Slide ruleA simple explanation of a clever calculator 
Dear reader, The other day I was talking to a lady who had many mathematical qualifications but had never seen a slide rule. This is a bit like an intergalactic astronaut admitting they weren't quite sure where Mars was let alone visited it! So here is a little page for all those people who missed out on a very clever and, in its day, essential calculating device. We better start with Logarithms. If you've heard the word and it frightens you  don't worry. They are simple and amazing. Then you can make your own slide rule in two minutes. 
Logarithms 
A logarithm is a magic number related to another number.
Don't worry how you calculate the magic number.
In the olden days we just looked up tables printed in books.
Today if you need a logarithm your scientific calculator or computer will tell you.
 
The magic is that if you add the logarithms from two numbers
you get the logarithm of the number you'd otherwise get by
multiplying the two original numbers.
Change the numbers outlined in red and recalculate to see what happens.
 
Here is a table of logarithms for numbers 1 to 99.
Read the tens down the side and the units across the top.
 
But what about all those numbers from minusasmuchasyouwant
to plus asmuchasyouwant? Surely you would need an enormous
book to deal with 4, 40, 4 thousand, 4 million and so on?
Lets see some more logarithmic cleverness:
See what's happening? If the logarithm of 10 is 1 then to multiply by ten add 1. (To divide, subtract). This means that a book of log tables only needed to cover 0.0001 to 0.9999 in 1000 lines of ten columns to be accurate to four decimal places.  
The logarithm of 24 is the logarithm of 2.4(0.38) plus 1 = (1.38)
Here is an example:

The slide rule 
Now you know how logarithms work you may be wondering if you can have a go.
Actually to sit down with a book of log tables is a little bit more complicated
than I have explained so far. For now let's stick with the rough and ready
explanation above. (You can find out about division and fractions later.)
But now let us look at the very clever pocket calculator called the slide rule.
If I want to multiply 2 by 3 I have to add the logs of these together. I could get my pencil and paper and log books out and do the sum log of 2 ( = 0.301) plus log of 3 ( = .4771) added together is .7782 which by looking in my tables is the log of 6.But suppose I had measured a line 3.01 centimetres and added another line onto the end of it 4.771 centimetres long. Now I have a line 7.782 centimetres long. If only I had a clever way of measuring out these distances... ... Here's how it's done. Print out this page then cut out the picture below. Then cut along the horizontal line so you have a top and bottom scale. 
Here is how to multiply 2 by 3: Move the bottom scale to
the right so that the 1 on the bottom scale matches the 2 on the top. Now
read along the bottom scale to the 3 and see what it says on the top
scale. 6! All we have done is added a log of 2 distance on the top
to a log of 3 distance on the bottom to get a log of 6 distance on the top.
Here is how to divide 9 by three: Move the bottom scale so that the 3 is underneath the 9. Now see what is above the 1 on the bottom scale. 3! So we subtracted distances to divide and add to multiply. What happens if we try to multiply 4 by 5? As for 2 times 3... Put the 1 of the bottom scale against the 4 of the top then read off what is above the 5. Oops. Run off the end of the top scale! Luckily we can use the same trick we used to save printing an enormous book of log tables to save having to make a very long slide rule. So long as we keep track of the tens we can still do all the multiplications and divisions we want then multiply the final result by the tens we took out earlier. Instead of putting the left hand 1 of the bottom scale against the 4 on the top, move the bottom slide to the left so that the right hand 1 is under the 4. Now read off above the 5. You should see 2 which stands for 20.

Tricks from the slide rule era...

Back of the envelopeYou might wonder how engineers 'of the old school' managed without modern calculators. Easy: You do rough sums with the figures but make sure you get the decimal point in the right place. For example: If our factory makes 2000 bobbins a week and we have 5 machines, how many is this per year per machine?With most engineering calculations you leave a Factor of Safety of perhaps four times so quick calculations are often going to be fine.


Reality checkHow can we check a calculation quickly?One way is to work out the units as a separate sum. Example: How far is the Sun from the Earth? If the speed of light 300000000 metres per second and it takes about 9 minutes for light to reach the Earth then perhaps the answer is 3 times 9 (with 8 zeros on the end) kilometres? Let's see: metres per second is metres divided by seconds. So our sum if worked out in units is: metres times minutes divided by seconds. is this the same as Km? Doesn't look quite right does it. Let's have another go:
