Babylonian system Theory

Read Katz Ch. 3.3 (pp. 60-66) and 3.9 (pp. 88-90). As you read the first part of Ch 3.3 also read Euclid Propositions II-1 to II-8 just the propositions not the proofs (the proofs are not particularly illuminating). For each one make sure you understand how the proposition is expressed by the diagram. [Quick note: Katz’ image for Elements II-5 (Figure 3.12 p. 62) is inaccurate—it is vertically stretched—the Euclid image in II-5 is better. ] Requirements: 2 The goal of this problem is for you to see how Euclid’s II-6 relates to the solving of the Babylonian system we discussed on Day 2: x – y = 7 and xy = 60. Solving this system boils down to completing the square to solve a quadratic equation. First refresh your memory about the Babylonian diagram for this system and how they completed the square to solve it. Read Euclid’s entire proof for II-6 referring to Katz pp. 61-62 to help. Reproduce the diagram and label the lengths and areas appropriately for the specific Babylonian system in (with x y 7 60 etc.). Explain how Euclid’s diagram amounts to completing the square. Use it to solve for x (and y) showing your steps. The following four of Euclid’s propositions have essentially the same diagram: Elements II-6 Elements VI-29 Data 59 and Data 84. They all boil down to solving the same quadratic equation. Is Euclid just repeating himself? Using what Katz says about these summarize why Euclid would see these propositions as accomplishing goals that are not identical. Read Proposition II-8. Translate the statement into an algebraic equation. Then reproduce the diagram and label the lengths and areas in the diagram using your variables. If it is not entirely clear from your labeling explain how the diagram illustrates the algebraic equation you wrote. Then verify the proposition using algebra (not Euclid’s way). The goal of this problem is for you to see how Euclid’s II-6 relates to the solving of the Babylonian system we discussed on Day 2: x – y = 7 and xy = 60. Solving this system boils down to completing the square to solve a quadratic equation. First refresh your memory about the Babylonian diagram for this system and how they completed the square to solve it. Read Euclid’s entire proof for II-6 referring to Katz pp. 61-62 to help. Reproduce the diagram and label the lengths and areas appropriately for the specific Babylonian system in (with x y 7 60 etc.). Explain how Euclid’s diagram amounts to completing the square. Use it to solve for x (and y) showing your steps.The following four of Euclid’s propositions have essentially the same diagram: Elements II-6 Elements VI-29 Data 59 and Data 84. They all boil down to solving the same quadratic equation. Is Euclid just repeating himself? Using what Katz says about these summarize why Euclid would see these propositions as accomplishing goals that are not identical.Read Proposition II-8. Translate the statement into an algebraic equation. Then reproduce the diagram and label the lengths and areas in the diagram using your variables. If it is not entirely clear from your labeling explain how the diagram illustrates the algebraic equation you wrote. Then verify the proposition using algebra (not Euclid’s way).