Over the past several posts we have worked towards an understanding of the rising times of arcs of the ecliptic at various latitudes. With this work complete, Ptolemy presents a table of these rising times in 10º intervals for select latitudes. Continue reading “Almagest Book II: Rising Time of the Ecliptic Tables”
Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs
With the previous theorem about the ascensional differences complete, it’s time to move on to determine how to figure out the rising time of arcs of the ecliptic for 10º segments at various latitudes using what Ptolemy promises to be a shortcut in the math. In the modern sense it really doesn’t seem to be much of a shortcut, but that’s because with the assistance of calculator’s, the equations we were using previously seem much less daunting. If it had to be done by hand, I’m sure it would be far more tedious.
Instead, Ptolemy reduces the number of calculations by going through the proof regarding ascensional differences as well as making use of some previously calculated values to avoid having to do other calculations.
To get started, Ptolemy revises the previous drawing, making it a bit simpler by removing the ecliptic and renaming a few of the points, as well as changing a few definitions.
Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs”
Almagest Book II: Ascensional Difference
Not content to simply figure out how long it would take a zodiacal constellation to rise at latitudes other than the equator, Ptolemy sets out to further divide the ecliptic into 10º arcs and he’s promised an easier method than what we’ve done previously. But before we can get there, Ptolemy gives a brief proof which he’ll make use of later.
To start, we begin with the vernal equinox on the horizon:
Data: Stellar Quadrant Observations – 12/23/18
It’s been almost three months since I’ve been able to get out and observe. It’s been a combination of assistants being sick, bad weather, bad moonphases, and starting a new job. But just before the holiday this year, we had a nice clear night. There was a near full moon, but as Padraig is planning on moving to Boston in the next year, I’m in need of someone else that wants to help. Fortunately, another member of my barony, Auga, volunteered to assist, so this night was mostly meant to get her trained on how things work and the moon was actually somewhat helpful as it made seeing what we were doing with setup and learning the instrument all the easier.
Continue reading “Data: Stellar Quadrant Observations – 12/23/18”
Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for Remaining Arcs
In the last post, we explored the rising time for one zodiacal sign which would comprise 30º of the ecliptic. Since the preliminary math is now out of the way, we can quickly do 60º of the ecliptic which constitutes Aries and Taurus. From there, we’ll be able to more quickly compute the remaining constellations as well. Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for Remaining Arcs”
Almagest Book II: Calculation of Rising Times at Sphaera Obliqua
Now that we’ve done a bit of preliminary theory, we’re ready to jump into an example problem. As with before, Ptolemy selects Rhodes at a latitude of 36º. Since we’ve been seeing several of the same diagrams here, I’m going to forego the buildup and give a mostly completed diagram:
Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua”
Astronomiae Instauratae Mechanica
When construction initially began on my quadrant, I had little to go on except the etching of the instrument and my own knowledge of how such an instrument should work. I knew the image came from a work of Tycho Brahe’s, Astronomiae Instaurate Mechanica (Instruments for the Restoration of Astronomy), but at the time was unable to find a translation to see if there was anything I was missing.
Fortunately, a few months ago, I finally located a translation of the vast majority of the book. However, the website seemed to lack the beautiful etchings that should accompany the text. However, scans of the original work were available through the NASA ADS. Unfortunately, being scans of a book over 400 years old, not all of the images were in good condition.
Still, I have worked to piece the text back together with the images, working over the past few months to digitally clean the images, removing smudges and stains, and reproducing the text as a digital copy best matching the original format as possible.
I now provide it as a .pdf for those interested.
Almagest Book II: Symmetry of Rising Times – Arcs of the Ecliptic Equidistant from the Same Solstice
In the last post, we proved that two arcs of the ecliptic that are equidistant from the same equinox rise in the same amount of time. In this post, we’ll prove something similar for what happens with arcs of the ecliptic equidistant from the same solstice.
It’s been awhile since I’ve been able to update the blog with anything from the Almagest. As noted in the last post from the book, this section is not one of the better written ones. Indeed, it’s taken me the better part of a month to really work out how the diagram is put together.
Ultimately the trouble stemmed from the fact that it’s not a single diagram; it’s actually two pasted together1, so instead of throwing it all at you at once like Ptolemy did, let’s work through each piece in turn before pasting it together.
To begin, let’s start with a simple celestial sphere:
Data: Stellar Quadrant Observations – 9/22/18 (Crucible at the Crossroads)
This weekend was Crucible at the Crossroads here in my home Barony of Three Rivers. I taught two classes, one on a general overview of medieval astronomy, and one introducing practical observational astronomy. In addition, it was the autumnal equinox on Saturday, so some minor adjustments were made to the quadrant to allow for solar angle observations. Lastly, I was called up in court for a few awards related to the project.
More on all of that beneath the fold. Continue reading “Data: Stellar Quadrant Observations – 9/22/18 (Crucible at the Crossroads)”
Almagest Book II: Symmetry of Rising Times – Arcs of the Ecliptic Equidistant from the Same Equinox
In this next chapter, Ptolemy’s goal is to
show how to calculate, for each latitude, the arcs of the equator… which rise together with [given] arcs of the ecliptic.
To do this, we’ll do a bit of convenient math, breaking the full ecliptic into its traditional 12 parts. However, since these signs are not of equal size, Ptolemy takes an even 30º for each sign, beginning with Aries, then Taurus, etc…
The first goal will be to prove that
arcs of the ecliptic which are equidistant from the same equinox always rise with equal arcs of the equator.