In the next chapter, Ptolemy gives some configurations of stars that he considers important. These are in addition to stars being on straight lines with one another which we encountered previously. He outlines three general categories.
Configurations on the Sphere
The first of these Ptolemy describes as
those considered with respect to the planets, sun and moon, or the parts of the zodiac alone.
In short, configurations of this category are purely in the celestial sphere.
This category gets subdivided.
The first category is
for all stars in general, when the fixed star and planet come to be on the same circle through the poles of the ecliptic, or on circles which are different, but at intervals [of a regular polygon] with three, four, or six angles.
The first half of this is pretty straightforward. Ptolemy is telling us that it is significant when a planet is on the same great circle through the pole of the ecliptic as a particular star – i.e., it has the same ecliptic longitude.
He doesn’t say why, but the answer is that this is a tie in to astrology which helps the second half make more sense. There, he imagines the plane of the ecliptic, inscribed with a triangle, square, and hexagon. If the great circle through any of the points on those figures and a planet then touch a fixed star, it is considered significant, astrologically.
This concept is known as the aspects with the three figures being referred to as the trine, quartile, and sextile respectively. You can read more on it here.
Essentially, if a star doesn’t have the same ecliptic longitude, but has an ecliptic longitude that’s a multiple of $120º$, $90º$, or $60º$ away from the ecliptic longitude of the planet, it’s considered significant1.
Next, Ptolemy calls attention to
some stars in particular, those for which one of the planets can pass directly below it (these stars fixed in that narrow band of the zodiac containing the latitudinal motions of the planets) – for these, [configurations] with the five planets concern their apparent contacts or their occultations, and with the sun and moon, their last visibilities, conjunctions and first within the rays of [one of] the luminaries and begins to become invisible.
In other words, Ptolemy takes note of which stars have the same ecliptic longitude as the extrema of the latitudes for the various luminaries.
And there’s several ways those stars can be related to those luminaries.
The first he gives is their “contacts or their occultations” which Ptolemy defines as “when [the fixed star] is covered by the centre of [one of] them.”
A true planetary occultation of a naked eye star would be very rare indeed as their angular diameter is so very small. However, at some distance away, the background star would be unable to be distinguished from the planet. Essentially, our visual resolution would cause them to look as if they were one object. This is what Ptolemy refers to instead of a true occultation, defined by when the star was actually behind the object.
He defines “last visibility” as “the situation when a star falls within the rays of [one of] the luminaries and begins to become invisible”.
In other words, this is the hairy edge of being able to resolve the background star.
Configurations with respect to the Earth alone
The second broad category Ptolemy describes as those
configurations of the fixed stars with the earth alone
which he notes are often referred to as “cardines” and which Ptolemy divides into four sub-categories – the ascendant, the culminating above the earth, the descendant, and the culminating below the earth.
Again, these are all used in astrology, but these are also positions which we have encountered previously. Ascendant describes stars that are rising. Upper culmination is stars at their highest point above the horizon, when they are on the meridian. Descendant is setting. Lower culmination is the stars reaching their lowest point below the horizon which we cannot see, but is also on the meridian.
These definitions are not provided here by Ptolemy. He instead gives us a run on paragraph describing the motions of the sky from different latitudes.
He begins with the situation in which the observer is on the equator:
Now, in the region where the [celestial] equator is in the zenith, all the fixed stars rise and set once in every revolution reach culmination above the earth, and one culmination below the earth; for in that situation, the pole of the equator lie on the horizon, and do not make any parallel circles, either always visible or always invisible.
Next, he describes the motions at the terrestrial poles:
And in the regions where [one of] the [celestial] poles is in the zenith, none of the fixed stars either rises or sets. For in that situation, the [celestial] equator assumes the position of the horizon, and one of the hemispheres into which it divides [the heavens] rotates always above the earth, while the other rotates always below the earth. Hence, each star repeats the same type of culmination twice in one revolution, some reaching culmination above the earth twice, the others culmination below the earth twice.
This is where things start getting a bit funny because at the poles, stars neither rise, nor set. Thus, ascendant and descendant positions are not defined.
Similarly, at the true terrestrial pole, stars will also have no motion up or down in the sky – they are entirely parallel to the horizon.
However, Ptolemy isn’t necessarily describing the true pole. Rather, he is describing the “region” near the pole.
In that case, some stars very near the horizon may rise or set, but most stars will still be circumpolar, never rising or setting. In this instance, they still have a highest and lowest point, but never will their lowest point be below the earth. Thus, Ptolemy chooses to describe both culminations as “above the earth” even though one is higher than the other, hence the “twice” he describes.
Similarly, there will be stars that are always below the horizon and while one will be closer to the horizon than the other, they will again both be below the horizon.
Next, he discusses the intermediate latitudes:
But at the other, intermediate, terrestrial latitudes, some of the [parallel] circles are always visible, and some are always invisible; so the stars cut off between these and the pole neither rise nor set, and perform two culminations in each revolution; those stars in the region which is always visible [culminate twice] above the earth, and those in the region which is always invisible [culminate twice] below the earth. The remaining stars, which lie on parallels greater [than the always visible and invisible parallels], both rise and set, and culminate once above the earth and once below the earth in each revolution.
For these intermediate latitudes, Ptolemy is telling us that some stars are circumpolar while other rise and set.
For these stars the time [of travel] from any one of the cardines back to the same one, is the same at every place: it comprises one revolution, to the senses2. The time from one cardine to the one diametrically opposite is the same at every place when one considers the meridian [passage], since it comprises half a revolution.
In this segment, Ptolemy is saying that the time it takes for a star to go from one of these cardines through the other three and back to the same one, is always the same. Although Ptolemy doesn’t state it, this is the definition of a sidereal day and it doesn’t matter where on the earth you are – the amount of time is always consistent.
When one considers the horizon [passage], it is again constant where the equator is in the zenith: each of the two intervals [from rising to setting and from setting to rising] comprises half a revolution, since in that case, all the parallel circles are bisected, not only by the meridian, but also by the horizon. However, at all other terrestrial latitudes, if one takes separately the time spent above the earth and the time spent below the earth [by a star], neither is the same for all stars [at a given latitude]; nor is the time spent above the earth for any particular star equal to the time it spends below the earth, except for those stars which happen to lie precisely on the equator; for the latter is the only circle which is bisected by the horizon at sphaera obliqua too, whereas all the other [parallels] are divided [by the horizon] into arcs which are neither similar nor equal.
While the time to go through the full cycle of cardines may be the same everywhere, the length between successive ones is not unless the observer it at sphaera recta3. The reason is that the great circle of the horizon is not perpendicular to the celestial equator at those latitudes, and thus, does not evenly bisect the north and south celestial hemispheres.
Furthermore, in accordance with this, the time from rising or setting to one or other of the culminations is equal to the time from the same culmination to setting or rising, since the meridian bisects those segments of the parallels which are above and below the earth; but the times from rising or setting to the two [opposite] culminations are unequal at sphaera obliqua, but equal at sphaera recta, since only in the latter situation are the whole segments [of the parallel circles] above the earth equal to the segments below the earth. Hence, for sphaera recta, [heavenly bodies] which culminate simultaneously always rise and set simultaneously too (in so far as their motion about the poles of the ecliptic is imperceptible4; but, for sphaera obliqua, [heavenly bodies] which culminate simultaneously neither rise nor set simultaneously, but the more southerly ones always rise later and set sooner than the more northerly.
In this final section, Ptolemy extends this to discuss the rising and setting times of stars as unequal.
Toomer has a summary that I think is quite good:
If $a$ is the time from rising to upper culmination, $b$ from upper culmination to setting, $c$ from setting to lower culmination, and $d$ from lower culmination to rising, then $a = b$ and $c = d$ but (at sphaera obliqua) $a \neq c$ and $b \neq d$.
Configurations with respect to the Earth and Celestial Sphere
The last broad category Ptolemy defines is
The accepted configurations of the fixed stars considered with respect to the earth and at the same time, to the planets or the parts of the zodiac.
This again gets divided into a couple sub-categories.
The first is the
risings, culminations, or setting which are simultaneous with those of one of the planets or with some part of the zodiac.
Ptolemy doesn’t give any more information on this sub-category which seems fair as these are fairly straightforward in concept. These are essentially the phaenomena that we discussed when investigating the history of the star catalog.
The second sub-category is
configurations with respect to the sun, which are of nine types.
Yikies. That’s a lot of sub-sub-categories.
The first type of configuration is that called “dawn easterly”, when the star is on the eastern horizon together with the sun.
In short, if the star lies on the great circle of the horizon for the sun on a given day, from the observer’s latitude, then it gets this classification.
However, Ptolemy breaks this down even further.
One variety of this is called “dawn invisible later rising”, when the star, which is just at last visibility, rises immediately after the sun; another is called “dawn true simultaneous rising”, when the star arrives at the eastern horizon precisely the same time as the sun; the third is called “dawn visible earlier rising”, when the star, which is jut at first visibility, rises before the sun.
Essentially, these distinctions allow for the possibilities that the star in question is not exactly on the great circle of the horizon at the same time as the sun, but is a bit ahead of or behind it.
The second type of configuration is that called “dawn culmination”, when the sun is on the eastern horizon while the star is at the meridian, either above or below the earth.
Again, this is straightforward: When the sun rises, the star is on the meridian.
Of this too, there are varieties: one is called “dawn invisible later culmination”, when the star culminates immediately after sunrise; a second is called “dawn true simultaneous culmination”, when the star culminates at the same time as the sun rises; and the third is called “dawn earlier culmination”, when the star culminates immediately before sunrise.
These varieties again give the wiggle room for a star not being exactly on the meridian as the sun rises.
The third type of configuration is that called “dawn westerly”, when the sun is on the eastern horizon and the star on the western. This too has varieties: one is called “dawn invisible later setting”, when the star sets immediately after sunrise; a second is called “dawn true simultaneous setting”, when the star sets at exactly the same time as the sun rises; and the third is called “dawn visible earlier setting”, when the sun rises immediately after the star has set.
If you’re not getting the pattern here, Ptolemy is effectively comparing the sun at one caridine with stars at the other and then allowing for the wiggle room. So I’ll just go through the rest without commentary:
The fourth type of configuration is that called “meridian easterly”, when the sun is on the meridian and the star is on the eastern horizon. This too has varieties: one during the day and invisible, when the sun is culminating above the earth as the star is rising; the other during the night and visible, when the sun is culminating below the earth as the star is rising.
The fifth type of configuration is that called “meridian culmination”, when the sun and star both reach the meridian at the same time. This too has varieties: two are during the day and invisible, when the sun is culminating above the earth and the star is either culminating above the earth together with the sun, or else culminating below the earth opposite it; and two during the night, when the sun is culminating below the earth; one of these is invisible, when the star too culminates below the earth together with the sun, and the other is visible, when the star culminates above the earth opposite it.
The sixth type of configuration is that called “meridian westerly”, when the sun is on the meridian and the star is on the western horizon. This too has varieties: one during the day and invisible, when the sun is culminating above the earth as the star is setting; the other during the night and visible, when the sun is culminating below the earth as the star is setting.
The seventh type of configuration is that called “evening easterly”, when the sun is on the western horizon and the star on the eastern. This again has varieties: one is called “evening visible later rising”, when the star rises immediately after the sun has set; another is called “evening true simultaneous rising”, when the star rises at the same time as the sun sets; the third is called “evening invisible earlier rising”, when the sun sets immediately after the star has risen.
The eighth type of configuration is that called “evening culmination”, when the sun is on the western horizon and the star is on the meridian either above or below the earth. This too has varieties: one is called “evening later culmination”, when the star culminates immediately after sunset (when the culmination is above the earth, this is visible); another is called “evening true simultaneous culmination”, when the star culminates at the same time as the sun sets; the third is called “evening invisible earlier culmination”, when the sun sets immediately after the star has culminated.
The ninth type of configuration is that called “evening westerly”, when the star is on the western horizon together with the sun. This too has varieties: one is called “evening visible later setting”, when the star, just at last visibility, sets immediately after the sun; another is called “evening true simultaneous setting”, when the star sets at exactly the same time as the sun; and the third is called “evening invisible earlier setting”, when the star, which is just at first visibility, sets [just] before the sun.
And that’s it for this chapter.
In the two chapters remaining in Book VIII, Ptolemy will explore how we determine some of these configurations, discussing again the mathematics of the celestial sphere.
- I’ll pause to note that in the astrology I’ve seen, I don’t tend to find interest in actual stars, but what points are on the zodiac at these intervals.
- Toomer suggests that this phrase “to the senses” is tacked on because Ptolemy is aware this isn’t strictly true as precession changes this but in an imperceptible manner over a single revolution.
- Which is to say, on the equator. It’s been awhile since we’ve used these terms.
- Toomer’s interpretation of this passage is that Ptolemy is including planets in this statement which suggests that he considers planets to be a type of star, which is consistent with other passages.