Astronomy 110 Lecture 7 Days of week were named for Sun, Moon, and visible planets Impact of Greek Thought: Philosophical Basis By 600BCE common Greek view was that: • Universe was rational • It followed natural laws • Investigation and critical thought encouraged and valued • Emphasis on process Sounds rational but: Pythagorean Paradigm (~500BCE) • Earth unmoving, central • Planets move on circular orbits with uniform motion 1. Mathematical astronomy is good for our souls 2. the passionate love, “of the discipline of things that are always what they are.” 3. Ptolemy refers to his mathematical accounts as likenesses or images. [Models] Plato (~400BCE) • Laws of the universe came from reason not experiment. • No relationship between Physics and Mathematics • Earth at the center of the universe • Heavenly bodies on tightly packed spheres centered on the Earth • Heavens must be “perfect” : objects moving on perfect spheres or in perfect circles. Aristotle (~350BCE) The most sophisticated geocentric model Sufficiently accurate to remain in use for 1,500 years. Ptolemy’s major work named Hè Megalè Syntaxis (Greek) or the Almagest (Arabic) Originally a mathematical model to predict planetary positions Ptolemy of Alexandria (A.D. 100-170) From an Icelandic manuscript written between 1747 and 1752 (Wikipedia). Ptolemy’s Almagest First Latin Translation, George Trebizond, ca. 1481. First Arabic Translation 9th Century by Rabban al-Tabari, (the Rabbi of Tabaristan) astronomer and physician. Some highlights of the history of math and astronomy from: http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Chronology/full.html About 500BC The Babylonian sexagesimal number system is used to record and predict the positions of the Sun, Moon and planets. (See this History Topic.) About 450BC Greeks begin to use written numerals. (See this History Topic.) About 360BC Eudoxus of Cnidus develops the theory of proportion, and the method of exhaustion. About 340BC Aristaeus writes Five Books concerning Conic Sections. About 330BC Autolycus of Pitane writes On the Moving Sphere which studies the geometry of the sphere. It is written as an astronomy text. About 300BC Euclid gives a systematic development of geometry in his Stoicheion (The Elements). He also gives the laws of reflection in Catoptrics. About 290BC Aristarchus of Samos uses a geometric method to calculate the distance of the Sun and the Moon from Earth. He also proposes that the Earth orbits the Sun. About 250BC In On the Sphere and the Cylinder, Archimedes gives the formulae for calculating the volume of a sphere and a cylinder. In Measurement of the Circle he gives an approximation of the value of π with a method which will allow improved approximations. In Floating Bodies he presents what is now called "Archimedes' principle" and begins the study of hydrostatics. He writes works on two- and three- dimensional geometry, studying circles, spheres and spirals. His ideas are far ahead of his contemporaries and include applications of an early form of integration. About 235BC Eratosthenes of Cyrene estimates the Earth's circumference with remarkable accuracy finding a value which is about 15% too big. About 230BC Eratosthenes of Cyrene develops his sieve method for finding all prime numbers. (See this History Topic.) About 200BC Possible earliest date for the classic Chinese work Jiuzhang suanshu or Nine Chapters on the Mathematical Art. (See this History Topic.) About 180BC Date of earliest Chinese document Suanshu shu (A Book on Arithmetic). (See this History Topic.) 127BC Hipparchus discovers the precession of the equinoxes and calculates the length of the year to within 6.5 minutes of the correct value. His astronomical work uses an early form of trigonometry. About 150BC Hypsicles writes On the Ascension of Stars. In this work he is the first to divide the Zodiac into 360 degrees. About 20 Geminus writes a number of astronomy texts and The Theory of Mathematics. He tries to prove the parallel postulate. (See this History Topic.) About 110 Menelaus of Alexandria writes Sphaerica which deals with spherical triangles and their application to astronomy. About 150 Ptolemy produces many important geometrical results with applications in astronomy. His version of astronomy will be the accepted one for well over one thousand years. About 250 The Maya civilization of Central America uses an almost place-value number system to base 20. (See this History Topic.) 263 By using a regular polygon with 192 sides Liu Hui calculates the value of π as 3.14159 which is correct to five decimal places. (See this History Topic.) About 400 Hypatia writes commentaries on Diophantus and Apollonius. She is the first recorded female mathematician and she distinguishes herself with remarkable scholarship. She becomes head of the Neo-Platonist school at Alexandria. About 460 355 Zu Chongzhi gives the approximation /113 to π which is correct to 6 decimal places. (See this History Topic.) 499 Aryabhata I calculates π to be 3.1416. He produces his Aryabhatiya, a treatise on quadratic equations, the value of π, and other scientific problems. 575 Varahamihira produces Pancasiddhantika (The Five Astronomical Canons). He makes important contributions to trigonometry. 594 Decimal notation is used for numbers in India. This is the system on which our current notation is based. (See this History Topic.) 628 Brahmagupta writes Brahmasphutasiddanta (The Opening of the Universe), a work on astronomy; on mathematics. He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and compute square roots. About 700 Mathematicians in the Mayan civilization introduce a symbol for zero into their number system. (See this History Topic.) About 775 Alcuin of York writes elementary texts on arithmetic, geometry and astronomy. About 810 House of Wisdom set up in Baghdad. There Greek and Indian mathematical and astronomy works are translated into Arabic. About 810 Al-Khwarizmi writes important works on arithmetic, algebra, geography, and astronomy. In particular Hisab al-jabr w'al-muqabala (Calculation by Completion and Balancing), gives us the word "algebra", from "al-jabr". From al-Khwarizmi's name, as a consequence of his arithmetic book, comes the word "algorithm". About 900 Abu Kamil writes Book on algebra which studies applications of algebra to geometrical problems. It will be the book on which Fibonacci will base his works. 920 Al-Battani writes Kitab al-Zij a major work on astronomy in 57 chapters. It contains advances in trigonometry. About 970 Abu'l-Wafa invents the wall quadrant for the accurate measurement of the declination of stars in the sky. He writes important books on arithmetic and geometric constructions. He introduces the tangent function and produces improved methods of calculating trigonometric tables. About 990 Al-Karaji writes Al-Fakhri in Baghdad which develops algebra. He gives Pascal's triangle. About 1000 Ibn al-Haytham (often called Alhazen) writes works on optics, including a theory of light and a theory of vision, astronomy, and mathematics, including geometry and number theory. He gives Alhazen's problem: Given a light source and a spherical mirror, find the point on the mirror were the light will be reflected to the eye of an observer. About 1020 Ibn Sina (usually called Avicenna) writes on philosophy, medicine, psychology, geology, mathematics, astronomy, and logic. His important mathematical work Kitab al-Shifa' (The Book of Healing) divides mathematics into four major topics, geometry, astronomy, arithmetic, and music. About 1050 Hermann of Reichenau (sometimes called Hermann the Lame or Hermann Contractus) writes treatises on the abacus and the astrolabe. He introduces into Europe the astrolabe, a portable sundial and a quadrant with a cursor. 1072 Al-Khayyami (usually known as Omar Khayyam) writes Treatise on Demonstration of Problems of Algebra which contains a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. He measures the length of the year to be 365.24219858156 days, a remarkably accurate result. 1093 Shen Kua writes Meng ch'i pi t'an (Dream Pool Essays), which is a work on mathematics, astronomy, cartography, optics and medicine. It contains the earliest mention of a magnetic compass. .