Kepler / Astronomy and Mathematics

[Three laws of planetary motions]

Kepler (1571-1630) pigeonholed an extensive observational data on planets left behind by an astronomer Tyco Brahe (1546-1601) to refine on Copernican theory, establishing the famous Kepler's three laws:

• Law 1: A planet will move in an elliptic orbit one of whose focal points is the sun;
• Law 2: A radius linking a planet and the sun will generate the same area per the same time (law of equal areas);
• Law 3: Ratios of the square of a planetary orbital period to the cube of a mean distance between the planet and the sun are constant.
However, these laws were found out only from the data and not mathematically deduced from some specific ground; Eager as he is to do so, Kepler did not have any mathematics meeting such purpose in his days.

Law 2 was demonstrated after Kepler's long deliberation as follows:
The area solved can be approximated to the sum of integrals of sectors.

Thus, Area varies as the sum of radius vectors.

[Solid geometry of tun]

Kepler published his book Solid geometry of tun in 1615.
In his book, Kepler introduced 87 solid patterns, or volumes generated by revolving sections of circular cone (circle, ellipse, parabola and hyperbola) about various lines, and solved these volumes using the argument of infinitesimal.

How to calculate an area of circle.
Divide a circle into sectors. Divide as many times as possible.

Next, expand these sectors on a line. Isosceles triangle are obtained.

Transform these Isosceles triangles to gather their apexes to Center of circle , and we get a right triangle identical in area with an area of the circle.

Thus, the area of the circle comes to be .

[What was criticized by other mathematicians those days]

• Anderson (1582-1619)
Kepler should never do such a thing as finding fault with the ancient person worthy of sincere respect. While Archimedes did not suppose that circle can be expanded to triangle, you yourself gave the ground to do so in expanding your apple to a section of circular cylinder. What kind of sages can understand such type of transformation? ------
• Glysin(1577-1643),I don't know a correct spelling of his name.
I believe that this method by Kepler has its great value in finding geometric laws and solving the problems, but will never recommend it as a demonstration if there is another method ascertained by geometricians. This vaguer method of transformation can find acceptance with those being familiar to ordinary demonstrations by Archimedes and Euclid? In any case, I would never change my brain.

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