The Greek Age of Mathematics
c Ken W. Smith, 2012
Last modified on February 15, 2012
Contents
2 The Greek Age
2.1 Thales, the Pythagoreans and the first Greek proofs . . . . . . . .
2.1.1 The Pythagoreans . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Pythagorean numerology . . . . . . . . . . . . . . . . . . .
2.1.3 Commensurable numbers . . . . . . . . . . . . . . . . . . .
2.1.4 The impact of noncommensurability . . . . . . . . . . . . .
2.1.5 Two types of Pythagorean triples . . . . . . . . . . . . . . .
2.1.6 Polygonal numbers . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Euclidean tools . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Euclidean calculator . . . . . . . . . . . . . . . . . . .
2.2.2 The quadratic formula . . . . . . . . . . . . . . . . . . . . .
2.2.3 The Fundamental Theorem of Algebra and Euclid’s lemma
2.2.4 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Euclid’s Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 The fabric of mathematics . . . . . . . . . . . . . . . . . . .
2.3.2 The preamble to the Elements . . . . . . . . . . . . . . . .
2.3.3 Propositions from Book I . . . . . . . . . . . . . . . . . . .
2.3.4 Euclid’s Proposition I.47 . . . . . . . . . . . . . . . . . . . .
2.4 Elements, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Books II, III, IV . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Triangles and their centers . . . . . . . . . . . . . . . . . .
2.4.3 Books V and VI . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Books VII, VIII, IX . . . . . . . . . . . . . . . . . . . . . .
2.4.5 Book X . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.6 Books XI, XII, XIII . . . . . . . . . . . . . . . . . . . . . .
2.5 The Greeks after Euclid . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Eratosthenes and the Greek view of the universe . . . . . .
2.5.2 Chords and trigonometry . . . . . . . . . . . . . . . . . . .
2.5.3 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.4 Apollonius and conic sections . . . . . . . . . . . . . . . . .
2.5.5 Volumes and Areas . . . . . . . . . . . . . . . . . . . . . . .
2.6 Conclusion of the Greek Age . . . . . . . . . . . . . . . . . . . . .
2.6.1 Diophantus and Diophantine equations . . . . . . . . . . .
2.6.2 Primitive Pythagorean Triples . . . . . . . . . . . . . . . .
2.6.3 The Euclidean algorithm . . . . . . . . . . . . . . . . . . .
2.6.4 The three problems of antiquity . . . . . . . . . . . . . . . .
2.6.5 Pappus, Theon and the commentators . . . . . . . . . . . .
2.6.6 Truth & Falsehood . . . . . . . . . . . . . . . . . . . . . . .
2.6.7 References for Greek mathematics . . . . . . . . . . . . . .
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
2
5
5
5
7
8
8
10
10
11
11
12
13
13
14
15
18
21
21
21
22
22
23
23
24
24
25
26
26
27
34
34
34
35
37
37
38
38
2
The Greek Age
Following Wikipedia’s article on ancient Greece we will set the dates of ancient Greece from 750 BC to
592 AD, a period of 13 21 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 philosophers and mathematicians, people who, with leisure time, asked “big questions” about life, science and
the universe. During this time, we see the rise of two city-states (“polis”) Athens and Sparta. It is in
Athens that the philosopher Plato built his Academy and other philosophers such as Socrates, Sophocles and Aristotle lectured and built their schools and groups of disciples. Much of western (European
and American) civilization owes it worldview and political system to those developed in Athens during
the classical period of Greece.
The classical period ended with the unification of Greece by Alexander the Great, the creation of the
city 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 of
his empire to the conquest of Greece by the Romans. It was a time of expansion of Greek culture and
philosophy.
After 146 BC Greece was controlled by Rome but the Greek ideals and Greek language permeated
the Roman empire. The Greco-Roman empire persisted through emperors Julius and Augustus Caesar
and into the rise of Christianity, until in 330 AD, Constantine changed the empire forever by officially
making it “Christian.” The new “Christian” culture replaced the Greek culture and the Greek influence
declined. (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 529
AD.
Axiomatic systems
The Greeks during their time took mathematics to a higher level, far exceeding the works of the Babylonians and Egyptians. In particular, the Greeks included mathematics within a broad rational approach
to philosophy and scientific investigation. Mathematical “facts” were created by careful reasoning and
logical argument. One was allowed to ask, “Why is that true?” and to expect a careful explanation as
an answer.
A natural result of the Greek rational approach to mathematics was an axiomatic system based on
a collection of foundational axioms on which all other arguments rely. The axiomatic system was best
displayed 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 system
underlies 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 in
Miletus?) is the first recorded philosopher. His emphasis on reasoning and concepts made mathematics
part of his philosophical thinking.
He was followed by Pythagoras (possibly a student of Thales) whose teachings on geometry and
number permanently changed our understanding of mathematics. Pythagoras probably spent some time
in Egypt, learning and then extending the Egyptian understanding of mathematics. He started a school
in southern Italy, a school that was both mathematical and cultic, with rigid religious beliefs and secret rituals. His followers, who worshiped mathematics (“All is number!”), significantly extended the
2
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 side
then 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 angles
A, 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 “straight
line” as is the sum of angles B and C. Thus A is a “straight line” less the value of C and B is also
a “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” property
that all straight lines give the same angle and the “logic” property that “equals subtracted from
equals gives equals.”
3
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 C
with 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’s
identify the magnitude of these angles by α. By a similar argument, the angles ∠BCO and ∠CBO
are 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
and
2β + ∠BOC = π
Therefore, adding these last two equations we have
2α + 2β + ∠AOB + ∠BOC = 2π
Substitution equation 1 into equation 2 we have
2α + 2β + π = 2π
and so
2α + 2β = π
and upon dividing by two we have
α + β = π/2.
But α + β is the magnitude of the angle C and so C is a right angle. 2
4
(2)
2.1.1
The Pythagoreans
Pythagoras followed Thales and may have been a student. Apparently Pythagoras traveled from Samos
to Egypt and then to Crotona in southern Italy where he started a commune or cult which worshipped
number. Many early mathematical statements are attributed to the Pythagoreans including the main
theorem named after Pythagoras.
Along with worshipping ”number”, the Pythagoreans held a variety of mystical (religious) beliefs
including the transmigration of the soul and possible release from the perpetual cycle of reincarnation
through various purification rites. They were vegetarians (apparently for a variety of reasons) and recognized the relationship between numbers and music.
The Pythagoreans were the first to prove the theorem now named after them. They may have had
several different proofs. A dissection proof of the Pythagorean theorem appears on page 81 of Eve’s book
and page 105 of Burton. (See also this website.) There are many proofs of the Pythagorean theorem at
the cut-the-knot webpages.
The Pythagorean Theorem is equivalent (logically) to the parallel postulate. (More on the parallel
postulate later.)
Some of the main subjects of the Pythagoreans show up in the medieval collection of subjects in early
universities. 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 to
particular numbers.
One form of Pythagorean attribution to numbers associated a number with the sum of its proper
factors. For example, the number 10 was associated with 1+2+5 = 8. Since the sum of factors is smaller
then 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 factors
were 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 of
284 is 1 + 71 + 2 + 142 + 4 = 220. So 284 does dominate 220 in some sense? (But then what are the sum
of 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 the
top. 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-th
triangular number.)
The Pythagorean mysticism about numbers is echoed in superstitious systems of numerology in
which 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 be
expressed as multiples of some (extremely small) number. From their viewpoint, given a length x and
another length y, there was some small length, maybe an“atom”, that allowed one to write both lengths
5
as an integer multiple of that “atom”. If we call the atom α then we could find integers M and N such
that x = M α and y = N α.
Some of their arguments about computations and relations among lengths assumed this “commensurability” between pairs of numbers.
Let’s explore this
√ idea. Type two numbers into your calculator, like an approximation for π and an
approximation for 2, say
x = 3.141592653589793 and y = 1.4142135623730950.
Then we might make the “atom” be α = 0.000000000000001 and write x = 3141592653589793α and
y = 14142135623730950α.
Or another example: take
99
355
and y =
.
x=
113
70
With fractions it is easy to find the “atom” – just get a common denominator! In this case the
denominators are relatively prime so the common denominator is the product of the two denominators.
So we write
x=
355 · 70
24850
99
99 · 113
11187
355
=
=
and y =
=
=
.
113
113 · 70
7910
70
70 · 113
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 held
sway 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 until
we eventually get exact division, with no remainder. At this point we have found the atom (the last
remainder) 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 it
did ... 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 and
page 84 of Eves.) Draw a square with sides of length 1. The square root of 2 is represented by the
diagonal. 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.
6
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 , B1 C1 ,
√
BC1 all have the same length (which in modern notation is 1 − 2.) If the lengths of AB and AC have
a 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. Now
we do this again on the smaller square with vertices A, B1 and C1 . We create the point P so that the
length of C1 P 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 “incommensurable” numbers?
2.1.4
The impact of noncommensurability
The Greeks of Pythagoras’ day believed that pairs of numbers were commensurable, that is, could be
written
√ as integer multiples of some common standard, some common atom. To discover that numbers
like 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 = M α and y = N α. What
were we to do if there was no “atom” α?
Zeno
More 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. (In
Greek this smallest piece might be called an ”atom”. Or we might use a Latin word and call such a small
indivisible piece a “quantum”.)
For those who believed that length or time were built out of smallest atoms or quanta, Zeno provided
the Arrow Paradox. In this paradox, an arrow is flying through the air. Let’s examine a single
quantum of time. In that small quantum, the arrow must be motionless. (If not, then it is moving
through the quantum, in one location at the start of the quantum and in another at the end. This
violates the indivisibility of the quantum.) But if the arrow is motionless at each quantum of time then
it 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 smaller
and smaller pieces, Zeno had the paradox of Achilles and the tortoise. In this paradox, the tortoise
is given a head start and Achilles tries to pass it. But Achilles first runs to the place that the tortoise was
at the beginning of the race. By then the tortoise has moved on a little to a new location. So Achilles
continues on to that location. But the tortoise has moved on. And so on....
7
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 read
a much later paradox – Thomson’s lamp – in the same vein.
Eudoxus
The 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 proportion
x : y by saying that x : y and a : b are the same proportion if
1. the existence of integers M, N such that M x > N y implies M a > N b;
2. the existence of integers M, N such that M x = N y implies M a = N b;
3. the existence of integers M, N such that M x < N y implies M a < N b.
In other words, the relationship between M x and N y is always identical to the relationship between
M a and N b.
(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 of
the following equations. If one started with an odd square, say m2 = 2k − 1 then one would begin with
the relation
(2k − 1) + (k − 1)2 = k 2 .
(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 Pythagorean
triples are on pages 107-109 in Burton and page 82 in Eves.) These pairs of Pythagorean triples fit
into the collection of primitive Pythagorean triples that were discovered much later and recorded by
Diophantus.
2.1.6
Polygonal numbers
The Pythagoreans (and the Greeks after them) had a fascination with numbers that could be organized
into some shape. For example, if one places four blocks in a row on the floor and then three blocks above
them, followed by a third row with two blocks and a fourth row with one block, then the blocks form a
triangle of four rows and ten blocks. Since 10 = 4 + 3 + 2 + 1, 10 was a triangular number. I will write
t4 = 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.
8
(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 dabbled
with 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.
9
2.2
The Euclidean tools
Most of what we know about the early Greek mathematics comes from the mathematician Euclid who
taught 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 (says
Wikipedia.) There the Greeks built the first university and the first library. The famous university and
library at Alexandria lasted almost a thousand years, from 331 BC to 641 AD when it was sacked by the
Arabs.
Euclid was the head of the Mathematics Department there in its early days, around 300 BC. He
wrote numerous works on mathematics. His most famous book, one of the most widely published books
in history, is The Elements. Today we tend to call it a “geometry” book. But in his day, it was about all
of 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. To
multiply two numbers, one viewed the numbers as lengths and drew a rectangle with those lengths for
its sides.
In order to do calculations, one needed the right tools. For the Greeks, this meant that one had to
construct 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 r
about the point C. A simple compass was used for this. (There is more on compass and straightedge
constructions 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 or
overlapping segments.
then numbers like 2, 3 and 4
For example, if one represents unity (1) by a short length:
were 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
c
a a+c
and are
two parallel lines cutting across an angle. This creates similar triangles so the ratios ,
b b+x
x
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 length
of 35 then we draw this figure with segments of length a = 3, b = 5 and c = 1. The parallel lines will then
mark off a length x = 35 .
In the figure below (Eves, p. 88) the big right triangle with short legs of length a and x, on the left
side of the circle is similar to the right triangle with short legs x and b on the right side of the triangle.
(Why?)
10
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 35 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
q
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 and
a − 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 in
x(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 to
x2 + ax + b2 = 0 and x2 + ax − b2 = 0. (See the material on this from the discussion of Babylonians and
Egyptians, 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 elementary
grades.
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 his
famous book, The Elements. (More about this book later.)
Please include the Fundamental Theorem of Arithmetic (FTA) and Euclid’s Lemma in your collection
of mathematical tools!
11
2.2.4
Quadrature
The Greek view of number was visual, geometric. Our modern bias distinguishes their visual imagery
and argument (“geometry”) from “algebra” and our modern mathematics have an excessive emphasis on
algebra instead of geometry. For this reason it is difficult for us to move back in time and see mathematics
in 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 were
interested in constructing a square with a given area.
For example, they sought to “understand” the circle by constructing a square with an area equal to
that of the circle. We now speak of their attempts to “square” the circle, but for the Greeks, the square
was 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 did
a 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 certain
area. 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 Hippocrates
of Chios in about 450 B.C. Hippocrates, in an attempt to square the circle, shows how to square a slice
of 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 circle
using Euclidean tools.
12
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, became
the textbook for anyone who claimed to be educated, and probably went through close to a thousand
different 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 and
Congress.
The style of this textbook is revolutionary. Everything is carefully and logically argued, beginning with
a set of 23 definitions, five “postulates” and five “common notions”. Using these items as a foundational
layer, Euclid builds an edifice of mathematics, one brick at a time, each result (called a Proposition) is
argued from previous results.
Some good sources: Eves, Chapter 5 is devoted to Euclids Elements. Burton has a nice introduction
to Elements in sections 4.1 and 4.2. David Joyce’s webpage provides The Elements online. I recommend
that 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 students
memorize these results in middle school or high school without understanding the meaning or explanation
of the terms or computations.
(I am reminded of the memorization of spells by the characters in the Harry Potter novels – here
at Wikipedia is a list of the many things a student – Hermione Granger? – would have to memorize if
he/she were hoping to excel at Hogwarts.)
In mathematics we can (should) organize our understanding and formulae into categories. I have
taken the suggestions of a recent Evolution of Mathematics (MATH 4367) class and created a list of
mathematical 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 the
concepts 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,
13
(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 Pythagorean
theorem 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 the
lengths 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 a
work in which each result was carefully reasoned from the previous results. This work comes down to us
as 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 basic
truths, 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 lay
out 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. These
postulates 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 sum
to less than 180 degrees then those two lines, when extended indefinitely, will eventually meet on
that side.
The five common notions are axioms of logic – claims about logic and what is allowed in the logical
arguments.
1. Things equal to the same thing are equal to each other. (Algebraically: if x = y and x = z then
y = z.)
14
2. “If equals are added to equals, the wholes are equal.” (Algebraically: if x = y and a = b then
x + a = y + b)
3. “If equals are subtracted from equals, the remainders are equal.” (Algebraically: if x = y and a = b
then x − a = y − b)
4. Objects that coincide with one another (that is, one can be moved to lie identically on top of the
other) 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 and
that the line we extend is extended uniquely.
One thing he assumes unconsciously are that if circles are close enough that the distance between
their 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 the
circle. These “true” statements are never proven and never explicitly assumed. Later mathematicians
would add these as axioms.
Although we may struggle to put these axioms into a modern context, nine of them are pretty obvious
and 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 simply
in its length. And it is clear that the axiom bothered Euclid in some sense, for he avoids the use of the
axiom whenever possible.
The strange axiom in the list is Euclid’s Fifth Postulate. Imbedded in the fifth postulate is a claim
about the geometry of the universe. Also imbedded in the fifth postulate is a hint of the need to grapple
with 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 copied
graphics from that website in the material below.)
Proposition 1, to construct an equilateral triangle. Here we have an elegant use of the axioms. (And
this is the first place Euclid assumes something he has not explicitly given as an axiom! What does he
assume?)
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 knew
but avoided. (Why?)
15
Here Euclid marks off equal line segments AF and AG and then argues from SAS (the previous
proposition!) that the triangles ∆F AC and ∆GAB are congruent.
Therefore F C ∼
= BG and since BC = CB, the triangles ∆F BC 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. In
proving 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 construct
an isosceles triangle in such a way that the result of Proposition 5 conflicts with the assumption in
Proposition 6.
This is a nice use of “proof by contradiction” (or, in Latin, “reductio ad absurdum”.)
Proposition 9. How to bisect an arbitrary angle.
16
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 describes
them on page 154.
Proposition 27. “If two lines are cut by a transversal so as to form a pair of congruent alternate interior
angles, then the lines are parallel.”
17
Proposition 29. “A transversal falling on two parallel lines makes the alternate interior angles congruent
to one another, the corresponding angles congruent, and the sum of the interior angles on the same side
of 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” proof
or simply “Euclid’s Proof.”
We begin with a right triangle (right angle at C)
C
A
B
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 letters
at all the vertices. I have suppressed the various letters.
C
A
B
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 mistranslation
of a Greek term meaning ”Insect.” Burton calls this “the mousetrap proof.”
18
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 square
on the hypotenuse is the sum of the areas of the squares on the shorter sides.
C
A
B
We
are
the
the
make this argument by looking at pairs of triangles. We construct a triangle whose base and height
the same as that of the dark blue square. Then we construct a triangle whose base and height are
same as that of the light blue square. The areas of these triangles must then be half of the areas of
corresponding squares.
Then we will argue that these triangles are congruent, so their areas are equal and so the areas of the
corresponding squares are equal.
19
C
A
C
B
A
B
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 at
A.)
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 the
triangles in orange.
7. Therefore the area of the square on the hypotenuse is (visually!) the sum of the areas of the squares
on 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?
20
2.4
Elements, Part II
Euclid’s Elements contained thirteen “books”. The first book had 48 propositions on plane geometry and
we discussed them in the previous section. Here we briefly summarize the other books. All thirteen books
were used by teachers and students of mathematics for the next 2000 years, through various translations
and editions.
A translation of these books is worth reading. Since Euclid did not have algebra, everything is written
in 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 is
subtracted from the second and the last numbers equal to the first, then the excess of
the 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 II
This 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 III
Book 3 has 37 propositions dealing with circles and related things such as tangents, secants, chords,
etc.
Book IV
This 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 be
obtained 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 operations
of rotation, reflection, and dilation. Consequently this invariance is a necessary property for any point
aspiring 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. The
circumcenter and incenter are the centers of circumscribed and inscribed circles. The centroid (center of
mass, 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 common
point of intersection of the altitudes. I believe the first time this is mentioned is in Archimedes’ Book of
Lemmas 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 López (University of Puerto Rico at Rio Piedras) writes “In book IV of
Euclid’s Elements the notable points of the triangle, the incenter and the circumcenter are discussed. In
the process of constructing the inscribed and the circumscribed circles it is proven that these notable points
are concurrency points of the corresponding cevians of the triangle. Giovanni Ceva (1648-1734) stated
21
his famous result during the second have of the 17th century or the beginning of the 18th. Presumably he
must 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 of
the mathematician when examining these objects; Euclid’s arguments are exclusively geometric, there is
absolutely no algebra. Our modern viewpoint (due to Descarte, Fermat and Pascal) places coordinates
on these objects and uses algebra. Think about the effect this point of view has on the mathematical
approach!
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. Proposition 10 constructs the 36-72-72 triangle in preparation for constructing a regular pentagon; this triangle
will also show up in the golden ratio. The book ends with Proposition 16 where a regular 15-gon is
constructed.
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 material
was developed by Eudoxus.
Book V
Book 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 VI
Book 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 the
study is of the integers, including factoring numbers, divisibility, prime numbers and “Euclid’s Lemma.”
Book VII
The 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 had
trouble stating it [the FTA] in ... modern form as he did not use the product of more than 3 numbers.”
Book VII
Book 8 has proportions on number theory including geometric progressions and their sum.
Book IX
This 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).
22
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 XI
Wikipedia: “Book 11 generalizes the results of Books 16 to space: perpendicularity, parallelism, volumes of parallelepipeds.”
Book XII
In 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 XIII
This 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 all
congruent and are regular convex polygons. This result, attributed to Plato, gives us the five Platonic
solids.
The drawings below are from http://www.uwgb.edu/DutchS/symmetry/platonic.htm.
23
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 the
creation 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 became
part of the Ptolemy dynasty after Alexander’s death in 323 BC. During Euclid’s lifetime the Greeks
developed the city of Alexandria and its great university, eventually creating a “museum” devoted to the
Muses. This museum eventually included the great Library of Alexandria which retained thousands of
papyrus 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 the
son 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 on
the day of the summer solstice, the sun’s shadow at high noon differed by about 7.2 degrees between the
two sites, then the diameter of the sphere was about 50 times the distance between the two cities. (See
Burton, page 187.) His estimate for the circumference of the earth was then (in modern language) close
to 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 with
Alexandria as the center. For his day, the map is quite good; it has much of Africa and Europe and some
of Asia.
He became chief librarian at the University of Alexandria. He was named Pentathlus (apparently
because 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 the
shape of the universe.
Aristarchus of Samos (Greek, ca 310 - 230 BC) put forward a heliocentric (sun-centered) hypothesis
for the solar system.
He measured the distance from the earth to the sun. He took the triangle with the earth, sun and
moon as vertices and measured the angle of the sun in this triangle, when the moon was at first quarter
and so the moon formed a right angle. Since, in his measurement, the vertex formed by the sun in this
triangle was 3 degrees (or less), he estimated that the sun was about 19 times as far from the earth as
the 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 also
estimated 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 0o 100 . This means that since the cotangent
of 0o 100 is about 400 then the distance to the sun is 400 times the distance to the moon and the diameter
of 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 moves
around the sun, the stars should appear to change positions when measured six months apart. Since the
stars did not change positions, either the sun was not the center of the solar system or the stars were
very far away!
We now know the stars are indeed very far away!
24
Hipparchus (Greek, ca 180 - 125 BC) was an astronomer who developed a table of chords (trig!) which
were 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. These
measurements were used later by European astronomers.
Claudius Ptolemy (Greek, ca. 85 - ca 165 AD) was an astronomer and mathematician. (Do not confuse
his 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 angles
of a given circle by half-degree intervals from 1/2 to 180, expressed in sexagesimal notation. He wrote
definitive work on astronomy in 150 AD, called Syntaxis Mathematica (Mathematical Collection.) This
work was later called the Magiste or “Greatest” and eventually, the Almagest. Book 6 on the theory of
eclipses gives π = 377/120 = 3; 8, 30 ≈ 3.1416. Books 7 and 8 catalogued 1028 stars. (He also kept track
of the planets.)
He wrote on map projections (stereographic projection), optics, and music. The work on map projections 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.
Trigonometry
The 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 basic
trigonometry.
Look at trig identities, small sided triangles and eventually DeMoivre’s theorem.
The universe, as the Greeks understood it
Aristarchus 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 considerable
under-estimate.) If the stars are immeasurably further away – hundreds, if not thousands of times further
than the sun – then the universe is hundreds of millions of miles across. Aristarchus knew that the universe
was immense!
Our current knowledge of the universe
We 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 the
nearest galaxy, the Andromeda Galaxy. Further away are approximately 100 billion galaxies, the furthest
being 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 about
1023 miles.
Numbers of this magnitude are beyond human comprehension. One might attempt to get a good idea
of the size of the universe by looking at the
1. Powers of Ten video
or
25
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 mathematician of all time – was Archimedes (287 BC – 212 BC.) It is possible that he spent some time at the
University 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 in
212 BC.
We now know that Archimedes wrote numerous mathematical works. Many of these were recopied
and edited by Greeks after him; quite a number of his works have been lost and we know only about
them by references to them from other Greek writers.
One of the most interesting surviving works is the Archimedes Palimpset, rediscovered in 1906. This
ancient 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 later
scraped 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 on
and I will move the earth.” Other physics investigation included the water screw, used in irrigation.
p
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 solar
system 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 apparently
given 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 rudimentary
limit 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 of
Euclid. 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
26
normals) of the three types of conics. The famous problem of Apollonius is: given 3 circles to construct
the 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 (nonconstructible) 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 area
and 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 was
true. They argued that the area of a circle is the same as that of a triangle with base equal to the
circumference and height equal to the radius. (So C = 21 Cr.) 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 the
circle.
We adopt the formula for the area of a triangle,
A=
1
bh
2
(5)
(where b represents the length of the base and h the height of the triangle) and note that these
triangles all have height equal to the radius of the circle and their bases form the circumference.
So the area of the circle is simply
1
A = Cr
(6)
2
27
That’s it. Done!
Of course it is now customary to write the circumference in terms of the radius, so C = 2πr (by
the definition of π) and so A = 21 (2πr)r = πr2 .
But if we want to be really careful about our argument, we have to worry about the fact that
we really used an infinite collection of triangles. Does our argument about the triangles filling up
the circle still make sense? Here, from Eves, p. 381, is the main idea in Archimedes conclusion
regarding “exhaustion” of the circle.
Consider the triangle formed by joining the center O of the circle to the points A and B on the
circle. 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 draw
a sector AOB with a large central angle, below, so that the difference between the sector and the
triangle 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 part
of the triangle 4AOB. Will this region ever be counted in our construction? We double the total
number of triangles in our circle by bisecting the angle ∠AOB to create the point M on the circle
and new smaller triangles 4AOM and 4BOM . Now the area in yellow (above) is included in
those triangles but we are still missing the area in orange. Since the area of 4AM B is absorbed
by these new triangles and that area is half of the rectangle formed by A, R, S and B then more
than half of the excess has been removed. So each time we bisect our triangles and double the
28
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 interior
of the circle! (More precisely, given any point on the interior of the circle, there is only a finite
number 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 three
dimensional 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 of
the 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 πr2 h.
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 the
additional property that plane cross-sections parallel to the base and at the same height are always
equal. 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 a
little to one side. Would that change the volume? If at each height the horizontal cross-section did
not 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 explicitly
used it during Renaissance times. But the concept was known at least to Archimedes and probably
widely accepted.
4. The volume of a pyramid
The Greeks could explain the volume of a pyramid, after doing that work, could compute the volume
of a cone.
Everyone learns, at some point – at least in calculus – that the volume of a cone is 31 times the area
of the base (πr2 ) times the height. But why? This is because, first, more generally, the volume of
a pyramid with a triangular base is 13 times the area of the base times the height.
We will work through the argument which appears (essentially) in Proposition 7, Book XII of
Euclid’s Elements. Consider a three dimensional solid in the shape of a prism, with a triangular
base. 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 three
parallel sides so that the triangle at the top is the congruent to the base triangle.
29
We then cut out a triangular base pyramid by slicing along the plane through the three points C 0 , A
and 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 the
triangular base, leaving a figure with a rectangular backing and a triangular top (on the right).
30
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 the
first since it has a long triangular side and height perpendicular to that. In this manner, the figure
on the right has been cut into two pieces with equal volume. But the volume of each of these two
pieces is equal to the first pyramid so we have decomposed the prism into three equal parts.
31
Thus the volume of a triangular based pyramid is one-third the area of the base times the height,
that is
1
V = Ah
(7)
3
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 formula
above and so for any pyramid with a polygonal base, we still have
V =
1
Ah
3
But what about a cone? A cone is a “pyramid” with a circular base. We return to the Greek view
of 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 = 31 Ah. Since now we know
that the area of the circle is 12 Cr then the volume of a cone is 31 ( 12 Cr)h which in modern notation
is 13 πr2 h.
6. The volume of a sphere
Archimedes’s argument for the volume of a sphere used Cavalieri’s principle. Archimedes showed
that one could view cross-sections of the top half of a sphere as plane figures with the same area as
the 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.)
32
Since the volume of the cylinder-with-cone-removed is πr2 h − 13 πr2 h = 32 πr2 h and since the height
of the cylinder is also r, the cylinder-with-cone-removed (on the left) has volume 23 πr3 . Therefore
the volume of the sphere (on the right) must be twice that,
V =
4 3
πr .
3
(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 pyramid
has height h and the base is a square with sides of length b while the top is a square with sides of
length a then the volume is V = 31 h(a2 + ab + b2 ).)
The knowledge of the volume of a frustum dates back to the Babylonians and Egyptians; the
Egyptians correctly computed the volume of the frustum of a square pyramid in the Moscow papyrus
(Eves, p. 55.)
33
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 work
also included some elementary algebra, including the first recorded syncopated algebra.
His problems requiring only (positive) rational solutions led to problems with integer solutions after
one multiplied by a common denominator. So modern “Diophantine” equations are equations which
require 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 K γ for “square” and “cube”
of number and ∆γ ∆, ∆K γ , KK γ stood for square-square, square-cube and cube-cube of number. So he
would write K γ λ for 35x3 . (See Burton, page 219.) Diophantus also had symbols for addition and
subtraction. Negative numbers were not used, but one could subtract positive numbers from a larger
quantity. I do not believe there was an equal sign.
He wrote Arithmetica, On Polygonal Numbers and Porisms. Later in Europe, Regiomontus translated
Arithmetica. This work was a book on algebraic number theory, discussing sums of squares and writing
difference of cubes as sum of cubes. (See sample problems in page 181 of Eves.) Fermat’s marginal notes
were based on reading Diophantus’ work.
A number of Diophantus’s problems are on worksheet 5. The algebra is relatively simple, but the
emphasis 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’s
examine primitive Pythagorean triples (PPTs).
Choose integers u, v, with u > v. Set a = u2 −v 2 , b = 2uv, and c = u2 +v 2 . This will be a Pythagorean
triple. (Check this!)
Theorem 1. (Due to Euclid, 300 BC.) The Pythagorean triple (a = u2 − v 2 , b = 2uv, c = u2 + v 2 ) is
primitive 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 − v 2 , b = 2uv, c = u2 + v 2 ) is primitive (that is, no
positive integer but 1 divides all three terms) and show that this implies GCD(u, v) = 1 and one of the
integers is even.
Part 2. Then we assume that GCD(u, v) = 1 and one of the integers u, v is even and show that this
implies 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 second
part 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 − v 2 , b =
2uv, c = u2 + v 2 . By the Fundamental Theorem of Arithmetic (due to the Greeks!) there is a prime p
dividing 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.
34
If p = 2 then since p divides a, it must divide u2 − v 2 . But if exactly one of u, v is even and the other
is odd then u2 − v 2 is odd. So if p = 2 divides a = u2 − v 2 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 + v 2 that p must be a
common 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 − v 2 to be
positive, we will assume that u > v. We also assume that u, v have no common factors and exactly one
is 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 − v 2 and b = 2uv or b = u2 − v 2 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 − kv 2 , b = 2kuv, c = ku2 + kv 2 ) 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 integers
The 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 z̄ = 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
p
||z|| = a2 + b2 .
Note that z z̄ = a2 + b2 so we may also write the length in the simple form
√
||z|| = z z̄.
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 be
factored 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 is
one of the oldest mathematical algorithms.
It is also one of the most applicable. The algorithm provides a systematic way to find the greatest
common divisor GCD of two integers and provide additional important information about the relationship
between the GCD and the two integers involved.
Modern technology uses a variety of algorithms based on modular arithmetic including the public-key
encryption RSA algorithm. Many of these algorithms in turn rely on the Euclidean Algorithm as an
algorithm 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
35
set of polynomials is similar. The concepts here may be generalized to any algebraic system which obeys
the 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 particular
interested in the largest such divisor, the greatest common divisor of both a and b. (The set of all
common 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 of
them. 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 this
case (since −1 and 1 divide every integer) we say that a and b “have no common divisors.” Equivalently
we 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, if
d is the greatest divisor of a and b and r is the remainder (guaranteed by the division algorithm) upon
division 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 common
divisor 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 pair
of 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 this
case q = 7 and the remainder is r = 23436 45805.) Now we want the GCD of b and r, which are smaller
numbers. Since the size of the positive integers is dropping, we may repeat this step, replacing a pair of
integers
ai and bi
by
bi 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 just
bk (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 write
a = a(1) + b(0), and b = a(0) + b(1), and, at each stage, write the new number in the form as + bt for
integers s and t. Since each step in our algorithm involves computing ai −qbi , we may think of this process
as 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 the
quotient in the current step of the division algorithm), bi = as + bt, s, and t. In the last two rows, we
compute the GCD of 12347 and 0, and so the GCD of the original two numbers must be 12347. The
process 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. Here
12347 = a(−398721) + b(3188882).
36
Table 1: Euclidean algorithm in tabular form
−q
-7
-1
-449
-42
-5
-4
-214
as + bt
a = 8435225645419
b = 1054696161403
1052352515598
2343645805
55549153
10581379
2642258
12347
0
s
1
0
1
-1
450
-18901
94955
-398721
85421249
t
0
1
-7
8
-3599
151166
-759429
3188882
-683180177
The equation
GCD(a, b) = as + bt
(9)
is called Bezout’s Identity. The Euclidean Algorithm not only finds the GCD of a and b but it also
finds 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 and
spent considerable energy trying to find solutions. These are now called the Three Classical Problems of
Antiquity. 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 ),
2.6.5
√
3
2, and π, by the Euclidean construction tools.
Pappus, Theon and the commentators
A number of later Greek teachers are noted for their commentaries on Euclid and the earlier Greek
mathematicians. During this period there was not a great deal of mathematical exploration; instead
writers synthesized the earlier works, revised them and annotated them. From these later writers we
learn the history of the Greek golden age.
Pappus of Alexandria (Greek, ca. 300 AD) wrote commentaries on Euclid’s work and on Ptolemy’s
work.
He wrote Mathematical Collection, ”a veritable mine of rich geometric nuggets” which has also provided us with much of our understanding of Greek mathematical history.
He discussed the method of Apollonius for writing and working with large numbers and commented
on 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 many
centuries afterwards. He also created an edition of Ptolemy’s works. Along with being a mathematician
and scholar in his own right, he is also remembered as the father of Hypatia.
37
Hypatia (Greek, died 415 AD) is the first recorded mathematician who was a woman. She was the
daughter of Theon of Alexandria. She studied mathematics, medicine, philosophy and wrote commentaries on Diophantus’ Arithmetica and Apollonius’ Conic Sections. She traveled and lectured, was praised
by Synesius of Cyrene (later bishop of Ptolemais). Hypatia never married. She was leader of neo-Platonic
school of philosophy and defended paganism against the new religion, Christianity, sweeping the Roman
Empire at the time. Because of her religious activity, the new patriarch, Cyril of Alexandria had her
attacked, tortured and killed. Her death is recognized as the end of the creative days of the University
of 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 and
indeed are very much a part of mathematics.)
3. (Euclid) The parallel postulate. (Some geometries – including the geometry of space-time in our
universe – 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.
38