Study Guide for Exam 2

Study Guide for Exam 2
Math 330: History of Mathematics
November 7, 2005.
What follows is a list of topics that might be on the exam. Of course, the test will only contain
a selection of these: there is simply too much to put on one exam. (Thus, some of the topics will
appear on the final instead, and some topics will appear on this exam and the final).
History is full of names of people, place names, and dates. Don’t worry: the only personal
names, place names, or dates you need to memorize are given in this study guide. (Footnotes and
parenthetical remarks in this study guide are given for information and clarification. You do not
need to known them for the exam.)
Greek Mathematicians
(The following list skips Thales and Pythagoras who were covered on the previous test, and some
later Greek mathematicians who lived during Roman times).
1. Later Pythagoreans including the semi-legendary Hippasus. They included women as well as
men. The Pythagoreans were instrumental in the early development of Greek mathematics.
For example, legend has it that Hippasus discovered incommensurables (and something about
the dodecahedron). According to legend, much of Pythagorean knowledge was kept secret,
and Hippasus was expelled from the Pythagoreans for disclosing some of their secrets (or
taking credit for them). Also according to legend, Hippasus died at sea for this impiety.
2. Hippocrates of Chios. (Not the famous physician from Cos from about the same time and
who came up with the “Hippocratic oath”.) Active in Athens about the time of Socrates.
Author of the first surviving fragment of Greek mathematics. In it, he showed that a certain
lune is equal in area to a certain triangle. Thus he showed that some curved regions could be
squared. He was apparently trying to square the circle.
Hippocrates was the first to write an Elements of geometry (now lost). He was also interested
in duplicating the cube. He did not solve it completely, but he did reduce the problem to the
problem of two mean proportionals: find x and y with
a : x = x : y = y : 2a.
This idea was later used by Archytas, Menaechmus and Eratothenes to find a solution.
3. Hippias the sophist. Lived in Athens about the time of Socrates. Invented of the quadratrix
which can be used to to trisect angles (or multisect angles) or to square the circle. Hippias
himself probably used the quadratrix for trisecting an angle.
4. Theaetetus of Athens. One of the most important mathematicians of Plato’s Academy1 He
was a good friend of Plato. Thought to have come up with some of the ideas incorporated
into Euclid’s Elements, especially those related to incommensurable (irrational) lengths and
the Platonic solids.
5. Eudoxus. The most famous mathematician associated with Plato’s Academy. Thought
to have come up with some of the ideas incorporated into Euclid, especially the theory of
proportion. He also invented the method of exhaustion, and he used that method to prove
that the volume of a cone is one third the base area times the height. Likewise he proved that
the volume of the pyramid is one third the base area times the height. This last formula was
known to the Egyptians, but Eudoxus seems to be the first to have proved it.2 Archimedes
later applied the method of exhaustion to figure out the volume of a sphere, and to study
other geometric volumes, areas, and centers of gravities.
6. Menaechmus and Dinostratus. Brothers who were associated with Plato’s Academy.
Menaechmus3 is famous for his discovery of the conic sections (ellipses, parabolas, and hyperbolas). Menaechmus used conics to solve the Delian problem (duplicating the cube).
Dinostratus is given credit for showing that the quadratrix4 of Hippias can be used to square
the circle.
7. Archytas from Southern Italy. A later Pythagorean. Also a general and statesman. Lived
around the time of Plato; in fact, he was a friend of Plato, and so was possibly an important
link between the Pythagoreans and the mathematicians of Plato’s Academy.
He duplicated the cube by the method of two mean proportions. His method for constructing
two mean proportionals is based on a finding a point of intersection between a a certain cone,
cylinder, and torus (of inner radius zero), and using this point to form a “bouncing triangle”.
(Apparently, in his time, the Pythagoreans were no longer secretive about their knowledge.)
8. Euclid. The first major mathematician who lived in Alexandria (but he might have been
educated in Athens). He lived after Alexander the Great (about 300 B.C.), and in his
time, the main center of mathematics shifted from Athens to Alexandria5 . The most famous
mathematics book of all times is Euclid’s Elements.
9. Archimedes. Active in Syracuse in Sicily, before Sicily became part of the Roman Empire.
Syracuse was conquered by Rome in 212 BC on the day that Archimedes died. He was
Active after 400 BC, died young of battle wounds in 369 BC.
Eudoxus was also famous as an astronomer, and worked on the Delian problem. He developed an ingenious
theory of planetary motion based on a complicated arrangement of 27 nested rotating spheres. He also wrote a book
on geography.
Alexander the Great is said to have asked his tutor Menaechmus to show him an easy way to learn geometry. He
responded “O king, for travelling through the country there are private roads and royal roads, but in geometry there
is one road for all”.
In Latin, quadratrix means “square forming”.
Ptolomy, the successor to Alexander the Great in Egypt, is said to have asked Euclid to show him a shorter way
to learn geometry than the Elements. Euclid responded that there was no royal road to geometry.
amazingly talented in both pure mathematics and applied mathematics including engineering
and early physics. You should know the following things he did: (i) Established the basics of
hydrostatics and mechanics including the law of the lever and the law concerning the weight of
a submerged object. (ii) Found areas, volumes, and centers of gravity associated with curved
lines and surfaces and gave proofs using the method of exhaustion. (iii) Approximated π
as 22/7 using inscribed and circumscribed 96-gons (he actually showed 3 10
71 < π < 3 7 ) and
demonstrated that with enough skill and patience π could be approximated to arbitrarily high
precision by using inscribed and circumscribed n-gons with n sufficiently large.6 (iv) Proved
that the area of a circle was equal to the area of a certain triangle; describe this triangle.7
(v) Discovered and proved that the volume of a sphere was 2/3 that of the circumscribed
cylinder. What did he want on his tombstone? (vi) Discovered and proved that the surface
area of a sphere was 2/3 that of the circumscribed cylinder. (vii) Showed that his spiral could
be used to square the circle. (viii) Came up with a creative way to trisect an angle using a
straightedge with two markings.
Also be able to answer the following: How did he trisect an angle? According to Archimedes
hydrostatic principle, how much weight does a submerged object lose? What did he say when
he discovered this law? What did this have to do with a crown allegedly of gold? How did
Archimedes help in the defense of Syracuse? How did Archimedes die? What was inscribed
on his tomb? Derive the modern formula for the area of the circle and the volume of the
sphere using Archimedes result.
10. Eratosthenes. The head librarian at Alexandria. A contemporary of Archimedes (in fact
some of Archimedes’ work was addressed to Eratostheses). He did work in geography, astronomy, poetry, history, and other fields as well as mathematics. His nickname was beta because
he was the second best at everything he did. (He was also called pentathlos because he did
not excel at any one thing, but was more like a well-rounded athelete.)
He figured out a way to estimate the circumference of the earth by comparing the sun’s
shadow at two different locations. His estimate was fairly accurate. (He developed the Sieve
of Eratosthenes: a way to find prime numbers.) He also invented a device that allowed him
to solve the problem of finding two mean proportionals by forming a “bouncing triangle”.
11. Apollonius of Perga. Active in Alexandria (and Pergamum). A few decades younger than
Archimedes. Famous for his work on conic sections.8
12. Pappus. Lived in Alexandria which was then part of the Roman Empire.9 The last great
Greek geometer. Wrote excellent commentary on earlier geometry with some original results
of his own. His commentaries are often the only source we have for information on earlier
mathematicians. (His main work was the Synagoge or Collection). By his time, advanced
Geometry had almost died out. He tried to revive interest in Geometry.
Archimedes and other ancient mathematicians did not use the letter π, instead they talked directly about the
ratio of the circumference to the diameter or they referred to other related ratios.
This fact seems to have been known, but probably not proved, before Archimedes.
Apollonius was also famous for his work in astronomy.
Pappus lived around AD 300.
Historical People and Events
You will need to know the following, including the dates.
1. Socrates. Philosopher and teacher of Plato. Active in Athens after the Persian-Greek war.
He died in 399 BC.
2. Plato. Follower of Socrates. Founded the Academy in Athens. Not a great mathematician
himself, but advocated the philosophic value of mathematical study. According to legend he
had the following motto written above his Academy: let no one ignorant of geometry enter.
3. Aristotle. Taught by Plato, and lived in Athens. Interested in physics, logic, biology,
mathematics, and just about everything else. One of the tutors of Alexander the Great.
4. Alexander the Great from Macedon. Conquered Greece, the Persian Empire, Egypt, and
more. He died in 323 BC. His death is the beginning of the Hellenistic Period. During this
period the center of culture was the city of Alexandria in Egypt. Hellenistic culture was really
a mix of Greek, Persian, Egyptian, Babylonian, Hebrew, and other cultures. Aristotle was
Alexander’s personal tutor. Alexander died young in 323 BC. His generals divided his empire
into several smaller kingdoms. One of his generals was King Ptolomy of Egypt.
5. Alexandria. A city in Northern Egypt founded by Alexander the Great. It had a famous
Museum (a kind of university) and Library. The Ptolomies made it into the pre-eminant
cosmopolitan hellenistic city. It replaced Athens as the center of culture and learning.
Summary of Dates
Know the following dates and their significance.
• 399 BC (Death of Socrates)
• 323 BC (Death of Alexander the Great)
• 212 BC (Death of Archimedes. Date that Syracuse was taken by the Romans)
Study the website html/m330/extranotes/Greekwords.html
I will give you the Greek word, and I will ask you for its meaning in Ancient Greek. Tell me both
the mathematical and the non-mathematical meanings, and other important information.
Mathematical Topics
1. Pythagorean Number Theory. Illustrate with a picture the fact that two times the nth
trianglular number Tn is an n by n + 1 rectangular number (called an “oblong number”).
Conclude that Tn = n(n + 1)/2. (Hint: just illustrate the case n = 3 or n = 4. The general
case obviously works in the same way.)
2. Pythagorean Number Theory. In a similar way, illustrate with a picture the fact that
the sum of two consecutive triangular numbers is a square number.
3. Pythagorean Number Theory. Illustrate with a picture how the nth square number is
the sum of the first n odd numbers. Hint: successive gnomons. (Similarly, the oblong number
n(n + 1) is the sum of the first n even numbers.)
4. Pythagorean Number Theory. When you write the odd numbers in a triangular pattern,
what is the sum of the nth row? Answer: the nth cube.10 Give some examples. 7+9+11 = 27
or 13 + 15 + 17 + 19 = 64 = 43 .
5. Pythagorean Number Theory. Why are 6 and 28 perfect? Are there any known odd
perfect numbers? Answer: no.
6. The Pythagorean Theorem. Give a proof of the Pythagorean Theorem involving similar
triangles. Also draw a picture of the Pythagorean Theorem involving three squares attached to
the triangle. State the theorem in terms of adding areas of squares. What is the generalization
of the Pythagorean theorem used by Hippocrates? Who knew the Pythagorean theorem (but
probably not the proof) before Pythagoras: the Babylonians or the Egyptians? Answer: the
7. Pythagorean Triples. What is a Pythagorean triple? Examples include 3, 4, 5 and 5, 12, 13.
These are misnamed since the Babylonians produced them over a thousand years before
Pythagoras. Sketch a 4 by 4 square with the gnomon of size 9 added to illustrate 32 + 42 = 52 .
This gnomon method can be used to produce arbitrarily large Pythagorean triples.11
8. Ruler and Compass Constructions. Know some of the ruler and compass constructions
presented in class. In particular, study 1-9, 17, and 20 in the following link: html/m330/extranotes/compass.html
Prove that the geometric mean construction actually works. Be able to construct the golden
section and the Pentagon.
9. Golden Ratio. A key to constructing the pentagon is to construct two segments whose ratio
is the golden ratio (the Greeks used the term extreme and mean ration: the name golden ratio
This fact can be used to show that 13 + 23 + . . . + n3 = n(n + 1)/2 which is a beautiful formula relating
triangles, squares, and cubes.
In general, take an odd square number g = a2 , let b = (g − 1)/2. Then when you add the gnomon of size g to
the size b square you produce a size b + 1 square. Thus a2 + b2 = (b + 1)2 . For example, if g = 25 then a = 5 and
b = (25 − 1)/2 = 12. The larger square that includes the gnomon has size 13, so 52 + 122 = 132 . The Babylonians
must have had another method since they produced triples that do not arise by gnomons. We discussed one possible
Babylonian method in class.
is modern.) Let x be a length longer than y. We say that x and y are in the golden ratio if
x : y = y : (x − y). Assume for convenience that y = 1, and show that if x : y is in the golden
ratio then
x : 1 = 1 : (x − 1). Derive from this a quadratic equation satisfied by x. Show that
1+ 5
x = 2 is the positive root of the equation.
10. Three Problems of Antiquity. What are the three problems? We know today (as a result
of field theory, Galois theory, and the transcendence of π) that these problems cannot be
solved with an unmarked ruler and compass. However, if one uses other curves and tools,
then one can solve them. Some of the best Greek mathematics resulted from their attempts
to solve these problems. We have made these problems a major theme of our class.
11. The Delian Problem or Duplicating the Cube. One of the Three Problems of Antiquity.
Hippocrates reduced the problem to that of finding two mean proportionals: find segments
x, y such that a : x = x : y = y : 2a. Explain why x3 = 2a2 in this case. The more general
problem of two mean proportionals is to find x, y such that a : x = x : y = y : b. Here
a, b are two given segments. Show how this problem leads to the intersection of two conics.
This was done by Menaechmus. Describe briefly the solution of Eratosthenes and Archytas
with “bouncing triangles”. Eratosthenes produced his “bouncing triangle” with a mechanical
sliding device, while Archytas used the intersection of a cylinder, cone, and torus (of inner
radius zero).
12. Trisecting an Angle. One of the Three Problems of Antiquity. Describe Archimedes
solution using a ruler with two marks. Explain why it works. Describe Hippias’s solution
using his quadratrix (but don’t prove that it works).
13. The Lune of Hippocrates. Explain how the lune is defined on a 45-45-90 right triangle.
Sketch it. Why is it called a lune? Prove that the area of the lune is one half the area of the
triangle. (This implies that the lune can be squared. This was the first curved region that we
know of that was successfully squared. It can be thought of as a step on the way to squaring
the circle.)
14. Squaring the Circle. One of the Three Problems of Antiquity. This reduces as follows.
Theorem If you can find two segments AB and CD such that AB : CD = π : 1 or AB : CD =
2π : 1 or AB : CD = π/2 : 1, etc. then you can square the circle. This is a consequence of
the theorem on the area of the circle proved by Archimedes (but probably discovered much
Show how Dinostratus constructed two segments with ration π/2 : 1 using the quadratrix.
Show how Archimedes constructed two segments with ratio π/2 : 1 using his spiral.
In polar coordinates, the spiral of Archimedes is r = aθ. The tangent line through the point
P with θ = π/2 is not to hard to find with modern methods, and its intersection with the
x-axis at Q is easily found. What can you say about QO : P O where O is the origin?
In mixed coordinates, the quadratrix is y = π2 θ. So in polar coordinates it is r sin θ =
Solve for r. What is the limit of r as θ → 0? How does that help square the circle?
π θ.
15. The Method of Exhaustion. In elementary geometry, two areas or volumes are shown to
be equal by cutting up the first region and reassembling it to be equal to the second region.
However, curved areas and general solids cannot always be studied in this way. The method
of exhaustion is a more sophisticated method based on lower and upper approximations of
the region, and using these approximations to show that the area or volume cannot be bigger,
nor smaller, than the proposed answer. Eudoxus initiated this method, and it was used by
Euclid. Archimedes also used this method.12
16. The Elements. The most famous mathematics book of all time. Written by Euclid (around
300 BC). About half of it is devoted to plane geometry including the Pythagorean theorem
and the construction of the pentagon. Some of it deals with algebra, in a geometric style
(including methods for solving quadratic equations). Some of it deals with number theory
(where it proves that there are an infinite number of primes, gives the Euclidean Algorithm,
and discusses even perfect numbers). Some of it deals with solid geometry. It gives Eudoxus’
theory of proportion and uses his method of exhaustion. It gives Theaetetus’ theory of
incommensurability. The Elements culminate in the constructions of the Platonic solids. Are
conics covered in the Elements? No, but Euclid wrote another book about conics (which is
now lost).
17. Volumes of Cones and Pyramids. The volume of such solids is the height times 1/3 the
area of the base. Some civilizations knew this, at least for pyramids, before Eudoxus’ time
(but some got it wrong). However, Eudoxus is the first to prove it (using the method of
exhaustion). A proof can be found in Euclid’s Elements.
18. Area of Circles. Archimedes wrote a treatise on the circle. In it he (i) approximated π as
22/7 by approximating a circle with inscribed and circumscribed 96-gons, and (ii) used the
method of exhaustion to prove that the area of a circle is equal to a triangle whose base is the
circumference of the circle, and whose height is the radius of the circle. Derive the modern
formula A = πr2 from this theorem. Hint: Use the fact that c = 2πr.
19. Approximations of π. Archimedes gave an approximation for π by using inscribed and
circumscribed 96-gons. This gives the famous 22/7 approximation for π. (His actual estimate
was 3 71
< π < 3 17 ). Even more important than the actual numbers, he showed a method,
which with enough patience and with a large enough n-gon, gives approximations to any
desired degree of accuracy.
20. Volume and Surface Areas of Spheres. Archimedes showed that the volume of a sphere
is 2/3 times the volume of the circumscribed cylinder. Show that this leads to the modern
formula V = 34 πr3 . Hint: even before Archimedes, it was known that the volume of the
cylinder was given by the base area times the height. He also showed that the surface area
of a sphere was 2/3 time the surface area of the circumscribed cylinder.
21. Conics. How were they defined before Apollonius? Answer: In terms of slicing a cone with
a plane perpendicular to a line going from the vertex to the base. You get parabolas if the
vertex angle is right, ellipses if the angle is acute, and hyperbolas if the angle is obtuse.
How did Apollonius define them? In terms of a fixed double cone, but with different slicing
planes. Some planes give ellipses or even circles, some give parabolas, some give hyperbolas.
What is one difference between Apollonius and earlier authors? In Apollonius the hyperbola
It is called “exhaustion”, not (only?) because it is tiring, but because the inscribed approximations take away or
“exhaust” as much of the areas or volume as you need for the proof.
has two branches, while earlier it just had one. Who wrote the most famous book about
them? Apollonius. Who invented them? Menaechmus. What was the original motivation
for inventing them? Finding two mean proportionals to solve the Delian problem. Did
Euclid study them? Yes, but not in the Elements. What further applications do they have?
1. perspective: circles in perspective appear to be ellipses, or even hyperbolas or parabolas,
2. paraboloids can be used to focus sound and light waves, 3. Kepler later discovered that
the paths of planets and comets are approximately conic sections. 4. Galileo later discovered
that the paths of projectiles are approximated by a parabola. (Not only do slices of cones
give conic sections, but slices of cylinders give ellipses as well.)