Now that we have completed the above discussion, we will first set out, for each of the five planets, the smallest period in which it makes an approximate return in both anomalies, as computed by Hipparchus.
As we’ve done with the sun and moon, we will now focus on the periods associated with the planets.
While Ptolemy cites Hipparchus, he quickly notes that:
These [periods] have been corrected by us, on the basis of the comparison of their positions which became possible after we had demonstrated their anomalies, as we shall explain at that point. However, we anticipate and put them here, so as to have the individual mean motions in longitude and anomaly set out in a convenient from for the calculations of anomalies.
This passage hard to parse at this point, but what Ptolemy is saying here is that the values Ptolemy is about to give us have been corrected from those of Hipparchus. He is presenting them, with corrections already added, even though he will not explain what those corrections are until a later chapter. This way, he doesn’t have to rewrite these large tables later.
That being said, Ptolemy assures us that even if we didn’t adopt these corrections,
it would in fact make no noticeable difference in those calculations even if one used more roughly computed mean positions.
Before laying out the tables, he takes a bit of care to explain his definitions although they haven’t particularly changed since we last used the terms.
[B]y “motion in longitude” [we mean] the motion of the centre of the epicycle around the eccentre, and by “anomaly” the motion of the body around the epicycle.
With this out of the way, he begins by giving the returns for each of the planets, starting from the outermost:
[F]or Saturn, $57$ returns in anomaly correspond to $59$ solar years (as defined by us, i.e., returns to the same solstice or equinox), plus about $1 \frac{3}{4}$ days, and $2$ revolutions [in longitude] plus $1;43º$;
Let’s pause here and consider where these numbers came from. In particular, there’s three pieces here. The first is the number of [tropical] years which Neugebauer calls $N$. The second is the number of integer rotations of the planet in longitude, which Neugebauer refers to as $R$. Lastly, we have the number of anomalistic periods (i.e., the number of rotations about the epicycle), which Neugebauer refers to as $A$.
Although neither Ptolemy, Toomer, or Neugebauer go about how each of these were derived, Neugebauer explains that the initial values were almost certainly inherited from the Babylonians. Ptolemy cites Hipparchus as his source, but Toomer suggests that Ptolemy was simply unaware of the true origin of them.
However, the values that Ptolemy inherited from the Babylonians were not the ones given here. As noted above, Ptolemy has already incorporated his adjustments to the figures he inherited. Taking Saturn as an example here, the Babylonian values had $57$ returns in anomaly corresponding to $59$ years even, and $2$ revolutions even. The additional days and fractions of a revolution were the adjustments added by Ptolemy that we will explore later1.
Moving on to the other two outer planets:
[F]or Jupiter, $65$ returns in anomaly correspond to $71$ solar years (as defined above) less about $4 \frac{9}{10}$ days, and to $6$ revolutions of the planet from a solstice back to the same solstice, less $4 \frac{5}{6}º$;
[F]or Mars, $37$ returns in anomaly correspond to $79$ solar years (as defined by us) plus about $3;13$ days, and to $42$ revolutions of the planet from a solstice back to the same solstice, plus $3 \frac{1}{6}º$;
I’ll pause here again to address a comment that Ptolemy slid in when discussing Saturn in which he stated
[I]n the case of the $3$ planets which are always overtaken by the sun2, the number of revolutions of the sun during the period of return is always, for each of them, the sum of the number of revolutions in longitude and the number of returns in anomaly of the planet.
In other words, for the outer planets: $N = R + A$.
Ptolemy doesn’t give any justification for this3, so I’ll leave it alone for now.
Moving on to the interior planets,
[F]or Venus, $5$ returns in anomaly correspond to $8$ solar years (as defined by us) less about $2;18$ days, and to a number of [longitudinal] revolutions of the planet equal to that of the sun, $8$, less $2 \frac{1}{4}º$;
[F]or Mercury, $145$ returns in anomaly correspond to $46$ of the same kind of years plus about $1 \frac{1}{30}$ days, and to a number of [longitudinal] revolutions which is, again, equal to that of the sun, $46$, plus $1º$.
With this in hand, Ptolemy divides the change in anomaly by the number of days it takes to achieve such motion to begin deriving the mean anomalistic motion.
It’s been awhile since we’ve tackled any of these, so let’s remind ourselves where things are coming from.
Ptolemy told us that, for Saturn, there were $57$ returns in anomaly. That means the planet went about its epicycle $57$ times for a total motion of $20,520º$.
That motion, he tells us, was over the course of $59$ years plus $1 \frac{3}{4}$ days. We need to convert this to number of days which requires us recalling Ptolemy definition of a year4 which was $365;14,48$ days.
If we multiply that by the number of years and then add on the remaining $1 \frac{3}{4}$ days, we get $21,551;18$ days.
Next we divide the total motion by the amount of time it took to get the average daily motion in anomaly to be $0;57,07,43,41,43,40 \frac{º}{day}$.
Ptolemy repeats this for the other planets getting the following anomalies:
Saturn: $0;57,07,43,41,43,40 \frac{º}{day}$
Jupiter: $0;54,09,02, 46,26,00 \frac{º}{day}$
Mars: $0;27,41,40,19,20,58 \frac{º}{day}$
Venus: $0;36,59,25,53,11,28 \frac{º}{day}$
Mercury: $3;06,24,06, 59, 35, 50 \frac{º}{day}$
He then divides each of these by $24$ to get the hourly motion, multiplies by $30$ to get the monthly motion, and then by $365$ for the motion in an Egyptian year, and then by $18$ for the motion in $18$ year increments. In each of these, he subtracts out full revolution.
Next, Ptolemy turns his attention towards the mean motion in longitude, but proceeds in a different manner.
We can also find the mean motions in longitude corresponding to the above without reducing the number of [longitudinal] revolutions to degrees and dividing them by [the number of days in] the period set out above for each planet.
In other words, Ptolemy will use a different method from the one we used above where we simply divided the number of anomalistic periods by the length of time.
For Venus and Mercury, it is obvious that we [determine the mean motion in longitude] this by taking the same mean motions as we set out previously for the sun
This statement might not be immediately obvious, but what Ptolemy is stating here is that Mercury and Venus have a mean motion equal to that of the sun. Using Neugebauer’s notation from above: $N = R$.
This is because these two planets have a maximum elongation from the sun due to them being interior planets; their motion is tied to the sun because of it.
Thus, we don’t need to do anything further for these and can simply refer to the solar mean motion tables for their longitudinal mean motions. Easy enough.
For the exterior planets, Ptolemy gives another shortcut,
[F]or the other three planets, [we determine their longitudinal mean motion] by taking the difference between the [mean motion in] anomaly and the corresponding solar [mean] motion for each individual entry.
Let’s try this out for Saturn.
For a single day, the mean motion in anomaly was $0;57,07,43,41,43,40 \frac{º}{day}$.
And the solar mean motion is $0;59,08,17,13,12,31 \frac{º}{day}$.
Subtracting the former from the latter I get: $0;02,00,33,31,28,51 \frac{º}{day}$, which is precisely what Ptolemy has5.
This is then repeated for Jupiter and Mars, and then multiplied to determine the hourly, monthly, yearly, and 18 year period motions of the planets.
As with before, Ptolemy collects these into a large table for all of the planets which is presented in the next chapter.
- And perhaps they’ll shed some more light on how these were derived.
- I.e., the superior planets – Mars, Jupiter, and Saturn.
- Nor does Neugebauer.
- He defines it as “$365$ days, plus a fraction which is less than $\frac{1}{4}$ by about $\frac{1}{300}$”.
- It’s also quite close to what I get by dividing things out in the same manner as we did for the mean anomalistic motion. I suspect Ptolemy did this because addition and subtraction are far easier in sexagesimal than multiplication and division.