The tilted sky
In the previous chapter, we worked out the rising times of the ecliptic at sphaera recta: the view of the sky from the Earth's equator, where the celestial equator passes directly overhead and both north and south celestial poles sit exactly at each side of the horizon. That was the geometrically simplest case. Now Ptolemy turns to sphaera obliqua: the sky as seen from any latitude away from the equator, where the horizon is tilted with respect to the equatorial plane. Crucially, this is where everyone Ptolemy was aware of lived.
The difference between sphaera recta and sphaera obliqua is entirely a matter of where you stand. The celestial sphere itself does not change; the stars, the equator, the ecliptic, and the poles are all in the same positions. What changes is the horizon.
Figure 10 — The same celestial sphere, seen from different latitudes. Use the slider or the city buttons to move the observer north or south.
In the upper panel, you can see the observer's position on the globe and how the horizon plane tilts relative to the celestial sphere. In the lower panel, you can see what this observer actually sees: the parallel arcs of the equator, the tropics, and the ecliptic change their inclination across the sky as latitude changes. The diagram shows local noon on the September equinox, with the Sun on the meridian; the gnomon at the centre of the horizon casts a shadow that grows as the observer moves away from the equator.
At the equator (sphaera recta), the celestial poles sit on the horizon, the equator passes through the zenith, and the gnomon casts no shadow at noon on the equinox because the Sun is directly overhead. Move the slider to see how all of this changes at other latitudes.
At the equator, the horizon is a great circle passing through both celestial poles, and it cuts every parallel (the equator, the tropics, and all circles in between) exactly in half. This means that every star spends exactly half of each daily rotation above the horizon and half below it, and every day of the year is twelve hours long, regardless of where the Sun is on the ecliptic.
Move north or south, and the horizon tilts. From the observer's point of view, the celestial lines all appear to tilt across the sky; but they have not shifted on the sphere, because the Earth is negligibly small compared with the distance to the stars (as discussed in Chapter 3). The parallel circles, which were all split evenly at sphaera recta, are now cut unevenly: some arcs spend more time above the horizon than below, others less. This is why days and nights are unequal, and why they change with the seasons.
Ptolemy's known world
Ptolemy had never been to the equator, and as far as we know, neither had any astronomer whose observations he could draw on. He never saw the south celestial pole. His treatment of sphaera recta in the previous chapter was entirely theoretical, extrapolated as a necessity from what he could see. Now he turns to sphaera obliqua, which is the reality for every observer he actually knew of.
Ptolemy believed the entire inhabited world occupied roughly one quarter of the Earth's surface: the region north of the equator and spanning less than half the full circle of longitude. He offered two proofs of this:
- For latitude, he pointed out that noon shadows on the equinox day always point north, never to the south; this means every known observer is north of the equator. (This is indeed the opposite in the southern hemisphere, where sundials have to be constructed with counter-clockwise numbers.)
- For longitude, he noted that the same lunar eclipse, observed from the westernmost and easternmost known places, never shows a local-time difference of more than twelve hours; since twelve hours corresponds to half a daily rotation (half of the earth), the known world spans less than 180° of longitude, fitting within a single quarter of the globe.
In practice, the astronomical world available to Ptolemy was far smaller even than this quarter. The widest longitude span for which he had actual eclipse records was between modern-day Erbil in Iraqi Kurdistan and Tunis in Tunisia. These two cities are separated by only about 23° of longitude, barely a twelfth of the full circle. His theoretical framework was global, but his data came from a narrow strip of the ancient Mediterranean and Near East.
The investigations that follow in Book II all concern the geometry of sphaera obliqua: how the lengths of days and nights, the noon shadows, the rising times, and the angles between the great circles all depend on the observer's latitude.