How did Kepler arrive at his equation without calculus?

[alemsalem]

The Kepler equation is: M = E - e Sin(E); where M is the mean anomaly and E the eccentric anomaly and e is the eccentricity.

we get it from integrating dr/r' and changing variables to E, how did Kepler get it without calculus and using only his laws.

Another thing how did he see his second law (areal velocity) without differentiation.

 

[stevenb]

The Kepler equation is: M = E - e Sin(E); where M is the mean anomaly and E the eccentric anomaly and e is the eccentricity.

we get it from integrating dr/r' and changing variables to E, how did Kepler get it without calculus and using only his laws.

Another thing how did he see his second law (areal velocity) without differentiation.

Even more amazingly, he did it without computers to manipulate, calculate and visualize the data he was working with.

Bottom line, he did it all with brains and sweat. That's how geniuses always do the amazing things they do.

 

[alemsalem]

Even more amazingly, he did it without computers to manipulate, calculate and visualize the data he was working with.

Bottom line, he did it all with brains and sweat. That's how geniuses always do the amazing things they do.

Cool, I want to be a genius!

 

[Michel_vdg]

Cool, I want to be a genius!

99% hard work and 1 good idea ... I once read that the basic idea he had, was that of a row boat, and how the boat goes the fasted where the oar is in the water, more or less

[PLAIN]http://img593.imageshack.us/img593/5549/rowboat.jpg

 

[pmacias]

Kepler arrived at his equation without using calculus by utilizing his laws of planetary motion and his observations of planetary orbits. His first law states that planets move in elliptical orbits with the sun at one focus, and his second law states that the line connecting a planet to the sun sweeps out equal areas in equal times. By combining these two laws and using geometric principles, Kepler was able to derive the relationship between the mean anomaly and the eccentric anomaly, which is represented by the equation M = E - e Sin(E). This equation allows us to determine the position of a planet along its orbit at a given time.

As for his second law, Kepler was able to observe the equal areas being swept out by a planet in equal times without the use of differentiation. He used a method called the "law of areas" which states that the area swept out by a planet in a given time is directly proportional to the time and the distance from the planet to the sun. This can be represented mathematically as A/t = 1/2r^2, where A is the area, t is the time, and r is the distance from the planet to the sun. This allowed Kepler to see the relationship between the speed of a planet and its distance from the sun, without the need for calculus.

Overall, Kepler's laws of planetary motion were based on observations and geometric principles, rather than calculus. It was not until later on that his laws were mathematically explained and derived using calculus by scientists such as Isaac Newton. However, Kepler's laws still remain fundamental in our understanding of planetary motion and have greatly contributed to the field of astronomy.