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Pp. 539-542 of volume 10-1 of Gauss's werke include entry 97 in Gauss's diary:

I have found new exact formulas for the parallax of the Moon.

as well as the formulas themselves (which were rather found in a different place than Gauss's diary) and an excerpt from his 1799 letter to the Prussian cartographer Karl Ludwig von Lecoq who was assisted by Gauss in his trigonometric survey of Westphalia. It is not a well known fact, but Gauss's first astronomical activity was done in this context (this was two years before his calculation of Ceres orbit), and Lecoq wrote in his "report on the trigonometric survey of Westphalia" that "In the astronomical part of the survey, the calculations of Gauss were of great help to me". It seems that determining the exact geographic location of points by means of astronomical observations was quite a common procedure in the 18th century.

I have not included Gauss's formulas here (the formulas are quite complicated) since I want to understand more about the context of the "lunar parallax problem" and not to immerse myself immediately in the mathematical details. I know that the ancient Greek astronomer Hipparchus used a kind of lunar parallax calculation during a solar eclipse in order to determine the distance to the Moon (he did it by comparing the details of the total eclipse and the partial eclipse observed in two cities whose distance was known). But I don't understand how it is possible that a complete description of the lunar parallax was not established yet in the end of the 18th century (if it was well calculated then I don't think that Lecoq should have needed Gauss's help), a time when spherical trigonometry was quite ripe and a lot of data was collected on the Moon's orbital parameters.

Also, I'm not sure I understand the complexity of this problem - Parallax in general is the angular shift in observed position of a distant object when the observer changes his location; so the description of the lunar parallax angle $\alpha$ seems to me to be very simple - $\mathbb{tan\alpha} = \frac{2R_E}{d}$ (here $R_E$ is the Earth radius and $d$ the Earth - Moon distance).

So I would like to understand what meanings were attached to the notion of lunar parallax in the 18th century, and to understand what was the main difficulty in the related problems.

Update

I now understand that determining the lunar parallax was crucial for exact calculations of the observer's geographic location - especially in surveys of "small regions" in global terms (such as Westphalia) where the accuracy needed is much higher (I guess the accuracy needed is 10's or 100's times higher than in more global surveys). This is because the change in observed lunar position consists of the change in zenith direction (of the earthly observer) and the parallactic change due to the change in observer location. That is, when an observer changes his earthly location he can deduce his geographic location directly from the change in the background of fixed stars, but if he uses the Moon as a reference (which is much closer to us than fixed stars) he has to remember that the change in observed lunar position is not only the result of his zenith rotation but also of the Moon's parallax.

Therefore, to avoid errors one has to separate the parallax part from the first part in order to determine the change in zenith direction (which amounts to determining geographic location). I don't know why 18th century surveys had to use the Moon instead of some fixed stars, but in any case, if one uses the Moon, they are led to an interesting geometric problem.

Although the magnitude of parallax correction $\pi '$ can be easily determined by observing the Moon altitude $a$ - as in terry-s's answer, it satisfies the equation $\mathbb{sin} (\pi ') = \mathbb{cos}(a) \cdot \mathbb{sin}(\pi) = \mathbb{cos}(a) \cdot (\frac{R_E}{d})$ (sine theorem in euclidean trigonometry) - determining the direction in which the parallax correction should be applied (in order to find the exact direction of observer's zenith) seems to much more complicated as it depends on the lunar declination (which changes from 18 to 28 degrees and back during a month), the hour angle, as well as the observer's latitude and longitude.

Thus, in terms of data obtained by observing the Moon's position in the sky, the direction of parallax correction seems to be a complicated function of the Moon's altitude and azimuth as well as the timing of the observation, so determining the observer's latitude $\theta$ and longitude $\varphi$ by this method turns into a complicated trigonometric equation in unknowns $\theta,\varphi$ and coefficients that depend on the lunar position (azimuth, altitude and day of the month). This explains the apparent complexity of Gauss's formulas.

Question

Apart from deciphering Gauss's formulas, I also fail to understand why cartographers needed to use the Moon's position (a process that is prone to complexities) instead of using some fixed stars? Is it just because the Moon is the simplest celestial body to find at night?

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Lunar parallax is a composite problem. Two parts of it are related by the 'horizontal parallax', $\pi$ in the source you cited and linked. It is evaluated in one part of the problem and applied in the other.

First part of the problem (not addressed by Gauss in your linked source) is to evaluate at any time the moon's horizontal parallax, i.e. its parallax when observed on the horizon. This quantity depends on its continually varying distance from the earth. It is defined as part of lunar theory, and (nowadays) by an ephemeris, e.g. by part of the current JPL development ephemeris based on lunar laser-ranging data to which the lunar theory or equations of motion are fitted. The daily horizontal parallaxes at 0h TT are given in the Astronomical Almanac. In Gauss's time the moon's current horizontal parallax was already provided by the usual almanacs, or it could be obtained more laboriously from the fundamental tables of the time, which were actually used to calculate the almanacs, e.g. Mayer's tables as improved by Mason (one of the the surveyors of the 'Mason-Dixon' line).

Finding the horizontal parallax is the part of the problem that depends on the moon's distance from the earth. The basic expression for the (horizontal) parallax given in the question was not quite correct, because the horizontal parallax is conventionally defined as the difference of view-direction towards the moon's center as between an observer on the earth's surface where the moon is on the horizon, and another observer located on the line between the moon and the geocenter (i.e. for whom the moon is at the local zenith) -- where the current parallax becomes zero. So the (horizontal) parallax angle when defined in this standard way becomes $\alpha$ where $$\mathbb{\sin \alpha} = \frac{R_E}{d}$$ and the symbols are as defined in the equation in the original question.

The second part of the problem is the one addressed by Gauss's entry linked in the question, it is to find the actual current parallax at a given altitude or zenith distance of the moon, given the horizontal parallax for the current time. Gauss made his formulae general in the (limited) sense that they would take as their inputs the moon's current horizontal parallax and then (in order to calculate its current altitude) also its declination (from the ephemeris), plus its hour angle (calculated from the Right Ascension given in the ephemeris and the current local sidereal time). These quantities are combined trigonometrically to produce the parallax at current altitude. This current parallax varies rather simply with apparent altitude or equivalent zenith distance, it is the horizontal parallax affected by a factor proportional to the sine of the apparent zenith distance. (Current parallax is zero for an object in the zenith, and at its greatest, equal to the horizontal parallax, for an object on the horizon.)

The effect of lunar parallax is always to depress the (apparent) altitude, thus increasing the moon's apparent zenith distance. If in doubt, just consider the line from geocenter to moon (zero parallax for an observer along that line). Next, consider where that line is for an observer with the moon on the horizon -- answer, a long way below the observer's feet. It will be apparent that at the observer's horizon, the observer will have to look for the moon a lttle downwards compared with the line of view to the moon from the geocenter which is far below the observer's feet.

Many of the relationships are more fully described in the various Explanatory Supplements to the astronomical almanacs.

Gauss, in the entry linked in the question, also addressed a separate problem, that of evaluating the augmentation of the moon's apparent diameter. The moon's apparent diameter, when (hypothetically) seen from the geocenter, is provided by lunar theory and current ephemeris data. But for a real observer, the moon's linear distance will be different and therefore also its apparent diameter. E.g. for an observer with the moon at the zenith the moon will be less distant than the standard assumes, because the observer is then standing 'on top of' the earth's radius, so that the moon is that much closer, and its apparent diameter larger. Gauss calculates this 'augmented diameter' (Vergroesserter Diameter), because it is useful in connection with observations that are based, as usual, on one of the moon's limbs (edges) although the desired final quantity should be referred to the moon's center. (It should also be noted that Gauss takes as his reference distance the moon's distance when seen on the horizon, not quite the same as the current standard geocentric distance.)

edit: A more modern and fully-explained treatment of lunar parallax that should be useful for comparison can be found for comparison at www.hathitrust.org, starting (for the lunar parallax) at page 76 of the 1960s Explanatory Supplement to the Astronomical Almanac and the American ephemeris and nautical almanac. Earlier and later pages give useful information on related questions.

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  • $\begingroup$ thanks for this answer [+1]! It did help me to gain some insights into these problems. Just to make sure I understood it - if $\pi$ is the horizontal parallax, than its "augmented parallax" (denoted $\pi '$ in Gauss's formulas) equals (according to sine theorem in euclidean trigonometry): $\frac{R_E}{sin \pi '} = \frac{d}{sin a}$ where $a$ is the observed altitude of the moon. The main point of difficulty is to express $a$ in terms of other parameters. Is what I wrote correct? $\endgroup$
    – user2554
    Apr 10, 2023 at 10:37
  • $\begingroup$ Just a correction to my comment - it should be $90-a$ (when expressed in degrees) instead of $a$. $\endgroup$
    – user2554
    Apr 10, 2023 at 10:57
  • $\begingroup$ And some of Gauss's notation is still unclear to me - what is the meaning of "cotangent of right ascension of the parallax" that Gauss wrote its formula on p.539? and what is $d'$ (Gauss defines $d$ to be the declination of moon)? $\endgroup$
    – user2554
    Apr 10, 2023 at 13:04
  • $\begingroup$ @user2554 : I don't see any 'augmented parallax' in the fmlae you quoted. What's augmented is not the parallax but the moon's apparent diameter big-delta, due to how far the observer's location approaches the moon. Some of Gauss's quantities are also not defined in the brief page you linked. Difference of standards may mean that his relations do not transfer directly to current use. I will try and find and post an online access to one of the Explanatory Supplements, even the 1960s one should help a lot on parallax if the 1990s-2000s editions turn out to be copyright-locked. $\endgroup$
    – terry-s
    Apr 10, 2023 at 14:02
  • $\begingroup$ @user2554 : I'm amending to add a link to a more current and available treatment of lunar parallax where all the quantities are defined. I hope that helps. -- But I'm afraid I can't see the meaning of the symbols you used in the equation given in your first comment, so can't comment on its correctness. $\endgroup$
    – terry-s
    Apr 10, 2023 at 14:25

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