[This 6 inch rule uses fewer minor divisions.]
In order to understand just why 2.12 is set where it is (figure 2),
notice that the interval from 2 to 3 is divided into 10 large or major
divisions, each of which is, of course, equal to one-tenth (0.1) of the
amount represented by the whole interval. The major divisions are in
turn divided into 5 small or minor divisions, each of which is one-fifth
or two-tenths (0.2) of the major division, that is 0.02 of the
whole interval. Therefore, the index is set above
2 + 1 major division + 1 minor division = 2 + 0.1 + 0.02 = 2.12.
In the same way we find 3.16 on the C scale. While we are on this
subject, notice that in the interval from 1 to 2 the major divisions are
marked with the small figures 1 to 9 and the minor divisions are 0.1 of
the major divisions. In the intervals from 2 to 3 and 3 to 4 the minor
divisions are 0.2 of the major divisions, and for the rest of the D (or
C) scale, the minor divisions are 0.5 of the major divisions.
Reading the setting from a slide rule is very much like reading
measurements from a ruler. Imagine that the divisions between 2 and 3 on
the D scale (figure 2) are those of a ruler divided into tenths of a
foot, and each tenth of a foot divided in 5 parts 0.02 of a foot long.
Then the distance from one on the left-hand end of the D scale (not
shown in figure 2) to one on the left-hand end of the C scale would he
2.12 feet. Of course, a foot rule is divided into parts of uniform
length, while those on a slide rule get smaller toward the right-hand
end, but this example may help to give an idea of the method of making
and reading settings. Now consider another example.
Example 3a: 2.12 * 7.35 = 15.6
If we set the left-hand index of the C scale over 2.12 as in the last
example, we find that 7.35 on the C scale falls out beyond the body of
the rule. In a case like this, simply use the right-hand index of the C
scale. If we set this over 2.12 on the D scale and move the runner to
7.35 on the C scale we read the result 15.6 on the D scale under the
hair-line.
Now, the question immediately arises, why did we call the result 15.6
and not 1.56? The answer is that the slide rule takes no account of
decimal points. Thus, the settings would be identical for all of the
following products:
Example 3:a: 2.12 * 7.35 = 15.6
b: 21.2 * 7.35 = 156.0
c: 212 * 73.5 = 15600.
d: 2.12 * .0735 = .156
e: .00212 * 735 = .0156
The most convenient way to locate the decimal point is to make a mental
multiplication using only the first digits in the given factors. Then
place the decimal point in the slide rule result so that its value is
nearest that of the mental multiplication. Thus, in example 3a above, we
can multiply 2 by 7 in our heads and see immediately that the decimal
point must be placed in the slide rule result 156 so that it becomes
15.6 which is nearest to 14. In example 3b (20 * 7 = 140), so we must
place the decimal point to give 156. The reader can readily verify the
other examples in the same way.
Since the product of a number by a second number is the same as the
product of the second by the first, it makes no difference which of the
two numbers is set first on the slide rule. Thus, an alternative way of
working example 2 would be to set the left-hand index of the C scale
over 3.16 on the D scale and move the runner to 2.12 on the C scale and
read the answer under the hair-line on the D scale.
The A and B scales are made up of two identical halves each of which is
very similar to the C and D scales. Multiplication can also be carried
out on either half of the A and B scales exactly as it is done on the C
and D scales. However, since the A and B scales are only half as long as
the C and D scales, the accuracy is not as good. It is sometimes
convenient to multiply on the A and B scales in more complicated
problems as we shall see later on.
A group of examples follow which cover all the possible combination of
settings which can arise in the multiplication of two numbers.
Example
4: 20 * 3 = 60
5: 85 * 2 = 170