*The Demarcation Problem was what brought me to this site, as it happens.*Right.

Then, you'll happy to find out that Kepler fudged/faked/falsified the entire set of data in the New Astronomy.

This data in turn lead to the law of universal gravitation published by Newton.

This law was then used to derive the nonlinear differential equations approach to orbital mechanics.

*Kepler published his first law of planetary motion based on the data gathered by Tycho*

Brahe in 1609. The law states that planets orbit the sun in ellipses with the sun at one focus.“Almost 400 years later, William H. Donohue undertook the task of translating

Kepler’s 1609 Astronomia Nova into the English New Astronomy (Donohue 1992)

when in the course of his work he redid many of Kepler’s calculations, he was

startled to find some fundamental inconsistencies with Kepler’s reporting of these

same calculations (Donohue 1988). Writing of Donohue’s pathbreaking work in

The New York Times, William Broad (1990) summarized Donahue’s findings

saying that although Kepler claimed to have confirmed the elliptical orbit by

independent observations and calculations of the position of Mars, in fact Kepler

derived the data from the theory instead of the other way around . . .

“But a close study of Kepler’s New Astronomy . . . shows that the plotted points

[he used] do not fall exactly on the ellipse (of course, measurements rarely fall

exactly on a theoretical curve because they usually have random error sources

incorporated into them.) Curtis Wilson (1968), however, carries error argument

further. The lack of precision inherent in the method . . . would have forced Kepler

to use the plotted points only as a guide to his theorizing . . .

“After detailed computational arguments Donahue concluded the results

reported by Kepler . . . were not at all based on Brahe’s observational data; rather

they were fabricated on the basis of Kepler’s determination that Mars’s orbit was

elliptical. Donahue reasons that Kepler must have gone back to revise his earlier

calculations that were made prior to his understanding that the orbit of Mars was

actually elliptical. Thus, anyone who cared to check Kepler’s tables would find

numbers that are consistent with the elliptical orbit [he] postulated for Mars and

would be inclined to believe that the numbers represented observational data. In

fact, they were computed from the hypothesis of an elliptical orbit and then

modified for measurement error; such data, if they were truly observations, would

be prime facie evidence of the theories’ correctness.

“So Donahue . . . realized that the theory was not obviously derivable from the

observations, . . . ‘Not only would the numbers be confused, but Kepler saw clearly

that no satisfactory theory could come from such a procedure. . . [Instead], he chose

a short cut.’ He became so convinced of what drove these physical processes that he subjectively projected his personal nonobservational-based belief onto the reporting scene to convince others in the scientific community of the validity of his theories.”

Thus, the very first law of planetary motion was built not on observation but on theory

and the mathematics was then employed to prove the theory not test it.

http://adsabs.harvard.edu/full/1988JHA....19..217DKepler's fabricated figures, by W.H. Donohue

The scholar, William H. Donahue, said the evidence of Kepler's scientific fakery is contained in an elaborate chart he presented to support his theory.

The discovery was made by Dr. Donahue, a science historian, while translating Kepler's master work, ''Astronomia Nova,'' or ''The New Astronomy,'' into English. Dr. Donahue, who lives in Sante Fe, N.M., described his discovery in a recent issue of The Journal of the History of Astronomy.

The fabricated data appear in calculated positions for the planet Mars, which Kepler used as a case study for all planetary motion. Kepler claimed the calculations gave his elliptical theory an independent check. But in fact they did nothing of the kind.

''He fudged things,'' Dr. Donahue said, adding that Kepler was never challenged by a contemporary. A pivotal presentation of data to support the elliptical theory was ''a fraud, a complete fabrication,'' Dr. Donahue wrote in his paper. ''It has nothing in common with the computations from which it was supposedly generated.''

But when Dr. Donahue started working through the method to make sure he understood the basis for Kepler's chart, he found his numbers disagreeing with those of the great astronomer. After repeatedly getting the wrong answers for the numbers displayed on Kepler's chart, Dr. Donahue started trying other methods. Finally, he realized that the numbers in the chart had been generated not by independent calculations based on triangulated planetary positions, but by calculations using the area law itself.

''He was claiming that those positions came from the earlier theory,'' Dr. Donahue said. ''But actually all of them were generated from the ellipse.''

Thus, the notion that a planet orbits the Sun in an elliptical orbit was a simple fabrication, based on fudged data.

*In fact, a strong argument for the validity of Newton’s laws of motion and gravity was that they could be used to derive Kepler’s laws.*But the entire Nova Astronomia was faked/falsified, each and every entry.

Mathematics applied to deterministic problems in the natural sciences (C.C. Lin/L.A. Segel), chapter 2: Deterministic systems and ordinary differential equations (pg. 36-70)

To accomplish a mathematical formulation, we adopt a polar coordinate system (r, θ) with the sun as the origin.

The second law of Kepler then states that, following the orbit (r(t), θ(t)) of a planet,

r2dθ/dt = h

The first law of Kepler states that the orbit can be described by the simple formula,

r = p/(1 + ecosθ)

Then one can show that the acceleration in the radial direction is

ar = d2r/dt2 - r(dθ/dt)2 = -h2/pr2

Thus the acceleration is inversely proportional to the square of the radial distance.

Newton, by combining the above results with his second law of motion, was led to formulate the present form of the law of universal gravitation.

This, in turn, leads to a system of N particles in gravitational interaction; e.g., the solar system comprising the sun and the nine major planets.