In the final chapter of Book VIII, Ptolemy discusses the first and last visibilities of stars. Before diving into the text, let’s first take a moment to think about what this means.
The description is, in some ways, self evident – we’re considering when, based on the movement of the sun, are stars first and last visible.
Recall that, when viewed from inside the celestial sphere (i.e., on earth), the sun appears to move west to east along the ecliptic. Thus, as time passes, stars that were washed out by the sun will, at some point, become visible because they have risen while the sun has moved far enough along the ecliptic that it will be sufficiently far below the horizon as to not wash the star out in the morning glow before the sun actually crosses the horizon. This point in time is known as their helical rising and describes when they are first visible in the sky throughout the year.
These stars will, of course, initially be only briefly visible as the daily rotation of the sun will quickly cause them to be lost in the morning glow, but as the sun continues to progress eastward along the ecliptic1, the star will be visible for longer.
Similarly, we can ask at what point the sun has moved sufficiently close to a star that, even after the sun has set, the star will be lost in the evening glow and thus, it is effectively invisible for the season until its next helical rising.
In case you haven’t been checking the dates on these posts, you may well notice that it’s been nearly a month and a half since my last post. It’s because this section has been quite challenging as I find Ptolemy’s descriptions here are an absolute mess. This is probably among the least sensical sections I’ve yet encountered. I think you’ll see why as we go through Ptolemy’s commentary.
First off, we might be tempted to assume that the discussion in the last chapter would be all we would need to cover this topic. After all, the sun’s position is limited to the ecliptic. Thus, if we wanted to know if a star would be visible, we could use the technique discussed in the last post to determine which point of the ecliptic would rise/set with it. From there, we could calculate the position of the sun on the ecliptic to see whether it would be above or below the horizon.
But as I hinted at in my introduction, Ptolemy fully recognizes that the sky does not go from dark to light the instant the sun crosses the horizon.
Thus , Ptolemy opens this chapter by cautioning the reader against trying to approach the problem in such a manner:
[I]n the case of the first and last visibilities [of the fixed stars], we find that the geometrical method expounded [above], using only their position [in latitude and longitude], is no longer adequate. For it is not possible to find the size of the arc by which the sun must be removed below the horizon in order for a given star to have its first or last visibility by methods similar to the geometrical procedures by which, e.g., one demonstrates the point on the ecliptic with which that star rises.
The result of this is that the star must be further removed from the solar position to have its first/last visibility.
For that arc [the arcus visionis] cannot be the same for all stars nor the same for a given star at all places [on earth], but varies according to the magnitude of the star, its distance in latitude from the sun, and the change in the inclinations of the ecliptic [with respect to the horizon].
Ptolemy will get to defining this more explicitly later, but one of the core concepts of this section is the arcus visionis – the arc of visibility. This is not a phrase used by Ptolemy explicitly, but is one that arises later. We’ll explore a rigorous definition shortly, but for now, we can just think of it as the angle the sun needs to be below the horizon before a star on the horizon is visible.
With this concept in mind, Ptolemy delineates three factors we should consider when discussing the visibility of stars:
1) The magnitude of the star
2) The distance in [ecliptic] latitude from the sun
3) The inclination of the ecliptic with respect to the horizon
So let’s explore each of these.
The Magnitude of the Star
It should go without saying that fainter stars will need to be further from the sun as they are more easily lost in the twilight.
To help illustrate, Ptolemy describes the following diagram:
In this, circle $ABGD$ is the meridian for horizon $BED$ with zenith at $H$. Circle $AEG$ is the ecliptic and point $Z$ is some hypothetical position of the sun which is below the horizon.
[I]t is clear that, given a star rising simultaneously with point $E$ of the ecliptic, if a star of greater magnitude has its first visibility when the sun is at a distance of $arc \; EZ$ below the earth, a star of lesser magnitude, even one at an equal distance in latitude from the sun, will have its first visibility when the sun is at a greater distance than $arc \; EZ$2, and [thus] the effect of its rays is weaker.
Ptolemy is really just restating what we said above: Faint stars get washed out in sunrise/sunset first. Thus, the sun will need to be further below the horizon for the same stars to have their first visibility.
However, this passage is where I start to find some trouble with Ptolemy’s description. This is because $arc \; EZ$ is a very odd way to approach this problem. As I state in my footnote, this isn’t the arcus visionis; It’s the arc of the ecliptic below the horizon and doesn’t necessarily tell you much about how far below the horizon the sun is since that will also be dependent on the third factor Ptolemy mentions – the inclination of the horizon to the ecliptic.
Regardless, if we only consider one latitude, it will suffice as a proxy for the time being.
Distance in Ecliptic Latitude of the Star
[F]or stars of equal magnitude, if a star which is closer in latitude to point $E$ has its first visibility at a distance [of the sun from the horizon] of $arc \; EZ$, a star which is farther than that [from point $E$ in latitude] will have its first visibility at a lesser [solar] distance. For, given the same distance of the sun below the horizon, the rays in the vicinity of the ecliptic and of the sun itself are denser than those farther away.
This is a bit hard to parse despite the concept being straightforward. The way I envisioned this was by adding two stars to the circle of the horizon in the above diagram. One just above $Z$, and the other much more towards $D$. This second one would have the prescribed greater distance in ecliptic latitude as it is further from the ecliptic.
What Ptolemy is saying is that, if the first of these was having its first visibility when the sun was at $Z$, the one further away would have had its first visibility sometime earlier in the season.
This is because the sky does not brighten evenly; it brightens most near where the sun is rising. Stars further away, even if crossing the horizon at the same time as a star near the point the sun rises, will thus need the sun to be further below the horizon.
Inclination of the Ecliptic with Respect to the Horizon
[I]n the case of the stars of equal magnitude which rise at equal distances in latitude [from the sun], the more the ecliptic is inclined to the horizon, [thus] making $\angle DEZ$ smaller, the greater the [solar] distance $arc \; EZ$ at which the star will have its first visibility.
This section is where things start falling apart in my opinion. I’ll do my best to try to explain what I think Ptolemy was trying to state here, but ultimately, I’ve been unable to find an interpretation of these passages that fully makes sense.
First, let’s draw in a star on the horizon at some ecliptic latitude.
Now, as Ptolemy described, let’s decrease $\angle DEZ$ and see what happens.
Here, we can see that the point of the horizon at the same ecliptic latitude has moved towards $D$, thereby moving it further away from the sun and making it visible earlier.
The last bit of Ptolemy’s sentence, about the “[solar] distance, $arc \; EZ$ at which the star will have it’s first visibility” is a bit odd. As best I can tell, what Ptolemy is getting at is that, in order to have the two stars both have their first visibility on the same date, in the second image, we’d need to move the sun at $Z$, further towards $G$ to make up for the fact that the star’s position moved that direction as well. This would increase the length of $arc \; EZ$.
Ptolemy evidently knew this passage would be troublesome as he tries to explain this further with a new diagram in which
…we also draw in the [great] circle $H \Theta ZK$ through the poles of the horizon and the sun at $Z$, which will, obviously, be perpendicular to the horizon…
Here’s the image Ptolemy is describing.
This is essentially the same as the previous diagram except for the addition of the great circle through the poles of the horizon, making it an altitude circle.
Ptolemy now considers a star where that circle meets the horizon, at $\Theta$.
In this image,
For if… [we consider this diagram] … the [vertical] distance of the sun below the earth will always remain equal to $Z \Theta$ for the same star, since, for an equal interval so taken, the [effect of] the rays above the earth will be similar; but if $arc \; \Theta Z$ is kept constant, $arc \; EZ$ will, as we said, become less as the ecliptic is raised towards a perpendicular position, and greater as it is more inclined [to the horizon].
The arc described here, $arc \; \Theta Z$ is the definition of the arcus visionis – the altitude of the sun below the horizon and must be measured along this line (albeit without there necessarily being a star at $\Theta$), since altitude is defined as perpendicular to the horizon.
However, this is where things really seem to fall apart in my reading.
Let’s look at the first part of this where Ptolemy states
…the [vertical] distance of the sun below the earth will always remain equal to $Z \Theta$ for the same star…
This is flatly untrue. Let’s take a look at what happens as we increase the inclination of the horizon and ecliptic (i.e., increase $\angle DEZ$).
In this image, I’ve kept the point of the star on the horizon, $\Theta$ consistent since Ptolemy clearly states that this must be for the same star.
However, the shift of the horizon, and thus its altitude circle, means that points $E$ and $Z$ move down the ecliptic, thus increasing the length of $arc \; \Theta Z$. Ultimately, there is no way to change the inclination of the horizon and maintain the length of this arc – the more parallel the altitude circle gets to the ecliptic, the longer the arc gets.
Furthermore, it’s not clear that $arc \; EZ$ changes in any meaningful way.
However, I think there’s an alternative way to interpret this which is as follows:
In this image, I’ve not altered $arc \; \Theta$ to be along the altitude circle for the newly oriented horizon. Rather, I’ve kept it the same, which seems to follow Ptolemy’s statement that “the effect of the rays above the earth will be similar” since this would keep the star the same distance from the sun.
Furthermore, in this case, we clearly see that $arc \; EZ$ does decrease as described.
However, this interpretation simply cannot be correct as we’re about to start setting up some Menelaus configurations, and if we do not keep $arc \; Z \Theta$ as part of an altitude circle, they no longer work correctly.
I believe that Neugebauer also noticed this issue as he has the following to say3:
Ptolemy makes the simplifying assumption that [$arc \; Z \Theta$] as determined for [the initial latitude] is practically independent of the geographical latitude, or rather, he hopes that this is so. For more northerly locations, the decreasing values of [the inclination, i.e., increasing $\angle DEZ$] would require increasing [$arc \; Z \Theta$] but this might be compensated for by the diminishing brightness of the sun in northern climates such that the previous [$arc \; Z \Theta$] might again suffice.
We’ll encounter Ptolemy’s claims about the brightness of the sun at different latitudes shortly.
Effects
So now that we’ve explored these three effects, where does that leave us in our quest to determine the first and last visibilities?
What Ptolemy essentially tells us is that we can’t answer the question purely from mathematics, but that we will need to have an observation of when these occur at one latitude, but then we can use that information to mathematically determine the distance between the point of the ecliptic on the horizon and the position of the sun for other locations. At least, if we hold to the previously mentioned problematic assumption that $arc \; Z \Theta$ is somehow constant.
This is where Ptolemy discusses the brightness of the sun at different latitudes Neugebauer noted.
As he puts it:
[W]e need observations for each individual fixed star in order to determine the [required] distance of the sun below the earth as measured along the ecliptic. And if even the distance vertical to the horizon [ie., $arc \; \Theta Z$] does not remain the same for the same stars at all locations on the earth, because the rays of similar density do not have the same obscuring effect in the thicker air of the more northerly terrestrial latitudes4, we will need observations, not merely at one terrestrial latitude, but at each of the others alike. However, if the arc corresponding to $Z \Theta$ remains constant everywhere on earth for the same stars (as seems likely, since the fixed stars too must be affected by the variation in the atmosphere in the same way as the rays [of the sun] are), the distances observed at a single terrestrial latitude will suffice us to determine those at the other latitudes: [we can do this] by geometrical methods, whether the variation in the inclination of the ecliptic is due to the terrestrial location or to the demonstrated motion of the sphere of the fixed stars towards the rear with respect to [the ecliptic].
To illustrate, Ptolemy refers back to the previous figure and sets up a Menelaus configuration as pictured below.
From this, we can state that:
$$\frac{Crd \; arc \; 2AB}{Crd \; arc \; 2BH} = \frac{Crd \; arc \; 2AE}{Crd \; arc \; 2EZ} \cdot \frac{Crd \; arc \; 2Z\Theta}{Crd \; arc \; 2\Theta H}$$
As usual, let’s go through the pieces.
First off, arcs $BH$ and $\Theta H$ are both $90º$ since they are from the zenith to the horizon.
Second, $arc \; EZ$ is given since, for any date, we can calculate the position of the sun at $Z$, and we have previously shown that we can calculate the point of the ecliptic lying on the horizon, $E$.
Similarly, since we know the point $E$ of the ecliptic, we also know where it will culminate at $A$ and thus know $arc \; AE$.
That leaves us with only the unknown we’re after, $arc \; Z\Theta$.
Once this [$arc \; Z\Theta$] has been found, and provided that it remains the same for all locations, we can use it to derive the amounts of $arc \; EZ$ at [all] other terrestrial latitudes from the same considerations.
Ptolemy continues to use the same diagram and Menelaus configuration, but this time uses the other form of the equation:
$$\frac{Crd \; arc \; 2HB}{Crd \; arc \; 2AB} = \frac{Crd \; arc \; 2H\Theta}{Crd \; arc \; 2Z \Theta} \cdot \frac{Crd \; arc \; 2 ZE}{Crd \; arc \; 2EA}$$
In this, we’re staring with $arc \; Z\Theta$ as known since we just calculated it for one location previously.
Now, we’re going to assume that the position of the horizon is different, but somehow $arc \; Z\Theta$ is preserved.
As with before $arc \; HB$ is $90º$ as it’s the horizon to the zenith.
Additionally, we can determine $arc \; AB$, which is the distance between the culminating point of the ecliptic and the horizon. This one requires looking at the point of the ecliptic that is culminating and first referring to the Table of Inclinations (between the ecliptic and celestial equator). From that, we can then recall that the distance between the horizon and the celestial equator will be $90º$ minus the observer’s latitude. Knowing those two pieces allow us to find $arc \; AB$ and from that, we can also find $arc \; AH$ since it’s $90º – arc \; AB$.
We can determine $arc \; HT$ as being $90º + arc \; Z \Theta$.
Lastly, we’ve already discussed how to find $arc \; EA$ in the last Menelaus configuration.
Thus, the only piece we’re left with the $arc \; EZ$ which is what Ptolemy wanted to find.
Last Visibility
Ptolemy then quickly tells us that this same procedure can practically be used to determine the same arc for the last visibility. The only difference will be that
the ecliptic will be drawn on the other side [of circle $BED$], in accordance with the way it is inclined when the horizon of [$arc$] $BD$ is taken as the western part.
Conclusions
Having laid out a potential path for discussing this topic, Ptolemy declines to go any further stating,
We think that the above suffices as an indication of the methods in this type of theoretical investigation, enough [at least] so that it cannot be said that we have neglected this topic. However, seeing that the computation of this kind of prediction is of great complexity, not only because of the great number of different terrestrial latitudes and the inclinations of the ecliptic involved, but also because of the sheer multitude of the fixed stars; seeing, too, that, in respect of the actual observations of the phases it is laborious and uncertain, since [differences between] the observers themselves and the atmosphere in the regions of observation can produce variation in and doubt about the time of the first suspected occurrence, as has become clear, to me at least, from my own experience and from the disagreements in the kind of observations; seeing, furthermore, that because of the motion [through the ecliptic] of the sphere of fixed stars, even for the individual terrestrial latitudes the simultaneous risings, culminations, and settings cannot remain forever identical with the present ones, which would take such a vast amount of numerical and geometrical computations to calculate, we have decided to dispense with such a time-consuming operation. For the time being, we content ourselves with the approximate [phases] which can be derived either from earlier records or from actual manipulation of the [star] globe for any particular star.
In other words, Ptolemy states he’s laid the foundation for the first and last visibility for those interested in going further, but he has no personal interest in trying to present the first and last visibility for each star at the various latitudes.
Finally, Ptolemy refers back to the impetus for the discussion of this topic in the first place (meteorological astrology), and doubts whether it has any real value:
Moreover, we notice that the prognostications concerning the states of the atmosphere derived from first or last visibilities (if indeed one assigns these as the cause [of changes in the weather], and not rather the positions [of the sun] in the ecliptic), are almost always approximations, and do not exhibit a perfect regularity and invariability; it seems that this causal factor has only general application, and derives its strength, no so much from the actual times of the first or last visibility, as from the configurations with respect to the sun, taken as intervals in round numbers, and, in part, the inclinations of the moon at those configurations.
Here, Ptolemy suggests that the weather isn’t driven so much by the stars as it is by the sun and the latitude – Entirely correct statements.
- Or, to look at it a different way, the star rises 4 minutes earlier each night.
- Note that $arc \; EZ$ is not the arcus visionis although it is related as we’ll see.
- Neugebauer gives different names to the points in the diagram. I have given the equivalent ones from Toomer in their place for consistency.
- This is an interesting claim from Ptolemy. We haven’t discussed the “thickness” of air anywhere else in the Almagest. However, this is a rather imprecise term. If we’re referring to the density of the atmosphere, it is more dense at more northern latitudes due to the lower average temperatures. But the higher temperatures near the equator result in the atmosphere expanding and thus being deeper. So it’s hard to imagine precisely what Ptolemy is trying to suggest here.