Chapter 68

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Kepler, bringing his Astronomia Nova to a close, has one more task to conquer astronomy and establish his astrophysical system. The task is to understand the changes of the ecliptic plane. This is a question that had inspired him for many years: a paradox that could in no way be accounted for by Tycho and the other two systems.

“It was said in chapter 14 that in any one period of Mars whatever, the obliquity or inclination of Mars's plane to the plane of the ecliptic remains fixed. There is, however, some doubt whether this obliquity is the same, and fixed, for all ages.” (p. 633)

Throughout the course of a night, there is very little movement of the stars in relation to each other, and the most change that someone may expect to observe is perhaps of the second motion of the planets, with the moon being the fastest. Taking observations of any significant change in the planets takes days and weeks. And to notice the retrograde motion of Mars, one must wait 2 years. For us, the next time we will be able to observe Mercury transiting the sun will be in 2016!

Some motions however, are impossible for someone to notice within their lifetime. And it is only through the continuation of civilization over millennia that mankind can develop a sufficient edifice to allow the individual scientist to make a discovery of a universal physical principle.

It is this process that allows Kepler to discover something that at his time was absolutely impossible to observe: the equatorial plane of the rotating solar body that Kepler names “the Royal Road (Via Regia).”

Tycho Brahe, using data from Ptolemy, finds that the latitude of the otherwise fixed stars changes. Why would this happen? Think about it for a second. Remember the animation in chapter 62, the latitude is the angular distance from the star to a dropped-down point on the ecliptic plane. If the latitude of a fixed star does not change throughout the year, but it does change over a 1,400 year period, what must be the cause of this?

Read the second paragraph on page 633 again:

“in the region of the summer solstice, the latitudes of the northern stars increased and those of the southern stars decreased; and in turn, in the region of the winter solstice, the latitudes of the northern stars decreased and those of the southern stars increased. As one goes from these places towards the equinox points, the alteration of the latitudes diminishes, until near the equinox points there is none at all.” (p. 633)

This is what Tycho “saw.”


The animation below is a more fleshed-out illustration of the diagram in the chapter. The buttons will add more illustration to it, but before you use them, try to determine which region is the summer solstice, which the winter solstice and which the equinoxes.

Kepler knows that not only do the latitudes of the fixed stars change, but they do so at different speeds. If one were to think of how a sine function gradually decreases in the rate of its change as it approaches 90 degrees, while a cosine function does the opposite, changing very slightly at first, but gradually speeding up, then one has a good first approximation understanding of how the latitudes of the fixed stars change differently.

Kepler determines that the cause is the motion of the intersection (or the node) of the ecliptic plane and the Royal Road.

In this way, Kepler is able to determine something extremely profound, so pay attention at this point in the chapter (which is the end of page 635). If, in our animation, the greatest change of latitude occurs at the node, and the greatest observed change occurs around the summer solstice, then, the nodes are near the solstices. If the equinoxes are in Aries and Libra, than the nodes are in Cancer and Capricorn which is direction of the line of Apsides!

Kepler reminds us:

“Now the sun’s apogee, or the earth’s perihelion, is at 5½° Cancer, and thus by chapter 57, the diameter of power, causing the eccentricity, points at the sun when the earth is at 5½° Aries. But also, by chapter 63, the diameter of power that causes the latitude points at the sun when the earth is at the limit, which is in Aries by the present chapter 68. Therefore, by the same chapter 63, both powers can be effected by the same diameter of the earth’s body. Hence one may argue plausible that this indivisible circle or mean ecliptic and the true one known to us coincide at 5½° Cancer and Capricorn.” (p. 636)

Not only that! But Kepler mentions that the aphelia of Mars, Jupiter and Saturn have similar characteristics. Kepler showed in chapter 63 (on page 616) that the apsides and nodes do not coincide for the superior planets, being off by significant amounts, or even 90 degrees. Yet it is possible that the new determinations of nodes, based on finding the royal road, could change this, and allow a single cause to account for both the apsides and the latitudes.

Later, Kepler explores the precession of the equinoxes. Here is an animation illustrating the note on the margin of page 638:

In attempt to resolve the problems presented, Kepler finds problems with Ptolemy’s observations, and is thus unable to make a full and final conclusion. Despite the problems, Kepler is optimistic about the future, and wishes for us to discover what he was unable:

“Therefore, as long as we are wanting suitable observations from antiquity, circumstances compel us to leave this discussion of the notion of the nodes, along with many other matters, to posterity, if, indeed, it should please God to vouchsafe the human race a length of time in this world sufficient to work through such remaining questions thoroughly.” (p. 640)


Now, On To the Future!