The concept of dual transitions was outlined in two articles published in this journal [1,2]. The purpose of this article is to demonstrate this concept on the example of works of Kepler.
In the evolution from Ptolemaic system to heliocentric system Copernicus accomplished just the first dual transition by changing the roles of Earth and Sun. Sun from being the center of an epicycle became the center of a deferent. And Earth from being the center of a deferent became the center of an epicycle. Still all planets were assumed to move along two circles, epicycle and deferent.
Kepler accomplished several more dual transitions to make the heliocentric system viable:
Firstly, he replaced two circles with one ellips, i.e. replaced two lines with one center, with one line with two centers.
Secondly, instead of searching for a relationship between the measure of what a point (planet) traveled and the time it took, he started searching for a relationship between the measure of what the line (connecting Sun and planet) traveled and the time it took. The measure of what a point traveled is distance, whereas the measure of what a line traveled is area it swept. Having accomplished this dual transition from point to line, he found his second law.
The third dual transition of Kepler requires a longer explanation. The movement of each planet is characterized by several parameters, e.g. orbital period, average orbital radius, etc. Let for planet number i the parameters be {p_{ij}}. Initially in his book "Cosmographic mystery" Kepler set the goal of finding relationships between the same parameters of different planets, i.e. for every parameter number j find a relationship
f_{j}(p_{1j}, p_{2j}, ..., p_{nj}) = 0,
where n the number of planets.
But search for such relationships yielded no results. This caused him to accomplish the third dual transition by switching to searching for the dual relationships
f_{i}(p_{i1}, p_{i2}, ..., p_{im}) = 0,
where m is the number of parameters.
In other words, instead of searching for relationships between the same parameters of different planets he switched to searching for relationships between different parameters of the same planet. As a result he found that
p_{i1}^{2} = k x p_{i2}^{3}
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