How were logs and roots calculated before calculators?

[dpm]

I'm not sure whether this is the correct forum, so I apologise if it's in the incorrect forum.

Anyway, when studying A level maths a few years ago, we came across a technique for calculating roots that my teacher claimed was used before calculators were invented. I can't remember the actual technique used, but I remember thinking at the time that it was particularly clever. He also claimed that before calculators, mathematicians were forced to invent other, similarly clever techniques to work out their logs and roots etc.

So, what were these techniques? How exactly were the logs and roots calculated before modern computing machines?

 

[Andre]

Slide rules,
http://www.sphere.bc.ca/test/sruniverse.html

Everybody had a slide rule very close at hand.

This is how it wurx:
http://www.hpmuseum.org/sliderul.htm

 

[Math Is Hard]

How is it possible that BobG hasn't shown up in this thread yet?

 

[masudr]

There is an algorithm to calculate square roots to any required degree of accuracy with merely pen and paper alone. I learned it from a book published in about 1920 aimed at 16 year olds when I was about 10; the book is also no longer in print. Anyway, my point is, I can't really remember it, but if someone really wanted to, I could half-remember and half-work it out. Or maybe someone else knows it too.

 

[SGT]

There is an algorithm to calculate square roots to any required degree of accuracy with merely pen and paper alone. I learned it from a book published in about 1920 aimed at 16 year olds when I was about 10; the book is also no longer in print. Anyway, my point is, I can't really remember it, but if someone really wanted to, I could half-remember and half-work it out. Or maybe someone else knows it too.

Suppose you want the squareroot of 55. It must lie between 7 and 8.
Try 7.5. $$\frac{55}{7.5} = 7.333$$
Now, calculate the mean:
$$\frac{7.5 + 7.333}{2} = 7.41666$$
and use it in the new iteration:
$$\frac{55}{7.41666} = 7.41573$$
$$\frac{7.4166 + 7.41573}{2} = 7.416198468$$
Using the calculator of Windows, you get for the squareroot of 55 the value 7.416198487...

 

[hypermorphism]

This "divide-and-average" method can be derived using the more general http://planetmath.org/encyclopedia/NewtonsMethod.html of finding zeroes of nonlinear functions.

 

[masudr]

These algorithms are slow; a faster one realizes that
$$(10a+b)^2 = 100a^2 + 20ab + b^2$$
and works out the digits of the square root based on pairs of digits of the square, by finding a $b$ such that $b(20a+b)$ matches whatever is left (having subtracted the first term already).