BISECTION OF ECCENTRICITY

The notion of “bisection of eccentricity” has been briefly mentioned in earlier sections. Here, we provide a more detailed discussion of this concept and aim towards clarifying the double meaning it holds. Most of the content presented in this section is not new but having the information on bisection of eccentricity grouped in one place should boost our understanding of it significantly.

We recall that on Ptolemy’s planetary model, he had added a third device called the “equant point” to the intermediate deferent-epicycle model such that the width of the retrograde loops could fit exactly onto it. The angular velocity of a body on orbit, when observed from the equant point, is constant. Figure 14 is repeated below for easy reference.

Figure 14

As a result of the addition, two eccentricities were defined. Taking the radius of the deferent to be a, e1 represented the eccentricity of the Earth E with respect to centre C and e2 represented the eccentricity of the equant point X with respect to C. That is,

The total eccentricity refers to the sum of e1 and e2. In this case, by “bisection of eccentricity”, it refers to the situation where Ptolemy places the equant point on the mirror image of E along the line of apsides such that e1 = e2 is obtained.

The other meaning of “bisection of eccentricity” arises when we turn towards comparing Kepler’s eccentricity value with Ptolemy’s (or Hipparchu’s) value. The former is half the value of the latter because of the difference in the way Kepler and Ptolemy had measured the separation between Sun and Earth on their respective solar models. A similar explanation to that in the section "Ptolemy's models", together with figure 19, is given below. 

Figure 19

C represents the centre of the sun’s orbit, EP denotes where Ptolemy positions the earth, EK denotes where Kepler positions the earth and XK marks Kepler’s equant point. Consider radius of the sun’s orbit to be a, then the intervals between XK, C, EK and EP would each be .

Both Ptolemy and Kepler had calculated the solar eccentricity as the ratio of the separation between Earth and centre of Sun’s orbit to the orbital radius of the Sun. However, with reference to Figure 19, we see that Kepler would have measured that particular separation as whereas Ptolemy would have measured it as The resulting value of eccentricity found by Kepler is thus equivalent to the bisected value of Ptolemy’s eccentricity.

In summary, we see that “bisection of eccentricity” either refers to Ptolemy splitting the total eccentricity exactly into two on his planetary theory, or refers to Kepler dividing the solar eccentricity as defined in Ptolemy’s (or Hipparchus’) solar theory. In Heilbron’s book, the discussion has focused on the second definition.