When I was at school we had to do Mathematics so we had some quite difficult ‘sums’ to do – without mobile phone apps and even without calculators. How did we manage? (I will get to computers later but I have to start back at school.)
I should be honest here. I like Maths. I did ‘A’ level Pure Mathematics and Applied Mathematics at school, then ‘S’ level Mathematics, then I went to University and did a degree in Mathematics. Now I play around with spreadsheets for all sorts of things. But like everyone else, back in the fifties and sixties I had to do it the hard way. (If you read on you will see that it got easier in the seventies.)
Mental arithmetic at school was important. From very early days at school the times tables were taught by rote – from two times two is four up to twelve times twelve. We did these back in my days at Grange Hill from the age of about seven.
We used our knowledge of tables to do relatively simple sums. At junior school we learned about decimals, subtracting large numbers, long multiplication and long division, all done just by using pencil and paper methods. I won’t go into the details but before Secondary School we knew how to work out: 3456 x 789, and we knew how to find 3456 ÷ 789 as a decimal. (We didn’t do sums quite that hard but we knew the method.)
At Ilford County High School we needed a quicker way to work out arithmetic. (Of course the same was true at all other schools.) Probably in the Third Form, we did logarithms, a relatively easy and quick way to multiply by using addition. I have considered at length whether to try to explain the way that logarithms work – and I have decided against it. It would take quite some time to do it clearly and I suspect that WordPress would fail with the mathematical notation.
I will show you an example. The oldies can sit back, reminisce and gloat – thinking: ‘Yes, I remember those books. We did that. The young generation have it too easy nowadays.’ And the youngsters can just think: ‘What! Surely no-one actually had to add up three-digit numbers!’
I will give actually give two examples.
A printed list in a little book allowed you to look up every number from 1.01, 1.02, 1.03 all the way to 9.98, 9.99, to find the logarithm of that number. Ok we will call them just Logs. (The battered SMP booklet in the picture is probably from the eighties but the numbers inside were the same!)
Here are the first ones. Log(1.01) is 0.004, Log(1.02) is 0.009, Log(1.03) is 0.013. (I have simplified it by using three figures. The tables coped with four. The Log of 1 is 0.000 and the Log of 10 is 1.000. You don’t need tables for them.)
If you wanted to know 2 x 3, you looked up 2.00 in the book of tables. It’s 0.301. Then you looked up Log(3.00), which is 0.477 (Believe me, I knew both of those. They were part of what we learned at school. I didn’t have to look them up!) Then you wrote down a little sum and added them up:
Now we have to get the answer by looking up anti-logarithms. We want a number with a Log of 0.778, so we turn to another page in the book, look up Antilog(0.778) and the answer is – Surprise, Surprise! – 6!
So we complete the table.
Because we always show our working, we write 2 x 3 = 6.00 (using logs) and put the table by the side.
That was easy. You all knew that 2 x 3 was 6, so here is just one more example, not quite so obvious.
12.34 x 567.8 = ???
You add the right hand column first, then look up 0.845 in the Antilog tables. So the answer is 7007. [You will either know, or will have worked out by now, that the bit before the decimal point in a logarithm tells you whether its 7.007 or 70.07 or 700.7 or 7007. It gets more complicated for 0.7007 or 0.07007!]
We did everything with those books of Log Tables. As well as multiplication (as shown above, by adding Logs) we could do division, (subtracting Logs) powers (multiplying Logs – using Logs), and sines, cosines and other trigonometry (using other tables in that little book).
It was the way calculations were done in a world without calculators.
A year or two later we learned to use slide rules, which were just physical devices using the same methods as logarithms. You can see in the picture three wooden graduated rulers fitted together. The middle one slides along. In the diagram it is set up to multiply by two – so you can see how it shows that 2 x 2 = 4 and 2 x 3 = 6. For points in between you count along subdivisions, so you can also read 2 x 1.5 = 3.0 or 2 x 2.3 = 4.6 . The method was not quite as accurate as Logs but it was easier and quicker.
(It doesn’t matter if you haven’t followed the last few paragraphs. If will give you a sense of wonder at how clever we used to be.)
We come now to something I thought at first would be simple – just mention calculators. They were non-existent in the sixties; appeared in the early seventies; were cheap enough for general use by the mid-seventies; and were allowed in schools by the late eighties, replacing all that work with log tables and slide rules.
But then I realised that these cumbersome machines are now so outdated that most youngsters have never seen one and wouldn’t even recognize one! You probably have a Calculator app on your phones. The picture above shows an early calculator. It’s small strip of LED (light emitting Diodes) at the top was all you had then to show what was happening, with physical buttons to enter data.
The picture shows a scientific calculator, which could do trigonometry and other functions. Early basic models just did addition, subtraction, multiplication and division.
All you really need to know that these things just didn’t exist in the early sixties. At schools we had to manage with pencil and paper methods, log tables and slide rules.
A Diversion about Tractors
Why am I talking about tractors?
Well, for about six months after I left school, in 1966, I worked in the Accounts Department of Ford Motor Company at their Tractor factory at Basildon, down the road a bit from Ilford. Looking back on that period now, it is amazing that businesses ever managed their accounts with such primitive technology. Most of the time I was sorting through bits of paper, matching invoices to delivery notes so that we only paid for goods actually received.
In theory, every invoice was matched by its proof of delivery and that happened some of the time. More often numbers and dates did not match and there were all sorts of bits of paper to split, rearrange and match. Difficult situations were referred to ‘the Auditors,’ always said with an air of trepidation. I was just a humble clerk – not even that, just a ‘temp.’
[For those who want to reminisce about tractors, there were four models then of different sizes: Dextra, Super Dextra, Major and Super Major. They were bright blue. The one shown in the picture is a Dextra, the smallest model. Every part that arrived at the factory had a part number. Most started C5NN, which meant they were painted bright blue. Engines just had a long number. Delivery notes always seemed to include part number EZE 35, wooden pallets, which were returnable and free.]
At the Ford Tractor works, I dealt mainly with internal transfers from Belgium. If I remember correctly, the turnover just for this was about a million pounds a month, a lot of money in those days. It was before the hand-held calculators shown earlier so all we had were some slow, bulky adding machines and these ‘Facit’ machines for multiplication.
You had to manually set up both numbers, then wind the handle several times to get the answer. It was entirely mechanical and so the machines were noisy. It was a miracle that accounts were published every months with figures bearing some approximation to what had actually happened.
Most people in businesses through Britain would not have ever seen something so technologically advanced as this.
They did have a computer – just one for the whole factory complex at Basildon – and I will come to that later. This was going to be blog about computers … Maybe next time.