Having laid out his order of the spheres, Ptolemy still has more to say as an introduction to planetary theory before moving on to a specific one.
First, he notes that he will be sticking to uniform circular motion “since these are proper to the nature of divine beings, while disorder and non-uniformity are alien [to such beings].”
Next, he declares that “no-one before us has yet succeeded in” coming up with a suitable theory1. He describes the difficulties that plague attempts to do so:
For, in investigations of the periodic motions of a planet, the possible [inaccuracy] resulting from comparison of [two] observations (at each of which the observer may have committed a small observational error) will, when accumulated over a continuous period, produce a noticeable difference [from the true state] sooner when the interval [between the observations] over which the examination is made is shorter, and less soon when it is longer.
In other words, if you attempt to derive parameters like the mean motion of a planet by using two observations that are close in time, the inherent error in those measurements is a larger proportion than if the time between observations is larger. This statement is true for deriving any of the time based parameters, but is especially bad for slow moving planets.
But we have records of planetary observations only from a time which is recent in comparison with such a vast enterprise: this makes prediction for a time many times greater [than the interval for which observations are available] insecure.
To me, this reads as Ptolemy admitting that he may suffer from the same problem. This is unsurprising for many of the planets (specifically the outer ones) since their orbital periods are so long.
[Secondly], in investigation of the anomalies, considerable confusion stems from the fact that it is apparent that each planet exhibits two anomalies, which are moreover unequal both in their amount and in the periods of their return: one [return] is observed to be related to the sun, the other to the position in the ecliptic; but both anomalies are continuously combined, whence it is difficult to distinguish the characteristics of each individually.
The two anomalies here should be largely unsurprising. Jumping into modern astronomy, one would certainly come from the fact that any planet’s orbit about the sun is elliptical. But superimposed on that from the point of view of an observer on Earth will be the motion of the Earth itself.
But Ptolemy is correct on the challenge here: How will we disentangle these two effect in order to be able to say for certain what each of their periods is? Ptolemy puts off answering that question for now and instead notes the poor quality of the observations available to him for such purposes.
[It is] also [confusing] that most of the ancient [planetary] observations have been recorded in a way which is difficult to evaluate, and crude. For [$1$] the more continuous series of observations concern stations and phases [i.e., first and last visibilities2. But detection of both of these particular phenomena is fraught with uncertainty: stations cannot be fixed at an exact moment, since the local motion of the planet for several days both before and after the actual station is too small to be observable; in the case of the phases, not only do the places [in which the planets are located] immediately become invisible together with the bodies which are undergoing their first or last visibility, but the times too can be in error, both because of the atmospherical differences and because of differences in the [sharpness of] vision of the observers.
Here, Ptolemy tells us that the observations left to him are mostly first and last visibilities which are problematic. It’s quite easy to miss when a planet is first or last visible. Even when you know to be looking for a planet in the morning glow or twilight, they’re easy to miss. Thus, even if an observer says that a planet was first or last visible on a certain date doesn’t mean they’re correct.
But there’s other problems:
[$2$] In general, observations [of planets] with respect to one of the fixed stars, when taken over a comparatively great distance, involve difficult computations an an element of guesswork in the quantity measured, unless one carries them out in a manner which is thoroughly competent and knowledgeable. This is not only because the lines joining the observed stars do not always form right angles with the ecliptic, but may for an angle of any size (hence one may expect considerable error in determining the position in latitude and longitude, due to the varying inclination of the ecliptic [to the horizon frame of reference]); but also because the same interval [between the star and planet] appears to the observer as greater near the horizon, and less near mid-heaven; hence, obviously, the interval in question can be measured as at one time greater, at another less than it is in reality.
I find this passage interesting as it gives some insight to how observations must have been conducted. If I were to try to observe planetary positions, I would attempt to observe a planet as it was near culmination by finding the angular distance from multiple stars with known positions3 using an instrument like a Jacob’s staff. Then, the planet would lie at the intersection of the circles of those diameters around each of the stars.
But what Ptolemy describes here indicates that observers were likely measuring angular distance along or at least parallel to the horizon near these first and last visibilities, in which case, Ptolemy is entirely correct that the line of the horizon would certainly not form a right angle with the ecliptic, thus making its ecliptic longitude poorly established, especially if the star lies further from the ecliptic.
I suspect that there could be some geometrical determination of the angle of the horizon with respect to the ecliptic at the time of observation to account for this, but Ptolemy does not discuss it any further.
It is also interesting that Ptolemy notes the effects of atmospheric distortion as a compounding problem near the horizon.
Ptolemy then suggests that these issues were the reason that
Hipparchus… did not even make a beginning in establishing theories for the five planets, not at least, in his writing which have come down to us.
Instead, Ptolemy notes that
All that [Hipparchus] did was to make a compilation of the planetary observations arranged in a more useful way, and to show by means of these that the phenomena were not in agreement with the hypotheses of the astronomers of that time.
Next, Ptolemy tells us why he believes that Hipparchus did not go further in one of the longest, most-rambling, run-on sentences I’ve yet seen. I’ll try to break it up.
For, we may presume, he thought that one must not only show that each planet has a twofold anomaly, or that each planet has retrograde arcs which are not constant, and are of such and such sizes (whereas the other astronomers had constructed their geometrical proofs on the basis of a single unvarying anomaly and retrograde arc);
Ptolemy first reiterates that the planets should have two anomalies but then adds a complication not previously mentioned in that, when a planet enters retrograde, the amount of time in which is is retrograde is not consistent.
nor [that it was sufficient to show] that these anomalies can in fact be represented either by means of eccentric circles or by circles concentric with the ecliptic, and carrying epicycles, or even by combining both, the ecliptic anomaly being of such and such a size, and the synodic anomaly of such and such [a size]
When deriving the solar and lunar models, Ptolemy took great care to demonstrate the equivalence of the epicyclic and eccentric models so long as certain parameters are the same4. Evidently Ptolemy feels that this will bear repeating for the planets as well and assumes that Hipparchus would not have wanted to deal with doing so.
(for these representations have been employed by almost all those who tried to exhibit the uniform circular motion by means of the so-called “Aeon-tables”, but their attempts were fault and at the same time lacked proofs: some of them did not achieve their object at all, the others only to a limited extent);
This section is the indication I mentioned in a footnote above indicating that there was likely some attempt at producing tables prior to Ptolemy. Toomer suggests that these tables were not based on a fully formed theory, but “by integer number of revolutions in some huge period, in which they all return to the beginning of the zodiac, and the planetary equations were calculated by a combination of epicycles or of eccentre and epicycle which was not reducible to a geometrically consistent kinematic model, i.e., to a class of Greek works which were the ancestors of the Indian siddhantas.”
I take what Toomer is saying here as being akin to a repetition in a cycle similar to the Saros cycle for eclipses.
but [we may presume], he reckoned that one who has reached such a pitch of accuracy and love of truth throughout the mathematical sciences will not be content to stop at the above point, like the others who did not care [about the imperfections]; rather, that anyone who was to convince himself and his future audience must demonstrate the size and the period of each of the two anomalies by means of well-attested phenomena which everyone agrees on, must then combine both anomalies, and discover the position and order of the circles by which they are brought about, and the type of their motion; and finally must make practically all the phenomena fit the particular character of the arrangement of circles in his hypothesis. And this, I suspect, appeared difficult even to him.
Here, Ptolemy again praises Hipparchus’ devotion to accuracy and completeness implying that Hipparchus would have settled for nothing less than a complete model as opposed to a repeating table.
The point of the above remarks was not to boast [of our own achievement].
Sure thing. We totally believe you.
Rather, if we are at any point compelled by the nature of our subject to use a procedure not in strict accordance with theory (for instance, when we carry out proofs using without5 further qualification the circles described in the planetary spheres by the movement [of the body, i.e.,] assuming that these circles lie in the plane of the ecliptic, to simplify the course of the proof); or [if we are compelled] to make some basic assumptions which we arrived at not from some readily apparent principle, but from a long period of trial and application, or to assume a type of motion or inclination of the circles which is not the same and unchanged for all planets; we may [be allowed to] accede [to this compulsion], since we know that this kind of inexact procedure will not affect the end desired, provided that is is not going to result in any noticeable error;
The above passage seem to me to be Ptolemy trying to preemptively reassure readers that may take objection to some of the methods he is about to use. For example, the portion about taking the circles to lie in the plane of the ecliptic is foreshadowing that Ptolemy will be doing precisely this, and ignoring the change in latitude for the time being. This should be unsurprising since we’ve seen him do the same thing with the moon’s path.
Ptolemy tells us that this won’t be a problem and, in general, the planets do stay confined to the ecliptic reasonably well. Mercury is the worst, which deviates by up to $7º$, followed by Venus at $3.4º$, Saturn at $2.5º$, Mars at $1.8º$, and Jupiter at $1.3º$.
But what of the other portion in which Ptolemy talks about the “qualification of the circles”? Here, I suspect Ptolemy is indicating that he does not intend to complete the models for the planets to the same degree he did for the sun and moon.
In those, he worked out the relative sizes of each, essentially determining scale. Ptolemy will not be doing the same for the planets.
He also talks about “a type of motion or inclination of the circles which is not the same and unchanged for all planets.”
In this, Toomer indicates that he is referring to some special applications he’s going to be using for Mercury.
and we know too that assumptions made without proof, provided only that they are found to be in agreement with the phenomena, could not have been found without some careful methodological procedure, even if it is difficult to explain how one came to conceive them (for, in general, the cause of first principles is, by nature, either non-existent or hard to describe);
This portion sounds to me a lot like Ptolemy suggesting that he can’t trace down every logical thread and defend all of them. In other words, there’s going to be some “trust me on this” coming up.
we know, finally, that some variety in the type of the hypotheses associated with the circles [of the planets] cannot plausibly be considered strange or contrary to reason (especially since the phenomena exhibited by the actual planets are not alike [for all]); for, when uniform circular motion is preserved for all without exception, the individual phenomena are demonstrated in accordance with a principle which is more basic and more generally applicable than that of similarity of the hypotheses [for the planets].
This passage reads to me as Ptolemy trying to defend one of this new introductions to astronomical theory that he knows isn’t going to go over well: The inclusion of a concept known as an “equant6 We’ll certainly discuss this more later, but to preview, this concept splits the “uniform” motion from the “circular” motion, having one of those appear at one point, and the other appear at another.
The observations which we use for the various demonstrations are those which are most likely to be reliable, namely [$1$] those in which there is observed actual contact or very close approach to a star or the moon, and especially [$2$] those made by means of the astrolabe instrument7. [In these], the observer’s line of vision is directed, as it were, by means of the sighting-holes on opposite sides of the rings, thus observing equal distances as equal arcs in all directions, and can accurately determine the position of the planet in question in latitude and longitude with respect to the ecliptic, by moving the ecliptic ring on the astrolabe, and the diametrically opposite sighting holes on the rings through the poles of the ecliptic, into alignment with the object observed.
Above we discussed the unreliability of the previous observations on which Ptolemy may have been tempted to use8. But here, Ptolemy tells us that he will favor two other types of observations.
The first is a close conjunction with a star or the moon. In the case of the former, the position can be taken as that of, or similar to, the fixed star near which it is passing. For the latter, the position of the moon could be calculated although Ptolemy would then need to consider the half degree diameter of the moon.
He goes further into praising the use of the astrolabe describing how it can be used, even when the planet is not directly on the ecliptic simply by rotating the ecliptic ring to meet it.
- Neugebauer talks a bit more about this in History of Ancient Mathematical Astronomy in which he indicates that Apollonius evidently had a model of planetary motion but notes that previous astronomers such as Eudoxus, Aristarchus, and Archimedes (i.e., just before Apollonius) “shows a lack of interest in empirical numerical data in contrast to the emphasis on the purely mathematical structure. It would therefore be a perfectly defensible position, in view of Ptolemy’s silence, to assume that Apollonius also was primarily interested only in the mathematical aspect of the theory of planetary motion and not in the numerical agreement with observational facts.”
As a bit of contrast, Toomer notes that there were certainly planetary theories that did make predictions prior to Ptolemy and cites Indian astronomy which was based on pre-Ptolemaic astronomy as well as a reference Ptolemy will make to “aeon-tables” below. However, Ptolemy clearly deems these attempts insufficient.
- I suspect this is why Ptolemy spent some time at the end of Book VIII addressing first and last visibilities.
Toomer notes that Ptolemy is likely referring to planetary observations from Babylonian astronomers “which are characteristically of this type.” He states that they are available to modern astronomers through the “diaries” (citing this paper), but Ptolemy probably knew them as having been passed down from Hipparchus.
- From the star catalog.
- This was done in III.3 (which I broke into three posts) and IV.5.
- This is indeed the translation Toomer provides but does not seem to make sense. I propose this portion of the sentence should have read, “when we carry out proofs
usingwithout further qualification of the circles…”. - This is not a term Ptolemy used but was introduced later. He referred to this as “the eccentre producing the mean motion.”
- Reminder that this is not what we tend to call an astrolabe, but is an armillary sphere.
- I.e., the first and last visibilities.