Diagonal Argument

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The Speed Law

Kepler initially believed in the inverse distance speed law. I’ve discussed how he argued that his whirlpool force hypothesis provided the physical explanation for Ptolemy’s equant. Now we’ll look at the transition to the area law.

First though a change in perspective. Kepler did not have the calculus concept of instantaneous velocity. Informally he did use terms like “swiftness” (celeritas). But for computations, he worked either with the delay: the time to traverse a given small length of the orbit. Or else, inversely, the length traversed during a given small time interval. Initially he used the delay. For this reason, several authors refer just to the “distance law” instead of the “inverse distance speed law”. I’ll do the same in this post.

Delay makes sense when we reflect on the astronomer’s goal: to predict when the planet will appear at a given position. The computational procedure: starting with the planet at aphelion, divide the orbit into many small segments; then add the delays for each segment, each delay proportional to the distance. Determine the constant of proportionality by doing this for the entire orbit, and comparing the result with the known period.

Kepler does this in Chapter 40, chopping the orbit of Mars (still assumed circular) into 360 parts, each subtending a central angle β of 1°. (See the left side of the figure below.)

Say r_i is the Sun-Mars distance at i degrees from aphelion. So Kepler is calculating the time for Mars’s arrival at that position with this formula:

t_n=∑_i=0ⁿ r_i Δβ / ∑_i=0³⁶⁰ r_i Δβ

Δβ is proportional to the length of the segment Δs, because β is a central angle. So each term is proportional to the delay.

You may recognize these sums as Riemann sums for integrals ∫r dβ. Of course Kepler did not have integral calculus to draw on, but he did have Archimedes. He wrote:

… since this procedure is mechanical and tedious … I looked around for other means. And since I knew that the points of the eccentric [i.e., the orbit of Mars] are infinite, and their distances are infinite, it struck me that all these distances are contained in the plane of the eccentric. For I had remembered that Archimedes, in seeking the ratio of the circumference to the diameter, once thus divided the circle into an infinity of triangles … Accordingly, instead of dividing the circumference, as before, I now cut the plane of the eccentric into 360 parts by lines drawn from the point where the eccentricity is reckoned [C in my diagram].

A word about what I will call the sunny side of presentism. Kepler did not have calculus in his toolbox, or even algebra, but dealt with geometrical close equivalents. It illuminates matters to place the modern¹ approach alongside his.

So, he is trying to find t as a function of s, the length traveled. His distance law becomes dt/ds∝r instead of ds/dt∝1/r. Thus t∝∫r ds. The area of a central sector (like ACM) is proportional to the length of the arc (AM here). Each of these central “triangles” has base ds=Rdβ and altitude R, thus area proportional to R²dβ.

Kepler now extends this to sectors from the sun, like ASM. The area of ASM, he argues, should furnish a measure of the sum of the Sun-Mars radii—that is, of the delay along the arc.

This area is easily computed: just divide ASM into the central sector ACM and the triangle △CSM. The eccentricity CS is eR, and △CSM has base eR and altitude R sinβ. If β is measured in radians, then the area is 1/2 R²(β+e sinβ).

Let’s see how integral calculus views this. The “infinite sum of radii” is ∫r dβ, or maybe ∫r dθ. (When the eccentricity is small, dβ≈dθ, so it shouldn’t matter much.) An infinitesimal element of area is 1/2 r² dθ (see the right side of the figure above). For small eccentricity r≈R, so

1/2 ∫r² dθ ≈ 1/2 R∫r dθ

Therefore the sum of the radii is proportional to the area, approximately.

Kepler recognized that his equivalence was not exact:

Nevertheless, my argument contains a paralogism, not, indeed, of great moment. It arises from this: that while Archimedes did indeed divide the circle into an infinity of triangles, they stood upon the circumference at right angles, so that their vertices were at the center of the circle C. But one cannot proceed in the same way with triangles standing upon the circumference with their vertices at S, because the circumference is intersected obliquely by the straight lines from S in all places other than aphelion and perihelion.

[I have edited this slightly to match my diagram.]

He discovered the inexactness from a brute-force calculation. He also provided a demonstration that the sum of the radii over the whole circle is greater than the area. See the figure below:

Pair the radii as indicated, SM₁ with SM₂ and CM₁ with CM₂. Here M₁ and M₂ are the two ends of a diameter. The triangle inequality tells us that SM₁+SM₂ is greater than the diameter M₁M₂. But the sum of the diameters should give us the area of the circle.

Note that if you drop a perpendicular from S to the diameter (the dotted line SP), then the sum PM₁+PM₂ will equal the diameter. Kepler called PM₁ and PM₂ the diametral distances. They were to play an crucial role in the discovery of the elliptical orbit.

A page or so after the passage quoted above, Kepler says:

[this method] agrees most precisely with the observations in the theory of the sun or earth. Nevertheless, it errs in two respects. First, it supposes that the orbit of the planet is a perfect circle, which, as will be demonstrated below in Ch.44, is not true. Second, it uses a plane [i.e., area] which does not exactly measure the distance of all points from the sun. Nevertheless, as if by a miracle, each of these exactly cancels the effect of the other, as is demonstrated below in Ch.59.

Translator Donahue notes: “Readers who hope to find a clear explanation in Ch.59 will be disappointed”, and he describes the treatment there as “no more than qualitative gropings towards a solution”.

The transition from the distance law to the area law

The transition involved several points.

(1) Suppose instead of having dt∝r, we let dt be proportional to the diametral distance. This is R+eR cosβ (see figure below):

So t is proportional to ∫(1+e cosβ)dβ=β+e sinβ. As noted above, the area is 1/2 R²(β+e sinβ). Thus t is proportional to area, if we replace the radial distance from the sun with the diametral distance.

(2) Kepler understood this geometrically. The correct formula for the area of each small triangle in our first figure (with vertex at the sun) is 1/2 base×altitude, not 1/2 base×side. But the altitude of the small triangle is the diametral distance. So the area precisely measures the “infinite sum of diametral distances”.

(3) Spoiler alert! Kepler ultimately concluded that the Mars-Sun distances are the diametral distances. Hence the area precisely measures the “infinite sum of distances to the ellipse”.

(4) Now we have an odd situation. We have the eccentric circle. The actual orbit is an ellipse inside it, tangent to the circle at the apsides. But the slogan “equal areas in equal times” refers to the areas of sectors of the circle, not the ellipse.

This is easily fixed. The ellipse is a “squished circle”, obtained from the enclosing circle by the transformation (x,y)↦((b/a)x,y). (Here b and a are the semi-minor and semi-major axes.) That means that proportional areas remain proportional, the times are still proportional to the elliptical areas swept out, and the slogan still works.

(5) We still have one problem. Kepler drew radius vectors from the sun to the orbit, and spaced them using the central angle. In other words, the areas are proportional to ∫r dβ, where r now is the Sun-Mars distance with Mars on the ellipse. But this disagrees with the distance law. That says dt/ds∝r, so t∝∫r ds. For the circle, ds=Rdβ, so the proportionalities hold using either dβ or ds. Not true for the ellipse: ds≠r dβ because the radius vectors are no longer perpendicular to the curve.

In Keplerian geometrical terms, his distance law says: for equal length segments, the delay is proportional to the radial distance. Yet here he is dividing the ellipse into equiangular segments. Not kosher!

But if we resolve ds into radial and transverse components—say ds_r and ds_θ—then ds_θ=r dβ. In fact r² dβ is twice the area element, so ∫r ds_θ=∫r² dβ is both the measure of the delay and twice the area of the sector.

Looked at another way, it is just the transverse speed, v_θ=ds_θ/dt, that is inversely proportional to the radial distance.

This makes sense physically. The “solar whirlpool” has no radial component of force. It acts purely transversely on the planet. Go way back to the evidence that suggested it: the speeds at the apsides are inversely proportional to the distances. But at the apsides there is no radial component.

Upshot: the physics agrees with the geometrical analysis. The area law reigns supreme.

From the Astronomia Nova to the Epitome

Ch.59 contains “Protheorems” to justify the elliptical orbit and the distance law. These demonstrate points (1)–(4) above.

I quoted Kepler earlier saying that his two errors canceled exactly. By this he means point (3), not a literal cancellation of quantities.

Ch.59 also shows that Kepler recognized the lack of equivalence between the distance law and the area law. He expresses it as I did above, that equal segments on the enclosing circle do not correspond to equal segments on the ellipse. He then gives a confusing argument why one should use the former. Accepting this change makes the distance law equivalent to the area law, as we’ve observed.

I think this counts as more than “qualitative gropings towards a solution”; Donahue undersells Kepler’s achievement. However, it still falls short of his final explanation, in the Epitome.

In that treatise, Kepler identifies two forces at work:

For it has been said above that, if the orbit of the planet is divided into the smallest equal parts, the times of the planet in them increase in the ratio of the distances between them and the sun. But this is to be understood not of all equal parts as such, but principally of [the apsides]….But in the case of the other parts which face the sun obliquely, this is to be understood only of that which in any of these parts belongs to the movement around the sun. For since the orbit of the planet is eccentric, therefore in order to form it two elements of movement are mingled together, as has been demonstrated already: one element comes from the revolution around the sun by reason of one solar virtue; the other comes from the libration towards the sun by reason of another solar virtue distinct from the first. [Translation by Aiton (p.88)]

Kepler’s “movement around the sun” refers to ds_θ, so Kepler is saying that for equal ds_θ’s, the dt’s are proportional to the distances. In other words, dt/ds_θ is proportional to r, or ds_θ/dt=v_θ is proportional to 1/r.

Let’s write F_θ for the whirlpool force and F_r for the libration force. As I said above, it makes sense that F_θ would be proportional to v_θ and not to v. The proportionality F_θ∝1/r formed a cornerstone of Kepler’s physics. Kepler regarded the change from ds to ds_θ as correcting an oversight, preserving the essential features of his system.

In the Epitome Kepler also returns to the matter of inverse square vs. inverse linear. The Epitome is written in a Q&A format. Here’s the quote:

But if the light is attenuated in the ratio of the squares of the intervals, i.e., in the ratio of the surfaces; why therefore does not the motor virtue too become weaker in the ratio of the squares rather than in the simple [ratio]?

Because the motor virtue has as subject a form from the solar body, not according as it is merely body but according as it is set in motion in a revolution around its immobile axis and poles.

Therefore even if the form from the solar body is attenuated in longitude and in latitude no less than the light is; nevertheless this attenuation contributes towards the weakening of the motor virtue only by reason of the longitude: for the local movement which the sun gives to the planets takes place only in longitude, wherein even the parts of the solar body are mobile, not also in latitude towards the poles of the body with respect to which the sun is immobile. [Translated by C.G. Wallis, who uses “form” to translate the Latin species, rendered “image” by Stephenson and left untranslated by Donahue.]

We saw in post 21 the key to the matter: the whirlpool force has only cylindrical symmetry, not spherical. Kepler shows awareness of the issue and its resolution. Stephenson puts it this way:

As we have seen in the Astronomia nova, Kepler believed this force to vary inversely with distance from the sun. He carefully justified this relation in the Epitome, arguing that it was in fact reasonable for the motive force to follow a different law than light. He based his argument upon a distinction between “the immaterial image of the solar body, flowing out to the planets and beyond”, and “its force or energy, which close at hand grasps and moves the planet”. The image, although immaterial, was the subject for the force. As it spread through all the universe, the image suffered attenuation just as did light, in the squared proportion of distance from the source. In contrast, the force was not a substance (not even an immaterial substance), but merely an attribute, a virtue or ability, which the solar image possessed when in contact with the body of a planet. In empty space, where the image was spreading out and thinning as it expanded, there was no motive virtue, and hence no question of its weakening. The virtue came into play only where the sun’s image moved through a planetary body. It acted in one direction only, the direction of the sun’s rotation around the zodiac, and it was weakened only insofar as its subject, the moving solar image, was thinned out in this one direction. The motive effect of the sun’s image, therefore, decreased in simple inverse proportion to distance from the sun. [Stephenson (pp.142–143)]

Before leaving the topic of the speed law and the whirlpool force, let’s cite again their relation to Newtonian physics. In their final form, Kepler’s assumptions read: F_θ∝1/r and F_θ∝v_θ. So v_θ∝1/r, that is, v_θr is constant. Angular momentum is mv_θr, which Newtonian physics tells us is conserved. Mathematical equivalence!

[1] Modern!? Only 300 years old instead of 400!

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The Vicarious Hypothesis

Part II of the Astronomia nova bears the title “Imitation of the Ancients”. Kepler obtains a model for Mars with an eccentric circle and an equant. But the eccentricity is not bisected. Instead, the center is closer to the equant than to the sun. See the figure below. (Notation: S=Sun, C=Center, E=Equant, r=radius, e_S=SC/r, and e_E=EC/r.)

As this post is a bit long, let me highlight the main points.

• The general model has parameters e_S and e_E. We always have e_S+e_E=2e, where e is the eccentricity.
• We get the best fit for heliocentric longitudes using the ratio e_S:e_E=5:3. This is quite close to Kepler’s vicarious hypothesis.
• We get the best fit for latitudes and “off to the side” observations with e_S=e_E, the famous “bisection of eccentricity”. It’s longitudes or latitudes, you can’t have both! At least with a circular orbit.
• Kepler determined the parameters for his vicarious hypothesis by an intricate double iteration, using four acronychal observations.
• The whirlpool force hypothesis requires bisection of eccentricity.
• Although Kepler ultimately rejected the vicarious hypothesis as the correct model, he continued to use it to compute heliocentric longitudes. This is an implicit appeal to his “zeroth law”.

A Modern Perspective

I will refer to the scheme of the figure above as the vicarious model; Kepler used this term for a particular choice of parameters. By the “elliptical model”, I mean the usual elliptical orbit with the planet obeying the area speed law. I assume throughout that the period is the same for both models.

Before considering what Kepler wrote and did, we’ll examine the two models with modern techniques.

Suppose all we care about are heliocentric longitudes. We want to choose the parameters in the vicarious model to match these as accurately as possible. We don’t care about latitudes or distances. Observationally, this means we look only at acronychal measurements of the geocentric longitude. For as noted in post 24, when the planet is in opposition, the heliocentric and geocentric longitudes are equal.

What parameters will make the vicarious model best approximate the elliptical model for heliocentric longitudes? Evans and Linton (p.179) derive the answer: e_S+e_E=2e where e is the eccentricity of the ellipse, and e_S:e_E=5:3.

Evans obtains this result several ways. He imposes three conditions on the motion. Let P be perihelion, and Q quadrature (i.e., the point such that PSQ is a right angle). First condition: the time from perihelion P to Q must be the same for both models. To an excellent approximation when e is small, this implies that e_S+e_E=2e.

Second condition: the angular speed at aphelion must be the same for both models. This gives the equation

;

retaining only terms up to order e², this becomes

e_S+e_E = 2e+2ee_S−⁵/₂e²

Third condition: the angular speed at perihelion must be the same for both models. By similar reasoning, we have

;

approximated to

e_S+e_E = 2e−2ee_S+⁵/₂e²

So we have three equations: e_S+e_E=2e, and the aphelion and perihelion angular speed results. Any two of these equations yields e_S:e_E=5:3.

Evens gives another derivation, using an expression for the heliocentric longitude measured from perihelion as a function of time. (Linton gives essentially the same derivation, but using aphelion instead of perihelion.) Let this be θ, and let ω be the average angular speed, i.e., 2π/T. Up to second order in e, the expression is

θ(t) = ωt+2e sin ωt+⁵/₄e² sin 2ωt

for the elliptical model, and

θ(t) = ωt+(e_S+e_E) sin ωt+½e_S(e_S+e_E) sin 2ωt

for the vicarious model. So first: when e_S+e_E=2e, the expressions agree to first order in e. Second: when also e_S:e_E=5:3, we have e_S=⁵/₄e, and so they agree to second order.

In short, the vicarious model gives very good heliocentric longitudes when e_S:e_E=5:3. To quote Evans: “In fact, with the precision achievable with naked-eye observations even in Kepler’s day, there is no perceptible error in the angular position anywhere around the orbit—as Kepler himself verified.”

Okay, what if we do care about distances and latitudes? Geocentric latitudes provide a means to determine distances. I’ll explain this in more detail below, but intuitively: the closer you are to Mars, the bigger the latitude looks. An optical effect, just like approaching a vertical pole and seeing it look taller. Viewing Mars out of opposition, “from the side”, furnishes another way to check actual distances.

Perihelion and aphelion are minimal and maximal Sun-Mars distances, so a vicarious model that tries to do a good job with distances will share the apsidal line with the elliptical model. That means they will also have the same center and same r, where r is the vicarious radius and the elliptical semi-major axis. In the elliptical model, the Sun-Mars distance at aphelion is r+e and at perihelion is r−e. In the vicarious model, these distances are respectively r+e_S and r−e_S. Conclusion: the vicarious model gives the correct Sun-Mars distance at aphelion and/or perihelion only when e_S=e. In other words, the eccentricity must be bisected.

The upshot: using the scheme of the above figure, you are faced with Scylla and Charybdis. You can have an excellent model for finding heliocentric longitudes at any time, all around the orbit. Just choose e_S:e_E=5:3. Or you can have a model for distances that is accurate at aphelion and perihelion. Choose e_S=e_E=e. You can’t have both.

One more observation: if we do bisect the eccentricity, then the two expressions for θ(t) will first disagree at order e². That is, in the term with the factor sin 2ωt. The maximum (absolute) value occurs when 2ωt=±π/2. Since 2ω=4π/T, that means at t=±T/8. For small eccentricity, this is close to the octants.

What Kepler Did

After some preliminaries, Kepler gets down to business in Chapter 16 of the Astronomia nova. It takes four parameters to specify a vicarious model: the two eccentricities e_S and e_E, the direction of aphelion, and a time when the planet is at aphelion (known as the epoch). Kepler used four acronychal observations to derive the parameters.

The left of the figure above is Kepler’s diagram; the right side removes some clutter. A is the sun, B is the center of the orbit, and C is equant. D, E, F, and G are the four acronychal observations. They give us heliocentric longitudes: the directions of the lines AD, AE, AF, and AG. The directions from the equant (CD, CE, CF, and CG) are called mean longitudes. Since we know the times of the observations, we would know the mean longitudes if we knew the epoch and the direction of aphelion. That’s because the planet moves uniformly when viewed from the equant, and we know the period. Referring to the right side of the figure, you can see that we’d have two angles of the triangle AGC; of course, this also holds for the other three triangles. Temporarily chose the distance unit so that AC has length 1. Then we can solve all these triangles and find the positions D, E, F, and G.

Kepler imposed two constraints. First, the four acronychals must lie on a circle. This is easily checked, since a quadrilateral is cyclic if and only if opposite angles add to 180°. Also, the center B of the circle must lie on the line segment AC. If these are met, then we can determine both the radius and the center of the circle. That gives us the two parameters of the vicarious model, AB/r=e_S and CB/r=e_E.

Unfortunately, there is no direct way to find the epoch and direction of aphelion from the acronychals. Kepler proceeded iteratively. Starting with an estimate for these two quantities, he computed the positions DEFG. If these failed the circle requirement, he adjusted the direction of aphelion. “Repeat until done”, i.e., until DEFG lie on a circle. Gingerich (Ch.22) calls this the “inner iteration”. Then Kepler checked that B lay on the line segment from A to C. If not, he adjusted the epoch. This is the “outer iteration”, since the aphelion direction must now be recomputed. Eventually the process converged on values with both requirements satisfied. Kepler’s final result: e_S:e_E=18564:11332≈4.91:3.

Gingerich (Ch.22) and Voelkel (pp.106–107,111,114–121) present the whole messy story, based on Kepler’s copious manuscripts. In contrast to the account in the Astronomia nova, Kepler began in 1600 by tackling the earth’s orbit (i.e., revising Tycho’s solar theory). In the same year he made his first assault on the orbit of Mars. The two investigations continued in a tangled fashion, with interruptions, until sometime in 1602.

Downfall: Down but not Out

In Chapter 18, Kepler proclaims victory:

You see then, O studious reader, that the hypothesis found by the method developed above, is able in its calculations to account, in turn, for the four observations upon which it was founded, but also to comprehend all the other observations within two minutes…

And Chapter 19 begins:

Who would have thought it possible? This hypothesis so closely in agreement with the acronychal observations, is nonetheless false…

Chapters 19 and 20 then show, “with great thoroughness and almost masochistic delight” (Koestler (p.322)) that the vicarious hypothesis disagrees with geocentric latitudes and observations out of opposition.

We’ve already seen all the ingredients. First, latitudes. Consider the Sun-Earth-Mars triangle (see figure above). At opposition this lies in a plane perpendicular to the earth’s orbit (the ecliptic). The inclination of Mars’s orbital plane is the angle Earth-Sun-Mars. The geocentric latitude is the supplement of the angle Sun-Earth-Mars. If you know the Earth-Sun distance, then you can solve the triangle and get the Sun-Mars distance.

Kepler does this in Chapter 19, and concludes that the Martian eccentricity is between 0.08000 and 0.09943; his vicarious hypothesis sets it at 0.11332. He also notes that “combined eccentricity” (i.e., from the sun to the equant) is about 0.18564; half that is 0.9282, or just about the mean between 0.08000 and 0.09943.

Chapter 20 looks at some longitudes when Mars at perihelion and aphelion, but Earth is “off to the side”—that is, away from opposition. Again the results refute the vicarious model, and fit a bisected eccentricity rather well.

Puzzling. Kepler anticipates the resolution: “the orbit of the star is not a perfect circle, but an oval…”

Although the vicarious hypothesis failed, it still provides Martian longitudes to an accuracy of 2′. Kepler uses it for this purpose in subsequent chapters. In other words, Kepler assumes that if the earth were in opposition, instead of off to the side, then the vicarious longitudes would be correct.

Note the implicit appeal to the zeroth law: No special treatment for the earth. The solar longitude of Mars can’t depend on Earth’s position. Okay, but why prefer the acronychal (opposition) data to the off-to-the-side data?

Answer: the longitudes in question are the Sun-Mars directions. An acronychal observation directly measures this, since the Earth-Mars and Sun-Mars directions are identical. This doesn’t hold for an off-to-the-side measurement. So we should trust the acronychal values, and the vicarious hypothesis gives those values with good accuracy around the whole of Mars’s orbit.

Feetnote

(1) The Mars model by Longomontanus, mentioned in post 24, used a Copernican epicyclet instead an equant (see post 20). But we can convert it into a nearly equivalent equant model; Evans shows how to translate the parameters. Under this transformation, the Longomontanus model had e_S:e_E=5:3.

(2) Kepler complains in Chapter 16:

If this wearisome method has filled you with loathing, it should more properly fill you with compassion for me, as I have gone through it at least seventy times at the expense of a great deal of time, and you will cease to wonder that the fifth year has now gone by since I took up Mars, although the year 1603 was nearly all given over to optical investigations.

Gingerich (Ch.21) wondered why it took Kepler 70 iterations. He programmed it, and it took the computer only 9 trials! He guessed initially (and wrongly) that “Kepler was horribly plagued by numerical errors”. But when he gained access to Kepler’s manuscripts (Ch.22), he found two causes: bad data, and Kepler’s treatment of redundant observations. Gingerich writes:

Thus we see that in the vicarious orbit solution Kepler worked always with the same four oppositions, but the results were repeatedly tested against additional oppositions. In the course of five years’ work, the reduction of the basic data was continually improved. His 70 iterations were spent (probably) in five separate solutions differing only in the values chosen for the initial times and angles.

Gingerich also notes a fine point I glossed over in my account:

Tycho Brahe had no operational way of knowing precisely when Mars was at opposition. Tycho’s raw observations only approximated the opposition places, and Kepler was obliged to correct and interpolate them to obtain the acronychal positions he required.

(3) The whirlpool force explanation for the equant requires bisecting the eccentricity (see post 4 and post 20). So in considering the vicarious model at all, Kepler tempered his commitment to this hypothesis.

Voelkel (pp.107–111) found confirmation in a document Kepler wrote around 1602. We’ve noted how the inverse distance speed law works tolerably well within an orbit, but fails when comparing different orbits (see post 20). In the document, Kepler notices his earlier errors in this matter.Voelkel suggests that this realization freed Kepler to study other divisions of eccentricity. It must have come as a relief when the latitude and off-side computations indicated that eccentricity should be bisected.

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Earth’s Orbit

Ptolemy’s solar model gives the sun an eccentric orbit, with uniform speed—no equant. Both Copernicus and Tycho adopted the uniform speed for the earth’s/sun’s orbit. As a consequence, they end up with twice the correct eccentricity. (See post 8.)

The Ptolemaic system further disguises the earth’s orbit as the epicycles of the outer planets and the deferent of Venus. (Mercury is sui generis.) The epicycles have neither equants nor eccentricities, which amounts to giving the earth uniform speed in a Sun-centered circular orbit. Venus’s deferent has both an eccentric circle and an equant.

Kepler of course would have none of this. No special treatment for the earth! He gave it an equant, with the center of the orbit situated midway between the equant and the true sun. So Kepler assigned the earth half its former eccentricity. This is known as the “bisection of eccentricity”.

How does this matter for Mars? Earth is our observing platform. If you are mistaken about your own celestial position, you will inevitably screw up how you interpret the data.

Kepler devised an ingenious triangulation scheme to determine the earth’s orbit. Mars has a period of 687 days. Observations separated by that interval thus involve Earth-Mars vectors from the variable position of the earth to a fixed position for Mars. Mentally reversing the direction of the vector gives us essential information about the earth’s orbit. I will look at how Kepler used this insight below. First, though, some more historical context.

Voelkel writes:

Kepler undertook to write the Astronomia nova in October or November 1602, long before the discovery of the ellipse and elucidation of the area law… At the time he resolved to write the book, he had written to Herwart von Hohenburg that his work on Mars had allowed him to solve the problem of the orbit of the sun (or earth). And in many ways, the Astronomia nova was as much about the earth’s orbit as it was about Mars’s orbit. [Voelkel (p.212)]

Kepler develops his theory of the earth in the first few chapters of Part III. Part II contains the vicarious hypothesis. In fact, this almost inverts the timeline. Not only the zeroth law, but the whirlpool force hypothesis pushed Kepler to investigate the earth’s orbit.

Presenting the vicarious hypothesis first served a rhetorical purpose. Kepler could say, “See? I tried to make the traditional scheme work. The failure of my Imitation of the Ancients, and the way in which it failed, indicate that there is something amiss with the existing model of Earth’s orbit.”

This innovation with the earth’s orbit met with considerable resistance. Tycho disliked it, as did Kepler’s pen pal Fabricius. Tycho felt, “If it ain’t broke, don’t fix it!” Likewise Fabricius, who wrote, “We ought not heedlessly to depart from Tycho’s well-developed solar hypothesis.” As Voelkel puts it:

…this unprecedented innovation, clashing as it did with Tycho’s solar theory and meeting Fabricius’s deep skepticism, needed an unusually sure foundation, especially because it was the ultimate justification for his physical astronomy. [Voelkel (p.236)]

The disapproval lasted—more so than for the elliptical orbit. The astronomer Giovanni Cassini, using a so-called heliometer (aka meridian line) at the Basilica of San Petronio, finally confirmed Kepler’s bisection in 1655. Heilbron (p.102–112) recounts the story. He writes:

Many astronomers objected to a solar equant. Thus began what Flamsteed, writing a generation after its resolution, called a “controversy … of no small moment”. It was this great and obscure controversy between adherents of Ptolemy’s traditional solar theory and proponents of Kepler’s “bisection of the eccentricity”, with its Copernican associations, that Cassini proposed to settle at San Petronio.

…to put the point in a few words (they are those of Astronomer Royal Flamsteed), “the Suns Excentricity is bisected as the Copernicans affirme.” This connection of ideas—the bisection of the eccentricity implies that the earth is a planet—became commonplace among Copernicans.

The Great Martian Catastrophe

In 1593 Tycho observed Mars. It was in the wrong position, by about 10 Moon diameters! The Ptolemaic and Copernican tables gave longitudes differing by about 5°, in opposite directions. The values remained off for several weeks. Gingerich called this “The Great Martian Catastrophe”.

Fixing the earth’s orbit reduced the size of the “great Martian catastrophe” tenfold. Specifically, it reduced the maximum longitudinal discrepancy from about 5° to about 0.5°, or 30′. Kepler’s area speed law plus the elliptical orbit reduced it further to about 2′.

Other simplification flowed from the modified earth’s orbit. Kepler called it “the Key to a Deeper Astronomy”.

The Two Methods

OK, now let’s look at how Kepler determined the position of Earth’s equant. The old theory had the earth move uniformly around the center of the orbit, which is displaced from the sun. Effectively, this makes the equant and the center identical. Kepler replaced this with the center midway between the sun and the equant. The figure below illustrates the first method, from Chapters 22 and 23. In Chapter 22 he shows that the equant is not the same as the center.

In the figure, S, C, and E are the sun, the center, and the equant; and O₁ and O₂ are two positions of the earth. Recall his triangulation trick: if the time interval between O₁ and O₂ is a multiple of 687 days, then Mars will occupy the same position M both times. The times of O₁ and O₂ are chosen so that the two angles θ are equal. Since motion is uniform about the equant, angles about the equant are measured by times. If O₁ and O₂ are separated by equal times from the aphelion and perihelion respectively, then the two angles will be equal.

Kepler also needed the line EM to be perpendicular to the line of apsides (the diameter in the figure). Again this is achieved by picking the right epochs. Altogether, this method requires observations at very particular times. We see the importance of Tycho’s treasure trove.

Now suppose that E and C were the same. Then the diagram would be symmetrical about the line EM, and angles α and β would be equal. (Mentally slide the line EM left until E coincides with C.) But Kepler found that α exceeded β by more than one degree. “Witness the great difference” he exclaimed.

In Chapter 23 he computes the relative positions of S, C, and E. The directions O₁M and O₂M are known from observation. That gives us the angles α and β. The times of the observations give him the angle θ (see above). So all relevant angles are known. I won’t go through the intricate trigonometry that computes the position of E in the line of apsides from this data. But intuitively, it is clear it uniquely determines the position: sliding E from perihelion to aphelion continuously increases α up from zero and decreases β down to zero, so there is exactly one spot where they have the right values.

Kepler found that the eccentricity (CE over the orbital radius) was 0.01834. Tycho’s value for the eccentricity was 0.03584. Half that is 0.01792, close enough to 0.01834. The bisection of eccentricity confirmed!

Chapter 24 sets out with the bold declaration, “Now that we have once made a hazard of this, we are buoyed by audacity and will begin to move more free on this battlefield.” Kepler proposes to take triplets of observations, separated by multiples of 687 days, and from them find the distances of the equant to three points of the orbit. Three points determine a circle. “If a fourth observation will be at hand, it will serve as a test.”

The figure above reproduces the diagram from Chapter 24. Here κ is the fixed position of Mars. The variable Earth positions are θ,η,ε,ζ. The dashed circle is the earth’s new orbit, with the new center β bisecting the eccentricity.

The observations give the Earth-Mars directions θκ, ηκ, εκ, and ζκ, but not the distances. Earth-Equant directions are given by the times (because uniform motion). Mars-Equant directions are given by Tycho’s theory of Mars (or by the vicarious hypothesis). Consider an Earth-Equant-Mars triangle, such as △ηακ. We have all the directions, and thus all the angles. The law of sines tells us all ratios of sides. In particular, it gives us the distance αη in terms of ακ. We have this for all four Earth positions, which means we have the distances αθ, αη, αε, αζ in terms of some common unit. (For another exposition of this material, see Stephenson (pp.53–55), Voelkel (pp.105–106), or Donahue (pp.45–50).)

The distances differed, again disproving the identity of the equant with the center. The results once more agreed with the bisection of eccentricity. Kepler now had empirical proof of one of his deductions from his whirlpool force hypothesis.

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Here Comes the (True) Sun

I mentioned earlier “Kepler’s zeroth law”, a corollary to the motto:

Refer everything to the true sun, and not the mean sun. Nor should the earth receive any special treatment.

Kepler plucked several fruits from this tree. Here are two fruits of the zeroth law.

Apsides

The solid circle is the true orbit for Mars. The dashed circle has the wrong apsidal line, passing through the mean sun instead of the true sun. X and Y are the positions of Mars in the two models at the same time. Note that the longitudes of X and Y differ little, whether viewed from the true sun or the mean sun. But the distances differ significantly.

Gingerich summarizes the situation nicely:

[T]he eccentricities assigned by Ptolemy to each of the planets are essentially vector sums of the earth’s and the planet’s eccentricities, with the result that in each case the line of apsides is slightly wrong…Copernicus did not bother to sort out the individual eccentricities, so each planetary eccentricity implicitly combines the earth’s. In addition, his apsidal lines are drawn through the mean sun… [Gingerich, Chap.19 “Kepler’s Place in Astronomy”]

So far as heliocentric longitudes are concerned, the wrong apsidal line won’t pose too serious a problem. See the figure above, which shows that solar distances are not so forgiving.

To understand this better, we need to look at acronychal risings. These are observations taken when the planet (here Mars) is in opposition to the sun. (See the figure below. ‘Acronychal’ means ‘night rising’.)

Key point: for an acronychal observation, the heliocentric and geocentric longitudes are the same. The astronomer can directly measure only geocentric longitudes. Except at opposition and conjunction, deducing the heliocentric longitude requires an assumption about the earth’s position. At conjunction, the brightness of the sun makes the planet invisible.

Kepler analysed the situation in Chapters 5 and 6. (These make slow reading; Stephenson (pp.31–39) gives a detailed discussion.)

This was a “must pass” test for the switch to the true sun. Tycho’s chief assistant Longomontanus (aka Christian Severinus) had worked out a model of Mars’s orbit that accounted for the acronychal data with an accuracy of 2 minutes, for a period of over 40 years. As Kepler explains in Chapter 7:

A hypothesis was invented which, it was proclaimed, represented all these oppositions within a distance of two minutes in longitude… It was only in the latitude at achronychal positions and also the parallax of the annual orb [i.e., observations out of opposition] that Christian got stuck… At the beginning there was great controversy between us as to whether it were possible to set up another sort of hypothesis which would express to a hair’s breadth so many positions of the planet, and whether it were possible for the former hypothesis to be false despite its having accomplished this so far over the entire circuit of the zodiac.

Chapter 6 showed the near agreement of the true sun and mean sun models for acronychal longitudes. But observations “from the side” (i.e., out of opposition) could amount to as much as 1° 20′, fourteen times the error at opposition and easily observable.

Latitudes

In post 13, I noted (following Swerdlow) that poor data bore most of the blame for Ptolemy’s complicated latitude theory. But having the orbital planes pass through the mean sun is just asking for trouble. The figure below illustrates the problem.

The diagram shows a cross-section of the three-dimensional geometry. O_aphelion and O_perihelion are Earth’s aphelion and perihelion, so the line through them is the ecliptic, and the mean sun is midway between O_aphelion and O_perihelion. The solid line at an angle is the orbital plane of Mars, with the inclination greatly exaggerated.

Shifting Mars’s orbital plane to pass through the mean sun, while keeping the inclination the same, gives the dashed line. Mars shifts from M to M; exactly where M lands depends on the longitude theory. Viewed from Earth’s aphelion its predicted latitude would be too large, from Earth’s perihelion too small, and from other points in the earth’s orbit also incorrect (except for two points). To fix this problem, one might naturally change the inclination, resulting in the dotted line. This dotted line represents a plane passing through the true position of Mars (M) and the mean sun. However, this fixes things only at one position M; at other points in Mars’s orbit, like M′, you’d need a different inclination. We see the cause of the rocking orbital plane.

This scheme still has a lot of leeway, since it takes three (non-collinear) points to determine a plane; an infinite number of planes pass through M and the mean sun. But it is impossible to avoid a varying inclination, simply because the mean sun does not lie in the actual orbital plane of Mars. Any plane passing through M and the mean sun will intersect Mars’s orbital plane in a line, which means it can correct matters for at most one other point of Mars’s orbit.

Copernicus’s latitude theory did not follow this scheme, but was essentially a transcription of the Ptolemy theory into heliocentric terms. It suffered from an additional drawback: the inclination depended on Earth’s position. This violates Kepler’s zeroth law.

Kepler showed if the orbital plane of Mars passes through the true sun, then you can use a fixed inclination. (The line of nodes does rotate very slowly. Kepler knew this.) He demonstrated this in three ways. I’ll outline only the first two.

Both involve finding the geocentric latitude for observations when this will equal the heliocentric latitude. When Mars is “at a limit”—that is, as far above or below the ecliptic as it gets—this will give the inclination.

Kepler noted that the highest accuracy is not required for these methods. Say M is Mars, and M′ is M projected onto the ecliptic. The distance MM′ doesn’t change that much for M near a limit.

For the first method, suppose that the earth and the sun are equidistant from Mars, hence likewise from M′. Then MM′ appears the same whether viewed from the sun or the earth. So the geocentric and heliocentric latitudes are equal. (See Kepler’s diagram, reproduced above. Here A=Earth, B=Sun, E=Mars (my M), and C=projection of E to the ecliptic (my M′).)

How do we know when Sun-Mars-Earth form an isosceles triangle? Given the ratio of the Sun-Earth and Sun-Mars sides, (i.e., the orbital radii), basic trigonometry tells us the Sun-Earth-Mars angle. It turns out that you can use any observation where that angle is between (roughly) 60° and 72°.

For the second method, Kepler noted that when two planes cut one another, any two lines drawn in the respective planes to a point on the line of intersection, and perpendicular to that line, always include the same angle. (See the diagram above, also Kepler’s.) So if B is the earth, F is Mars, and D is its projection to the ecliptic, then ∠ DBF is the inclination. Therefore we need an observation when the earth lies in the line of nodes, Mars is at a limit, and the sun and Mars appear at right angles from the earth. For this method, we make no assumptions about the orbital radii. However, it requires very special observations.

Kepler consistently found an inclination close to 1.8° with all three methods and many observations. He concluded that not only is the inclination fixed, but it’s rather small. Tycho’s assistants had made a mess of things, finding not only that the orbit of Mars wobbled, but was also “fractured”, with a maximum latitude of 4.6° at one limit and 6.4° at the other.

I noted at the beginning of post 13 that Ptolemy “decoupled” longitudinal computations from latitudinal ones. Now, the speed of planet differs from the speed of its projection to the ecliptic. Moreover, the ratio of these two speeds is not constant. Ptolemy could safely ignore this issue because the effect is small; that’s true because because the inclinations are small. But with Tycho’s much greater accuracy, the matter could no longer be swept under the rug. (It’s known as the “reduction to the ecliptic”.) So in cleaning up the latitude theory, Kepler laid a crucial foundation stone for everything else.

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