Study Guide for Exam 2
Math 330: History of Mathematics
November 21, 2007.
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Introduction
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.)
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Mathematicians
Our starting point in this unit is the Academy at Athens. We also cover the mathematicians at
Alexandria, Archimedes of Syracus in Sicily, and early Chinese and Indian mathematicians.
1. 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.1 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.
2. 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 Alexandria. The most famous mathematics book of all times is Euclid’s Elements.
King 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.
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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.
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3. 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
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
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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.2 (iv) Proved
that the area of a circle was equal to the area of a certain triangle; describe this triangle.3
(v) discovered the area of parabolic segments in terms of 4/3 the area of a certain triangle,
(vi) Discovered and proved that the volume of a sphere was 2/3 that of the circumscribed
cylinder. What did he want on his tombstone? (vii) Discovered and proved that the surface
area of a sphere was 2/3 that of the circumscribed cylinder. (viii) Showed that his spiral
could be used to square the circle. (ix) 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?
4. 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.4 His estimate was respectively accurate. In fact, his estimate was
more accurate than that found in Ptolomy. Ptolomy’s small estimate led Columbus to believe
that a sea voyage to India from Spain was practical.
5. Apollonius of Perga. Active in Alexandria (and Pergamum). A few decades younger than
Archimedes. Famous for his work on conic sections.5
6. Aristarchus (of Samos). Greek astronomer who lived several centuries before the more
famous Ptolomy.6 He proposed the heliocentric theory but Ptolomy and the other Greek astronomers’ geocentric theory won out, at least until Copernicus. Why did his contemporaries
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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.
3
This fact seems to have been known, but probably not proved, before Archimedes.
4
In addition: 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”.
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Apollonius was also famous for his work in astronomy.
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He lived before the base 60 system and the degree measure of angles were introduced to Alexandria from
Mesopotamia.
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reject the heliocentric theory? Answer: apparent lack of motion on earth and the apparent
fixed positions of stars. He estimated dn /dm where dn is the distance to the sun, and dm is
the distance to the moon. How did he do this?
7. Ptolomy. Lived in Alexandria which was then part of the Roman Empire. He was an
astronomer, mathematician, and geographer. His most famous book is commonly called the
Almagest (original name: Syntaxis Mathematica). This book gives a mathematical treatment
of planetary motion, and the motion of the sun and the moon. His geocentric theory did
not use simple circular orbits. What did it use? answer: epicyles. His complicated system
represents the observations quite well. It needed to be complicated to be somewhat accurate
since it was based on a flawed premise: the geocentric model of the solar system.
What was Ptolomy’s theorem on cyclic quadrilaterals? how do you prove it? What type of
trigonometric function did he use? answer: the chord function. How is it related to sin θ?
answer: in the unit circle Chord(θ) = 2 sin(θ/2). Draw a picture illustrating this. Today we
use a unit circle. Ptolomy used a circle of radius of 60 for convenience. If you do not use the
unit circle, you multiply chords, and half chords, by the radius r. He had a table of chords
for every half a degree. (He also cataloged the position of over a 1000 stars. Ptolomy also
listed estimated latitudes and longtitudes of major known locations on the earth.)
8. Heron. Lived in Alexandria which was then part of the Roman Empire. He was mostly an
applied mathematician and engineer. His works describe very interesting mechanisms using
pneumatics, that is air pressure, steam, or water pressure, as well as other mechanisms. (He
developed war engines, fire engines, fancy puppets, fancy toys, wind organs, coin-operated
machines, sundials, fancy tricks with mirrors, etc.) What is Heron’s formula for a triangle?
How is it a limiting case of Brahmagupta’s formula?
9. Pappus. Lived in Alexandria which was then part of the Roman Empire.7 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.
10. Diophantus. Lived in Alexandria which was then part of the Roman Empire. First to use
algebraic symbolism instead of just words. His book, the Arithmetica was quite sophisticated,
certainly the most sophisticated algebra in the Greek speaking world. Solving equations for
him requires methods quite a bit different than the standard methods taught today in algebra.
This is mainly because he didn’t just want real number solutions, but wanted rational or
integer solutions. His work later inspired Fermat to invent the study of number theory. There
is a modern branch of number theory called Diophantine equations.
11. Hypatia and the commentators Hypatia was more than just a mathematician. She
was also a neo-Platonist philosopher and was known as a charismatic teacher. She wrote
commentary on earlier mathematicians. At this time, scholars were trying to keep alive the
earlier scientific and mathematical culture and so would write commentaries to make the
earlier works accessible, and to add their opinions to them. Hypatia was the last major
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Pappus lived around AD 300.
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intellectual figure of Alexandria and the Roman Empire. (Even after the western Roman
Empire fell, Byzantine mathematicians wrote commentaries to keep the tradition alive.)
12. Liu Hui and Nine Chapters on the Mathematical Art. Liu Hui wrote (in 263 AD)
commentary and additions to the Chinese classic: Nine Chapters on the Mathematical Arts.
He lived right after the Han dynasty split up into smaller states.
The Nine Chapters on the Mathematical Arts (Jiuzhang suanshu) was a practical textbook
of mathematics dating from the Han dynasty or earlier. It was used for many hundred
years train civil servants who had to take tests to get their positions. The Nine Chapters
is the most important early work of Chinese mathematics. Its influence was so strong that
it might be called the “Euclid of China”, but without proofs. It consists of word problems
(246 problems, surveying, trade, taxation, distribution problems, construction of canals, and
other problems you might see in as word problems in a textbook today.) Topics: Arithmetic
including rules for common fractions. Areas. Proportion, percentages and exchange rates.
Square and cube roots. Volumes (but not the volume of a sphere). Linear and quadratic
equations. Pythagorean theorem (which the Chinese called the Gougu rule), and examples
of Pythagorean triples. Similar triangles.
Some interesting features; use of matrices to solve systems of linear equations (up to six equations and six unknowns). The use of negative numbers, especially for matrix manipulations.
Correct rules for adding and subtracting negative numbers.
Liu Hui supplied explanations, some of which can be regarded as proofs, for much of the
Nine Chapters. He also pointed out that 3, the value of π used in the book, was not a
good approximation and gave an excellent approximation (3.14159) using n-gons (in fact a
3072-gon) in a manner similar to Archimedes.
13. Aryabhata (476 - about 550). Indian mathematician and astronomer. Like most Indian
mathematicians, his mathematics and trigonometry was written as part of an astronomy
book. He wrote a poem (the Aryabhatiya) in Sanskrit on astronomy containing a section
on mathematics (118 verses total, 33 devoted to mathematics. These verses contain 66 rules
of mathematics.) He used the idea of sine instead of the idea of chord. The Indian word
for sine means “half-chord”. He worked on Diophantine problems that were motivated by
cycles occurring in astronomy. For example, he used a form of the Euclidean Algorithm. He
developed a sine table (his circle had radius 3438 and circumference 21600 which is the same
as the earlier Greek astronomer Hipparchus who lived before Ptolomy). He also worked on
quadratic and indeterminate equations.
14. Brahmagupta (died 670). Indian mathematician and astronomer (head of an astronomical
observatory). What is Brahmagupta’s formula for cyclic quadrilaterals? He was an early
user of negative numbers and zero, and gave rules for them. In modern notation: a + 0 = a,
a − 0 = a, a0 = 0, 0 − 0 = 0, product or quotient of two negatives is a positive, product or
quotient of a positive with a negative is a negative, 0/a = 0. He was the first that I know of
to give rules for multiplying and dividing negative numbers. He even has rules that we don’t
quite agree with today: 0/0 = 0.
Interestingly enough, later mathematicians did not always accept negative numbers. For
example, Arabic mathematicians avoided negative numbers, and treated zero as just a place
holder.
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Brahmagupta used syncopated notations like Diophantus. His works were in Sanskrit verse.
He was interested in Diophantine problems. (His works discuss geometry, algebra, Pythagorean
triples, Diophantine equations including 8x2 + 1 = y 2 and 61x2 + 1 = y 2 with solution
x = 226153980 and
√ y = 1766319049, sine functions. Even has the formula for the first n
squares. He used 10 for π.)
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Historical People and Events
You will need to know the following, including the dates.
1. 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.
2. Alexandria. A city in Northern Egypt founded by Alexander the Great. It had a famous
Museum (a kind of university) and Library. The Ptolomies were a dynasty of kings that made
Alexandria it into the pre-eminant cosmopolitan Hellenistic city. It replaced Athens as the
center of culture and learning.
3. The Roman empire. The major cities included Rome, of course, but also Constantinople
(or Byzantine, today Istanbul) in Asia Minor (today Turkey). However, Alexandria remained
a center of mathematics and culture. In the western half of the Roman Empire, people spoke
Latin. (In fact, Spanish, French, Italian, Portuguese and even Romanian all descend directly
from Latin. These languages are called Romance Languages.) In the eastern half of the
Roman Empire, including Alexandria, people continued to write in Greek.
4. 476: The fall of the (western) Roman Empire. The eastern Roman Empire lasted for
almost 1000 years after this, and evolved into the Byzantine Empire. Rome was conquered
by Germanic tribes.8
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Summary of Dates
Know the following dates and their significance.
• 323 BC Death of Alexander the Great
• 300 BC Approximate date of the Elements of Euclid.
• 212 BC Death of Archimedes. Date that Syracuse was taken by the Romans
• 250 Approximate date of Liu Hui in China and Diophantus in Alexandria.
• 499 Date when Aryabhata composed his poem (the Aryabhatiya).
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England was also conquered by Germanic tribes, the Angles and the Saxons. The English language is a Germanic
language, but it has a huge amount of French and Latin influences as well. In fact, English draws from many languages
for its vocabulary. We have seen the importance of Greek words to our English vocabulary.
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Numerals, Letters and Words
1. Study the handout
public.csusm.edu/aitken html/m330/extranotes/GreekWordsPrinter.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.
2. Be able to write Chinese rod numerals. For the Chinese rod numerals see the Chinese numerals
entry at Mac-Tutor.
www-groups.dcs.st-and.ac.uk/∼history/HistTopics/Chinese numerals.html
3. Be able to write Roman numerals up to 6000. See page 68–69 of Math through the Ages.
4. Be able to write Mayan numbers up to 7199. Did the Mayans use zero?
5. How did Diophantus write x? How about x2 , x3 , x4 , x5 , x6 ?
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Mathematical Topics
1. 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. See my
essay on constructing numbers for more information.
2. Trisecting an Angle. One of the Three Problems of Antiquity. Describe Archimedes
solution using a ruler with two marks. Describe Archimedes solution using a spiral. See my
essay on constructing numbers for more information. (From the readings you learned about
Hippias’s solution using his quadratrix. You don’t need to know about Hippias’s solution in
the current test).
3. Squaring the Circle. One of the Three Problems of Antiquity, also known as the quadrature
of the circle problem. It is closely related to the problem of rectifying a circle. Know the
relationship between these two problems. See my essay on constructing numbers for more
information.
4. The Spiral of Archimedes. Be able to find the tangent line to the spiral. Explain how
this can be used to rectify a certain circle. Know that it was Archimedes who first developed
this idea. See my essay on constructing numbers for more information.
5. Modern Perspective on Constructing Numbers. The modern perspective on construction problems is via closure properties: closure under addition, multiplication, roots, etc. It
uses real numbers, a concept foreign to the ancient Greeks. (It also uses the theory of fields,
but that is too advanced for this class). I may ask some questions based on my essay on
constructing numbers.
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6. The Method of Exhaustion. In really 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.9
7. 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. 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).
8. 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.
9. 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.
10. Area of Parabolic Segments. Archimedes proved that a parabolic segment has area 4/3
times the triangle with the same base and height.
11. 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.
12. Conics. Who wrote the most famous book on conics? Apollonius. Did he invent conics?
No, they were studied earlier. What was the original motivation for inventing them? Finding
two mean proportionals to solve the Delian problem. How did Apollonius’ approach differ
from his predecessors? He used a double cone, while earlier only regular cones were used.
He considered different slicing planes. Some planes give ellipses or even circles, some give
parabolas, some give hyperbolas. Since Apollonius used the double cone, his hyperbolas have
two branches, while earlier hyperbolas just had one. Did Euclid study them? Yes, but not in
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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 by contradiction.
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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.10 (Not only do slices of cones give conic sections, but slices of
cylinders give ellipses as well.)
13. Systems of Linear Equations. All cultures could handle one equation and one unknown,
or two equations and two unknowns. The Chinese classic The Nine Chapters handled up to
six equations and six unknowns. It used matrix manipulations to solve such systems. These
manipulations are very similar to what we call “Gaussian Elimination”.
14. Cyclic Quadrilaterals. Can you state Ptolomy’s theorem? Can you prove it? Can you
state Brahmagupta’s formula? Can you relate it to Heron’s formula?
15. 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
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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. Later mathematicians improved on this bound. However, until
modern times, the method used was essentially Archimedes’ but with more and more sides
on the polygon. Liu Hui, independently of Archimedes, used to same idea to find an excellent
approximation (3.14159 using a 3072-gon, but only with inscribed polygons).11
16. Diophantine Problems. Write 16 as the sum of two squares. Solution (using modern
symbolic algebra): let x be one of the squares and let (2x − 4) be the other.
16 = x2 + (2x − 4)2 = x2 + 4x2 − 16x + 16.
So
0 = 5x2 − 16x = x(5x − 16).
Thus x = 16/5. The other number is 2x − 4 = 32/5 − 20/5 = 12/5. Know that the Indian
mathematicians solved Diophantine problems.
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Also conics solve many loci problems.
Chinese mathematicians were very good at approximating π, and later Arabic mathematicians were very good as
well. Zu Chongzhi approximated to seven decimal places with 355/11 in the year 480, a record that was not surpassed
for almost 1000 years. The Arab mathematician al-Kashi calculated π accurately to 14 digits in 1430, a record that
was not surpassed for over a century and a half. For practical need, one rarely needs more accuracy than Al-Khashi.
However, determined people kept going. Notable steps after Al-Khashi include 35 places in 1596, 100 places in 1706,
and 200 places in 1824. Today, computers can calculate billions of digits of π. In fact, calculating π used to be
a
√ method of testing computers. Brahmagupta had an interesting, but not very accurate, irrational approximation:
10. Earlier in India, Aryabhata gave a much better value: 31416/10000.
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