before we went digital

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Although the digital revolution of the later years of the 1900s produced electronic calculators which made complex calculations quick and easy, there were effective calculating devices before then. One was the slide-rule.

The photo shows the fairly standard plastic slide rule which I used regularly in the 1960s and 1970s. I understand that similar ones existed from the early 1900s, but they were probably made of ivory.

Like most slide rules, my slide rule consists of a wide ruler, a narrower one which slides along the central groove of the larger one and a spring-loaded cursor. By sliding the narrow ruler to a suitable position, and reading off from one of the other scales, all sorts of complex calculations can be made quickly and easily. The cursor increases accuracy by enabling markings on the two scales to be lined up more precisely.

The next section gives a simple example of how slide rules work.

Let's take a simple example of multiplying two numbers:

Suppose you want to multiply 2 by 3. Yes, that's so simple that you know the answer anyway, but this is to show the principle.

Move the sliding scale (B in the photograph) until its 1 is lined up with the first number, in this case 2 - see the above picture.

Then look along the sliding scale to the 3 and move the cursor precisely over it to read off the number it is aligned with on the fixed scale. You can clearly see that the 3 is lined up with the 6. So 3 × 2 = 6

By sliding the cursor to various other positions you can also readily see that other values of 2 times any number can also be read off, for example 2 x 2 = 4 and 2 x 4 = 8 etc.

Using the cursor to mark intermediate positions enables several numbers to be multiplied together in one go.

Where the numbers contain decimals, for example if 3.59 is to be multiplied by 4.62, the cursor becomes more important for accurate reading, although three significant figures are all that can normally be achieved with a standard slide rule.

Slide rules cannot put in powers of ten or a decimal point. So you need to do a rough answer in your head to see what these should be. For example a multiplication like:

732.8 × 926400

Needs to be treated as:

7 x 10^{2} x 9 x 10^{5}

This comes to 63 X 10^{7}
which is
better expressed as

6.3 X 10^{8}

So when the slide rule gives the result as 6.79 you know that the actual result must be

6.79 × 10^{8}

This result is accurate to 3 significant figures which is all that is normally required. In fact it is in some ways unfortunate that calculators give so many significant figures because they are normally meaningless in real situations.

Division is the reverse of multiplication in that the cursor is placed over the number to be divided and the number that is to divide it is slid along so that the two are lined up. Then the answer is read of from the position of the 1 (or the 10 if the 1 is off the scale).

So 6 ÷ 3 = 2 could be calculated from the second of the above two photos.

Multiplication and division on a slide rule work by sliding the central rule along so that numbers on the two scales are effectively added or subtracted. To accommodate this, the scales are not uniform, but are what are known as log scales (or logarithmic scales). When I was at school in the 1950s, we did not have slide rules. Instead we were expected to use booklets of tables, which were generally known as 'log books' or 'logs' even though they included many other functions.

Mathematical explanations of logarithms are available on the internet, but it is enough to say here that a logarithmic scale is non-linear, as can be seen from the photographs, such that multiplication and division can be achieved through addition and subtraction.

It will not have escaped your notice that there are other scales on the slide rule. These encompass sines, cosines and more advanced functions.

Slide rules came in different lengths to give different accuracies and there were even circular ones.