Homework 4: Euclid's Elements

Notes

• In order to get full credit in the open ended questions, you need to spend some time reflecting upon and writing your answers. In most of these problems, the probability that a one-sentence answer will get full credit is close to 0 (and so will be what you learn from the exercise)
• Make sure you show all your work on the problems that require it so. Otherwise, even if you give a correct answer, if you do not explain how you obtained it, you'll get very little or no credit.
• It would be great if you discussed ideas with your classmates. The write-up, however, must be done individually.
• Recall that the slides of the lectures can be found here.

Problems

Suggestion: Play Euclidea

1. In each of the next two questions below, determine the side length of the square, explain how you obtained the solution and provide a drawing with measurements marked.
   1. Divide an isosceles triangle with base of length 8 and height of length 9 into polygonal pieces from which a square can be assembled.
   2. Consider a right triangle T whose legs have length 2 and 25. Divide T into polygonal pieces from which a square can be assembled. (Hint: Break first the triangle into pieces that can be reassembled into a rectangle.)
   You can use cm or any other unit of your choice. "Make" your own solutions in paper or GeoGebra. Explain how you obtained the solution and provide photos of the pieces you made, in paper or in GeoGebra. (If you complete this successfully, you will have produced quadratures of an isosceles and a right triangle!)
2. Divide an isosceles triangle with base of length 8 and height of length 9 into polygonal pieces from which a square can be assembled. What is the side length of this square? (You can use cm or any other unit of your choice.) “Make” your solution on paper or in GeoGebra. Explain how you obtained the solution and provide photos of the pieces you made or created in GeoGebra. (If you complete this successfully, you will have produced a quadrature of an isosceles triangle!)
3. In the proof of "Proposition IX.20: Prime numbers are more than any assigned multitude of prime numbers", given three primes a new one is defined. Find four concrete examples of this procedure, assuming that one of the given primes is 41 and the other one is 2. (You need to choose the third prime).
4. You are given three points on a line, A, B and C, so that B is between A and C and the length of BC is 1 (in some unit). Describe how to construct a segment of length √(length of AB). (Hint: the magic words here are "geometric mean") using straightedge and compass. Implement your construction in GeoGebra and include a screenshot of your work in GeoGebra.
5. Is it possible to construct a square with the same area of a given circle, only using straight-edge and compass? Why or why not?
6. Euclid Elements is the earliest extant example of axiomatic mathematics. Describe its structure and how the axiomatic method depends on such a structure. (One or two paragraphs will suffice.)
7. Recall that in class we discussed the three impossible problems of antiquity.
   1. Explain which of these problems, in your opinion, is the hardest to tackle and why.
   2. One of these problems, the trisection of the angle, is equivalent to constructing (with compass and straightedge) a segment whose length is the solution of a certain cubic equation. The other two problems are equivalent to constructing two numbers. Explain what are these two numbers and how they relate to the corresponding problem.
8. Explain in your own words the statement and proof of Proposition I.1 of Euclid's Elements. (You can start by "Given a segment, draw a circle with center one of its endpoints.") Explain whether there is any gap in the proof, from a modern point of view.
9. When and where (approximately) did the first mathematical proof we know of appear?
10. State two equivalent (but distinct) forms of Euclid’s fifth postulate.
11. Explain the Proposition I of Euclid's Elements in your own words. Do not use labels of points or segments. (For instance, you can write, "given a segment, draw a circle with center one of its endpoints and…") Explain whether there is any gap in the proof, from a modern point of view.
12. Given a segment of length 1 (of some unit, say inches), describe the steps to construct a segment of length √2. Extra credit: Implement your construction in GeoGebra and include a photo of your work.
13. Is it possible to construct a square with the same area as a given circle, using only straight-edge and compass? Give a justification for your answer.
14. What is the most important feature of Greek mathematics, beginning with Thales, which we did not find in earlier cultures? What makes this feature important? Why did Plato put the sign "let no one ignorant of geometry enter" at the door of his academy?
15. Extra credit: In Proposition 14 of Book 2 of Euclid’s Elements, a construction for squaring a “rectilinear figure A” is given. Perform the construction in this GeoGebra activity starting with a rectangle BCDE (instead of a general rectilinear figure A). That is, in the fourth sentence of the proof (“Then, if BE equals ED, then that which was proposed is done, for a square BD has been constructed equal to the rectilinear figure A.”). Construct a square of side length ED, and check in GeoGebra using the Area tool that the areas of rectangle BCDE and the square of side ED are equal. Send a screenshot for credit.

Sample Quiz 1

1. In each of the next two questions below, determine the side length of the square, explain how you obtained the solution and provide a drawing with measurements marked.
   1. Divide an isosceles triangle with base of length 8 and height of length 9 into polygonal pieces from which a square can be assembled.
   2. Consider a right triangle T whose legs have length 2 and 25. Divide T into polygonal pieces from which a square can be assembled. (Hint: Break first the triangle into pieces that can be reassembled into a rectangle.)
   You can use cm or any other unit of your choice. Include a drawing of your solutions in your answers.
2. Is it possible to construct a square with the same area of a given circle, only using straight-edge and compass? Give a justification to your answer.
3. In the proof of "Proposition IX.20: Prime numbers are more than any assigned multitude of prime numbers.", given three primes a new one is defined. Find four concrete examples of this procedure, assuming that one of the given primes is 41 and the other one is 2. (You need to choose the third prime).
4. Explain in your own words the statement and proof of Proposition I.1 of Euclid's Elements. (You can start by "Given a segment, draw a circle with center one of its endpoints.") Explain whether there is any gap in the proof, from a modern point of view.