Perhaps the errors in Chapter 47 came from wrongly using the area as a measure of time. After all, Kepler has not yet considered area as a true measure of time, but only as a convenient way of adding an infinite number of distances. The true principle in his mind is that the distance from the sun determines the rate of motion. In this chapter, Kepler uses a sum-of-distances approach, rather than an area approach for measuring time. This has several advantages:
“And so with renewed preparations, I settled down again upon the eggs…”
A Difficult Calculation
Kepler’s path to implementing this idea is quite difficult, and will probably require several readings-through, and perhaps the use of a spreadsheet program to try to replicate Kepler’s method. (The way to be sure that you know what Kepler is doing, is to do it yourself!)
Whew! That’s a lot of work! And no one anomaly can be known without working through all the others leading up to it. Kepler:
“I can’t imagine anyone reading this not being overcome by the tedium of it even in the reading. So the reader may well judge how much vexation we (my calculator and I) derived hence, as we thrice followed this method through the 180° of anomaly, changing the eccentricity each time.”
There is a problem in all of this…
Determining the ovoid distance
To properly account for the optical magnification of each portion of the oval orbit, we must first know the total length of the orbit. Otherwise, the magnification would be incorrect and the equated anomaly for mean anomaly of 180° would not be 180°.
Just as Kepler sought the quadrature of the area of the ovoid in chapter 47, he must now determine its circumference. To this end, he uses the diagram on the left. The semi-circles DR and HK are centered on B, with radii BD and BH corresponding to the semi-major and and semi-minor axes, respectively, circumscribed outside and inscribed within, the green dotted ellipse DR. The circumference of the ovoid must be greater than HK and less than DR, but where in the middle between the two is the correct circumference located?
Kepler now draws two “mean” semi-circles: DK (in blue, centered on I) and OP (in red, centered on B). Circle DK has a radius (and thus a circumference) which is the arithmetic mean between the two bounding circles, while circle OP has a radius (and circumference) which is the geometric mean between the two. (Do you know how to construct OP?)
Kepler argues that it is correct to use the arithmetic mean circle DK. He reasons thus: the geometric mean circle has the same area as the ellipse, while the arithmetic mean circle has a larger radius (and area). But among shapes covering a given area, it is the circle which has the least circumference, and thus the geometric mean circle, having an area equal to the ellipse, must have too short a circumference. This leaves him with the arithmetic-mean semicircle, whose length is 179°23′40″ where the semicircle KR is 180°.
This arithmetic length is only an estimate. Kepler must try out hypothesized lengths of the oval and work out the entire 180° to determine whether he has chosen the right length for the oval. If he has, he will get an equated anomaly of 180° for a mean anomaly of 180°.
“As for me, I… by a most laborious and dogged calculation found… that where the perfect semicircle is 180°, the oval is 179°14′15″.”
With a length for the oval, Kepler can now implement his method. (For a degree-by-degree working-through, click here for a Microsoft Excel spreadsheet.) This is the resulting motion:
Now Kepler can implement his tedious method, and compare his results with the “veracious vicarious,” his “index of truth.”
These errors, especially the fact that the 90° value is so far off, indicate to Kepler that the eccentricity (9165) is too big. Redoing his entire lengthy calculation with an eccentricity of 9230, he arrives at:
As in chapter 47, the planet moves too quickly in the middle longitudes. But note the reasonably good results -- this is the closest that Kepler has come to matching the accuracy of the vicarious hypothesis since he began trying to implement his physical causes in the current Part IV.
“So, when I saw that as the physical causes introduced in chapter 45 were the more skillfully, and the more fittingly advocated, for the implementation of the theoretical foundations of the calculation, I came ever nearer to the true equations furnished by the vicarious hypothesis of ch. 16, I greatly congratulated myself, and was confirmed in the opinion of chapter 45.