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.
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.
ProblemsKepler raises four objections against the use of this method:
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.
ProblemsKepler raises three difficulties to the use of this method:
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.”
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.