A Lesson On Spirals in the exercise below has been attributed to theodorus of cyrene (~465399 BC) . Share information with students about theodorus of cyrene (see first http://courses.wcupa.edu/jkerriga/Lessons/A Lesson on Spirals.html
Extractions: A Lesson on The Root Spiral Kate Long The Shipley School Klong@shipleyschool.org Objectives Practice with compass and straight edge Explore a geometric representation of square roots, deepening understanding Introduce students to spirals, curves that are seldom studied in traditional textbooks Develop an awareness of the historical context for the study of irrational numbers and spirals Recognize spirals in nature and appreciate the mathematics inherent in them Historical Perspectives Theaetus , Socrates makes reference to Theodorus proving that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 were irrational. Plato raises the question "Why did Theodorus stop at 17?". One possible answer is that the 17 hypotenuse belongs to the last triangle that does not overlap the figure. In 1958 E. Teuffel proved several more interesting facts about the root spiral. If the procedure for generating the spiral is continued indefinitely so that the figure overlaps, no two hypotenuses will coincide. In other words, they will never lie directly on top of each other. Also, if the sides of unit "one" length are extended forever, they will not pass through any of the other vertices of the total figure. The Spiral of Theodorus approximates the Logarithmic Spiral. By the early 1600s the logarithmic spiral was being studied in depth. See Historical Perspectives II in Appendix A for additional information.
Math History - Pre-historic And Ancient Times About 425BC, theodorus of cyrene shows that certain square roots are irrational.This had been shown earlier but it is not known by whom. http://lahabra.seniorhigh.net/pages/teachers/pages/math/timeline/MpreAndAncient.
Extractions: Prehistory and Ancient Times Middle Ages Renaissance Reformation ... External Resources About 30000BC Palaeolithic peoples in central Europe and France record numbers on bones. About 25000BC Early geometric designs used. About 4000BC Babylonian and Egyptian calendars in use. About 3400BC The first symbols for numbers, simple straight lines, are used in Egypt. About 3000BC Babylonians begin to use a sexagesimal number system for recording financial transactions. It is a place-value system without a zero place value. About 3000BC Hieroglyphic numerals in use in Egypt. About 3000BC The abacus is developed in the Middle East and in areas around the Mediterranean. A somewhat different type of abacus is used in China. About 1950BC Babylonians solve quadratic equations.
Personal Names search TmiWeb for this person theodorus of cyrene. Born 465 BC; died 398 BC.Theodorus was a tutor of Plato and Theaetetus and is best remembered by http://euromusicology.cs.uu.nl:6334/dynaweb/info/persinfo/persons/@Generic__Book
Extractions: Expand Search T Born: ; died: . Born: c. 1500; died: 1557. Italian mathematician. He was self-taught. In 1521, he was teaching mathematics in Verona and in 1534 he went to Venice. By 1541, he had achieved the remarkable triumph of solving the cubic equation. Tartaglia, general trattato Born: ; died: . Unidentified Greek writer about music, cited in Pontio, Dialogo , p. 4REM; FW Mentioned in Baldi, Cronica Mentioned in Baldi, Cronica Name variants: Born: 190 BC; died: 158 BC. Roman poet Adelphoe Andria Heautontim. Hecyra Phormio Name variants: Lived: fl. 8th-7th c. BC. Greek poet and musician, important in the development of classical Greek music and poetry. Upon somewhat doubtful evidence, Terpander is credited with having completed the octave and adding the sixth and seventh strings to the kithara. The muse of dancing, often seen dancing with her lyre and a plectrum.
Personal Names Theaetetus was a disciple of Socrates and studied with theodorus of cyrene.He taught at some time in Heraclea (located in presentday southern Italy). http://euromusicology.cs.uu.nl:6334/dynaweb/info/persinfo/persons/@Generic__Book
Extractions: Expand Search T Theaetetus Born: c. 417 BC; died: 369 BC. Athenian mathematician who had a significant influence on the development of Greek geometry. Theaetetus was a disciple of Socrates and studied with Theodorus of Cyrene. He taught at some time in Heraclea (located in present-day southern Italy). Biography by Baldi (from Cronica
History Of Mathematics: Greece 425); theodorus of cyrene (c. 425); Socrates (469399); Philolaus of Croton (dc390); Democritus of Abdera (c. 460-370); Hippasus of Metapontum (or of http://aleph0.clarku.edu/~djoyce/mathhist/greece.html
History Of Mathematics: Chronology Of Mathematicians 425) *SB *MT; theodorus of cyrene (c. 425); Socrates (469399); Philolaus ofCroton (dc 390) *SB; Democritus of Abdera (c. 460-370) *SB *MT 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
PSIgate - Physical Sciences Information Gateway Search/Browse Results theodorus of cyrene Born 465 BC in Cyrene (now Shahhat, Libya) Died 398 BC inCyrene (now Shahhat, Libya) Previous (Chronologically) Next Biographies http://www.psigate.ac.uk/roads/cgi-bin/search_webcatalogue.pl?term1=Libya&limit=
Chapter 16: Archimedes geometry teacher, Hippocrates of Chios, who tried to fit together all therules, and theodorus of cyrene, who discovered many of the irrationals. http://www.anselm.edu/homepage/dbanach/arch.htm
Extractions: During the 4th century B.C., Greek geometry burst its bonds and went on to the tremendous discoveries of the "age of giants." And Greek culture, too, burst from the mainland of Hellas and spread to most of the eastern Mediterranean. Both developments were connected with the romantic figure of Alexander the Great. After Plato's time, teachers and alumni from the Academy had gone on to found schools of their own. In particular, Plato's most famous associate, the great philosopher Aristotle, had set up the Lyceum in Athens, and started the systematic classification of human knowledge. And Aristotle's most renowned pupil was the warrior king Alexander of Macedon, who tried to conquer the world. In thirteen years, Alexander extended his rule over Greece proper, and Ionia, Phoenicia, Egypt, and the vast Persian domains as far as India. Then he died, and his empire broke up. But throughout those far-flung lands, he had founded Greek cities and planted the seeds of Greek civilization-the Greek language, Greek art, and, of course, Greek mathematics. Mathematicians traveled with his armies. And there is even a
Notes And Queries -- Table Of Contents ( 1920, S12-VI [106]) EDWARD BENSLY theodorus of cyrene Notes and Queries 1920 s12VI 158;doi10.1093/nq/s12-VI.106.158-e PDF. HFB COMPSTON theodorus of cyrene http://nq.oxfordjournals.org/content/vols12-VI/issue106/index.dtl
Extractions: To see an article , click its [Full Text] or [PDF] link. To review many abstracts , check the boxes to the left of the titles you want, and click the 'Get All Checked Abstract(s)' button. To see one abstract at a time , click its [Abstract] link. W. ROBERTS LORD CALEDON'S PICTURES FROM THE GERINI GALLERY
Notes And Queries -- Table Of Contents ( 1920, S12-VI [103]) HEG EVANS theodorus of cyrene Notes and Queries 1920 s12VI 91;doi10.1093/nq/s12-VI.103.91-a PDF. AMB IRWIN CHESS THE KNIGHT S TOUR http://nq.oxfordjournals.org/content/vols12-VI/issue103/index.dtl
Gazalé, M.: Number: From Ahmes To Cantor. What was the Ladder of theodorus of cyrene and how did the ancient Greeks calculatesquare roots with such extraordinary proficiency? http://www.pupress.princeton.edu/titles/6794.html
Extractions: From Ahmes to Cantor Shopping Cart Table of Contents We might take numbers and counting for granted, but we shouldn't. Our number literacy rests upon centuries of human effort, punctuated here and there by strokes of genius. In his successor and companion volume to Gnomon: From Pharaohs to Fractals This abundantly illustrated book, remarkable for its coherency and simplicity, will fascinate all those who have an interest in the world of numbers. Number will be indispensable for all those who enjoy mathematical recreations and puzzles, and for those who delight in numeracy. Table of Contents Subject Areas: VISIT OUR MATH WEBSITE Main Selection of the Library of Science Book Club
The History Of Freethought And Atheism Finally we can say with confidence that theodorus of cyrene was an atheist fromthe contents of his work On the Gods. Roman Skeptics http://www.positiveatheism.org/india/s1990c25.htm
Extractions: from "An Anthology of Atheism and Rationalism" The idea that atheism has a long and colorful history may strike some people as unlikely. It is true, however, and those few people who have made an effort to study that history have found enough there to fill several thick volumes (and most of them barely scratched the surface). There have undoubtedly always been unbelievers in the current religious belief of an area or a people. Before the invention of writing, however, we know of no specific unbelievers who left any record of their unbelief. In ancient India, there was a group of freethinkers known as the Lokayatas (before 600 B.C., although remnants of this group are found in India as late as the 14th century.) In China, both Confucius and Lao Tsze can be viewed as freethinkers in relation to the religion(s) prevalent in China at the time. Early Greek Freethinkers The first real freethinkers or atheists who seem to have produced work specifically dealing with religion in a negative way were found in ancient Greece. Although most of these works have not survived to the present day (and charges of atheism were often politically motivated and really not based upon fact), we know that Anaxagoras was accused of impiety and forced to leave Greece. He supposedly held that the sun was a red-hot body and that the moon was a physical object which was larger than Greece. He did not, however, apparently make an attack on the popular religious beliefs.
Ce-Cl: Positive Atheism's Big List Of Quotations Diagoras the Melian and theodorus of cyrene entirely believed there were no suchbeings. But they who have affirmed that there are Gods, have expressed such http://www.positiveatheism.org/hist/quotes/quote-c1.htm
Extractions: Celsus (ca. C.E. 170), quoted from Antony Flew, Atheistic Humanism, p. 17 This is one of their [the Christians'] rules. Let no man that is learned, wise, or prudent come among us: but if they be unlearned, or a child, or an idiot, let him freely come. So they openly declare that none but the ignorant, and those devoid of understanding, slaves, women, and children, are fit disciples for the God they worship.
Body 5.15 Hippocrates, On Regimen 5.16 Ps.Aristotle, Problems 5.17 Diogenes the Cynic5.18 Bion of Borysthenes 5.19 theodorus of cyrene 5.20-5.25 Zeno of Citium http://www.utexas.edu/courses/cc348hubbard/toc.html
From Ken.Pledger@vuw.ac.nz (Ken Pledger) Newsgroups Sci.math Shortly after then, theodorus of cyrene generalized the result by proving theirrationality of the square roots of 3, 5, , 15, but got stuck at sqrt(17) http://www.math.niu.edu/~rusin/known-math/98/sqrt_irrat
Extractions: From: Ken.Pledger@vuw.ac.nz (Ken Pledger) Newsgroups: sci.math Subject: Irrationals (was Re: please don't laugh...) Date: Wed, 09 Dec 1998 15:05:55 +1200 In article Newsgroups: sci.math Subject: Re: how to prove: if x is not a perfect square, sqr (x) is irrational, x positive integer Date: Sat, 12 Dec 1998 23:12:34 -0500 On Sat, 12 Dec 1998, TS wrote: :Date: Sat, 12 Dec 1998 20:40:37 GMT :From: TS
Biographical List Of Names (TH - THN) Compiled By GIGA theodorus of cyrene, Greek philosopher (fl. 340 BC) BUY AMAZON BOOK Theodotus,Samos rhetorician (- 43 BC) - READ QUOTES (1) BUY AMAZON BOOK http://www.giga-usa.com/gigaweb1/quotes2/qubioth.htm
Chronological Author List "Before 300 BC" Compiled By GIGA theodorus of cyrene, Greek philosopher (fl. 340 BC) BUY AMAZON BOOK ChuangTzu (Zhuangzi), Chinese philosopher (369 BC - 286 BC) - READ QUOTES (3) BUY http://www.giga-usa.com/quotes/lists/quayb300.htm
All Sides Of The Story Finally we can say with confidence that theodorus of cyrene was an atheist fromthe contents of his work On the Gods. Another current of thought that lasted http://www.teachingreligion.com/atheism/history.html
Extractions: Agnosticism Atheism Buddhism Christianity ... Links Some men have always disbelived in gods or supreme powers. The only problem is that the phenomenon of atheism could not be easily described early on, as primitive languages had no way to symbolize negation, or existence. Hence saying "gods don't exist" would be a daunting task indeed. The arrival of phonetic language changes that. Atheistic views started to emerge in India, then Greece. India : Probably the first sign of skeptic thought comes from the Rig-Veda, a text which is thought to have been written around 1000 BC. The philosophy promoted in it could be said to be atheistic by omission, as shows us this creation hymn : "Who knows for certain? Who shall here declare it? Whence was it born and whence came this creation? The gods were born after this world's creation. Then, who can know from whence it has arisen? None know whence creation has arisen and whether he has or has not produced it. He who surveys it in the highest heaven, he only knows, or happily, he may know not". Around 500 BC, Buddhism, inspired by the Rig-Veda, became a theistic philosophy. Jainism, an atheistic religion, also began around that time.
PHILTAR - Compendium Of Philosophers/T theodorus of cyrene (465398 BC). A brief introduction to his life and work.Theon of Smyrna (c70-c135). An introduction to his work http://philtar.ucsm.ac.uk/compendium_of_philosophers/t/
Extractions: Links to materials by and/or about over a thousand philosophers from thousands of years from all over the world from A to Z This compendium contains entries large and small, single or multiple, on hundreds of philosophers. Links vary in size from a few lines of biography to the whole of the Summa Theologica. Sometimes you are directed to a site which has further links. In that case there is no guarantee that all the further links will work, but enough work to make a visit worthwhile. This compendium does not provide links to philosophers own home pages. A list of them can be found here A B C ... Z Tagore, Rabindranath (1861-1941) A small Tagore website with a few links
