One challenge in crafting a catalog of stars is the question of when to stop.
Ptolemy’s star catalogue lists all visible stars of magnitude 1, 2 or 3, most stars of 4th magnitude, many of the 5th and some of the 6th ; a few others are cited as ‘dim’.
But there are hundreds and hundreds of stars of 6th , 5th, even 4th magnitude which Ptolemy doesn’t include. Some of these might be defined by tertiary and quaternary starlines, but most remain remarkably unexceptional.
Ptolemy’s aim is not a comprehensive map of the stars, but a useful map.
To that end, one has to draw a line somewhere.
What’s hard to fathom is where Ptolemy draws his line: 1022 stars.
Hmmm. . . That’s an odd number.
Our first reaction is – why not 1000? There’s a nice, round number, a power of ten. Look through the catalog, and you’ll easily find a couple of dozen dim stars that might be omitted without being missed.
Why 1022?
Our second reaction is – why not 1024?
There’s a thread of secrecy and mystery which runs through the history of Greek mathematics, which back in the day ventured into philosophy, at times into outright magical thinking. (Perhaps that’s always been true of advanced mathematics.) Geometry and mathematics at times yield results which border on the mystical; Pythagoras and his adherants conflated logic and reason and magic and art. For instance, Greek mathematicians discovered mathematical ratios which resulted in pleasant musical assonance; the fretboard of a modern guitar, the tuning of a ‘well-tempered klavier’ have roots in Pythagorean principles of mathematical harmony.
We can safely presume that Pythagoreans knew a sequence of numbers familiar today: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2056. . . . the powers of two.
Most people of the time wouldn’t have known much more than the first few numbers in this series. As the world became more complex and interconnected by trade over land and sea, the arithmetic of inventories and accounting evolved. Ten-fingered humans the world around developed decimal arithmetics for these purposes.
Powers of two are familiar to users of modern electronics – computers, cameras, phones, and other devices with digital brains. While decimal accounting works well on paper, the physics and geometry of computer chip design scales not by tens, but by factors of two. (Your ob’t s’vnt uses USB thumb drives of 512 MB, 4 GB, 32 GB for data transfer and backup.)
Attuned to these powers of two, 1022 immediately jumped off the page as an odd number, an undistinguished count.
Why not add two more to the list to reach 1024 stars – 2 to the 10th power?
We propose that Ptolemy skewed his accounting, to hide the number 1024 – a very Pythagorean sort of thing to do.
Below, we show where he concealed a pair of stars.
* * * * * * *
Perhaps you’d like to try to solve this yourself.
You’ll want access to a copy of Ptolemy’s star catalog (Almagest VII.v – VIII.i). [LINK]
And you’ll want some planetarium software. [LINK]
But if you simply want to proceed to the solution, then click here [LINK].