Geometrical Astronomical Exercises

Given uniform circular motion and an eccentric solar orbit, determine the ratio of the distance between the centers of the eccentric and ecliptic circles and the radius of the eccentric circle, and where the Sun's apogee is, for the following situations:

1. Spring, summer, fall, and winter all are 915/16 days long.

2. Spring and summer are 1825/8 days long, and fall and winter are 0 days long.

3. Spring and summer are 13631/32 days long, and fall and winter are 4521/32 days long.

4. Spring and fall are 915/16 days long, summer is 1825/8 days long, and winter is 0 days long (no Christmas here!).

stop at this like the rest; but that anyone who was to persuade himself and those in touch with him would have to demonstrate the magnitude and periods of each of the anomalies [two anomalies in each planet's position: one relative to the Sun's position, the second relative to the ecliptic] by clear and consistent appearances; and, putting both together, he would have to find out the position and order of the circles by which these anomalies are produced and the mode of their movement and finally show about all the appearances to be consistent with the peculiar property of this hypothesis of the circles. (Almagest, IX 2)

New here in Greek geometrical astronomy is insistence on a quantitative science. Astronomers must demonstrate quantitatively the size and the period of each planet's position relative to the Sun's position and in the ecliptic. And they must also demonstrate the size and period of each planet's retrograde arc, by means of well-attested observations.

The quantitative attitude may not command as dramatic an appearance as have scientific discoveries and theories throughout history. Nonetheless, the quantitative attitude is an indispensable foundation of modern science and of Western civilization. The transformation from qualitative to quantitative occurred in the study of planetary motions.

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