Hitherto our attention has been confined to a consideration of the
primary divisions of the scales. The same principle of graduation is,
however, used throughout; and after what has been said, this part of the
subject need not be further enlarged upon. Some explanation of the
method of reading the scales is necessary, as facility in using the
instrument depends in a very great measure upon the dexterity of the
operator in assigning the correct value to each division on the rule. By
reference to Fig. 5, it will be seen that each of the primary spacings
in the several scales is invariably subdivided into ten; but since the
lengths of the successive primary divisions rapidly diminish, it is
impossible to subdivide each main space into the same number of parts
that the space 1–2 can be subdivided. This variable spacing of the
scales is at first confusing to the student, but with a little practice
the difficulty is soon overcome.
With the C or D scale, it will be noticed that the length of the
interval 1–2 is sufficient to allow each of the 10 subdivisions to be
again divided into 10 parts, so that the whole interval 1–2 is divided
into 100. The shorter main space 2–3, and the still shorter one 3–4,
only allow of the 10 subdivisions of each being divided into five parts.
Each of these main spaces is therefore divided into 50 parts. For the
remainder of the scale each of the 10 subdivisions of each main space is
divided into two parts only; so that from the main division 4 to the end
of the scale the primary spaces are divided into 20 parts only.
In the upper scales A or B, it will be found that—as the space 1–2 is of
only half the length of the corresponding space on C or D—the 10
subdivisions of this interval are divided into five parts only.
Similarly each of the 10 subdivisions of the intervals 2–3, 3–4, and 4–5
are further divided into two parts only, while for the remainder of the
scale only the 10 subdivisions are possible, owing to the rapidly
diminishing lengths of the primary spacings.
The values actually given on the rule run from 1 to 10 on the lower
scales and from 1 to 100 on the upper scales, and, as explained on page
9, all factors are brought within these ranges of values by multiplying
or dividing them by powers of 10. By following this plan, we virtually
regard each factor as merely a series of significant figures, and make
the necessary modification due to the “powers of 10” when fixing the
position of the decimal point in the answer.
Many, however, find it convenient in practice to regard the values on
the rule as multiplied or divided by such powers of 10 as may be
necessary to suit the factors entering into the calculation. If this
plan is adopted, the values given to each graduation of the scales will
depend on that given to the left index figure (1) of the lower scales,
this being any multiple or submultiple of 10. Thus IL on the D scale may
be regarded as 1, 10, 100, 1000, etc., or as 0·1, 0·01, 0·001, 0·0001,
etc.; but once the initial value is assigned to the index, the ratio of
value must be maintained throughout the whole scale. For example, if 1
on C is taken to represent 10, the main divisions 2, 3, 4, etc., will be
read as 20, 30, 40, etc. On the other hand, if the fourth main division
is read as 0·004, then the left index figure of the scale will be read
as 0·001. The figured subdivisions of the main space 1–2 are to be read
as 11, 12, 13, 14, 15, 16, 17, 18 and 19—if the index represents 10,—and
as corresponding multiples for any other value of the index.
Independently considered, these remarks apply equally to the A or B
scale, but in this case the notation is continued through the second
half of the scale, the figures of which are to be read as tenfold values
of the corresponding figures in the first half of the scale.
The reading of the intermediate divisions will, of course, be determined
by the values assigned to the main divisions. Thus, if IL on D is read
as 1, then each of the smallest subdivisions of the space 1–2 will be
read as 0·01, and each of the smallest subdivisions of the spaces 2–3 or
3–4 as 0·02, while for the remainder of the scale the smallest
subdivisions are read as 0·05. In the A or B scale the subdivisions of
the space 1–2 of the first half of the scale are (if IL = 1) read as
0·02, 0·04, etc.; for the divisions 2–3, 3–4, and 4–5, the smallest
intervals are read as 0·05 of the primary spaces, and from 5 to the
centre index of the scale the divisions represent 0·1 of each main
interval. Passing the centre index, which is, now read as 10, the
smallest subdivisions immediately following are read 10·2, 10·4, etc.,
until 20·0 is reached; then we read 20·5, 21·0, 21·5 22·0, etc., until
the figured main division 5 is reached. The remainder of the scale is
read 51, 52, 53, etc., up to 100, the right-hand index.
Further subdivision of any of the spaces of the rule can be effected by
the eye, and after a little practice the operator will become quite
expert in estimating any intermediate value. It affords good practice to
set 1 on C to 1·04, 1·09, etc. on D, and to read the values on D, under
4, 6, 8, etc. on C. As the exact results are easily calculated mentally,
the student, by this means, will receive better instruction in
estimating intermediate results than can be given by any diagram.
Some rules will be found figured as shown in Fig. 5; in others, the
right-hand upper scales are marked 10, 20, 30, etc. Again, others are
marked decimally, the lower scales and the left-hand upper scales being
figured 1, 1·1, 1·2, 1·3 ... 2·5, etc. The latter form has advantages
from the point of view of the beginner.
The method of reading the A and B scales, just given, applies only when
these scales are regarded as altogether independent of the lower pair of
scales C and D. Some operators prefer to use the A and B scales, and
some the C and D scales, for the ordinary operations of proportion,
multiplication, and division. Each method has its advantages, as will be
shown, but in the more complex calculations, as involution and
evolution, etc., the relation of the upper scales to the lower scales
becomes a very important factor.
The distance 1–10 on the upper scales is one-half of the distance 1–10
on the lower scales. Hence any distance from 1, taken on the upper
scales, represents _twice the logarithm_ which the same distance
represents on the lower scales. In other words, the length which
represents log. N on D, would represent 2 log. N on A; and, conversely,
the length which represents log. N on A, would represent (log. N)/(2) on
D.
Now we have seen (page 8) that multiplying the log. of a number by 2
gives the log. of the square of the number. Hence, above any number on D
we find its _square_ on A, or, conversely, below any number on A, we
find its _square root_ on D. Thus, above 2 we find 4; under 49, we find
7 and so on. Obviously the same relation exists between the B and C
scales.