The Pythagorean Theorem - Educational Outreach

---High above the earth, we see the moon hanging inthe evening sky at an angle and inclination that variesaccording to latitude, date, and time. The ancient Greekswere the first to measure the distance to the moon and themoon’s diameter. They actually did this using severaldifferent ways. We will follow suit by employing the modernLaw of Sines, utilizing it one of the same geocentric setupsthat the Greeks used 2000 years ago in order to make thesetwo astronomical determinations. Today, our instrument ofchoice would be a theodolite in order to make the neededprecision angular measurements. The Greeks actually hadan ancient version of the same instrument, called anastrolabe, which allowed them to make the neededmeasurements in their day. Figure 4.11 shows our setupfor both measurements.Obelisk for measuring the Moon’sshadow in the absence of an astrolabeBF0.23620 014.47228 0 0.23620 0 Moon14.47228 0A 3959miles0.5D0Surface distanceEarthfrom B to D is1000milesCMoonFigure 4.11: Measuring the MoonIn order to measure the distance to the moon, firstpick two points B and D on the earth’s surface a knowndistance apart. Figure 4.11 suggests 1000 miles, roughlythe distance that the Greeks used. Both points need tohave either the same latitude or the same longitude. In theUnited States, two points of equal latitude (such as 40 0 N)are probably a tad easily to locate than two points of similarlongitude. Point D should correspond to a time of nightwhere the full moon is straight overhead or squarely in themiddle of the ecliptic as sighted by use of an astrolabe withthe sighting point squarely in the center of the moon’s disk.150