For occasional requirements, the method described on page 45 of
determining powers and roots other than the square and cube, is quite
satisfactory. When, however, a number of such calculations are to be
made, the process may be simplified considerably by the use of what are
known as _log.-log._, _logo-log._, or _logometric_ scales, in
conjunction with the ordinary scales of the rule. The principle involved
will be understood from a consideration of those rules for logarithmic
computation (page 8) which refer to powers and roots. From these it is
seen that while for the multiplication and division of numbers we _add_
their logarithms, for involution and evolution we require to _multiply_
or _divide_ the logarithms of the numbers by the exponent of the power
or root as the case may be. Thus to find 3^{2.3}, we have (log. 3) × 2·3
= log. _x_, and by the ordinary method described on page 45 we should
determine log. 3 by the aid of the scale L on the back of the slide,
multiply this by 2·3 by using the C and D scales in the usual manner,
transfer the result to scale L, and read the value of _x_ on D under 1
on C. By the simpler method, first proposed by Dr. P. M. Roget,[8] the
multiplication of log. 3 by 2·3 is effected in the same way as with any
two ordinary factors—_i.e._, by adding their logarithms and finding the
number corresponding to the resulting logarithm. In this case we have
log. (log. 3) + log. 2·3 = log. (log. _x_). The first of the three terms
is obviously the _logarithm of the logarithm_ of 3, the second is the
simple logarithm of 2·3, and the third the _logarithm of the logarithm
of_ the answer. Hence, if we have a scale so graduated that the
distances from the point of origin represent the logarithms of the
logarithms (the log.-logs.) of the numbers engraved upon it, then by
using this in conjunction with the ordinary scale of logarithms, we can
effect the required multiplication in a manner which is both expeditious
and convenient. Slightly varying arrangements of the log.-log. scale,
sometimes referred to as the “P line,” have been introduced from time to
time, but latterly the increasing use of exponential formulæ in
thermodynamic, electrical, and physical calculations has led to a
revival of interest in Dr. Roget’s invention, and various arrangements
of rules with log.-log. scales are now available.
_The Davis Log.-Log. Rule._—In the rule introduced by Messrs. John Davis
& Son Limited, Derby, the log.-log. scales are placed upon a separate
slide—a plan which has the advantage of leaving the rule intact for all
ordinary purposes, while providing a length of 40 in. for the log.-log.
scales.
In the 10 in. Davis rule one face of the slide, marked E, has two
log.-log. scales for numbers greater than unity, the lower extending
from 1·07 to 2, and the upper continuing the graduations from 2 to 1000.
On the reverse face of the slide, marked -E, are two log.-log. scales
for numbers less than unity, the upper extending from 0·001 to 0·5, and
the lower continuing the graduations from 0·5 to 0·933. Both sets of
scales are used in conjunction _with the lower or D scale of the rule_,
which is to be primarily regarded as running from 1 to 10, and
constitutes a scale of exponents. In the 20 in. rule the log.-log.
scales are more extensive, and are used in conjunction with the upper or
A scale of the rule (1 to 100); in what follows, however, the 10 in.
rule is more particularly referred to.
It has been explained that on the log.-log. scale the distance of any
numbered graduation from the point of origin represents the log.-log. of
the number. The point of origin will obviously be that graduation whose
log.-log. = 0. This is seen to be 10, since log. (log. 10) = log. 1 = 0.
Hence, confining attention to the E scale, to locate the graduation 20,
we have log. (log. 20) = log. 1·301 = 0·11397, so that if the scale D is
25 cm. long, the distance between 10 and 20 on the corresponding
log.-log. scale would be 113·97 ÷ 4 = 28·49 mm. For numbers less than 10
the resulting log.-logs. will be negative, and the distances will be
spaced off from the point of origin in a negative direction—_i.e._, from
right to left. Thus, to locate the graduation 5, we have
log. (log. 5) = log. 0·699 = ̅1·844; _i.e._, −1 + 0·844 or −0·156;
so that the graduation marked 5 would be placed 156 ÷ 4 = 39 mm. distant
from 10 in a _negative_ direction, and proceeding in a similar manner,
the scale may be extended in either direction. In the -E scale, the
notation runs in the reverse direction to that of the E scale, but in
all other respects it is precisely analogous, the distance from the
point of origin (0·1 in this case) to any graduation _x_ representing
log. [-log. _x_.]. It follows that of the similarly situated graduations
on the two scales, those on the -E scale are the _reciprocals_ of those
on the E scale. This may be readily verified by setting, say, 10 on E to
(R.H.) 1 on D, when turning to the back of the rule we find 0·1 on -E
agreeing with the index mark in the aperture at the right-hand extremity
of the rule.
In using the log.-log. scales it is important to observe (1) that the
values engraved on the scale are definite and unalterable (_e.g._, 1·2
can only be read as 1·2 and not as 120, 0·0012, etc., as with the
ordinary scales); (2) that the upper portion of each scale should be
regarded as forming a prolongation to the right of the lower portion;
and (3) that immediately above any value on the lower portion of the
scale is found the 10th power of that value on the upper portion of the
scale. Keeping these points in view, if we set 1·1 on E to 1 on D we
find over 2 on D the value of 1·1^2 = 1·21 on E. Similarly, over 3 we
find 1·1^3 = 1·331, and so on. Then, reading across the slide, we have,
over 2, the value of 1·1^{2 × 10} = 1·1^{20} = 6·73, and over 3 we have
1·1^{3 × 10} = 1·1^{30} = 17·4. Hence the rule:—_To find the value of
x^n, set x on E to 1 on D, and over n on D read x^n on E._
With the slide set as above, the 8th, 9th, etc., powers of 1·1 cannot be
read off; but it is seen that, according to (2) in the foregoing, the
missing portion of the E scale is that part of the upper scale (2 to
about 2·6) which is outside the rule to the left. Hence placing 1·1 to
10 on D, the 8th, 9th, etc., powers of 1·1 will be read off _on the
upper part_ of the E scale. In general, then,
If _x_ on the _lower_ line is set to 1 on D, then _x^n_ is read directly
on that line and _x_^{10_n_} on the upper line.
If _x_ on the _upper_ line is set to 1 on D, then _x^n_ is read directly
on that line and _x_^{_ⁿ⁄₁₀_} on the lower line.
If _x_ on the _lower_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read
directly on that line and _x^n_ on the upper line.
If _x_ on the _upper_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read
directly on that line and _x_^{_ⁿ⁄₁₀₀_} on the lower line.
These rules are conveniently exhibited in the accompanying diagram (Fig.
14). They are equally applicable to both the E and -E scales of the 10
in. rule, and include practically all the instruction required for
determining the _n_th power or the _n_th root of a number. They do not
apply directly to the 20 in. rule, however, for here the relation of the
lower and upper scales will be _x^n_ and _x_^{100_n_}.
EX.—Find 1·167^{2·56}.
Set 1·167 on E to 1 on D, and over 2·56 on D read 1·485 on E.
EX.—Find 4·6^{1·61}.
Set 4·6 on upper E scale to 1 on D, and over 1·61 on D read 11·7
(11·67) on E.
EX.—Find 1·4^{0·27} and 1·4^{2·7}.
Set 1·4 on E to 10 on D, and over 2·7 on D read 1·095 = 1·4^{0·27} on
lower E scale and 2·48 = 1·4^{2·7} on upper E scale.
[Illustration: FIG. 14.]
EX.—Find 46^{0·0184} and 46^{0·184}.
Set 46 on upper E scale to 10 on D, and over 1·84 on D read 1·073 on
lower E scale and 2·022 (2·0228) on upper E scale.
EX.—Find 0·074^{1·15}.
Using the -E scale, set 0·074 to 1 on D, and over 1·15 on D read 0·05
on -E.
The method of determining the root of a number will be obvious from the
preceding examples.
EX.—Find ^{1.4}√(17) and ^{14}√(17).
Set 17 on E to 1·4 on D, and over 1 on D read 7·56 on upper E scale
and 1·224 on lower E scale.
EX.—Find ^{0·031}√(0·914).
Set 0·914 on -E to 3·1 on D, and over 10 on D read 0·055 on upper -E
scale.
When the exponent _n_ is fractional, it is often possible to obtain the
result directly with one setting of the slide. Thus to determine
1·135^{¹⁷⁄₁₆} by the first method we find ¹⁷⁄₁₆ = 1·0625, and placing
1·135 on E to 1 on D, read 1·144 on E over 1·0625 on D. By the direct
method we place 1·135 on the E scale on 1·6 on D, and over 1·7 on D read
1·144 on E. It will be seen that since the scale D is assumed to run
from 1 to 10 we are unable to read 16 and 17 on this scale; but it is
obvious that the _ratios_ (1·7)/(1·6) and (17)/(16) are identical, and
it is with the ratio only that we are, in effect, concerned.
Since an expression of the form _x_^{-_n_} = (1)/(_x^n_) or
((1)/(_x_))^{_n_}, the required value may be obtained by first
determining the reciprocal of _x_ and proceeding as before. By using
both the direct and reciprocal log.-log. scales (E and -E) in
conjunction however, the required value can be read directly from the
rule, and the preliminary calculation entirely avoided. In the Davis
form of rule, the result can be read on the -E scale, used in
conjunction with the D scale of the rule, _x_ on E being set to the
index mark in the aperture in the back of the rule.
EX.—Find the value of 1·195^{−1·65}.
Set 1·195 on E to the index in the left aperture in the back of the
rule, and over 1·65 on D read 0·745 on the -E scale.
It may be noted in passing that the log.-log. scale affords a simple
means for determining the logarithm or anti-logarithm of a number to any
base. For this purpose it is necessary to set the base of the given
system on E to 1 on D, when _under_ any number on E will be found its
logarithm on D. Thus, for common logs., we set the base 10 on E to 1 on
D, and under 100 we find 2, the required log. Similarly we read log. 20
= 1·301; log. 55 = 1·74; log. 550 = 2·74, etc. Reading reversely, over
1·38 on D we find its antilog. 24 on E; also antilog. 1·58 = 38;
antilog. 1·19 = 15·5, etc.
For logs. of numbers under 10 we set the base 10 to 10 on D; hence the
readings on D will be read as one-tenth their apparent value. Thus log.
3 = 0·477; log. 5·25 = 0·72; antilog. 0·415 = 2·6; antilog. 0·525 =
3.·35, etc.
The logs. of the numbers on the lower half of the E scale will also be
found on the D scale; but a consideration of Fig. 14 will show that this
will be read as _one-tenth_ its face value if the base is set to 1 on D,
and as _one-hundredth_ if the base is set to 10.
For natural, hyperbolic, or Napierian logarithms, the base is 2·718. A
special line marked ε or _e_ serves to locate the exact position of this
value on the E scale, and placing this to 1 on D we read log._{_e_} 4·35
= 1·47; log._{_e_} 7·4 = 2·0; antilog._{_e_} × 2·89 = 18, etc. The other
parts of the scale are read as already described for common logs.
Calculations involving powers of _e_ are frequently met with, and these
are facilitated by using the special graduation line referred to, as
will be readily understood.
If it is required to determine the power or root of a number which does
not appear on either of the log.-log. scales, we may break up the number
into factors. Usually it is convenient to make one of the factors a
power of 10.
EX.—3950^{1·97} = 3·95^{1·97} × 10^{3 × 1·97} = 3·95^{1·97} ×
10^{5·91}.
Then 3·95^{1·97} = 15, and 10^{5·91} (or antilog.) 5·91 = 812,000.
Hence, 15 × 812,000 = 12,180,000 is the result sought.
Numbers which are to be found in the higher part of the log.-log. scale
may often be factorised in this way, and greater accuracy obtained than
by direct reading.
The form of log.-log. rule which has been mainly dealt with in the
foregoing gives a scale of comparatively long range, and the only
objection to the arrangement adopted is the use of a separate slide.
_The Jackson-Davis Double Slide Rule._—In this instrument a pair of
aluminium clips enable the log.-log. slide to be temporarily attached to
the lower edge of the ordinary rule, and used, by means of a special
cursor, in conjunction with the C scale of the ordinary slide. In this
way both the log.-log. and ordinary scales are available without the
trouble of replacing one slide by the other. Since the scale of
exponents is now on the slide, the value of _x^n_ will be obtained by
setting 1 on C to _x_ on E and reading the result on E under _n_ on C.
By using a pair of log.-log. slides, one in the rule and one clamped to
the edge by the clips, we have an arrangement which is very useful in
deducing empirical formulæ of the type _y_ = _x^n_.
_The Yokota Slide Rule._—In this instrument the log.-log. scales are
placed on the face of the rule, each set comprising three lines. These,
for numbers greater than 1, are found above the A scale while the three
reciprocal log.-log. lines are below the D scale. Both sets are used in
conjunction with the C scale on the slide. Other features of this rule
are:—The ordinary scales are 10 in. long instead of 25 cm. as hitherto
usual; hence the logarithms of numbers can be read on the ordinary scale
of inches on the edge of the rule. There is a scale of cubes in the
centre of the slide and on the back of the slide there is a scale of
secants in addition to the sine and tangent scales.
[Illustration: FIG. 15.]
_The Faber Log.-log. Rule._—In this instrument shown in Fig. 15, the two
log.-log. scales are placed on the face of the rule. One section,
extending from 1·1 to 2·9, is placed above the A scale, and the other
section, extending from 2·9 to 100,000, is placed below the D scale.
These scales are used in conjunction with the C scale of the slide in
the manner previously described. The width of the rule is increased
slightly, but the arrangement is more convenient than that formerly
employed, wherein the log.-log. scales were placed on the bevelled edge
of the rule and read by a tongue projecting from the cursor.
[Illustration: FIG. 16.]
Another novel feature of this rule is the provision of two special
scales at the bottom of the groove, to which a bevelled metal index or
marker on the left end of the slide can be set. The upper of these
scales is for determining the efficiency of dynamos and electric motors;
the lower for determining the loss of potential in an electric circuit.
_The Perry Log.-log. Rule._—In this rule, introduced by Messrs. A. G.
Thornton, Limited, Manchester, the log.-log. scales are arranged as in
Fig. 16, the E scale, running from 1·1 to 10,000, being placed above the
A scale of the rule, and the -E or E^{−1} scale running from 0·93 to
0·0001, below the D scale of the rule. These scales are read in
conjunction with the B scales on the slide by the aid of the cursor.
The following tabular statement embodies all the instructions required
for using this form of log.-log. slide rule:—
When _x_ is greater than 1.
_x^n_ Set 1 on B to _x_ on E; over _n_ on B read _x^n_ on E
_x_^{-_n_} Set 1 on B to _x_ on E; under _n_ on B read _x_^{-_n_} on
E^{−1}
_x_^{_ⁱ⁄ₙ_} Set _n_ on B to _x_ on E; over 1 on B read _x_^{_ⁱ⁄ₙ_} on
E
_x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E; under 1 on B read _x_^{_⁻ⁱ⁄ₙ_}
on E^{−1}
When _x_ is less than 1.
_x^n_ Set 1 on B to _x_ on E^{−1}; under _n_ on B read _x^n_ on
E^{−1}
_x_^{-_n_} Set 1 on B to _x_ on E^{−1}; over _n_ on B read _x_^{-_n_}
on E
_x_^{_ⁱ⁄ₙ_} Set _n_ on B to _x_ on E^{−1}; under 1 on B read
_x_^{_ⁱ⁄ₙ_} on E^{−1}
_x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E^{−1}; over 1 on B read
_x_^{_⁻ⁱ⁄ₙ_} on E
If 10 on B is used in place of 1 on B, read _x_^{_ⁿ⁄₁₀_} in place of
_x^n_ on E, and _x_^{-_ⁿ⁄₁₀_} in place of _x_^{-_n_} on E^{−1}. If 100
on B is used, these readings are to be taken as _x_^{_ⁿ⁄₁₀₀_} and
_x_^{-_ⁿ⁄₁₀₀_} respectively.
In rules with no -E scale the value of _x_^{-_n_} is obtained by the
usual rules for reciprocals. We may either determine _x^n_ and find its
reciprocal or, first find the reciprocal of _x_ and raise it to the
_n_th power. The first method should be followed when the number _x_ is
found on the E scale.