Before continuing into the math portion of this book, a brief interlude is necessary to explore how Ptolemy does his math. Chiefly, he uses the sexagesimal system which is a base 60 (as opposed to our base 10). The reason for this is that 60 has a large number of factors, which means it’s ideal for quick math since it you can make lots of fractions out of it.
This may sound odd at first, but consider that in some respects, it’s one we already use for telling time. There are 60 minutes to an hour and 60 seconds to a minute. In fact, that’s where the word “second” for measuring time comes from as it was the second division of the whole number.
Similarly, in modern astronomy we also see some of this sexagesimal system coming into play when we talk about arcminutes and arcseconds.
To help clear things up, let’s take an example using modern astronomical notation: The bright star Sirius is located at a declination of -16º42m58s.
The whole unit does not need to be converted, so we can leave the -16º alone. The 42m is the first division, so it’s 42/60. The 58s is the second division so it’s 58/602. Writing it all together and going through the math:
$$Dec = -\left(16 + \frac{42}{60} + \frac{58}{60^2}\right)$$
$$Dec = -\left(16 + \frac{42}{60} + \frac{58}{3600}\right)$$
$$Dec = -\left(16 + .70 + .016\right)$$
$$Dec = -16.716^\circ$$
While modern astronomy uses decimals once you start getting into seconds, there’s no reason that the divisions would have to stop there. It could easily continue on to a third division (1/216,000) and beyond.
So far, I’ve given examples in the modern minutes and seconds, but for more general applications of the sexagesimal system these terms do not apply. Instead, the order of divisions are separated by commas after the whole number is separated by a semi-colon. So to reuse the example above, Sirius’ coordinates would be written as -16;42,58.
Longer decimals, such as the sexagesimal approximation of 1/17 would be written as 0;3,31,45,52,56,28,14,7… where each successive number would be divided by 60 to one higher power. In other words:
$$\frac{0}{60^0} + \frac{3}{60^1} + \frac{31}{60^2} + \frac{45}{60^3} + \frac{52}{60^4} + \frac{56}{60^5} + \frac{28}{60^6} + \frac{14}{60^7} + \frac{7}{60^8}$$
This value is actually very accurate given that the value of the denominator, $60^8$, in the last place I wrote here is ~1.6796 x 1014! But Ptolemy promises not to get too hung up with such accurate representations. Rather he says,
Since we always aim at a good approximation, we will carry out multiplication and divisions only as far as to achieve a result which differs from the precision achievable by the senses by a negligible amount.
In other words, he only intends to worry about such things to the limit of observational error in his time.
Due to the use of this system, Ptolemy also defines his unit circle somewhat differently. Whereas typically we define the radius of the unit circle to be 1, Ptolemy divides it into 60 parts (so a diameter of 120 parts). In many respects, this is equivalent since we’re in a base 60 system and 60/60 = 1. He merely pre-divides it so he can effectively start at the first division.
Another thing Ptolemy does, is defines the circumference of his circle to be 360 parts. This means that an arc on the circumference of a circle will have the same number of units as the central angle which it subtends.
In general, I prefer to work in decimals, but since the purpose of this project is to become familiar with period methodology, I will be tending to display my numbers in sexagesimal. That being said, on the backside, I find it a difficult system to carry out multiplication and division in, and as such, I carry out the math by converting and using a calculator, and then convert back to sexagesimal to display the answer.
Lastly, as a minor note about my own notation, we’re about to be dealing a lot in line segments. These are generally written as two letter combinations of the endpoints. So if a line started a A and ended at B, it would be written $\overline{AB}$ (where the overline denotes that it’s a line segment as opposed to an arc). However, there’s no real difference between where a line starts and ends (unless it’s a vector which is well beyond the mathematics required here), so $\overline{AB} = \overline{BA}$. Ptolemy obviously didn’t care about the order of the letters since they’re equivalent, but to make things visually easier on myself (and hopefully you as well), I will often put the letters in alphabetical order. I do make exceptions to this when a specific ordering makes more sense in terms of the diagram. I simply wish to point this out for those that are less familiar with such geometrical notation.