This very brief description will give some idea of the chief instruments and methods used, and when we see how very rough and elementary they were, and remember that the Greeks had to work out their observations without algebra, or decimal notation, we are amazed at their results, and their far-reaching ambitions. [Sidenote: Eratosthenes B.C. 276-194.] Already in the very early days of the Museum, Eratosthenes, a celebrated geographer, made a bold attempt to utilize observations of the sun measuring the size of the earth. It was known that in Syene (the modern Assuan) on the day of the summer solstice at noon no shadows were thrown, and the bottoms of wells could be seen: evidently therefore the sun was in the zenith. Eratosthenes found that the sun’s distance from the zenith in Alexandria at noon on the same day was 7° 12′, or one-fiftieth of the circumference of the heavenly sphere, consequently the two towns must be distant from one another (assuming them to be nearly in the same meridian) one fiftieth of the circumference of the earth. The distance from Alexandria to Meroe was known, and from Meroe to Syene had been paced by the king’s professional pacers; the whole was 5000 stadia. 50 times 5000 = 250,000. The figure always quoted by the ancients is however 252,000. If the stadium used by Eratosthenes was the measure generally used for long distances which have been paced, this estimate is equal to 24,662 miles, only about 200 miles less than the modern value. It was partly by luck that Eratosthenes got such a good result, for he was evidently only working with round numbers, and the extra 2000 stadia seem to have been added in order to make one degree equal to exactly 700 stadia. But in any case it was a highly creditable performance. [Sidenote: Euclid _c._ B.C. 300.] [Sidenote: Apollonius _c._ B.C. 270.] There were celebrated mathematicians and geometers at Alexandria, whose work was most useful to astronomy, such as Euclid, and Apollonius of Perge. The latter is specially mentioned by Ptolemy in connection with the new theory of “moveable eccentrics,” which was invented to account for the varying brightness of the planets, as well as their peculiar movements. Fig. 23 explains this theory. Let P A be a great revolving circle upon which Mars is fixed. (In the hands of the Alexandrian mathematicians the spheres almost disappear, and they deal practically only with circles.) If the earth were at its centre, as Eudoxus demanded, Mars must always be at the same distance, but if we make the circle eccentric to Earth, by putting its centre at C while Earth is at E, then the distance and consequently the brightness will constantly vary, and Mars will be brightest when at perigee P (point nearest Earth), and faintest when in apogee A (point furthest from Earth).[49]