The encounter of Astrology and Mathematics. A glimpse of the minds of the Greeks. Christina Che ung " Nov 5, 2013_CU HK I

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1 The encounter of Astrology and Mathematics A glimpse of the minds of the Greeks Christina Che ung I~ " Nov 5, 2013_CU HK I

Mathematics ~ To the ancient Greeks we owe the notion of mathematics as a form of theoretical knowledge, a body of propositions in a deductively ordered sequence.

2 Mathematics ~ To the ancient Greeks we owe the notion of mathematics as a form of theoretical knowledge, a body of propositions in a deductively ordered sequence. ~ The transition from praxis to theorla - from mathematics as techne (techniques for dealing with practical activities) - to mathematics as eplsteme and gnests (a form of pure knowledge).

3 Mathematics - Pythagoras In around 530 B.C., Pythagoras and his followers numbers had a significance which went far beyond their function as a calculating device. There was, they argued, a mathematical reality underlying the physical world, "num bers in all things," and this could be grasped by reason and observation.

Pythagoras - Numbers in All Things To give a single example, take a string stretched across a sounding box and pluck a note from it. Halve the length of the string and try again.

4 Pythagoras - Numbers in All Things To give a single example, take a string stretched across a sounding box and pluck a note from it. Halve the length of the string and try again. The note is one octave higher. There appears to be some kind of relationship between sound and number. The Italian Leonardo Fibonacci, for instance, discovered around 1200 A. D. that certain numbers, 3,5,8,13,21,34,55,89, reoccur throughout nature. The petals of almost all flowers occur in one of Fibonacci's number. All snowflakes are different but all have a six-fold symmetry.

$r>; r-; f\ r-; I, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.!fo;,.l+n.21 V V V V V + + + + +,,! 89 " " 2- ~ 5 so.ror lht lou t di,11b' tht Ifi'l" tntt ll11pl)'.$

5 Pythagoras - Numbers in All Things The Fibonacci Sequence Explained 1 + I = ~ T li~ Ffbonacct Seq uence =[J] Explained = ~ 3 +5 ~ 1ID 5 +8 ~ 1UI ~ 1m r-. r>; r-; f\ r-; I, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.!fo;,.l+n.21 V V V V V ,,! 89 " " 2- ~ 5 so.ror lht lou t di,11b' tht Ifi'l" tntt ll11pl)'.dd tilt',wo pr ttf't'dlr,~ temu, F = (11-1) + (11-2) ~ ~

6 Pythagoras - Numbers in All Things The Fibonacci Sequence Example...

7 Pythagorean Theorem...

8 ... Pythagoras theorem A A' + H' = C'... b

9 Pythagoras

10 Plato

11 Euclid ~ Euclid (sometime between 325 and 250 B.C.), brings us to what one may call the ~go lde n age " of Greek geometry. ~ Eucli d's fame rests on his celebrated thirteen-book textbook mathematics, the Elements, which brought together mu ch that already had been discovered and loosely proved but in a precise and syste matic form. ~ The first six books, covered topi cs in plane geometry, remain ed the standard textbook in th e field for European and North Ameri can schools until th e early twentieth cent ury. ~ It was proved to be one of th e most influential books in history, the foundation of mathematical training for nearly two th ousand years.

12 --- Euclid's methods of exhaustion 'f-- - S T

Euclid's reduction ad absurdam Th e famous exam ple is the proof of whether th e number of prime numbers (num bers which can only be divided by themselves or one) are infinite.

13 Euclid's reduction ad absurdam Th e famous exam ple is the proof of whether th e number of prime numbers (num bers which can only be divided by themselves or one) are infinite. One sta rts by assum ing th e opposite th at there is a finite number of prime number and m akes a list of all of them one knows. Then th ey are multiplied together and one is added to the total. Th e new total must either be a prime number or not. If it is not, th en th ere must be, logically, once th e "one" has been added, some prime number not yet listed which divides into it. If it is a prime number then there is another prime number to add to th e list. Either way th ere is a prime number to add to the first list. If one th en continues by multiplying in th e new prime number, adding one again to th e total. The same must apply. Th ere must always be further prime numbers and so the number of prime numbers is infinite.

14 Euclid - the Father of Geometry...

15 Elements of Geometry

16 Euclid's elements -".. T n.~...u~,, C C,, U c t,..._..../. G a _-. _...,.,. r. ~ ":' _.. :..~"::.=-:.~ --_...- ' _ h. -..._-_._.. "...n," " 1,_...- -_-, _ _... - ~ D

17 Euclid's elements - Postulates and Axioms...,--...,.,.V...TElj......_ ~. ""...r.- _ ""',...In... &om.., _ -- m, -,".'.'-'....::,.. " lo.ct>..., T... _... _... oqool.....,....._.._..._..., " ~ a+b_l800 "--- Then' _l 2...epar_ ".JkIIU,- "

Archimedes ~ Archimedes' basic project in geometry was continuous with the work of Eudoxus: to determine the measurements of geometric figures via the "exhaustion" technique.

18 Archimedes ~ Archimedes' basic project in geometry was continuous with the work of Eudoxus: to determine the measurements of geometric figures via the "exhaustion" technique. ~ This technique led him to his formulation of TI. The ratio of the circumference of any circle to its diameter is less than 3 1 / 7 but greater than 31 / 71. ~ Archimedes' work, in the determination of the surface area volume of a sphere, for instance, anticipates the integral calculus.

19 Archimedes ~ Part of Archimedes' success lay in the ability to envisage mechanical ways of dealing with problems and use these as a foundation for finding mathematical proofs of them. He was laying the foundation of applied mathematics. ~ One of the solids which fascinated Archimedes and others was the cone, and it was a successor of Archimedes, Apollonlus, from Perga in Pamphylia (active c B.C.) who took the work in this area to its further extreme. Apollonius went on to make cuts across the cone and named some of the resulting shapes, the ellipse, the parabola, and the hyperbola for the first time. The questions he posed still, apparently, challenge mathematicians today.

20 Archimedes

21 Mathematics and Astronomy ~ Mathematics and Science met in the study of the stars. In the field of applied mathematics the interest of Hellenistic scientists was confined chiefly to astronomy.

A subject of fascination ~ They were always a subject of fascination with the Greeks as they had been for other civilizations including the Babylonians and Egyptians.

22 A subject of fascination ~ They were always a subject of fascination with the Greeks as they had been for other civilizations including the Babylonians and Egyptians. By the fourth century the conventional wisdom underlay all Greek astronomy was that there was an order in the heavens and that a circular motion was the natural way for this to be expressed. The problem was that even elementary observations of the stars showed that their movements were irregular but here again the Greeks assumed that even irregularities operated in an orderly way.

23 Cosmos as a universal sympathy ~ It was commonplace of Hellenistic belief that the cosmos was connected in a "universal sympathy" with one part being able to influence another. The sun was seen to influence the seasons and the moon the state of the tides. It was one step from here to argue that human life might be affected by the stars and the Greek astronomers accepted this as a possibility. Astrology and astronomy were therefore not seen to be in conflict.

24 The picture of the universe - Generally accepted ~ In early 4 t h century B.C. the picture of the universe generally accepted among educated Greeks was as follows: the earth is at the centre, spherical and motionless; at the outer edge is a sphere on which the fixed stars are located, which rotates about the earth daily; in between are the sun, the moon, and the planets, which also rotate about the earth, but with different motions and in different directions.

25 The Contributions of Greeks to Astronomy ~ It is in the early 4 th century B.C. that the most original and profound Greek contribution to astronomy appears. Eudoxus, a mathematician of the first rank, who in addition to his profound contributions to geometry who devised the first geometrical model to explain the apparent motions of the heavenly bodies. As a theoretical construct, Eudoxus' system was extraordinary, but when confronted with observable astronomical facts it was clearly defective.

26 Aristachus of Samos Arlstachus of Samos (c. 280 B.C.), working at the Museum in Alexandri a during th e first half of the third century B.C., had learn ed from earlier scientists that the earth turns on its axis, and he declared th e revolutionar y theory that the sun is some three hundred times lar ger than th e earth. He argued against Ari stotle's view that the earth is at th e centre of the universe. Instead, he proposed the heliocentric theor y, that is, th e earth and th e other planets revolve around the sun. This noti on was so contrar y to Greek reli gious beliefs that the Stoic philosopher Clea nthes called for the prosecution of the astronomers for impiety. Th eir ability to decide th e issue on th e basis of observation was limited by th e lack of a telescope and other instruments. Th e heliocentric th eory was revived eighteen hundred years later by th e grea t Poli sh astronomer Co pernicus and confirmed by Galileo 's use of th e telescope.

Hlpparchus ~ In the 2 nd century B.C., Hlpparchus, whose working life was spent in Nicaea and Rhodes from about 150 to 125 B.C., made enormous progress to the reform of Greek astronomy to predictive mathematical science.

27 Hlpparchus ~ In the 2 nd century B.C., Hlpparchus, whose working life was spent in Nicaea and Rhodes from about 150 to 125 B.C., made enormous progress to the reform of Greek astronomy to predictive mathematical science. ~ Hlpparchus took from the Babylonians the idea of computing astronomical positions by means of tables, but he adapted it to the Greek idea of representing the phenomena by geometric models. For the first time among the Greeks, Hlpparchus was able to compute pure eclipse phenomena. He was also the first to calculate an essentially correct lunar distance, of about sixty earth radii, which was far below the true value.

The change of direction to horoscopic astrology No successor capable proof properly appreciating and developing Hipparchus's achievements appeared for almost three hundred years, but certain aspects

28 The change of direction to horoscopic astrology No successor capable proof properly appreciating and developing Hipparchus's achievements appeared for almost three hundred years, but certain aspects of his world were enthusiastically seized on. In particular, the enormous expansion in the Greco-Roman world of horoscopic astrology (which depends on the computation of the positions of the heavenly bodies at birth or more other critical time) in the century after Hlpparchus is intimately connected with the change of direction he had introduced into Greek astronomy. It was mainly for this purpose that planetary tables were constructed, based on epicyclic and eccentric models, but in a fashion that was neither mathematically consistent nor logically defensible.

Ptolemy ~ Ptolemy, carried out his observations between A.D. 127 and 141, com pleted his great astronomical work, known as the Almagest, in about A.D. 150.

29 Ptolemy ~ Ptolemy, carried out his observations between A.D. 127 and 141, com pleted his great astronomical work, known as the Almagest, in about A.D ~ Almagest is a masterpiece that was designed to present the whole of mathematical astronomy (as the Greeks understood the term) in a logical and comprehensive fashion. ~ As the measurement of the stars became more precise, tables of the relative positions of the stars to each other on specific dates could be recorded and so began the idea of creating a personal horoscope based on the day of an individual's birth. ~ Astrology was supported by the Stoics and was popular among all classes in the Roman emperor.

30 Almagest as a standard work on astronomy ~ There were some serious shortcomings in the work of Ptolemy, even by the standards of antiquity. Despite these imperfections, Almagest was so superior to anything preceding it that it soon established itself as the standard work on astronomy, a position it occupied for over a thousand years in both Europe and the Middle East.

31 The Ptolemaic System

32 Dawn -7 Polemy

33 ... Copernicus, Tycho Brahe, and Kepler ~ Almost all planetary tables down to the beginning of the modern era were based on Ptolemy's elements. It was not until the 16 th century, with the work of Copernicus, Tycho Brahe, and Kepler, that Greek astronomy, as formulated by Ptolemy, finally became obsolete.

34 Heliocentric Universe...

35 ... Copernicus - Heliocentric Model -...,.

36 Nicolaus Copernicus ---

37 Kepler

38 Newton ~ Einstein ---

39 What is a Mathematician? Two people are traveling in a balloon over unknown territory. "Hey!" they call out to a passerby, "where are we?" The person looks carefully for a moment at them and yells back, "You're in a balloon!" "He must be a mathematician," says one of the travellers to the other. "Why is that?" asks the other. "First, he thought awhile before answering. Second, his answer is absolutely precise. And third, it's utterly useless."

40 Crucial Achievements of the Greek scientists ~ Greek scientists regarded such applications to the needs of life as beneath their dignity. They were men of knowledge whose purpose was to understand the world as it was, not to change it. Their crucial achievement was to create methods of acquiring and assessing knowledge and here their legacy was immense.

How the Greeks Used Geometry to Understand the Stars

previous index next How the Greeks Used Geometry to Understand the Stars Michael Fowler, University of Virginia 9/16/2008 Crystal Spheres: Plato, Eudoxus, Aristotle Plato, with his belief that the world