The Third Bunch of Ways to Multiply

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Let’s multiply!  (The First Bunch had Egyptian, Babylonian, lattice, and current methods, and the Second Bunch had Greek, Napier and Genaille-Lucas, plus links to Vertically and Crosswise Multiplication and finger multiplication.)

We’ll just do two more today, starting with one I forgot we’d already talked about:

(13) The Method of the Cups, which was described by Friar Juan Diez in 1556 (see  this post from last year).  It’s done digit by digit, but multi-digit numbers aren’t always written on a single line: if you want a glimpse of what it’s like, here’s the image for 875×978.

1556 Multiplication by Friar Diez

(14) Prosthaphaeresis, or Multiplication with Trig Tables. This method dates back to the late 1500s, although logarithms essentially rendered this method obsolete.

Here’s the basic idea:  remember the sum and difference formulas for sine and cosine?  We’ll use those.  For example, since:

cos(a+b)=cos(a)cos(b)-sin(a)sin(b)

and

cos(a-b)=cos(a)cos(b)+sin(a)sin(b)

then by adding those together, we get

cos(a+b)+cos(a-b) = 2cos(a)cos(b)

which leads to the formula

cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b)

And that’s the formula that we’ll use, although by also looking at sine we could just have easily come up with a different formula that had a mixture of sines and cosines.    When the last step of dividing by 2 is ignored, the resulting formulas are called the Werner formulas.

Here’s how we’ll do it.  Suppose we want to multiply 875 and 978, because those were the numbers that appeared in the previous example.  We’re going to be using a table of cosine values (like this one).  This table actually shows both sine and cosine, but cosine is read on the right.

Trig 0

Notice that 0.9997 is cos(1.4°), for example — we’re adding 0.2° to the angle for each column, going right to left.

Cosine values always fall between 0 and 1, so we need to scale our numbers to be between 0 and 1.  Thus 875 becomes 0.875 and 978 becomes 0.978, and we’ll have to multiply our final answer by 103·103=106.

Remember that our formula is cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b).  In this case, cos(a) is 0.875 and cos(b) is 0.978.  We need to find a and b.

Trig 1

Remember that we read the angles on the right.  Since 0.875 is between 0.8746 and 0.8763, our angle a must be between 28.8° and 29°.  We’ll use 29°, since that gives the best estimate.

Trig 2

Likewise, we need to find b knowing that cos(b)=0.978.  From the chart we can see that b is between 12.0° and 12.2°, but basically it’s 12°.  Yay!  We’re halfway there!

We have a and b,  so the sum a+b is 41° and the difference a-b is 17°.  Now we need to know cos(a+b) and cos(a-b).

Trig 3

This part is easy!  The cosine of 41° is just 0.7547

Trig 4

and the cosine of 17° is 0.9563.

Now we can plug all this into our formula!  Remember how 0.875 was cos(a) and 0.978 was cos(b)?  Using:

cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b)

we get

0.875 \cdot 0.978 = \frac{1}{2}\cdot 0.7547 + \frac{1}{2} \cdot 0.9563

which simplifies down to 0.8555.  Multiplying that by the 106 from earlier gives 875×978 as approximately 855500.   We’re only off by 250, which feels like a lot but it’s less than ½% of the answer, so in the grand scheme of things it’s pretty good, and with more accurate tables we could have done even better.

Next up:  Log tables and slide rules!

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2 Responses to “The Third Bunch of Ways to Multiply”

  1. Sue Says:

    Dear goddess! Did people really do that?!

  2. Ξ Says:

    Sue, yes, apparently they did. I think this method might become efficient if the numbers were several digits long and you had a really good table: looking up numbers would be faster than other kinds of calculations, and apparently some tables gave values not just to the nearest minute, but down to every second of a degree.

    (I think there were also some calculations that came up in astronomy where you’d already be dealing with the sine or cosine of angles and you then had to multiply them, so in those specific cases these formulas would be much more natural to use.)

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