Kepler and Aristarchus in my opinion are the two greatest Scientist's in history. Aristarchus was a Greek philosopher that taught a Heliocentric model of our solar system but made the mistake of using perfectly circular orbits which made his model less accurate at predictions than the Geocentric Ptolemaic model that Aristotle defended. The real reason the Geocentric model was taught was because it was more accurate not because of religion. Copernicus credited Aristarchus with inspiring his model of our solar system, in which Copernicus added epicycles to all the planets orbits around the Sun. These small epicycles of Copernicus I believe anticipated the elliptical orbits. If you make the period of revolution of the small epicycle once per revolution around the Sun the actual path traveled is an ellipse.
We might wonder how Kepler derived his famous 3rd law? If we take the logarithm of both sides of the relationship; Log(T^2)=Log (ka^3) where k is an arbitrary dimensioning constant, we get 2 Log(T) = 3Log(a) + Logk which when plotted Log v. Log gives a straight line Log(T) = 3/2 Log(a) + C where C is the ordinate intercept. The important parameter here is 3/2 which directly leads to his third law. The 3/2 coefficient is independent of the base of the Logs used. There were natural, base-2 and Briggs base-10 log tables in use then. Kepler used this Logs technique on known planetary data and published in Kepler’s Ephemerides (1620) and explicitly dedicates his work to Napier (Logarithms). This is what is called an 'empirical' law (obtained from observational data), it can be derived theoretically from Newton's Law of Universal Gravitation (the Inverse Square Law). But this is cart and horse stuff as Newton used Kepler's Laws to formulate his later on!
Kepler and Aristarchus in my opinion are the two greatest Scientist's in history. Aristarchus was a Greek philosopher that taught a Heliocentric model of our solar system but made the mistake of using perfectly circular orbits which made his model less accurate at predictions than the Geocentric Ptolemaic model that Aristotle defended. The real reason the Geocentric model was taught was because it was more accurate not because of religion. Copernicus credited Aristarchus with inspiring his model of our solar system, in which Copernicus added epicycles to all the planets orbits around the Sun. These small epicycles of Copernicus I believe anticipated the elliptical orbits. If you make the period of revolution of the small epicycle once per revolution around the Sun the actual path traveled is an ellipse.
We might wonder how Kepler derived his famous 3rd law?
If we take the logarithm of both sides of the relationship;
Log(T^2)=Log (ka^3) where k is an arbitrary dimensioning constant,
we get 2 Log(T) = 3Log(a) + Logk
which when plotted Log v. Log gives a straight line
Log(T) = 3/2 Log(a) + C where C is the ordinate intercept.
The important parameter here is 3/2 which directly leads to his third law.
The 3/2 coefficient is independent of the base of the Logs used. There were natural, base-2 and Briggs base-10 log tables in use then.
Kepler used this Logs technique on known planetary data and published in Kepler’s Ephemerides (1620) and explicitly dedicates his work to Napier (Logarithms).
This is what is called an 'empirical' law (obtained from observational data), it can be derived theoretically from Newton's Law of Universal Gravitation (the Inverse Square Law). But this is cart and horse stuff as Newton used Kepler's Laws to formulate his later on!
Thanks for that Tom! A bit beyond my meagre mathematical abilities, but I'm sure it's very useful!
It's not Pahtolomy...
It's pronounced Tol o mee