All items carry our Total Satisfaction Guarantee . If you are dissatisfied with any item you purchase, simply advise us and return it within 15 days for replacement or refund. Eukleides (Euclid) c. 330 - 275 B.C.E Eukleides (Euclid of Alexandria), although little is known about his life, is likely the most famous teacher of mathematics of all time. His treatise on mathematics, The Elements , endured for two millennia as a principal text on geometry. The Elements commences with definitions and five postulates . The first three postulates deal with geometrical construction, implicitly assuming points, lines, circles, and thence the other geometrical objects.

47. Mathematics Geometry - Lesson Plans Webquests euclid s geometry History and Practice Alex Pearson, The Episcopal Academy, Merion, Pennsylvania euclid s geometry History and Practice euclid S http://www.edhelper.com/cat210.htm

Mathematics Geometry Return to edHelper.com Lesson Plans, Webquests, and Worksheets. Search: Categories Mathematics General (+93 WebQuests!) Arts (+42 WebQuests!) Computers Dance Drama Economics (+23 WebQuests!) Environment Geography Health (+27 WebQuests!) History Holidays Journalism Language (+173 WebQuests!) Languages Literature Mathematics (+93 WebQuests!) Music Physed Reading Science (+375 WebQuests!) Social Studies (+399 WebQuests!) Thematic Webquests (+146 WebQuests!) Math Worksheets ClassBuilder Gradebook Grading can now be more convenient with ClassBuilder! Maintain a gradebook, build exams, create lessons, and more. Version 2.0 now includes Palm Support! Sites for Teachers Ranks some of the popular educational sites. Lesson Plans Spelling Worksheets Vocabulary Worksheets Algebra Worksheets ... Spelling and Vocabulary Mathematics Geometry Math Worksheets Index Decimals Integers ... Grade 10 Mathematics Geometry Lessons! 3-D Drawing and Geometry - Cathi Sanders 3-D Drawing and Geometry - Cathi Sanders All about 2D shapes Beautiful Bugs - Joanne Caniglia, Diane Raney; Explorer ... Bubble Geometry > > Bubble Geometry - Science Museum of Minnesota Bubble Geometry Have you ever seen a square bubble? Experiment with bubbles. Create bubble wands out of found objects (straws, pipe cleaners, strawberry baskets and coat... buckyballs,

48. 10.8. Euclid (330?-275? B.C.) euclid replied, There is no royal road to geometry and sent the king to study. euclid s fame comes from his writings, especially his masterpiece Elements. http://www.shu.edu/projects/reals/history/euclid.html

10.8. Euclid (330?-275? B.C.) IRA Euclid is one of the most influential and best read mathematician of all time. His prize work, Elements , was the textbook of elementary geometry and logic up to the early twentieth century. For his work in the field, he is known as the father of geometry and is considered one of the great Greek mathematicians. Very little is known about the life of Euclid. Both the dates and places of his birth and death are unknown. It is believed that he was educated at Plato's academy in Athens and stayed there until he was invited by Ptolemy I to teach at his newly founded university in Alexandria. There, Euclid founded the school of mathematics and remained there for the rest of his life. As a teacher, he was probably one of the mentors to Archimedes Personally, all accounts of Euclid describe him as a kind, fair, patient man who quickly helped and praised the works of others. However, this did not stop him from engaging in sarcasm. One story relates that one of his students complained that he had no use for any of the mathematics he was learning. Euclid quickly called to his slave to give the boy a coin because "he must make gain out of what he learns." Another story relates that Ptolemy asked the mathematician if there was some easier way to learn geometry than by learning all the theorems. Euclid replied, "There is no royal road to geometry" and sent the king to study. Euclid's fame comes from his writings, especially his masterpiece

49. Properties Of Space euclid s geometry had done more than help architects and cartographers. The confidence in euclid s geometry was starting to be undermined. http://scholar.uwinnipeg.ca/courses/38/4500.6-001/Cosmology/Properties-of-Space.

Click here to download a Printable pdf version of this page. Oh, for heavens sake, Norman! You act as if you have never seen a hole in the time-space continuum before. Three properties of space will be discussed: Geometry, Topology and Dimensionality Euclidean and Non-Euclidean Geometry In one area of human inquiry there had long existed a quiet confidence in our ability to fathom something of the ultimate truth about the universe. People thought that if this success was possible in one area of inquiry then perhaps it was true in others. The source of this confidence was the age-old study of geometry that Euclid and the ancient Greeks had placed upon a firm logical foundation. Euclidean geometry is a geometry where the Pythagoras Theorem for triangles holds. The theorem gives the distance-squared between two points (c in the diagram) as the sum of the squares of the other two sides (a and b ). Any space where this Euclidean distance function holds is said to be spatially flat.

50. EUCLID, The Elements Three works by euclid have not survived Porisms possibly an ancient version of analytic geometry. Surface Loci ? Pseudaria ? http://www.math.tamu.edu/~don.allen/history/euclid/euclid.html

Next: About this document EUCLID Euclid is known to almost every high school student as the author of The Elements , the long studied text on geometry and number theory. No other book except the Bible has been so widely translated and circulated. From the time it was written it was regarded as an extraordinary work and was studied by all mathematicians, even the greatest mathematician of antiquity Archimedes, and so it has been through the 23 centuries that have followed. It is unquestionably the best mathematics text ever written and is likely to remain so into the distant future. Euclid Little is known about Euclid , fl. 300BC, the author of The Elements . He taught and wrote at the Museum and Library at Alexandria, which was founded by Ptolemy I. Almost everything about him comes from Proclus' Commentary , 4th cent AD. He writes that Euclid collected Eudoxus' theorems, perfected many of Theaetetus', and completed fragmentary works left by others. Euclid is said to have said to the first Ptolemy who inquired if there was a shorter way to learn geometry than the Elements: ...there is no royal road to geometry

51. Euclid It was only in the 19th century that the limitation of euclid s geometry as applied to space was first discovered by Nicholas Lobatchevsky, and later by http://www.angelfire.com/ks/learning/euclid.html

var cm_role = "live" var cm_host = "angelfire.lycos.com" var cm_taxid = "/memberembedded" **EUCLID** [334? - 280 B.C.] or [325 - 270 B.C.] By Arun Kumar Tripathi Darmstadt University of Technology, Germany WHO has not heard of Euclid during his first lessons in geometry in the School? He created the geometry of the universe on which Newton built his laws of gravitation and motion. It was only in the 19th century that the limitation of Euclid's geometry as applied to space was first discovered by Nicholas Lobatchevsky, and later by Einstein who completely repudiated the Euclidean geometry in his Relativity Theory. However, Euclidean geometry and the Newtonian laws are the one which are most valid on earth. The great Greek mathematician Euclid, whose book on geometry has struck fear into teenagers' hearts for two millennia. Although he apparently studied at Plato's Academy in Athens, Euclid's home was Alexandria, Egypt, where he worked during the reign of King Ptolemy one of many Egyptian kings of the same name. (By the way, Ptolemy was not the astronomer Claudius Ptolemaeus known for the "Ptolemaic system" that identified the Earth as the center of the universe.) EARLY DAYS Euclid seemed to have studied in Plato's Academy, the then best known school of Mathematics a "Cambridge of Greece". He was believed to be a "Phoenician" with a "Greek outlook". It was the period when Alexander of Macedonia, after his world conquest, had established the township of Alexandria in Egypt. Ptolemy, the governor of Alexandria in Egypt, was a great learned man and he founded the great university of Alexandria which surpassed even Plato's Academy. There Euclid was invited to teach geometry.

52. Euclid than by learning all the theorems. euclid replied, There is no royal road to geometry and sent the king to study. THE ELEMENTALS. http://www.crystalinks.com/euclid.html

EUCLID (325 BC- 265 BC) Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements . The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. For his work in the field, he is known as the father of geometry and is considered one of the great Greek mathematicians. Very little is known about the life of Euclid. Both the dates and places of his birth and death are unknown. It is believed that he was educated at Plato's academy in Athens and stayed there until he was invited by Ptolemy I to teach at his newly founded university in Alexandria. There, Euclid founded the school of mathematics and remained there for the rest of his life. As a teacher, he was probably one of the mentors to Archimedes Little is known of Euclid's life except that he taught at Alexandria in Egypt. According to Proclus (410-485 A.D.) in his Commentary on the First Book of Euclid's Elements , Euclid came after the first pupils of Plato and lived during the reign of Ptolemy I (306-283 B.C.). Pappus of Alexandria (fl. c. 320 A.D.) in his Collection states that Apollonius of Perga (262-190 B.C.) studied for a long while in that city under the pupils of Euclid. Thus it is generally accepted that Euclid flourished at Alexandria in around 300 B.C. and established a mathematical school there. Proclus also says that Euclid "belonged to the persuasion of Plato,'' but there exists some doubt as to whether Euclid could truly be called a Platonist. During the middle ages, Euclid was often identified as Euclid of Megara, due to a confusion with the Socratic philosopher of around 400 B.C.

53. Euclid And The Elements euclid s axiomatic approach to geometry is what caused it to eclipse other Elements written before it (such as that of Hippocrates of Chios). http://www2.sunysuffolk.edu/wrightj/MA28/Euclid/Essay.htm

Euclid and the Elements Very little is known about the life of Euclid. He taught and wrote at the Museum and Library of Alexandria (Greece) around 300 BCE. The government established the Museum as a place where scholars would meet and discuss ideas. The fellows received a stipend and were exempt from taxation. An anecdote about Euclid is that when Ptolemy requested a short cut to geometric knowledge, Euclid replied that there "is no royal road to geometry." Another story is that when a student asked what practical use studying geometry could be, Euclid ordered a slave to give the man a penny, since "he must make gain from what he learns." Euclid wrote at least ten books on subjects ranging from mathematics to optics. His Elements was a textbook that was a compilation of mathematical knowledge of the time. The thirteen books included sections on geometry, number theory, and solid geometry. No original copy of the Elements exists. Over the centuries, errors entered manuscripts, as well as addition and "clarifications." Modern editions are based on a revision by the Greek commentator Theon (approx. 400 AD). The first complete Latin (the international language of science) appeared in the eighth century. The first printed English translation appeared in 1482 (Campanus). The first complete English translated was the Billingsley translation (1571). The importance of the Elements lies not only in the mathematical content, but in the structure and organization of the book. Euclid's axiomatic approach to geometry is what caused it to eclipse other "Elements" written before it (such as that of Hippocrates of Chios). Euclid starts with basic ideas and builds systematically on them. "To the modern reader, the work is incredibly dull. There are no examples, there is no motivation, there are no witty remarks, there is no calculation. There are simply definitions, axioms and proofs."

54. Greek For Euclid These are misguided efforts that miss the point completely. euclid s geometry is not the only geometry. Even euclid knew that geometry http://www.du.edu/~etuttle/classics/nugreek/lesson17.htm

The postulates are the basis on which the whole structure of the Elements is built, and they must be carefully constructed. Euclid gave five postulates, of which the first three are of remarkable simplicity. The final two actually contain the essential specification of Euclidean space. Many commentators have moved them erroneously to common notions, or have tried to prove them from the other postulates. These are misguided efforts that miss the point completely. Euclid's geometry is not the only geometry. Even Euclid knew that geometry on the surface of a sphere was different, but he had a concrete idea of three-dimensional space that his geometry was to represent. In fact, it does so very well. The departures from Euclidean geometry in the neighbourhood of the Earth are extremely small, less than those arising from drawing figures on a portion of the Earth's surface close to us. Euclid's results, are, eminently, true in a very practical sense. The fourth postulate is equivalent to a statement that angles and distances are unchanged by an arbitrary rotation or translation in space, and the fifth postulate that space is "flat" in a sense well-known in Riemannian geometry. These postulates are stated by Euclid in a form that is applicable to the course of reasoning in the Elements, and allow the proof of the necessary results. It is idle to try to prove these postulates; non-Euclidean geometries are well-known. The ai)th/mata are the things demanded, the postulates. The third-person imperative has no expressed subject; it means "let it be conceded that". The verb

55. What Is Science? Summaries And Reviews By Joan Hughes and Experiment by Joan Hughes Joan Hughes Summary and Review of Science and Hypothesis by Henri Poincare Poincare demonstrates that euclid s geometry does not http://www.mdx.ac.uk/www/study/Science.htm

Joan Hughes is a retired experimental chemist investigating the epistemological theories of physical and social science. Click here for poems by Joan Hughes Social Science History by Andrew Roberts Empiricism, Theory and Imagination The Ideas of Locke, Hume and Wollstonecraft ... What is Science? Summaries and Reviews by Joan Hughes A summary from a Chemistry book Physical Chemistry by W. J. Moore, 1972 Science, Geometry, Imagination and Experiment A summary and review of: Science and Hypothesis by Henri Poincare Experience and Inference A summary and review of: Human Knowledge By Bertrand Russell (1948) Deductive Testing A summary and review of: The Logic of Scientific Discovery by Karl Popper (1934) Science, Technology and Engineering The word Originally, it just meant knowledge: scientia being Latin for knowledge. In medieval Europe, science might be theoretical understanding of truth as distinct from the moral understanding of one's conscience. Science could be any knowledge learnt by study. The different branches of study were also known as sciences. Medieval universities taught seven sciences, in two groups:

56. ABC Geometry - Words To Help You Think About Space, Shape, Measurement And The L The Greeks called this geometry, which means earth measurement. About 300BC, a Greek called euclid, who lived in Egypt, developed proofs of the geometric http://www.mdx.ac.uk/www/study/glonumge.htm

The ABC Study Guide, University education in plain English alphabetically indexed. Click here to go to the main index ABC Geometry Words to help you think about space, shape, measurement and the laws of thought. Geometry: The ancient Egyptians and Mesopotamians developed methods of measuring objects and calculating relations that they used to build monuments like the pyramids The Greeks called this geometry, which means earth- measurement. About , a Greek called Euclid , who lived in Egypt, developed proofs of the geometric rules that the Egyptians had devised. Euclid's proofs started from axioms and reasoned logically from them to conclusions. This has been seen by some philosophers as a model for what science (or part of Science) should be. Hobbes argued for a Social Science based on Euclidian methods. Poincare used Euclidian and other geometries to argue that science is based on imagination. The 3,4,5 rule Take three straight lines Make one 3 units long, make another 4 units long, make the other 5 units long.

57. Springer-Verlag - Geometry & Topology A guided reading of euclid s Elements leads to a critical discussion and rigorous modern treatment of euclid s geometry and its more recent descendants, with http://www.springeronline.com/sgw/cda/frontpage/0,10735,4-102-22-2013290-0,00.ht

58. Geometry Lesson Plans euclid s geometry History and Practice This series of interdisciplinary lessons on euclid s Elements was researched and written by Alex Pearson, a Classicist http://www.teach-nology.com/teachers/lesson_plans/math/geometry/

Best Sites Curriculum Daily History Downloads ... Professional Development Enter your email address for FREE weekly teaching tips! Home Teacher Resources Lesson Plans Math ... 3-D Drawing and Geometry - In this geometry unit you will see some different types of 3-D drawings, and learn how to do these drawings yourself. You will also explore many interesting careers that use these techniques, from architecture to movies. All about 2D shapes - Children will know the names and properties of a variety of 2D shapes. Children will be able to identify a right angle and parallel lines. Beautiful Bugs - Reinforces observation skills by providing an opportunity to create symmetric shapes or bugs. Develops visualization skills as well as provides the opportunity to link science and the arts through a sketching activity. Buckyballs - Students wondering why we study about polyhedral can find one reason by looking through the information in the links to Buckyballs.

59. Proclus Diadochus Was A Neoplatonist And The Head Of Plato's Academy Who Wrote A Proclus Diadochus was a neoplatonist and the head of Plato s Academy who wrote a commentary on euclid s geometry. You are here About http://ancienthistory.about.com/cs/proclusdiadochus/

zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About History Ancient / Classical History Home ... Trojan War Hero Achilles - Troy zau(256,152,145,'gob','http://z.about.com/5/ad/go.htm?gs='+gs,''); Atlas and Places - Where? Ancient Greece - Greek Ancient Rome - Romans ANE Egypt Persia Israel... ... Help zau(256,138,125,'el','http://z.about.com/0/ip/417/0.htm','');w(xb+xb); Stay Current Subscribe to the About Ancient / Classical History newsletter. Search Ancient / Classical History Proclus Diadochus Proclus Diadochus was a neoplatonist and the head of Plato's Academy who wrote a commentary on Euclid's geometry. Alphabetical Recent Up a category Proclus Diadochus Glossary entry on Proclus explaining origin of his name and his philosophical career. Philosophers Timeline Chronological list of Greek and Roman philosophers and mathematicians with dates. Early Geometry Section of Proclus' Commentary on Euclid's Geometry. Proclus Biography of Proclus, with a look at his contributions to geometry, astronomy, physics and theology. Proclus Encyclopedia Britannica article on Proclus calls him the last major Greek philosopher. As a neoplatonist he taught that thoughts are reality, and concrete "things" are merely appearances

60. Geometry: Euclid And Beyond today for euclidean geometry or just plane geometry and solid geometry to not mean synthetic geometry but rather a version of euclid s geometry with the http://www.math.cornell.edu/~dwh/papers/Euclid-review/Euclid-review.htm

Bulletin of the A.M.S Geometry: Euclid and Beyond by Robin Hartshorne, Springer-Verlag, New York, 2000, xi+526, ISBN 0-387-98650-2 Reviewed by David W. Henderson Introduction The first geometers were men and women who reflected on their experiences while doing such activities as building small shelters and bridges, making pots, weaving cloth, building altars, designing decorations, or gazing into the heavens for portentous signs or navigational aides. Main aspects of geometry emerged from three strands of early human activity that seem to have occurred in most cultures: art/patterns, building structures, and navigation/star gazing. These strands developed more or less independently into varying studies and practices that eventually were woven into what we now call geometry Art/Patterns: To produce decorations for their weaving, pottery, and other objects, early artists experimented with symmetries and repeating patterns. Later the study of symmetries of patterns led to tilings, group theory, crystallography, finite geometries, and in modern times to security codes and digital picture compactifications. Early artists also explored various methods of representing existing objects, and living things. These explorations led to the study of perspective and then projective geometry and descriptive geometry, and (in 20th Century) to computer-aided graphics, the study of computer vision in robotics, and computer-generated movies (for example, Toy Story). Navigation/star gazing: For astrological, religious, agricultural, and other purposes, ancient humans attempted to understand the movement of heavenly bodies (stars, planets, sun, and moon) in the apparently hemispherical sky. Early humans used the stars and planets as they started navigating over long distances; and they used this understanding to solve problems in navigation and in attempts to understand the shape of the Earth. Ideas of trigonometry apparently were first developed by Babylonians in their studies of the motions of heavenly bodies. Even Euclid wrote an astronomical work

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