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Tuesday, May 30th, 2017 10:51 am
After reading Kepler (and having already read some Copernicus, Galileo and Newton), I decided to go all the way back and read Ptolemy's Almagest. Well, some of it. The edition I read was "The Almagest: Introduction to the Mathematics of the Heavens" (2014), which adds some supplemental materials and only follows some of the main arguments of the book. Still, I figured this was good enough to mark Great Books #16 off as completed.

Beyond the challenge and pleasure of following a genuinely brilliant (if, you know, totally wrong) line of mathematical reasoning, what really stood out to me about the Almagest was the metrology. Ptolemy lived in a very different world than us, where angular measurements were the only ones with any accuracy, and even those were only good to about half a degree. He thought primarily in terms of geometry, which is much better at giving proportions between lines (AB:BC as DE:EF, etc) than it is at giving absolute answers like our algebraic thinking. The first section of the book, in fact, is him building up a table of chords. Given an arc angle, you could look up the length of the chord, with the answer based on a circle with a diameter of 120. Like a lot of older technical books, it's a fascinating what intellectual infrastructure that we take for granted had to be spelled out step by step. (I'm particularly thinking of the first bit of "De Re Metallica", where Agricola has to define the points of the compass, advocating for that as a better system than the names winds commonly used for directions at the time.)

One fascinating aspect to the numbers is that they're all sexagesimal. Even the lengths are given in fractions of 1/60 and 1/3600. He even goes beyond that, at one point saying "we will have the daily mean motion of the sun 0°59'8''17'''13''''12'''''31'''''' degrees." (The accuracy being imputed here is completely unfounded, as it is based on a measured length of the year of 365 days, plus 1/4, minus 1/300, "most nearly". That's adding roughly, in modern terms, 6 decimal places of precision!)

This raises the interesting question of what to call the units beyond 'minutes' and 'seconds'. We get 'minute' from the Latin minuta parta, "small part", and 'second' from secunda minuta parta, "second small part". So if you wanted to extend that, you could start with tertiam, quarta, quintus, sextus, etc. Working from there, applying a bit of linguistic sandpaper, we could reasonably deduce a series of terms like 'terts', 'quarts', 'quints' and 'sexts', had they ever been commonly used.

Which raises another interesting question -- why didn't they ever get used? I think it's a matter of scale. The places where we use sexagesimal are are very human-scale things, time and angles. The advantages of decimal for other measurements was more immediately useful, I suspect, when it was obvious that you might want to divide things into smaller and smaller sections. Money, for instance -- someone will always care about another decimal place down. But time and angles are already divided to a very fine degree with minutes and seconds. A 60th of a second (16 milliseconds) is right on the edge of human perception, and we didn't even develop the technology to accurately measure it until just a few hundred years ago. And an arc second is already a very tiny measurement, only just barely measurable with handheld instruments by experienced navigators using comparatively advanced sextants. So we stuck with the old sexagesimal for those, lacking a strong motive to switch.

Ptolemy had a few other quirks that I found intriguing. He very often would specify what size degree a solution was, usually formulated like "X degrees where four right angles form a circle, and 2X degrees where two rights angles form a circle". He did this so often that I have to think there was a specific reason for it. Maybe a constructive geometry trick where it is useful to start with twice the angular measurement? (That could be a real time saver, particularly when you remember that the values were originally given as sexagesimal Greek numerals. While these are a bit better than Roman numerals when it comes to arithmetic, even a seemingly simple operation such as multiplying by two could pose a problem.) However, I've yet to find an authoritative answer to this.

He also liked to specify right triangles as a series of chord lengths, as if it was circumscribed, assuming the hypotenuse/diameter had a length of 120. Again, I wonder if there is a constructive geometry trick using a divider and straight edge where this is particularly useful information.

All in all, a very interesting read. It's a beautiful system that he develops, bringing together hundreds of years of incredibly spotty astronomical records, each described using its own archaic calendaring system, and managing to pull a very elegant solution out of it. I didn't realize that he actually discuses the possibility of a heliocentric universe early on, before dismissing it as silly. It really drove home the intellectual seduction of the "perfect circular motion", given how complicated the full Ptolemaic system is. The epicycles are only the first level of complexity! The planets don't move regularly around the epicycle, they move so that their motion looks regular as seen from another place that isn't even the center of the Zodiac circle. (That is, the Earth.) And the circle the epicycles are moving on (their deferent) isn't centered on the Earth anyway. (This is all much more complicated for Mercury, btw.) And that's just for the longitudinal anomaly! To also account for the latitudinal anomaly, you have to add more circles that work kind of like gears, tilting the epicycle and its deferent separately up and down as they spin. All in all, you need something like 90 spinning circles to fully describe just the 7 planets (including the sun and the moon) they could see at the time. Whew -- and it still took hundreds of years for people to drop that idea, once it was rigorously challenged! Never underestimate the poisonous appeal of the wrong kind of beauty when it comes to science.

One of the better visualizations of a full Ptolemaic system that I found:

A good explanation of how you get from Ptolemy to Copernicus:

Warning: Following video links from this videos will quickly get you to serious, modern attempts at disproving the heliocentric model. It's a scary world out there. :(
Wednesday, May 31st, 2017 01:53 am (UTC)
Man, do I wish that YouTube had existed when I was doing Ptolemy in college. That would have made visualizing the Almagest a whole lot easier.

I also wish that I'd had a scientific calculator (this was in 1983; I didn't even know that such things existed) on the day when my instructor, whose math class I was failing, assigned us the task of doing a Ptolemaic proof, using different numbers. I had to do all those sexagesimal calculations by hand. At a certain point, I got stuck; I just could *not* get the numbers to work. So I invented a new section to the proof in order to bridge the gap.

Turned out that the instructor had given us the wrong numbers. I got an A in that class.