The arcs between the equator and the ecliptic
Ptolemy now knows two things: the ecliptic is a great circle tilted at 23½° to the equator, and the two circles cross at the equinox points. His next task is to work out, for every point along the ecliptic, how far that point is from the equator, measured in degrees of arc along a great circle through the poles (see diagram).
The reason Ptolemy needs this is practical. The Sun, Moon, and planets all move along the ecliptic. But the daily rotation, which determines when objects rise above and set below the horizon, happens along the equator. To predict when a given point of the ecliptic will rise above the horizon, Ptolemy must first be able to convert positions on the ecliptic into positions relative to the equator. The table he builds in this section is the first step towards calculating those rising times, which are the subject of the next chapter.
Figure 8 — The ecliptic (gold) and equator (blue) cross at the equinox. At the solstice, the arc between them is the full obliquity: 23½°. At intermediate points, the arc is somewhere in between. Finding its exact value for every point along the ecliptic is the problem Ptolemy sets out to solve.
The question is a geometric one. At the equinox, the ecliptic and equator cross, so the distance between them is 0°. At the solstice (90° as measured along the ecliptic from the equinox) the distance reaches its maximum: the full obliquity of 23½°.
But the relationship between the two is not proportional; moving 45° along the ecliptic does not bring you halfway (11¾°) from the equator. This is due to the geometry of circles intersecting on the surface of a sphere, which is different to how straight lines cross on a plane: the ecliptic pulls away from the equator "slowly" at first near the equinoxes, then more steeply, then slowly again as it levels off near the solstices.
With these tools and the known obliquity of 23½°, Ptolemy works through each degree of the ecliptic from 0° to 90° and produces a complete lookup table converting any position on the ecliptic into a distance, in degrees of arc, from the equator. The table can then give, for example: "something which we know to be X degrees along the ecliptic will be Y degrees away from the equator (above or below)".
Methods
To carry out the calculation, Ptolemy uses two tools he has prepared in earlier sections. The first is his table of chords (Book 1, Chapters 10-11): a lookup table that relates angles to straight-line lengths within a circle, serving the same purpose that a table of sines would serve today. The second (Book 1, Chapter 13) is a theorem from spherical trigonometry due to Menelaus of Alexandria (c. 70–140 CE), which relates the arcs of three intersecting great circles on a sphere.
Declination and the sine function
In modern astronomy, the quantity Ptolemy is tabulating is called declination as a function of ecliptic longitude. Ecliptic longitude is the angle measured along the ecliptic from the spring equinox where the ecliptic crosses the equator. Declination is the angular distance north or south of the celestial equator: the celestial equivalent of latitude on a map.
Today this conversion is a single formula: the declination δ of a point at ecliptic longitude λ is given by sin δ = sin ε · sin λ, where ε is the obliquity. One line of trigonometry replaces several pages of chord-table arithmetic.
Ptolemy did not have trigonometry in this form. The sine function was not yet in use in the Greek-speaking world; it would be developed by Indian mathematicians several centuries later. Instead, Ptolemy worked with chords: the straight-line distance across a circle subtended by a given angle. Chords are closely related to sines, but require more cumbersome calculations. Every step that a modern astronomer would handle with a single sine or cosine, Ptolemy handled by doubling an angle, looking up the chord, and dividing. His results are accurate to within a few arc-minutes of modern values.