Reading Questions

Chapter 1

Section 1.1
- What is meant by tallying? What are some early examples? Where have you seen this in your mathematical education?
- Tell us about the Ishango bone. When? Where? Interesting details?
- Tell us about Peruvian Quipus. When? What? Base? Interesting details?
- Tell us about the mechanics of ancient Mayan numerical systems. When? Where? Base? Priest system versus common people system?
- What seems to be a primary motivator for the Mayan numerical system? It has the earliest known use of what symbol?
- Summarize Burton’s conjectures regarding the forces that triggered numerical representation.
Section 1.2
- Roughly when did ancient Egyptians have a “fully developed” number system and what does our author mean by “fully developed”?
- What is the diffence between the heiroglyphic numerical representation and the heiratic (or demotic) numerical representation of ancient Egyptians? What were the advantages and disadvantages?
- Elaborate on the book’s assertion that the Egyptian representation was not positional. What does this mean?
- Tell us about the Greek alphabetic numerical system. When? Where? Base?
- What would a multiplication table look like for the Greek alphabetic system?
- What is meant by a ciphered numerical system?
Section 1.3
- What is the definition of “Babylonian” our text uses?
- The book’s discussion of the Babylonian numerical system covers roughly what time period?
- (in person students) Walk to Rasmuson Library main floor and find the cunieform tablet on display near the south entrance. (Ask for the Ur III cuniform tablet, if needed.) How old? Where? Reaction?
- (on line students) Go to cdli.ucla.edu and in the quick search menu on the left enter the CDLI no. for the UAF tablet: P491488. How old? Where? Reaction?
- Describe the Babylonian positional system. Base? Symbols? Positional? Additive?
- Make an argument for and against the assertion that ancient Babylonians had a zero.
- What base 10 number is represented by the base 60 number: 3,0,18;40
- Write the decimal number 408.333333 in base 60 using the textbook’s notation.
- What are some of the advantages of a base 60 system?
- The book’s discussion of Chinese mathematics covers roughly what time period?
- Compare the writing materials of the peoples we have read about in Chapter 1. (Egyptians, Babylonians, Chinese, Mayans, Incas). What would be the advantages and disadvanges?

Chapter 2

Section 2.1
- Describe the Rhind Papyrus. Size? When was it written? Author? Where was it found?
- Go to the British Museum’s website and look at the Rhind Papyrus. What do you observe?
Section 2.2
- Describe the Egyptian method of multiplication.
- Describe the Egyptian method of division.
- Describe the Egyptian method of representing fractions.
- What was the purpose of the Unit Fraction Table?
Section 2.3
- Describe at least two of the four problems mentioned in this section.
- This section ends with a subsection called “Egyptian Mathematics as Applied Arithmetic.” Summarize Burton’s justification for this heading.

Section 2.4
- This section covers the geometry of ancient Egyptian during what period? (See the last paragraph.)
- What may be the origins of geometry?
- On page 54 is a modern formulation of the Egyptian formula for the area of a circle. Use the Egyptian formula and our modern formula to calculate the area of a circle with radius 5 feet. What is the difference in calculated area?
- Tell us about the Moscow papyrus. Age? Size? Approximate location?
- Go to the Mathematical Association of America webpage and view pictures of the Moscow papyrus. Thoughts?
- List the geometric formulas mentioned in this section. (eg the area of a circle)
- Which one in the list above is really impressive? How would you calculate the same quantity?
- Given the list above, what other formulas must Egyptian have known.
- Tell us some facts about the Great Pyramid of Giza (Gizeh).
- Summarize Burton’s conclusions about Egyptian geometry during this period. Do you agree?

Section 2.5
- As you read through this section, make a list of the mathematical topics/problems mentioned. What is on that list?
- What time period is meant by “Old Babylonian”?
- How did Babylonians multiply? Divide?
- How did Babylonians represent fractions?
- What is a table of reciprocals and how was it used?
- What common fallacy does Burton suggest lead to the Babylonians’ in-depth study of the relationship between perimeter and area?
- Why did Babylonians solve multiple types of quadratic equations?
- Describe other ways in which the Babylonians’ approach to quadratic equations is different from our. (There is no observation that is too simple or obvious here.)
- Burton asserts that the Babylonians “ outstripped “ the Egyptians. What is his argument? Do you agree?
- What we understand about the mathematics of any particular group of people is (obviously) dependent on the documentation that survives to be studied. Describe how the nature of the surviving documents impacts what we know about the mathematics of ancient Egyptians and Babylonians.

Section 2.6
- Was Pythagoras (585-500 bc) the first person to discover the Pythagorean Theorem?
- Go to the UCLA CDLI webpage and look at Plimpton 322. With what you have learned so far, can you read the right-most column?
- Describe what sort of table appears to be on Plimpton 322.
- What is meant by a primitive Pythagoran triple? Give an example of a Pythagorean triple that is primitive and one that is not.
- Give a short description of the two examples in the text where Babylonian mathematicians used Pythagorean triples.
- Explain how Egyptian rope-stretchers could have used rope to form right angles.
- On page 79, we see a scheme for estimating the square root of a^2 + b using the calculation a+b/2a. How good is this estimation? Calculate the error on the two examples in the text, namely: 18^2 + 21 and 10^2 + 5.

Chapter 3

Section 3.1
- How was the quantity sqrt(2) treated differently by Babylonian and Greek mathematicians?
- Who are the Phoenicians? Where? When?
- Describe some of the triggers for Greek expansion in the Mediterranean. What are the consequences for mathematics?
- Page 85 describes the beginning of the use of coin currency in Greek cities around 700 bc. How does this impact mathematics?
- Compare the distribution of learning among ancient Egyptians, Babylonians and Greeks.
- Compare the political structure of the ancient Egyptians, Babylonians and Greeks.
- Give us Thales of Miletus’ details. When? Where? Place Thales in comparison to the authors of the Rhind Papyrus and Plimpton 322.
- List at least 5 mathematical accomplishments attributed to Thales.
- By what mechanism do we know anything about the mathematics of Thales?
Section 3.2
- Give us Pythagoras’ details. When? Where?
- Tell us some nonmathematical things about Pythagoras and his followers.
- Page 94 of our text describes a text by Nicomachus of Geresa with (translated) title Introduction to Arithmetic. Look at copies of this at the MAA webpage or the University of Cambridge digital library. What are the dates of the physical copies shown? What does this have to do with Pythagoras?
- What is a triangular number? A square number?
- Give us Zeno’s details. When? Where?
- Describe the paradox of Achilles and the tortoise and explain why we no longer view the situation as paradoxical.

Section 3.3

Remind us of the approximate date and location of Pythagoras, Eucid, Theon, and Eudoxus.
Elaborate on the difference between the assertion that a mathematician or mathematical culture “knows the Pythagorean theorem” and the assertion that the person or culture “constructed Pythagorean triples.”
Several strategies for constructing Pythagorean triples are mentioned in this section. Compare the strategy attributed to Pythagoras (middle of page 107) and the one attributed to Euclid (bottom of page 5). What sort of triples do they construct?
What is meant by incomensurable quantities? Give a concrete example of two incomensurable quantities.
Elaborate on the difference between finding good approximations of sqrt(2) and knowing sqrt(2) is irrational.
Explain the reasoning behind Burton’s assertion on page 116 that for the next 2000 years, Geometry “served as the basis for almost all rigorous mathematical reasoning.”

How would you explain what an irrational number is to an elementary school child?
Can you find a combinatorial proof that 1+2+3+ … + n = (n+1) choose 2? Note that the RHS counts the number of 2-element subsets of a set with n+1 elements. (This question is strictly for fun.)

Section 3.4

Tell us about Hippocrates of Chios.
What are the Three Construction Problems of Antiquity?
Given a circle of radius r, what is the side length of the square of the same area?
Given a rectangle with side lengths a and b, what is the side of the square with the same area?
Given a cube with side length s, what is the side length of a cube with double the volume?
Given a square with side length s, what is the side of a square with double to area?
What is meant by “straight edge and compass construction”?
Roughly when was it determined that the Three Construction Problems did not have straight edge and compass constructions in the strict sense?

Section 3.5

Tell us about Hippias.
What was the original meaning of “sophist”?
What are some reasons the quadratrix is an interesting curve?
What is the origin of the word “Academy”?

Section 4.1 and 4.2

Tell us about Alexandria and the Museum.
Tell us about Euclid.
Look at the first 5 Postulates of the Elements. What do you observe?
Look at the first 5 Common Notions of the Elements. Do all of the items in the list seem reasonable? Mathematically rigorous?
What is the modern understanding of the axiomatic approach used by Euclid in the Elements?

Section 4.3

What is our modern definition of a prime number and what is Euclid’s? (To see Euclid’s definition go here )
What does Euclid mean by a number in definition 2?
Give an example of two relatively prime numbers and two relatively composite numbers.
Find the prime factorization of 128 and 24.
Find the greatest common divisor of 128 and 24.
Find the least common multiple of 128 and 24.
What is the Fundamental Theorem of Arithmetic?
You get a gold star if you recognize the modern terminology for the first two propositions in Book VII.

Section 4.4

Give the basics on Eratosthenes. (pronounciation, dates, location)
Tell us about Eratosthenes’ Geographica.
Tell us about Eratosthenes’ mesolabium. What was it supposed to do?
What is the sieve of Eratosthenes?
What is the modern estimation of the circumference of the earth in miles?
Give us the basics of Claudius Ptolemy. (pronounciation, dates, location)
The next section (4.5) is about Archimedes. Find his dates and location and explain why Ptolemy is in section 4.4.
What is the topic of Ptolemy’s Almagest? What is its significance.
What is the topic of Ptolemy’s Geographical Dictionary and its significance?

Section 4.5

Remind us of Archimedes dates and location.
Tell us about some of the non-mathematical things for which Archimedes is known
State Proposition 1 using modern notation. (page 199)
When was the first use of the symbol pi to represent the ratio of circumference of a circle to its diamter.
If a circle has a radius of r, what is the perimeter of an inscribed hexagon and how did you get that value?
If a circle has a radius of r, what is the perimenter of a circumscribed hexagon and how did you get that value?
On page 200, the author recounts several improvements to the approximation of pi. Pick two to describe to us.
What was the point of Archimedes treatise called The Sand-Reckoner?
What is an Archimedean spiral?
Tell us about Apollonius (dates and location)
Tell us about Apollonius’ Conics.

Sections 5.1 and 5.2

Remind us roughly when Archimedes, Eratosthenes, and Apollonius lived compared to Ptolemy and Diophantus.
Describe some of the changes that occured in the Mediterranean region over the time period between Archimedes, Eratosthenes, and Apollonius lived compared to Ptolemy and Diophantus.
What do we know about Diophantus’ life?
Describe Diophantus’ algebraic notation.
Describe what type of problems appear in Diophantus’ Arithmetica.
What sorts of answers where acceptable to Diophantus?
What is meant by an indeterminant equation or system of eequations?
Describe in modern notation the problem: Divide 16 into the sum of two squares. How do you think a modern Calculus student would answer this question?

Section 5.3

For each mathematician below, remind us of approximately when and where they lived and list two mathematical accomplishments.

Arybhata
Brahmagupta
Mahavira
Bhaskara
Chang Ch’iu-chien
Sun-Tsu
Describe the Cattle of the Sun problem. When did it originate and when was it completely solved?

Section 5.4

Tell us about Pappus of Alexandria.
Tell us about Hypatia.
Recount the deterioriation and ultimate loss of the Museum in Alexandria.
Describe the role that Proclus (410-485 ce) and Boethius (475-524 ce) play in the history of Greek mathematics.

Section 5.5

For each mathematician below, remind us of approximately when and where they lived and list two mathematical accomplishments.

Mohammed ibn Musa al-Khowarizmi
Abu Kamil
Thabit ibn Qurra
Omar Khayyam

Section 6.1

The Dark Ages in Europe refers to roughly what time period?
Name two Hindu mathematicians and two Islamic mathematicians who lived during the same time period as the Dark Ages in Europe.
Our text asserts that essetially all ancient Greek text had been translated into Arabic by what time?
Explain the role of the Church and Church schools in development of education during the Dark Ages.
Who is Charlemagne?
Who Alcuin of York?
Remind us what was Boethius’ role in the history of mathematics.
When did Islamic Empire first enter Spain?
Tell us about Gerard of Cremona and Adelard of Bath.
When did the printing press arrive in Europe?
Describe the role of the Spanish city of Toledo and Southern Italy including Sicily in the development of mathematics in Europe.

Section 6.2

Tell us about Leonardo of Pisa.
Why are our numerals called Hindu-Arabic numbers.
Describe the importance of liber Abaci or Book of Counting
Tell us about Jordanus de Nemore.

Section 7.1

Our text opens with a litany of grim events in Europe during the late Middle Ages (1300-1500). Tell us about three of them.
Give our text’s working definition of The Renaissance.
Explain how the fall of Constantinople to the Turks in 1453 influenced European intellectual culture.
The invention of printing with movable type “revolutionized the transmission and dissemination of ideas”. Elaborate on this assertion from our text by giving three specific examples.
Printing requires something on which to print. Tell us about the differences in papyrus, parchment, and paper as material on which to write or print.
Tell us about Johannes Muller (1436-1476) and his text On Triangles of All Kinds.
Tell us about Luca Pacioli (1445-1514) and his text Summa de Arithmetica Geometria Proportioni et Proportionalita
Describe the early European universities (c 1200)
At the end of this seciton, our text asserts that by what date were translations of most Greek mathematical texts available in Europe?

Section 7.2

List three specific areas of mathematical progress made by European mathematicians in the 1500’s.
Tell us about the contributions of Robert Recorde in the development of mathematics.
Remind us of Nicolas Copernicus’ contribution to astronomy.
Tell us about the life of Nicolo Tartaglia.
Tell us about the mathematical and scientific work of Tartaglia.
Tell us about the life of Girolamo Cardano.
Tell us about the mathematics of Cardano.

Section 8.1

Summarize in a single sentence the message of the first two paragraphs of Section 8.1.
For each mathematician or scientist in the list below (a) identify the title and date of publication they authored, (b) give a rough description of the topic, and (c) a rough location.
1. Francois Vieta
2. Robert Recorde
3. Girolamo Cardano
4. Raphael Bombelli
5. Simon Stevin
6. John Napier
7. Johannes Kepler
8. Tycho Brahe
9. Galileo Galilei
10. Nicolaus Copernicus

Section 8.2

Tell us about the life of Rene Descartes.
Tell us about Descartes’ Discours de la Methode and La Geometrie.
In what class or grade in school did you learn how to sketch the graph of something like y=3x^2-9 by plotting points?
Tell us something about the life of Pierre de Fermat.
Compare the work of the two artists Duccio di Buoninsegna here and Raphael here.
1. For Duccio, look specifically at Annunciation, Disputation with the Doctors, and Flight into Egypt
2. For Raphael, look specifically at School of Athens and Wedding of the Virgin

How would you describe the differences to someone who hadn’t seen the pictures?

Section 8.3

Tell us something about Isaac Newton’s life prior to his arrival at Cambridge in 1661.
Describe some of the mathematical books that influenced Newton prior to his development of Calculus.
Page 390 of our text describes Isaac Barrow’s method for finding tangents to curves. He finds the slope, a/e, by substituting x-e for x and y-a for y and then ignoring certain terms. Use his method on the function y = x^2 +10
Remind us why Newton spent 1665-1666 at his home in Woolsthorpe and not at Cambridge.
Tell us the three great discoveries made by Newton duing his years of seclusion in Woolsthorpe.
When was Newton’s On the Methods of Series and Fluxions published?

Section 8.4

Tell us something about Gottfried Wilhelm Leibniz’s life.
Describe some of the mathematical texts that influenced Leibniz prior to his development of Calculus.
Flip casusally through pages with headings Leibniz’s Creation of the Calculus and Newton’s Fluxional Calculus (roughly pages 413-419). How to they compare to modern notation and usage?
Compare the differences in how Newton and Leibniz were treated at the end of their respective lives.
Tell us about Maria Agnesi and Emilie du Chatelet, mentioned at the end of this section. Why are they here?

For Monday 10 April

Read Section 9.1

The first paragraph suggests that probability theory originated from what two roots?
What is an annuity?
What was the role of John Gaunt (1620-1674) in the development of probability theory? When was his tract Natural and Political Observations Made upon the Bills of Mortality published?
What was the role of Christiaan Huygens and his tract On Reasoning in Games of Chance in the development of probability theory?
The use of dice in games of chance is commonplace. What is one of the oldest examples of dice?
Remind us about the life of Pierre de Fermat and his role in the development of Calculus.
Tell us about the life of Blaise Pascal.
Tell us about Antoine Gombaud, Chevalier de Mere.

For Wednesday 12 April

Read Section 11.1 for the history

What curve known to ancient Greeks made some skeptical about the truth of Euclid’s 5th axiom.
Remind us of Euclid’s definition of parallel lines. What other properties of parallel lines (in Euclidean geometry) do we know (or automatically assume)?
What is Playfair’s axiom, who is John Playfair, and why does this axiom bear his name?
What does is mean mathematically to say ``Statement X is equivalent to Euclid’s 5th Axiom”? How does one typically prove that two statements are equivalent?
List two statements that are equivalent to Euclid’s 5th axiom that do not have the words “parallel” or “line” in the statment.
Tell us a little about Girolamo Saccheri, Johann Lambert, Adrien-Marie Legendre.
Tell us a little about the life of John Bolyai (1802-1860)
Tell us a little about the life of Nicolai Ivanovitch Lobachevsy (1793-1856)

For Friday 14 April

Give us some interesting facts about Leonhard Euler.
Give us some interesting facts about Carl Friedrich Gauss.