>> [music] We've already thought a little bit about various ways that you can speed
up multiplication. We've seen quarter squares, log tables,
and even slide rules. There's another method that was super
popular in the quarter century before Napier unveiled logarithms to the world.
The method has the admittedly very pretentious name of prosthaphaeresis.
The idea is to use a cosine product identity, namely this formula.
The cosine of a times the cosine of b is the cosine of a plus b and the cosine of a
minus b, averaged together. Perhaps, somewhat surprisingly, you can
use this cosine product identity to multiply numbers.
Let's see how. So, let's try to use this trick to
multiply 0.17 and 0.37. Now, these numbers are so small, you just
multiply them out. But it'll demonstrate the trick that would
work on more complicated looking numbers. The formula that we're trying to use is
this cosine a times cosine b formula. I can only multiply together, cosines.
So, I should rewrite 0.17 and 0.37 as cosine a and cosine b.
Now, if I look at my table of inverse cosine and I look up 0.17, I find that
cosine of 1.39997 is close to 0.17. I can also use my table to compute arc
cosine of 0.37 to be about 1.19179. In other words, if I take cosine of this
number, I get really close to 0.37. Now, the formula that we're trying to use
is this formula that tells me to compute a plus b and a minus b.
So, given that, I've got these approximate values for a, I can compute a plus b to be
2.59176 and a minus b to be 0.20818. Here, I've got a table of cosine values
and if I look at 0.20, zero of column, first, second, third, fourth, fifth,
sixth, seventh, eighth column, I find out that cosine of 0.208 is about 0.97845.
And I can similarly use my table for cosine to approximate cosine of a plus b,
it's negative 0.85261. Now, the formula that we've got for the
product of cosine a and cosine b, tells us it's the average of cosine a plus b and
cosine a minus b. I've computed cosine of a minus b and
cosine a plus b, so all I have to do is take the average of these two numbers, and
if I average these two numbers, I get 0.06292.
Wow. I mean, look.
This average is really close to the actual product of 0.17 and 0.37, which is 0.0629.
We're managing to multiply together numbers by rewriting those numbers as
cosines and using this formula that tells me how to multiply cosines.
Instead of actually multiplying, I've replaced the multiplication with
additions, differences, table lookups, and an average.
Logarithms are obviously a huge improvement over this method.
But it's pretty fantastic to think that 400 years ago, humans were trying to
figure out how they could use trigonometric identities to do things like
multiply numbers. That required a tremendous amount of
creativity to think that the things about triangles would have anything to do with
multiplying numbers. In other words, trigonometry is not just
about triangles. You can do more than just understand
triangles if you really understand a whole bunch of trigonometry.
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