Study Guide for Exam 1 Math 330: History of Mathematics October 2, 2006. 1 Introduction What follows is a list of topics that might be on the exam. Of course, the test will only contain only 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. (Parenthetical remarks in this study guide are given for clarification. You do not need to memorize them for the exam.) 2 Ancient Mathematicians 1. Ahmes the scribe. Lived about 1650 BC, after the Middle Kingdom of Egypt. His is the earliest known name in the history of mathematics. Author of the famous Rhind-Ahmes Papyrus, the most extensive papyrus we have yet found. (Since Egyptian writing only recorded consonants, not vowels, we do not know exactly how to pronounce his name. Perhaps his name was Ahmose). 2. Thales from Miletus. Miletus was in Ionia, and Ionia was a Greek region on the coast of Asia Minor. Said to have predicted the eclipse of 585 BC (you don t need to memorize this date, but you do need to know that he lived in the 6th century BC). Credited with the first use of proof in mathematics. Said to have brought geometry to Greece from Egypt. Study the special study notes for Thales: public.csusm.edu/aitken html/m330/extranotes/thales.html 3. Pythagoras from Samos (a Greek island). Also lived in the 6th century, but later than Thales. He is said to have traveled to Egypt 1
and Babylon, and learned from priests. After coming back to the Greek world he founded a philosophical and religious society in Croton, Italy. He and his followers, called the Pythagoreans, were fascinated with mathematics, musical harmonies, and the concept of proportions and harmony in all aspects of life, and the universe or kosmos. The Pythagoreans included women as well as men. They discovered incommensurables, and at least some of the regular solids (including the dodecahedron). The first Greeks to make mathematics part of higher education. Mathematics, a term invented by Pythagoras, includes arithmetic, geometry, music, and astronomy. At first, their knowledge was kept secret. Pythagoras said that all is number. Discovered the simple mathematical ratios associated to harmonious notes in music. He is given credit for the first proof of the Pythagorean theorem (the theorem, but perhaps not the proof, was known to the Babylonians). Study the Pythagoras Fact page: public.csusm.edu/aitken html/m330/extranotes/pythagoras.html Later Pythagoreans including the semi-legendary Hippasus (as well as Archytas, but Archytas will be on the next test). The Pythagoreans were instrumental in the early development of Greek mathematics. For example, legend has it that Hippasus discovered incommensurables, or something about the dodecahedron and/or the pentagon). 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 one legend, Hippasus died at sea for this impiety. 4. 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. 2
(This idea was later used by Archytas, Menaechmus and Eratothenes to find solutions, but not with straightedge and compass alone.) 5. 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 just for trisecting an angle. 6. Democritus was the first Greek to state that the cone has volume equal to one-third the cylinder of the same dimensions, and that pyramids and tetrahedrons have volumes one-third that of the rectangular solids (boxes) and prisms of the same dimension. This agrees with the modern formulas for such volumes (which are easily derived using calculus). Archimedes says that Democritus discovered these theorems but didn t have a proof. Since Archimedes had a very high standard of proof, it could be that Democritus had a non-rigorous proof or argument that was improved later (by a famous mathematician named Eudoxus). The Egyptians most likely knew these facts over a thousand years before Democritus, at least for pyramids. Democritus was a philosopher, not just a mathematician. He is best known for advocating the atomic theory of matter. 3 Historical People and Events You will need to know the following, including the dates. 1. Beginning of urban civilization and writing in Egypt and Mesopotamia. Before 3000 BC. 2. Hammurabi. Was the ruler at the time of the height of Babylonian mathematics. Lived about 1800 BC. Hammurabi is famous for developing a legal code. 3. About the same time, 1800 BC, the Middle kingdom of Egypt. This is the period of Egyptian history responsible for most known Egyptian mathematics. Ahmes wrote his papyrus a bit later, about 1650 BC, after the Middle Egyptian period, but based on a previous manuscript from the Middle Egyptian period. 4. The first Olympic games in Greece: 776 BC. Also, approximately the time of the invention of the Greek alphabet. 3
5. The Persian-Greek war. Ended (490 BC) before the Golden age of Athens. 6. Pericles and the Golden age of Athens. After the Persian-Greek war. Pericles died of the plague. 7. Socrates. Philosopher and teacher of Plato. Active in Athens after the Persian-Greek war. He died in 399 BC. 4 Summary of Dates Know the following dates and their significance. 3000 BC 1800 BC (approximate date of height of Egyptian and Babylonian mathematics). 1650 BC (approximate date of height of the Rhind-Ahmes papyrus). 776 BC 6th century BC (500 to 600 BC, the period including Thales and Pythagoras) 6th century BC (400 to 500 BC, golden age of Athens) 399 BC (Death of Socrates) Also know that Thales came before Pythagoras. 5 Other Historical Questions Be able to answer the following: 1. What are the three major periods of ancient Egyptian history? 2. In which period were the pyramids made? 3. Which period was responsible for the height of Egyptian mathematics? [Middle Kingdom] 4. Name the two major Egyptian scrolls. 4
5. What were these scrolls made of? [Papyrus] 6. Which of these major Egyptian scrolls has the unit fraction table? [Rhind] 7. Which Egyptian scroll has the truncated pyramid problem? [Moscow] 8. What inscription led to the decipherment of hieroglyphics? [Rosetta Stone] 9. What is the major river of Egypt? What are the major rivers through mesopotamia? 10. What does mesopotamia mean? What do meso and potamos mean? 11. Know the difference between hieroglyphics and hieratic. 12. Which of these two scripts were used in the Ahmes and Moscow Papyri? [hieratic] 13. Which language were the original cuneiform written in? [Sumerian] 14. Which were cuneiforms written on? [Clay tablets] 15. Which language were the mathematical cuneiform and the other writers of the Old Babylonia period written in? [Akkadian] 16. Where is Ionia? 17. Where is Miletus? [In Ionia] 18. Where is Croton? What is it famous for? 19. Where is Samos? [It is an island off Ionia] 6 Numbers and Letters 1. Know the Greek alphabet: upper case, lower case, name, Latin equivalent. [See handout] 2. Know how to write sexigesimal (base 60) numbers in cuneiform, as well as our system (example: translate 12, 34; 14 into cuneiform. Translate cuneiform into our system). 3. Know how to write numbers up to 999 in hieroglyphics. 5
4. Know how to write numbers in Attic numerals (used in Athens until replaced by the Ionic numerals). 5. Know the Ionic number system (I will help you by partially filling up the table include the 3 archaic characters: digamma, koppa, and san). Write numbers up to 9999. (Numbers larger than that, and fractions, could be written in a variety of ways depending on the particular author). 7 Mathematical Topics 1. Egyptian Arithmetic. Egyptian multiplication and division using doubling. For example, multiply 104 and 5 using the method of doubling. Divide 261 by 9 with a similar method. 2. Egyptian Fractions. Understand Egyptian (or unit) fractions. Here are some problems that you should be able to do: (1) convert 7/9 or 14/5 into distinct unit fractions n. (2) multiply the fraction 5 by 14. (3) Add 2 3 4 and 3 4 7. (4) Divide 104 by 5 and write the answer in terms of distinct unit fractions. (5) Subtract 1 4 7 from 2 using the idea of red numbers (a common denominator). For these problems, you may use the unit fraction table found in the Ahmes-Rhind Papyrus. 3. Egyptian Geometry. Know the Egyptian method for finding the volume of the truncated pyramid (also called a frustum). Which papyrus was it in? Describe the hypothetical derivation of the method involving slicing into nine pieces. Also, describe a modern algebraic method of deriving (h/3)(a 2 + ab + b 2 ) assuming as known the formula V = (1/3)HB 2 for complete pyramids. 4. Egyptian Geometry. How did Ahmes calculate the area of a circle? Answer: divide the diameter by 9, multiply by 8, and square. What value of π results from this approximation? Justify your answer (using πr 2 ). 5. Babylonian Numerals. Be able to use base 60. Give examples of how the Babylonians replaced division by multiplication (using a table of reciprocals for hard situations). For example, to divide 42 by 5, just multiply 42 by 0; 12. Do additions and subtractions in base 60. You do not need to do an explicit multiplication in base 60, but you do need to know that (i) multiplication tables were used, and (ii) to divide you just multiply by the reciprocal. 6
6. Sexigesimal Numbers. Convert from base 60 to base 10. For example, what is 12, 30, 45 in base 10? What is 1; 23, 30 in base 10? Convert from base 10 to base 60. For example, what is 12/80 in base 60? What is 13451 in base 60? 7. Babylonian Numerals. Is the Babylonian method of writing numbers ambiguous? Can different numbers be written the same way? 8. Babylonian Algebra. Suppose that you are told s = x + y and a = xy but you are not told x and y. Show how to find x and y using the Babylonian method. Also: how do modern students find x and y? Show that the modern method leads to quadratic equation x 2 +a = sx. So in some sense the Babylonians could solve the quadratic equation x 2 + a = sx. On the other hand, we have no evidence that the Egyptians could solve quadratic equations involving three terms. If you know the area and perimeter of a rectangle, then the above can be used to find the sides. In fact, the language of the Babylonians suggest that they thought of a = xy as an area, and x, y as lengths. The Greeks could also solve this type of problem. So in some sense the Greeks had non-symbolic algebra too. (Euclid wrote geometric proofs for solving these types of problems). 9. Babylonian Algebra. Suppose that you are told d = x y and a = xy. Show how to find x and y using the Babylonian method. (I am using x and y for convenience: Babylonian algebra was not symbolic.) 10. Babylonian Algebra. Give a geometric justification for the above Babylonian Algebra. Start by drawing a square of area m 2 where m is the arithmetic mean of x and y. Show how this square can be cut and reassembled to give and area of a = xy plus 2 (where = x m = m y where x > y). Conclude that m 2 = a + 2. 11. Thales Theorem. This states that the angle in a semicircle is a right angle. Be able to state and prove it. (You may assume that the sum of the angles of the triangle adds up to 180 degrees or two right angles. Some attribute this theorem about the sum of the angles to Pythagoras.) 7
12. Commensurable and Incommensurable. What do these terms mean? How does this relate to rational and irrational numbers? Which society discovered incommensurable lengths. Show that the diagonal of a square is incommensurable with its side. 13. Pythagorean Geometry. Which Greeks used the pentagram as a symbol of heath their society? Find the angles of the pentagon and the pentagram. (You might want to use 36 degrees as a convenient unit). Show that the angle (3 units or 108 ) of a pentagon is trisected by the diagonals. (The diagonals are the sides of the star inside the pentagon. In other words, the pentagon has five diagonals). An interesting fact is that there are many similar triangles in the pentagram, and the ratio of certain lengths are equal to the golden ratio. 14. Pythagorean Music Theory. Who discovered that ratios of string lengths such as 1 : 2, or 2 : 3, or 3 : 4 produce harmonious notes? Know that 1 : 2 gives an octave, 2 : 3 gives a fifth, (and 3 : 4 give a fourth). 15. Pythagorean Number Theory. Illustrate with a picture the fact that two times the nth trianglular number T n is an n by n + 1 rectangular number (called an oblong number ). Conclude that T n = n(n + 1)/2. (Hint: just illustrate the case n = 3 or n = 4. The general case obviously works in the same way.) 16. 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. 17. Pythagorean Number Theory. Illustrate with a picture how the nth square is the sum of the first n odd numbers. 18. Pythagorean Ratios. Given two numbers, find the arithmetic mean, geometric mean, and harmonic mean. Give a word problem that can be solved by using the harmonic ratio. You should be able to construct arithmetic and geometric means using compass and straightedge (but you do not need to find harmonic means in this way). 19. Pythagorean Number Theory. Why are 6 and 28 perfect? Are there any known odd perfect numbers? Answer: no. (The oldest unsolved problem in mathematics is to show that no odd perfect numbers exist). 8
20. 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 Babylonians]. 21. 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. 22. Ruler and Compass Constructions. Know the ruler and compass constructions presented in class. These are listed as 1-9, 13, 15, and 17 in the following link: public.csusm.edu/aitken html/m330/extranotes/compass.html Prove that the geometric mean construction actually works using similar triangles. Be able to construct the (golden) section, and from that the Pentagon. 23. Golden Ratio. A key to constructing the pentagon is to find the unique dividing point on a given line segment such that whole segment : big part = big part : small part. The ratios that occur in the above equality were called the extreme and mean ratio, and the division point is called the section: (the name golden ratio is modern). If x is the whole length, and y is the big part, then we can write the above ratio as 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/(x 1). Show that x = 1+ 5 2 is the positive root of the equation. 24. Three Famous Problems of Antiquity. List and describe these 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 a straightedge and compass alone. However, if one uses other curves and techniques, then one can solve them. Some of the 9
best Greek mathematics resulted from their attempts to solve these problems and the curves and techniques they introduced. We will make these problems a major theme of our class. 25. The Lune of Hippocrates. We will focus on the lune that can be built out of semi-circles on a 45-45-90 right triangle. (There are other lunes, but not all of them can be squared). Sketch it. Why is it called a lune? Prove that the area of this particular lune is half the area of the triangle. This implies that the lune can be squared (since any triangle 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. 26. The Delian Problem or Duplicating the Cube. One of the three famous problems. How is this connected with the plague? Hippocrates reduced the problem to that of finding two mean proportionals: find segments m 1, m 2 such that a : m 1 = m 1 : m 2 = m 2 : 2a. Show that m 3 1 = 2a2 in this case. 27. 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 the Greeks (but some got it wrong). Democritus is the first Greek mathematician/philosopher to point this out. He might have had a rough proof of it. We believe that the Egyptians knew this volume formula: our main evidence is the Moscow papyrus which gives an accurate calculation of the volume of a truncated pyramid. 10