Chapter 30

      It is true that in this way (that is, in associating the distance αζ of the angle δαζ with an angle which is as much smaller than δαζ, as δαζ is smaller than δβζ) a path is attached to the circuit of the earth (or sun) about α which is oval rather than exactly circular.
(pp. 363-364)

When we follow Kepler's instructions, we will cause the planet to move along an oval (non-elliptical) path.

  • First, Kepler says that we will associated "the distance αζ" with a new angle.  This length αζ is just the distance from the sun to the position of the planet if it moved in a circle.
  • Now, the planet will not be this distance from the sun at angle δαζ,but rather at "an angle which is as much smaller than δαζ as δαζ is smaller than δβζ.  So:
    δαζ - new angle = δβζ - δαζ.
  • To construct this angle from α, we will recreate the angle δβζ at α, by drawing a new line from α parallel to βζ.
  • Now, simply make a new line from α, by duplicating the difference of δβζ to δαζ.  Place the planet along this line, at distance αζ from the sun, and you have the orbit Kepler proposes:
To review, three lines come out of the sun at α:
  • One, with an angle at α equal to angle δβζ is not connected to anything
  • δαζ is connected to the red point on the circle
  • the third line is connected to the blue location of the planet on the oval path.
The differences between these angles are the same.  Further, the length from the sun to the red point is the same as the length from the sun to the blue point.

Thanks to Ms. Rodarte:
     A .pdf of the actual values given in Chapter 30.