Almagest Book X: Correction for Equant – Third Opposition

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”

Almagest Book X: Preliminary determination of Mars’ Apogee & Eccentricity – Part 2

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”

Almagest Book X: Observations and Introduction to Iterative Calculations for Apogee and Eccentricity

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”

Almagest Book X: Position of Venus About the Epicycle on Dec 15/16 138 CE

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”

Almagest Book X: The Location of Venus’ Apogee

Despite the second chapter of Book X ostensibly being about the epicycle of Venus, Ptolemy instead continues discussing the line of apsides. Specifically, we determined that the line of apsides runs through the points $25º$ into Taurus and Sagittarius, but have not determined which is the apogee and which is the perigee1.

In this post, we’ll follow along as Ptolemy determines which of these two points is which. Continue reading “Almagest Book X: The Location of Venus’ Apogee”