Part IV

      To reduce the matter to essentials, we may say: the generation of an elliptical orbit of Mars was recognized by Kepler's measurements to be the result of what Gottfried Leibniz was to make his unique discovery: his definition of the differential of the infinitesimal calculus. Simply said: the notion of the infinitesimal which Kepler presented to “future mathematicians,” was a reflection of the observed consistency of the fact, that the area subtended by the sweep of the orbit of Mars, relative to the Sun, varied in an ordering of “equal areas swept, during equal times.” In other words: the elliptical orbit did not determine the motion of Mars; rather, the relevant, perfectly infinitesimal principle of physical action, generated the elliptical orbit of this specific characteristic, the characteristic of equal areas swept within equal times.

Here Kepler applies physics to the motion of Mars, developing three main topics: measuring the effect of graviation, determining the true non-circular shape of Mars's path, and discovering how that path is dynamically created.

Kepler begins in chapter 41 by telling you that the determination of Mars's characteristics from three observations, which would be conclusive were the planet a circle, creates different outcomes depending on which three observations you choose. Since a circle is wholly determined by three points, perhaps it does not move in a circle. He'll come to this later, after testing the circle more. He uses another method in chapter 42 to determine the correct eccentricity and size of Mars's orbit, and then applies it in chapter 43 where he tests his implementation of the area-time property of physical gravitation to determine the longitudinal position of Mars. His results are off by 8' compared to the vicarious hypothesis. He concludes in chapter 44 that Mars must not move in a circle, and offers a proposal for the shape of its non-circular motion in chapter 45.

Now, he must apply this non-circular hypothesis of the planetary path to determine whether his hypothesis is correct. In chapter 46, he has to again bring in the vicarious hypothesis for longitudinal positions to test the lengths generated by the hypothesis of chapter 45. Then, in chapter 47, he develops a way to attempt to measure the area of the oval path to apply area-time to it. This he finds to be off by 6', which he surmises might be an error in the way he determined the areas. So, in chapter 48, he uses the sum of the 360 distances of the planet from the sun at each 1° of its path. This is very difficult, but eventually produces results that are only about 3' off.

I greatly congratulated myself, and was confirmed in the opinion of chapter 45

But, the methods he had been employing begged the question -- he had to assume the path of the planet to measure its distance from the sun and thence its speed, but its speed determined its path as well. He returns, in chapter 49, more directly to the process of chapter 45, but finds himself 8' off again.

We immediately seized upon a certain quantity for the oval, solely on account of the elegance of the physical causes and the graceful uniformity of the epicylic motion, which was falsely given credence.

Does this oval motion even give the correct distances? Kepler answers this question in chapter 51, along with a confirmation of the direction of aphelion. Using the data he develops there, he proves conclusively that the apparent sun must be the center for the world in chapter 52. He then develops more sun-earth-Mars distances in chapter 53, and applies all the distances he knows to accurately determine the ratio of the orbits of earth and Mars, and the eccentricity of Mars's orbit in chapter 54. Comparing these accurate observational distances with the distances created by the circular and oval (ch.45) hypotheses, Kepler finds that while the circle is too large, the oval overcompensated and creates lengths that are too short. The "middle course" is presented in chapter 55, and a means by which it can actually be created as a process, rather than a shape, is presented in chapter 56.

Now, the question of the natural principles that could cause such a reciprocation are taken up in the immense chapter 57. A correction to eliminate the puffed-cheekedness is made in chapter 58. With this final change to the orbit, Kepler shows in chapter 59 that the area of the circular area swept out is not only close to the sum of the distances from the planet, but is exactly this sum for the ellipse. He now applies all that he knows of the orbit of Mars in chapter 60, which includes the Kepler Problem posed to "future mathematicians."