I recently learned about the existence of the Prosthaphaeresis algorithm [0], an even more ancient and obsolete trigonometric calculation technique. It allows you to use a trig table to do calculations when you don't have a log table. To calculate a x b, find arccos(a) and arccos(b), then use the product-to-sum formula: cos(a) x cos(b) = 1/2 x [cos(a + b) + cos(a - b)]. For example, to calculate 4294967296 x 2147483648. First, scale both numbers down by a factor of 1e10, a ≈ 0.4295, b ≈ 0.2147. Next, do inverse trig table lookup: x = arccos(a) ≈ 1.1269, y = arccos(b) ≈1.3544. Now, sum and difference: y + x = 2.4813, y - x = 0.2275. Finally, do trig table lookup and scale up, (cos(2.4813) + cos(0.2275)) x 1/2 x 1e20 ≈ 9221026434613711000. The true answer is 9223372036854775808, error is less than 0.03%. It was used by astronomers in the 16th century before logarithm was invented.
I learned it while reading the Wikipedia article Slide Rule, it mentions that two math educators David B. Sher and Dean C. Nataro invented a Prosthaphaeresis slide rule in 2004 - it's what a slide rule may look like in an alternative universe. Unfortunately, I still haven't seen it - I couldn't find the original paper, it doesn't seem to have a DOI, can't use Sci-Hub.
Prosthaphaeresis works because an angle measure inherently is a type of logarithm. It’s the logarithm of a rotation, instead of the logarithm of a scale. Like other kinds of logarithms, this converts multiplication (composition of rotations) to addition (addition of angle measures).
If you want to multiply two numbers, you can treat them as rotations, take the logarithm of each (i.e. find the angle measure), add the two, then take the exponential.
Is there a more general definition of logarithms than being the inverse of the exponential function?
Mathworld says Prosthapharesis formulas "convert a product of functions into a sum or difference" [0], which resembles a property of logarithms (i.e. log(ab) == log(a) + log(b)). However I don't quite get how angle measurement itself is a type of logarithm.
Starting for Euler's formula you can write cos(x) = (exp(ix) + exp(-ix))/2 so the cosine is in some sense a "type of exponential". If the cosine is a type of exponential then its inverse (the function that takes you to the angle measurement) is inherently a "type of logarithm".
This is a very handwavey statement. I would say a more precise statement is that the proper exponential and log maps are the "correct" thing to use to map multiplication to adidtion and vice-versa. They are exactly the maps that do this properly. The fact that cosine is a sum of exponentials means that you can write the inverse of cosine as a (slightly ugly) formula in terms of logarithms (arcos(z) = -i log(sqrt(1-z^2) + z). This is the sense in which the map to angle measures is a "type of log".
This method is rather more cumbersome (due to rescaling and then converting the factors to rotations as an intermediate step) but fundamentally based on the same concept as:
There is an axiomatic approach that defines a logarithm as any function f(x) such that f(x•y) = f(x) + f(y). But it's easy to prove that this reduces to the usual definition for logarithms on the real numbers.
This is funny because when I was in school being taught navigation, we were told that celestial navigation would basically tell you where you were half an hour ago, because doing all the math with trig tables and a slide rule or log tables (forced to learn how to do it without electronic calculators), it would take about 30 minutes to reduce a star sight to a position on Earth surface.
So now we use that algorithm and you've just calculated where the ship was a few hours ago :-)
Is that really true? Maybe it takes 30 min to calculate from first principles, but I'd expect you could have a book of tables that tells you with just one or two lookups.
From first principles would take a lot longer. 30 minutes is roughly what I remember to reduce about 4 stars. I could be off a bit since it's been a really long time and I forget all the steps, but I remember a lot of table lookups and interpolations to get to a final fix.
I learned it while reading the Wikipedia article Slide Rule, it mentions that two math educators David B. Sher and Dean C. Nataro invented a Prosthaphaeresis slide rule in 2004 - it's what a slide rule may look like in an alternative universe. Unfortunately, I still haven't seen it - I couldn't find the original paper, it doesn't seem to have a DOI, can't use Sci-Hub.
[0] https://en.wikipedia.org/wiki/Prosthaphaeresis