!

" . Mathematics centered in Athens in the fourth century, B.C.E., and later Hellenic Greek developments. We now consider the growth of mathematics that took place in Athens. during the 4th century B.C.E. This is the time of the famous Athenian philosophers Socrates (469 to 399 B.C.E.), Plato (429 to 347 B.C.E.) and Aristotle (384 to 322 B.C.E.) Socrates was an Athenian and an inspirational philosopher who lived his philosophy. He stepped aside from the questions of cosmology and turned his attention to mankind. He was interested in moral and social questions which led him to want careful definitions of concepts like justice and beauty. Some philosophers already maintained that ideas like justice meant one thing in one place and a different thing in another. He rejected this sort of philosophy.

He believed that knowledge is already in the student, and the task of the teacher, given a student who desires to learn, is to ask the right questions to draw forth the knowledge from the student. This was a process of dialectic, with Socrates asking questions and the student giving answers. If the answers were not carefully done, they suggested more questions and so on. Sometimes the process did not lead to a final solution. When it did, a true universal definition would come out of it. Since this process often goes from various examples to a general definition, we see the use of inductive reasoning going on here. He looked for universal principles that were not relative to particular times or places. Socrates stood up to ruling authorities in accordance with his principles on several occasions, refusing to go along with hasty trials of people who were being scapegoated. He finally was put on trial for introducing new religious practices and of corrupting the young. He was really tried for insisting that people think critically and for themselves. This threatened those in power. He was defiant in his own defense and was judged guilty and a sentence of death proposed. He was allowed to propose an alternative sentence. If he had proposed exile they would have accepted it. He proposed that he be given free meals and a small fine. He was then sentenced to death by a large majority and had to drink hemlock juice, a poison. One of Socrates’ students was the next Athenian intellectual light, Plato, who later studied with the Pythagorean, Archytas in Tarentum. Plato knew Socrates from a fairly early age and admired him deeply. He was present at the trial, and ended up suspicious of democracy as the best form of government. He was the founder of the famous Academy in Athens. During his time at the Academy Theaetetus the geometer (about 417 to 369 B.C.E.), Aristotle and Eudoxus studied there, and both Aristotle and Theaetetus taught there. Plato followed Socrates in his philosophical thought and left aside questions of cosmology to concentrate on matters more directly concerned with man. He created the system of ideal forms which is portrayed in the Allegory of the Cave, in book VII of his written work, The Republic. This theory of ideal forms as the basis of reality, in which what we sense is illusion, may be seen as a synthesis of Heraclitus’ and Parmenides’ thought. Plato wrote another dialog titled Theaetetus, named after his contemporary, among the 25 dialogs that we have which he wrote.

He taught the quadrivium of Archytas at the Academy and was an initiator of a search for an axiomatic foundation for mathematics. This fits well with the theory of ideal forms, conceptually. He insisted on construction with straight-edge and compass in geometry. He insisted on careful definitions and that they be used carefully in proofs. He is attributed with discovering the five regular solids which are called the Platonic solids. He probably did not do this himself. It is probable that the Pythagoreans could construct the cube, tetrahedron and octahedron conceptually, and that Theaetetus did this for the dodecahedron and the icosahedron. In a regular solid, all of the faces must be the same regular polygon. In the Timaeus, another of his written works, he shows how to construct these solids and associates the cube with earth, the octahedron with air, the tetrahedron with fire, the icosahedron with water, and the dodecahedron with the universe (perhaps recalling that the zodiac has 12 constellations or signs.) His written works show more connection with, and knowledge of, mathematics than what we know of Socrates. Supposedly, over the entrance to Plato’s Academy was written “Let no man ignorant of geometry enter here.” We see that the legacy of the Pythagorean horror of irrational numbers and a stress on careful, logical proof is leading the Greeks to base their mathematics on geometry. The use of alphabetic characters to represent numbers did not help them understand numbers, either.

The third intellectual light of Athens was Aristotle, who came from Stageira, in Thrace. At about 17 years of age he joined the Academy in Athens. The teachings of Plato had a permanent influence on Aristotle’s ideas. But he also went his own way in the matter of science and thus was more sympathetic to the old Ionian cosmologists. He accepts the theory of ideal forms, but in a modified way. He held that one had to learn the concrete objects which the senses perceive in order to appreciate the ideal forms.

He constructed a logic which was an analysis of proof statements. This subject has evolved into the present day propositional logic found in mathematical logic or foundations of mathematics courses. He eventually left the Academy and was retained by Philip of Macedon to tutor his son, Alexander from the age of 13 until he ascended the throne at the age of 26. Alexander became Alexander The Great. Ten years later Aristotle founded his own school, the Lyceum, in Athens. He was more mathematical than Plato. He rejects the use of a whole infinity in argument and in mathematics. The concept of all of the integers is meaningless to him. He allowed the use of potential infinity, such as using as large a set of integers as you need for your purposes. His name is connected to the thirteen semiregular solids which generalize the 5 regular solids by allowing several different regular polygons to be faces in one solid. He extends Plato’s thoughts on axiomatizing mathematics by considering axioms as more fundamental sorts of underpinnings of all argumentation and postulates as being more limited and pertaining to just one subject. He makes a distinction between number and magnitude. A number is a rational number. A magnitude is a different mental object, and is a length of time, the length of a line, the area of a figure or the volume of a solid. Calculating with magnitudes was a different activity than calculating with numbers for Aristotle. The solution to this problem of calculation with magnitudes was provided at first by Theaetetus, then more satisfactorily by Eudoxus of Cnidus. Aristotle wrote on physics, and accepted Eudoxus’ epicyclic theory of the cosmos as a fact, whereas Eudoxus intended the theory as a sort of explanation, via a mathematical model. Aristotle believed that there were actual mechanical spheres out in

space with the planets, sun, moon and stars fixed onto them. This influenced astronomy for close to 2,000 years. One of the best mathematicians produced by the Greeks was Eudoxus, who came from Cnidus, a city in southeast Ionia. His dates are ~391 to 338 B.C.E. He created a usable theory by which the Greeks could calculate with incommensurables or magnitudes as though they were numbers, using proportions. He also tightened up the Pythagorean spherical astronomy by adding smaller spheres attached to the larger spheres and which could be used to explain the retrograde motion of the planets. This epcyclic system had 27 spheres in its description of the heavens. He pioneered the method of exhaustion in proving mathematical statements. We consider one example below. Here is the proof, by Eudoxus, of the fact used by Hippocrates above in his squaring of the lune, that the areas of two circles are in the same ratio as the squares of their diameters. Note that neither the areas nor the diameters are numbers to these Greeks. The definitions of ratios of magnitudes according to Eudoxus form the beginnings of Book five of Euclid’s Elements. We will not repeat them here. Instead we proceed with modern notation. We let

!

P1

n denote an inscribed n " gon in circle C

1 with diameter d

1 and

!

P2

n denote an inscribed n " gon in circle C

2 with diameter d

2.

The earlier Greeks could prove that

!

area of P1

n( )area of P

2

n( ) =

d1( )

2

d2( )

2, using similar figures.

We did this in class.

Eudoxus states that

!

area of C1( )

area of C2( )

is larger than, less than, or equal to

!

d1( )2

d2( )2

.

Notice the logical underpinning here, a trichotomy. He then shows that the larger and smaller choices lead to contradictions, so the two must be equal. Here is a second point of logic. He does not directly show the two expressions are equal, but rules out all other possibilities. Here is how it goes for smaller:

if

!

area of C1( )

area of C2( )

< d

1( )2

d2( )

2 then

!

area of C1( ) "

d2( )

2

d1( )

2 < area of C

2( )

say

!

area of C2( ) " area of C

1( ) #d

2( )2

d1( )

2 = $ , some small number.

Choose the number n so that

!

area of C1( ) " area of P

1

n( ) < # and

!

area of C2( ) " area of P

2

n( ) < # . so

!

area of C2( ) " # < area of P

2

n( ) The earlier Greeks could construct this geometry carefully and find such an n.

Since

!

area of P2

n( ) > area of C2( ) " # = area of C

1( ) $d

2( )2

d1( )

2

dividing by

!

area of P2

n( ) , we have

!

1 > area of C

1( )area of P

2

n( )"d

2( )2

d1( )

2 =

!

=

area of C1( ) "

d2( )

2

d1( )

2

area of P1

n( ) "d

2( )2

d1( )

2

= area of C

1( )area of P

1

n( ) > 1

But this inequality amounts to 1 > 1, which is a contradiction. So the “smaller than” option cannot be true. Eudoxus eliminates the “larger than” option in a similar way, so the “equal” option is true. This is good practice for the real analysis course in our Mathematics Department. It amounts to an argument about a limit. Homework IV

1. Explain Hippocrates’ squaring of the lune? What did he do? How did he do it? 2. What is the quadratrix of Hippias? How can you use it to trisect an arbitrary angle? 3. Use an inscribed hexagon, and a circumscribed hexagon, in the style of Eudoxus to give bounds on the value of the number

!

" . (For math credit, do the above and then use dodecagons. ) Worksheet #6 in the course website has a useful diagram for this.

4. Use the following diagram, with some elementary geometrical theorems to prove the Pythagorean theorem. Early Greeks may have done

this. a b a b b a b a

The large square has all 4 sides equal to a + b, as shown. One of the a-b triangles

has hypotenuse of length c.

a. Prove the inner quadrilateral is a square. b. Then prove the Pythagorean Theorem

!

a2 + b

2 = c

2 We now come to the study of Euclid (330 to 270 B.C.E.) about whom we know

very little for sure. It is plausible that he studied in Athens at Plato’s Academy and went to Alexandria around 300 B.C.E., after the Museum (the house of the Muses) was a going concern. This seems to be where he spent most of his life working. He wrote several works that we know, including the Phaenomena, a work on spherical geometry, the Optics on perspective and the Data for those who had complete mastery of the first 6 books of the Elements. It is the 13 books of the Elements for which Euclid is famous. This has been a source of geometrical knowledge for students of mathematics for 2,000 years now. We outline some of the important mathematics in it.

Book I is on triangles and ends with Theorem 47. (the Pythagorean Theorem) If a right triangle has legs a and

b and hypotenuse c then

!

a2 + b

2 = c

2

and Theorem 48 If a triangle has sides a, b and c and

!

a2 + b

2 = c

2 then the angle opposite side c is a right angle.

Book II is on areas and we have Proposition 11. To cut a given straight line so that the rectangle on the whole and one segment is equal to the square on the other segment. a x b

!

ab " ax = xb2

(This is called division into mean and extreme ratio.) Proposition 14. Given any polygonal figure, construct a square with the same area. Book III is on circles, chords and angles.. Proposition 18. The radius of a circle forms a right angle with the tangent to the circle at the circumferential end of the radius. Proposition 20 An angle at the center of a circle is half the size of an

angle inscribed in the circumference if the angles subtend the same arc on the circumference.

Proposition 31 An angle inscribed in a semicircle is a right angle.

Book IV is on circles, inscribed and circumscribed polygons Proposition 11 To inscribe a regular pentagon in a given circle. Book V is on Eudoxus’ theory of proportions and magnitudes. (needed for Book VI)

Book VI is on Similar figures. Proposition 25 Construct a rectilineal figure which is at once similar to a given rectilineal figure and equal in area to another given rectilineal figure. Book VII is on Number Theory. Definitions of even, odd, prime, composite and perfect numbers. Proposition 2. The Euclidean algorithm for finding the greatest common

divisor of two integers. Example. Find the g.c.d. of 159 and 35. 159 = 4(35) + 19 35 = 1(19) + 16 19 = 1(16) + 3 16 = 5(3) + 1 3 = 3(1), so 1 is the g.c.d., the numbers are relatively prime. Back substitution in the equations just worked through will then exhibit

the g.c.d. as a sum of integer multiples of the original two numbers. In this case 1 = r(159) + s(35). find r and s: 1 = 16 – 5(3) 1 = 16 – 5(19 – 16) = 16(6) – 5(19) 1 = (35 – 19)6 – 5(19) = 35(6) – 11(19)

1 = 6(35) – 11(159 – 4(35)) = 50(35) – 11(159) so r = (-11) and s = 50

Book VIII is about squares, cubes and large sets of numbers. Book IX is more Number Theory Proposition 14 (The Fundamental Theorem of Arithmetic) Any integer

can be factored into powers of prime numbers in only one way, (disregarding the order in which we write the powers of the primes.) Proposition 20 (the infinity of prime numbers) Prime numbers are more than any assigned multitude of prime numbers. (Note the phrasing to avoid any actual infinity.) Proposition 36 (all even perfect numbers) If

!

2n

" 1 is a prime number then

!

2n " 1

2n

" 1( ) is a perfect number. ( The mathematician Leonard Euler proved that this formula gives all even perfect numbers in a posthumously published paper in 1849.)

Book X is on commensurable numbers.

Lemma 1 (before Proposition 29) All primitive Pythagorean triples are of the form a = 2rs, b =

!

r2 + s

2 and c =

!

r2 " s

2, where r and s are natural numbers, r - s is odd and r and s have no common factors. (Primitive means the three numbers have no common factor. )

Proposition 2. (Commensurability of both rational and irrational numbers)

If, when one continually measures off the smaller of two magnitudes from the other, and then measures off the remainder from the first, and keeps on going back and forth, one finds that the remainder never is a multiple of the remainder before it, then the two magnitudes are incommensurable.

Book XI is on solid geometry, lines, perpendiculars, planes, parallelepipeds. Book XII is on ratios of areas and volumes of pyramids, cones, circles and

spheres. Proposition 2.Circles are to one another as the squares on their diameters.

Book XIII is the climax. The five regular solids are inscribed in spheres. Propositions 13(tetrahedron), 14(octahedron), 15(cube), 16(icosahedron), & 17(dodecahedron). Also a proof that there are only those 5 regular solids.

One important fundamental at the very start of the Elements is the parallel postulate. Euclid must have known that there was something different about it because he holds off from using it for 28 propositions in Book I. Only in proposition 29 of Book I does he break down and use it, because he has to. Here is the parallel postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles , the two straight lines, if extended far enough, will meet on that side on which the angles are less than two right angles. (Notice that he avoids using infinity and never has to say the lines never meet.) Attempts were made throughout history to prove this postulate from the others which have a more basic character to them. No one succeeded. Attempts ultimately led to the creation of non-Euclidean geometries in the 19th century. Homework V Constructions in Greek Geometry In the five constructions below, use a separate sheet of paper for each problem, and give yourself plenty of room to work. Use only one side of each sheet of paper. Use a dark pencil or pen. Do not erase (except for mistakes) but show all construction lines so that I can see what you did. That is, where you put the point of the compass and what arcs or lines you drew. Label the parts of your diagram carefully. Include a step by step description of your construction at the end of each construction. See me for help or hints. You may get other help, and work with other students is encouraged, but hand in your own work which is not a copy of someone else’s.

Buy a compass if you don’t already have access to one. I have ten sets of rulers and compasses in my office. If you want to drop by and borrow a set, they are there to use. Please restrict yourselves to one set per group of students who plan to work together. 1.Euclid’s first proposition (Theorem) states that, given any straight line segment, an equilateral triangle exists having that segment as one of its three congruent sides. Draw a straight line segment a few inches long, and construct the predicted triangle. Use only a compass and a straight edge. You can’t use the measuring marks on the ruler. 2. Draw any triangle that has three different lengths for its three sides. Use your compass and straight edge to make a congruent copy of it. 3. Draw any angle, as long as it is not a right angle. Use your compass and straight edge to make a congruent angle. 4. Draw a line L and a pint P not on the line L. Construct a line which passes through P and is parallel to L. 5. Draw three large triangles that look different from each other to you, one each on separate sheets of paper. For the first triangle, construct the three perpendicular bisectors of the three sides. For the second, construct the three angles bisectors of the three angles of the triangle, for the third, construct the three medians. On each page write the appropriate definition of perpendicular bisector, angle bisector or median. On each page the three lines you drew should intersect in one point, if you extend the lines far enough. The intersection point may be inside or outside of the triangle. 6. a. Draw a triangle with no right angles in it. Construct a rectangle with the same area

as this triangle. b. Construct a square with the same area as this rectangle. A somewhat forlorn figure in Greek thought is the astronomer Aristarchus, from Samos, who lived around 270 B.C.E. He posited a much larger universe than his contemporaries, and made the outrageous statement that the earth was not the center of the universe. He stated that the sun was at the center, and that the earth rotated around the sun. He was hooted at, since, if the earth goes around the sun, then we should see parallax for a distant star, like the change in position of a person relative to an index finger, when viewed from one eye versus the other. The Greeks had no idea of the distances involved, and the angle of parallax was only measured carefully for the star 61 Cygnus by Friedrich Wilhelm Bessel in December of 1838. The angle is 0.314 arcseconds, the size of a taxicab in Manhattan when viewed from Mexico City. The only ancient follower of Aristarchus’ ideas of whom we are aware was a Persian astronomer, Seleucus of Seleucia, about 300 years after Aristarchus died. We next look at the life and mathematical work of Archimedes (287 B.C.E. to 212 B.C.E.) of Syracuse in Sicily, considered the most original ancient mathematician.

Archimedes used physical experiments to discover mathematical theorems, but always proved things carefully before claiming the result as true. He was a mathematical physicist, as well as a mathematician. He discovered the Law of Levers, the Principle of Compound Pulleys, and the Archimedean Buoyancy Principle. The simple water pump called the water snail was a discovery of Archimedes, and is still in use in some countries to aid in irrigation systems. An example of his elegant genius is his procedure for trisecting any angle. Recall the quadratrix of Hippias, which shows the level of complication mathematicians before Archimedes used to trisect an angle. He only needs one length on his straight edge, say two points marked on it. This barely exceeds the Platonic law of straight edge and compass as the only tools allowed in construction.

Here is his construction. The angle ABC is to be trisected. He uses his two-point length to draw a circle of that radius, centered at the vertex of the angle. He extends the line BC through the circle to form line GC. Now he lays his straight edge at point F and pivots it about this point until line segment ED has length FB, the radius of the circle he drew. Angle EDB is one-third of angle ABC.

E

F

B C

A

G D

Here is why this is so. Angle ABC is exterior to triangle BDF, so it equals angle BFE plus angle BDE. Since angle BFE equals angle BEF, and angle BEF is exterior to triangle BED, we see that angle ABC equals angle BDE plus 2(angle BDE) = 3(angle BDE) and we are done. Archimedes basically performed integral calculus calculations to find the area enclosed by a finite part of a parabola. He proved carefully that the volume of a sphere of

radius r is given by

!

V = 4"r3

3and that the surface area of this same sphere is given by

!

S = 4"r2. He used an inscribed regular polygon of 96 sides in a circle to show that

!

310

71 < " < 3

1

7, a method that was used for about 1,000 more years to get better

estimates of the number

!

" . The result of which he was most proud states that the volume of a cylinder circumscribed about a sphere has volume 3/2 that of the sphere. He asked that this be engraved on his tombstone. Even though actual infinity was a taboo concept in Greece, recently uncovered papers of Archimedes indicate that he used this idea. He certainly was interested in very large numbers, and took Aristarchus’ model for the size of the inverse to calculate the number of grains of sand needed to fill up the universe. He invented a better system of representing large numbers than the ordinary Greek representation of numbers to do this.

The City of Syracuse in Sicily had rebelled against Rome after Hannibal defeated the Romans at Cannae, during the second Punic war. Archimedes was slain by a Roman soldier as the Romans were conquering his native Syracuse. Archimedes is said to have been doing geometry in his home, when the soldier ordered him to stop and come. Lost in thought, Archimedes disobeyed, and even though the commanding general had ordered the troops to spare Archimedes, this non-communicative interaction lost him his head.

We now turn to Eratosthenes from Cyrene in Libya, who lived from 276 to 195

B.C.E. He went to Alexandria in 246 and was the head of the prestigious Library at the Museum there by 235. The method for finding prime numbers, the Sieve of Eratosthenes is attributed to him. He wrote on geography and used a system of latitudes and longitudes in this work. He calculated the size of the circumference of the earth by noting that the sun cast no shadow at noon in Syene, now Aswan, 5,000 stades south of Alexandria. He measured the angle made by the sun’s rays with a stake planted in the ground on the same date at noon in Alexandria. In the picture below, the angle CAB within the earth is the same angle as the angle EBD between the stake, BD, and the sun’s

rays EB. Using the geometry of angles in circles,

!

360o

angle BAC =

circumference

5,000.

One usage of Egyptian Stades gives a value of 24,662 miles, only 226 miles shy of our average circumference estimate. This is giving him the benefit of several doubts, but his estimate is a good one for sure.

Alexandria

Syene

Sun's rays

A

B

D

E

C

Apollonius, who lived from 270 to 174 B.C.E., worked in Alexandria and came

from Pergamum in Asia Minor and was the Euclid of Conics. There was a library at Pergamum which attempted to rival the institution in Alexandria. He wrote a treatise on Conics in 8 books. In these works he coins the terms parabola, hyperbola and ellipse. It is reported that he expressed large numbers in a place-value system with a base of 10,000. He also estimated

!

" to a more accurate number than Archimedes, but probably using the same method. He also wrote on astronomy, and improved on Eudoxus’ theory of epicycles, by adding in an off-center circle into the model.

Hipparchus (175 to 125 B.C.E.) is the father of observational astronomy. He compiled a careful table of 850 stars. His observations tied theory and practice together. He compiled tables of chords of angles to help locate stars in the sky and, in doing so, began to create trigonometry. In 127 B.C.E. he went to Rhodes and stayed there. He made estimates of the distances from the earth to the sun and the moon. He introduced the division of the circle into

!

360o into Greek intellectual circles.

Homework VI 1. Give a proof that Archimedes’ construction of the trisection of an angle works, using the diagram above. That is, fill in the written sketch of a proof that appears after the diagram, with some explanatory prose. 2. Use the worksheet on the sieve of Eratosthenes to find all of the 168 prime numbers less than or equal to 1,000. (There are 25 primes less than 100, 46 less than 200, 62 less than 300, 78 less than 400, 95 less than 500, 109 less than 600, 125 less than 700, 139 less than 800, 154 less than 900 and 168 less than 1,000.) 3. Two problems solved by Apollonius concerning tangent circles are the

following. Suppose you have three circles in the plane. a. Construct an example in which there cannot be a fourth circle tangent to your

three circles. b. Construct an arrangement of three circles in the plane which have eight other

circles tangent to all three original circles. Draw the original 3 and the 8 tangent circles.

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. 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.

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.

In 338 B.C.E. Philip of Macedon conquered all of Greece except Sparta. His son, Alexander the Great, swept across the known world from 336 B.C.E.to 323 B.C.E. when he died in Babylon at the age of 33 from a sudden fever. He established the city of Alexandria as his central city in the delta of the Nile River in Egypt. The Museum there housed a huge library in a time when books of any kind were extremely valuable. The museum was a place of scholarship dedicated to knowledge. After the death of Alexander the Egyptian portion of his empire came to be controlled by his general, Ptolemy who had Alexander’s body brought to Alexandria for burial, and who founded the Museum.

The Ptolemaic dynasty ruled Egypt for the next 300 years and carried forth the intellectual tradition of Greece. The other grand empire that grew from Alexander’s was Seleucid Persia. Seleucus was a son of one of Philip of Macedon’s generals and he served with distinction during Alexander’s campaign in India in 326 B.C.E. He ended up ruling the area of Mesopotamia and Persia after Alexander’s death. By 129 B.C.E. the Romans had conquered all of Macedonia, Greece and Pergamum. By 30 B.C.E. Octavian Caesar had also conquered Seleucid Persia and Egypt. There is much less support for basic research during the Roman rule, and in Roman society. IV (Addendum on the Roman era) The founding of Rome is taken to be 753 B.C.E. During the sixth century B.C.E. the Etruscans came to power and had an elected king and a senate for a governing body. In 509 B.C.E. a republic was formed, with no king and a senate. From 265 to 146 B.C.E. the Romans engaged in the three Punic wars against Carthage, which had Phoenician roots, culturally. During the second Punic war a famous Carthaginian general was Hannibal, who marched an army, including African elephants through Spain, France, and over the Alps into northern Italy where he defeated the Romans in three battles. They never beat him in Italy, but attacked Carthage, and he had to go back. The famous Roman general, Scipio Africanus did defeat him there, at the battle of Zama. In 48 B.C.E. Julius Caesar engaged Egyptian forces in the Nile delta and the resulting fire destroyed the library but not the whole museum. An indication of the cavalier attitude of the Romans towards intellectual pursuits is the fact that at that time, Mark Antony gave the entire library at Pergamum to Cleopatra, to make up for the accident. Here is a paraphrase of a statement from the famous Roman orator and consul (the highest elected office in Rome), Cicero: The Greeks held the geometer in the highest regard; thus nothing made more progress among them than mathematics. But we have decided that the usefulness of this subject is in measuring and counting and we pursue mathematics no farther. During a stay in Sicily Cicero found the tomb of Archimedes, with Archimedes’ favorite theorem carved into the stone. The tomb was overgrown with brush. The most famous astronomer-mathematician of antiquity is Claudius Ptolemy who was born in upper Egypt near Akhmin, and worked at the Museum in Alexandria. His dates are 100 to 178 A.D. His first name is Roman and Ptolemy is the name of Alexander’s general who took over the western part of his empire when he died. Ptolemy’s work was highly original, both in astronomy and in mathematics. His book on astronomy is composed of 13 books on the level of Euclid’s Elements. It is called The Almagest in the west. This is a transliteration of the Arabic “al majasti,” meaning the greatest. This work includes new astronomical observations, an improved lunar theory,

and a masterful blend of previous models for the heavens. It is another earth-centered epicyclic model, but one that is quite accurate. He develops trigonometry further and has the equivalent of the identities

sin(A + B) = sin(a)cos(B) + cos(A)sin(B) and

!

sin2" =

(1 # cos(2"))

2

He calculates very good approximations to

!

2 and works with irrational numbers. He wrote a work on geography in which he lists 8,000 places by longitude and latitude. He proved the following Ptolemy’s Theorem. If ABCD is any quadrilateral inscribed in a circle, then the sum of

the products of the opposite sides is equal to the product of the diagonals. See the picture below: (AC)x(BD) = (AD)x(BC) + (AB)x(CD)

B

D

A

C

This contains the equivalent of the trigonometric identity Sin A – B) = sin(A)cos(B) – sin(B) cos(A) A contemporary of Ptolemy, Galen (~129-216 B.C.E.) from Pergamum, investigated the human body and worked to establish a rational basis for practicing medicine. Roman law prohibited autopsy or dissection of human bodies, so he worked with animals. He made advances, particularly in the methods he used. He wrote important works on medicine. Another Hellenic Greek named Hero worked at the Museum between 150 and 250 A.D. He wrote and worked on mathematics that we would call more applied in nature. We have 13 books of his that have survived the centuries of loss. He wrote on the mathematics of mirrors, air pressure, surveying and numerical mathematics. In particular, the algorithm we used in finding square roots in our Babylonian worksheets is first recorded in his writings. He also records a formula for finding the cube root of a number. He also has the formula for the area of a triangle for which you can’t measure the

altitude, like a triangle surrounding a lake or a mountain. If s = (a + b + c)/2, where a, b and c are the sides of our triangle, then A =

!

s(s " a)(s " b)(s " c) . Here we call s the semiperimeter. A towering figure in this later period of Hellenistic mathematics is Diophantus, the father of algebra, who worked about 250 A.D. His book, the Arithmetica, was used by many key European mathematicians in their researches in the 1600’s and afterwards and by the earlier mathematicians of both India and Islam. He tries to solve equations by finding integer solutions, or rational number solutions. His efforts at solving equations of the form

!

y2 = 1 + a " x

2 where a is not a perfect square led to the work of Brahmagupta and Bhaskara in India as we shall see later. These equations are called Pell’s equations after an English mathematician, John Pell (1611-1685), who had little to do with them.

Diophantus allowed no negative or irrational solutions in his work. He studies algebraic equations, free from motivation in geometry. He uses notation for the powers of the unknown,

!

"#

for x2 (dunamis or power),

!

"#

for x3 (Kubos or cube),

!

"#" for

!

x4 , and so on up to the sixth power. For the ancients, there was no geometric

analog to any power beyond the third. He has notation for the negative powers of the

unknown by adding the character chi to the power as in

!

x"2

= 1

x2

= #$% .

He uses these notations to change determinate (have a finite set of solutions) and indeterminate (have infinitely many solutions) arithmetic problems into algebraic equations, which he then solves. He uses substitutions and eliminates unknowns in equations, manipulations that do not have geometric interpretations.

Another mathematician who worked in Alexandria from about 300 to 330 A.D. is

the geometer Pappus. We know of 8 books that he wrote and have evidence that he wrote at least 12. He improves on Ptolemy’s lunar astronomy, lists the 13 Archimedean semi-regular solids and reputedly proved the

Pappus’ Theorem If one rotates a planar region about a disjoint line, the volume

swept out is equal to the area of the region multiplied by the distance that the center of gravity of the region moves in the rotation. Pappus defined the concept of cross-ratio, which is a property of line segments

that remains the same under projection from a point. This becomes an important idea in the projective geometry of the 17th century, which grew out of artists’ use of perspective in the 15th century. Here is an example:

(AE)(FG)/(AG)(EF)

=

(AB)(CD)/(AD)(BC)

CBA

E

F

G

D

In 392 or 393 A.D. the library in the temple of Serapis, connected to the

Museum, was burned by a Christian mob. Theon of Alexandria was a teacher and scholar of Greek and Egyptian

knowledge who lived from 335 to 405 A.D. He published an edition of Ptolemy’s Almagest and an edition of Euclid’s Elements that was the best edition until the 18th century. His daughter, Hypatia, exceeded her father as a scholar. She corrected his edition of Ptolemy’s Almagest and published the corrected version, she wrote a commentary on Apollonius’ Conics and a commentary on Diophantus’ Arithmetica. She was a neoplatonist in her philosophy, mixing together mysticism with the ideas of Plato and Aristotle. Synesius of Cyrene, who became a Christian bishop, wrote favorably of her. But, in 412 A.D. there was a struggle over who would be elected bishop of Alexandria. She backed Orestes who lost out to Cyril. During Lent of 415 A.D. she was accused of witchcraft and was murdered brutally by a mob. She is the first woman mathematician of whom we have knowledge.

Homework VII 1. Use Ptolemy’s Theorem: If the inscribed quadrilateral in a circle is a rectangle then the

Pythagorean Theorem is a special case of Ptolemy’s Theorem. 2. Use the formula of Hero to calculate the area of the following triangles with the

indicated lengths of the sides:

a. 73, 51 and 26 b. 102, 146, 52 3. Problem modeled on Diophantus’ Arithmetica: The sum of two numbers is 30 and the sum of their squares is 500. What are the numbers?

(Hint: represent the numbers as 15 + x and 15 – x.) 4. Find one solution where both x and y are integers, to the Diophantine equation 7x + 5y = 11. (If you can find more, do so.)