In prescribing this program of computer animations of physical-economic processes, I avoided the usual error of economic modelling, by taking Johannes Kepler's approach to the discovery of the principle of universal gravitation as a precedent.
− Lyndon LaRouche, Deficits as Capital Gains: How to Capitalize a Recovery, December 19, 2005
Ostensibly dealing with the second inequality, Part III is the birth of modern physical science. The process that guided Kepler through Part II with the proven impossibility of the equant here bears physical fruit.
You may be surprised that Kepler announces that there is an equant in the theory of the earth, since he has just refuted its existence in Part II. It is worth knowing how Ptolemy, Copernicus, and Brahe dealt with the earth-sun motion. Ptolemy uses equants for all the planets, but not for the sun: it moves at a uniform speed (although it does have an epicycle). Copernicus, ever following Ptolemy, has the earth move uniformly, not introducing the double epicycle he applies to the other planets. Brahe is as Ptolemy, with an equant-free sun. Furthermore, if you look for the earth in Copernicus's table of the planets, you won't find it! Although it supposedly moves, Copernicus only tells you of the motion of the sun in the sky, rather than any cause for the earth's motion.
Against this privileged treatment of the earth, Kepler offers Part III. He begins by demonstrating from observations in chapter 22 that the earth is not centered around its point of uniform motion (that is, it does not move as a simple eccentric around a center), but, in fact, shows in chapter 23 that, like Mars and the other planets, the earth has a bisected eccentricity. To give a clear demonstration of this, he does something unique: he watches the earth from Mars in chapter 24. Since it is difficult to watch the earth while standing on it, Kepler uses the earth-sun-Mars movement to develop a very clear course for the earth in the heavens. He goes on to conclusively show that the center of the earth's motion is the midpoint between the true sun and the point of uniform motion. Thus, the earth, too, conclusively appears to have a bisected eccentricity. He spends chapters 25, 26, and 27 determining, without any assumptions, the eccentricity and aphelion direction of the earth, and confirms his values in chapter 28. He then develops a method for determining distances, and presents a table of oval earth-sun distances in chapter 30. Now, some might object to Kepler that the old, simple eccentric models made correct tables of equations, and therefore, changing the model will upset the correct tables. In chapter 31, Kepler eliminates this objection.
But why has Kepler been using an equant for so many chapters? Aren't equants false? Listen to this statement, which he demonstrates in chapter 32:
First, the reader should know that in all hypotheses constructed according to this Ptolemaic form, however great the eccentricity, the speed at perihelion and slowness at aphelion are very closely proportional to the lines drawn from the center of the world to the planet.What could cause this change of speed?
Now it is an axiom in natural philosophy of the most common and general application that of those things which can occur at the same time and in the same manner, and which are always subject to like measurements, either one is the cause of the other or both are effects of the same cause.
Therefore,... the cause of this intensification and weakening resides in... the center of the world [the sun].After noting light's similarity to this power, light is dismissed as the direct medium for this cause.
The remaining possibility, then, is that, just as light, which lights the whole earth, is an immaterial species of that fire which is in the body of the sun, so this power which enfolds and bears the bodies of the planets, is an immaterial species residing in the sun itself, which is of inestimable strength.He then considers magnetism:
For, by the demonstration of the Englishman William Gilbert, the earth itself is a big magnet... It is therefore plausible, since the earth moves the moon through its species and is a magnetic body, while the sun moves the planets similarly through an emitted species, that the sun is likewise a magnetic body.
Further, in chapter 37, he writes that the earth has a force that "retains the moon, like a sort of chain, which would be there even if the moon did not circle the earth at all." After taking up the possibility of motions proper to each of the planets in chapter 38, and the difficulty they would encounter in acting to cause the planets to move in circles (in chapter 39), Kepler applies his principle of gravitation to the determination of the motion of the earth. In the profound chapter 40, Kepler develops a technique for applying his everywhere-different physical principle to the path of the planet. He develops area as the measure of time of motion.