~ 2. Who was Euclid? How might he have been influenced by the Library of Alexandria?

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1 CD Reading Guide Sections 5.1 and What was the Museum of Alexandria? ~ 2. Who was Euclid? How might he have been influenced by the Library of Alexandria? --)~ 3. What are the Elements of Euclid? What impact did they have on the history of mathematics? 4. How did Euclid define right angles and perpendicular lines? 5. How did Euclid define a circle? What was the diameter of a circle? 6. How did Euclid define parallel lines? 7. What is the purpose of Euclid's postulates and common notions? What is the difference between the two? 8. In what sense do Euclid's first three postulates limit his geometry? 9. What is Euclid's fifth postulate? Why was it controversial? 10. How would we express the common notions in modern terms?

2 0 Discussion Questions Sections 5.1 and What familiar result is expressed by Proposition 5.27 Which of the common notions does the proof rely on? 2. Draw a diagram to illustrate what Proposition 5.3 says. How does it differ from Proposition 5.47 ~ 3. Why is Proposition 5.4 true? --')"""\ 4. Do section 5.2 exercises r 2, 5, 6, and 7.

3 Reading Guide Sections 5.3 and What does Proposition 5.10 say? What should the diameter of a parallelogram be? 2. How can Proposition 5.12 be used to compute the area of a parallelogram? 3. What additional result do you need before Proposition 5.13 can be used to compute the area of a triangle? 4. What are the complements about the diameter of a parallelogram? What does Euclid claim to be true about them? 5. What is the gnomon of a parallelogram? What are the parallelograms on the diameter? 6. Translate Propositions 5.17 and 5.18 into modern algebraic identities. -) 7. Draw figures and use modern notation to illustrate what Propositions 5.21 and 5.22 say. One of these contains an error. Which is it?

4 Discussion Questions Sections 5.3 and Rewrite Proposition 5.7 in modern terms. Why is it true? Try to answer in a Euclidean style. 2. Show how Proposition 5. 7 is used to prove Proposition Finish the proof of Proposition 5.9. ~ 4. Draw a sequence of figures to illustrate Euclid's proof of the Pythagorean Theorem. 5. Do section 5.3 exercise Translate Propositions 5.19 and 5.20 into modern algebraic identities. 7. Do section 5.4 exercise lab in base 10 (rather than the suggested base 60) and carry your results to the second decimal place. 8. Do section 5.4 exercise Do section 5.4 exercises 6 and 7. Which algebraic identities are being proven geometrically? --)10. Do section 5.4 exercise 10.

5 Reading Guide Sections 5.5 and ~'"'. In the third book of the Elements, Euclid considers the intersection of two lines and the segments cut off on them by a circle. Describe the different possible cases. Which of these cases are addressed by Propositions 5.26 and 5.27? 2. What are the six types of angles that Euclid considered? 3. Rewrite Proposition 5.28 using modern language. 4. What is the difference between the antecedent and the consequent of a ratio? 5. Describe how to alternate, separate, combine, convert, and compound ratios. 6. Rewrite Propositions 5.44 and 5.45 in modern terms.

6 Discussion Questions Sections 5.5 and What does Proposition 5.24 say? Why is it true? ~ 2. Do section 5.5 exercises 5, 6, 7, 9, and!lo. 3. What do Propositions 5.36 and 5.37 say in modern language? How can the former be used the prove the latter? 4. Do section 5.6 exercises 7, 8, and Given a number (length) L, show how to compute (construct) L 2 and JI geometrically.

7 Reading Guide Sections 5.9 and 5.10 ~ 1. What is the axiom of Archimedes? Why does it require that the quantity subtracted be more than half of the given, or remaining, magnitude at each step of the process? ~ 2. How would we state Proposition 5.55 today? 3. What is the difference between a pyramid and a prism? 4. What is Book XIII of the Elements about?

8 Discussion Questions Sections 5.9 and 5.10 ~ 1. Why is Proposition 5.55 true? ~ 2. How does Proposition 5.55 suggest the existence of 1r? How can it be translated into an area formula for circles? 3. How can our modern formula for the volume of a pyramid be derived from Proposition 5.59?

9 Reading Guide and Discussion Questions Chapter 6, Part 1 The prompts marked with an asterisk (*) below are reading guide questions. *l. What was Archimedes' formula for the area of a circle? How accurate is it? 2. How did Archimedes prove his formula for the area of a circle? Provide diagrams to illustrate the main points of the proof. *3. What makes Proposition 6.2 "flawed"? How inaccurate is it? ~ *4. ~ *7. What claim does Proposition 6.3 make about 1r? The proof of Proposition 6.3 involves the construction of a two 96-sided regular polygons. How are they used? Why 96 sides? Do section 6.1 exercise 3 using Archimedes' formula for the area of a circle together with the result that circles are to each other as the squares on their diameters. How did Archimedes define spheres, cones, and cylinders? 8. Show how Archimedes formula for the surface area of a cone of height h and base radius r translates into the modern formula 1rrJr 2 + h Translate Archimedes' formula for the area of a frustrum of a cone into a modern formula. How accurate is the formula? *10. Show how Archimedes' formulas for the surface area and volume of a sphere translate into modern formulas. *11. Translate the Theorem of Archimedes into modern notation and verify that it is true. 12. How do Archimedes formulas for areas and volumes differ from Euclid's?

10 Reading Guide and Discussion Questions Chapter 6, Part 2 The prompts marked with an asterisk (*) below are reading quide questions. *l. What is noteworthy about Archimedes' Q11,adrat11,re of the Parabola? *2. How does Archimedes define the diameter and vertex of a parabola? What is a segment of a parabola? *3. Draw pictures to describe what Propositions 6.13, 6.14, and 6.15 say. Label the ordinates of the parabola on your diagram for Proposition How does Proposition 6.15 imply our modern formula for a parabola? 5. Proposition 6.20 should say that A+ B + C Z +! Z =! A. Use modern notation and algebra to show why this is true. Why did Archimedes need this result? --~ *6. Create a diagram to illustrate Proposition Use calculus to verify that it is true for the segment of the parabola y = l - x 2 cut off by the x-axis. *7. What was "the method" Archimedes used to find results to prove? *8. Who was Apollonius? What kinds of problems did he solve in Tangencies? What was important about Plane Loci? *9. How did Apollonius' definition of a cone differ from that of his predecessors?

11 Reading Guide and Discussion Questions Chapter 7, Part 1 The prompts marked with an asterisk (*) below are reading guide questions. ~ * 1. Who was Ptolemy? For what body of work is he best known? *2. Write Heron's formula using a, b, and c for the sides of a triangle, and s for the quantity (a+b+c)/2. Read the paragraphs in sections 7.1, 7.2, and 7.3 that provide historical context, and then for each of the propositions listed below, copy the statement of the result, and include one or more diagrams or modern formulas that illustrate what it says. *3. Proposition 7.1 ~ *4. Propisition 7.2 ~ *5. Proposition 7.3 ~ *G. Proposition 7.4 (see section 7.2 exercise 6 to understand how to find AC) *7. Proposition 7.5 _-=) 8. How do each of these propositions help you to fill in Ptolemy's table of chords? Which chords still cannot be computed? 9. How did Ptolemy compute the chord of 1? The questions below are based on a video what will be shown in class. If you must miss class, or don't get all of the answers while watching the video, you can find what you need in the written transcript How is trigonometry typically used today? What was it originally used for? 11. How did Ptolemy and other early astronomers picture the solar system differently than we do today? 12. Which features of the night sky allowed early astronomers to predict the seasons and other irn portant events? 13. Why was it important to be able to predict the seasons? 14. What was the ecliptic? How was it determined? 15. What is Rn armillary sphere? 16. What W<'l'C the "stars that move"? How did they appear to move? Why was their motion rousidered important?

12 Reading Guide and Discussion Questions Chapter 7, Part 2 The prompts marked with an asterisk (*) below are reading guide questions. ~ *1. *2. *3. ~ *4. Who was Diophantus? How old was he when he died? What is syncopated notation? What are the species of an unknown quantity? Create a table that compares our modern algebraic notation with of Diophantus and the Diophantine-style notation suggested by the author of our textbook. Translate problems 7.3 and 7.4 and their solutions into modern language and notation. ~ *5. What is a Diophantine equation? *6. Translate problem 7.5 into modern language and notation. *7. Translate problem 7.6 into modern language and notation. *8. What is the Pell equation? What did Diophantus prove about it? 9. What did Diophantus consider "absurd" about the solution to problem 7.9? --=, 10. Do section 7.4 exercises 2, 3ab, and,t 1'

13 Reading Guide and Discussion Questions Chapter 8, Part 2 The prompts marked with an asterisk (*) below are reading guide questions. ~ * 1. What "key factor" distinguished Indian arithmetic from that of the Greeks? ~ *2. Who was Brahmagupta? In The Opening of the Universe, he gave the correct rules for adding, subtracting, and multiplying by zero. Why were such rules necessary ( as opposed to obvious)? *3. What was the significance of Brahmagupta's "fortunes" and "debts"? *4. What is commercial arithmetic? How did it play a role in the development of mathematics in India? *5. What was the rule of three? 6. Use modern algebraic notation to show why the solution to Problem 8.8 makes sense. ~ *7. Which text does the author of our textbook consider to be the "pinnacle" of medieval Indian mathematics? Who wrote it? What kind of mathematics does it contain? ~ 8. To what quadratic equation does Problem 8.10 correspond? Use algebra to solve this equation. Why did Bhaskara reject one of the solutions? ~9. Show how Bhaskara would have computed multiple solutions to the equation ax 2 + b = y 2 starting with a = 11 and b = 5. How would he have transformed them into solutions to the equation 1 lx = y 2? Show how your calculations can be used to approximate vrr. 10. What is a combinatorial statement? Where does the earliest such statement come from? How did the statement's author get the number 63? How can this number be determined from the "arithmetic triangle"? *11. How can the entries of the arithmetic triangle be determined individually without constructing the whole table? Who gave the correct formula for doing so? *12. What is the significance of the jiva?

14 Reading Guide and Discussion Questions Chapter 9, Part 1 The prompts marked with an asterisk (*) below are reading guide questions. * 1. Where does the word "zero" come from? *2. Why do we write our numbers with the units position on the right and the highest power of ten on the left? *3. What is the grating method of multiplication? 4. Why does the grating method work? ~ *5. What was the "House of Wisdom"? Which notable mathematician used to work there? ~ *6. Where does the word "algorithm" come from? Where does the word "algebra" come from? *7. What distinguishes the style of al Khwarizmi's work from that of Diophantus or the Babylonians? 8. In Completion and Restoration, al Khwarizmi considers three types of "simple" equations and three types of "compound" equations. How would we describe these types using modern notation? 9. Why did al Khwarizmi work with different types of equations rather than writing all equations using the same form (such as ax 2 +bx+ c = O)? ~ 10. Use al Khwarizmi's method to solve the problem one square and five roots amount to twenty-four dirhems. To what modern equation does this problem correspond? Draw a picture that shows geometrically why al Khwarizmi's solution works. 11. Use modern algebraic notation to show why the solution to Problem 9.2 works. 12. Show geometrically why al Khwarizmi's procedure for computing 10 minus 1 times 10 minus 1 requires the product of two negatives to be positive. ~ 13. Do section 9.2 exercises 3 and 4.

15 Reading Guide and Discussion Questions Chapter 10, Part 1 The prompts marked with an asterisk (*) below are reading guide questions. * 1. During the medieval period, progress in mathematics was made primarily by mathematicians in India and the Islamic world. The small advances made in Europe came primarily from religious scholars. Why? *2. What does the author of our textbook consider to be "one of the most important events of the medieval period"? *3. Why is the twelfth century considered to be the "century of translation" in the history of mathematics? *4. Where does our word "sine" come from? ~ *5. Between roughly 1100 and 1300 the "key elements of modern mathematics" entered Europe. What where these key elements and were did they come from? ~ *6. Who was Leonardo of Pisa? What is the name of his most famous work? What was its purpose? *7. How did Fibonacci represent fractions? Why was this system useful? 8. Do section 10.2 exercise 2ab using Fibonacci's procedure for division. Why does this procedure work? *9. What is the Fibonacci sequence? Where does it come from? *10. --!) 11. ~ 12. ~ How did mathematicians of the medieval period deal with negative numbers? How did their treatment of them differ from that of the Greeks? What does Proposition 10.1 say? Why is it true? What does Proposition 10.3 say? Why is it true? Give an example to illustrate what Proposition 10.4 says. Why does it work? What are congruous numbers? \ivhat must be true about them?

16 @ Reading Guide and Discussion Questions Chapter 10, Part 2 The prompts marked with an asterisk ( *) below are reading guide questions. *1. Who wrote Regarding the Given Numbers? What kinds of mathematical results does it contain? 2. Translate the statement and the proof of Proposition into modern language and notation *5. ~ Translate the statement and the proof of Proposition 10.8 into modern language and notation. Translate the statement and the proof of Proposition 10.9 into modern language and notation. Who was Ben Gershon? What did he contribute to the history of mathematics? Translate Propositions and into modern language and notation. Propositions and together provide an early example of proof by induction. How? ~ 8. Do section 10.3 exercise 1. *9. Who was Nicholas Oresme? What was his "great contribution"? 10. What does the author of our textbook mean when he says that "Oresme skipped Cartesian graphs and went straight to the definite integral"? 11. What is Merton's rule? How can Oresme's "figuration of qualities" be used to illustrate it? 12. How did Oresme show that it is possible for an object to increase to an infinite velocity, yet travel a finite distance? 13. How could he show that it is possible for an object to travel for an infinite amount of time, yet to travel a finite distance? 14. Do section 10.4 exercises lab and 2ab.

17 @ Reading Guide and Discussion Questions Chapter 11, Part 1 The prompts marked with an asterisk (*) below are reading guide questions. *1. Which book did for trigonometry what Euclid's Elements did for geometry? Who wrote it? Most of this book deals with "solving" a triangle. What does it mean to "solve" a triangle? *2. Who was Tycho Brahe? How did he use the results in Regiomantus' On Triangles? Why did this result in "one of the first significant cracks in the Ptolemaic system of the universe"? 3. Do section 11.1 exercise 1 using Rule How can you check that your answer is correct? *4. How did the printing press create "the perfect environment for an explosive growth in science and mathematics"? *5. Who was Rheticus? What did he contribute to the history of mathematics? 6. What does Rule 11.2 say, and why does it work? 7. Do section 11.2 exercise 2a. How would we state this problem using modern algebraic notation? How would we solve it? What is the advantage of Chuquet's method? *8. When and by whom was our familiar algebraic and arithmetic notation invented? *9. Who was Tartaglia? For what is he known? ~ *10. ~ 11. *12. ~ *13. Why is Tartaglia's procedure of solving cubic equations known as "Cardano's method"? What formula for x (in terms of p and q) corresponds to Cardano's rule for x 3 +px = q? What did Cardano consider to be "impossible roots"? How did he deal with them? According to the author of our textbook, "the practical value of solutions to the cubic and biquadratic equation was nonexistent." Why? If there was no practical value in the solutions, then why did people seek (and fight over) them? Do section exercises 1 a and 6a. i

18 @ Reading Guide and Discussion Questions Chapter 12, Part 1 The prompts marked with an asterisk (*) below are reading guide questions. *l. What is analytic geometry? By whom was it invented? *2. Who began our modern use of letters near the end of the alphabet (like x, y, z) to represent unknowns, and letters near the front (like a, b, c) to represent parameters? Who was he? For what else is he known? 3. How did Descartes solve geometric construction problems? Show how he solved Problem ~ *4. Who was Pierre de Fermat? Why was he working with the Plane Loci of Apollonius? What "fundamental principle" of analytic geometry did he observe?...--:, 5. How do Propositions 12.1 and 12.2 correspond to locus problems? Where are the familiar x- and y-coordinates in these propositions? *6. Why first observed the Fundamental Theorem of Algebra? What does this result say in modern terms? *7. Who first referred to imaginary numbers? What made them "imaginary"? *8. Who was Marin Mersenne? *9. Given an example to illustrate the statement of Fermat's Lesser Theorem. ~ *10. What is a Fermat number? Why was Fermat interested in numbers of this form? What is the difference between a Fermat prime and a Mersenne prime? *11. What was Fermat's Last Theorem? Where did it come from? ~ 12. Do section 12.2 exercises 3 and L. ~ *13. Why did Galileo Galilei decide it was impossible to apply equalities and inequalities to infinite quantities? *14. What observations about ellipses were contained in Johann Kepler's New Stereometry? Why did Kepler need these results? *15. Who was Buenaventura Cavalieri? 16. In modern terms, we would say Cavalieri proved that la x 2 dx Proposition correspond to this statement? 1 :1 + Il How does 3

19 Reading Guide and Discussion Questions Chapter 13, Part 1 The prompts marked with an asterisk (*) below are reading guide questions. *l. Why should John Wallis also be considered one of the founders of analytic geometry? *2. How did Wallis use the oo symbol? *3. Why did Wallis determine ratios of volumes and sums, rather than computing these quantities directly? 4. How did Isaac Barrow ;tate the Fundamental Theorem of Calculus? How would we write his observations in modern notation? How does the modern statement correspond to Barrow's? ~ *5. Who was Isaac Newton? How did he get interested in mathematics? ---::s;> i..6. What were Newton's four major contributions to math and physics? *7. Rewrite Proposition 13.5 with modern notation. *8. How did Newton's approach to integration differ from that of Wallis? *9. What two general problems did Newton introduce in On the Method of Fluxions and Infinite Series? How would we state these problems more generally in terms of functions, derivatives, and integrals? *10. What is the difference between a fluent and a.fiuxion? *11. What was one of the first problems to which Newton applied the method of fiuxions? *12. Where did the systematic use of our modern xy-coordinate system appear for the first time? *13. What is significant about Newton's Principia? *14. Rewrite Lemma 13.2 using modern language and notation. ~ *15. How did Newton state the product rule? How did he prove it? ~ 16. What was George Berkeley's objection to Newtons' proof of the product rule? How would we reconcile this objection today? ~ d 7. What was it "the work of Leibniz, and not that of Newton, that would drive analysis during the eighteenth century"?

Once they had completed their conquests, the Arabs settled down to build a civilization and a culture. They became interested in the arts and

Once they had completed their conquests, the Arabs settled down to build a civilization and a culture. They became interested in the arts and The Islamic World We know intellectual activity in the Mediterranean declined in response to chaos brought about by the rise of the Roman Empire. We ve also seen how the influence of Christianity diminished