In this paper, three sets of solar and planetary models have been brought into discussion, namely, by Hipparchus, Ptolemy and Kepler. Ptolemy’s equant theory for the planets is an amazingly close approximate to Kepler’s planetary model. In this section, we aim to find out mathematically how this has been possible. Prior to that, we would derive the equations defining the position of a body in its orbit in each of the three models. The defining parameters of the body’s position are the true anomaly and the radius vector. In finding out the equations of these parameters, we would also make use of the eccentricity. Information on which this section is based mostly on Evans’ article entitled “The division of the Martian eccentricity from Hipparchos to Kepler: A history of the approximations to Kepler motion” and shall be complemented with brief notes from other books as recorded in the reference section.

As a remark, in Heilbron’s book, the value of eccentricity in Ptolemy’s planetary theory is e whereas that in Kepler’s is However, in most mathematical textbooks, the former is usually taken as 2e while the latter as 2e. For the rest of the paper, we wish to identify with Heilbron’s notation. Hence, the form of any equation involving eccentricity will be maintained as that found in general textbooks but the eccentricity values will be altered to correspond with Heilbron’s and written in parentheses without expansion.

As described under "Kepler's Laws", the equation for the elliptical orbit is


where r is the planet’s distance from the Sun, θ is the true anomaly and e is the eccentricity.

By means of the binomial theorem, the above equation may be expanded as the following:




Also derived in "Kepler's Laws" is the condition of constant areal velocity, that is,

When (1) is substituted into the above, and the resulting differential equation for θ(t) is expanded and integrated through order e2, we obtain


where w is the angular velocity.

Equations (2) and (3) are the defining equations for a body on the Keplerian model. We next derive similar equations for a body on an eccentric orbit with equant. Figure 23 is the reference diagram for the derivation steps that follow.

Figure 23

The equation of the orbit in polar coordinates for Z is that of a circle eccentric to C, which is,

Hence, by binomial theorem,


Since mean anomaly M is equivalent to wt and q = θM Þ θ = M + q therefore, θ = wt + q.

Applying Sine Rule to triangle OZX, we have

This gives



By substituting the expression of r for an eccentric with equant model into this equation, and expanding to second order in e, we get


For Hipparchus’ solar theory, e1 = e and e2 = 0. On Ptolemy’s equant theory, he does a bisection of eccentricity by putting e1 = e2.  Hence each is of value . By substituting the two sets of eccentricity values into equations (3) and (4), we find the corresponding true anomaly and radius vector equations as shown below. In addition, equations (2) (include up to second order terms in e and set the semimajor axis as unity) and (3) have also been added into the table for comparison purposes.



True anomaly, θ(t)

Radius vector, ρ(t)


e1 = e


e1 = e and e2 = e


Seeing the data, suppose we ignore the second order terms, then the first discrepancy arises in the radius vector or Hipparchus’ model. Consider the true anomalies of Ptolemy and Kepler. The difference between them is a mere that amounts to a maximum of 8 min of arc. Considering the low level of precision in instruments used to make measurements of celestial bodies in ancient days, it would have been hard to recognize that this minute difference points towards an error in the model. It is no wonder that so much controversy had occurred amongst the astronomers then, in their desire to conclude which planetary theory was the true and exact one.

In fact, Ptolemy’s and Kepler’s models have approximated each other so well that up to first order terms in e, the empty focus of the Keplerian ellipse is indistinguishable form the equant point in Ptolemy’s model. The working steps that follow provide the mathematics behind such an approximation.

Figure 24

Referring to Figure 24, FZ makes an angle f with the line of apsides AP; q and r have their usual meanings. Take FZ = h. By properties of ellipses, FZ = AP - OZ. That is, h = 2a - ρ or if the semimajor axis has been set to unity, h = 2a - ρ. In addition, We aim to find an expression for to represent the motion of the planet as observed from the empty focus. Applying Sine Rule to triangle OFZ, we have,


Upon substituting the Keplerian expressions for θ(t) and ρ(t) which were found earlier in this section, and expanding, we yield the following:

Therefore, if we only consider terms in e up to first order, is equivalent to wt which in turn gives constant motion of the planet Z when observed at the empty focus F. This is why Kepler’s empty focus is said to behave as Ptolemy’s equant point, to order e.