History of Mathematics Homework 3 Due Wednesday, October 6

1. Beginning, as did Archimedes, with a regular hexagon inscribed in a circle, use an Archimedean recursion algorithm to find either p12 and P12 or a12 and A12. What value of π would be implied by the arithmetic mean of your answers? 2. Prove the theorem concerning the “circle of Apollonius;” that is, show that the locus of points whose distance from two fixed points are unequal but are in fixed ratio, is a circle. 3. Solve the “Problem of Apollonius” for the case of two points and a line. In other words, describe a compass-and-straightedge construction that will accomplish the following: Given two points and a line not containing either of the points, to draw a circle passing through the two points and tangent to the line. 4. How can one account for the fact that the period of the rise of Greek trigonometry was a time of decline in Greek geometry? 5. Solve the problem of Diophantus in which it is required to find two numbers such that their sum is 10 and the sum of their cubes is 370. 6. Solve the following problem from the Greek Anthology of Simplicius (fl. 520): if one pipe can fill a cistern in one day, a second in two days, a third in three days, and a fourth in four days, how long will it take all four running together to fill it?

History of Mathematics Homework 3 Due Wednesday, October 6

1. Beginning, as did Archimedes, with a regular hexagon inscribed in a circle, use an Archimedean recursion algorithm to find either p12 and P12 or a12 and A12. What value of π would be implied by the arithmetic mean of your answers? 2. Prove the theorem concerning the “circle of Apollonius;” that is, show that the locus of points whose distance from two fixed points are unequal but are in fixed ratio, is a circle. 3. Solve the “Problem of Apollonius” for the case of two points and a line. In other words, describe a compass-and-straightedge construction that will accomplish the following: Given two points and a line not containing either of the points, to draw a circle passing through the two points and tangent to the line. 4. How can one account for the fact that the period of the rise of Greek trigonometry was a time of decline in Greek geometry? 5. Solve the problem of Diophantus in which it is required to find two numbers such that their sum is 10 and the sum of their cubes is 370. 6. Solve the following problem from the Greek Anthology of Simplicius (fl. 520): if one pipe can fill a cistern in one day, a second in two days, a third in three days, and a fourth in four days, how long will it take all four running together to fill it?