Homework due October 3, 1995
1. Archimedes' calculation of pi.
(a) What is the distance around the perimeter of a regular hexagon inscribed in a circle of radius 1 (so the points lie on the circle)?
(b) What is the distance around the perimeter of a regular hexagon escribed around a circle of radius 1 (so the sides of the hexagon touch the circle)?
(c) By generous use of Pythagoras' Theorem (which you also needed in (b)) find a better approximation to pi, as Archimedes did, by finding the distance around the perimeter of a regular 12-sided figure inscribed in a circle of radius 1.
(d) How much better an approximation to pi does the 12-sided figure give compared with the 6-sided one? Knowing pi = 3.1416…, does going from 6 to 12 cut the error approximately in half, or a quarter or what? What would you guess that going to a 24-sided figure would do? Archimedes actually went to 96. What kind of accuracy do you think this gave him: one part in 100, one part in 1,000, 10,000 or what?
2. It is sometimes suggested that the darkening of the sky during Christ's crucifixion might have been a solar eclipse. According to the Gospels, the crucifixion occurred on the afternoon after the Jewish Passover meal, which was set on the 15th day of the month Nisan. The Jewish calendar is lunar, so that each month begins with a new moon. So is the suggestion possible or not? Give your reasoning. (From M. J. Crowe, Theories of the World from Antiquity to the Copernican Revolution, Dover.)
3 (a) Show with a diagram why Polaris is never visible at sea level in the Southern hemisphere.
(b) Suppose there is a volcanic island in the Pacific, 90 miles south of the equator, and on a clear night Polaris is just visible on the horizon from the very top of the mountain. Approximately how high is the mountain?
4. Explain the following phenomena in both the Ptolemaic and the Copernican systems:
(a) The time between the autumnal equinox and the winter solstice is not the same as that between the vernal equinox and the summer solstice.
(b) The planets Venus and Mercury never get more than a certain angle away from the Sun.
(c) The planet Mars sometimes loops backwards in its path relative to the background of fixed stars.