_Scales._—Not the least important feature of the modern slide rule is
the provision of the special scales on the under-side of the slide, and
by the use of which, in conjunction with the ordinary scales on the
rule, a large variety of trigonometrical computations may be readily
performed.
Three scales will be found on the reverse or under-side of the slide of
the ordinary Gravêt or Mannheim rule. One of these is the evenly-divided
scale or scale of equal parts referred to in previous sections, and by
which, as explained, the decimal parts or mantissæ of logarithms of
numbers may be obtained. Usually this scale is the centre one of the
three, but in some rules it will be found occupying the lowest position,
in which case some little modification of the following instructions
will be necessary. The requisite transpositions will, however, be
evident when the purposes of the scales are understood. The upper of the
three scales, usually distinguished by the letter S, is a scale giving
the logarithms of the sines of angles, and is used to determine the
natural sines of angles of from 35 minutes to 90 degrees. The notation
of this scale will be evident on inspection. The main divisions 1, 2, 3,
etc., represent the degrees of angles; but the values of the
subdivisions differ according to their position on the scale. Thus, if
any primary space is subdivided into 12 parts, each of the latter will
be read as 5 minutes (5′), since 1° = 60′.
_Sines of Angles._—To find the sine of an angle the slide is placed in
the groove, with the under-side uppermost, and the end division lines or
indices on the slide, coinciding with the right and left indices of the
A scale. Then over the given angle on S is read the value of the sine of
the angle on A. If the result is found on the left scale of A (1 to 10),
the logarithmic characteristic is −2; if it is found on the right-hand
side (10 to 100), it is −1. In other words, results on the right-hand
scale are prefixed by the decimal point only, while those on the
left-hand scale are to be preceded by a cypher also. Thus:—
Sine 2° 40′ = 0·0465; sine 15° 40′ = 0·270.
Multiplication and division of the sines of angles are performed in the
same manner as ordinary calculations, excepting that the slide has its
under-face placed uppermost, as just explained. Thus to multiply sine
15° 40′ by 15, the R.H. index of S is brought to 15 on A, and opposite
15° 40′ on S is found 4·05 on A. Again, to divide 142 by sine 16° 30′,
we place 16° 30′ on S to 142 on A, and over R.H. index of S read 500 on
A.
The rules for the number of integers in the results are thus determined:
Let N be the number of integers in the multiplier M or in the dividend
D. Then the number of integers P, in the product or Q, in the quotient
are as follows:—
When the result is found to the right of M or D, │P = N − 2│Q = N
and in the same scale │ │
When the result is found to the right of M or D, │P = N − 1│Q = N + 1
and in the other scale │ │
When the result is found to the left of M or D, and│P = N − 1│Q = N + 1
in the other scale │ │
When the result is found to the left of M or D, and│P = N │Q = N + 2
in the same scale │ │
If the division is of the form (20° 30′)/(50), the result cannot be read
off directly on the face of the rule. Thus, if in the above example 20°
30′ on S, is placed to agree with 50 on the right-hand scale of A, the
result found on S under the R.H. index of A is 44° 30′. The required
numerical value can then be found: (1) By placing the slide with all
indices coincident when opposite 44° 30′ on S will be found 0·007 on A;
or (2) In the ordinary form of rule, by reading off on the scale B
opposite the index mark in the opening on the under-side of the rule.
The above rules for the number of integers in the quotient do not apply
in this case.
If it is required to find the sine of an angle simply, this may be done
with the slide in its ordinary position, with scale B under A. The given
angle on scale S is then set to the index on the under-side of the rule,
and the value of the sine is read off on B under the right index of A.
Owing to the rapidly diminishing differences of the values of the sines
as the upper end of the scale is approached, the sines of angles between
60° and 90° cannot be accurately determined in the foregoing manner. It
is therefore advisable to calculate the value of the sine by means of
the formula:
Sine θ = 1 − 2 sin^2 (90 − θ)/(2).
To determine the value of sin^2 (90 − θ)/(2). With the slide in the
normal position, set the value of (90 − θ)/(2). on S to the index on the
under-side of the rule, and read off the value _x_ on B under the R.H.
index of A. Without moving the slide find _x_ on A, and read under it on
B the value required.
EX.—Find value of sine 79° 40′.
Sine 79° 40′ = 1 − 2sin^2 5° 10′.
But sine 5° 10′ = 0·0900, and under this value on A is 0·0081 on B.
Therefore sine 79° 40′ = 1 − 0·0162 = 0·9838.
The sines of very small angles, being very nearly proportional to the
angles themselves, are found by direct reading. To facilitate this, some
rules are provided with two marks, one of which, a single accent (′),
corresponds to the logarithm of (1)/(sine 1′) and is found at the number
3438. The other mark—a double accent (″)—corresponds to the logarithm of
(1)/(sine 1″) and is found at the number 206,265. In some rules these
marks are found on either the A or the B scales; sometimes they are on
both. In either case the angle on the one scale is placed so as to
coincide with the significant mark on the other, and the result read off
on the first-named scale opposite the index of the second.
In sines of angles under 3″, the number of integers in the result is −5;
while it is −4 for angles from 3″ to 21″; −3 from 21″ to 3′ 27″; and −2
from 3′ 27″ to 34′ 23″.
EX.—Find sine 6′.
Placing the significant mark for minutes coincident with 6, the value
opposite the index is found to be 175, and by the rule above this is
to be read 0·00175. For angles in seconds the other significant mark
is used; while angles expressed in minutes and seconds are to be first
reduced to seconds. Thus, 3′ 10″ = 190″.
_Tangents of Angles._—There remains to be considered the third scale
found on the back of the slide, and usually distinguished from the
others by being lettered T. In most of the more recent forms of rule
this scale is placed near the lower edge of the slide, but in some
arrangements it is found to be the centre scale of the three. Again, in
some rules this scale is figured in the same direction as the scale of
sines—viz., from left to right,—while in others the T scale is reversed.
In both cases there is now usually an aperture formed in the back of the
left extremity of the rule, with an index mark similar to that already
referred to in connection with the scale of sines. Considering what has
been referred to as the more general arrangement, the method of
determining the tangents of angles may be thus explained:—
The tangent scale will be found to commence, in some rules, at about
34′, or, precisely, at the angle whose tangent is 0·01. More usually,
however, the scale will be found to commence at about 5° 43′, or at the
angle whose tangent is 0·1. The other extremity of the scale corresponds
in all cases to 45°, or the angle whose tangent is 1. This explanation
will suggest the method of using the scale, however it may be arranged.
If the graduations commence with 34′, the T scale is to be used in
conjunction with the right and left scales of A; while if they commence
with 5° 43′ it is to be used in conjunction with the D scale.
In the former case the slide is to be placed in the rule so that the T
scale is adjacent to the A scales, and, with the right and left indices
coinciding, when opposite any angle on T will be found its tangent on A.
From what has been said above, it follows that the tangents read on the
L.H. scale of A have values extending from 0·01 to 0·1; while those read
on the R.H. scale of A have values from 0·1 to 1·0. Otherwise expressed,
to the values of any tangent read on the L.H. scale of A a cypher is to
be prefixed; while if found on the R.H. scale, it is read directly as a
decimal.
EX.—Find tan. 3° 50′.
Placing the slide as directed, the reading on A opposite 3° 50′ on T
is found to be 67. As this is found on the L.H. scale of A, it is to
be read as 0·067.
EX.—Find tan. 17° 45′.
Here the reading on A opposite 17° 45′ on T is 32, and as it is found
on the R.H. scale of A it is read as 0·32.
As in the case of the scale of sines, the tangents may be found without
reversing the slide, when a fixed index is provided in the back of the
rule for the T scale.
We revert now to a consideration of those rules in which a single
tangent scale is provided. It will be understood that in this case the
slide is placed so that the scale T is adjacent to the D scale, and that
when the indices of both are placed in agreement, the value of the
tangent of any angle on T (from 5° 43′ to 45°) may be read off on D, the
result so found being read as wholly decimal. Thus tan. 13° 20′ is read
0·237.
If a back index is provided, the slide is used in its normal position,
when, setting the angle on the tangent scale to this index, the result
can be read on C over the L.H. index of D.
The tangents of angles above 45° are obtained by the formula: Tan. θ =
(1)/(tan. (90 − θ)). For all angles from 45° to (90° − 5° 43′) we
proceed as follows:—Place (90 − θ) on T to the R.H. index of D, and read
tan. θ on D under the L.H. index of T. The first figure in the value
thus obtained is to be read as an integer. Thus, to find tan. 71° 20′ we
place 90° − 71° 20′ = 18° 40′ on T, to the R.H. index of D, and under
the L.H. index of T read 2·96, the required tangent.
The tangents of angles less than 40′ are sensibly proportional to the
angles themselves, and as they may therefore be considered as sines,
their value is determined by the aid of the single and double accent
marks on the sine scale, as previously explained. The rules for the
number of integers are the same as for the sines.
Multiplication and division of tangents may be quite readily effected.
EX.—Tan. 21° 50′ × 15 = 6.
Set L.H. index of T to 15 on D, and under 21° 50′ on T read 6 on D.
EX.—Tan. 72° 40′ × 117 = 375.
Set (90° − 72° 40′) = 17° 20′ on T to 117 on D, and under R.H. index
of T read 375 on D.
_Cosines of Angles._—The cosines of angles may be determined by placing
the scale S with its indices coinciding with those of A, and when
opposite (90 − θ) on S is read cos. θ on A. If the result is read on the
L.H. scale of A, a cypher is to be prefixed to the value read; while if
it is read on the R.H. scale of A, the value is read directly as a
decimal. Thus, to determine cos. 86° 30′ we find opposite (90° − 86°
30′) = 3° 30′ on S, 61° on A, and as this is on the L.H. scale the
result is read 0·061. Again, to find cos. 59° 20′ we read opposite (90°
− 59° 20′) or 30° 40′ on S, 51 on A, and as this is found on the R.H.
scale of A, it is read 0·51.
In finding the cosines of small angles it will be seen that direct
reading on the rule becomes impossible for angles of less than 20°. It
is advisable in such cases to adopt the method described for determining
the _sines_ of the _large_ angles of which the complements are sought.
_Cotangents of Angles._—From the methods of finding the tangents of
angles previously described, it will be apparent that the cotangents of
angles may also be obtained with equal facility. For angles between 5°
45′ and 45°, the procedure is the same as that for finding tangents of
angles greater than 45°. Thus, the angle on scale T is brought to the
R.H. index of D, and the cotangent read off on D under the L.H. index of
T. The first figure of the result so found is to be read as an integer.
If the angle (θ) lies between 45° and 84° 15′, the slide is placed so
that the indices of T coincide with those of D, and the result is then
read off on D opposite (90 − θ) on T. In this case the value is wholly
decimal.
_Secants of Angles._—The secants of angles are readily found by bringing
(90 − θ) on S to the R.H. index of A and reading the result on A over
the L.H. index of S. If the value is found on the L.H. scale of A, the
first figure is to be read as an integer; while if the result is read on
the R.H. scale of A, the first _two_ figures are to be regarded as
integers.
_Cosecants of Angles._—The cosecants of angles are found by placing the
angle on S to the R.H. index of A, and reading the value found on A over
the L.H. index of S. If the result is read on the L.H. scale of A, the
first figure is to be read as an integer; while if the result is found
on the R.H. scale of A, the first _two_ figures are to be read as
integers.
It will be noted that some of the rules here given for determining the
several trigonometrical functions of angles apply only to those forms of
rules in which a single scale of tangents T is used, reading from left
to right. For the other arrangements of the scale, previously referred
to, some slight modification of the method of procedure in finding the
tangents and cotangents of angles will be necessary; but as in each case
the nature and extent of this modification is evident, no further
directions are required.