Tracing the history of medieval astronomy
Back in IX.3, Ptolemy noted that the planetary periods
have been corrected by us, on the basis of the comparison of their positions which became possible after we had demonstrated their anomalies, as we shall explain at that point.
As a reminder, what he was telling us then, was that he had made corrections to the periods provided to him by Hipparchus. We’ll now go over those corrections for Mars. Continue reading “Almagest Book X: Correction to the Periodic Motions”
In the previous two posts, we’ve been looking at how the apparent position of the center of a superior planet’s epicycle shifts position along the various circles. Specifically, we’ve been looking at how the angle between oppositions differs between the eccentre of mean distance and the equant circle.
In this post, we’ll do the same for the third opposition Ptolemy gave us. Unsurprisingly, the process will be largely the same as in the past two posts. Continue reading “Almagest Book X: Correction for Equant – Third Opposition”
Having gotten our preliminary estimate for the eccentricity and position of apogee
let us investigate the differences which can be derived from them in the ecliptic arcs which we week to determine at each of the oppositions [in turn].
In this post, we’ll cover Ptolemy’s method for the first opposition. Continue reading “Almagest Book X: Correction for Equant – First Opposition”
In the last post, we began the process of determining Mars’ eccentricity and the position of apogee. The technique is fundamentally similar to the three eclipse method used for the moon, but the presence of the equant is throwing a wrench in things.
Specifically, right at the beginning of the last post we had to make an assumption about $\angle BDG$ that we know was incorrect. We’re not quite ready to address that, but for now we’ll work on finishing finding the line of apsides as well as the eccentricity as if that assumption were correct. Then, in future posts, we’ll go about trying to iteratively correct that assumption. Continue reading “Almagest Book X: Preliminary determination of Mars’ Apogee & Eccentricity – Part 2”
In the last Almagest post, Ptolemy explained that for the models of the superior planets, we’ll be needing to use oppositions. So with that explained, we’ll begin by working to develop the model for Mars by looking at its eccentricity and apogee. However, this isn’t going to be nearly as straightforward as it has been previously. As we’ll see, we’re missing some of the angles and arcs that would be needed for this. Thus, Ptolemy is going to use an iterative method to close in on them. Continue reading “Almagest Book X: Observations and Introduction to Iterative Calculations for Apogee and Eccentricity”
Having completed the preliminary models for the inferior planets, Mercury and Venus, Ptolemy now turns his attention towards the superior ones: Mars, Jupiter, and Saturn. Continue reading “Almagest Book X: Preliminary Comments Regarding the Models for Superior Planets”
Now that we’ve worked out the mean speed of Venus about the epicycle, we can now determine where it was on the epoch date. Continue reading “Almagest Book X: Venus’ Epoch Position”
Continuing with our process of finding the position of Venus about the epicycle at two widely discrepant points in time, we’ll now look at one from Timocharis. Continue reading “Almagest Book X: Position of Venus About the Epicycle on Oct 10/11 271 BCE”
Now that we’ve worked out the basic parameters of Venus’ epicycle and eccentre, we’ll need to determine the motion about the epicycle1.
As we did for Mercury, we’ll determine the position at two points in time and then use that to calculate the speed. In this post, we’ll concentrate on determining the position for the first of these observations. Continue reading “Almagest Book X: Position of Venus About the Epicycle on Dec 15/16 138 CE”
Now that we’ve sorted out the size of the epicycle, Ptolemy next step will be to determine the point about which the eccentre rotates as he doesn’t want to assume it rotates about the center of the eccentre2. Continue reading “Almagest Book X: The Equant of Venus”