(a) The Greek astronomer Aristarchus, in the 3rd century BCE, is thought to have been the first person to have deduced the distance from the Earth to the Moon, using observations

of lunar eclipses. Aristarchus first observed that the diameter of the Moon was approximately one-third of the diameter of the Earth's shadow, at the distance of the Moon. He also knew that the angular diameters of the Moon and Sun, as seen from the Earth, are almost exactly equal, based on observations of solar eclipses. With the aid of a sketch of the geometry of a lunar eclipse, show that these two observations imply that the physical diameter of the Moon is approximately one-quarter of the diameter of the Earth. (b) Now using the fact that the angular diameter of the Moon is close to 0.5°, or a little under 0.01 radians, estimate the distance of the Moon in units of the Earth's diameter ᎠᏯ . (c) About 19 centuries later, the Italian-French astronomer Giovanni Domenico Cassini attempted to estimate the distance to the Sun by combining observations of the planet Mars when it was closest to the Earth, against the background of distant stars, from two widely-separated places on Earth. (He used data from Paris and French Guyana in South America, having sent out an expedition for this purpose.) From these data he calculated that the difference in the direction to Mars would have been approximately 40 arcseconds, if it had been possible to observe it from the Earth's North and South poles simultaneously. [Recall that there are 3600" in 1°, or 206, 265" in one radian.] Assuming that the distance to Mars at the time was 0.5 AU, use this result to estimate the distance to the Sun (i.e., 1 AU) in terms of the Earth's diameter De. In fact, both of these estimates were quite accurate, for their times, and each became the standard value used by later astronomers for a century or more. Nowadays, we know the distance to the Moon accurate to a few centimeters, thanks to laser retro-reflectors left on its surface by the Apollo missions. The AU was not measured accurately until the advent of radar astronomy and interplanetary spacecraft in the 1960s, though many more attempts were made in the interim, mostly using transits of Venus to gauge the diameter of the Sun.