Zodiac rising times for someone at the equator
Now that we have a table translating positions on the ecliptic into positions relative to the equator, we can put it to use to answer a deceptively simple question: how long does it take for a given stretch of the ecliptic to rise above the horizon? The answer depends on where the observer is standing on the earth, so we begin with the simplest case: an observer standing on the geographical equator.
The celestial equator at the geographical equator
Where one stands on the earth determines the angle at which the celestial equator crosses the sky. In the Ptolemaic geocentric model, the entire sky is a hollow sphere that rotates around the celestial poles; in the Copernican model, the earth rotates within a stationary sphere. Either way, the observer's latitude determines the inclination of the equator relative to the horizon.
In Copernican heliocentric reality, the geographical equator (which is itself the widest middle section of the spherical earth) traces a line along the sky when the earth rotates. On astronomical scales, one's latitude does not change one's distance from the geographical equator (everyone is always at the centre of the celestial sphere), but it does determine one's inclination towards the equator.
For someone in Europe, the ground is a plane tilted relative to the plane of the earth's equator, and the celestial equator crosses the sky at a corresponding slant. But for someone on the geographical equator (in Macapá in Brazil, Kampala in Uganda, or Pontianak in Indonesia, for example) the ground is exactly perpendicular to the axis of rotation. The celestial equator rises from the eastern horizon exactly vertically, passes through the zenith directly overhead, and descends vertically to the western horizon.
At such a location, the celestial equator stands straight up from the horizon, passing directly overhead through the zenith. This configuration has traditionally been called sphaera recta: the upright sphere.
The simplicity of sphaera recta
This is the simplest case to analyse, because the symmetry is clean. For every observer on earth, the horizon divides the celestial equator exactly in half. But for someone at the geographical equator, the horizon also divides every other great circle into two equal halves, including the ecliptic. That means a full quadrant of the ecliptic (90°, or three zodiac signs) must rise in the same time as a full quadrant of the equator (also 90°). The question is only how that time is distributed among the three signs within the quadrant. The same relationships repeat for each of the four quadrants.
To answer "how long," Ptolemy needs a clock. His clock is the equator itself. The equator rotates at a perfectly uniform rate (one full turn every day), so the number of degrees of equator that pass above the horizon in a given interval of time is a direct measure of that time. The equator is a circle of 360°, so if 15° of the equator cross the horizon, one twenty-fourth of a day has passed (since 360° ÷ 24h = 15° per hour). In traditional astronomy, these units are called equinoctial time-degrees.
The equator is the only circle on the celestial sphere that rotates at a truly uniform rate, which makes it ideal as a timekeeping device. The ecliptic, by contrast, is useless as a clock. Because it is tilted relative to the axis of rotation, different parts of the ecliptic cross the horizon at different speeds. Thirty degrees of ecliptic near the equinox might take less time to rise than thirty degrees near the solstice. This is precisely the problem at hand: the ecliptic's "clock" runs unevenly, so we must always translate ecliptic motion into equator motion (equinoctial time-degrees) to know how much real time has elapsed for something moving along the ecliptic (such as a zodiac constellation, or one of the "wanderers": the Moon or the planets).
The quantitative question, then, is this: when a given number of degrees of the ecliptic rises above the horizon, how many degrees of the equator rise with it?
Lines intersecting on a sphere
The ecliptic and the equator are both great circles on the celestial sphere, and a great circle is the straightest possible line on a curved surface. But two "straight lines" on a sphere behave differently from two straight lines on a flat surface. On a flat surface, two crossing lines always maintain the same angle between them. On a sphere, the angle between two great circles changes continuously as you move along either one. This can be seen when the celestial sphere is spread out as a flat map: if the equator is kept as a straight horizontal line, the ecliptic appears as a wave that crosses it at the equinox points and then curves away to either side, reaching its greatest distance at the solstices.
Using the table from the previous chapter together with the geometry of the sphere, Ptolemy calculates the rising times for each zodiac sign in the first quadrant (from the vernal equinox to the summer solstice).
Figure 9 — At sphaera recta, the equator (blue) stands vertical and the ecliptic (gold) crosses it at 23½°. Use the slider to advance the daily rotation and watch the first three zodiac signs rise from the eastern horizon (left). Black lines connect each sign boundary on the ecliptic to its corresponding point on the equator, showing that equal 30° arcs of the ecliptic correspond to unequal arcs of the equator: 27.9°, 57.8°, and 90.0° cumulatively.
The first sign (Aries, from 0° to 30° of the ecliptic) rises with 27;50° of the equator. The second sign (Taurus, from 30° to 60°) rises with 29;54°. The third sign (Gemini, from 60° to 90°) rises with 32;16°. The sum of all three is exactly 90°, as it must be: one full quadrant of the ecliptic rises with one full quadrant of the equator.
Ptolemy also calculates rising times for every 10° section of the ecliptic, noting that for arcs smaller than 10°, a simple proportional estimate is accurate enough. He presents the results in a table beginning at the equinox: one of many numerical tables in the Almagest.
Modern astronomy: right ascension
What Ptolemy is computing here is, in modern terms, the right ascension of each point on the ecliptic. Right ascension is the angular distance measured eastward along the celestial equator from the vernal equinox to the point where a great circle through the poles drops down from a given position on the ecliptic.
Ptolemy's rising-time table at sphaera recta is essentially a table of right ascensions, expressed in degrees. Modern astronomers use the same quantity but conventionally express it in hours, minutes, and seconds (where 1 hour = 15°) rather than degrees. A star at right ascension 6h 00m is at 90° from the equinox, which is the same as saying it rises with 90° of the equator at sphaera recta.
The connection is direct: at sphaera recta (and only at sphaera recta), the rising time of a point on the ecliptic equals its right ascension. At any other latitude, the rising time differs from the right ascension by an amount that depends on the observer's latitude and the object's position on the ecliptic.