Chapter 46

Kepler has made a hypothesis in chapter 45 about how the planet can be made to change its distance from the sun, but it is necessary to put this hypothesis into numbers to see whether it is correct. Kepler makes three attempts in this chapter to determine how the oval hypothesis of chapter 45 will make the planet move.

The "fictitious" circle

Rather than using a constantly rotating epicycle, Kepler introduces a fictitious eccentric with which he measures the distances of the planet from the Sun at given times. The distances given by the eccentric and the epicycle are equivalent, as has been shown and used in chapters 2, 39, and 40.

First attempt

Here the time elapsed since the planet was at aphelion is measured by angle δβε, placing the planet at ε on our fictitious eccentric. This means that the distance of the planet from the sun is αε. Yet, since the planet is moving slowly near aphelion, it will be at this distance αε before it reaches ε. This earlier location of the planet at this time is μ.

To determine when we reach μ Kepler uses his physical cause, reasoning that the area δαε exceeds area δβε to the same extent that arc δε exceeds arc δμ.

Why is this? Think -- the area δαε represents a sum of distances of the planet over the time represented by either area δβε or arc δε. To the extent that this is larger, the planet should have moved proportionally less over the expended period of time, moving only arc δμ.

Problems

1. The sum of the distances of the planet from the sun is not exactly measured by the area. See chapter 40 for a full consideration of this difference.
2. Although in chapter 40 the area was measured on a circle that corresponded to the motion of the planet, here, our fictitious eccentric has its circumference measuring out equal times. In chapter 40 it was fine to add up the distances, since each one corresponded to times that could be added together -- the times to move equal units of arc on the orbit. But now, since distances added along the circumference are spaced according to equal times, it would be incorrect to simply add them, although Kepler says that this difference will be very small. See the helpful aside for more.
3. Let's say we calculated the two areas, and have performed our proportion to determine what arc δμ ought to be. If we draw this from the center, we would have angle δβμ. But it is not possible to construct any arbitrary angle or any division of the circle. For more on this, see Book I of the Harmonies of the World.
4. We cannot use a circular angle δβμ to measure what is really an oval arc δμ.

Second attempt

Perhaps a more direct path can be taken to determine planet's position. To avoid the second objection of the first method, we can measure the area swept out, not on the fictitious eccentric, but along the planet's true path, in the manner of chapter 40. Kepler:

“However, on the true path of the planet, the plane between the arc of the path and the sun α is likewise the true measure of the time during which the planet is found on the arc lying above it, by chapter 40.”

Area εβδ is a measure of the time, moving uniformly along the circumference of our fictitious eccentric. So if area μαδ can be made equal to it, then the position μ would be the correct location for the given time.

When we look at the two areas laid atop each other, we see that most of the area is in common to the two triangles. We need simply find where to cut line βε at η to put μ in the right place. This would be the cut that removes εημ from triangle εβδ and adds an equal area αηβ to triangle μαδ.

Problems

1. It is still true that an area (an approximation) is not equal to the sum of the distances (the true measure of time, in Kepler's view).
2. It is not possible to construct this division corresponding to the desired area. Kepler writes, in words almost identical to his posing of the "Kepler Problem" in chapter 60 that:

   “There is no geometrical way showing how to cut a given semicircle in a given ratio with a straight line drawn from a given point on the diameter.”

3. Since the planetary locations, such as μ do not lie on the circumference of the circle, it would be an error to measure the areas along the circle.

Third attempt

Kepler's difficulties in trying to use his oval hypothesis for distances in conjuction with his physical principles of distance-time and area-time lead him to a contrivance:

“Since geometry has left us destitute, in order that we may have a description of the line which has been born to us out of the theory of chapter 56, let us go seek the assistance of a contrivance by fetching our vicarious hypothesis from chapter 16, which places the lines... at which the planet stands at the correct zodiacal places at the correct times, combining it with the present... theory of chapter 45.”

To combine these two ideas, Kepler uses two circles. The dashed circle has point C as its center and D as its equant, according to the vicarious hypothesis. This circle will be used to determine the planet's zodiacal location as seen from the sun (point H). The additional, solid circle centered on point B, uses the eccentricity of chapter 42 (or the bisected eccentricity proposed for all planets in part III). This circle will be used to give us point F at the distance from the sun according to the hypothesis of chapter 45. By swinging the length AF up to the line AH, a new point (red) is created -- the position of the planet according to the hypothesis of chapter 45.

“Thus the line AG, constituted by two manifestly false hypotheses, is nevertheless true in its zodiacal longitude, and its length is consonant with the hypothesis of chapter 45.”

Note that this animation, like the diagram in Kepler's book, places C in the middle between B and D, which is not correct -- but placing C at its correct position, nearer to B, made the animation hard to see.

What shape does this combined motion give to the orbit of the planet? Kepler writes that it is not elliptical, but instead is egg-shaped (oval), and, as Kepler says, it is narrower at the bottom than at the top. You can see this more easily if you increase the eccentricity in the animation.

But mustn't there be some way of calculating the planetary positions directly from Kepler's physical principles and the hypothesis of chapter 45? Kepler continues his search in chapter 47.

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