Chapter C: N. P (4)

EX.—Find the amount of £500 at 5 per cent. for 6 years, with compound
interest.

Set L.H. index of C to £1·05 on D, and read at the index on the
scale of equal parts on the under-side of rule, 0·0212. Multiply by
6, we obtain 0·1272, which, on the scale of equal parts, is placed
to the index in the notch at the end of the rule. Then opposite 500
on C read £670 on D, the amount required, including compound
interest.

MISCELLANEOUS CALCULATIONS.

To calculate percentages of compositions.

Set weight (or volume) of sample on C, to weight (or volume) of substance considered, on D; then under index of C read required percentage on D.

EX.—A sample of coal weighing 1·25 grms. contains 0·04425 grm. of ash.
Find the percentage of ash.

Set 1·25 on C to 0·04425 on D, and under index on C read 3·54, the
required percentage of ash on D.

Given the steam pressure P and the diameter _d_ in millimetres, of the throat of an injector, to find the weight W, of water delivered in lb. per hour from W = (_d_^2√̅P)/(0·505).

Set 0·505 on C to P on A; bring cursor to _d_ on C and index of C to cursor. Then under _d_ on C read delivery of water on D.

To find the pressure of wind per square foot, due to a given velocity in miles per hour.

Set 1 on B to 2 on A, and over the velocity in miles per hour on D read pressure in lb. per square foot on B.

To find the kinetic energy of a moving body.

Set 64·4 on B to velocity in feet per second on D, and over weight of body in lb. on B read kinetic energy or accumulated work in foot-lb. on A.

TRIGONOMETRICAL APPLICATIONS

_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.

THE SOLUTION OF RIGHT-ANGLED TRIANGLES.

From the foregoing explanation of the manner of determining the trigonometrical functions of angles, the methods of solving right-angled triangles will be readily perceived, and only a few examples need therefore be given.

Let _a_ and _b_ represent the sides and _c_ the hypothenuse of a right-angled triangle, and _a_° and _b_° the angles opposite to the sides. Then of the possible cases we will take

(1.) Given _c_ and _a_°, to find _a_, _b_, and _b_°.

The angle _b_° = 90 − _a_°, while _a_ = _c_ sin _a_° and _b_ = _c_ sin _b_°. To find _a_, therefore, the index of S is set to _c_ on A, and the value of _a_ read on A opposite _a_° on S. In the same manner the value of _b_ is obtained.

EX.—Given in a right-angled triangle _c_ = 9 ft. and _a_° = 30°. Find
_a_, _b_, and _b_°.

The angle _b_° = 90 − 30 = 60°. To find _a_, set R.H. index of S to 9
on A, and over 30° on S read _a_ = 4·5 ft. on A. Also, with the slide
in the same position, read _b_ = 7·8 ft. [7·794] on A over 60° on S.

(2.) Given _a_ and _c_, to determine _a_°, _b_°, and _b_.

In this case advantage is taken of the fact that in every triangle the sides are proportional to the sines of the opposite angles. Therefore, as in this case the hypothenuse c subtends a right angle, of which the sine = 1, the R.H. index (or 90°) on S is set to the length of _c_ on A, when under _a_ on A is found _a_° on S. Hence _b_° and _b_ may be determined.

(3.) Given _a_ and _a_°, to find _b_, _c_, and _b_°.

Here _b_° = (90 − _a_°), and the solution is similar to the foregoing.

(4.) Given _a_ and _b_, to find _a_°, _b_°, and _c_.

To find _a_°, we have tan. _a_° = _a_/_b_, which in the above example will be (4·5)/(7·8) = 0·577. Therefore, placing the slide so that the indices of T coincide with those of D, we read opposite 0·577 on D the value of _a_° = 30°. The hypothenuse _c_ is readily obtained from _c_ = _a_/(sin _a_°).

THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES.

Using the same letters as before to designate the three sides and the subtending angles of oblique-angled triangles, we have the following cases:—

(1.) Given one side and two angles, as _a_, _a_°, and _b_°, to find _b_, _c_, and _c_°.

In the first place, _c_° = 180° − (_a_° + _b_°); also we note that, as the sides are proportional to the sines of the opposite angles, _b_ = (_a_ sine _b_°)/(sine _a_°) and _c_ = (_a_ sine _c_°)/(sine _a_°).

Taking as an example, _a_ = 45, _a_° = 57°, and _b_° = 63°, we have _c_° = 180 − (57 + 63) = 60°. To find _b_ and _c_, set _a_° on S to _a_ on A, and read off on A above 63° and 60° the values of _b_ (= 47·8) and _c_ (= 46·4) respectively.

(2.) Given _a_, _b_, and _a_°, to find _b_°, _c_°, and _c_.

In this case the angle _a_° on S is placed under the length of side _a_ on A and under _b_ on A is found the angle _b_° on S. The angle _c_° = 180 − (_a_° + _b_°), whence the length _c_ can be read off on A over _c_° on S.

(3.) Given the sides and the included angle, to find the other side and the remaining angles.

If, for example, there are given _a_ = 65, _b_ = 42, and the included angle _c_° = 55°, we have (_a_ + _b_) ∶ (_a_ − _b_) = tan. (_a_° + _b_°)/(2) ∶ tan. (_a_° − _b_°)/(2). Then, since _a_° + _b_° = 180° − 55° = 125°, it follows that (_a_° + _b_°)/(2) = (125°)/(2) = 62° 30′.

By the rule for tangents of angles greater than 45°, we find tan. 62° 30′ = 1·92. Inserting in the above proportion the values thus found, we have 107 ∶ 23 = 1·92 ∶ tan. (_a_° − _b_°)/(2). From this it is found that the value of the tangent is 0·412, and placing the slide with all indices coinciding, it is seen that this value on D corresponds to an angle of 22° 25′. Therefore, since (_a_° + _b_°)/(2) = 62° 30′, and (_a_° − _b_°)/(2) = 22° 25′, it follows that _a_° = 84° 55′, and _b_° = 40° 5′. Finally, to determine the side _c_, we have _c_ = (_a_ sin _c_°)/(sin _a_°) as before.

PRACTICAL TRIGONOMETRICAL APPLICATIONS.

A few examples illustrative of the application of the methods of determining the functions of angles, etc., described in the preceding section, will now be given.

To find the chord of an arc, having given the included angle and the radius.

With the slide placed in the rule with the C and D scales outward, bring one-half of the given angle on S to the index mark in the back of the rule, and read the chord on B under twice the radius on A.

EX.—Required the chord of an arc of 15°, the radius being 23 in.

Set 7° 30′ on S to the index mark in the back of the rule, and under
46 on A read 6 in., the required length of chord on B.

To find the area of a triangle, given two sides and the included angle.

Set the angle on S to the index mark on the back of the rule, and bring cursor to 2 on B. Then bring the length of one side on B to cursor, cursor to 1 on B, the length of the other side on B to cursor, and read area on B under index of A.

EX.—The sides of a triangle are 5 and 6 ft. in length respectively,
and they include an angle of 20°. Find the area.

Set 20 on S to index mark, bring cursor to 2 on B, 5 on B to cursor,
cursor to 1 on B, 6 on B to cursor, and under 1 on A read the area =
5·13 sq. ft. on B.

To find the number of degrees in a gradient, given the rise per cent.

Place the slide with the indices of T coincident with those of D, and over the rate per cent. on D read number of degrees in the slope on T.

As the arrangement of rule we have chiefly considered has only a single T scale, it will be seen that only solutions of the above problem involving slopes between 10 and 100 per cent. can be directly read off. For smaller angles, one of the formulæ for the determination of the tangents of submultiple angles must be used.

In rules having a double T scale (which is used with the A scale) the value in degrees of any slope from 1 to 100 per cent. can be directly read off on A.

To find the number of degrees, when the gradient is expressed as 1 in _x_.

Place the index of T to _x_ on D, and over index of D read the required angle in degrees on T.

EX.—Find the number of degrees in a gradient of 1 in 3·8.

Set 1 on T to 3·8 on D, and over R.H. index of D read 14° 45′ on T.

Given the lap, the lead and the travel of an engine slide valve, to find the angle of advance.

Set (lap + lead) on B to half the travel of the valve on A, and read the angle of advance on S at the index mark on the back of the rule.

EX.—Valve travel 4½in., lap 1 in., lead ⁵⁄₁₆in. Find angle of advance.

Set 1⁵⁄₁₆ = 1·312 on B to 2·25 on A, and read 35° 40′ on S opposite
the index on the back of the rule.

Given the angular advance θ, the lap and the travel of a slide valve, to find the cut-off in percentage of the stroke.

Place the lap on B to half the travel of valve on A, and read on S the angle (the supplement of the _angle of the eccentric_) found opposite the index in the back of the rule. To this angle, add the angle of advance and deduct the sum from 180°, thus obtaining the _angle of the crank_ at the point of cut-off. To the cosine of the supplement of this angle, add 1 and multiply the result by 50, obtaining the percentage of stroke completed when cut-off occurs.

EX.—Given the angular advance = 35° 40′, the valve travel = 4½in., and
the lap = 1 in., find the angle of the crank at cut-off and the
admission period expressed as a percentage of the stroke.

Set 1 on B to 2·25 on A, and read off on S opposite the index, the
supplement of the angle of the eccentric = 26° 20′. Then 180° − (35°
40′ + 26° 20′) = 118° = the crank angle at the point of cut-off.
Further, cos. 118° = cos. 62° = sin (90° − 62°) = sin 28°, and placing
28° on S to the back index, the cosine, read on B under R.H. index of
A, is found to be 0·469. Adding 1 and placing the L.H. index of C to
the result, 1·469, on D, we read off under 50 on C, the required
period of admission = 73·4 per cent. on D.

The trigonometrical scales are useful for evaluating certain formulæ. Thus in the following expressions, if we find the angle _a_ such that sin. _a_ = _k_, we can write:—

(_k_)/(√1 − _k^2_) = tan. _a_; (√1 − _k^2_)/(_k_) = cot. _a_; √(1 −
_k^2_) = cos. _a_; etc.

In the first expression, take _k_ = 0·298. Place the slide with the sine scale outward and with its indices agreeing with the indices of the rule. Set the cursor to 0·298 on the (R.H.) scale of A, and read 17° 20′ on the sine scale as the angle required. Then under 17° 20′ on the tangent scale, read 0·312 on D as the result.

SLIDE RULES WITH LOG.-LOG. SCALES.

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.

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.

_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.

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.

EX.—3·45^{−1·82} = 0·105.

Set 1 on C to 3·45 on E, and under 1·82 on C read 9·51 on C. Then set
1 on B to 9·5 on A, and under index of A read 0·105 on B.

When _x_ is less than 1 the second method is more suitable.

EX.—0·23^{−1·77} = ((1)/(0·23))^{1·77} = 4·35^{1·77} = 13·5

Set 1 on B to 0·23 on A, and under index of A read (1)/(0·23) = 4·35
on B.

Set 1 on C to 4·35 on E, and under 1·77 on C read 13·5 on E.

As with the Davis rule, the exponent scale C will be read as ⅒th its face value if its R.H. index (10) is used in place of 1.

SPECIAL TYPES OF SLIDE RULES.

In addition, to the new forms of log.-log. slide rules previously described, several other arrangements have been recently introduced, notably a series by Mr. A. Nestler, of Lahr (London: A. Fastlinger, Snow Hill). These comprise the “Rietz,” the “Precision,” the “Universal,” and the “Fix” slide rules.

THE RIETZ RULE.—In this rule the usual scales A, B, C, and D, are provided, while at the upper edge is a scale, which, being three times the range of the D scale, enables cubes and cube roots to be directly evaluated and also _n_^{³⁄₂} and _n_^⅔.

A scale at the lower edge of the rule gives the mantissa of the logarithms of the numbers on D.

THE PRECISION SLIDE RULE.—In this rule the scales are so arranged that the accuracy of a 20 in. rule is obtainable in a length of 10 in. This is effected by dividing a 20 in. (50 cm.) scale length into two parts and placing these on the working edges of the rule and slide. On the upper and lower margins of the face of the rule are the two parts of what corresponds to the A scale in the ordinary rule; while in the centre of the slide is the scale of logarithms which, used in conjunction with the 50 cm. scales on the slide, is virtually twice the length of that ordinarily obtainable in a 10 in. rule. The same remark applies to the trigonometrical scales on the under face of the slide. Both the sine and tangent scales are in two adjacent lengths, while on the edge of the stock of the rule, below the cursor groove, is a scale of sines of small angles from 1° 49′ to 5° 44′. This is referred to the 50 cm. scales by an index projection on the cursor.

If C and C′ are the two parts of the scale on the slide and D and D′ the corresponding scales on the rule, it is clear that in multiplying two factors 1 on C can only be set directly to the upper scale D; while 10 on C′ can only be set directly to the lower scale D′. Hence if the first factor is greater than about 3·2, the cursor must be used to bring 1 on C to the first factor on D′. Similarly, in division, numerators and denominators which occur on C and D′ or on C′ and D cannot be placed in direct coincidence but must be set by the aid of the cursor.

Any uncertainty in reading the result can be avoided by observing the following rule: _If in setting the index_ (1 _or_ 10) _in multiplication, or in setting the numerator to the denominator in division, it is necessary to cross the slide, then it will also be necessary to cross the slide to read the product or quotient._

THE UNIVERSAL SLIDE RULE.—In this instrument the stock carries two similar scales running from 1 to 10, to which the slide can be set. Above the upper one is the logarithm scale and under the lower one the scale of squares 1 to 100. On the edge of the stock of the rule, under the cursor groove, is a scale running from 1 to 1000. An index projecting from the cursor enables this scale to be used with the scales on the face of the rule, giving cubes, cube roots, etc.

On the slide, the lower scale is an ordinary scale, 1 to 10. The centre scale is the first part of a scale giving the values of sin _n_ cos _n_, this scale being continued along the upper edge of the slide (marked “sin-cos”) up to the graduation 50. On the remainder of this line is a scale running from right to left (0 to 50) and giving the value of cos^2_n_. In surveying, these scales greatly facilitate the calculations for the horizontal distance between the observer’s station and any point, and the difference in height of these two points.

On the back of the slide are scales for the sines and tangents of angles. The values of the sines and tangents of angles from 34′ to 5° 44′ differ little from one another, and the one centre scale suffices for both functions of these small angles.

THE FIX SLIDE RULE.—This is a standard rule in all respects, except that the A scale is displaced by a distance (π)/(4) so that over 1 on D is found 0·7854 on A. This enables calculations relating to the area and cubic contents of cylinders to be determined very readily.

THE BEGHIN SLIDE RULE.—We have seen that a disadvantage attending the use of the ordinary C and D scales, is that it is occasionally necessary to traverse the slide through its own length in order to change the indices or to bring other parts of the slide into a readable position with regard to the stock. To obviate this disadvantage, Tserepachinsky devised an ingenious arrangement which has since been used in various rules, notably in the Beghin slide rule made by Messrs. Tavernier-Gravêt of Paris. In this rule the C and D scales are used as in the standard rule, but in place of the A and B scales, we have another pair of C and D scales, displaced by one-half the length of the rule. The lower pair of scales may therefore be regarded as running from 10^{_n_} to 10^{_n_ + 1}, and the upper pair as running from √(10) × 10^{_n_} to √(10) × 10^{_n_ + 1}. With this arrangement, _without moving the slide more than half its length_, to the left or right, it is always possible to compare _all values between_ 1 _and_ 10 _on the two scales_. This is a great advantage especially in continuous working.

Another commendable feature of the Beghin rule is the presence of a reversed C scale in the centre of the slide, thus enabling such calculations as _a_ × _b_ × _c_ to be made with one setting of the slide. On the back of the slide are three scales, the lowest of which, used with the D scale, is a scale of squares (corresponding to the ordinary B scale), while on the upper edge is a scale of sines from 5° 44′ to 90°, and in the centre, a scale of tangents from 5° 43′ to 45°. On the square edge of the stock, under the cursor groove, is the logarithm scale, while on the same edge, above the cursor groove, are a series of gauge points. All these values are referred to the face scales by index marks on the cursor.