# How did Copernicus find the nodes of Mars? what were they?

I'm trying to find the location of the nodes of Mars (i.e., the point of intersection between the ecliptic and Mars's plane [rather I should say Mars average plane]) given by Copernicus. According to the Model, as I understand it, they should cut the ecliptic and be 180 degrees apart.

The location of the nodes is important to have in order to calculate the latitude of Mars for any given point on its orb. However, I suspect that because Copernicus was using the Mean Sun, the nodes - by observation - would not be exactly 180 degrees apart. I ask where are the Martial nodes according to Copernicus, and how he determined their location.

Basically my question can be applied to Tycho Brahe as well, and I would appreciate if someone has the numbers?

Mars's latitudes with Copernicus: Ancient theories of the planets' latitude did not give the same prominence as is now usual to the 'nodes' of the inclined planetary orbital planes, i.e. to their intersections with the plane of the ecliptic. Instead, what one sees in Ptolemy and the Ptolemaic tables, as well as in work of some later authors (including Copernicus's 'De Revolutionibus') is an emphasis on the 'limits' of latitude, north and/or south (see G J Toomer's 1984 translation of Ptolemy's 'Almagest' e.g. at p.633).

Copernicus, too, discusses the northern limit. In book 6, chapter 1, of 'De Revolutionibus', Copernicus wrote that the northern limit of latitude for the orbit of Mars, for his time*, occurred at 27° of Leo. (That amounts to 147° of ecliptic longitude.) *The year applicable to this figure has been identified as 1523, based on a manuscript note by Copernicus (see Swerdlow (1984) reference linked below, at p.498).

In the case of a more modern heliocentric orbit, such as Kepler's, the relation of the two kinds of measure is simple enough. The northern limit occurs at 90 degrees of the orbit onwards from the ascending node, and the descending node is located 90 degrees onwards from the northern limit.

Copernicus in book 6 chapter 1 also explained the nodes in that way, implying clearly that for Mars the longitudes of the ascending and descending nodes were (for his time) at 57° and 237°. But the complications of Copernicus's latitude theory mean that the point of view from which these measures are taken is neither the earth nor the sun, but rather the geometrical center of the earth's (assumedly circular and eccentric) orbit around the sun. In his theory, and partly from this cause, the maximum north and south latitudes did not need to be numerically equal (celestial latitudes were always measured as deviations from the apparent course of the sun around the ecliptic or zodiac, but in Copernicus's theory of planetary latitudes the sun was not placed in the plane of the planetary orbits).

These complications among others are discussed by N M Swerdlow in "Mathematical Astronomy in Copernicus's De Revolutionibus" (1984), Part 1, at ch.6 (pp.483-538). Swerdlow calls Copernicus's theory of planetary latitudes the 'most deeply flawed' part of 'De Revolutionibus', and describes evidence suggesting that it was revised and finalised in haste shortly before publication, leaving a number of inconsistencies. He also mentions that this part of Copernicus's work has attracted few studies.

Mars's latitudes with Tycho Brahe: In the question, analogous queries are raised about the latitudes of Mars according to Tycho Brahe. The situation there is rather different than with Copernicus: Tycho published, besides his astronomical observations, theories for the motion of the sun and moon, but not theories for the other planets. It is possible that the available account closest to Tycho's point of view is in the (non-Keplerian) account given in 1622 by Tycho's former assistant, Christian Soerensen or Severinus, known as 'Longomontanus', in his book 'Astronomia danica'. As with Copernicus, the theories of planetary latitude given by Longomontanus are complicated. They seem to share much with Copernicus, and the numbers closest to those sought by the question are in 'Liber secundus', chap.14, on page 265. Longomontanus gives for Mars a longitude at 18° of Aquarius (i.e. longitude 318°) for the southern limit of latitude. He also gives 'intersections' which appear to be the ascending and descending nodes for Mars's orbit, at 18° of Taurus and Scorpio. The greatest latitudes he gives are unequal to north and south (4°33' and 6°42'), so he clearly did not share Kepler's views. Longomontanus' work is in Latin and I am not aware of any translation.

Copernicus's 'De Revolutionibus' book 6 (on planetary latitudes, with examples based on Mars) is translated on p.307-330 in Edward Rosen, "Nicholas Copernicus on the Revolutions. Translation and commentary", (Polish Scientific Publishers, Warsaw/Cracow; also Macmillan; also Johns Hopkins, Baltimore; 1978).

• Wow. Thanks very much for the elaborate answer. glancing at the refs, it seems indeed N. C was not using observations but interpolation from Ptolemy tables. More notes: (1): regards: "In his theory, and partly from this cause, the maximum north and south latitudes did not need to be numerically equal" -- I suspect they were equal in the model (not in observation); it looks he forces the discrepancy between the North and South to match - see p.500). (2): with respect to Longomontanus, just to note those numbers of Lat(6 for South/ 4 for North) are from Earth. from Sun we have 1.50
– d_e
Feb 15 at 10:16
• (3): N M Swerdlow doing very good job on explaining the theories of Latitude. And as he notes, not many bother to do it. I've already encountered him when I was trying to learn about the Ptolemaic theory of Latitude (which Copernicus pretty much adapts to his heliocentric model) in another question
– d_e
Feb 15 at 10:27