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.
Multiplying using a slide rule
Slide rule with its cursor over the first number to be multiplied with
the sliding scale slid along so that its 1 is in the same position.
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 the its 1 is lined up
with the first number, in this case 2 - see the top picture.
Slide rule with its the sliding scale in the same position but with its
cursor slid along to be over the second number to be multiplied so that
it also shows the result of the multiplication.
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
× 2 = 4 and 2 × 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 do not put in a decimal point. So for example a multiplication
732.8 × 926400
Needs to be treated as:
7.33 x 102 × 9.26 x 105
The slide rule gives the result of 7.33 x 9.26 as
67.9, ie 6.79 × 101
So multiplying the 10s together by adding their indices gives an answer
6.79 × 108
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
Dividing using a slide rule
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 two photos.
The principle involved - log scales
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
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.
To multiply four digit numbers together, we had to look up the numbers
that were their logs,
and add these log numbers together. Then we would look up the result in the antilog
table, which would give the answer to the multiplication. Division was
similar except that the logs had to be subtracted. This probably sounds
longwinded by today's standards where calculators are everywhere, but we
were used to it and regarded it as a lot more straightforward than long
multiplication and long division. Only occasionally did we spare a thought
for the individuals who had to work out the logarithms for so many numbers.
Other slide rule functions
It will not have escaped your notice that there are other scales on the
slide rule. These encompass sines and cosines and more advanced functions.
Slide rules came in different lengths to give different accuracies and
there were even circular slide rules.