Major Access = Averate Distance over Eccentric Anomaly

[emstone]

Hi,

This is my first post and I am truly perplexed. I am reading principia in a class and it seems that Newton has taken Kepler's law which requires the average distance over the eccentric anomaly that he measured to determine the period, or vise-versa. Newton has substituted the major-axis for the average distance in this proportionality.

Now, I have done Appolonius before but once again, not from a textbook, only the original source so go easy on me.

I searched the web to try to understand why this is true but everywhere just says it is true! Can anyone shed some light on why the major axis is equal or proportional to the average distance from the center of a body in an elliptical orbit?

 

[emstone]

I think I figured this out in my sleep intuitively. All points that aren't at the ends of the major axis of thh ellipse are balanced as the shape is symmetrical. If you take the mean distance from the focii to the ends of the major axis and divide then I think you get half of the major axis!