Having laid out the general structure of the milky way, Ptolemy next describes the process of “the construction of the solid globe and the delineation of the constellations”.
He begins by describing what color to make the globe and describes it as
somewhat deep, so as to resemble, not the daytime, but rather the nighttime sky, in which the stars actually appear.
Ptolemy then stars laying out the coordinate grid with the ecliptic and its poles:
We take two points on it precisely diametrically opposite, and with these as poles draw a great circle: this will at all times be in the plane of the ecliptic.
In this step, he states we should mark the ecliptic and its poles.
Next, he begins laying out the lines of ecliptic longitude.
At right angles to the latter and through [the poles of the ecliptic] we draw another [great] circle, and starting from one of the intersections of this with the first circle we divide the ecliptic into the [conventional] $360$ degrees, and write by it the numbers at intervals of however many degrees seems convenient.
Ptolemy is not stating that we need to draw $360$ individual circles to divide up the ecliptic longitudes, but rather, only as many as seems fitting for the space allowed.
Then we make, from a tough and unwarped material, two rings with rectangular cross-section, accurately turned on a lathe in all dimensions: one should be smaller [than the other], and fit closely to the globe on the whole of its inner surface, while the other should be a little larger than this. In the middle of the convex face of each ring we draw a line accurately bisecting its width. Using these lines as guides we cut out one of the latitudinal sections defined by the line over half the circumference, and divide [each of] the semi-circular recessed sections [thus created] into $180$ degrees.
This part gets rather confusing as Ptolemy isn’t telling us where he’s going with this. So I’ll pause and post a picture from Neugebauer’s History of Ancient Mathematical Astronomy.
What Ptolemy is going to be doing is creating two rings which can be rotated around the globe. In this image, Neugebauer has drawn them as only half-rings, but since they can be rotated around, this would work either way.
The inner ring will be used to measure ecliptic latitude and will be used to lay out the stars on the globe.
The outer ring will act as a meridian. Although Ptolemy tells us that it should be graduated, he isn’t particularly clear on how. Since the definition of a meridian requires that it go through the celestial poles, that means this ring could be used to measure declination. In that case, we could create the graduations starting at $0º$ beginning $90º$ from the poles (i.e., on the celestial equator) and going up to $90º$ at each pole. Or we could do it the other way around and measure the angular distance from the north celestial pole (i.e., polar distance). I prefer the former.
Turning back to Ptolemy’s description, he tells us that the rings should have a “rectangular cross-section.” If that doesn’t make sense, think of cutting the ring radially. The shape of the cut is the cross section. In this case, Ptolemy says it should be rectangular which would be like a washer as opposed to a torus.
Although Ptolemy doesn’t say how much larger the outer ring should be, we can estimate by its function that its inner edge should match the outer edge of the smaller ring the same way the smaller ring’s inner edge should “fit closely to the globe on the whole of its inner surface”.
The portion about cutting the ring on its convex face is also rather confusing but Toomer offers a clear explanation in a footnote again pointing to a figure from HAMA:
Here, we can see that half of the ring has been cut away and the scale inscribed on this inner portion. This is because, if we affix the ring to the globe at point $P$, then the line around this ring that is directly above the globe is that center line and placing the markings on the outer edge would cause the user to observe in the wrong place over the globe’s surface.
When this is done, we take the smaller of the rings as the one which will always represent the circle through both poles, that of the equator and that of the ecliptic, and also through the solstical points ([this circle runs] along the plane surface of the above mentioned recessed section), and, boring holes through the middle of it at diametrically opposite points at the ends of the recessed section, we attach it, by mans of pins [through those holes], to the poles of the ecliptic which we took on the globe, in such a way that the ring can revolve freely over the whole spherical surface.
In short, Ptolemy is saying to affix the inner ring to the poles of the ecliptic.
Since it is not reasonable to mark the solstical and equinoctial points on the actual zodiac of the globe (for the stars depicted [on the globe] do not retain a constant distance with respect to these points), we need to take some fixed staring-point in the delineated fixed stars. So we mark the brightest of them, namely the star in the mouth of Canis Major [Sirius], on the circle drawn at right angles to the ecliptic at the division forming the beginning of the graduation, at the distance in latitude from the ecliptic towards its south pole recorded [in the star catalogue]. Then, for each of the other fixed stars in the catalogue in order, we mark the position by rotating the ring with the graduated recessed face about the poles of the ecliptic: we turn the face of its recessed section to that point on the [globe’s] ecliptic which is the same distance from the beginning of the numbered graduation (at Sirius) as the star in question is from Sirius in the catalogue; then we go to that point on the graduated face which we have [thus] positioned which is, again, the same distance from the ecliptic as the star is in the catalogue, either towards the north or the south pole of the ecliptic as the particular case may be, and at that point we mark the position of the star; then we apply to it a spot of yellow colouring (or, for some stars, the colour they are noted [in the catalogue] as having), of a size appropriate to the magnitude of each star.
Here, Ptolemy is telling us that, because the ecliptic longitude drifts slowly westward1, we shouldn’t affix a set coordinate system since it will change over time.
If you’re feeling like this contradicts the previous instructions regarding drawing on the celestial equator, you’re not alone.
Initially, I thought these could be reconciled by interpreting the above instructions to indicate that Ptolemy only intended for lines of longitude to be drawn on, but the coordinates not be attached.
This would seem consistent with how Ptolemy tells the reader to position the stars. He says not to use a fixed coordinate system, but rather, to determine all positions relative to Sirius. To do that, we would need to count the lines of longitude to determine the difference in longitude, even if we didn’t necessarily know what those lines of longitude were at a given time.
However, I fail to be able to square this explanation with Ptolemy’s statement above that we should “write by [the lines of longitude] the numbers”. I can only interpret this to mean that the lines of longitude should be explicitly defined.
Regardless, this section does give a clear method by which to lay out the stars and provides instructions that they be colored2 and the size varied based on the magnitude.
As for the configurations of the shapes of the individual constellations, we make them as simple as possible, connecting the stars within the same figure only by lines, which moreover should not be very different in colour from the general background of the globe. The purpose of this is, [on the other hand], not to lose the advantages of this kind of pictorial description, and [on the other] not to destroy the resemblance of the image to the original by applying a variety of colours, but rather to make it easy for us to remember and compare when we actually come to examine [the starry heaven], since we will be accustomed to the unadorned appearance of the stars in their representation on the globe too.
In short, Ptolemy doesn’t want us muddying up the globe too much and wants it to be a tool used to learn the sky as it appears. If we add too much to it, we may feel lost when trying to observe the actual night sky.
Thus, he recommends the lines of the constellation be drawn on only lightly. Neugebauer interprets these lines as the “boundaries of the constellations” however, I believe Ptolemy is telling us to draw the “stick figure” of the constellation as he clearly states we should be “connecting the stars”. What stars would represent the constellations boundaries?
We also, then, mark the location of the Milky Way on [the globe], in accordance with its positions, arrangements, densities and gaps as described [in VIII.2].
There’s not much to expand upon here. Ptolemy suggests the inclusion of the milky way. The only comment I can add is that evidently this was not necessarily common practice. In the Introduction to the Phaenomena, Geminus tells us “The Milky Way is not inscribed on most spheres.”
Then we attach the larger of the rings, which will always represent a meridian, to the smaller ring which fits around the globe, on poles coinciding with those of the equator. These points [the poles of the equator] are, in the case of the larger, meridian [ring] attached, again, at the diametrically opposite ends of the recessed and graduated face (which will represent the [section of the meridian] above the earth); but in the case of the smaller ring, [which passes] through both poles, they will be fixed at the ends of diametrically opposite arcs which stretch the $23;51º$ of the obliquity from each of the poles of the ecliptic. We leave small solid pieces in the recessed parts of the rings, to receive the bore holes for the attachments [of the pins representing the poles].
Here, Ptolemy tells us we should affix the larger of the two rings to the smaller one at a point $23;51º$3 from the pivots for the inner ring.
To do so, we will need to leave the full width of the smaller ring at those points (as opposed to cutting it to the center line) so there’s actually metal surrounding the point we would drill the holes to pin the larger ring.
This larger ring will be the meridian ring. But immediately there seems to be a problem.
The meridian is defined as the great circle that goes through the poles of the celestial equator and since the poles about which this ring can be rotated are attached to the inner ring and not the actual poles on the globe, there is no guarantee that the poles of this ring would actually be at the celestial poles. Thus, I take exception to Ptolemy’s statement that this ring will “always” be a meridian.
However, Ptolemy tells us how to set things up to ensure that it will be:
Now the recessed face of the smaller of the rings must, clearly, always coincide with the meridian through the solstical points. So on any occasion [when we want to use the globe], we set it to that point of the ecliptic graduation whose distance from the starting-point defined by Sirius is equal to the distance of Sirius from the summer solstice at the time in question (e.g., at the beginning of the right of Antonius, $12 \frac{1}{3}º$ in advance). Then we fix the meridian ring in position perpendicular to the horizon defined by the stand [of the globe]4, in such a way that it is bisected by the visible surface of the [base], but can be moved round in its own plane: this is in order that we may, for any particular application, raise the north pole from the horizon by the appropriate arc for the latitude in question, using the graduation of the meridian [to place the ring correctly].
Another image from Neugebauer will help:
Here, we can see Neugebauer’s version of the stand. Basically a flat table with a bowl cut out. At opposite ends, there would be a piece that extends inwards with a groove in which the meridian ring, $m$, could be placed to hold the globe such that the meridian would be perpendicular to the surface, which represents the horizon. It could then be rotated so that point $N$ is the appropriate angle5 above the horizon which would be easy to determine using the graduations on the same ring.
However, that still doesn’t necessarily mean that $N$ is at the north celestial pole.
To ensure it is, we would first rotate the inner ring so it goes through the summer solstice. Since the solstice moves and therefore isn’t marked, Ptolemy tells us that we should determine its location by measuring from Sirius using our knowledge of Sirius’ distance from the summer solstice, which he reminds us is $12 \frac{1}{3}º$ in advance for the epoch he uses. Applying the constant of precession we can advance this forwards or backwards in time as needed.
Doing this places places the ecliptic poles and solstices6 all on the same line and also has the result of putting $N$ at the position of the north celestial pole for that moment.
That way, so long as the inner ring is kept fixed with respect to the globe, the user is free to rotate the globe (so long as the inner ring is rotated with it, keeping $N$ fixed) about the celestial poles allowing the user to see how the sky would behave from the latitude set with respect to the horizon.
We shall suffer no disadvantage from our inability to mark the equator and the solstical points on the globe itself. For since the face of the meridian is graduated, the point between the poles of the equator which is $90º$ of the quadrant distant from both will be equivalent to the points on the equator, while the points $23;51º$ distant from that point will be equivalent to the points on the two solstical circles, the one to the north to those on the summer solstical circle, and the one to the south to those on the winter solstical circle. Thus, when any star is rotated with the primary, east-to-west rotation to the graduated face of the meridian, we can again, by means of that same graduation, determine its distance from the equator or the solstical circles, as measured on the great circle through the poles of the equator.
Here, Ptolemy tells us that, even though we don’t have the celestial equator drawn on, we can still figure out the things we might want.
If we want to determine the point on the ecliptic which are the equinoxes, we can rotate the globe (in the stand and keeping the latitude ring fixed) until we find the point that the ecliptic falls at exactly $0º$ on the meridian ring (i.e., $90º$ from the celestial equator).
Similarly, we can find the solstices by rotating the globe to find the points at which the ecliptic lies directly beneath $\pm 23;51º$ on the meridian ring.
As a final note, my interpretation is that the outer ring should be constructed the same as the inner. That is to say, it is cut away on one side down the center line. However, Neugebauer offers another interpretation in which a channel is cut down the center line, allowing for a star to be sighted through it.
I personally do not like this solution as it makes it difficult to tell which star you’re looking at in the event that a few stars are close together.
Overall, I think this is a very clever sort of globe. Ptolemy tells us that Hipparchus had a celestial globe and I wonder how similar it would have been to this one. Furthermore, I wonder how possible it would be to add a ring to the base to allow for altitude and azimuth measurements to be reliably taken.
- Ptolemy would have described this as an eastward drift of the stellar sphere, but it shakes out the same.
- Neugebauer notes that Ptolemy tells the reader to refer to the descriptions in the star catalog for colors other than yellow. However, the only other color given is “red” or “reddish”. Nowhere does Ptolemy describe stars as “white” or “blue”. Neugebauer wonders if Ptolemy intended for stars described as “bright” to be this color. I find this unlikely since there are bright stars that are red. For example, α Tau, Ptolemy describes as both “bright” and “red”.
- As a reminder, this is the obliquity of the ecliptic Ptolemy uses which is a bit high.
- Ptolemy hasn’t actually told us anything about the stand, nor will he. However, he evidently intends for the globe to be placed in one. Grasshoff considers the possibility that this stand may have a channel cut out in which the meridian ring could be placed to ensure perpendicularity.
- I.e., the same as the latitude.
- Points $P$, $P’$ and $SS$ in the first image.