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>> [music] We've already thought a little
bit about various ways that you can speed

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up multiplication.
We've seen quarter squares, log tables,

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and even slide rules.
There's another method that was super

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popular in the quarter century before
Napier unveiled logarithms to the world.

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The method has the admittedly very
pretentious name of prosthaphaeresis.

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The idea is to use a cosine product
identity, namely this formula.

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The cosine of a times the cosine of b is
the cosine of a plus b and the cosine of a

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minus b, averaged together.
Perhaps, somewhat surprisingly, you can

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use this cosine product identity to
multiply numbers.

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Let's see how.
So, let's try to use this trick to

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multiply 0.17 and 0.37.
Now, these numbers are so small, you just

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multiply them out.
But it'll demonstrate the trick that would

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work on more complicated looking numbers.
The formula that we're trying to use is

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this cosine a times cosine b formula.
I can only multiply together, cosines.

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So, I should rewrite 0.17 and 0.37 as
cosine a and cosine b.

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Now, if I look at my table of inverse
cosine and I look up 0.17, I find that

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cosine of 1.39997 is close to 0.17.
I can also use my table to compute arc

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cosine of 0.37 to be about 1.19179.
In other words, if I take cosine of this

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number, I get really close to 0.37.
Now, the formula that we're trying to use

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is this formula that tells me to compute a
plus b and a minus b.

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So, given that, I've got these approximate
values for a, I can compute a plus b to be

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2.59176 and a minus b to be 0.20818.
Here, I've got a table of cosine values

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and if I look at 0.20, zero of column,
first, second, third, fourth, fifth,

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sixth, seventh, eighth column, I find out
that cosine of 0.208 is about 0.97845.

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And I can similarly use my table for
cosine to approximate cosine of a plus b,

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it's negative 0.85261.
Now, the formula that we've got for the

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product of cosine a and cosine b, tells us
it's the average of cosine a plus b and

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cosine a minus b.
I've computed cosine of a minus b and

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cosine a plus b, so all I have to do is
take the average of these two numbers, and

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if I average these two numbers, I get
0.06292.

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Wow.
I mean, look.

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This average is really close to the actual
product of 0.17 and 0.37, which is 0.0629.

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We're managing to multiply together
numbers by rewriting those numbers as

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cosines and using this formula that tells
me how to multiply cosines.

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Instead of actually multiplying, I've
replaced the multiplication with

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additions, differences, table lookups, and
an average.

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Logarithms are obviously a huge
improvement over this method.

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But it's pretty fantastic to think that
400 years ago, humans were trying to

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figure out how they could use
trigonometric identities to do things like

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multiply numbers.
That required a tremendous amount of

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creativity to think that the things about
triangles would have anything to do with

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multiplying numbers.
In other words, trigonometry is not just

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about triangles.
You can do more than just understand

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triangles if you really understand a whole
bunch of trigonometry.

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.