The Greek Age of Mathematicsc© Ken W. Smith, 2012

Last modified on February 15, 2012

Contents

2 The Greek Age 22.1 Thales, the Pythagoreans and the first Greek proofs . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 The Pythagoreans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Pythagorean numerology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Commensurable numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 The impact of noncommensurability . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.5 Two types of Pythagorean triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.6 Polygonal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The Euclidean tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 The Euclidean calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 The quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 The Fundamental Theorem of Algebra and Euclid’s lemma . . . . . . . . . . . . . 112.2.4 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Euclid’s Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 The fabric of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 The preamble to the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Propositions from Book I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Euclid’s Proposition I.47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Elements, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 Books II, III, IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Triangles and their centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.3 Books V and VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.4 Books VII, VIII, IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.5 Book X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.6 Books XI, XII, XIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 The Greeks after Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.1 Eratosthenes and the Greek view of the universe . . . . . . . . . . . . . . . . . . . 242.5.2 Chords and trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.3 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.4 Apollonius and conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.5 Volumes and Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Conclusion of the Greek Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6.1 Diophantus and Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . 342.6.2 Primitive Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6.3 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6.4 The three problems of antiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.6.5 Pappus, Theon and the commentators . . . . . . . . . . . . . . . . . . . . . . . . . 372.6.6 Truth & Falsehood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.6.7 References for Greek mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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2 The Greek Age

Following Wikipedia’s article on ancient Greece we will set the dates of ancient Greece from 750 BC to592 AD, a period of 13 1

2 centuries. The most relevant period for our studies – the time of greatest growth– occurs in the classical period of Greek history.

During the classical period of Greece (500-323 BC) we see the rise of an intelligentsia of philoso-phers and mathematicians, people who, with leisure time, asked “big questions” about life, science andthe universe. During this time, we see the rise of two city-states (“polis”) Athens and Sparta. It is inAthens that the philosopher Plato built his Academy and other philosophers such as Socrates, Sopho-cles and Aristotle lectured and built their schools and groups of disciples. Much of western (Europeanand American) civilization owes it worldview and political system to those developed in Athens duringthe classical period of Greece.

The classical period ended with the unification of Greece by Alexander the Great, the creation of thecity of Alexandria and then Alexander’s death in 323 BC.

The Hellenistic period (323-146 BC) extended from the death of Alexander and the division ofhis empire to the conquest of Greece by the Romans. It was a time of expansion of Greek culture andphilosophy.

After 146 BC Greece was controlled by Rome but the Greek ideals and Greek language permeatedthe Roman empire. The Greco-Roman empire persisted through emperors Julius and Augustus Caesarand into the rise of Christianity, until in 330 AD, Constantine changed the empire forever by officiallymaking it “Christian.” The new “Christian” culture replaced the Greek culture and the Greek influencedeclined. (During this time, the female mathematician, Hypatia, was murdered by a Christian mob.)The Greek age ended with the closing of the last neoplatonic academy by the emperor Justinian in 529AD.

Axiomatic systemsThe Greeks during their time took mathematics to a higher level, far exceeding the works of the Baby-

lonians and Egyptians. In particular, the Greeks included mathematics within a broad rational approachto philosophy and scientific investigation. Mathematical “facts” were created by careful reasoning andlogical argument. One was allowed to ask, “Why is that true?” and to expect a careful explanation asan answer.

A natural result of the Greek rational approach to mathematics was an axiomatic system based ona collection of foundational axioms on which all other arguments rely. The axiomatic system was bestdisplayed by Euclid when (in 300 BC) he wrote a comprehensive treatise on all the areas of mathematics,all the “elements” of mathematics. (We will explore Euclid’s Elements in a later section.)

Mathematics continues to be based on a careful axiomatic system. The concept of axiomatic systemunderlies our exploration of Greek history.

2.1 Thales, the Pythagoreans and the first Greek proofs

Beginning around 750 BC, the Greeks asked philosophical questions. “Why?” “Is it always this way?Is there a pattern? Is there an underlying principle?” The philosopher Thales (b. before 600 BC inMiletus?) is the first recorded philosopher. His emphasis on reasoning and concepts made mathematicspart of his philosophical thinking.

He was followed by Pythagoras (possibly a student of Thales) whose teachings on geometry andnumber permanently changed our understanding of mathematics. Pythagoras probably spent some timein Egypt, learning and then extending the Egyptian understanding of mathematics. He started a schoolin southern Italy, a school that was both mathematical and cultic, with rigid religious beliefs and se-cret rituals. His followers, who worshiped mathematics (“All is number!”), significantly extended the

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understanding of mathematics, creating a foundation of theorem and proof.

Thales proved a number of geometrical theorems. Five theorems attributed to Thales (see Eves p. 73,Burton, p. 87) are:

1. A circle is bisected by its diameter.

2. The base angles of an isoceles triangle are equal.

3. Vertical (opposite) angles formed by intersecting lines are equal.

4. The ASA congruence rule for triangles: If two triangles agree on two angles and an included sidethen the triangles are congruent.

5. Any angle inscribed in a semicircle is a right angle.

Let us prove the third and fifth of Thales’ theorems.

3. Theorem. Vertical (opposite) angles formed by intersecting lines are equal.

Proof. In modern terms, we label the four angles formed when two lines cross. Call the anglesA,B,C and D. (See the figure below.)

(This figure is from the Wikipedia webpage on vertical angles. It is in the public domain.)

We seek to prove that angles A and B are equal. However, the sum of angles A and C is a “straightline” as is the sum of angles B and C. Thus A is a “straight line” less the value of C and B is alsoa “straight line” less the value of C. Therefore A and B are equal.

In modern notation∠A+ ∠C = π = ∠B + ∠C.

Therefore∠A+ ∠C = ∠B + ∠C.

and subtracting ∠C from both sides gives

∠A = ∠B. 2

What results did we assume along the path of this proof? We assumed the “geometric” propertythat all straight lines give the same angle and the “logic” property that “equals subtracted fromequals gives equals.”

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5. Theorem. Any angle inscribed in a semicircle is a right angle.

(This result was apparently known to the Babylonians. But did they have a proof?)

Proof. Consider a triangle inscribed in a semicircle. Label the vertices of the triangle A,B and Cwith A and C on the diameter of the circle. Let O represent the center of the circle.

By result # 2 above (on isosceles triangles), the angles ∠ABO and ∠BAO are congruent. (Let’sidentify the magnitude of these angles by α. By a similar argument, the angles ∠BCO and ∠CBOare congruent. (Let’s identify the magnitude of these angles by β.)

(This figure is from this Wikipedia webpage on “Thales’” theorem. It is released into the public domain

by inductiveload.)

Now we look at the angles ∠AOB and ∠BOC. These add to a straight line (that is, π or 180o.)That is,

∠AOB + ∠BOC = π. (1)

But the sum of the angles of a triangle is “a straight line” also, that is

2α+ ∠AOB = π

and2β + ∠BOC = π

Therefore, adding these last two equations we have

2α+ 2β + ∠AOB + ∠BOC = 2π (2)

Substitution equation 1 into equation 2 we have

2α+ 2β + π = 2π

and so2α+ 2β = π

and upon dividing by two we haveα+ β = π/2.

But α+ β is the magnitude of the angle C and so C is a right angle. 2

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2.1.1 The Pythagoreans

Pythagoras followed Thales and may have been a student. Apparently Pythagoras traveled from Samosto Egypt and then to Crotona in southern Italy where he started a commune or cult which worshippednumber. Many early mathematical statements are attributed to the Pythagoreans including the maintheorem named after Pythagoras.

Along with worshipping ”number”, the Pythagoreans held a variety of mystical (religious) beliefsincluding the transmigration of the soul and possible release from the perpetual cycle of reincarnationthrough various purification rites. They were vegetarians (apparently for a variety of reasons) and rec-ognized the relationship between numbers and music.

The Pythagoreans were the first to prove the theorem now named after them. They may have hadseveral different proofs. A dissection proof of the Pythagorean theorem appears on page 81 of Eve’s bookand page 105 of Burton. (See also this website.) There are many proofs of the Pythagorean theorem atthe cut-the-knot webpages.

The Pythagorean Theorem is equivalent (logically) to the parallel postulate. (More on the parallelpostulate later.)

Some of the main subjects of the Pythagoreans show up in the medieval collection of subjects in earlyuniversities. These were the quadrivium: arithmetic (= number theory), geometry, music and spherics(= astronomy) and the trivium: grammar (writing/speaking), logic, rhetoric (convincing arguments).

2.1.2 Pythagorean numerology

Since the Pythagoreans worshiped number, it is not surprising that they attributed special meanings toparticular numbers.

One form of Pythagorean attribution to numbers associated a number with the sum of its properfactors. For example, the number 10 was associated with 1+2+5 = 8. Since the sum of factors is smallerthen 10 was “deficient”. On the other hand 12 was associated with 1+2+3+4+6 = 16. Thus 12 is“abundant”.

Number which like 6 = 1 + 2 + 3 or 28 = 1 + 2 + 4 + 7 + 14 were equal to the sum of their factorswere said to be “perfect”.

What is the sum of the factors of 284? Since 284 = 4 · 71 and 71 is prime then the sum of factors of284 is 1 + 71 + 2 + 142 + 4 = 220. So 284 does dominate 220 in some sense? (But then what are the sumof factors of 220?)

Numbers also had shapes. Triangular numbers were created by putting objects into a triangular pile.Imagine stacking a pile of logs, putting 3 logs on the bottom row, 2 more on the next row and 1 on thetop. The number of logs, 6=1+2+3 is a triangular number. Burton, p. 95, defines

tn = 1 + 2 + 3 + · · ·+ n

to be the n-th triangular number. Thus t3 = 6, t4 = 10, t5 = 15, etc. (Eves, p. 80, uses Tn for the n-thtriangular number.)

The Pythagorean mysticism about numbers is echoed in superstitious systems of numerology inwhich one believes that numbers control or influence one’s life.

2.1.3 Commensurable numbers

The early Greeks believed that numbers were “commensurable”, that is, any two numbers could both beexpressed as multiples of some (extremely small) number. From their viewpoint, given a length x andanother length y, there was some small length, maybe an“atom”, that allowed one to write both lengths

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as an integer multiple of that “atom”. If we call the atom α then we could find integers M and N suchthat x = Mα and y = Nα.

Some of their arguments about computations and relations among lengths assumed this “commensu-rability” between pairs of numbers.

Let’s explore this idea. Type two numbers into your calculator, like an approximation for π and anapproximation for

√2, say

x = 3.141592653589793 and y = 1.4142135623730950.

Then we might make the “atom” be α = 0.000000000000001 and write x = 3141592653589793α andy = 14142135623730950α.

Or another example: take

x =355

113and y =

99

70.

With fractions it is easy to find the “atom” – just get a common denominator! In this case thedenominators are relatively prime so the common denominator is the product of the two denominators.So we write

x =355

113=

355 · 70

113 · 70=

24850

7910and y =

99

70=

99 · 113

70 · 113=

11187

7910.

Set

α :=1

7910

and then x = 24850α and y = 11187α.

This concept, “commensurability” (or the existence of this “atom” α) seems very reasonable and heldsway for centuries.

There was even an algorithm for finding the “atom.” Here it is.If x is larger than y then find the largest integer q such that one can fit q copies of y into x. So we

require the largest integer q such that x− qy ≥ 0 but x− (q + 1)y < 0.Now write r = x − qy. Clearly r < y. If both x and y are multiples of this atom α then q − xy is

also, so r is. We then continue dividing r into y and finding a new remainder. We continue this untilwe eventually get exact division, with no remainder. At this point we have found the atom (the lastremainder) that allows us to show x and y are commensurable.

Unfortunately, there is nothing to assure us that this process ever finishes! Everyone believed that itdid ... but there was no proof that this “algorithm” had to stop.

Then someone found a counterexample – two lengths which were not commensurable!There are debates about the first pair found, but it was likely to be

√2 and 1. (There is also an

argument that the pair were instead the golden ratio and 1.)

Here is the argument that√

2 and 1 are not commensurable. (It appears on page 115 of Burton andpage 84 of Eves.) Draw a square with sides of length 1. The square root of 2 is represented by thediagonal. From a vertex of the square (say the lower left), mark off a segment of length 1 on the diagonal.In this figure from Eves, p. 84, that point is called B1, so that the segment CB1 has length 1.

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Now construct a perpendicular to CA at B1. That line intersects AB at C1.An argument about congruent triangles (using the Pythagorean theorem) shows that AB1, B1C1,

BC1 all have the same length (which in modern notation is 1−√

2.) If the lengths of AB and AC havea common atom, then the length of BC1 is a multiple of that atom and so then is the length of AC1.

We have shown that the sides AB1 and AC1 are just smaller multiples of this common atom. Nowwe do this again on the smaller square with vertices A,B1 and C1. We create the point P so that thelength of C1P is the length of AB1, etc.

But these figures are all squares, all similar but smaller, each imbedded in the other. At each stage,our process takes a square and creates a smaller square. So this process continues forever!

Discussion: What is the modern version of the statement about “commensurable” and “incommen-surable” numbers?

2.1.4 The impact of noncommensurability

The Greeks of Pythagoras’ day believed that pairs of numbers were commensurable, that is, could bewritten as integer multiples of some common standard, some common atom. To discover that numberslike√

2 were not commensurable with 1 was a bit of a shock.One of the problems with this is that the Greeks viewed “proportion” in terms of commensurability.The ratio of a length x to the length y was the fraction M/N where x = Mα and y = Nα. What

were we to do if there was no “atom” α?

ZenoMore questions about the structure of the number line were raised by the philosopher Zeno of Elea

around 450 BC. Suppose one viewed length as being built up of extremely small indivisible pieces. (InGreek this smallest piece might be called an ”atom”. Or we might use a Latin word and call such a smallindivisible piece a “quantum”.)

For those who believed that length or time were built out of smallest atoms or quanta, Zeno providedthe Arrow Paradox. In this paradox, an arrow is flying through the air. Let’s examine a singlequantum of time. In that small quantum, the arrow must be motionless. (If not, then it is movingthrough the quantum, in one location at the start of the quantum and in another at the end. Thisviolates the indivisibility of the quantum.) But if the arrow is motionless at each quantum of time thenit is motionless across any finite collection of them! Indeed, the arrow doesn’t move at all!

On the other hand, for those who believed that length and time could be infinitely divided into smallerand smaller pieces, Zeno had the paradox of Achilles and the tortoise. In this paradox, the tortoiseis given a head start and Achilles tries to pass it. But Achilles first runs to the place that the tortoise wasat the beginning of the race. By then the tortoise has moved on a little to a new location. So Achillescontinues on to that location. But the tortoise has moved on. And so on....

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If Achilles is to pass the tortoise, he must first do an infinite number of steps ... and this is impossible.Zeno had a similar paradox called the Dichotomy paradox, in which motion was divided in half and

then in half again, an infinite number of times.

The paradoxes of Zeno are available at Wikipedia. For those with interests in these things, also reada much later paradox – Thomson’s lamp – in the same vein.

EudoxusThe problem of proportion and incommensurable numbers was resolved by Eudoxus of Cnidus around

370 BC. Eudoxus defined proportion so that commensurability was not involved. He defined a proportionx : y by saying that x : y and a : b are the same proportion if

1. the existence of integers M,N such that Mx > Ny implies Ma > Nb;

2. the existence of integers M,N such that Mx = Ny implies Ma = Nb;

3. the existence of integers M,N such that Mx < Ny implies Ma < Nb.

In other words, the relationship between Mx and Ny is always identical to the relationship betweenMa and Nb.

(More on this later.)

2.1.5 Two types of Pythagorean triples

The ancient Greeks had at least two families of Pythagorean triples:

1. (2n+ 1, 2n2 + 2n, 2n2 + 2n+ 1)

and

2. (2n, n2 − 1, n2 + 1)

These were apparently created by picking a square, an odd or even square, and working with one ofthe following equations. If one started with an odd square, say m2 = 2k − 1 then one would begin withthe relation

(2k − 1) + (k − 1)2 = k2. (3)

Solve for k in terms of m and the values k − 1 and k will complete the triple.Similarly, if one had an even square, write it as m2 = 4k and notice that

(k − 1)2 + 4k = (k + 1)2. (4)

Solve for k in terms of m and then k − 1 and k + 1 complete the triple. (These pairs of Pythagoreantriples are on pages 107-109 in Burton and page 82 in Eves.) These pairs of Pythagorean triples fitinto the collection of primitive Pythagorean triples that were discovered much later and recorded byDiophantus.

2.1.6 Polygonal numbers

The Pythagoreans (and the Greeks after them) had a fascination with numbers that could be organizedinto some shape. For example, if one places four blocks in a row on the floor and then three blocks abovethem, followed by a third row with two blocks and a fourth row with one block, then the blocks form atriangle of four rows and ten blocks. Since 10 = 4 + 3 + 2 + 1, 10 was a triangular number. I will writet4 = 10 to denote the fourth triangular number; the smaller triangular numbers are t1 = 1, t2 = 1 + 2 =3, t3 = 1 + 2 + 3 = 6.

Here, from the Wikipedia webpage on triangular numbers are the first six triangular numbers.

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(Melchoir publishes this under the Creative Commons Attribution-Share Alike 3.0 Unported license.)

Similarly, the Greeks identified squares (1, 4, 9, 16, ...) – these were true squares! – and even dabbledwith pentagonal and hexagonal numbers.

More information is available on polygonal numbers and figurate numbers on the Wikipedia webpages.The teenage Gauss, two thousand years later, would prove that every positive integer is the sum of

three triangular numbers. His proof would stir him to choose math (and physics) as his field of study.

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2.2 The Euclidean tools

Most of what we know about the early Greek mathematics comes from the mathematician Euclid whotaught at the university in Alexandria around 300 BC.

Built in 331 BC by Alexander the Great, the city of Alexandria had 500,000 people by 300 BC (saysWikipedia.) There the Greeks built the first university and the first library. The famous university andlibrary at Alexandria lasted almost a thousand years, from 331 BC to 641 AD when it was sacked by theArabs.

Euclid was the head of the Mathematics Department there in its early days, around 300 BC. Hewrote numerous works on mathematics. His most famous book, one of the most widely published booksin history, is The Elements. Today we tend to call it a “geometry” book. But in his day, it was about allof mathematics. It was about the very “elements” of mathematics!

For Euclid and the Greeks, mathematics was about the real world. It was visual and concrete. Tomultiply two numbers, one viewed the numbers as lengths and drew a rectangle with those lengths forits sides.

In order to do calculations, one needed the right tools. For the Greeks, this meant that one had toconstruct the correct figure.

The Greeks began by constructing straight lines. (Easy – use a straightedge and connect two points!)They also constructed circles: fix a point C and a particular length r and spin an object of length rabout the point C. A simple compass was used for this. (There is more on compass and straightedgeconstructions at Wikipedia.)

These tools were all one needed (or so they thought) to create every number.We will investigate this concept in some detail.

2.2.1 The Euclidean calculator

We naturally understand concepts like a + b or a − b; these are easy to draw by adding segments oroverlapping segments.

For example, if one represents unity (1) by a short length: then numbers like 2, 3 and 4were represented by copies of that length along a straight line.

What other operations can we “see” in the real world? The figure below (Eves, p. 88, Figure 20) has

two parallel lines cutting across an angle. This creates similar triangles so the ratiosa

b,a+ c

b+ xand

c

xare

all the same.

In particular,ax = bc.

So the drawing above allows us to multiply and divide lengths. If, for example, we want to draw a lengthof 5

3 then we draw this figure with segments of length a = 3, b = 5 and c = 1. The parallel lines will thenmark off a length x = 5

3 .

In the figure below (Eves, p. 88) the big right triangle with short legs of length a and x, on the leftside of the circle is similar to the right triangle with short legs x and b on the right side of the triangle.(Why?)

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This means that a : x = x : b so x2 = ab and

x =√ab !

So this drawing allows us to take square roots. If, for example, we want to take the square root of 53 ,

we might take the length 53 drawn earlier above as our segment of length a and put down a segment of

length b = 1 next to it. Draw the circle at the midpoint of the line of length a+ b = 53 + 1 and the length

marked by x will be√

53 · 1.

2.2.2 The quadratic formula

The Greeks had a solution to this problem: given lengths a and b, construct a rectangle with sides x anda− x such that the area of this rectangle is the same as the area of the square with length b.

In other words, solve for x inx(a− x) = b2.

This is equivalent to solving the equation x2 − ax+ b2 = 0.They could also solve the problem

x(x− a) = b2.

This is equivalent to solving the equation x2 − ax − b2 = 0. Other constructions gave solutions tox2 + ax+ b2 = 0 and x2 + ax− b2 = 0. (See the material on this from the discussion of Babylonians andEgyptians, before the Greek Age, my notes, section 1.)

What is our modern solution to the quadratic formula? How is it different?

2.2.3 The Fundamental Theorem of Algebra and Euclid’s lemma

The Greeks knew that every integer can be factored uniquely into a product of primes. (For example,120 = 2 · 2 · 2 · 3 · 5; this factoring is unique up the ordering of the primes.)

This result is now called “The Fundamental Theorem of Arithmetic” and we teach it in the elementarygrades.

The Greeks could also prove the following result: If p is a prime and p divides a product of integers,AB then either p divides A or p divides B. We write this in symbols as

p|AB =⇒ (p|A) ∨ (p|B).

This very useful result is often called “Euclid’s Lemma” since Euclid proved it and used it in hisfamous book, The Elements. (More about this book later.)

Please include the Fundamental Theorem of Arithmetic (FTA) and Euclid’s Lemma in your collectionof mathematical tools!

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2.2.4 Quadrature

The Greek view of number was visual, geometric. Our modern bias distinguishes their visual imageryand argument (“geometry”) from “algebra” and our modern mathematics have an excessive emphasis onalgebra instead of geometry. For this reason it is difficult for us to move back in time and see mathematicsin the visual way that the Greeks did.

A modern application of mathematics is computation of “area under a curve” using integral calculus.The Greeks also needed to compute areas, but they thought of area as properties of squares, so they wereinterested in constructing a square with a given area.

For example, they sought to “understand” the circle by constructing a square with an area equal tothat of the circle. We now speak of their attempts to “square” the circle, but for the Greeks, the squarewas a natural way to describe an area problem.

The Greeks did not have integral calculus and their tools for finding area were limited. But they dida number of interesting problems such as constructing the area of the lune.

We use the term quadrature (from “quadratic”) to describe the construction of a figure with a certainarea. The Greek quadrature problems were a prelude to integral calculus.

One of the earliest interesting quadrature problems is the quadrature of the lune given by Hippocratesof Chios in about 450 B.C. Hippocrates, in an attempt to square the circle, shows how to square a sliceof the circle formed by the intersection of two circles.

(This figure, from the Wikipedia article on the lune, has been released into the public domain by Michael Hardy.)

It would take two thousand years for mathematicians to prove that one could not “square” the circleusing Euclidean tools.

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2.3 Euclid’s Elements

Euclid, teaching and writing in Alexandria, around 300 BC., wrote a number of books on mathematics.His most famous one is simply called Elements, that is, the “main points” of mathematics.

After the invention of the printing press in 14xx, this book was published throughout Europe, becamethe textbook for anyone who claimed to be educated, and probably went through close to a thousanddifferent editions (and millions of copies.) Abraham Lincoln said that he worked through it as a teenager,teaching himself how to think logically and it’s information was assumed in the halls of Parliament andCongress.

The style of this textbook is revolutionary. Everything is carefully and logically argued, beginning witha set of 23 definitions, five “postulates” and five “common notions”. Using these items as a foundationallayer, Euclid builds an edifice of mathematics, one brick at a time, each result (called a Proposition) isargued from previous results.

Some good sources: Eves, Chapter 5 is devoted to Euclids Elements. Burton has a nice introductionto Elements in sections 4.1 and 4.2. David Joyce’s webpage provides The Elements online. I recommendthat website as a nice resource; I have used material from it in the following discussion.

2.3.1 The fabric of mathematics

Mathematics today at first glance seems to be a collection of results and computations. Many studentsmemorize these results in middle school or high school without understanding the meaning or explanationof the terms or computations.

(I am reminded of the memorization of spells by the characters in the Harry Potter novels – hereat Wikipedia is a list of the many things a student – Hermione Granger? – would have to memorize ifhe/she were hoping to excel at Hogwarts.)

In mathematics we can (should) organize our understanding and formulae into categories. I havetaken the suggestions of a recent Evolution of Mathematics (MATH 4367) class and created a list ofmathematical results we need to learn and have attempted to break them down by certain basic categories.

1. Some things are “definitions,” that is, they simply involve the “naming” of objects:

(a) C = 2πr is the definition of the constant π.

(b) the sine of an angle is the ratio of the opposite side to the hypotenuse (and other trig definitions)

(c) the slope of a line,

(d) the concept of logarithm,

(e) i2 = −1,

(f) there are 360 degrees in a circle.

(g) Even the area of a rectangle and the volume of a cube are definitions (in some sense) of theconcepts of area and volume.

2. Some things are quick and immediate from these definitions:

(a) The area of a parallelogram (“push” a rectangle over a little and use the same formula)

(b) The area of a triangle (cut a parallelogram in half)

(c) The area of a trapezoid (move a piece around to form a rectangle)

(d) The volume of a cylinder (move an area – that of the base – through the third dimension)

(e) The slope of a secant line to a curve (apply the definition of slope)

3. But then there are real theorems, requiring some logic and computation:

(a) The Pythagorean theorem,

(b) the area of a circle,

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(c) The quadratic formula,

(d) the volume of a prism,

(e) the volume of a cone,

(f) the double angle formula for cosine and sine,

(g) triangle congruence theorems like SAS and SSS,

(h) The law of sines, the law of cosines,

(i) The derivative of xn, the derivative of sine,

(j) the sum of the angles of a triangle is 180o.

(k) Euler’s equation: eiθ = cos θ + i sin θ.

We are interested in how these results were discovered and how we know they are true.

4. Some things are immediate results (corollaries) of the theorems. For example, from the Pythagoreantheorem we immediately get

(a) a distance formula for points in the plane,

(b) identities like cos2 θ + sin2 θ = 1,

(c) 1 + tan2 θ = sec2 θ, etc.

From the Pythagorean theorem and statements about isosceles and equilateral triangles, we get thelengths of sides of a 45-45-90 triangle and a 30-60-90 triangle.

2.3.2 The preamble to the Elements

In 300 BC, the mathematician Euclid attempted to organize all the known results of mathematics into awork in which each result was carefully reasoned from the previous results. This work comes down to usas The Elements (of mathematics) and is the most influential mathematical work of all time.

Euclid’s approach is very logical. It is axiomatic.After 23 definitions, intended to precisely set down the mathematical terms, Euclid introduces five

postulates and five common notions. Today we would call these “axioms”. They are assumed to be basictruths, obvious in some sense. They are not to be proved.

In any logical argument or philosophical structure, we need a place to start. These ten axioms layout what we accept as true and provide the tools for the rest of the work.

Here are the five postulates (Burton, p. 146.) I will put them loosely into my own words. Thesepostulates are mathematical (or geometrical) axioms, as opposed to logical axioms.

1. One can draw a line between any two points.

2. Given a line segment, one can extend it indefinitely.

3. Given a point and a radius, one can construct a circle centered at that point, with the given radius.

4. Any pair of right angles are equal.

5. If we draw a line across two other lines and if on one side of that line, the two angles formed sumto less than 180 degrees then those two lines, when extended indefinitely, will eventually meet onthat side.

The five common notions are axioms of logic – claims about logic and what is allowed in the logicalarguments.

1. Things equal to the same thing are equal to each other. (Algebraically: if x = y and x = z theny = z.)

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2. “If equals are added to equals, the wholes are equal.” (Algebraically: if x = y and a = b thenx+ a = y + b)

3. “If equals are subtracted from equals, the remainders are equal.” (Algebraically: if x = y and a = bthen x− a = y − b)

4. Objects that coincide with one another (that is, one can be moved to lie identically on top of theother) are equal.

5. “The whole is greater than the part.” (Algebraically: if C = A+B then C > A.)

Commentators have noticed that Euclid occasionally assumes more than these axioms really provide.For example, in postulates 1 and 2, he really intends that one draw a unique line between two points andthat the line we extend is extended uniquely.

One thing he assumes unconsciously are that if circles are close enough that the distance betweentheir centers is less than the sum of the radii then the circles must intersect. (See the first proposition.)Similarly, if one draws a line from the center of a circle to the exterior of the circle, it must intersect thecircle. These “true” statements are never proven and never explicitly assumed. Later mathematicianswould add these as axioms.

Although we may struggle to put these axioms into a modern context, nine of them are pretty obviousand we would presumably agree on them.

But one of these axioms is not so obvious. The fact that the axiom is not obvious is detectable simplyin its length. And it is clear that the axiom bothered Euclid in some sense, for he avoids the use of theaxiom whenever possible.

The strange axiom in the list is Euclid’s Fifth Postulate. Imbedded in the fifth postulate is a claimabout the geometry of the universe. Also imbedded in the fifth postulate is a hint of the need to grapplewith infinity.

2.3.3 Propositions from Book I

(Again, a good resource for this material is David Joyce’s online edition of the Elements. I have copiedgraphics from that website in the material below.)

Proposition 1, to construct an equilateral triangle. Here we have an elegant use of the axioms. (Andthis is the first place Euclid assumes something he has not explicitly given as an axiom! What does heassume?)

Proposition 4. The Side-Angle-Side (SAS) property of triangle congruence.

Proposition 5. That the base angles of an isosceles triangle are equal. This proof is sometimes called“the bridge of fools”. The geometer Pappus had a more elegant proof, one which Euclid clearly knewbut avoided. (Why?)

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Here Euclid marks off equal line segments AF and AG and then argues from SAS (the previousproposition!) that the triangles ∆FAC and ∆GAB are congruent.

Therefore FC ∼= BG and since BC = CB, the triangles ∆FBC and ∆GCB are congruent.Finally, since ∠ABG = ∠ACF and ∠CBG = ∠BCF then ∠CBG = ∠BCF.

Proposition 6. The converse of proposition 5 – that a triangle with two equal angles is isosceles. Inproving Proposition 6, Euclid uses a “proof by contradiction” and also the result from Proposition 5.That is, he assumes that a triangle with two equal angles is not isosceles and then proceeds to constructan isosceles triangle in such a way that the result of Proposition 5 conflicts with the assumption inProposition 6.

This is a nice use of “proof by contradiction” (or, in Latin, “reductio ad absurdum”.)

Proposition 9. How to bisect an arbitrary angle.

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Proposition 10. How to bisect an arbitrary line segment.

Proposition 11. How to draw a perpendicular at a point.

Proposition 16. The exterior angle of a triangle is greater than either of the opposite interior angles.(Discussed on page 153 of Burton.)

We have two propositions about parallel lines which are converses of each other. Burton describesthem on page 154.

Proposition 27. “If two lines are cut by a transversal so as to form a pair of congruent alternate interiorangles, then the lines are parallel.”

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Proposition 29. “A transversal falling on two parallel lines makes the alternate interior angles congruentto one another, the corresponding angles congruent, and the sum of the interior angles on the same sideof the transversal congruent to two right angles.”

The concept of “parallel” is transitive:Proposition 30. If two lines are each parallel to a third line then the two lines are parallel.

And finally we have a famous result:Proposition 32. The sum of the interior angles of a triangle is 180 degrees.

2.3.4 Euclid’s Proposition I.47

This is Euclid’s first proof of the Pythagorean Theorem. It is sometimes called “The Bride’s Chair” proofor simply “Euclid’s Proof.”

We begin with a right triangle (right angle at C)

A B

C

and construct squares on the three sides. We drop a perpendicular from C through the hypotenuse AB,dividing the square on the hypotenuse into two pieces.

The proof in Euclid’s Elements, Book 1, proposition 47, is often drawn like this, below, with lettersat all the vertices. I have suppressed the various letters.

A B

C

This drawing makes the proof appear complicated and leads to names like ”The Peacock’s Tail” or(according to Cut-the-Knot.org), ”The Windmill.” Even the term ”Bride’s Chair” may be a mistranslationof a Greek term meaning ”Insect.” Burton calls this “the mousetrap proof.”

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But despite all the lines, the proof is really quite straightforward and worth knowing.

Euclid’s proof will argue that the colored areas, below, correspond in size and so the area of the squareon the hypotenuse is the sum of the areas of the squares on the shorter sides.

A B

C

We make this argument by looking at pairs of triangles. We construct a triangle whose base and heightare the same as that of the dark blue square. Then we construct a triangle whose base and height arethe same as that of the light blue square. The areas of these triangles must then be half of the areas ofthe corresponding squares.

Then we will argue that these triangles are congruent, so their areas are equal and so the areas of thecorresponding squares are equal.

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A B

C

A B

C

The steps of this proof are:

1. The area of the dark blue square is twice the area of the dark green triangle.

2. The area of the light blue rectangle is twice the area of the light green triangle.

3. But the two triangles are congruent by the side-angle-side congruence rule (the common angle is atA.)

4. Therefore the two triangles have the same area.

5. So the dark blue square and the light blue rectangle have the same area.

6. A similar argument can be made about the dark red square and the light red rectangle, using thetriangles in orange.

7. Therefore the area of the square on the hypotenuse is (visually!) the sum of the areas of the squareson the shorter sides.

There are possibly simpler (?) proofs of the Pythagorean Theorem.Euclid later provides an alternate proof of the Pythagorean Theorem, a “dissection” proof in which

copies of the right triangle are moved around to demonstrate the result.Other proofs involve arguments about similar triangles.

=⇒ So a question arises. Why did Euclid choose this proof in Book I?

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2.4 Elements, Part II

Euclid’s Elements contained thirteen “books”. The first book had 48 propositions on plane geometry andwe discussed them in the previous section. Here we briefly summarize the other books. All thirteen bookswere used by teachers and students of mathematics for the next 2000 years, through various translationsand editions.

A translation of these books is worth reading. Since Euclid did not have algebra, everything is writtenin prose. Here, for example, is the sum of a geometric progression, Prop. XII. 35.

“If as many numbers as we please are in continued proportion, and there issubtracted from the second and the last numbers equal to the first, then the excess ofthe second is to the first as the excess of the last is to the sum of all those before it.”

A translation of Euclid’s Elements is available at David Joyce’s website.

2.4.1 Books II, III, IV

Book IIThis book introduces the basic concepts and identities from algebra (like the distributive law and the

square of a+ b) but all from a geometric viewpoint.

Book IIIBook 3 has 37 propositions dealing with circles and related things such as tangents, secants, chords,

etc.

Book IVThis book had 16 propositions on constructions of regular polygons. Regular polygons with 4, 5 and

6 sides were constructed. (A regular polygon on three sides was the first proposition in Book I!)Euclid could not construct a polygon on 7 sides.A polygon on 15 sides was constructed, possibly for use in astronomy.

2.4.2 Triangles and their centers

“...the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can beobtained by simple constructions. Each of them has the property that it is invariant under similarity.In other words it will always occupy the same position (relative to the vertices) under the operationsof rotation, reflection, and dilation. Consequently this invariance is a necessary property for any pointaspiring to be a triangle center.” (From the Wikipedia article on triangle centers.)

There are a number of triangle centers and their early discovery is buried in Euclid’s Elements. Thecircumcenter and incenter are the centers of circumscribed and inscribed circles. The centroid (center ofmass, barycenter) is where lines joining vertices to midpoints meet.

Dan Kemp (Professor of Mathematics at South Dakota State University) says in a private email:“One of the notable points of the triangle that Euclid does not discuss is the orthocenter, the commonpoint of intersection of the altitudes. I believe the first time this is mentioned is in Archimedes’ Book ofLemmas in the proof of Proposition 6 ... where the concurrence property is invoked with the phrase ‘...by the properties of triangles...’. ”

The orthocenter is where the altitudes meet.

In the same discussion, Jorge Lopez (University of Puerto Rico at Rio Piedras) writes “In book IV ofEuclid’s Elements the notable points of the triangle, the incenter and the circumcenter are discussed. Inthe process of constructing the inscribed and the circumscribed circles it is proven that these notable pointsare concurrency points of the corresponding cevians of the triangle. Giovanni Ceva (1648-1734) stated

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his famous result during the second have of the 17th century or the beginning of the 18th. Presumably hemust have had some examples of notable points (points of concurrency of cevians).” -

The modern approach to showing that lines meet in a point involves Ceva’s theorem.

Euler would later show that the orthocenter, circumcenter and the centroid all lie on a common line,now called the Euler line. (See David Joyce’s work on the Euler line.)

We will explore these triangle centers in a worksheet. Of particular relevance is the viewpoint ofthe mathematician when examining these objects; Euclid’s arguments are exclusively geometric, there isabsolutely no algebra. Our modern viewpoint (due to Descarte, Fermat and Pascal) places coordinateson these objects and uses algebra. Think about the effect this point of view has on the mathematicalapproach!

In Book IV of Euclid’s Elements, propositions 2 through 5 deal with the incenter and circumcenter.After this the work moves on to inscribed and circumscribed squares, pentagons and hexagons. Proposi-tion 10 constructs the 36-72-72 triangle in preparation for constructing a regular pentagon; this trianglewill also show up in the golden ratio. The book ends with Proposition 16 where a regular 15-gon isconstructed.

Certainly an implied question in the text is the construction of regular polygons in general.

2.4.3 Books V and VI

These books resolved the issue of proportions, without relying on commensurability. Most of this materialwas developed by Eudoxus.

Book VBook V develops the critical concept of proportions, following Eudoxus. Everything is geometric,

without the assumption of commensurability.Wikipedia notes that ”Proposition 25 has as a special case the inequality of arithmetic and geometric

means.”

Book VIBook 6 uses proportions to develop a theory of similar figures.

2.4.4 Books VII, VIII, IX

Books 7-9 are over number theory. These are done from a geometrical viewpoint, of course. But thestudy is of the integers, including factoring numbers, divisibility, prime numbers and “Euclid’s Lemma.”

Book VIIThe famous Euclidean algorithm is in propositions 1 and 2. Propositions 30 and 32 include the

fundamental theorem of arithmetic, though, according to Wikipedia, “though Euclid would have hadtrouble stating it [the FTA] in ... modern form as he did not use the product of more than 3 numbers.”

Book VIIBook 8 has proportions on number theory including geometric progressions and their sum.

Book IXThis book concludes the material on number theory by giving Euclid’s famous proof that the set of

prime numbers are infinite (prop. 20), statements about sums of collections of even and odd numbers,the formula for the sum of a geometric series (prop. 35) and a method to construct even perfect numbers(prop. 36).

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2.4.5 Book X

Book X attempts to classify irrational numbers (“incommensurable magnitudes”)The Euclidean algorithm is repeated here in a more general form.Here we get square roots and square roots of square roots, and so on.

2.4.6 Books XI, XII, XIII

Books XI - XIII are on solid geometry and include the method of exhaustion.

Book XIWikipedia: “Book 11 generalizes the results of Books 16 to space: perpendicularity, parallelism, vol-

umes of parallelepipeds.”

Book XIIIn Book XII, the area of a circle is proven (by the method of exhaustion) in proposition 2, after first

preparing the way with a proposition on the area of polygons.Most of Book XII is on the volumes of cylinders, cones, pyramids. Using the method of exhaustion, it

is shown that the volume of a cone is one-third of the volume of the cylinder with same height and base.(We will look at this further, in a later section.)

Book XIIIThis book is about the five Platonic solids.Our world does not just contain two-dimensional plane figures. There are also three-dimensional

“solid” objects. The Greeks proved that there are only five convex objects which have faces which are allcongruent and are regular convex polygons. This result, attributed to Plato, gives us the five Platonicsolids.

The drawings below are from http://www.uwgb.edu/DutchS/symmetry/platonic.htm.

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2.5 The Greeks after Euclid

The end of the Classical Age of Greek history and the beginning of the Hellenistic Age is marked by thecreation of the great city of Alexandria and the death of Alexander the Great.

The city of Alexandria in north Egypt was created by Alexander the Great in 332 BC and becamepart of the Ptolemy dynasty after Alexander’s death in 323 BC. During Euclid’s lifetime the Greeksdeveloped the city of Alexandria and its great university, eventually creating a “museum” devoted to theMuses. This museum eventually included the great Library of Alexandria which retained thousands ofpapyrus rolls.

We will look at some of the mathematicians and scientists who lived during the Hellenistic age.

2.5.1 Eratosthenes and the Greek view of the universe

Eratosthenes (Greek, ca 230 B. C, Cyrene., died 194 B. C, Alexandria), was a mathematician, astronomer,geographer, historian, philosopher, poet and athlete. He moved to Alexandria from Athens to tutor theson of the ruler, Ptolemy III (a descendant of Alexander the Great.)

He is known for describing a sieve for finding prime numbers.He measured the earth, fairly accurately, understanding that it was a sphere. He estimated that the

distance between Syrene and Alexandria was about 5000 stadia (approximately 500 miles?) and since onthe day of the summer solstice, the sun’s shadow at high noon differed by about 7.2 degrees between thetwo sites, then the diameter of the sphere was about 50 times the distance between the two cities. (SeeBurton, page 187.) His estimate for the circumference of the earth was then (in modern language) closeto 25,000 miles; the correct figure is 24,000 miles.

His research into accounts from travelers and explored led him to draw a map of the world withAlexandria as the center. For his day, the map is quite good; it has much of Africa and Europe and someof Asia.

He became chief librarian at the University of Alexandria. He was named Pentathlus (apparentlybecause of his athletic ability) and also nicknamed Beta.

He created a mechanical solution to duplication of the cube.

Eratosthenes’s work on a map of the world was complemented by Greek work on the stars and theshape of the universe.

Aristarchus of Samos (Greek, ca 310 - 230 BC) put forward a heliocentric (sun-centered) hypothesisfor the solar system.

He measured the distance from the earth to the sun. He took the triangle with the earth, sun andmoon as vertices and measured the angle of the sun in this triangle, when the moon was at first quarterand so the moon formed a right angle. Since, in his measurement, the vertex formed by the sun in thistriangle was 3 degrees (or less), he estimated that the sun was about 19 times as far from the earth asthe moon. (The cotangent of 3 degrees is about 19.)

Using the fact that during a solar eclipse the moon just exactly covered the surface of the sun, he alsoestimated that the sun was 19 times the size of the moon.

Aristarchus’s measurement was made without the modern telescope and so 3 degrees was quite liberal.The true measurement of the angle formed by the sun is about 0o10′. This means that since the cotangentof 0o10′ is about 400 then the distance to the sun is 400 times the distance to the moon and the diameterof the sun is about 400 times the diameter of the moon. (Wikipedia has a nice article on this.)

Aristarchus also observed that if the sun is the center of the solar system then as the earth movesaround the sun, the stars should appear to change positions when measured six months apart. Since thestars did not change positions, either the sun was not the center of the solar system or the stars werevery far away!

We now know the stars are indeed very far away!

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Hipparchus (Greek, ca 180 - 125 BC) was an astronomer who developed a table of chords (trig!) whichwere extended by Ptolemy and laid the foundation for the Almagest.

He made very careful observations of stars from an observatory in Rhodes, a Greek island. Thesemeasurements were used later by European astronomers.

Claudius Ptolemy (Greek, ca. 85 - ca 165 AD) was an astronomer and mathematician. (Do not confusehis name with the names of the rulers of Egypt after the death of Alexandria.)

Ptolemy took Hipparchus’ table of chords and gave the lengths of the chords of all central anglesof a given circle by half-degree intervals from 1/2 to 180, expressed in sexagesimal notation. He wrotedefinitive work on astronomy in 150 AD, called Syntaxis Mathematica (Mathematical Collection.) Thiswork was later called the Magiste or “Greatest” and eventually, the Almagest. Book 6 on the theory ofeclipses gives π = 377/120 = 3; 8, 30 ≈ 3.1416. Books 7 and 8 catalogued 1028 stars. (He also kept trackof the planets.)

He wrote on map projections (stereographic projection), optics, and music. The work on map pro-jections was necessary so that he could accurately draw a map of the spherical globe on flat paper.(European mathematicians would later take up this study.)

He attempted to prove Euclid’s 5th postulate from the others.A mathematical theorem, ”Ptolemy’s theorem”, is named after him. It is about products of diagonals

of a “cyclic quadrilateral.”Ptolemy also made an excellent map of the earth.

2.5.2 Chords and trigonometry

Ptolemy’s trig tables (see Swetz, p. 189)See Eves 6.9 and 6.10.

TrigonometryThe astronomers measured chords and half-chords. Our modern sine function is their half-chord.

In order to measure the sphere of stars that seems to hang over our heads, they had to develop basictrigonometry.

Look at trig identities, small sided triangles and eventually DeMoivre’s theorem.

The universe, as the Greeks understood itAristarchus thought the earth was about 8,000 miles in diameter (correct), that the moon about 40

earth-diameters away (30 is more accurate) and that the sun is 600 earth-diameters away (a considerableunder-estimate.) If the stars are immeasurably further away – hundreds, if not thousands of times furtherthan the sun – then the universe is hundreds of millions of miles across. Aristarchus knew that the universewas immense!

Our current knowledge of the universeWe now know that our solar system is many billions of miles in diameter and that the nearest star is

over 4 light-years away and thus about 25 trillion miles away!Yet we live in the Milky Way galaxy, about 100,000 light years across and over 100 billion stars.

Beyond our galaxy, at over a million light years away (and so over six million trillion miles away) is thenearest galaxy, the Andromeda Galaxy. Further away are approximately 100 billion galaxies, the furthestbeing 13 billion light years away.

Our universe (at least what we can see of it) is about 15 billion light years in diameter, that is about1023 miles.

Numbers of this magnitude are beyond human comprehension. One might attempt to get a good ideaof the size of the universe by looking at the

1. Powers of Ten video

or

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2. this map of the digital universe.

(I recommend these short videos!)One more video on sizes of planets and stars

2.5.3 Archimedes

The most famous mathematician of the Hellenistic Age – and probably the most famous Greek mathe-matician of all time – was Archimedes (287 BC – 212 BC.) It is possible that he spent some time at theUniversity in Alexandria. He apparently knew Eratosthenes and successors to Euclid there.

He developed numerous physical systems to move weight, including catapults and levers.There is a famous story about his solution to determining the purity of a king’s gold crown.Archimedes was apparently born in Syracuse, Sicily and eventually settled there. He helped the

Greeks defend that city. He was apparently killed by a Roman soldier when they overran the town in212 BC.

We now know that Archimedes wrote numerous mathematical works. Many of these were recopiedand edited by Greeks after him; quite a number of his works have been lost and we know only aboutthem by references to them from other Greek writers.

One of the most interesting surviving works is the Archimedes Palimpset, rediscovered in 1906. Thisancient copy of the work of Archimedes was made on parchment, a popular leathery substitute for papyrus,before the invention of paper. The parchment writing, probably made in the early Middle Ages, was laterscraped off (erased) and written over, filled with prayers and Christian songs and stored in a monastery.Recently the underlying text has been recovered and is available now online.

Some of the achievements of Archimedes are:

1. His study of pulleys and levers led him to make the famous claim, “Give me a place to stand onand I will move the earth.” Other physics investigation included the water screw, used in irrigation.

2. The area of a triangle with sides a, b, c is A =√s(s− a)(s− b)(s− c) where s is the semiperimeter.

(This is now called Heron’s formula.)

3. Worked on the Quadrature and Trisection problems.

4. Calculated π using 96-sided polygons and showed that 22/7 < π < 223/71.

5. In the Sand Reckoner he refers to a suggestion of Aristarchus that the sun is the center of the solarsystem and develops a system for writing very large numbers.

6. The Cattle Problem is a problem given in that text which requires large integers and is apparentlygiven as a challenge by Archimedes.

7. He computed formulae for the surface area and volumes of the sphere, cylinder and cone.

8. He rigorously developed a method for finding area of objects involving “exhaustion”, a rudimentarylimit concept, a precursor to calculus.

2.5.4 Apollonius and conic sections

Apollonius of Perga (Greek, ca 262 B. C. - ca 190 B. C) was “The Great Geometer”. He was nicknamed“Epsilon”. (Recall that Eratosthenes was called Beta.)

Apollonius was born in Perga in Asia Minor. He went to Alexandria and studied under successors ofEuclid. Later he founded a university at Pergamum, patterned after Alexandria.

His most famous work is his book on Conic Sections, in eight volumes. There he names the ellipse,parabola, hyperbola as cross-sections of a cone. He also describes normals and evolutes (envelopes of

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normals) of the three types of conics. The famous problem of Apollonius is: given 3 circles to constructthe tangent circle. (Apollonius’s work Includes degenerate cases where one of the circles is a line.)

His works are related to us by Pappus. His works were read by Renaissance mathematicians.He worked on the duplication of the cube and had a method which used a rectangle and a (noncon-

structible) circle.He had a method for writing large numbers.

2.5.5 Volumes and Areas

Here is a summary of some of the major concepts and formulae the Greeks developed to compute areaand volumes.

1. The area of a circle

The Greeks had an expression for the area of a circle (see Eves 2.13a) and they knew why it wastrue. They argued that the area of a circle is the same as that of a triangle with base equal to thecircumference and height equal to the radius. (So C = 1

2Cr.) Their argument used a method of“exhaustion” to deal with an infinite process; this method had the seeds of the limit concept.

The circle could be viewed as a collection of triangles all with their bases on the perimeter of thecircle.

We adopt the formula for the area of a triangle,

A =1

2bh (5)

(where b represents the length of the base and h the height of the triangle) and note that thesetriangles all have height equal to the radius of the circle and their bases form the circumference.So the area of the circle is simply

A =1

2Cr (6)

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That’s it. Done!

Of course it is now customary to write the circumference in terms of the radius, so C = 2πr (bythe definition of π) and so A = 1

2 (2πr)r = πr2.

But if we want to be really careful about our argument, we have to worry about the fact thatwe really used an infinite collection of triangles. Does our argument about the triangles filling upthe circle still make sense? Here, from Eves, p. 381, is the main idea in Archimedes conclusionregarding “exhaustion” of the circle.

Consider the triangle formed by joining the center O of the circle to the points A and B on thecircle. The triangular piece 4AOB clearly misses some area of the sector cut out by the arc AB,notably the area of the circle above the line segment AB. To exaggerate the problem, I all drawa sector AOB with a large central angle, below, so that the difference between the sector and thetriangle 4AOB is large.

Redrawn using the figure in Eves, p. 381

Note that the sector AOB of the circle includes the areas in yellow and orange which are not partof the triangle 4AOB. Will this region ever be counted in our construction? We double the totalnumber of triangles in our circle by bisecting the angle ∠AOB to create the point M on the circleand new smaller triangles 4AOM and 4BOM . Now the area in yellow (above) is included inthose triangles but we are still missing the area in orange. Since the area of 4AMB is absorbedby these new triangles and that area is half of the rectangle formed by A,R, S and B then morethan half of the excess has been removed. So each time we bisect our triangles and double the

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number of triangles, the area remaining is less than half of the previous. If we do this indefinitely,we completely exhaust the circle and so our triangles “eventually” cover every point on the interiorof the circle! (More precisely, given any point on the interior of the circle, there is only a finitenumber of steps required in our process before the triangles include that point.

The area of a circle is explained here in animation.

2. The volume of a cylinder

The Greeks knew the volume of a cylinder (see Eves 6.2)

Volume is (in some sense) area moved in a third dimension, so if one takes a plane figure (a circle,triangle, rectangle...) and moves it along an axis perpendicular to that plane, one creates a threedimensional figure whose volume is the product of the area (of the base) and the length of the axis(height.) We often express this as ”V = Ah” where A is the area of the base and h is the height ofthe object. This only works if the cross-sections parallel to the base are all the same.

Thus the volume of a parallelepiped (box) is ` ·w · h and the volume of a circular cylinder is πr2h.

3. Cavalieri’s principle

The Greeks used “Cavalieri’s principle”

Suppose we have two 3-dimensional solids with the same base area and the same height, with theadditional property that plane cross-sections parallel to the base and at the same height are alwaysequal. Then the volumes of the two objects should be the same.

For example, imaging a sculpture created by placing thin horizontal plates one on top of the other– maybe a sculpture carved out of a deck of cards, for example. Then imagine pushing the plates alittle to one side. Would that change the volume? If at each height the horizontal cross-section didnot change, and the height stayed the same, the volume should just be the same. The horizontal“slabs” are unchanged.

This concept is now called the Cavalieri’s principle after an Italian mathematician who explicitlyused it during Renaissance times. But the concept was known at least to Archimedes and probablywidely accepted.

4. The volume of a pyramid

The Greeks could explain the volume of a pyramid, after doing that work, could compute the volumeof a cone.

Everyone learns, at some point – at least in calculus – that the volume of a cone is 13 times the area

of the base (πr2) times the height. But why? This is because, first, more generally, the volume ofa pyramid with a triangular base is 1

3 times the area of the base times the height.

We will work through the argument which appears (essentially) in Proposition 7, Book XII ofEuclid’s Elements. Consider a three dimensional solid in the shape of a prism, with a triangularbase. Its volume is just the area of the base times the height.

In the figure below, the triangular base is at the bottom, with vertices A,B,C; the prism has threeparallel sides so that the triangle at the top is the congruent to the base triangle.

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We then cut out a triangular base pyramid by slicing along the plane through the three points C ′, Aand B. This removes a triangular pyramid (in red) with base 4ABC and height h.

We can take the three dimensional prism and cut out a pyramid (in red, on the left) with thetriangular base, leaving a figure with a rectangular backing and a triangular top (on the right).

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But then we can cut that figure in two, from upper right to lower left, to create two more figures.The first object has the same triangular base and height; the second has the same volume as thefirst since it has a long triangular side and height perpendicular to that. In this manner, the figureon the right has been cut into two pieces with equal volume. But the volume of each of these twopieces is equal to the first pyramid so we have decomposed the prism into three equal parts.

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Thus the volume of a triangular based pyramid is one-third the area of the base times the height,that is

V =1

3Ah (7)

where A is the area of the triangular base.

5. The volume of a cone

What if my pyramid has a base which is not a triangle. Maybe it is a polygon with more sides?We can decompose a polygon with n sides into n− 2 triangles and so we can just apply the formulaabove and so for any pyramid with a polygonal base, we still have

V =1

3Ah

But what about a cone? A cone is a “pyramid” with a circular base. We return to the Greek viewof a circle as an infinite number of triangles. Divide up the base circle into “many” triangles

and then just apply the formula for the volume of all these pyramids, V = 13Ah. Since now we know

that the area of the circle is 12Cr then the volume of a cone is 1

3 ( 12Cr)h which in modern notation

is 13πr

2h.

6. The volume of a sphere

Archimedes’s argument for the volume of a sphere used Cavalieri’s principle. Archimedes showedthat one could view cross-sections of the top half of a sphere as plane figures with the same area asthe plane figures which are cross-sections of a cylinder with a cone removed. (See the figure, below,by Michael Hardy at the English Wikipedia project.)

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Since the volume of the cylinder-with-cone-removed is πr2h− 13πr

2h = 23πr

2h and since the heightof the cylinder is also r, the cylinder-with-cone-removed (on the left) has volume 2

3πr3. Therefore

the volume of the sphere (on the right) must be twice that,

V =4

3πr3.

(Archimedes’ view of this process is explained in Swetz, p. 180 or Eves, p. 385.)

7. Volume of a frustum

The Greeks knew the volume of a frustum of a pyramid (see Eves 2.14, also Burton section 2.3.)The frustum of a pyramid is a truncated pyramid, one with the “top” removed. (If the pyramidhas height h and the base is a square with sides of length b while the top is a square with sides oflength a then the volume is V = 1

3h(a2 + ab+ b2).)

The knowledge of the volume of a frustum dates back to the Babylonians and Egyptians; theEgyptians correctly computed the volume of the frustum of a square pyramid in the Moscow papyrus(Eves, p. 55.)

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2.6 Conclusion of the Greek Age

2.6.1 Diophantus and Diophantine equations

Diophantus was a Greek mathematician who wrote on problems requiring rational solutions. His workalso included some elementary algebra, including the first recorded syncopated algebra.

His problems requiring only (positive) rational solutions led to problems with integer solutions afterone multiplied by a common denominator. So modern “Diophantine” equations are equations whichrequire integer solutions. In one of his works he gives a method to find all integer Pythagorean triples.

In his development of syncopated algebra, ζ stood for “number”, ∆γ and Kγ for “square” and “cube”of number and ∆γ∆,∆Kγ ,KKγ stood for square-square, square-cube and cube-cube of number. So hewould write Kγλε for 35x3. (See Burton, page 219.) Diophantus also had symbols for addition andsubtraction. Negative numbers were not used, but one could subtract positive numbers from a largerquantity. I do not believe there was an equal sign.

He wrote Arithmetica, On Polygonal Numbers and Porisms. Later in Europe, Regiomontus translatedArithmetica. This work was a book on algebraic number theory, discussing sums of squares and writingdifference of cubes as sum of cubes. (See sample problems in page 181 of Eves.) Fermat’s marginal noteswere based on reading Diophantus’ work.

A number of Diophantus’s problems are on worksheet 5. The algebra is relatively simple, but theemphasis on integers adds a subtlety and Diophantus worked these all without symbolic algebra.

2.6.2 Primitive Pythagorean Triples

A Pythagorean triple (a, b, c) is primitive if the greatest common divisor (GCD) of a, b, and c is 1. Let’sexamine primitive Pythagorean triples (PPTs).

Choose integers u, v, with u > v. Set a = u2−v2, b = 2uv, and c = u2+v2. This will be a Pythagoreantriple. (Check this!)

Theorem 1. (Due to Euclid, 300 BC.) The Pythagorean triple (a = u2 − v2, b = 2uv, c = u2 + v2) isprimitive if and only if GCD(u, v) = 1 and exactly one of the integers u or v is even.

Sketch of Proof.We sketch the proof of this theorem and leave the details for the exercises. There are two parts to

the proof, due to the “if and only if” statement.

Part 1. First we assume that the PT (a = u2 − v2, b = 2uv, c = u2 + v2) is primitive (that is, nopositive integer but 1 divides all three terms) and show that this implies GCD(u, v) = 1 and one of theintegers is even.

Part 2. Then we assume that GCD(u, v) = 1 and one of the integers u, v is even and show that thisimplies that no positive integer greater than 1 divides all three terms (a, b, c).

In each case, it is easier to show the contrapositive statement. For example, let’s look at the secondpart of the theorem. Instead of proving

GCD(u, v) = 1 and exactly one of u, v is even =⇒ ¬(∃d ∈ Z, d > 1, d a common factor of a, b, c),

let us prove the equivalent statement:

(∃d ∈ Z, d > 1, d a common factor of a, b, c) =⇒ [(GCD(u, v) > 1) OR (u, v agree in parity).

Suppose there is an integer d, d > 1, where d is a common factor of all three terms a = u2 − v2, b =2uv, c = u2 + v2. By the Fundamental Theorem of Arithmetic (due to the Greeks!) there is a prime pdividing d and so there is a prime p dividing all three integers a, b, c.

There are two possibilities. Either p = 2 or p > 2.

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If p = 2 then since p divides a, it must divide u2 − v2. But if exactly one of u, v is even and the otheris odd then u2− v2 is odd. So if p = 2 divides a = u2− v2 then u and v are either both odd or both even.(That is, they “agree in parity.”)

What if p > 2? Can you show that if p divides both b = 2uv and a = u2 + v2 that p must be acommon factor of both u and v.? (This argument is finished in the exercises.)

Let us use Euclid’s result to generate some Pythagorean triples. Since we want a = u2 − v2 to bepositive, we will assume that u > v. We also assume that u, v have no common factors and exactly oneis even.

If u = 2, v = 1 then a = 3, b = 4, c = 5. This is the Pythagorean triple, (3, 4, 5).If u = 3, v = 2 then a = 5, b = 12, c = 13.What if u = 4, v = 1? Or u = 4, v = 3?How many Pythagorean triples can we create if we try to keep a, b, c all less than 100?

Theorem 2.Every PPT can be created by Euclid’s method; that is, if (a, b, c) is a PPT then there are integers u

and v with GCD(u, v) = 1 where either a = u2 − v2 and b = 2uv or b = u2 − v2 and a = 2uv.(This theorem is harder to prove and so we skip the proof here.)

Corollary to Theorem 2.Every Pythagorean triple has form (a = ku2− kv2, b = 2kuv, c = ku2 + kv2) for integers u, v, k where

u, v are relatively prime and exactly one of u, v is odd.There is a Wikipedia article on PTs and PPTs.

Gaussian integersThe element z = a+ bi where i2 = −1 and a and b are real numbers is said to be a complex number.

The conjugate of a complex number z = a+ bi is z = a− bi.We may graph complex numbers in the Cartesian plane by equating the point (a, b) with the number

z = a+ bi. The length of the complex number z is its distance from the origin, that is

||z|| =√a2 + b2.

Note that zz = a2 + b2 so we may also write the length in the simple form

||z|| =√zz.

The complex number z = a+ bi is a Gaussian integer if a and b are integers.The length of the Gaussian integer z = a+bi is an integer if and only if (a, b, ||z||) form a Pythagorean

triple. If (a, b, ||z||) is a primitive Pythagorean triple then the Gaussian integer z = a + bi cannot befactored further in the set of Gaussian integers. (More on this later.)

2.6.3 The Euclidean algorithm

The Euclidean Algorithm appears in Book VII in Euclid’s The Elements, written around 300 BC. It isone of the oldest mathematical algorithms.

It is also one of the most applicable. The algorithm provides a systematic way to find the greatestcommon divisor GCD of two integers and provide additional important information about the relationshipbetween the GCD and the two integers involved.

Modern technology uses a variety of algorithms based on modular arithmetic including the public-keyencryption RSA algorithm. Many of these algorithms in turn rely on the Euclidean Algorithm as analgorithm acting on the ring of integers or as an algorithm acting on a ring of polynomials.

Here we introduce the Euclidean algorithm for the integers. The Euclidean Algorithm on the

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set of polynomials is similar. The concepts here may be generalized to any algebraic system which obeysthe division algorithm; such rings are called Euclidean Domains.

We say that the integer d divides the integer a (written d|a) if there is an integer k such that a = dk.For example, −5|20 since 20 = (−5)(−4). So the divisors of 20 are−20,−10,−5,−4,−2,−1, 1, 2, 4, 5, 10, 20.

Given two integers a and b, we seek divisors d which divide both of these integers. We are in particularinterested in the largest such divisor, the greatest common divisor of both a and b. (The set of allcommon divisors of a and b is exactly the set of divisors of the greatest common divisor.)

Hereafter we abbreviate “greatest common divisor” of a and b by GCD or GCD(a, b).In order to avoid issue about ”size”, we will define the GCD of integers a and b as the positive integer

d that satisfies the following condition:

If c|a and c|b then c|d (8)

If a and b have common divisors (other than −1, 1) then there is a common prime p dividing both ofthem. The GCD of a and b is 1 if and only if the only common divisors of a and b are −1 and 1. In thiscase (since −1 and 1 divide every integer) we say that a and b “have no common divisors.” Equivalentlywe say a and b are relatively prime.

If an integer d divides both integers a and b then for any integer q, it divides a− qb. In particular, ifd is the greatest divisor of a and b and r is the remainder (guaranteed by the division algorithm) upondivision of a by q, then d also divides r.

Conversely, if d divides b and d divides r = a−qb then d also divides a. Therefore the greatest commondivisor of a and b is also the greatest common divisor of b and r.

This is the essence of the Euclidean algorithm. We replace a pair of integes a and b by a smaller pairof integers b and r and iterate the process until we reach the smallest possible pair of integers.

Suppose we are given two integers, a and b (for example, a = 843 52256 45419 and b = 105 46961 61403).Instead of hunting for divisors of a (8435225645419), we may divide the smaller number b (1054696161403)into the larger, and use the division algorithm to get a remainder r = a − qb, where 0 ≤ r < b. (In thiscase q = 7 and the remainder is r = 23436 45805.) Now we want the GCD of b and r, which are smallernumbers. Since the size of the positive integers is dropping, we may repeat this step, replacing a pair ofintegers

ai and bi

bybi and ri = ai − qbi

until we finally get a remainder of zero. Since, in the last step, the GCD of zero and an integer bk is justbk (every integer divides zero!), then the final integer bk is the GCD of a and b.

This algorithm carries more information than might be obvious at first glance. Suppose we writea = a(1) + b(0), and b = a(0) + b(1), and, at each stage, write the new number in the form as + bt forintegers s and t. Since each step in our algorithm involves computing ai−qbi, we may think of this processas an elementary row operation on a matrix of integers with rows of the form

(bi = as+ bt, si, ti

).

(See Wikipedia for more on matrices and elementary row operations.)

We will work out our example with a = 8435225645419 and b = 1054696161403 in detail.

Table 1, below, is the algorithm in tabular form. The four columns represent −q (where q is thequotient in the current step of the division algorithm), bi = as + bt, s, and t. In the last two rows, wecompute the GCD of 12347 and 0, and so the GCD of the original two numbers must be 12347. Theprocess of computing the GCD of a = 8435225645419 and b = 1054696161403 requires nine rows.

Notice that this algorithm always allows us to write the GCD of a and b in the form as + bt. Here12347 = a(−398721) + b(3188882).

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Table 1: Euclidean algorithm in tabular form

−q as+ bt s t

a = 8435225645419 1 0-7 b = 1054696161403 0 1-1 1052352515598 1 -7-449 2343645805 -1 8-42 55549153 450 -3599-5 10581379 -18901 151166-4 2642258 94955 -759429-214 12347 -398721 3188882

0 85421249 -683180177

The equationGCD(a, b) = as+ bt (9)

is called Bezout’s Identity. The Euclidean Algorithm not only finds the GCD of a and b but it alsofinds the integers s and t which satisfy Bezout’s Identity.

2.6.4 The three problems of antiquity

The Greeks had three problems which they couldn’t solve. They believed these could be solved andspent considerable energy trying to find solutions. These are now called the Three Classical Problems ofAntiquity. They were

1. to trisect an arbitrary angle

2. to double the cube

3. to square the circle

These essentially required that we construct cos(20o), 3√

2, and π, by the Euclidean construction tools.

2.6.5 Pappus, Theon and the commentators

A number of later Greek teachers are noted for their commentaries on Euclid and the earlier Greekmathematicians. During this period there was not a great deal of mathematical exploration; insteadwriters synthesized the earlier works, revised them and annotated them. From these later writers welearn the history of the Greek golden age.

Pappus of Alexandria (Greek, ca. 300 AD) wrote commentaries on Euclid’s work and on Ptolemy’swork.

He wrote Mathematical Collection, ”a veritable mine of rich geometric nuggets” which has also pro-vided us with much of our understanding of Greek mathematical history.

He discussed the method of Apollonius for writing and working with large numbers and commentedon the 13 semiregular polyhedra of Archimedes.

He proved the linear case of the cross-ratio theorem, a fundamental theorem of projective geometry.He gave a focus-directrix view of conic sections and a generalization of the Pythagorean theorem.

Theon was the last librarian of Alexander. His edition of the Elements was the standard edition for manycenturies afterwards. He also created an edition of Ptolemy’s works. Along with being a mathematicianand scholar in his own right, he is also remembered as the father of Hypatia.

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Hypatia (Greek, died 415 AD) is the first recorded mathematician who was a woman. She was thedaughter of Theon of Alexandria. She studied mathematics, medicine, philosophy and wrote commen-taries on Diophantus’ Arithmetica and Apollonius’ Conic Sections. She traveled and lectured, was praisedby Synesius of Cyrene (later bishop of Ptolemais). Hypatia never married. She was leader of neo-Platonicschool of philosophy and defended paganism against the new religion, Christianity, sweeping the RomanEmpire at the time. Because of her religious activity, the new patriarch, Cyril of Alexandria had herattacked, tortured and killed. Her death is recognized as the end of the creative days of the Universityof Alexandria.

She is listed as one of the women in the “Mathematical Pleiades”.

2.6.6 Truth & Falsehood

What things have we believed to be “obvious” but discovered they were not?

1. (Pythagoras?) Pairs of numbers are commensurable. (False: 1 and√

2 are not commensurable.)

2. (Zeno) One cannot add up an infinite number of terms. (False: infinite series may converge andindeed are very much a part of mathematics.)

3. (Euclid) The parallel postulate. (Some geometries – including the geometry of space-time in ouruniverse – disobey the parallel postulate.)

2.6.7 References for Greek mathematics

The material in this section follows

1. Chapters 3 to 6 of Eves,

2. Chapters 3 to 5 of Burton,

3. Chapter 3 to 8 of Kline,

4. Sections 2.10 to 2.28 of Grattan-Guiness,

5. Part 3, chapters 24-30 of Swetz.

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