MATH 15, CLS 15 — Spring 2020 Problem Set 1, Due 24 January 2020

MATH 15, CLS 15 — Spring 2020 Problem set 1, due 24 January 2020

General instructions for the semester: You may use any references you like for these assignments, though you should not need anything beyond the books on the syllabus and the speciﬁed web sites. You may work with your classmates provided you all come to your own conclusions and write up your own solutions separately: discussing a problem is a really good idea, but letting somebody else solve it for you misses the point. When a question just asks for a calculation, that’s all you need to write — make sure to show your work! But if there’s a “why” or “how” question, your answer should be in the form of coherent paragraphs, giving evidence for the statements you make and the conclusions you reach. Papers should be typed (double sided is ﬁne) and handed in at the start of class on the due date.

1. What is mathematics? Early in the video you watched, du Sautoy says the fundamental ideas of mathematics are “space and quantity”; later he says that “proof is what gives mathematics its strength.” English Wikipedia tells us “mathematics includes the study of such topics as quantity, structure, space, and change,” while Simple English Wikipedia says “mathematics is the study of numbers, shapes, and patterns.” In French Wikipedia´ we read “mathematics is a collection of abstract knowledge based on logical reasoning applied to such things as numbers, forms, structures, and transformations”; in German, “mathematics is a science that started with the study of geometric figures and calculations with numbers”; and in Spanish, “mathematics is a formal science with, beginning with axioms and continuing with logical reasoning, studies the properties of and relations between abstract objects like numbers, geometric figures, or symbols” (all translations mine). These descriptions don’t all say quite the same thing, but are they consistent with each other? What different impressions of mathematics do they give? Are there activities that would be considered mathematical under one of these definitions but not another? Discuss, in an essay of about 300–500 words.

2. Sources for the history of mathematics. Our knowledge of Egyptian mathematics depends mainly on a couple of hand-written texts on papyrus; for the Ahmes papyrus (or Rhind papyrus), we don’t even have the entire roll, and the pieces we do have are in several different museums. We know Babylonian mathematics principally from a collection of hand-written texts on clay tablets, and, once again, some are broken or otherwise damaged. • Are these sources sufficient to give us the whole picture of Egyptian or Babylonian mathematics? Explain. • What problems arise from the present state of the sources we have? • Nowadays, mathematicians write mathematics on computers, and the first publication of a new mathematical text is often at a web site called the arXiv (arxiv.org, pronounced “archive”). Later, the text may appear in a scholarly journal or a book, which is probably published on line and may or may not also appear in print. Suppose you are a historian of mathematics living in 3020 and looking back a thousand years to the 21st century. What sources will you find? Will they be complete? Will you be able to read them?

3. Explanations, like those in the ancient texts. Multiply 231 and 732. Don’t use a calculator or computer, but write out the multiplication by hand the way you were probably taught in third or fourth grade. Now, explain in words how to do that multiplication, or any multiplication like that one. How do you know you’ve got the right answer? And how do you know the method you used always gives the right answer (assuming you haven’t made a mistake)? MATH 15, CLS 15 — Spring 2020 Problem set 2, due 31 January 2020

1. Reading Egyptian numerals. Write the following numerals in the modern Western system (the Hindu-Arabic system).

a. 2||

b. 22 |||| 222 ||||

c. 3 22 d. 4322| e. 433||||

f. r |||

g. r 2|

h. r 22

2. Writing Egyptian numerals. Write the following numerals in the Egyptian system.

a. 37 b. 94 c. 125

d. 1852 e. 2020 f. 1/2 g. 1/10

h. 1/17

Problems 3, 4, and 5 are on the reverse. 3. Egyptian fractions. Here are some ordinary fractions. Write them as an Egyptian would, that is, as sums of unit fractions; you can also use 2/3 if it’s useful. First write as actual fractions, then as over-barred numerals. Write at least two of them using Egyptian numerals, too. For example, given 3/4 you would write 1/2 + 1/4 (actual fraction), then 2 4 (over-barred), then r r (Egyptian style). || |||| Remember you can’t use the same unit fraction twice!

a. 5/6 b. 5/8

c. 2/5 d. 2/15 e. 3/10

Optional additional question, a bit more challenging: There is always more than one way to write a fraction Egyptian-style. Write at least one of the numbers above as a different sum of Egyptian fractions. For example, 3/4 = 1/2 + 1/4 but you could also write it as 2/3 + 1/12 or 1/2 + 1/6 + 1/12, or any number of other ways. Is one version “better” than another? Why? (This means you have to decide what “better” means here, of course.)

4. Egyptian calculation. Multiply 231 and 732 using the Egyptian method: pick one or the other number to start with, match it up with 1, double each column, and so on. This is the same multiplication you did on the last problem set; which way is easier? Of course you got the same answer in this example; can you determine if the Egyptian method will always work? Can you say why or why not?

5. Social context of Egyptian mathematics. What does the content of the Ahmes Papyrus (and of the other Egyptian texts) tell us about who does mathematics in Egypt, and about what they do with it? Think about what kinds of problems appear in the sources, but also about what’s in the text other than problems. Discuss, in about 300–500 words. MATH 15, CLS 15 — Spring 2020 Problem set 3, due 7 February 2020

1. Egyptian geometry: volumes. Problem 41 of the Ahmes papyrus asks for the volume of a cylindrical granary whose diameter is 9 cubits and whose height is 10 cubits. Here is how Ahmes works it out.

Take away 1/9 of 9, namely 1; this makes 8. Multiply 8 times 8; this is 64. Multiply 64 times 10; this gives 640 cubic cubits. Add 1/2 of it to it; that makes 960, which is the content of the granary in khar. Take 1/20 of 960, which is 48: then 4800 hekat of grain go into the granary.

You’ll need to know that a khar is a measure of volume, 2/3 of a cubic cubit. There are 20 hekat to a khar, and as you know hekat is the usual volume measure for grain. What is Ahmes actually doing here? Where does he get the numbers he uses? His method does not actually give the right answer: why not? Is it pretty close? (You’ll need to compare his result to the result you’d get if you use the actual formula for the volume of a cylinder; if you don’t remember it, the Wikipedia article “Cylinder” will tell you.) Consider a similar problem (not in the papyrus): suppose the granary has a diameter of 3 cubits and a height of 6 cubits. Work out the calculation using the same method as in the papyrus, and then give the actual answer.

2. Babylonian numerals: the base–60 system. The following are Babylonian numerals in the conventional transliteration. Express them as ordinary numerals in the modern Western system.

a. 1, 2 b. 1; 2 c. 0; 1, 2

d. 1, 4, 7 e. 1, 4; 7 f. 1; 4, 7 g. 5, 34

Here are some ordinary numbers; write them in base-60 in the conventional transliteration. If you want, you can also draw the cuneiform symbols.

a. 35

b. 350 c. 65 d. 119 e. 5 1/2

f. 6 2/3

You will not be surprised to ﬁnd out that there are more problems on the reverse. 3. Babylonian numerals: cuneiform. Write the following cuneiform numerals in the conventional transliteration, and then as ordinary numerals. Assume they are all whole numbers.

The numeral is 3,5 if it represents a whole number, but the numbers transliterated as 3;5 and 0;3,5 would be written exactly the same way. For that matter, 3,5,0 would also be written this way. Return to the above list and interpret each number as a whole number plus a fraction (like 3;5): transliterate and then write in our system.

4. Babylonian arithmetic: base 60. In our base-10 system, you can look at a number and tell whether it is divisible by 2, or by 5. How do you know? What about base 60? Is there an analogous set of divisibility tests? Can you look at a base-60 number and say at once whether it is divisible by 2? What other possible factors can you notice immediately using a test of the same type? For concreteness, apply your tests to the following base-60 numbers. For example, if you were working in base 10 and you were given 15, you’d say it is divisible by 5, and it isn’t divisible by 2. (And you know this is correct because 15 is 3 × 5.) In base 60, of course, the list of possible factors will be different. • 1,5 • 2,6 • 5,15 • 4, 25 • 17,36 Why does this work? Then, how would you multiply a base-60 number by 60? How would you divide by 60? Multiply each of the numbers in the list above by 60, writing them in the standard transcription as here. What would happen if you wrote them in cuneiform?

5. Math as a social practice in the Babylonian empire. We have a text known as the “Super- visor’s Advice to a Young Scribe” in more than 30 copies on cuneiform tablets from the Old Babylonian period; you will ﬁnd it on p. 80–82 of the sourcebook. Eleanor Robson, in her headnote, says “although neither mathematics nor any other school subject is explicitly mentioned here, we see the wide range of numerate tasks the scribe of a large household might have carried out” (p. 80). Read the text, then discuss Robson’s observation, in an essay of at least 300–500 words. Look at the kinds of work the young man and the older man talk about. Which of these tasks might have involved calculations, measurements, or other mathematical activities? How much mathematics did a Babylonian house scribe need to know? You refer to the text by line number; the numbers are given at the start of each paragraph of the translation. For example, you might write The older scribe says his teacher didn’t brag, because “if he had vaunted his knowledge, people would have frowned” (l. 16–20). You do not need to add a bibliography to your problem set solutions listing the sourcebook. MATH 15, CLS 15 — Spring 2020 Problem set 4, due 14 February 2020

1. Division, Babylonian style. We saw that the Egyptians treated division as the inverse of multiplication. The Babylonians did this as well, in a slightly different way. They would write a ÷ b as 1 a × b , and then just multiply. Will this always work? Why? 1 Of course, to do this efﬁciently, you need to know the values of b for various numbers, and we know the Babylonians wrote up tables to refer to. Write up your own table of 1-over-number (reciprocals), in base 60 transcription. Here are the ﬁrst two entries: number reciprocal 2 0;30 3 0;20

1 Check that these two entries are correct. Then ﬁll in the rest up to 60 . Omit those that don’t come out even as base-60 fractions. Which numbers will those be, and why?

2. Approximating square roots. The square root of a positive whole number is either another positive whole number (as the square root of 25 is 5) or it’s irrational: you can’t write it as a fraction, or as a decimal that terminates or repeats (or a base-60 fraction either). So if we need to use a number like the square root of 26, we need to approximate it. Here’s how the Babylonians did it. √ √ Clearly √ 26 is√ a little bit bigger than 25 = 5, so start with 5 — call that approximation #1. We know that 26 × 26 = 26, by definition, and we know that 5 × 26 = 26. In that second equation, the √ 5 26 first term (5) is smaller than 26, so the second term ( 5 ) must be bigger. Split the difference: take 26 1 25+26 the average of these two numbers, (5 + 5 ) × 2 = 5×2 = 5.1 — call that approximation #2. Are we getting close? Well, 5.1 × 5.1 = 26.01, which is pretty good. If we wanted an even better approximation, we could repeat the process: split the difference between approximation #2 and 26 divided by that number. Work that out for 26: what is approximation #3? And if you square approximation #3, what do you get? (I got 26.000000962.) Summing up: approximation #1 is 5, approximation #2 is 5.1, and approximation #3 is the number you just calculated. √ √ Using the same procedure, find approximations #1, 2, and 3 for 10 and 15. Use ordinary modern notation, and you may want to use a calculator by the time you get to the third approximation. Can you explain why this process works? (You don’t need to give a rigorous proof here, just an explanation.) In other words, how do you know each successive approximation is closer to the actual value?

There are two more problems on the reverse. 3. Geometrical algebra. Here is a problem from a tablet now in the British Museum (BM 13901).

I summed the area and two thirds of the side so that it was 0;35. Find the side.

First, work this out using modern methods (and modern numerals). Then, work it out using the Babylonian method: start with a projection to make a rectangle, then re-arrange. Verify that you get the same answer as before.

4. Pi, Babylonian style. In some problems, the Babylonians calculate the area of a circle as 0; 5 × C2, where C is the circumference, and the diameter as 0; 20 × C. Write these with modern numerals (that is, base X rather than base LX). How do these Babylonian calculations relate to the modern formulas? These calculations are not exact (and the Babylonians knew that), but depend on replacing the exact number π with an approximation. What approximation for π are the Babylonians using? What are the advantages and disadvantages of using this approximation, compared to the Egyptian one or the modern one? (The Babylonians did have more precise approximations for π but this one also occurs frequently in the problem texts.) MATH 15, CLS 15 — Spring 2020 Problem set 5, due 21 February 2020

1. Babylonian geometrical algebra. Two more problems from BM 13901, each asking for the length of a side of a square.

Problem A. I summed the area and the side and one third of the side of a square so that it was 0;55. Find the side. Problem B. I subtracted the square-side from the area and the result was 14;30. Find the side.

Work out each of these problems using the Babylonian geometrical method. Then solve them using the modern method, with a quadratic equation. You will of course get the same answer by each method; if you don’t, then one of your solutions is incorrect and you should check your work.

2. Pythagorean triples. A “Pythagorean triple” is three numbers — let’s call them a, b, and c — such that a2 + b2 = c2. For example, 3, 4, 5 is a Pythagorean triple. Find at least two more Pythagorean triples. There’s an easy way involving tweaking the one you already have, but try also to ﬁnd one that’s not derived from 3, 4, 5. Obviously you could simply go to the Wikipedia article “Pythagorean triple,” but that misses the point: work this out for yourself. Once you’ve found a couple of additional Pythagorean triples, think about the following general method to generate them. Start with two odd numbers that don’t have any common factors (for example, don’t use 9 and 15 which are both divisible by 3). For concreteness, call the bigger one p and the smaller one q. Calculate p2 + q2, p2 − q2, and 2pq. What do you notice about these three numbers? Do this for at least 3 pairs of odd numbers p and q. What happens?

Slightly harder: can you ﬁgure out why this method always works? Optional, and even more challenging: will this method generate all possible Pythagorean triples? Explain.

There are two further problems on the reverse. 3. How to think about problems. Our approach to the sides–and–areas problems is fundamentally symbolic and algebraic, while the Babylonian approach is fundamentally geometric. While both approaches reach the same answers, the thought processes behind them are different. Compare how they work and how they lead you to think about these problems. Does one way make more sense to you than the other? Does one way seem simpler than the other? Be careful not merely to say “using equations is simpler because it’s what I’m used to”; rather, give each method a fair chance. Introductory texts like the Open University pamphlet you’ve read often say “the Babylonians could solve quadratic equations.” Comment on that claim in view of the analysis you’ve just done. Discuss all this in an essay of 300–500 words.

4. Praise of a king. Sulgiˇ reigned in Ur from about 2029–1982 BCE. His name is pronounced “shul-gee,” and the mark on the “s” is called a ha´cek.ˇ Sulgiˇ commissioned several hymns praising himself. In one, we read:

I am a king, offspring begotten by a king and borne by a queen. I, Sulgiˇ the noble, have been blessed with a favourable destiny right from the womb. When I was small, I was at the academy, where I learned the scribal art from the tablets of Sumer and Akkad. None of the nobles could write on clay as I could. There where people regularly went for tutelage in the scribal art, I qualiﬁed fully in subtraction, addition, reckoning, and accounting. The fair Nanibgal, Nisaba, provided me amply with knowledge and comprehension. I am an experienced scribe who does not neglect a thing.

—“Sulgiˇ B,” ETCSL 2.4.2.02, lines 11–20.

(If you’re curious, you can ﬁnd the complete hymn in the Electronic Text Corpus of Sumerian Liter- ature; there is a link in Canvas.) Jens Høyrup observes that this hymn shows how important “the scribal art” was — it’s one of many things this king brags about doing well. Yet the only mathematics he brags about is “subtraction, addition, reckoning, and accounting.” Høyrup’s comment is, “it appears that his ghost-writers knew of no other mathematics.” In other words, the poet who wrote this hymn may not have been aware of mathematics beyond the most elementary kinds of problems. We can’t tell from this text whether Sulgiˇ himself knew more math than his poet mentions: maybe he did, maybe he didn’t. What does all this imply about the place of mathematics both within what scribes learned and in the larger culture? Discuss, in about 300–500 words. MATH 15, CLS 15 — Spring 2020 Problem set 6, due 28 February 2020

1. Greek numerals: alphabetic system. Write the following alphabetic or Ionian numerals in the modern Western system (the usual Hindu-Arabic numerals).

a. λζþ d. τοþ g. υπαþ b. ογþ e. ριαþ c. τζþ f. ρµηþ h. ωιδþ

Then write the following in the Greek alphabetic system.

a. 15 d. 321 g. 987 b. 150 e. 213 c. 105 f. 132

2. Babylonian “quadratics”: synthesis. You’ve now done several problems of the Babylonian type involving the area and the side of a square. Devise another square-side problem of your own analogous to the ones you’ve worked on. Set it up so the numbers come out nicely. Pose it Babylonian- style, and give both Babylonian and modern solutions.

3. Approaches to the history of mathematics. You have seen Babylonian “geometric algebra” presented in geometric terms, with explanatory diagrams which don’t actually occur in the Babylonian sources. This is how scholars like Eleanor Robson, Jens Høyrup, and Joran¨ Friberg interpret the existing texts. On the other hand, Otto Neugebauer, in his important book The Exact Sciences in Antiquity (Providence: 1957; 2nd ed. New York: 1969; p. 42), says

It is easy to show that geometrical concepts play a very secondary part in Babylonian algebra, however extensively a geometric terminology may be used. It sufﬁces to quote the existence of examples in which areas and lengths are added, or areas multiplied, thus excluding any geometrical interpretation in the Euclidean fashion which seems so natural to us.

All of these scholars are reading the same texts, and all are reading correctly, but they are drawing different conclusions. Why might that happen? Which ideas implicit in the Babylonian texts seem more important to the two groups of scholars? Discuss, in an essay of about 300–500 words.

One more problem appears on the reverse. 4. Mathematics as a social practice in classical Athens. Cuomo quotes (p. 17–18) a passage from Aristophanes’ play Wasps, produced in 422 BCE at an Athenian festival. Here is a slightly longer excerpt, in a livelier translation. In the play, the father is “addicted” to jury duty, a civic duty for Athenians as it is for us. The father enjoys the power that comes from voting to convict criminals, or people he doesn’t like, and he doesn’t mind the juror’s wages, either. The son is trying to convince his father to stop treating jury service as a hobby. The father is an ordinary Athenian, not an aristocrat and not highly educated.

Son: Then listen, pop, and relax your frown a bit. First of all, calculate roughly, not with counters but on your fingers, how much tribute we receive all together from the allied cities. Then make a separate count of the taxes and the many one-percents, court dues, mines, markets, harbors, rents, proceeds from confiscations. Our total income from all this is nearly 2000 talents. Now set aside the annual payment to the jurors, all 6000 of them, “for never yet have more dwelt in this land.” We get, I reckon, a sum of 150 talents. Father: So the pay we’ve been getting doesn’t even amount to a tenth of the revenue! Son: It certainly doesn’t. Father: In that case, where is the rest of the money routed? Son: To the politicians. You choose them to rule you, father, because you’ve been but- tered up by their slogans. And then they extort fifty-talent bribes from the allied cities. All these guys and their flunkies hold office and draw salaries, while you’re content if someone gives you those three obols, the ones you earned by your own hard work, rowing and soldiering and laying siege.

— Wasps 655–670, 682–685; translation J. Henderson, slightly adapted

What does this passage tell you about the mathematical knowledge of a typical Athenian? What assumptions is Aristophanes making about his audience? (The play was competing for a prize, so he wants to be sure that everyone in the audience understands the points and gets the jokes.) As you’ve read, many ofﬁces in Athenian government are given out by lot, so most Athenian men would have taken a turn at one of the lower-level jobs; what sort of mathematical knowledge would they need to do so? Discuss all this, in the usual 300–500 words. MATH 15, CLS 15 — Spring 2020 Problem set 7, due 6 March 2020

1. Conﬂicting sources: Socrates according to Xenophon. Cuomo quotes and summarizes various dialogues in which Plato’s “Socrates” character says mathematics is essential for philosophers, because it encourages abstract thought (p. 25); mathematical knowledge is innate in everyone’s mind and can be retrieved if someone asks you the right questions (p. 27–29); yet mathematical knowledge, like any other knowledge, can be dangerous if mis-used (p. 41–42); in the end, though, the philosophers who are to rule the ideal state must know advanced, abstract mathematics at the highest levels (p. 43, quoting Republic 525c–527a). Xenophon, a contemporary of Plato and another follower of Socrates, described Socrates and his teachings in a text called Memorabilia. According to him,

Socrates also taught how far a well-educated man should make himself familiar with any given subject. For instance, he said that the study of geometry should be pursued until the student was competent to measure a parcel of land accurately in case he wanted to take over, convey or divide it, or to compute the yield; and this knowledge was so easy to acquire, that anyone who gave his mind to mensuration knew the size of the piece and carried away a knowledge of the principles of land measurement. He was against carrying the study of geometry so far as to include the more complicated ﬁgures, on the ground that he could not see the use of them. Not that he was himself unfamiliar with them, but he said that they were enough to occupy a lifetime, to the complete exclusion of many other useful studies. . . . He also recommended the study of arithmetic. But in this case as in the others he recommended avoidance of vain application; and invariably, whether theories or ascertained facts formed the subject of his conversation, he limited it to what was useful. — Xen. Mem. 4.7.2–3, 8.

Clearly Plato and Xenophon put different ideas into the mouths of their “Socrates” ﬁgures. What does this mean for us if we are trying to determine what Socrates, Plato, or Xenophon actually believed about the uses of mathematics?

2. Measuring land. Cuomo says (p. 67) that administrators in Greek Egypt would estimate the area of a four-sided field by averaging opposite pairs of sides and multiplying the results. So if the a+c b+d sides of the field are a, b, c, and d going around clockwise, the estimate is 2 × 2 . Cuomo says this always over-estimates the actual area. (In fact, we know ancient Egyptians were using this estimation long before the Greeks showed up. We can also be pretty sure that the Greeks knew this method does not give the exact area.) What are the advantages of having a single rule that you can use for a field of any shape, so long as it has four sides? Now, verify what Cuomo says about over- estimating. First of all, show that this method gives the right answer if the field is a rectangle, so that a = c and b = d. Now what if the field is a parallelogram, so a = c and b = d, but the corners aren’t square? What’s the actual area, and does the Egyptian method produce the same number, a bigger one, or a smaller one? Then consider a trapezoid, where sides a and c are parallel but not the same length. What happens now? Optional: What about a general quadri- lateral, like the one in the diagram? Can you prove Cuomo’s contention that the method never gives a smaller answer than the actual area? 3. Accounting. Cuomo quotes a portion of a letter from the third century BCE, p. 69. Here is a fuller text. You will also find a link in Canvas to the Duke Databank of Documentary Papyri listing for this papyrus, including Greek text, this translation, and a picture.

Eukles to Anosis, greeting. I learn that you have deposited in the record ofﬁce the accounting of the pottery without bringing into it even the breakage which has occurred through the donkey drivers, and, in the matter of the pay still due in the case of the potters, that you have entered 8 drachmas per hundred pots instead of the 6 drachmas pay which had been given them; and that, too, although I am unable to obtain from them even what they agree (to furnish); and that, up to the present, you have not handed in to the record ofﬁce the accounting of the hogs ready for slaughter; and that, on the whole, you have begun to act like a scoundrel. Here there is a gap in the papyrus. If therefore, any of these charges are true you do not appear to be to blame, but I (appear so) to myself. Nevertheless I have written regarding these matters to Lykophron and to Apollonios that, if they discover any discrepancies from the accounts handed in by you, they write to me immediately in order that I, being present in person, may have my case judged in relation to you. For it is right that you, who are entering excess charges and are not reporting the totals correctly throughout the accounts, should pay the discrepancies, not I. For the future I will try, although I have often been fooled by you, to change my attitude and plan better. Good-bye.

— P. Col. 4 88, at Columbia University; 243 BCE, from the town of Philadelphia in Egypt. Translation from DDBDP.

This letter is followed on the same papyrus by another letter from Eukles to Dionysios:

To Dionysios. You will do me a favor by compelling Anosis also to deposit the same accounts, all of them, which Apollonios, my secretary, is handing in so that if any differences are recorded, our case in relation to him may be decided immediately.

The papyrus seems to have been Eukles’s reference copy, or his drafts, of letters that he sent. Anosis is the scribe for the town of Philadelphia, and Eukles is the town administrator, so Anosis works for him. Cuomo merely says that the letter shows that “accounts must be produced regularly and checked carefully,” which is true, but we can learn more than that. What do these two letters tell you about the mathematical needs of a town administrator? Discuss, in at least 300–500 words. MATH 15, CLS 15 — Spring 2020 Problem set 8, due 27 March 2020

Observe that this problem set is not due until after spring break. We will not have class on Friday 13 March.

1. The Sieve of Eratosthenes. Describe the sieve method, and explain why it works. Now go one step further. A standard optimization for this technique is to start crossing off numbers with the square of the prime you’ve just circled. For 2, this doesn’t make any difference: the ﬁrst number you cross off would be 4 either way. The next prime is 3, and in the standard method you’d cross off 3 + 3 = 6, then 6 + 3 = 9, and so on. If you start with the square of 3 instead, what happens? What about 5? In general, if you start from the square of the prime instead of counting directly from that prime, you’re ignoring all the numbers between the prime and its square for this step. Does that still work, or are you going to miss crossing off something you should have crossed off? Explain.

2. Euclid’s Algorithm. Using Euclid’s method, ﬁnd the greatest common divisor of each of the following pairs of numbers.

a. 1034, 99

b. 2401, 28 c. 4095, 98

3. Geometry by Euclid’s rules. Using only a ruler and a compass, as Euclid assumes, work through the following constructions. Remember, you can only use the ruler to make straight lines, not to measure — pretend it doesn’t have any markings on it. And the only thing the compass can do is make circles.

a. Given a point and a radius, draw the circle with that radius and with its center at the point.

b. Given two points, draw the line between them. c. Given two points, draw the line between them, and draw an equilateral triangle with that line as one of its sides. (Hint: use two circles, one centered at each of your given points.) d. Given a line segment, cut it in half. (Hint: you can use the triangle you just ﬁgured out how to make.)

e. Given a line segment, draw another one perpendicular to it; it doesn’t matter where the new line segment intersects the given one. (Hint: try right in the middle.)

The ﬁrst two require no explanation. For the last three, though, after you’ve drawn the ﬁgure, explain how you know it’s right.

There is one more problem on the reverse. 4. The community of Hellenistic mathematicians. Many of the works of Archimedes, and of other mathematicians as well, are written as letters to other mathematicians. For example, The Method is set out as a letter to Eratosthenes. Here is an excerpt.

Archimedes to Eratosthenes, greeting. I sent you on a former occasion some of the theorems I have discovered, merely writing them out and inviting you to discover the proofs, which at the moment I did not give. The enunciations of the theorems which I sent were as follows. Here follow the theorems, which I omit. These theorems are different from those I have told you about before, for in those theorems, we were comparing the sizes of conoids and spheroids and segments of them with the sizes of cones and cylinders. And none of those figures has yet been found equal to a solid figure bounded by planes. On the other hand, each of the figures under discussion here, bounded by planes and surfaces of cylinders, is in fact equal to one of the solid figures bounded by planes. I have written the proofs of these theorems in this book which I am now sending to you. And because I think you are a serious student, an eminent philosopher, and an admirer of mathematical inquiry, I thought it appropriate to write out for you and explain in detail in the same book a certain particular method that will let you get started investigating some of the problems in mathematics using mechanics. I believe this procedure is even useful for the actual proofs of the theorems. Certain things first became clear to me when I used the mechanical method — though they had to be demonstrated properly by geometry afterwards since the mechanical investigation does not give an actual proof. But of course it is easier to supply the proof after we’ve used the mechanical method to get some knowledge of the questions than it is to find the proof with no previous knowledge. This is a reason why, in the case of the theorems the proof of which Eudoxus was the first to discover (namely, that the cone is a third part of the cylinder, and the pyramid of the prism, having the same base and equal height), we should give a lot of the credit to Democritus who was the first to claim this was true, though he didn’t prove it. I myself have made the first discovery of the theorem I publish here, using the mechanical method. I think it’s appropriate to write up the method because it will be useful to other people: someone, now or later, will be able to use this method to discover other theorems that have not yet occurred to me.

— Translation adapted from Heath, The Method of Archimedes, Cambridge: 1912.

What does this introduction tell you about the attitudes to mathematical activity among the group of mathematicians that Archimedes and Eratosthenes both belong to? What kinds of things do they care about? How do they approach their work? To what extent are they competing with each other, and to what extent are they collaborating? Explain, in the usual 300–500 words.