## The Hipparchan Solar Model

The third book of the Almagest is devoted to the study of the motion of the Sun and presents the theory that Ptolemy attributed to Hipparchus. In one year the Sun travels 360 degrees along the zodiac, beginning from a given fixed star, passing through the 12 constellations of the ecliptic, and returning again to the same fixed star. This is known as the sidereal year (from sidus, Latin for "star") because the motion is measured with respect to the fixed stars. If the motion of the Sun is measured relative to the first point of Aries—one of the two points of intersection of the ecliptic and the celestial equator—one obtains another measure for the length of the year, the so-called tropical year. The two years are almost the same, differing only by a very small amount. The slight discrepancy is the result of an effect known as the precession of the equinoxes, first detected and measured by Hipparchus.

The primary characteristic of the Sun's annual motion around the ecliptic is that it takes place with variable velocity: in the spring and summer it moves more quickly along the ecliptic than it does in the fall and winter. As we saw in chapter 2, the Babylonians possessed tables that tabulated the variable solar motion using arithmetic functions of some sophistication. For each year and the beginning of each month these tables tabulated the position of the Sun along the ecliptic, given in terms of the constellation of the zodiac and the number of degrees (from 0 to 30) from the beginning point of the constellation. A comparison of the Hipparchan theory (as reported by Ptolemy) with extant Seleucid Babylonian tables dating from around 300 to 500 b.c. establishes beyond doubt that Hipparchus used these tables in the construction of his theory.

Hipparchus explained the variable solar motion by assuming that the Sun moves about the Earth on a circle whose center is displaced slightly with respect to the Earth. The Earth lies close to the center but is not at the center itself. In figure 3.3 the Sun moves on the circle whose center is at E. The observer on Earth is located at Z. The distance ZE (measured as a fraction of the radius) is known as the solar eccentricity. The direction of the line DA gives a second parameter that, together with the eccentricity, fixes the solar model. The Sun moves uniformly along the circle, but because it is observed from the Earth at a point offset slightly from the center, it appears to be moving more quickly when it is at D than when it is at A. The variable solar velocity is therefore explained as an optical phenomenon resulting from the way the moving Sun is observed from the Earth against the celestial sphere.

Hipparchus was able to compute tables of the Sun's position along the ecliptic as a function of time. He did so using a powerful new mathematical tool, developed by him and later extended by Ptolemy, known as trigonometry. The basic object of study is the triangle, and the central problem is to find a given side or angle, given that one knows the values of two other sides or angles. The angles are measured in degrees, with 360 degrees making up a whole circle and 90 degrees making up a right angle. The most complete exposition of trigonometry is contained in the first book of Ptolemy's Almagest, which explains how to construct a table of chords—a version of what we would call a table of sines—that gives the chords of angles from 0 to 180 degrees in one-half-degree increments.

Using Hipparchus's solar model, Ptolemy produced a table that enabled one to go from the mean, or average, position of the Sun to its true position in the sky. For a given one-year period one considers the position of the Sun at equally spaced intervals along the circle ABGDA. For each of these positions B one uses trigonometry to go from the angle AEB to the angle AZB. The first year begins at some specified point in time called the epoch. To find the position of the Sun at any future time, one determines the number of years elapsed since epoch, calculates the time elapsed since the beginning of the current year, determines for this time the position of the mean Sun, and then uses the table to go from this datum to the true position of the Sun.

Scientific cosmology may be said to have begun with the Hipparchan solar model. First, this model is causal, explaining the phenomena in question—the variable motion of the Sun—from two hypotheses: that the motion of the Sun takes place uniformly in a circle and that this motion is observed from Earth at a point that is slightly offset from the center of the circle. Second, in the Hipparchan model the motion of the celestial body is no longer assumed to

Figure 3.3: Hipparchus's solar model.

take place on a two-dimensional manifold embedded in the celestial sphere. Instead, the Sun follows a trajectory in which its distance from the Earth constantly varies, and a crucial third dimension signifying depth in space is implied in this conception. Third, the phenomenon in question and the accompanying cosmological assumptions are combined with suitable mathematical tools to provide a precise quantitative analysis of the motion that allows systematic comparison with observation and the basis for prediction. In summary, the Hipparchan model provides a coherent and causal explanation that yields an empirically adequate account of the Sun's motion. It was the first contribution to cosmology in the sense of what we would today call a science and pointed the way to the later work of Ptolemy.

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