KEPLER
DAY 42
Chapter 40
AN IMPERFECT METHOD FOR COMPUTING THE EQUATIONS FROM THE
PHYSICAL HYPOTHESIS, WHICH NONETHELESS SUFFICES FOR THE THEORY
OF THE SUN OR EARTH
KEPLER’S THREE LAWS OF PLANETARY MOTION.
Kepler is famous for having discovered the three laws of planetary motion. All three are of
great significance for astrophysics, and Newton reasons from all three of them to the law of
universal gravitation. These laws are as follows:
Law 1. Each planet moves on an ellipse which has the Sun at one focus.
Law 2. The line from each planet to the Sun sweeps out equal areas in equal times.
Law 3. Comparing any two planetary orbits, the squares of their periods are as the cubes of
their mean distances from the Sun.
Law 1 will be established by the end of the Astronomia Nova––at any rate, in the case of
Mars (and it may be done similarly with the remaining planets). Law 3 is stated in Kepler’s
Epitome of Copernican Astronomy 4.3, and in other places besides, but we will not concern
ourselves with establishing that particular law. But in Chapter 40 of Astronomia Nova, which
we are now considering, Kepler establishes Law 2, although almost as an aside—it is
approached as a means for computing “the physical part of the equation,” i.e. the angle
which is traversed around the equant-point in a given time. Kepler’s second law is very
significant, however, not only as a link to Newton’s physics of centripetal forces, but also as
a break with past thinkers. The ancient astronomers believed in some kind of uniformity of
motion underlying the apparent irregularity in the movements of the planets. Modern
astrophysicists agree with that general premise. But the ancient astronomers sought this
regularity in perfectly uniform circular motion, whereas for modern astrophysics that would
be a very special and particular case. More often, the celestial bodies do not sweep out equal
ANGLES in equal times around any point at all, but instead they sweep out equal AREAS in
equal times around the center of the centripetal force producing their orbits. Uniform angular
velocity has been replaced by uniform areal velocity.
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WHAT IS KEPLER DOING?
In the present chapter, Kepler is concerned principally with
C
Earth’s orbit.
The SUMMARY for Ch. 40 makes plain what he is
G
aiming at: “A method by which the physical part of the
equation, that is, the elapsed time of a planet over any arc of the
Q Equant
eccentric, may be found from the distances of the points of its
B Center
arc from the Sun.” The title reflects this, too.
Why does he want this? If we supply Q as the equantA Sun
point in Kepler’s diagram, then ∠BGQ is the “physical part of
the equation,” and ∠GQC is a measure of the TIME. This will
tell us where a planet should be at a given time, in longitude.
What he wants to do is show that his physical theory produces
D
consequences for the TIME, that is, it dictates––and dictates
correctly––what the angles about Q should be at any given time.
So he needs to reason synthetically, to reason forward, from causes to effects,
beginning from his physical hypotheses, to arrive at a way of calculating, from these, what
the angles are around Q (or what the angles AGB, BGQ are, which will be as good as GQC).
Given the time that has elapsed since the planet was at C, we would have the angle
CQG, and given the eccentricities (i.e. the values of AB and BQ in terms of the orbital
radius) we could easily calculate, by trigonometric methods, the values of the angles BGQ
and AGB. But that is not all Kepler is doing here.
THE STARTING POINT
In Ch. 32, Kepler showed that the speeds at aphelion and perihelion are inversely as
the distances from the Sun. He claims to have shown also that this rule is very nearly true
throughout the orbit (the real rule for the speeds at two locations on the orbit is that they are
inversely as the perpendiculars dropped from the Sun to the tangents at those two points).
Since for Kepler the speeds are inversely as the distances from the Sun, and since the
speeds are also inversely as the times they take to go an equal linear distance (the greater the
uniform speed, the shorter the time for it to go a given distance), therefore the times are
directly as the distances from the Sun.
That reasoning gets us from physical ideas to the result that the times are as the solar
distances. But the problem with the distances is that there is an infinity of them in the
eccentric orbit, and no time is spent at any one of those distances from the Sun.
So how do we find the time spent in any given arc of the eccentric orbit? We “ADD
UP ALL THE DISTANCES” in very short, very many, arcs. The total distance, or rather
AREA, will be proportional to the time spent sweeping out that area. This is the way we
stumble into Kepler’s Second Law. He is still so convinced of the importance of an equant,
that he brushes past the second law as a pure means to an end! Really, it is the only thing that
makes any physical sense (as Newton will later show), and which gives us a brand new way
of clocking the motion of the planet. We now have an area-clock, as opposed to an angleclock.
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FIRST [HINTED] ARGUMENT THAT THE AREA IS PROPORTIONAL TO THE TIME
(i.e. THE 2nd LAW), FOR THE SAKE OF CALCULATING, FROM PHYSICAL CAUSES,
“THE EQUATIONS,” i.e. THE ANGLES AGB etc.
Kepler hints at an argument where he says it “seemed to me ... that
by computing the area CAH or CAE I would have the sum of the infinite
distances in CH or CE.”
First, without calculus (but in anticipation of it), we assume
something which is strictly speaking impossible yet somehow intuitively
persuasive: Area GAC is composed of an infinitude of distances drawn
G
from A to arc GC. We assume that the area of the circular sector GAC is
somehow the sum of an infinity of lines. And area HAG is likewise
composed of an infinity of distances drawn from A to arc HG.
C
A
Thus
All the distances in arc HG = Area HAG
All the distances in arc GC Area GAC
C
Or
bit
G
But he said before (not quite correctly) that the distances from the
Sun are inversely as the speeds of the planet when it is at those
distances; but times are also inversely as speeds. Hence the times, or
delays, of the planet (at points? well, at very tiny arcs!) are directly
as the distances. So, since the times of the planet in each tiny pointarc are as the distances from the Sun (which distances it has during
its time in each teensy arc), hence
All the times in little arcs of HG = Area HAG
All the times in little arcs of GC Area GAC
i.e.
H
Q
E
B Center of orbit
A
D
Total time in arc HG = Area HAG
Total time in arc GC Area GAC
And that is Kepler’s 2nd Law, i.e. that the areas swept out around A, the Sun, are as the times.
Kepler SUGGESTS this argument, but then seems to give ANOTHER ARGUMENT
a little further on! (More on this in a moment.)
ONWARD TO THE ANGLES. . .
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Equant
Sun
Kepler is interested in getting from this new time-rule to
C
the rule for angles around the equant, i.e. a way of calculating
G
them. He says “Thus the area CGA becomes a measure of the
elapsed time or mean anomaly [movement] corresponding to the
H
arc of the eccentric CG.” That is, the planet is sweeping out
Q (Actual Equant)
equal AREAS in equal times around A just as it sweeps out equal
E
B (Center)
ANGLES in equal times around Q, the equant. So the AREAS
around A will be proportional to the ANGLES around Q.
A Sun
There is an occasion of confusion in Kepler’s diagram—
I
the arcs and angles around B are equal, but B is not the equantpoint, and so the times of these equal arcs and angles are not
K
equal. So why does he draw them? This is his SECOND
D
ARGUMENT, or rather his first and only explicit one, and which
is easily missed since it is so imperfectly expressed. It goes like this: All the sectors around
B, standing on equal arcs, are equal, and they are as the angles around B. These are NOT
swept out in equal times, since that would be true of the angles around Q. But if areas around
B are as angles around B, that suggests that where angles are as times around one point (Q),
perhaps areas are as times around another point (A). We are already expecting A to be a
significant center of uniformity somehow. Also, because of the equant, it takes the planet
more than half the time of the semicircle to cover half the semicircle, i.e. to complete arc
CGHE. But look! The AREA standing on that arc from A is also greater than half the whole
area of the semicircle! By a few suggestive hints like this, we draw the general conclusion
that A is the right point around which the planet will sweep out areas proportional to times:
“Therefore . . . as the area CDE is to half the periodic time . . . so are the areas CAG, CAH to
the elapsed times on CG and CH.”
Now the sector CGA has two components:
Area CGA = Sector CGB + rBGA
so
rBGA = Area CAG – Sector CGB
But area CAG is proportional to the TIME, and therefore
to the mean anomaly (i.e. to angular motion around the equantpoint on the eccentric equant-circle), i.e. to ∠CQG.
And area CGB is proportional to the eccentric anomaly,
i.e. to ∠CBG (sectors are as the angles, in a circle).
Hence the remainder, rBGA, must be proportional to the
difference between the angles CQG and CBG, i.e. to angle BGQ,
or “the physical part of the equation.”
So if we know the triangle BGA, i.e. its angles and its
sides and hence also its area, then we know both (1) the “optical
part” of the equation, i.e. ∠AGB, and also the “physical part” of
the equation, i.e. ∠BGQ (since the area of rBGA is proportional
to that angle).
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C
G
H
Q (Actual Equant)
E
I
K
D
B
(Center)
A
Sun
Hence Kepler concludes: “Thus the knowledge of this one triangle [i.e. rBGA]
provides both parts of the equation” for the movement GAC.
And that concludes Kepler’s correlation of his physical theory with the angular
measurement of planetary motion.
AFTERNOTE. This is a perfect example of DISCOVERY-MODE thinking. It is finding the
true from the true-enough-but-still-false! Look how many false things go into finding this
true thing (the 2nd Law):
1. A false orbit (circle).
2. A false rule for the velocities (as inverses of distances from the Sun).
3. A false motive (to discover how to find angles around the equant, a point we have
no general reason to believe exists).
4. All the false stuff about composing things of indivisibles. (Genuine methods of
calculus do not require us to compose continuous things out of an infinity of indivisibles.)
NOTE: Kepler’s 3rd Law, that the cubes of the major axes of planetary orbits are in the same
ratio as the squares of the orbital periods, can be verified just by plugging in the numbers:
Period in Days
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Mean Solar Distance in Miles
88
225
365
687
4333
10759
36,000,000
67,200,000
92,900,000
141,500,000
483,300,000
885,200,000
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