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The Ptolemaic theory of the latitude of the planets is dealt on in Book 13 of the Almagest. This theory (which I was struggling to find info about online) is considered quite complicated, but luckily this question is not about the exactness of this theory, but rather can be reduced to this: according to this theory, does the local maximum of the latitude fall exactly at the opposition?

It seems to be the case, since Tycho Brahe (who followed Ptolemy in this regard if I'm not mistaken) was clearly surprised (as Kepler quotes in his Astronomia Nova) when his observations proved that the local maximum does not fall exactly on the opposition, but can be some days away.

But if this is the case, how come Ptolemy allows this in his system? I think this is indeed within observational error (probably less than 10m of arc), but on the other hand, it can be observed relatively each day: how come the ancients could not see that the maximum latitude does not fall at the opposition?

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  • $\begingroup$ Longomontanus in his book (1622) seems, after bring a correct latitude method (apart from distances maybe), quotes Kepler on this p.268. [quotes from Astronomia Nova] He seems to deal with it. I beleive he tries to explain why Tycho was wrong here. (he first shows the correct by the correct method number how they fit observation). link here $\endgroup$
    – d_e
    Feb 18, 2023 at 15:22

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tl;dr: No. According to Ptolemy's model, the external planets will not get their maximal latitude exactly at the opposition. However, his model makes the maximum latitude to occur closer to the opposition than it really is.


Ptolemy had different models for the external and internal planets. This answer considers only external planets - as this was the issue at hand in the question about Mars.

Swerdlow N.M. (2005) Ptolemy’s Theories of the Latitude of the Planets in the Almagest, Handy Tables, and Planetary Hypotheses. is very interesting paper about the latitude theory of Ptolemy. It does manage to present the theory quite in a neat and relatively simple manner:

(1) The earth is at O, through which passes the nodal line of the eccentric, which is inclined to the plane of the ecliptic at an angle i1; N is the northern limit, near apogee, S is the southern limit, near perigee, and the midpoint of NS is M at an eccentricity e from O. M and e are the center of the eccentric and eccentricity projected into the line joining the limits. (2) When the center of the epicycle is at N or S, it is inclined to the plane of the eccentric in the line of sight, with the perigee to the north at N and to the south at S, so that the latitude βo at opposition Po is greater than the latitude βc at conjunction Pc. It is found from observation that the difference between βo and βc is so large that the epicycle is also inclined to the plane of the ecliptic by i2 and thus to the plane of the eccentric by i1 + i2. (3) When the center of the epicycle is at the ascending node or descending node , it lies in the plane of the ecliptic so the planet has no latitude wherever it is located. Hence as the epicycle moves from the limit to the node, i2 decreases from its maximum to zero, and as it moves to the next limit i2 again increases to its maximum. Ptolemy treats i1 + i2 as a single inclination of the epicycle to the plane of the eccentric. But since i1 may be taken as a fixed inclination, holding the epicycle parallel to the plane of the ecliptic, leaving i2 alone variable, which we believe a clearer way of showing the variable inclination, we have divided the inclination of the epicycle into two components, the fixed i1 and the variable i2.

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So, in my words, naively, we would except that the epicycle should be parallel to the ecliptic (i2=0 fixed). if this is the case, we would expect to see at every point (of the center of the epicycle) on the eccentric that the latitude at the opposition is bigger than the latitude at the conjunction: This is trivial and because at opposition the planet is closer than conjunction. [the paper uses the term "perigee [of the epicycle]" at point (2) (at least this is how I understand it): the meaning is the point of the epicycle the closet to Earth, which the planet occupies at opposition of course.].

Now, the thing is that Ptolemy observations (and interpolations) showed that the latitude gap between the opposition and conjunction is bigger than expected - hence Ptolemy gave it an extra push and to make the epicycle even more inclined in respect the the eccentric; but because when the center of epicycle is on the nodes, the planet is always found to have latitude of 0 no matter where it is located on the epicycle it means that at the nodes this extra tilt (i2) is 0. So at the nodes i2=0 but at the limits it received the maximum. So i2 is not fixed and its running along the eccentric.

Now that we understand the model, we can turn back to the question: do we get the maximum latitude exactly at the opposition? In reality the latitude is basically a function of two argument: (1) the proximity to the limit; (2) the distance from Earth. At opposition the argument (2) receive it minimum - hence the contribution to the latitude is maximal, but argument (1) might receive a stronger value before/after the opposition if it becomes closer to the limit. This why it is simple to see that the maximal latitude is not exactly on the opposition (only very close to it). To express that mathematically: $f(\lambda) = i1* \frac{1}{p}\sin(\alpha) + d(\phi)$. where $\lambda$ is the latitude. $\alpha$ is the distance from the node (assuming equal speed on the eccentric - which is wrong assumption but close enough to reality). and $p$ is the period of the planet, where Earth period =1; and where $\phi$ is the angles left for the planet to reach the "perigee of the epicycle". So $d(\phi)$ will donate the contribution of the distance to the latitude which is a function of $\phi$ which determined the distance (given the contestant Radii of Earth (Sun for Ptolemy) orb and the planet). When this function gets it maximum for a given location on the eccentric (\alpha); we know that when $\phi=0$, $d(\phi)$ receives its maximum values since the distance to Earth is the shortest, so $d'(\phi)$=0 when $\phi=0$. we are left with $f'(\lambda) = i1* \frac{1}{p}\cos(\alpha)$ which is not equal to 0 (except at the limit of course) hence we see that when $\phi=0$ (i.e., at opposition) we are not getting maximal latitude.

But in the model of Ptolemy because he had given (wrongly doing so) a tilt for the epicycle with respect to the ecliptic in the direction of the line of sight - it means that when the planet getting closer to Earth on the epicycle (as $\phi$ approaching $0$), it accrues latitude up till $+i2$ when it reaches the "perigee of the epicycle" at the opposition and $\phi=0$. So now the latitude function is as follows: $f(\lambda) = i1* \frac{1}{p}\sin(\alpha) + d(\phi) + i2*\cos(\phi)$. i2 itself, as we saw, is a function of $\alpha$, but it doesn't matter for our analysis , because we see that at opposition when $\phi=0$ the derivative is the same as before and is not equal to $0$. Hence, also in Ptolemy model we should not receive the maximum latitude exactly at opposition, but we can see how it comes closer to the opposition as basically the model of Ptolemy strengthen the influence of $\phi$ on the latitude. Moreover, Ptolemy not merely increase the influence of $\phi$, it was done at expense of the influence of i1 - because in reality (in terms of Ptolemy) i2=$0$. so Ptolemy numbers for i1 where smaller than reality for him to keep $i1+i2$ about right.


We are left in the Question of why Tycho was surprised. Let's see how Kepler presents the attitude of Tycho (from Ch. 66 of Astronomia Nova tranlsation William Donhue):

When I later had come to him in Bohemia, and frequently inquired about how the latitudes are arranged, he answered ... and recounted many other things. Of this present matter, he said most emphatically, ‘this is remarkable, that the latitudes reach their maximum before or after opposition to the sun.’

Tycho is reportedly wrote that "The cause of this [large gap between opposition and maximum latitude] needs to be looked into carefully" (Ch. 66 brought by Kepler).

I would have liked to give Tycho the credit to be surprised only by the extant of this large gap and not that there is a gap. Maybe it is so, but the account of Kepler in this chapter seems to suggest the expectation of being exactly at the opposition not only by Tycho but by other ancients.

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  • $\begingroup$ My equations are not accurate. $i1*sin(\alpha)$ is not the latitude of the center of the epicycle at angle $\alpha$ from the node. rather it is $\arcsin(\sin(i1) * \sin(\alpha))$. this is because we are dealing in spherical trigonometry. On small angel of i1 this us of low-effect, but at any case it does not change the analysis. So is the case with i2. $\endgroup$
    – d_e
    Mar 26, 2022 at 12:08
  • $\begingroup$ link to part of the paper which is open for reading: books.google.com/… $\endgroup$
    – d_e
    Mar 28, 2022 at 11:15

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