Mathematics in Mesopotamia
Important note: On the problems that require computations, the steps to find the answer must be included in your submission. An answer with little or no explanantion will receive little or no credit.
- 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?
- When and where (approximately) did the first mathematical proof we know of appeared?
- State two equivalent (but distinct) forms of Euclid’s fifth postulate.
- Explain the Proposition I of Euclid's Elements in your own words. Do not use labels of points of segments (For instance, you can write, give 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.
- Given a segment of length 1 (of some unit, say inches) describe the steps to construct a segment of length √2 (square root of 2). Extra credit: Implement your construction in Geogebra and include a photo your work.
- Is it possible to construct a square with the same area of a given circle, only using straight-edge and compass? Give a justifcation to your answer.
- Divide 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 our choice) “Make” your solution in paper or Geogebra. Explain how you obtained solution and provide photos of the pieces you made or in Geogebra. (If you complete this succesufully, you would have produced a quadrature of an isosceles triangle!)
- Extra Credit: On the Proposition 14 of Book 2 of Euclid 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 rectilinar 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 coequalnstructed 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 to grade.