IV Greek Mathematics Greek mathematics is a larger body of scholarship than what came before. It, of course, had the benefit of the Mesopotamian and Egyptian advances in scientific and mathematical knowledge. The main mathematical subjects we will study from the Greek heritage are: The Greek search for explanations of physical reality

The Greek use of logic and axioms

Greek geometry Greek number theory Greek astronomy

!

" . Pre-Socratic Greek thought.

We begin in Ionia, on the coast of Asia Minor, in the City of Miletus. The first important Greek philosopher and thinker of whom we have record is Thales of Miletus (~625 to 547 B.C.E.) He is said to have worked to explain the events in the world on the basis of logical reasoning about reality, rather than resorting to supernatural reasons. He visited Egypt, and lived in Miletus, a city with overland contact to Mesopotamia which was ruled by Cyrus the Great of Persia from 550 onwards. Aechaemenids were ruling Persia from 650 B.C.E. forwards, and there was a road from Persia to Sardis in Ionia, dating from Assyrian times which was at least partly in place before Cyrus the Great completed his Royal Road. Thales thus was exposed to two deep and venerable cultures with somewhat differing mathematics and science. Reconciling differences in the values of

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" , to choose a small point, might lead one to try to come up with careful argument for one, the other, or something new. Miletus was a cosmopolitan city with Greek, Mesopotamian, Persian, Egyptian and Phoenician people in its institutions. Points of view on most subjects must have been varied. Discussions about what are the true causes of natural events and of mathematics were not the exclusive purview of Thales.

Thales posited that the fundamental building block of all of reality is water. The various forms of water could account for liquids, solids and gases. His cosmology posited a large disc-shaped earth floating on water. When the water got rough enough it caused the earth to shake, causing earthquakes. Supposedly Thales sought after proofs of theorems in geometry. But we have no surviving writings of his.

A student of Thales, Anaximander of Miletus (~610 to 546 B.C.E.), made an outline of geometry according to Herodotus, the Greek historian. He replaced Thales’ water as the fundamental unit of reality by “the boundless.” Worlds come out of the

boundless, exist for a while, and then are absorbed back into the boundless. Anaximander was partial to Mesopotamian mathematics, whereas Thales seems to have favored Egyptian mathematics. After this time, with Cyrus the Great conquering Ionia, intellectual life suffered and innovative Greek thought began to migrate away from its Ionian birthplace.

Cyrus was a tolerant ruler, but the men he assigned to local rule were often

despotic. It was into this world that Pythagoras (~560 B.C.E. to 480 B.C.E.) was born. He was originally from Samos, he studied in Miletus, possibly with Anaximander and then traveled to Egypt. While he was in Egypt, the Persian king Cambyses conquered Egypt in 525 B.C.E. and Pythagoras went to Babylon for 7 years. During these times he learned the mysticism, mathematics and science of both cultures. He returned to Samos, but was exiled by the tyrant Polycrates. He emigrated and went to Italy, where there were Greek colonies. He settled in Croton, in southeastern Italy and founded a society of men and women who devoted their lives to philosophy, mathematics, mysticism and secrecy. All results discovered by the Pythagoreans were attributed to Pythagoras himself. The Pythagoreans believed in metempsychosis, that souls migrate to other bodies when we die, including the bodies of animals. As a result of this belief, they were strict vegetarians.

They believed that number was the basis of reality, possibly due to such discoveries as the harmonics gotten by varying the length of the string on a musical instrument and the observation that the motions of the planets could be described using numbers. Constellations of stars looked geometric. Their cosmology posited a spherical earth, surrounded by heavenly spheres that rotated around the earth, containing the sun, moon, 5 visible planets, and one for the distant stars. There was a problem, due to the recognized retrograde (backwards) motions of the planets from time to time. So the planets were loosely attached to the rotating heavenly sphere. They then correlated the numbers in music with the distance of the spheres from the central earth, so that each sphere had a characteristic tone, the music of the spheres.

It is interesting to notice that the Greeks seem to have been influenced by the Egyptian number representations more than the better Babylonian place-value system. The Greeks had a base ten number system and used alphabetic characters for the numbers. We only have evidence for this sort of representation of numbers from about 450 B.C.E. so that the Pythagoreans may have thought of number in terms of geometry, arrays of points. They dealt with ratios of two integers, and did not develop an arithmetic of fractions.

Number for the Pythagoreans meant what we call rational numbers, the ratio of 2 positive integers. The discovery of

!

2 and the realization that it is not a rational number was devastating for them. It involved proving that

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2 can not ever be expressed as a quotient of 2 integers, exactly. This sort of very careful mathematical argument is new to the world with Greek society. Previously mathematics was in the service of architects and practical people, so that such matters as differing values for

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" , which were each close enough to serve for all practical purposes, caused no grief. The horror which the discovery of an irrational (not a ratio) number caused is still with us today in the use of the words “rational” and “irrational” to describe certain human behaviors.

The Pythagoreans associated certain sequences of integers with geometric figures. They defined the triangular, square and pentagonal numbers, and proved various relationships among them.

The triangular numbers are

!

T1 = 1, T

2 = 1 + 2 = 3, T

3 = 1 + 2 + 3 = 6 ,

and in general

!

Tn

= 1 + 2 + 3 + " " " + n = n(n + 1)

2. These are the numbers of

points which can be arranged in equilateral triangular arrays of points. The square numbers are

!

S1 = 1, S

2 = 4, S

3 = 9, and S

n = n

2. These are the numbers of points which can be put into square arrays of points. We call (n x n ) n squared because of this association of the Pythagoreans. Similarly the pentagonal numbers are

!

P1 = 1, P2 = 1 + 4 = 5, P3 = 1 + 4 + 7 = 12, Pn

= 1 + 4 + " " " + (3n # 2)

=

!

n(3n " 1)

2

They defined the idea of prime number, perfect number, amicable pairs of

numbers and worked on “Pythagorean” triples of numbers which had been listed in Babylonian tables 1,000 years before Pythagoras.

A prime number is a multiplicative atomic building block of numbers. It is any number which is divisible only by 1 and itself. 1 is not considered to be a prime number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and there are 25 prime numbers less than 100. We go to worksheet # 4 The Sieve of Eratosthenes, at this point.

A perfect number is a number which is exactly equal to the sum of all of its proper divisors. Proper divisors are any divisors except the number itself. So 6, 10 and 28 are perfect numbers. The Pythagoreans knew these and also 496 and 8,128

Amicable numbers are a pair of numbers, each of which is exactly equal to the sum of the proper divisors of the other. The Pythagoreans only knew of the pair 220 and 284.

The Pythagoreans are the originators of the subject of Number Theory, a branch of current mathematics. A prominent Pythagorean, Archytas of Tarentum (~438 B.C.E.to 347 B.C.E.) also from Italy established a curriculum of studies known as the quadrivium which consisted of Geometry, Number Theory, Music and Astronomy. Both Plato and Eudoxus were students of his. Homework III 1. Adapt the proof by Aristotle that

!

2 is an irrational number to prove that

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11 is also an irrational number.

2. (Math credit) Prove that the nth triangular number equals

!

n(n + 1)

2

3. Show, using a Pythagorean argument involving arrays of points, that the nth

square number is a sum of two triangular numbers.

4.Show that 496 is a perfect number. 5. Show that 220 and 284 are amicable numbers. 6. (Math credit) Prove that the nth pentagonal number is equal to the nth square

number plus the (n – 1)st triangular number.

One remaining voice emanates from the Persian-dominated Ionian city of Ephesus, Heraclitus (~500 B.C.E.). He was an acerbic and brilliant philosopher who criticized the great minds of his time, including Homer and Pythagoras. He believed that diversity made up a cosmic unity, and is quoted as stating that “You cannot step twice into the same river, for fresh waters are ever flowing in upon you.” The unity that is diversity is a huge tension. He is frequently summed up, a little too quickly, as believing that everything is change. In keeping with his belief in a dynamic tension, he posits that the basis of all reality is fire.

We now consider the school of the Eleatics, from Elea in southwest Italy. Parmenides (~515 B.C.E. to 450 B.C.E.) is the first eminent thinker of whom we have documentation from this school. He argues that change and plurality are illusions, reality is, cannot change or be added to, and always was. It is the student and defender of Pearmenides, Zeno of Elea (~490 B.C.E. to 425 B.C.E.), who has a lasting effect on mathematics. Zeno uses logic to show that change is impossible. Some of his arguments have been answered in a way, but careful analysis shows his questions remain.

One of his arguments is that, to go from point A to point B, you have to go half the distance from A to B. Then you have to go the next quarter, then the next eighth, and

so on, which is an infinite number of smaller and smaller time intervals which can never add up. We now can add these up, and Zeno seems to have known that

!

1 + 1

2 +

1

4 +

1

8 + " " " +

1

2n

+ " " " = 2 ,

so that the real question begins to revolve around the continuity or discrete nature of space and time. Another argument of Zeno which refutes the possibility of change is to ask when the change takes place. If you know where an object is at each instant of time, when does its position change?

Another mathematical thinker was Democritus (~460 B.C.E. to 379 B.C.E.), from Abdera in Thrace, now northeastern Greece. He is credited with developing the first atomic theory of reality, that there are basic atoms of many elements, which come

together to form reality. He worked on mathematics and use the formula

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V = "r

2h

3 for

the volume of a right circular cone. He probably got this formula from the Egyptians, and we have no documentation that he could prove that it was a true, precise formula.

By this time, there were at least three unsolved mathematics problems circulating in the world of Greek thinkers. One was to square the circle: Given a circle with its area construct, using only a straight-edge and a compass, a square with the same area. A second was to double the cube: Given a cube with its volume construct, using only a straight-edge and compass, a cube with exactly twice the volume. The third was to trisect an arbitrary angle: Given any angle, use only a straight-edge and a compass to construct an angle with one-third the size. In all of these problems, straight-edge means no marks on your ruler.

An ingenious proof which squares a lune, was created by Hippocrates of Chios (460 B.C.E. to 380 B.C.E.).

He is not the famous Hippocrates of Cos (~460 to 370 B.C.E.), an early practitioner of medicine, after whom the Hippocratic oath, which all medical doctors now take, is named. This medicinal Hippocrates started clinical medicine and worked with rational reasons for illness like diet and the environment, not mystical, religious, punitive forces.

The mathematician Hippocrates argued in the following way:

O

I

AC

B

G

H

The above large circle ABC has a right triangle inscribed in it with vertex directly above the center, O. A smaller circle is drawn with diameter BC. The part of this smaller circle which lies out of the larger circle, is the lune (shaped like a crescent moon). We name it lune BHCI. The proof that one can square this lune is as follows. By the Pythagorean Theorem

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AC2

= AB2

+ BC2

= 2BC2

, because AB = BC.

Hippocrates knew that the areas of two circles are in the same proportion as the squares on their diameters, although he had no proof of this. Early Greek mathematicians knew this for the inscribed polygons in such circles. Eudoxus was the first to prove this carefully for circles. We would express it as

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1

2" Area of circle ABC( )

1

2Area of circle BHC( )( )

=

1

2"#

4" AC

2

1

2"#

4" BC

2

= 2BC

2

BC2

= 2

We see that the area of 1/4 of circle ABC, CIBO, equals the area of the semicircle

BHCG. The little slice CGBI is part of both areas, so subtract it from both. What is left on the outside of the large Circle ABC is our lune, and what is left on the inside of circle ABC is the triangle COB. So these two have equal areas. It was then easy for the Greeks to construct a square with area equal to the area of this triangle.

But you cannot use this to square the circle, and Hippocrates seems to have known this. Others inappropriately thought that his construction would work for any lune, but it doesn’t. Another gifted mathematician from this era was Hippias of Elis (~400 B.C.E.). He worked outside the box and trisected an arbitrary angle but using more tools than a straight-edge and compass. He constructed a curve called the quadratrix. A picture is below.

D

E

A

F

C

B

K L

Here is Hippias’ description. The radius AE of the circle with center at A rotates clockwise at a circular speed so as to reach line AB at the same time that the horizontal line CD drops from position CD to line AB. Then the intersections of the dropping line CD and the radius AE form a locus of points curving down from point C. The point F is a sample point on this curve. This curve is the quadratrix of Hippias. To trisect the angle BAE, drop a perpendicular from F to line AB. Cut this segment into thirds. Draw a horizontal line through the point marking the bottom third on this vertical line. This horizontal line cuts the quadratrix in a point G. The angle BAG trisects the angle BAE.

The equation in modern polar coordinates is

!

r = 2a"

# sin" where a is the radius of

the circle, and (r,

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" ) are the coordinates of the points on the quadratrix. Notice that when

!

" = 0, the limit is subtle, but then r = 2/

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" ~ 0.6366198, the foot of the quadratrix. Theodorus of Cyrene (~465 B.C.E. to 399 B.C.E. in what is now Libya ) improved the understanding of irrational numbers and showed that the square roots of the numbers 3 through 17, excepting 4, 9 and 16, are irrational. His student Theaetetus of Athens (~417 B.C.E. to 369 B.C.E.) went to the next level by using the Euclidean division algorithm for finding a greatest common divisor of two integers to find a common, if irrational, factor of two irrational numbers. Thus such numbers are commensurable. For instance the numbers

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27, 75 and 12 all are rational multiples of

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3 . This is an advance, conceptually. This advance shows up as proposition 2 of book X in Euclid’s Elements. Theaetetus’ definition of equality of irrational numbers was too cumbersome to work easily with. Here is an example of the ordinary Euclidean algorithm, which was clearly well known before Euclid. Consider the problem of finding the greatest common divisor of 159 and 36. We take as many multiples of the smaller number, 36 which are less than,or equal to, the larger number, 159; 159 = 4(36) + 15. Continue on, using 36 and 15, 36 = 2(15) + 6, continue with 15 and 6 15 = 2(6) + 3 6 = 2(3), and 3 is the G.C.D. the greatest common divisor of 159 and 36.

Here is a timeline of the history that parallels the mathematical developments described above.

The oldest civilization in the Aegean Sea area was that of ancient Crete, called the Minoan civilization. It began with neolithic Anatolian peoples migrating to Crete. This happened around 6500 B.C.E. to 6000 B.C.E. After 5000 B.C.E. there is evidence, in the form of pottery, of an emerging social structure.

The era from 3100 B.C.E. to 2100 B.C.E. is called the Early Minoan Age. There is documentation of settlements along the coast, and of burials in tombs and caves.

The time from 2100 B.C.E. to 1700 B.C.E. is called the Middle Minoan Age. The evidence shows an unstratified culture with no landlords or class distinctions. Burials are uniform, and the houses are not unequal in size and furnishings. The Minoans of this era traded with Egypt, Syria and Asia Minor. Around 1900 B.C.E.there was social upheaval with migrations into Crete from Greece and Asia Minor. The form of government changed and a king began to rule. Palaces were built in the communities and a system of paved roads connected the towns. There were sewers for the elite. Around 1700 B.C.E. powerful forces, probably an earthquake, destroyed the palaces.

The Late Minoan Age extends from 1700 B.C.E. to 1420 B.C.E. The palaces were rebuilt quickly and there were villas of the elite scattered around the villages. The central city and palace at Knossos were built in this era. There was a very elite upper class, and women played a powerful role in the society. The influence of the

Minoans spread to the Peloponesse, and influenced the emerging Mycenaean Greeks who were coming to dominate mainland Greece.

The huge volcanic eruption of Thera, the volcano on the Island of Santorini, must have harmed the Minoan fleet and port cities with large waves and possibly caused harvest failures there from ash and climate change. Ash from Thera is in the Greenland ice cap. The warrior society of the Mycenaeans was connected with the Minoan society, and one finds evidence of increasing militarism on Crete by around 1450 B.C.E. The Mycenaeans were Indo-Europeans who settled in the northern Peloponesse, were a warrior society and came to dominate mainland Greece by 1550 B.C.E. Around 1420 B.C.E. they conquered Crete. The early written language of these Mycenaeans is called Linear B, and is modeled on the Minoan language Linear A, which has never been translated. Linear B is a syllabary, not an alphabet. Of some interest is the invention of the first alphabet by either the Phoenicians or the Ugaritic people (Ugaru was an ancient city on the coast of Northern Syria) around 1400 B.C.E. We have cuneiform literature from Ugarit. This civilization had connections to Egypt and Cyprus. The prehistory of a settlement here starts around 6000 B.C.E. There is a list of Ugaratic kings from 1850 to 1180 B.C.E. By 1100 B.C.E. the Dorians swept over Greece and Crete. They became one of the largest ethnic groups in Greek society, and included the people called Spartans. These invasions came at a time when Mycenaean society became disrupted. The so- called mysterious “Sea Peoples” also invaded the area and seem to have damaged all of the established eastern Mediterranean societies at this time, including Egypt. The identity of the “Sea Peoples” is disputed. It may have included various different ethnic groups. There was a terrific famine over a stretch of years in Anatolia around this time, perhaps leading to the dissolution of the Hittite empire.

At the time of the Dorian invasions, the dialect of Greek known as Ionian Greek spread across the Aegean from mainland Greece to the coast of Asia Minor, which is now Turkey. This coastal area was called Ionia. Homer wrote the Iliad and the Odyssey in old Ionic during the 700’s B.C.E.. We have no real records of society in Greece from 1060 B.C.E. until around 800 B.C.E. when the Greeks began using the Phoenician alphabet and the city-states began to form. At this point the center of innovative thought is documented best from Ionia. Around 500 B.C.E. after the Persian conquest of Ionia the center shifts to Magna Graecia, as the Greek colonies in Italy came to be called. Around 300 B.C.E. the center shifts again, this time to Athens.