1. 1 Presented at Central University of Finance and Economics 中央财经大学 Beijing by 卜若柏 Robert Blohm Chinese Economics and Management Academy 中国经济与管理研究院 http://www.blohm.cnc.net April 20, 2008 2008 年 4 月 20 日 Greek Mathematics & Astronomy

2. 2 Early Greek Mathematics and Astronomy  Area of greatest objective pre-eminence. Greek art is a matter of subjective taste.  Derived more from Egypt than Babylon simple numerical rules in mathematics records of observations in astronomy  King of Egypt asked Thales ( 泰勒斯 ) for height of pyramid Thales waited for sun to rise to where his shadow was long as he was tall At that moment he measured the pyramid’s shadow equal to its height  Laws of perspective studied by geometer Agatharcus ( 阿加塔库斯 ) to paint scenery for Aeschuylus’ ( 伊斯奇鲁斯的 ) plays.

3. 3 Early Greek Mathematics and Astronomy  problem. The Delian ( 德洛斯的 ) Problem. How could temples make a statue 2-times bigger than the original? Doubling the dimensions made the statue 8-times bigger. Athenians consulted the Oracle at the Temple of Apollo ( 亚波罗神寺 ) at Delos ( 德洛斯 ) in the hope of stopping a plague ravaging their country. They were advised to double Apollo’s ( 亚波罗神 ) altar that had the form of a cube. As a result of several failed attempts to satisfy the god, the pestilence only worsened & at the end they turned to Plato to solve the problem. Plato’s Academy worked on it for centuries. We have no such arithmetical device as a continued fraction nor a series of side-and-diagonal numbers by which we can extract so easily and so accurately as we can. http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html : http://www.cut-the-knot.org/arithmetic/antiquity.shtml

4. 4 Early Greek Mathematics and Astronomy  : first irrational number to be discovered (known to Pythagoreans). Algorithm developed to approximate its value by constructing two series (a “dyad”) {a 0, a 1, a 2,…,a n } representing a side, and {b 0, b 1, b 2,…,b n } the “diagonal” (hypotenuse) of an isoceles right triangle: with the respective components a n, b n related by the equation  Pythagorean theorem where square of “rational” hypotenuse b n 2 and of correct hypotenuse differ by 1] 2a n 2 = b n 2  2a n 2 - b n 2 =   b n /a n =, ---> as n increases where “1” is called “unity” or a “monad” 2 n a 1 2  a b a

5. 5 Early Greek Mathematics and Astronomy (cont.d)  : first irrational number to be discovered (known to Pythagoreans). Algorithm developed to approximate its value by constructing two series (a “dyad”) {a 0, a 1, a 2,…,a n } representing a side, and {b 0, b 1, b 2,…,b n } the “diagonal” (hypotenuse) of an isoceles right triangle(cont.d) and from which a n = a n-1 + b n-1, and b n = 2a n-1 + b n-1 is derived to generate the series from a 1 = 1, b 1 = 1:  2a n-1 2 = b n-1 2   (4a n-1 2 - 2a n-1 2 ) + 4a n-1 b n-1 = (2b n-1 2 - b n-1 2 ) + 4a n-1 b n-1   2a n-1 2 + 4a n-1 b n-1 + 2b n-1 2 = 4a n-1 2 + 4a n-1 b n-1 + b n-1 2    a n-1 + b n-1 ) 2 = (2a n-1 + b n-1 ) 2   2a n 2 = b n 2   

6. 6 Early Greek Mathematics and Astronomy (cont.d)  : first irrational number to be discovered (known to Pythagoreans). Algorithm developed to approximate its value by constructing two series (a “dyad”) {a 0, a 1, a 2,…,a n } representing a side, and {b 0, b 1, b 2,…,b n } the “diagonal” (hypotenuse), of an isoceles right triangle(cont.d) Socrates called the “diagonal” series the “rational” diagonal Unity was considered both the start of the series and the “monad” difference 1 Plato considered 1, unity, as infinity where the 2 series merged, i.e. at n =

7. 7 Early Greek Mathematics and Astronomy (cont.d)  : first irrational number to be discovered (known to Pythagoreans). Algorithm developed to approximate its value by constructing two series (a “dyad”) {a 0, a 1, a 2,…,a n } representing a side, and {b 0, b 1, b 2,…,b n } the “diagonal” (hypotenuse), of an isoceles right triangle(cont.d) series derivable identities: b n = a n + a n-1 = 2b n-1 + b n-2 a n = ½(b n + b n-1 ) = 2a n-1 + a n-2 side 1 2 5 12 29 diagonal 1 3 7 17 41 n 1 2 3 4 5 side diagonal

8. 8  Early Greek Mathematics and Astronomy (cont.d)  Fibonacci ( 斐波那契 ) series: algorithm that approximates the value of “the Golden Mean” ( 黄金分割, 黄金比 ) by constructing an alternative two series {a 0, a 1, a 2,…,a n } and {b 0, b 1, b 2,…,b n } by setting a n = a n-1 + b n-1, and b n = a n-1 + 2b n-1, with a 1 = 1, b 1 = 1 with the respective components a n, b n related by the equation   b n 2 - (2a n + b n ) 2 =   a n /b n =  by alternate excess and deficit of 4  as n increases http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html http://mathworld.wolfram.com/FibonacciNumber.html

9. 9  Early Greek Mathematics and Astronomy (cont.d)  Fibonacci series: algorithm that approximates the value of “the Golden Mean” by constructing an alternative two series {a 0, a 1, a 2,…,a n } and {b 0, b 1, b 2,…,b n } by setting a n = a n-1 + b n-1, and b n = a n-1 + 2b n-1, with a 1 = 1, b 1 = 1 the “Golden Mean” is defined in Euclid’s Elements, (Book VI, Def. 3) as the ratio AB:AC = AC:CB determining the “Golden Mean” point C on any line segment AB http://babbage.clarku.edu/~djoyce/java/elements/bookVI/defVI3.html

10. 10 Early Greek Mathematics and Astronomy (cont.d)  Fibonacci series: algorithm that approximates the value of “the Golden Mean” by constructing an alternative two series {a 0, a 1, a 2,…,a n } and {b 0, b 1, b 2,…,b n } by setting a n = a n-1 + b n-1, and b n = a n-1 + 2b n-1, with a 1 = 1, b 1 = 1 (cont.d) series (c n = c n-3 + 2c n-2 ) derivable b n = a n + b n-1 (c n = c n-1 + c n-2 ) identities: a n = b n-3 + 2a n-2 = 2a n-1 + a n-2 +…+ a 1 side 1 2 5 13 34 diagonal 1 3 8 21 55 n 1 2 3 4 5 http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html n 1 2 3 4 5 6 7 8 9 10... c n 1 1 2 3 5 8 13 21 34 55...

11. 11 Early Greek Mathematics and Astronomy (cont.d)  Fibonacci series: algorithm that approximates the value of “the Golden Mean” by constructing an alternative two series {a 0, a 1, a 2,…,a n } and {b 0, b 1, b 2,…,b n } by setting a n = a n-1 + b n-1, and b n = a n-1 + 2b n-1, with a 1 = 1, b 1 = 1 (cont.d) the ratio of the side to the diagonal of a regular pentagon is the "golden ratio” (Euclid’s 欧几里德的 Elements 几何原本, Book VI, Prop. 11) http://babbage.clarku.edu/~djoyce/java/elements/bookIV/propIV11.html

12. 12 Early Greek Mathematics and Astronomy (cont.d)  Fibonacci series: algorithm that approximates the value of “the Golden Mean” by constructing an alternative two series {a 0, a 1, a 2,…,a n } and {b 0, b 1, b 2,…,b n } by setting a n = a n-1 + b n-1, and b n = a n-1 + 2b n-1, with a 1 = 1, b 1 = 1 (cont.d) The Golden Mean itself is the numerical equivalent of the geometrical construction of dividing a line in 'extreme and mean ratio', as a preliminary to the construction of a regular pentagon: that again being the half-way house to the final triumph, perhaps the ultimate aim, of Euclidian or Pythagorean geometry, the construction of the regular dodecahedron, Plato's symbol of the Cosmos itself. http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html

13. 13 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers “Of the two series which thus begin alike and then part company, the one leads to the square-root of 2 or the hypotenuse of an isosceles right-angled triangle, and the other leads to the Divine or Golden Section. These are the two famous surds or 'irrational numbers' of antiquity, and they are also the two pillars of Euclidian geometry.” http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html

14. 14 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers (cont.d) “It is inconceivable that the Greeks should have been familiarly acquainted with the one and yet unacquainted with the other of these two series, so simple, so interesting and so important, so similar in their properties and so closely connected with one another. Between them they arithmetize what is admittedly the greatest theorem, and what is probably the most important construction, in all Greek geometry. Both of them hark back to themes which were the chief topics of discussion among Pythagorean mathematicians from the days of the Master himself; and both alike are based on the arithmetic of fractions, with which the early Egyptian mathematicians and doubtless the Greek also were especially familiar. Depend upon it, the series which has its limit in the Golden Mean was just as familiar to them as that other series whose limit is.” http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html

15. 15 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers (cont.d) http://www.salomoni.it/davide/theology/blog/images/durer_melancholy.jpg Albrecht Durer ( 阿尔布雷特 · 丢勒 ) 1514 Melancholy Contemplating Geometry Polyhedron of much modern mathematical speculation:  6 pentagon side faces and 2 triangle faces (top and bottom)  cube truncated at top and bottom corners  combination dodecahedron (pentagon faces) and (its dual, the) icosahedron (triangle faces)

16. 16 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers (cont.d) “ These two methods, of finding the value of and the value of the Golden Mean, are by no means mere rough approximations, but they actually lead, more easily and quickly than does our modern arithmetic, to results of extreme accuracy. In the case of the side and diagonal numbers we need go no farther than the tenth place in the table (as can be done in less than two minutes) to get a fraction which is equivalent to the value of to six places of decimals!.” http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html

17. 17 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers (cont.d) “ The technical phrase 'excess or defect' is sometimes used, especially by Aristotle, in a sense which is obviously not the arithmetical one, though it must be more or less analogous thereto. A single instance must suffice. In the first chapter of the Historia Animalium ( 动物史 ), Aristotle tells us that, within the limits of a 'genus', such as Bird or Fish, the difference between one form or species and another is of the nature of 'excess or defect'; that their corresponding parts differ in property or accident, or in the degree to which they are subject to this or that property or accident, or in number or in magnitude - in short always, after some fashion or other, in the way of excess or defect. This statement is neither more nor less than a foreshadowing of our own comparative morphology. Aristotle would regard each species of bird much as a modern morphologist does; he would recognize the correspondence or homology of their several parts; and he saw, better perhaps than many morphologists do, how the differences between these corresponding parts are essentially quantitative differences, or 'differences of degree'.” http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html

18. 18 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers (cont.d) “ Aristotle was thinking, more Platonico, of all the fowls of the air as mere visible forms or [shapes], mere imperfect representations of, or approximations to, their prototype the ideal Bird. Just as we study the rational forms of an irrational number and through their narrowing vista draw nearer and nearer to the ideal thing, but always fail to reach it by the little more or the little less: so we may, as it were, survey the whole motley troop of feathered things only to find each one of them falling short of perfection, deficient here, redundant there: all with their inevitable earthly faults and flaws. Then beyond them all we begin to see dimly a bird such as never was on sea or land, without blemish, whether of excess or defect: it is the ideal Bird, the [model which is set in heaven].” http://www-groups.dcs.st-and.ac.uk/~history/Extras/Thompson_irrationals.html

19. 19 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers (cont.d) D'Arcy Thompson 家德阿塞 · 燕卜逊 a Scotsman, lived from 1860 to 1948 D'Arcy Thompson combined skills in a way that made him unique. He was a Greek scholar, a naturalist and a mathematician. He was the first biomathematician. Find out more at: http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Thompson_D'Arcy.html

20. 20 Early Greek Mathematics and Astronomy (cont.d)  Zeno of Elea ( 阿基里斯的芝诺 ). “Infinitesimals” generate paradoxes. Tortoise and Hare (race): tortoise moves 1 meter per minute, hare moves 10 meters per minute. Tortoise starts 10 meters ahead of hare Paradox: Hare cannot pass tortoise. Solution: not before 11.11111… = 11 1/9 minutes. Infinite number of ever-smaller intervals still gives a finite result. 1 minute meters 1011 1.1 minute meters 11 11.1 1.11 minute meters 11.1 11.11

21. 21 Early Greek Mathematics and Astronomy (cont.d)  Zeno of Elea. “Infinitesimals” generate paradoxes. (cont.d) The arrow. Space-time intervals are divisible into instants of time where the arrow is motionless. Time intervals are a series of instants of time. Paradox: arrow cannot move. Solution: not unless an interval is an infinite number of instants. At any instant there is an infinite number of ever smaller time intervals around that instant.

22. 22 Early Greek Mathematics and Astronomy (cont.d)  Eudoxus ( 攸多克索 ) Extended theory of proportion to irrational numbers. Most important proposition in Euclid’s ( 欧几 里德的 ) Elements ( 几何原本 ) and needed for “Method of Exhaustion” ( 穷尽法 )  Arithmetic theory of proportion: ad = cb applicable only to rational numbers  Irrational numbers had been studied by Theodorus ( 狄奥多  罗斯 ), Theaetetus ( 泰阿泰德 ) and Democritus ( 德谟克里特 ). Led to:  Eudoxus’ Geometric theory (Archimedes’ 阿几米德的 Axiom):  na mb nc md, for multiples n & m:  applicable also to irrational numbers  repeated in Euclid: triangles and  parallelograms which are under  the same height are to one  another as their bases ><>< ><>< http://babbage.clarku.edu/~djoyce/java/elements/bookVI/propVI1.html

23. 23 Early Greek Mathematics and Astronomy (cont.d)  Eudoxus (cont.d) “Method of Exhaustion” ( 穷尽法 ).  Applied by Archimedes ( 阿几米德 ) in “squaring the circle” to prove the area of a circle is equal to that of a right- angled triangle having the two shorter sides equal to the radius of the circle and its circumference respectively 1st approximation by taking upper and lower limits determined by inscribed and circumscribed squares Area is 1/2 of same rectangle a a Inscribed square Circumscribed square

24. 24 Early Greek Mathematics and Astronomy (cont.d)  Eudoxus (cont.d) “Method of Exhaustion” (cont.d)  Applied by Eudoxus: 1. The area of square EFGH is exactly half of that of square ABCD. 2. Since the circle lies entirely inside ABCD, it follows that EFGH covers more than half of the area of the circle. 3. The area of triangle EPF is exactly half of that of rectangular area ELKF. 4. By adding rectangular areas ELKF to the four sides of the square, the circle's area is completely contained. So, by going to the octagon from the square, the triangles EPF are covering more than half of the area of the circle that lay outside the square, or more than 3/4 of the circle’s area. The 16- sided inscribed polygon covers more than 7/8 ths of the circle's area, and so on.

25. 25 Early Greek Mathematics and Astronomy (cont.d)  Eudoxus (cont.d) “Method of Exhaustion” (cont.d)  Applied by Antiphon ( 安提丰 ) to calculate . Archimedes carried it to a 96-sided polygon to get 22/7 = 3 1/7 ) in Russell’s book] 1 1   · d  ·  p   http://galileoandeinstein.physics.virginia.edu/lectures/greek_math.htm p: perimeter of polygon

26. 26 Early Greek Mathematics and Astronomy (cont.d)  Eudoxus (cont.d) “Method of Exhaustion” (cont.d)  Applied by Archimedes ( 阿几米德 ) in “squaring the parabola” Lemma. If ABS is an Archimedian triangle: 1. The median MS is parallel to the axis of the parabola, 2. A 1 and B 1 are the midpoints of the sides AS and BS of ABS, 3. O is the midpoint of MS. Proof 1. There exists a point F determining A'B'F such that (1.1) lines AS and BS serve as perpendicular bisectors of sides A'F and B'F, (1.2) [by Theorem ?] the perpendicular from S to side A'B' bisects A'B’, & (1.3) [by Theorem ?] the line SM is parallel to the sides AA' and BB' of trapezoid ABB'A' and therefore passes through the midpoint M of side AB. 2. Triangles AA 1 O and BB 1 O are also Archemedean. So the first part applies: the medians from A1 and B1 are parallel to the axis of the parabola. [By Theorem ?] Those medians serve as midlines in triangles AOS and BOS. 3. [By Theorem ?] A 1 B 1 is a midline in ABS. It therefore cuts in half any cevian from S, MS in particular. http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesTriangle.shtml

27. 27 Early Greek Mathematics and Astronomy (cont.d)  Eudoxus (cont.d) “Method of Exhaustion” (cont.d)  Applied by Archimedes in “squaring the parabola” (cont.d) http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesTriangle.shtml By the lemma and Archimedes’ Axiom, a parabola divides the area of an outer Archimedes triangle in ratio 2:1 to the inner Archimedes triangle, therefore area ABS = 2 x area ABO and, therefore, the area of the parabolic segment AB equals 2/3 of the area of outer Archimedes triangle ABS and 4/3 of the area of inner Archimedes triangle ABO. Proof: Let area of triangle ABS = 1. Archimedes filled the parabolic segment with triangles, whose areas are easily arranged into a geometric series whose sum he already knew. The first triangle in the series is the inner triangle ABO. From the Lemma and Archimedes’ Axiom, area ABO = 1/2, the first term of the series. By Archimedes Axiom Area(A 1 B 1 S) = 1/4. Therefore, by Archimedes Axiom, area(AA 1 O) + area(BB 1 O) = 1/4. AA 1 O and BB 1 O are outer Archimedes triangles with inner Archimedes triangles (filled pink triangles) inside the parabola segment. The combined area of the filled inner triangles is half that of their outer Archimedes progenitors: 1/2·1/4=1/8. This is the second term of the progression. The third term comes from four smaller outer Archimedes triangles with the total area of 1/4 of the two preceding Archimedes triangles (which was 1/4). The combined area of their inner triangles is therefore 1/2·1/4·1/4, etc. Continuing this process, the total, which is the area of the parabola segment, is 1/2 (1 + 1/4 + 1/4·1/4 +...) = 2/3.

28. 28 Early Greek Mathematics and Astronomy (cont.d)  Archimedes ( 阿几米德 ) Volume of paraboloid Volume of hemisphere http://www.math.tamu.edu/~dallen/masters/Greek/archimed.pdf Volume of cylinder  r 3 Volume of hemisphere 2/3  r 3 + Volume of cone 1/3  r 3  (r 2 -h 2 ) =  r 2 -  h 2 Volume of paraboloid 1/2  r 3 Volume of cone 1/3  r 3 http://acm.cs.binghamton.edu/contests/2006-02-09/statements/html/Problem4-ArchimedesContest.html http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=631&pf=1 =

29. 29 Archimedes defined a unit called a "myriad" equal to 10 000 (same as Chinese unit 万 wan) and used powers of 100 000 000 (same as Chinese unit 亿 yi)

30. 30 Early Greek Mathematics and Astronomy (cont.d)  Euclid’s ( 欧几里德的 ) Elements ( 几何原本 ) Content was not original. Order of propositions and logical structure were Euclid’s. Foundation for the axiomatic method and mathematical reasoning. Provides the basis for constructing the regular solids assumed in Plato’s Timaeus ( 蒂迈欧篇 ). Assumptions deemed unquestionable were undermined by the relativity theory of gravitation (curved space geometry). Utility not expected  Ignored by Romans, except for mention by Cicero ( 西塞罗 ).  11th century: first still extant Latin translation  Conic sections were useless until the 17th century when Parabolic motion of projectiles discovered by Galileo, making them key to warfare Elliptic motion of planets discovered by Kepler ( 开普勒 ) making them key to astronomy.

31. 31 Greek-style proof not provided in Chinese calculation of height and distance of a distant object) In the 海盗算经 Haidao Suanjing (Sea Island Mathematical Manual written by 刘徽 Liu Hui in 263 AD) book of the 十部算经 (Ten Mathematical Classics Shibu Suanjing) introduced at the Imperial Academy, and included in the civil service examinations, by the 隋朝 (Sui) and 唐朝 (Tang) Dynasties. P1 and P2 are poles 5 pu high and 1000 pu apart. When viewed from X at ground level, 123 pu behind P1, the summit S of the island is in line with the top of P1. Similarly when viewed from Y at ground level, 127 pu behind P2, the top of the island is in line with the top of P2. Calculate the height a of the island and its distance b from P1. [Note: 1 pu is about 2 metres.] Suppose the poles are of height h and the distance between the poles is d. The height of the island: 1255 pu; distance from P1 to the island: 30750 pu. d h h b a http://www-history.mcs.st-and.ac.uk/HistTopics/Ten_classics.html by angular congruence of triangles by angular congruence of triangles This proof/derivation is ignored Only these two formulae/conclusions are stated for memorization

32. 32 http://www.math.brown.edu/~banchoff/STG/ma8/papers/mcecil/images/curved_space.jpg

33. 33 Early Greek Mathematics and Astronomy (cont.d)  Astronomy Many centuries of observations by Babylonians and Egyptians  Morning and evening star (Venus) thought to be different planets  Discovered cycle of eclipses. Reliable prediction of lunar eclipses.  Babylonians divided right angle into 90 degrees and the degree into 60 minutes. Anaximander’s ( 阿那克西曼德的 ) hypothesis  that the earth floats freely, not supported on anything, but is still immovable since, because it is at the center, it has no reason to move in one direction or another,  rejected by Aristotle because of the absurd conclusion that a man placed in a circle with food at different equal distances would starve to death for being unable to choose. [Buridan’s ( 布理当的 ) ass] Pythagoras was first to think of the earth as spherical, for aesthetic reasons.

34. 34 Early Greek Mathematics and Astronomy (cont.d)  Astronomy Anaxagoras discovered that the moon shines by reflected light, projecting the earth’s circular shadow. This supported the Pythagorean’s hypothesis of the earth as a sphere. Pythagoreans thought  the morning and evening stars (Venus) are identical;  all the planets move in circles around a “central fire”, but not the sun similarly to Copernican ( 哥白尼式的 ) hypothesis and an extraordinary emancipation from anthropocentric thinking; but dropped this hypothesis shortly after the time of Plato; and behind the earth relative to the Mediterranean Sea;  the moon turns a constant face to the earth, as the earth turns a constant face to the “central fire”;  there is a counter-earth at the same distance from the “central fire”because an eclipse of the moon sometimes occurs when both moon and sun are above the horizon, and another body’s shadow must be being cast on the moon; it completed 10 heavenly bodies (5 planets, earth, counter earth, central fire, sun, & moon) and 10 was a mystic number.

35. 35 Early Greek Mathematics and Astronomy (cont.d)  Astronomy Concluded that the sun must be much larger than the earth Heraclides of Pontus ( 滂土斯的赫拉克利德 )  discovered that Venus and Mercury revolve around the sun;  supposed that the earth rotates on its own axis once every 24 hours Aristarchus of Samos ( 撒摩的亚里士达克 )  first advanced the complete Copernican ( 哥白尼式的 ) heliocentric hypothesis both that all the planets, including earth, revolve in circles around the sun the earth rotates rotates on its axis once every 24 hours.  held that the fixed stars and the sun remain unmoved  was considered impious for “putting the hearth of the universe (the  earth) in motion” Hipparchus ( 希巴古 ) led rejection of the heliocentric hypothesis  considered the greatest living astronomer  wrote systematically on trigonometry and discovered precession of the equinoxes  did many exacting measurements and extensive cataloging.  adopted & improved the theory of epicycles named later for astronomer Ptolemy ( 托勒密 ). Unscientific for introducing complicating hypotheses to “save the (homocentric) theory” of planetary motion.

36. 36 Early Greek Mathematics and Astronomy (cont.d)  Astronomy (cont.d) Measurement  Accurate measurement of earth’s diameter mean distance of the moon  Underestimation of the sun’s size and distance Geometric, not dynamic. No concept of force  introduced by Newton  banished from Einstein’s ( 爱因斯坦的 ) General Relativity Theory ( 普遍相对论 ) that reverts to geometry  Roman conquest ended scientific discovery Symbolized in murder of Archimedes by Roman soldier Under Roman domination  Greeks lost self-confidence  acquired paralyzing respect for their predecessors.

37. 37 Measuring the circumference of the earth On a certain date a stick placed in the ground at Syene cast no shadow, whereas a stick at Alexandria has a small shadow. http://abyss.uoregon.edu/~js/images/Eratosthenes.gif

38. 38 The simple model of the revolutions of spheres centered around the earth could not explain all astronomical phenomena. In particular, planets were observed to wander across the fixed fields of stars over time; mostly they wandered in one direction, but occasionally they seemed to reverse course. To explain this strange retrogradation, Aristotle claimed that planets were attached, not directly to deferents, but to smaller spheres called epicycles. The epicycles were themselves attached to the deferents; the simultaneous revolution of both sets of spheres created an occasional apparent reversal of the planets' motions across the skies of the Earth. Ptolemy further modified this model to more accurately reflect observations by placing epicycles upon epicycles, creating an extraordinarily complicated--but fairly accurate--depiction of the cosmos. He also displaced the Earth from the center of the universe, claiming that, while Earth was enclosed by the celestial spheres, the spheres actually revolved around a point called an eccentric, which was near the Earth but not quite on it. http://www.websters-online-dictionary.org/definition/GEOCENTRIC+THEORY

39. 39 The Sun and Moon revolve about the Earth and the stars are fixed to the surface of a transparent sphere which rotates westwards with a period of one sidereal day. Mercury and Venus move in circles, called epicycles, whose centres are fixed on the line joining the Sun to the Earth. The rest of the planets also move in epicycles, whose centres themselves move in large circles centred on the Earth. http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_ptolemy.html#figure24 http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/ptolemy.gif http://abyss.uoregon.edu/~js/images/e1.gif

40. 40 http://www.shpltd.co.uk/palmieri-galileo.pdf

41. 41 http://www.shpltd.co.uk/palmieri-galileo.pdf

42. 42 http://www.shpltd.co.uk/palmieri-galileo.pdf

43. 43 http://www.shpltd.co.uk/palmieri-galileo.pdf Next slide Galileo was particularly impressed by the observation of Venus changing from the crescent shapes to a full round disk as it receded from the Earth. This confirmed that Venus' orbit extended to the opposite side of the Sun from the Earth, a result predicted by Copernicus, and in contradiction of Ptolemy's theory that Venus moved in a circular orbit about a moving point between the Earth and the Sun. The crescent phases would be expected under either theory, but the small full round disk would be seen only if Venus orbited the Sun. http://www.pacifier.com/~tpope/Venus_Page.htm

44. 44 http://www.pacifier.com/~tpope/VenusPhases_1609-1611.JPG Photos of Venus obtained with the Celestron 8 inch telescope (rightt) and with the one-inch aperture Galilean telescope (left). The Galilean picture is an average of eleven 1/200th sec exposures taken on April 25, 2004 at 7:11 pm PDT, about an hour before sunset. The 8-inch picture is a single 1/125th sec exposure, reduced in size to match the Galilean photo. A similar sequence taken seventeen days later on May 12, 2004 at 6:05 pm PDT. The Galilean picture is an average of ten 1/160th sec exposures, while the 8-inch picture is a single 1/250th sec exposure taken 10 minutes later. At the time of these photos, Venus was nearing its closest approach to the Earth. Not only its change in shape, but also the increase in apparent diameter, from 34 arc- seconds on April 25th to 44 arc-seconds on May 12th, is clearly evident.

45. 45 http://www.shpltd.co.uk/palmieri-galileo.pdf The changing angular diameter of Venus as revealed by the telescope was also significant, for the Copernican system predicted dramatic changes in the distance to Venus during the year, yet the naked eye had been able to perceive only small changes in brightness. Finally, the fact that the telescope revealed Venus to be less than fully illuminated refuted the then current (and quite plausible) idea that the planets might shine by their own self- generated light, rather than by the reflection of sunlight. http://www.pacifier.com/~tpope/Venus_Page.htm

46. 46 http://abyss.uoregon.edu/~js/images/epicycle_deferent.gif In the Middle Ages, when Arabian astronomers had accumulated more accurate observations of the planets, that the validity of the Ptolemaic model began to be questioned. Only by further complicating the model by adding epicycles to epicycles and by tilting the orbits was it possible to explain the latest data. http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_ptolemy.html

47. 47 http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/retrograde.gif Copernicus correctly stated that the farther a planet lies from the Sun, the slower it moves around the Sun. When the Earth and another planet pass each other on the same side of the Sun, the planet appears to retrace its path for a short while (which is known as retrograde motion) and then continue in its original direction (which is known as prograde motion). http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_ptolemy.html

48. 48 http://www.pacifier.com/~tpope/Galilean_Moons_of_Jupiter_2PB280064.jpg Galileo’s observations of Jupiter’s moons demonstrated that a planet could have moons circling it that would not be left behind as the planet moved around its orbit. http://csep10.phys.utk.edu/astr161/lect/history/galileo.html http://www.pacifier.com/~tpope/Saturn_Overlay.gif Galileo’s view of Saturn Galileo’s telescope http://www.sciencemuseum.org.uk/images/object_images/535x535/10315150.jpg

49. 49 Despite Galileo’s discoveries undermining the Ptolemaic system, Copernicus’ system itself still had two drawbacks to overcome: (1) lack of better predictive power, (2) continued need for epicycles because of circular orbits, and (3) an apparent observational refutation. (1) Measurements could not help accurately predict information about the planets' positions & events like transits any better than the Ptolemaic system. http://answers.yahoo.com/question/index?qid=20080207062245AAlmPLz http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_ptolemy.html (2) The Copernican model, with it assumption of uniform circular motion, still could not explain all the details of planetary motion on the celestial sphere without epicycles. The difference was that the Copernican system required many fewer epicycles than the Ptolemaic system because it moved the Sun to the center. http://www.cartage.org.lb/en/themes/sciences/Astronomy/Thestars/stellardistances/TheParallaxMethod/TheCoper nicanModel/TheCopernicanModel.htm (3) The Copernican system itself seemed refuted by one important observation. If the earth did move, then one ought to be able to observe the shifting of the fixed stars due to parallax. In reality, the stars are so far away that this motion is undetectable without careful telescopic observations, but the lack of parallax was considered the death of any non-geocentric theory and the paralax was not confirmed until the 1838 century. http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_ptolemy.html The geocentric (Ptolemaic) model of the solar system is still of interest to planetarium makers, as, for technical reasons, a Ptolemaic-type motion for the planet light apparatus has some advantages over a Copernican-type motion. http://en.wikipedia.org/wiki/Ptolemaic_system

50. 50 Parallax is larger for closer objects If the earth were actually on an orbit around the sun, why wasn't a parallax effect observed? That is, as illustrated in the adjacent figure, stars should appear to change their position with the respect to the other background stars as the Earth moved about its orbit, because of viewing them from a different perspective (just as viewing an object first with one eye, and then the other, causes the apparent position of the object to change with respect to the background). This objection is valid, but failed to account for what we now know to be the enormous distances to the stars. As illustrated in the following figure, the amount of parallax decreases with distance. The parallax effect is there, but it is very small because the stars are so far away that their parallax can only be observed with very precise instruments. Indeed, the parallax of stars was not measured conclusively until the year 1838. http://www.cartage.org.lb/en/themes/sciences/Astronomy/Thestars/stellardistances/TheParallaxMethod/TheCoper nicanModel/TheCopernicanModel.htm