D.

To calculate simple interest.

Set 1 on C to rate per cent. on D; bring cursor to period on C and 1 on C to cursor. Then opposite any sum on C find simple interest on D.

For interest per annum.

Set R.H. index on C to rate on D, and opposite principal on C read interest on D.

EX.—Find the amount with simple interest of £250 at 8 per cent., and for a period of 1 year and 9 months.

Set 1 on C to 8 on D; bring cursor to 1·75 on C, and 1 on C to cursor; then opposite 250 on C read £35, the interest, on D. Then 250 + 35 = £285 = the amount.

To calculate compound interest.

Set the L.H. index of C to the amount of £1 at the given rate of interest on D, and find the logarithm of this by reading on the reverse side of the rule, as explained on page 46. Multiply the logarithm, so found, by the period, and set the result, on the scale of equal parts, to the index on the under-side of the rule; then opposite any sum on C read the amount (including compound interest) on D.

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.

[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.

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.

THE ANDERSON SLIDE RULE.—The principle of dividing a long scale into sections as in the Precision rule, has been extended in the Anderson slide rule made by Messrs. Casella & Co., London, and shown in Fig. 17. In this the slide carries a scale in four sections, used in conjunction with an exactly similar set of scale-lines in the upper part of the stock. On the lower part of the stock is a scale in eight sections giving the square roots of the upper values. In order to set the index of the slide to values in the stock, two indices of transparent celluloid are fixed to the slide extending over the face of the rule as shown in the illustration. As each scale section is 30 cm. in length, the upper lines correspond to a single scale of nearly 4 ft., and the lower set to one of nearly 8 ft. in length, giving a correspondingly large increase in the number of subdivisions of these scales, and consequently much greater accuracy.

In order to decide upon which line a result is to be found, sets of “line numbers” are marked at each end of the rule and slide and also on the metal frame of the cursor. In multiplication, the line number of the product is the sum of the line numbers of the factors if the left index is used, or 1 more than this sum if the right index is used. The illustration shows the multiplication of 2 by 4. The left index is set to 2 (line number, 1), and the cursor set to 4 on the slide (line number, 2); hence, as the left index is used, the result is found on line No. 3. Similar rules are readily established for division. The column of line numbers headed 0 is used for units, that headed 4 for tens, and so on; one column is given for tenths, headed −4. The square root scale bears similar line numbers, so that the square root of any value on the upper scales is found on the correspondingly figured line below.

[Illustration: FIG. 17.]

THE MULTIPLEX SLIDE RULE differs from the ordinary form of rule in the arrangement of the B scale. The right-hand section of this scale runs from left to right as ordinarily arranged, but the left-hand section runs in the reverse direction, and so furnishes a reciprocal scale. At the bottom of the groove, under the slide, there is a scale running from 1 to 1000, which is used in conjunction with the D scale, readings being referred thereto by a metal index on the end of the slide. By this means cubes, cube roots, etc., can be read off directly. Messrs. Eugene Dietzgen & Co., New York, are the makers.

THE “LONG” SLIDE RULE has one scale in two sections along the upper and lower parts of the stock, as in the “Precision” rule. The scale on the slide is similarly divided, but the graduations run in the reverse direction, corresponding to an inverted slide. Hence the rules for multiplication and division are the reverse of those usually followed (page 30). On the back of the slide is a single scale 1–10, and a scale 1–1000, giving cubes of this single scale. By using the first in conjunction with the scales on the stock, squares may be read, while in conjunction with the cube scale, various expressions involving squares, cubes and their roots may be evaluated.

HALL’S NAUTICAL SLIDE RULE consists of two slides fitting in grooves in the stock, and provided with eight scales, two on each slide, and one on each edge of each groove. While fulfilling the purposes of an ordinary slide rule, it is of especial service to the practical navigator in connection with such problems as the “reduction of an ex-meridian sight” and the “correction of chronometer sights for error in latitude.” The rule, which has many other applications of a similar character, is made by Mr. J. H. Steward, Strand, London.

LONG-SCALE SLIDE RULES

It has been shown that the degree of accuracy attainable in slide-rule calculations depends upon the length of scale employed. Considerations of general convenience, however, render simple straight-scale rules of more than 20 in. in length inadmissible, so that inventors of long-scale slide rules, in order to obtain a high degree of precision, combined with convenience in operation, have been compelled to modify the arrangement of scales usually employed. The principal methods adopted may be classed under three varieties: (1) The use of a long scale in sectional lengths, as in Hannyngton’s Extended Slide Rule and Thacher’s Calculating Instrument; (2) the employment of a long scale laid in spiral form upon a disc, as in Fearnley’s Universal Calculator and Schuerman’s Calculating Instrument; and (3) the adoption of a long scale wound helically upon a cylinder, of which Fuller’s and the “R.H.S.” Calculating Rules are examples.

FULLER’S CALCULATING RULE.—This instrument, which is shown in Fig. 18, consists of a cylinder _d_ capable of being moved up and down and around the cylindrical stock _f_, which is held by the handle. The logarithmic scale-line is arranged in the form of a helix upon the surface of the cylinder _d_, and as it is equivalent to a straight scale of 500 inches, or 41 ft. 8 in., it is possible to obtain four, and frequently five, figures in a result.

Upon reference to the figure it will be seen that three indices are employed. Of these, that lettered _b_ is fixed to the handle; while two others, _c_ and _a_ (whose distance apart is equal to the axial length of the complete helix), are fixed to the innermost cylinder _g_. This latter cylinder slides telescopically in the stock _f_, enabling the indices to be placed in any required position relatively to _d_. Two other scales are provided, one (_m_) at the upper end of the cylinder _d_, and the other (_n_) on the movable index.

[Illustration: FIG. 18.]

In using the instrument a given number on _d_ is set to the fixed index _b_, and either _a_ or _c_ is brought to another number on the scale. This establishes a ratio, and if the cylinder is now moved so as to bring any number to _b_, the fourth term of the proportion will be found under _a_ or _c_. Of course, in multiplication, one factor is brought to _b_, and _a_ or _c_ brought to 100. The other factor is then brought to _a_ or _c_, and the result read off under _b_. Problems involving continuous multiplication, or combined multiplication and division, are very readily dealt with. Thus, calling the fixed index F, the upper movable index A, and the lower movable index B, we have for _a_ × _b_ × _c_:—Bring _a_ to F; A to 100; _b_ to A or B; A to 100; _c_ to A or B and read the product at F.

The maximum number of figures in a product is the sum of the number of figures in the factors and this results when all the factors except the first have to be brought to B. Each time a factor is brought to A, 1 is to be deducted from that sum.

For division, as _a_/(_m_ × _n_), bring _a_ to F; A or B to _m_; 100 to A; A or B to _a_; 100 to A and read the quotient at F.

[Illustration: FIG. 19.]

The maximum number of figures in the quotient is the difference between the sum of the number of figures in the numerator factors and those of the denominator factors, _plus_ 1 for each factor of the denominator and this results when A has to be set to all the factors of the denominator and all the factors of the numerator except the first brought to B. Each time B is set to a denominator factor or a numerator factor is brought to A, 1 is to be deducted.

Logarithms of numbers are obtained by using the scales _m_ and _n_ and hence powers and roots of any magnitude may be obtained by the procedure already fully explained. The instrument illustrated is made by Messrs. W. F. Stanley & Co., Limited, London.

THE “R.H.S.” CALCULATOR.—In this calculator, designed by Prof. R. H. Smith, the scale-line, which is 50 in. long, is also arranged in a spiral form (Fig. 19), but in this case it is wrapped around the central portion of a tube which is about ¾in. in diameter and 9½in. long. A slotted holder, capable of sliding upon the plain portions of this tube, is provided with four horns, these being formed at the ends of the two wide openings through which the scale is read. An outer ring carrying two horns completes the arrangement.

One of the horns of the holder being placed in agreement with the first factor, and one of the horns of the ring with the second factor, the holder is moved until the third factor falls under the same horn of the ring, when the resulting fourth term will be found under the same (right or left) horn of the holder, at either end of the slot. In multiplication, 100 or 1000 is taken for the second factor in the above proportion, as already explained in connection with Fuller’s rule; indeed, generally, the mode of operation is essentially similar to that followed with the former instrument.

The scale shown on one edge of the opening in the holder, together with the circular scale at the top of the spiral, enables the mantissæ of logarithms of numbers to be obtained, and thus problems involving powers and roots may be dealt with quite readily. This instrument is supplied by Mr. J. H. Steward, London.

THACHER’S CALCULATING INSTRUMENT, shown in Fig. 20, consists of a cylinder 4 in. in diameter and 18 in. long, which can be given both a rotary and a longitudinal movement within an open framework composed of twenty triangular bars. These bars are connected to rings at their ends, which can be rotated in standards fixed to the baseboard. The scale on the cylinder consists of forty sectional lengths, but of each scale-line that part which appears on the right-hand half of the cylinder is repeated on the left-hand half, one line in advance. Hence each half of the cylinder virtually contains two complete scales following round in regular order. On the lower lines of the triangular bars are scales exactly corresponding to those on the cylinder, while upon the upper lines of the bars and not in contact with the slide is a scale of square roots.

[Illustration: FIG. 20.]

By rotating the slide any line on it may be brought opposite any line in frame and by a longitudinal movement any graduation on these lines may be brought into agreement. The whole can be rotated in the supporting standards in order to bring any reading into view. As shown in the illustration, a magnifier is provided, this being conveniently mounted on a bar, along which it can be moved as required.

SECTIONAL LENGTH OR GRIDIRON SLIDE RULES.—The idea of breaking up a long scale into sectional lengths is due to Dr. J. D. Everett, who described such a gridiron type of slide rule in 1866. Hannyngton’s Extended Slide Rule is on the same principle. Both instruments have the lower scale repeated. H. Cherry (1880) appears to have been the first to show that such duplication could be avoided by providing two fixed index points in addition to the natural indices of the scale. These additional indices are shown at 10′ and 100′ in Fig. 21, which represents the lower sheet of Cherry’s Calculator on a reduced scale. The upper member of the calculator consists of a transparent sheet ruled with parallel lines, which coincide with the lines of the lower scale when the indices of both are placed in agreement. To multiply one number by another, one of the indices on the upper sheet is placed to one of the factors, and the position of whichever index falls under the transparent sheet is noted on the latter. Bringing the latter point to the other factor, the result is found under whichever index lies on the card. In other arrangements the inventor used transparent scales, the graduations running in a reverse direction to those of the lower scale. In this case, a factor on the upper scale is set to the other factor on the lower, and the result read at the available index.

[Illustration: FIG. 21.]

PROELL’S POCKET CALCULATOR is an application of the last-named principle. It comprises a lower card arranged as Fig. 21, with an upper sheet of transparent celluloid on which is a similar scale running in the reverse direction. For continued multiplication and division, a needle (supplied with the instrument) is used as a substitute for a cursor, to fix the position of the intermediate results. A series of index points on the lower card enable square and cube roots to be extracted very readily. This calculator is supplied by Messrs. John J. Griffin & Sons, Ltd., London.

CIRCULAR CALCULATORS.

Although the 10 in. slide rule is probably the most serviceable form of calculating instrument for general purposes, many prefer the more portable circular calculator, of which many varieties have been introduced during recent years. The advantages of this type are: It is more compact and conveniently carried in the waistcoat pocket. The scales are continuous, so that no traversing of the slide from 1 to 10 is required. The dial can be set quickly to any value; there is no trouble with tight or ill-fitting slides. The disadvantages of most forms are: Many problems involve more operations than a straight rule. The results being read under fingers or pointers, an error due to parallax is introduced, so that the results generally are not so accurate as with a straight rule. The inner scales are short, and therefore are read with less accuracy. Special scale circles are needed for cubes and cube roots. The slide cannot be reversed or inverted.

[Illustration: FIG. 22.]

[Illustration: FIG. 23.]

THE BOUCHER CALCULATOR.—This circular calculator resembles a stem-winding watch, being about 2 in. in diameter and ⁹⁄₁₆in. in thickness. The instrument has two dials, the back one being fixed, while the front one, Fig. 22 (showing the form made by Messrs. W. F. Stanley, London), turns upon the large centre arbor shown. This movement is effected by turning the milled head of the stem-winder. The small centre axis, which is turned by rotating the milled head at the side of the case, carries two fine needle pointers, one moving over each dial, and so fixed on the axis that one pointer always lies evenly over the other. A fine index or pointer fixed to the case in line with the axis of the winding stem, extends over the four scales of the movable dial as shown. Of these scales, the second from the outer is the ordinary logarithmic scale, which in this instrument corresponds to a straight scale of about 4¾in. in length. The two inner circles give the square roots of the numbers on the primary logarithmic scale, the smaller circle containing the square roots of values between 1 and 3·162 (= √(10)), while the other section corresponds to values between 3·162 and 10. The outer circle is a scale of logarithms of sines of angles, the corresponding sines of which can be read off on the ordinary scale.

On the fixed or back dial there are also four scales, these being arranged as in Fig. 23. The outer of these is a scale of equal parts, while the three inner scales are separate sections of a scale giving the cube roots of the numbers taken on the ordinary logarithmic scale and referred thereto by means of the pointers. In dividing this cube-root scale into sections, the same method is adopted as in the case of the square-root scale. Thus, the smallest circle contains the cube roots of numbers between 1 and 10, and is therefore graduated from 1 to 2·154; the second circle contains the cube roots of numbers between 10 and 100, being graduated from 2·154 to 4·657; while the third section, in which are found the cube roots of numbers between 100 and 1000, carries the graduations from 4·657 to 10.

What has been said in an earlier section regarding the notation of the slide rule may in general be taken to apply to the scales of the Boucher calculator. The manner of using the instrument is, however, not quite so evident, although from what follows it will be seen that the operative principle—that of variously combining lengths of a logarithmic scale—is essentially similar. In this case, however, it is seen that in place of the straight scale-lengths shown in Fig. 4, we require to add or subtract arc-lengths of the circular scales, while, further, it is evident that in the absence of a fixed scale (corresponding to the stock of the slide rule) these operations cannot be directly performed as in the ordinary form of instrument. However, by the aid of the fixed index and the movable pointer, we can effect the desired combination of the scale-lengths in the following manner. Assuming it is desired to multiply 2 by 3, the dial is turned in a backward direction until 2 on the ordinary scale lies under the fixed index, after which the movable pointer is set to 1 on the scale. As now set, it is clear that the arc-length 1–2 is spaced off between the fixed index and the movable pointer, and it now only remains to add to this definite arc-length a further length of 1–3. To do this we turn the dial still further backward until the arc 1–3 has passed under the movable pointer, when the result, 6, is read under the fixed index. A little consideration will show that any other scale length may be added to that included between the fixed and movable pointers, or, in other words, any number on the scale may be multiplied by 2 by bringing the number to the movable pointer and reading the result under the fixed index. The rule for multiplication is now evident.

_Rule for Multiplication._—_Set one factor to the fixed index and bring the pointer to 1 on the scale; set the other factor to the pointer and read the result under the fixed index._

With the explanation just given, the process of division needs little explanation. It is clear that to divide 6 by 3, an arc-length 1–3 is to be taken from a length 1–6. To this end we set 6 to the index (corresponding in effect to passing a length 1–6 to the left of that reference point) and set the pointer to the divisor 3. As now set, the arc 1–6 is included between 1 on the scale and the index, while the arc 1–3 is included between 1 on the scale and the pointer. Obviously if the dial is now turned forward until 1 on the scale agrees with the pointer, an arc 1–3 will have been deducted from the larger arc 1–6, and the remainder, representing the result of this operation, will be read under the index as 2.

_Rule for Division._—_Set the dividend to the fixed index, and the pointer to the divisor; turn the dial until 1 on the scale agrees with the pointer, and read the result under the fixed index._

The foregoing method being an inversion of the rule for multiplication, is easily remembered and is generally advised. Another plan is, however, preferable when a series of divisions are to be effected with a constant divisor—_i.e._, when _b_ in (_a_)/(_b_) = _x_ is constant. In this case 1 on the scale is set to the index and the pointer set to _b_; then if any value of a is brought to the pointer, the quotient _x_ will be found under the index.

_Combined Multiplication and Division_, as (_a_ × _b_ × _c_)/(_m_ × _n_) = _x_, can be readily performed, while cases of continued multiplication evidently come under the same category, since _a_ × _b_ × _c_ = (_a_ × _b_ × _c_)/(1 × 1) = _x_. Such cases as _a_/(_m_ × _n_ × _r_) = _x_ are regarded as (_a_ × 1 × 1 × 1)/(_m_ × _n_ × _r_) = _x_; while (_a_ × _b_ × _c_)/(_m_) = _x_ is similarly modified, taking the form (_a_ × _b_ × _c_)/(_m_ × 1) = _x_. In all cases the expression must be arranged so that there is _one more factor in the numerator_ than _in the denominator_, _1’s being introduced as often as required_. The simple operations of multiplication and division involve a similar disposition of factors, since from the rules given it is evident that _m_ × _n_ is actually regarded as (_m_ × _n_)/(1), while (_m_)/(_n_) becomes in effect (_m_ × 1)/(_n_). It is important to note the general applicability of this arrangement-rule, as it will be found of great assistance in solving more complicated expressions.

As with the ordinary form of slide rule, the factors in such an expression as (_a_ × _b_ × _c_)/(_m_ × _n_) = _x_ are taken in the order:—1st factor of numerator; 1st factor of denominator; 2nd factor of numerator; 2nd factor of denominator, and so on; the 1st factor as _a_ being set to the index, and the result _x_ being finally read at the same point of reference.

EX.—(39 × 14·2 × 6·3)/(1·37 × 19) = 134.

Commence by setting 39 to the index, and the pointer to 1·37; bring 14·2 to the pointer; pointer to 19; 6·3 to the pointer, and read the result 134 at the index.

It should be noted that after the first factor is set to the fixed index, the _pointer_ is set to each of the _dividing_ factors as they enter into the calculation, while the _dial_ is moved for each of the _multiplying_ factors. Thus the dial is first moved (setting the first factor to the index), then the pointer, then the dial, and so on.

_Number of Digits in the Result._—If rules are preferred to the plan of roughly estimating the result, the general rules given on pages 21 and 25 should be employed for simple cases of multiplication and division. For combined multiplication and division, modify the expression, if necessary, by introducing 1’s, as already explained, and subtract the sum of the denominator digits from the sum of numerator digits. Then proceed by the author’s rule, as follows:—

_Always turn dial to the_ LEFT; _i.e._, _against the hands of a watch_.

_Note dial movements only; ignore those of the pointer._

_Each time 1 on dial agrees with or passes fixed index_, ADD _1 to the above difference of digits_.

_Each time 1 on dial agrees with or passes pointer_, DEDUCT _1 from the above difference of digits_.

Treat continued multiplication in the same way, counting the 1’s used as denominator digits as one less than the number of multiplied factors.

EX.—(8·6 × 0·73 × 1·02)/(3·5 × 0·23) = 7·95 [7·95473+].

Set 8·6 to index and pointer to 3·5. Bring 0·73 to pointer (noting that 1 on the scale passes the index) and set pointer to 0·23. Set 1·02 to pointer (noting that 1 on the scale passes the pointer) and read under index 7·95. There are 1 + 0 + 1 = 2 numerator digits and 1 + 0 = 1 denominator digit; while 1 is to be added and 1 deducted as per rule. But as the latter cancel, the digits in the result will be 2 − 1 = 1.

When moving the dial to the left will cause 1 on the dial to pass _both_ index and pointer (thus cancelling), the dial may be turned back to make the setting.

It will be understood that when 1 is the _first_ numerator, and 1 on the dial is therefore set to the index, no digit addition will be made for this, as the actual operation of calculating has not been commenced.

In the Stanley-Boucher calculator (Fig. 23) a small centre scale is added, on which a finger indicates automatically the number of digits to be added or deducted; the method of calculating, however, differs from the foregoing. To avoid turning back to 0 at the commencement of each calculation, a circle is ground on the glass face, so that a pencil mark can be made thereon to show the position of the finger when commencing a calculation.

_To Find the Square of a Number._— Set the number, on one or other of the square root scales, to the index, and read the required square on the ordinary scale.

_To Find the Square Root of a Number._—Set the number to the index, and if there is an _odd_ number of digits in the number, read the root on the inner circle; if an even number, on the second circle.

_To Find the Cube of a Number._—Set 1 on the ordinary scale to the index, and the pointer (on the back dial) to the number on one of the three cube-root scales. Then under the pointer read the cube on the ordinary scale.

_To Find the Cube Root of a Number._—Set 1 to index, and pointer to number. Then read the cube root under the pointer on one of the three inner circles on the back dial. If the number has

1, 4, 7, 10 or −2, −5, etc., digits, use the inner circle. 2, 5, 8, 11 or −1, −4, etc., „ „ second circle. 3, 6, 9, 12 or −0, −3, etc., „ „ third circle.

_For Powers or Roots of Higher Denomination._—Set 1 to index, the pointer to the number on the ordinary scale, and read on the outer circle on the back dial the mantissa of the logarithm. Add the characteristic (see p. 46), multiply by the power or divide by the root, and set the pointer to the mantissa of the result on this outer circle. Under the pointer on the ordinary scale read the number, obtaining the number of figures from the characteristic.

_To Find the Sines of Angles._—Set 1 to index, pointer to the angle on the outer circle, and read under the pointer the _natural sine_ on the ordinary scale; also under the pointer on the outer circle of the back dial read the _logarithmic sine_.

THE HALDEN CALCULEX.—After the introduction of the Boucher calculator in 1876, circular instruments, such as the Charpentier calculator, were introduced, in which a disc turned within a fixed ring, so that scales on the faces of both could be set together and ratios established as on the slide rule. Cultriss’s Calculating Disc is another instrument on the same principle. The Halden Calculex, of which half-size illustrations are given in Figs. 24 and 25, represents a considerable improvement upon these early instruments. It consists of an outer metal ring carrying a fixed-scale ring, within which is a dial. On each side of this dial are flat milled heads, so that by holding these between the thumb and forefinger the dial can be set quickly and conveniently. The protecting glass discs, which are not fixed in the metal ring but are arranged to turn therein, carry fine cursor lines, and as these are on the side next to the scales a very close setting can be made quite free from the effects of parallax. This construction not only avoids the use of mechanism, with its risk of derangement, but reduces the bulk of the instrument very considerably, the thickness being about ¼in.

On the front face, Fig. 24, the fixed ring carries an outer evenly-divided scale, giving logarithms, and an ordinary scale, 1–10, which works in conjunction with a similar scale on the edge of the dial. The two inner circles give the square roots of values on the main scales as in the Boucher calculator. On the back face, Fig. 25, the ring bears an outer scale, giving sines of angles from 6° to 90° and an ordinary scale, 1–10, as on the front face. The scales on the dial are all reversed in direction (running from right to left), the outer one consisting of an ordinary (but inverse) scale, 1–10, while the three inner circles give the cube roots of values on this inverse scale. As the fine cursor lines extend over all the scales, a variety of calculations can be effected very readily and accurately.

[Illustration: FIG. 24.]

[Illustration: FIG. 25.]

SPERRY’S POCKET CALCULATOR, made by the Keuffel and Esser Company, New York (Fig. 26), has two rotating dials, each with its own pointer and fixed index. The S dial has an outer scale of equal parts, an ordinary logarithmic scale, and a square-root scale. The L dial has a single logarithmic scale arranged spirally, in three sections, giving a scale length of 12½in. The pointers are turned by the small milled head, which is concentric with the milled thumb-nut by which the two dials are rotated. The gearing is such that both the L dial and its pointer rotate three times as fast as the S dial and pointer. All the usual calculations can be made with the spiral scale, as with the Boucher calculator, and the result read off on one or other of the three scale-sections. Frequently the point at which to read the result is obvious, but otherwise a reference to the single scale on the S dial will show on which of the three spirals the result is to be found.

[Illustration: FIG. 26.]

_The K and E Calculator_, also made by the Keuffel and Esser Company, is shown in Figs. 27 and 28. It has two dials, of which only one revolves. This, as shown in Fig. 27, has an ordinary logarithmic scale and a scale of squares. There is an index line engraved on the glass of the instrument. The fixed dial has a scale of tangents, a scale of equal parts and a scale of sines, the latter being on a two-turn spiral. The pointers, which move together, are turned by a milled nut and the movable dial by a thumb-nut, as in Sperry’s Calculator, Fig. 26.

[Illustration: FIG. 27.]

[Illustration: FIG. 28.]

SLIDE RULES FOR SPECIAL CALCULATIONS.

ENGINE POWER COMPUTER.—A typical example of special slide rules is shown in Fig. 29, which represents, on a scale of about half full size, the author’s Power Computer for Steam, Gas, and Oil Engines. This, as will be seen, consists of a stock, on the lower portion of which is a scale of cylinder diameters, while the upper portion carries a scale of horse-powers. In the groove between these scales are two slides, also carrying scales, and capable of sliding in edge contact with the stock and with each other.

This instrument gives directly the brake horse-power of any steam, gas, or oil engine; the indicated horse-power, the dimensions of an engine to develop a given power, and the mechanical efficiency of an engine. The calculation of piston speed, velocity ratios of pulleys and gear wheels, the circumferential speed of pulleys, and the velocity of belts and ropes driven thereby, are among the other principal purposes for which the computer may be employed.

[Illustration: FIG. 29.]

THE SMITH-DAVIS PIECEWORK BALANCE CALCULATOR has two scales, 11 feet long, having a range from 1d. to £20, and marked so that they can be used either for money or time calculations. The scales are placed on the rims of two similar wheels and so arranged that the divided edges come together. The wheels are mounted on a spindle carried at each end in the bearings of a supporting stand. The wheels are pressed together by a spring, and move as one.

To set the scales one to the other, a treadle gear is arranged to take the pressure of the spring so that when the fixed wheel is held by the left hand the free wheel can be rotated by the right hand in either direction. When the amount of the balance has been set to the combined weekly wage the treadle is released locking the two wheels together, when the whole can be turned and the amounts respectively due to each man read off opposite his weekly wage. The Smith-Davis Premium Calculator is on the same principle but the scales are about 4 feet 6 inches long and the wheels spring-controlled. Both instruments are supplied by Messrs. John Davis & Son, Ltd., Derby.

THE BAINES SLIDE RULE.—In this rule, invented by Mr. H. M. Baines, Lahore, four slides carrying scales are arranged to move, each in edge contact with the next. The slides are kept in contact and given the desired relative movement one to the other, by being attached (at the back), to a jointed parallelogram. On this principle which is of general application, the inventor has made a rule for the solution of problems covered by Flamant’s formula for the flow of water in cast-iron pipes:—V = 76·28_d_^{⁵⁄₇}_s_^{⁴⁄₇}, in which _s_ is the sine of the inclination or loss of head; _d_ the diameter of the pipe in inches and V the velocity in feet per second. The formula Q = AV is also included in the scope of the rule, Q being the discharge in cubic feet per second and A the cross sectional area of the pipe in square inches.

FARMAR’S PROFIT-CALCULATING RULE.—The application of the slide rule to commercial calculations has been often attempted, but the degree of accuracy required necessitates the use of a long scale, and generally this results in a cumbersome instrument. In Farmar’s Profit-calculating Rule the money scale is arranged in ten sections, these being mounted in parallel form on a roller which takes the place of the upper scale of an ordinary rule. The roller, which is ¾in. in diameter, is carried in brackets secured to each end of the stock, so that by rotating the roller any section of the money scale can be brought into reading with the scale on the upper edge of the slide and with which the roller is in contact. This scale gives percentages, and enables calculations to be made showing profit on turnover, profit on cost, and discount. The lower scale on the slide, and that on the stock adjacent to it, are similar to the A and B scales of an ordinary rule. The instrument is supplied by Messrs. J. Casartelli & Son, Manchester.

CONSTRUCTIONAL IMPROVEMENTS IN SLIDE RULES.

The attention of instrument makers is now being given to the devising of means for ensuring the smooth and even working of the slide in the stock of the rule. In some cases very good results are obtained by slitting the back of the stock to give more elasticity.

In the rules made by Messrs. John Davis & Son, a metal strip, slightly curved in cross section as shown at A (Fig. 30), runs for the full length of the stock to which it is fastened at intervals. Near each end of the rule, openings about 1 in. long are made in the metal backing through which the scales on the back of the slide can be read. To prevent warping under varying climatic conditions both the stock of the rule and the slide are of composite construction. The base of the stock is of mahogany, while the grooved sides, firmly secured to the base, are of boxwood. Similarly the centre portion of the slide is of mahogany and the tongued sides of boxwood. Celluloid also enters into the construction, a strip of this material being laid along the bottom of the groove in the stock. A fine groove runs along the centre of this strip in order to give elasticity and to allow the sides of the stock to be pressed together slightly to adjust the fitting of the slide. As a further means of adjustment the makers fit metal clips at each end of the rule, so that by tightening two small screws the stock can be closed on the slide when necessary.

[Illustration: FIG. 30.]

[Illustration: FIG. 32.]

[Illustration: FIG. 31.]

In the rule made by the Keuffel and Esser Company of New York, one strip is made adjustable (Fig. 32).

THE ACCURACY OF SLIDE RULE RESULTS.

The degree of accuracy obtainable with the slide rule depends primarily upon the length of the scale employed, but the accuracy of the graduations, the eyesight of the operator, and, in particular, his ability to estimate interpolated values, are all factors which affect the result. Using the lower scales and working carefully the error should not greatly exceed 0·15 per cent. with short calculations. With successive settings, the discrepancy need not necessarily be greater, as the errors may be neutralised; but with rapid working the percentage error may be doubled. However, much depends upon the graduation of the scales. Rules in which one or more of the indices have been thickened to conceal some slight inaccuracy should be avoided. The line on the cursor should be sharp and fine and both slide and cursor should move smoothly or good work cannot be done. Occasionally a little vaseline or clean tallow should be applied to the edges of the slide and cursor.

That the percentage error is constant throughout the scale is seen by setting 1 on C to 1·01 on D, when under 2 is 2·02; under 3, 3·03; under 5, 5·05, etc., the several readings showing a uniform error of 1 per cent.

A method of obtaining a closer reading of a first setting or of a result on D has been suggested to the author by Mr. M. Ainslie, B.Sc. If any graduation, as 4 on C, is set to 3 on D, it is seen that 4 main divisions on C (40–44) are equal in scale length to 3 main divisions on D (30–33). Hence, very approximately, 1 division on C is equal to 0·75 of a division on D, this ratio being shown, of course, on D under 10 on C. Suppose √(4·3) to be required. Setting the cursor to 4·3 on A, it is seen that the root is something more than 2·06. Move the slide until a main division is found on C, which exactly corresponds to the interval between 2 and the cursor line, on D. The division 27–28 just fits, giving a reading under 10 on C, of 74. Hence the root is read as 2·074. For the higher parts of the scale, the subdivisions, 1–1·1, etc., are used in place of main divisions. The method is probably more interesting than useful, since in most operations the inaccuracies introduced in making settings will impose a limit on the reliable figures of the result.

For the majority of engineering calculations, the slide rule will give an accuracy consistent with the accuracy of the data usually available. For some purposes, however, _logarithmic section paper_ (the use of which the author has advocated for the last twenty years) will be found especially useful, more particularly in calculations involving exponential formulæ.

APPENDIX.

NEW SLIDE RULES—FIFTH ROOTS, ETC.—THE SOLUTION OF ALGEBRAIC EQUATIONS—GAUGE POINTS AND SIGNS ON SLIDE RULES—TABLES AND DATA—SLIDE RULE DATA SLIPS.

THE PICKWORTH SLIDE RULE.—In this rule, made by Mr. A. W. Faber, the novel feature is the provision of a scale of cubes (F) in the stock or body of the rule. From Fig. 33 it will be seen that the scale is fixed on the bevelled side of a slotted recess in the back of the rule. The slide carries an index mark, which is seen through the slot and can be set to any graduation of the scale; in its normal position it agrees with 1 on the scale. The C scale on the face of the rule is divided into three equal parts by two special division lines, marked II. and III., which, together with the initial graduation 1 of the scale, serve for setting or reading off values on the D scale. Similar division lines are marked on the D scale.

[Illustration: FIG. 33.]

In using the rule for cubes or cube roots the slide is drawn to the right, this movement never exceeding one-third of the length of the D scale. With this limited movement, and with a single setting of the slide, the values of ∛_̅a_, ∛(_a_ × 10), and ∛(_a_ × 100)) (_a_ being less than 10 and not less than 1) are given simultaneously and without any uncertainty as to the scales to use or the values to be read off.

_To Find the Cube of a Number._—The marks II. and III. on D divide that scale into three equal sections. If the number to be cubed is in the first section, I. on C is set to it; if in the second section, II. on C is set to it; if in the third section, III. on C is set to it. Then, under the index mark on the back of the slide will be found the significant figures of the cube on the scale F. If I. on C was used for the setting, the cube contains 1 digit; if II. was used, 2 digits; if III. was used, 3 digits. If the first figure of the number to be cubed is not in the units place, the decimal point is moved through _n_ places so as to bring the first significant figure into the units place, the cube found as above, and the decimal point moved in the _reverse direction_ through 3_n_ places.

_To Find the Cube Root of a Number._—The index mark is set to the significant figures of the number on scale F, and the cube root is read on D under I., II. or III. on C, according as the number has 1, 2 or 3 digits preceding the decimal point. Numbers which have 1, 2 or 3 figures preceding the decimal point are dealt with directly. Numbers of any other form are brought to one of the above forms by moving the decimal point 3 places (or such multiple of 3 places as may be required), the root found and its decimal point moved 1 place for each 3-place movement, but in the _reverse direction_.

THE “ELECTRO” SLIDE RULE.—In this special rule for electrical calculations, made by Mr. A. Nestler, the upper scales run from 0·1 to 1000, and are marked “Amp.” and “sq. mm.” respectively. The lower scale on the slide running from 1 to 10,000 is marked M (metres), while the lower scale on the rule (0·1 to 100) is marked “Volt.” The latter scale is so displaced that 10 on M agrees with 0·173 on the Volt scale. The four factors involved are the current strength (in Amp.); the area of a conductor (in sq. mm.); the length of the conductor (in metres); and the permissible loss of potential (in volts). Having given any three of these, the fourth can be found very readily. On the back of the slide are a scale of squares, a scale of cubes and a single scale corresponding to the D scale of an ordinary rule. Hence, by reversing the slide, it is possible to obtain the 2nd, 3rd and 4th powers and roots of numbers. In another form of the rule, the scale of metres is replaced by one of yards, while instead of the area of the conductor in sq. mm., the corresponding “gauge” sizes of wires are given.

THE “POLYPHASE” SLIDE RULE.—This instrument, made by the Keuffel & Esser Company, New York, has, in addition to the usual scales, a scale of cubes on the vertical edge of the stock of the rule, while in the centre of the slide there is a reversed C scale; _i.e._, a scale exactly similar to an ordinary C scale but with the graduations running from right to left. The rule is specially useful for the solution of problems containing combinations of three factors and problems involving squares, square roots, cubes, cube roots and many of the higher powers and roots. It is specially adapted for electrical and hydraulic work.

THE LOG-LOG DUPLEX SLIDE RULE.—The same makers have introduced a log-log duplex slide rule, in which the log-log scale is in three sections, placed one above the other, these occupying the position usually taken up by the A scale. These scales are used in the manner already described (page 86), but some advantage is obtained by the manner in which the complete log-log scale is divided, the limits being _e_^{¹⁄₁₀₀} to _e_^⅒ (on Scale L.L. 1); _e_^⅒ to _e_ (on Scale L.L. 2); and _e_ to _e_^{10} (on Scale L.L. 3), _e_ being the base of natural or hyperbolic logarithms (2·71828). In this way a total log-log range of from 1·01 to 22,000 is provided, meeting all practical requirements. These log-log scales are read in conjunction with a C scale placed at the upper edge of the slide. A similar C scale, but reversed in direction, is placed at the lower edge of the slide, this having red figures to distinguish it readily. The adjacent scale on the body of the rule is an ordinary D scale, and under this is an equally-divided scale giving the common logarithms of values on D. In the centre of the slide is a scale of tangents.

It will be understood that a “duplex” rule consists of two side strips securely clamped together at the two ends, forming the body of the rule, the slide moving between them; hence both front and back faces of the rule and slide are available, graduations on the one side being referred to those on the other by the cursor which extends around the whole. In this instrument, the scales on the back face are the ordinary scales of the standard rule with the addition of a scale of sines which is placed in the centre of the slide. It will be evident that this instrument is capable of dealing with a very wide range of problems involving exponential and trigonometrical formulæ.

SMALL SLIDE RULES WITH MAGNIFYING CURSORS.—Several makers now supply 5 in. rules having the full graduations of a 10 in. rule, and fitted with a magnifying cursor (Fig. 34). This forms a compact instrument for the pocket, but owing to the closeness of the graduations it is not usually possible to make a setting of the slide without using the cursor. This, of course, involves more movements than with the ordinary instrument. It is also very necessary to use the magnifying cursor in a _direct_ light, if accurate readings are to be obtained. If these slight inconveniences are to be tolerated, the principle could be extended, a 10 in. rule being marked as fully as a 20 in., and fitted with a magnifying cursor. The author has endeavoured, but without success, to induce makers to introduce such a rule.

The magnifying cursor, supplied by Messrs. A. G. Thornton, Limited, has a lens which fills the entire cursor. It has a powerful magnifying effect, and the change from the natural to the magnified reading is less abrupt than with the semicircular lens.

[Illustration: FIG. 34.]

THE CHEMIST’S SLIDE RULE.—A slide rule, specially adapted for chemical calculations, has been introduced recently by Mr. A. Nestler. In this instrument the C and D scales are as usually arranged; but, in place of the A and B scales, there are a number of gauge points or marks denoting the atomic and molecular weights of the most important elements and combinations. The scales on the back of the slide are similarly arranged, so that by reversing the slide the operations can be extended very considerably. The rule finds its chief use in the calculation of analyses. Thus, to find the percentage of chlorine if _s_ grammes of a substance have been used and the precipitate of Ag.Cl. weighs _a_ grammes, we have the equation, _x_ = (Cl.)/(Ag.Cl.) × (_a_)/(_s_). Hence, the mark Ag.Cl. on the upper scale of the slide is set to the mark Cl. on the upper scale of the rule, when under _a_ on the C scale is found the quantity of chlorine on D. By setting the cursor to this value and bringing _s_ on C to the cursor, the percentage required can be read on C over 10 on D.

The rule is also adapted to the solution of various other chemical and electro-chemical calculations.

THE STELFOX SLIDE RULE.—This rule, shown in Fig. 35, has a stock 5 in. long, fitted with a 10 in. slide jointed in the middle of its length by means of long dowels. By separating the parts the compactness of a 5 in. rule is obtained. The upper scales on the rule and slide resemble the usual A and B scales. The D scale on the lower part of the stock is in two sections, the second portion being placed below the first, as shown in the illustration. The centre scale on the slide corresponds to the usual C scale, while on the lower edge of the slide is a similar scale, but with the index (1) in the middle of its length. The arrangement avoids the necessity of resetting the slide, as is sometimes necessary with the ordinary rule, and in general it combines the accuracy of a 10 in. rule with the compactness of a 5 in. rule; but a more frequent use of the cursor is necessary. This rule is made by Messrs. John Davis & Son, Limited, Derby.

[Illustration: FIG. 35.]

ELECTRICAL SLIDE RULE.—Another rule by the same makers, specially useful for electrical engineers, has the usual scales on the working edges of the rule and slide, while in the middle of the slide is placed a scale of cubes. A log-log scale in two sections is provided; the power portion, running from 1·07 to 2, is found on the lower part of the stock, and the upper portion, running from 2 to 10^3, on the upper part of the stock. The uppermost scale on the stock is in two parts, of which that to the left, running from 20 to 100 and marked “Dynamo,” gives the efficiencies of dynamos; that on the right, running from 20 to 100 and marked “Motor,” gives the efficiencies of electric motors. The lowest scale on the stock, marked “Volt,” gives the loss of potential in copper conductors. The ordinary upper scale on the stock is marked L (length of lead) at the left, and KW (kilowatts) at the right; the ordinary upper scale on the slide is marked A (ampères) and mm^2 (sectional area) at the left, and HP (horse-power) at the right. Additional lines on the cursor enable the electrical calculations to be made either in British or metric units.

THE PICOLET CIRCULAR SLIDE RULE.—A simple form of circular calculator, made by Mr. L. E. Picolet of Philadelphia, is shown in Fig. 36. It consists of a base disc of stout celluloid on which turns a smaller disc of thin celluloid. A cursor formed of transparent celluloid is folded over the discs, and is attached so that the friction between the cursor and the inner disc enables the latter to be turned by moving the former. By holding both discs the cursor can be adjusted as required. The adjacent scales run in opposite directions, so that multiplication and division are performed as with the inverted slide in an ordinary rule. The outer scale, which is two-thirds the length of the main scale, enables cube roots to be found. Square roots are readily determined and continuous multiplication and division conveniently effected. Modified forms of this neatly made little instrument are also available.

[Illustration: FIG. 36.]

OTHER RECENT SLIDE RULES.—Among other special types of slide rule, mention should be made of the _Jakin_ 10 in. rule for surveyors, made by Messrs. John Davis & Son, Limited, Derby. By the provision of a series of short subsidiary scales, the multiplication of a sine or tangent of an angle by a number can be obtained to an accuracy of 1 in 10,000. The _Davis-Lee-Bottomley_ slide rule, by the same makers, has special scales provided for circle spacing. The division of a circle into a number of equal parts, often required in spacing rivets, bolts, etc., and in setting out the teeth of gearwheels, is readily effected by the aid of this instrument. The _Cuntz_ slide rule is a very comprehensive instrument, having a stock about 2¼ in. wide, with the slide near the lower edge. Above the slide are eleven scales, referable to the main scales by the cursor. These scales enable squares and square roots, cubes and cube roots, and areas and circumferences of circles to be obtained by direct reading. A much more compact instrument could be obtained by removing one-half the scales to the back of the rule and using a double cursor.

[Illustration: FIG. 37.]

In one form of 10 in. rule, supplied by Mr. W. H. Harling, London, the body of the rule is made of well-seasoned cane, with the usual celluloid facings. The rule has a metal back, enabling the fit of the slide to be regulated. This backing extends the full length of the rule, openings about 1 in. long being provided at each end, enabling the scales on the back of the slide to be set with greater facility than is possible with the notched recesses usually adopted. The author has long endeavoured, but without success, to induce makers to fit windows of glass or celluloid in place of the notched recesses. This would allow the graduation of the S and T scales to be set more accurately, and enable both to be used at each end of the rule—an advantage in certain trigonometrical calculations. It would have the further advantage of permitting each alternate graduation of the evenly-divided or logarithm scale to be placed at opposite sides of one central line, enabling the reading to be made more accurately and conveniently.

Many special slide rules have lately been devised for determining the time necessary to perform various machine-tool operations and for analogous purposes, while attention has again been given to rules for calculating the weights of iron and steel bars, plates, etc.

THE DAVIS-STOKES FIELD GUNNERY SLIDE RULE.—This rule, which is adapted for calculations involved in “encounter” and “entrenched” field gunnery, is designed for the 18 pr. quick-firing gun. The upper and lower portions of the boxwood stock are united by a flexible centre of celluloid, thus providing grooves front and rear to receive boxwood slides. Each of the nineteen scales is marked with its name, and corresponding scales are coloured red or black. The front edge is bevelled and carries a scale of 1 in 20,000. The rule solves displacement problems, map angles of sight, changes of corrector and range corrections for changes in temperature, wind and barometer, etc. A special feature for displacement calculations is the provision of a 50 yd. sub-base angle scale, by which the apex angle is read at one setting.

THE DAVIS-MARTIN WIRELESS SLIDE RULE.—In wireless telegraphy it is frequently necessary to determine wave-length, capacity or self-induction when one or other of the factors of the equation, λ = 59·6√(LC) is unknown. The Davis-Martin wireless rule is designed to simplify such calculations. The upper scale in the stock (inductance) runs from 10,000 to 1,000,000; the adjacent scale on the slide (capacity) runs from 0·0001 to 0·01 but in the reverse direction. The lower scale on the stock (wave-length) runs from 100 to 1000, giving square roots of the upper scale; while on the lower edge of the scale are several arrows to suit the various denominations in which the wave-length and capacity may be expressed.

IMPROVED CURSORS.—In some slide-rule operations, notably in those involved in solving quadratic and cubic equations, it not infrequently happens that readings are obscured by the frame of the cursor. Frameless cursors have been introduced to obviate this defect. A piece of thick transparent celluloid is sometimes employed, but this is liable to become scratched in use. Fig. 37 shows a recent form of frameless glass cursor made by the Keuffel & Esser Company, Hoboken, N.J., which is satisfactory in every way.

Cursors having three hair lines are now fitted to some rules, the distance apart of the lines being equal to the interval 0·7854–1 on the A scale.

THE DAVIS-PLETTS SLIDE RULE.—In this rule a single log.-log. scale and its reciprocal scale are arranged opposite the ordinary upper log. scale. Thus, common logarithms can be read directly, while by taking advantage of the properties of characteristics and mantissas of common logarithms, the scale can be extended indefinitely. As 10 is the highest number on the log.-log. scale, it is carried down to within 0·025 of unity. The reading of log.-log. values above 10 is effected in a very simple manner. There is also a scale in the centre of the slide which, used in conjunction with the upper log. scale enables the natural logarithm of any number between 0·0001 and 10,000 to be read direct, while any number on the upper log. scale can be multiplied or divided by _e^x_ if the latter is between these limits. On the back of the slide are scales for all circular and hyperbolic functions, these being used in conjunction with the upper log. scales.

THE CROMPTON-GALLAGHER BOILER EFFICIENCY CALCULATOR has a stock in the thickness of which is a slot admitting a chart which can be moved at right angles to the two separate slides. On the bevelled edge of one slide, the graduations are continued so as to read against curves on the chart, through an opening in the stock.

THE DAVIS-GRINSTED COMPLEX CALCULATOR.—This slide rule is of considerable service in connection with calculations involving the conversion of complex quantities from the form _a_ + _j_ _b_ to the form R∠θ, and _vice versa_. The usual process of conversion necessitates repeated reference to trigonometrical tables, and is both tedious and time-taking. The Complex Calculator enables the conversions to be effected without reference to tables and with the minimum expenditure of time and labour.

The rule, which is about 16 in. long, has five scales. The upper one (A) is an ordinary logarithmic scale thrice-repeated. The adjacent scales on the slide comprise (1) a logarithmic scale of tangents (B) ranging from 0·1° to 45°, and (2) a logarithmic scale of secants (C) from 0° to 45°. The lower scales D and E are identical with the A scale, and are provided to enable multiplication, etc., to be performed without the need for a separate slide rule. Readings can be transferred from A to the lower scales by means of the cursor.

In using the rule to convert _a_ + _j_ _b_ to R∠θ, the index (45°) of the B scale is set to the larger component and the cursor to the smaller component, on scale A. Then θ (or its complement if _b_ is greater than _a_) is read on B under the cursor. The cursor is then set to θ on the C scale, and R is read on A under the cursor. The rule is made by Messrs. John Davis & Son, Limited, Derby.

THE SOLUTION OF ALGEBRAIC EQUATIONS.

The slide rule finds an interesting application in the solution of equations of the second and third degree; and although the process is essentially one of trial and error, it may often serve as an efficient substitute for the more laborious algebraic methods, particularly when the conditions of the problem or the operator’s knowledge of the theory of equations enables some idea to be obtained as to the character of the result sought. The principle may be thus briefly explained:—If 1 on C is set to _x_ on D (Fig. 38), we find _x_(_x_) = _x_^2 on D under _x_ on C. If, however, with the slide set as before, instead of reading under _x_, we read under _x_ + _m_ on C, the result on D will now be _x_(_x_ + _m_) = _x_^2 + _mx_ = _q_. Hence to solve the equation _x_^2 + _mx_ − _q_ = 0, we reverse the above process, and setting the cursor to _q_ on D, we move the slide until the number on C under the cursor, and that on D under 1 on C, _differ by m_. It is obvious from the setting that the _product_ of these numbers = _q_, and as their difference = _m_, they are seen to be the roots of the equation as required. For the equation _x_^2 − _mx_ + _q_ = 0, we require _m_ to equal the _sum_ of the roots. Hence, setting the cursor as before to _q_ on D, we move the slide until the number on C under the cursor, and that on D under 1 on C, are _together equal to_ _m_, these numbers being the roots sought. The alternative equations _x_^2 − _mx_ − _q_ = 0, and _x_^2 + _mx_ + _q_ = 0 are deducible from the others by changing the signs of the roots, and need not be further considered.

[Illustration: FIG. 38.]

EX.—Find the roots of _x_^2 − 8_x_ + 9 = 0.

Set the cursor to 9 on D, and move the slide to the right until when 6·64 is found under the cursor, 1·355 on D is under 1 on C. These numbers are the roots required.

The upper scales can of course be used; indeed, in general they are to be preferred.

EX.—Find the roots of _x_^2 + 12·8_x_ + 39·4 = 0.

Set the cursor to 39·4 on A, and move the slide to the right until we read 7·65 on B under the cursor, and 5·15 on A over 1 on B. The roots are therefore −7·65 and −5.15.

With a little consideration of the relative value of the upper and lower scales, the student interested will readily perceive how equations of the third degree may be similarly resolved. The subject is not of sufficient general importance to warrant a detailed examination being made of the several expressions which can be dealt with in the manner suggested; but the author gives the following example as affording some indication of the adaptability of the method to practical calculations.

EX.—A hollow copper ball, 7·5 in. in diameter and 2 lb. in weight, floats in water. To what depth will it sink?

The water displaced = 27·7 × 2 = 55·4 cub. in. The cubic contents of the immersed segment will be (π)/(3)(3_r_ _x_^2 − _x_^3), _r_ being the radius and _x_ the depth of immersion. Hence (π)/(3)(3_r_ _x_^2 − _x_^3) = 55·4, and 11·25_x_^2 − _x_^3 = 52·9.

To solve this equation we place the cursor to 52·9 on A, and move the slide until the reading on D under 1 and that on B under the cursor together amount to 11·25. In this way find 2·45 on D under 1, with 8·8 on B under the cursor _c_, _c_, as a pair of values of which the sum is 11·25. Hence we conclude that _x_ = 2·45 in. is the result sought.

With the rule thus set (Fig. 39) the student will note that the slide is displaced to the right by an amount which represents _x_ on D, and therefore _x_^2 on A; while the length on B from 1 to the cursor line represents 11·25 − _x_. Hence the upper scale setting gives _x_^2(11·25 − _x_) = 11·25_x_^2 − _x_^3 = 52·9 as required.

[Illustration: FIG. 39.]

When in doubt as to the method to be pursued in any given case, the student should work synthetically, building up a simple example of an analogous character to that under consideration, and so deducing the plan to be followed in the reverse process.

SCREW-CUTTING GEAR CALCULATIONS.

The slide rule has long found a useful application in connection with the gear calculations necessary in screw-cutting, helical gear-cutting, and spiral gear work.

SINGLE GEARS.—For simple cases of screw-cutting in the lathe it is only necessary to set the threads per inch to be cut to the threads per inch in the guide screw (or the pitch in inches in each case, if more convenient). Then any pair of coinciding values on the two scales will give possible pairs of wheels.

EX.—Find wheels to cut a screw of 1⅝ threads per inch with a guide screw of 2 threads per inch.

Setting 1·625 on C to 2 on D, it is seen that 80 (driver) and 65 (driven) are possible wheels.

COMPOUND GEARS.—When wheels so found are of inconvenient size, a compound train is used, consisting (usually) of two drivers and two driven wheels, the product of the two former and the product of the two latter being in the same ratio as the simple wheels. Thus with 60 and 40 as drivers, and 65 and 30 as driven, we have, (60 × 40)/(65 × 30) = (2400)/(1950) = (2)/(1·625) as before.

With the slide set as above, values convenient for splitting up into suitable wheels are readily obtainable. Thus, (1600)/(1300); (2400)/(1950); (4000)/(3250); (4800)/(3900) are a few suggestive values which may be readily factorised.

SLIDE RULES FOR SCREW-CUTTING CALCULATIONS.—Special circular and straight slide rules for screw-cutting gear calculations have long been employed. For compound gears these usually entail the use of six scales, two on each of the two slides and two on the stock. The upper scale on the stock may be a scale of threads per inch to be cut, the adjacent scale (on the upper slide) a scale of threads per inch in the guide screw. Setting the guide screw-graduation to the threads to be cut, the lower slide is adjusted until a convenient pair of drivers is found in coincidence on the central pair of scales, while a pair of driven wheels are in coincidence on the two lower scales.

Some years ago, a slide rule was introduced by which compound gears could be obtained with a single slide. Assuming the set of wheels usually provided—20 to 120 teeth advancing by 5 teeth—the products of 20 × 25, 20 × 30, etc., up to 115 × 120 were calculated. These products were laid out along each of the two lower scales. The upper scales were a scale of threads per inch to be cut and a scale of the threads per inch of various guide screws. Setting the guide screw-graduation to the threads to be cut, any coinciding graduations on the lower scales gave the required pairs of drivers and driven wheels.

FRACTIONAL PITCH CALCULATIONS.—The author has long advocated the use of the slide rule for determining the wheels necessary for cutting fractional pitch threads, and it is gratifying to find its value in this connection is now being appreciated. For the best results a good 20 in. rule is desirable, but with care very close approximations can be found with an accurate 10 in. rule. In any case a magnifying cursor or a hand reading-glass is of great assistance.

EX.—Find wheels to cut a thread of 0·70909 in. pitch; guide screw, 2 threads per inch.

To 0·70909 on D, set 0·5 (guide screw pitch in inches) on C. To make this setting as accurately as possible, the method described on page 112 may be used. Set 10 on C to about 91 on D, and note that the interval 77–78 on C represents 0·91 of the interval 70–71 on D. Set the cursor to 78 on C and bring 5 to the cursor. The slide is then set so that 5 on C agrees with 7·091 on D.

Inspection of the two scales shows various coinciding factors in the ratio required. The most accurate is seen to be (55 on C)/(78 on D). These values may be split up into (55 × 50)/(65 × 60) to form a suitable compound train of gears.

GAUGE POINTS AND SIGNS ON SLIDE RULES.

Many slide rules have the sign (Prod.)/(−1) at the right-hand end of the D scale, while on the left is (Quot.)/(+1.) It is somewhat unfortunate that these signs refer to rules for determining the number of digits in products and quotients, which are used to a considerable extent on the Continent, and conflict with those used in this country. By the Continental method the number of digits in a product is equal to the sum of the digits in the two factors, if the result is obtained on the LEFT _of the first factor_; but if the result is found on the RIGHT of the first factor, it is equal to this sum − 1. The sign (Prod.)/(−1) the _right_-hand end of the D scale provides a visible reminder of this rule.

Similarly for division:—The number of digits in a quotient is equal to the number of the digits in the dividend, minus those in the divisor, if the quotient appears on the RIGHT _of the dividend_, and to this difference + 1, if the quotient appears on the LEFT of the dividend. The sign (Quot.)/(+1) at the _left_-hand end of the D scale provides a visible reminder of this rule.

The sign

+ⵏ– ⟵ⵏ⟶ –ⵏ+

found at both ends of the A scale is of general application but of questionable utility. It is assumed to represent a fraction, the vertical line indicating the position of the decimal point. If the number 455 is to be dealt with in a multiplication on the lower scales, we may suppose the decimal point moved two places to the left, giving 4·55, a value which can be actually found on the scale. If we use this value, then to the number of digits in this result, as many must be added as the number of places (two in this case) by which the decimal point was moved. If the point is moved to the right, the number of places must be subtracted. Similarly, in division, if the decimal point in the divisor is moved _n_ places to the left, then _n_ places must be subtracted at the end of the operation; while if the point is moved through _n_ places to the right, then _n_ places must be added. The sign referred to, which, of course, applies to all scales, completely indicates these processes and is submitted as a reminder of the procedure to be followed by those using the method described.

The signs π, _c_, _c′_, and M are explained in the Section on “Gauge Points,” p. 53.

On some rules additional signs are found on the D scale. One, locating the value (180 × 60)/(π) = 3437·74 and hence giving the number of minutes in a radian, is marked ρ′. Another, representing the value (180 × 60 × 60)/(π) = 206265, and hence giving the number of seconds in a radian is marked ρ″. A third point, marked ρ_{˶}, placed at the value (200 × 100 × 100)/(π) = 636620, is used when the newer graduation of the circle is employed.

These gauge points are useful when converting angles into circular measure, or _vice versa_, and also for determining the functions of small angles.

A gauge point is sometimes marked at 1146 on the A and B scales. This is known as the “Gunner’s Mark,” and is used in artillery calculations involving angles of less than 20°, when, for the purpose in view, the tangent and circular measure of the angle may be regarded as equal. For this constant, the angle is taken in minutes, the auxiliary base in feet, and the base in yards. The auxiliary base in feet on B is set to the angle in minutes on A when over 1146 on B is the base in yards on A. The value (1)/(1146) = (π × 3)/(180 × 60).

TABLES AND DATA.

MENSURATION FORMULAE.

Area of a parallelogram = base × height.

Area of rhombus = ½ product of the diagonals.

Area of a triangle = ½ base × perpendicular height.

Area of equilateral triangle = square of side × 0·433.

Area of trapezium = ½ sum of two parallel sides × height.

Area of any right-lined figure of four or more unequal sides is found by dividing it into triangles, finding area of each and adding together.

Area of regular polygon = (1) length of one side × number of sides × radius of inscribed circle; or (2) the sum of the triangular areas into which the figures may be divided.

Circumference of a circle = diameter × 3·1416.

Circumference of circle circumscribing a square = side × 4·443.

Circumference of circle = side of equal square × 3·545.

Length of arc of circle = radius × degrees in arc × 0·01745.

Area of a circle = square of diameter × 0·7854.

Area of sector of a circle = length of arc × ½ radius.

Area of segment of a circle = area of sector − area of triangle.

Side of square of area equal to a circle = diameter × 0·8862.

Diameter of circle equal in area to square = side of square × 1·1284.

Side of square inscribed in circle = diameter of circle × 0·707.

Diameter of circle circumscribing a square = side of square × 1·414.

Area of square = area of inscribed circle × 1·2732.

Area of circle circumscribing square = square of side × 1·5708.

Area of square = area of circumscribing circle × 0·6366.

Area of a parabola = base x ⅔ height.

Area of an ellipse = major axis × minor axis × 0·7854.

Surface of prism or cylinder = (area of two ends) + (length × perimeter).

Volume of prism or cylinder = area of base × height.

Surface of pyramid or cone = ½(slant height × perimeter of base) + area of base.

Volume of pyramid or cone = (⅓)(area of base × perpendicular height).

Surface of sphere = square of diameter × 3·1416.

Volume of sphere = cube of diameter × 0·5236.

Volume of hexagonal prism = square of side × 2·598 × height.

Volume of paraboloid = ½ volume of circumscribing cylinder.

Volume of ring (circular section) = mean diameter of ring × 2·47 × square of diameter of section.

SPECIFIC GRAVITY AND WEIGHT OF MATERIALS.

METALS. ─────────────────────┬───────────────┬───────────────┬─────────────── METAL. │ Specific │ Weight of 1 │ Weight of 1 │ Gravity. │Cub. Ft. (Lb.).│Cub. In. (Lb.). ─────────────────────┼───────────────┼───────────────┼─────────────── Aluminium, Cast │ 2·56│ 160│ 0·0927 Aluminium, Bronze │ 7·68│ 475│ 0·275 Antimony │ 6·71│ 418│ 0·242 Bismuth │ 9·90│ 617│ 0·357 Brass, Cast │ 8·10│ 505│ 0·293 „ Wire │ 8·548│ 533│ 0·309 Copper, Sheet │ 8·805│ 549│ 0·318 „ Wire │ 8·880│ 554│ 0·321 Gold │ 19·245│ 1200│ 0·695 Gun metal │ 8·56│ 534│ 0·310 Iron, Wrought (mean) │ 7·698│ 480│ 0·278 „ Cast (mean) │ 7·217│ 450│ 0·261 Lead, Milled Sheet │ 11·418│ 712│ 0·412 Manganese │ 8·012│ 499│ 0·289 Mercury │ 13·596│ 849│ 0·491 Nickel, Cast │ 8·28│ 516│ 0·300 Phosphor Bronze, Cast│ 8·60│ 536·8│ 0·310 Platinum │ 21·522│ 1342│ 0·778 Silver │ 10·505│ 655│ 0·380 Steel (mean) │ 7·852│ 489·6│ 0·283 Tin │ 7·409│ 462│ 0·268 Zinc, Sheet │ 7·20│ 449│ 0·260 „ Cast │ 6·86│ 428│ 0·248 ─────────────────────┴───────────────┴───────────────┴───────────────

MISCELLANEOUS SUBSTANCES. ────────────┬──────────┬────────── SUBSTANCE. │ Specific │Weight of │ Gravity. │1 Cub. In. │ │ (Lb.). ────────────┼──────────┼────────── Asbestos │ 2·1–2·80 │·076-·101 Brick │ 1·90 │ ·069 Cement │2·72–3·05 │·0984-·109 Clay │ 2·0 │ ·072 Coal │ 1·37 │ ·0495 Coke │ 0·5 │ ·0181 Concrete │ 2·0 │ ·072 Fire-brick │ 2·30 │ ·083 Granite │ 2·5–2·75 │·051-·100 Graphite │ 1·8–2·35 │·065-·085 Sand-stone │ 2·3 │ ·083 Slate │ 2·8 │ ·102 Wood— │ │ Beech │ 0·75 │ ·0271 Cork │ 0·24 │ ·0087 Elm │ 0·58 │ ·021 Fir │ 0·56 │ ·0203 Oak │ ·62-·85 │·025-·031 Pine │ 0·47 │ ·017 Teak │ 0·80 │ ·029 ────────────┴──────────┴──────────

ULTIMATE STRENGTH OE MATERIALS. ──────────────────┬────────────┬────────────┬────────────┬──────────── MATERIAL. │ Tension in │Compression │Shearing in │ Modulus of │lb. per sq. │ in lb. per │lb. per sq. │ Elasticity │ in. │ sq. in. │ in. │ in lb. per │ │ │ │ sq. in. ──────────────────┼────────────┼────────────┼────────────┼──────────── Cast Iron │ 11,000 to│ 50,000 to│ │ 14,000,000 │ 30,000│ 130,000│ │ to │ │ │ │ 23,000,000 „ aver.│ 16,000│ 95,000│ 11,000│ Wrought Iron │ 40,000 to│ │ │ 26,000,000 │ 70,000│ │ │ to │ │ │ │ 31,000,000 „ aver.│ 50,000│ 50,000│ 40,000│ Soft Steel │ 60,000 to│ │ │ 30,000,000 │ 100,000│ │ │ to │ │ │ │ 36,000,000 Soft Steel aver.│ 80,000│ 70,000│ 55,000│ Cast Steel aver.│ 120,000│ │ │ 15,000,000 │ │ │ │ to │ │ │ │ 17,000,000 Copper, Cast │ 19,000│ 58,000│ │ „ Wrought │ 34,000│ │ │ 16,000,000 Brass, Cast │ 18,000│ 10,500│ │ 9,170,000 Gun Metal │ 34,000│ │ │ 11,500,000 Phosphor Bronze │ 58,000│ │ 43,000│ 13,500,000 Wood, Ash │ 17,000│ 9,300│ 1,400│ „ Beech │ 16,000│ 8,500│ │ „ Pine │ 11,000│ 6,000│ 650│ 1,400,000 „ Oak │ 15,000│ 10,000│ 2,300│ 1,500,000 Leather │ 4,200│ │ │ 25,000 ──────────────────┴────────────┴────────────┴────────────┴────────────

POWERS, ROOTS, ETC., OF USEFUL FACTORS. _n_ │(1)/(_n_)│ _n_^2 │ _n_^3 │ √_̅n_ │ (1)/(√_̅n_) │ ∛_̅n_ │ (1)/(∛_̅n_) ────────────────┼─────────┼───────┼──────────┼──────┬┴───────────┬─┴────┬──┴───────── π = 3·142 │ 0·318│ 9·870│ 31·006│ 1·772│ 0·564│ 1·465│ 0·683 2π= 6·283 │ 0·159│ 39·478│ 248·050│ 2·507│ 0·399│ 1·845│ 0·542 (π)/(2) = 1·571 │ 0·637│ 2·467│ 3·878│ 1·253│ 0·798│ 1·162│ 0·860 (π)/(3) = 1·047 │ 0·955│ 1·097│ 1·148│ 1·023│ 0·977│ 1·016│ 0·985 (4)/(3)π = 4·189│ 0·239│ 17·546│ 73·496│ 2·047│ 0·489│ 1·612│ 0·622 (π)/(4) = 0·785 │ 1·274│ 0·617│ 0·484│ 0·886│ 1·128│ 0·923│ 1·084 (π)/(6) = 0·524 │ 1·910│ 0·274│ 0·144│ 0·724│ 1·382│ 0·806│ 1·241 π^2 = 9·870 │ 0·101│ 97·409│ 961·390│ 3·142│ 0·318│ 2·145│ 0·466 π^3 = 31·006 │ 0·032│961·390│29,809·910│ 5·568│ 1·796│ 3·142│ 0·318 (π)/(32) = 0·098│ 10·186│ 0·0095│ 0·001│ 0·313│ 3·192│ 0·461│ 2·168 _g_ = 32·2 │ 0·031│1036·84│ 33,386·24│ 5·674│ 0·176│ 3·181│ 0·314 2_g_ = 64·4 │ 0·015│4147·36│ 267,090│ 8·025│ 0·125│ 4·007│ 0·249 ────────────────┴─────────┴───────┴──────────┴──────┴────────────┴──────┴────────────

HYDRAULIC EQUIVALENTS.

1 foot head = 0·434 lb. per square inch. 1 lb. per square inch = 2·31 ft. head. 1 imperial gallon = 277·274 cubic inches. 1 imperial gallon = 0·16045 cubic foot. 1 imperial gallon = 10 lb. 1 cubic foot of water = 62·32 lb. = 6·232 imperial gallons. 1 cubic foot of sea water = 64·00 lb. 1 cubic inch of water = 0·03616 lb. 1 cubic inch of sea water = 0·037037 lb. 1 cylindrical foot of water = 48·96 lb. 1 cylindrical inch of water = 0·0284 lb. A column of water 12 in. long 1 in. square = 0·434 lb. A column of water 12 in. long 1 in. diameter = 0·340 lb. Capacity of a 12 in. cube = 6·232 gallons. Capacity of a 1 in. square 1 ft. long = 0·0434 gallon. Capacity of a 1 ft. diameter 1 ft. long = 4·896 gallons. Capacity of a cylinder 1 in. diameter 1 ft. long = 0·034 gallon. Capacity of a cylindrical inch = 0·002832 gallon. Capacity of a cubic inch = 0·003606 gallon. Capacity of a sphere 12 in. diameter = 3·263 gallons. Capacity of a sphere 1 in. diameter = 0·00188 gallon. 1 imperial gallon = 1·2 United States gallon. 1 imperial gallon = 4·543 litres of water. 1 United States gallon = 231·0 cubic inches. 1 United States gallon = 0·83 imperial gallon. 1 United States gallon = 3·8 litres of water. 1 cubic foot of water = 7·476 United States gallons. 1 cubic foot of water = 28·375 litres of water. 1 litre of water = 0·22 imperial gallon. 1 litre of water = 0·264 United States gallon. 1 litre of water = 61·0 cubic inches. 1 litre of water = 0·0353 cubic foot.

───────────────────────────────────────────────────────────────────────── EQUIVALENTS OF POUNDS AVOIRDUPOIS. ─┬───────┬────────────┬────────────────┬────────────────┬──────────────── │ 10 │ 100 │ 1000 │ 10,000 │ 100,000 ─┼───────┼────────────┼────────────────┼────────────────┼──────────────── │qr. lb.│cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb. 1│ 0 10│ 0 3 16│ 0 8 3 20│ 4 9 1 4│ 44 12 3 12 2│ 0 20│ 1 3 4│ 0 17 3 12│ 8 18 2 8│ 89 5 2 24 3│ 1 2│ 2 2 20│ 1 6 3 4│ 13 7 3 12│133 18 2 8 4│ 1 12│ 3 2 8│ 1 15 2 24│ 17 17 0 16│178 11 1 20 5│ 1 22│ 4 1 24│ 2 4 2 16│ 22 6 1 20│223 4 1 4 6│ 2 4│ 5 1 12│ 2 13 2 8│ 26 15 2 24│267 17 0 16 7│ 2 14│ 6 1 0│ 3 2 2 0│ 31 5 0 0│312 10 0 0 8│ 2 24│ 7 0 16│ 3 11 1 20│ 35 14 1 4│357 2 3 12 9│ 3 6│ 8 0 4│ 4 0 1 12│ 40 3 2 8│401 15 2 24 ─┴───────┴────────────┴────────────────┴────────────────┴────────────────

TRIGONOMETRICAL FUNCTIONS.

RIGHT-ANGLED TRIANGLES.

[Illustration: [Right-angled Triangle]]

Sin. A = (_a_)/(_b_) Sec. A = (_b_)/(_c_) Tan. A = (_a_)/(_c_)

Cos. A = (_c_)/(_b_) Cosec. A = (_b_)/(_a_) Cotan. A = (_c_)/(_a_)

Versin. A = (_b_ − _c_)/(_b_). Coversin. A = (_b_ − _a_)/(_b_).

───────┬─────────┬───────────────────────────────────────────────────── Given. │Required.│ Formulæ. ───────┼─────────┼───────────────────────────────────────────────────── _a_,_b_│ A,C,_c_ │Sin. A = (_a_)/(_b_) Cos. C = (_a_)/(_b_) _c_ = │ │ √((_b + a_)(_b − a_)) │ │ _a_,_c_│ A,C,_b_ │Tan. A = (_a_)/(_c_) Cotan. B = (_a_)/(_c_) _b_ = │ │ √(_a_^2 + _c_^2) │ │ A,_a_ │C,_c_,_b_│ C = 90° − A _c_ = _a_ × Cotan. A _b_ = │ │ (_a_)/(Sin. A) │ │ A,_b_ │C,_a_,_c_│C = 90° − A _a_ = _b_ × Sin. A _c_ = _b_ × Cos. │ │ A │ │ A,_c_ │C,_a_,_b_│ C = 90° − A _a_ = _c_ × Tan. A _b_ = │ │ (_c_)/(Cos. A) │ │ ───────┴─────────┴─────────────────────────────────────────────────────

OBLIQUE-ANGLED TRIANGLES.

_s_ = ½(_a + b + c_)

[Illustration: [Oblique-angled Triangle]]

───────────┬─────────┬───────────────────────────────────────────────── Given. │ │ Formulæ. ───────────┼─────────┼───────────────────────────────────────────────── A,B,C,_a_ │ Area= │(_a_^2 × Sin. B × Sin. C) ÷ 2 Sin. A A,_b_,_c_ │ „ │½(_c_ × _b_ × Sin. A) _a_,_b_,_c_│ „ │√(_s_(_s_ − _a_)(_s_ − _b_)(_s_ − _c_)) ───────────┼─────────┼───────────────────────────────────────────────── Given. │Required.│ Formulæ. ───────────┼─────────┼───────────────────────────────────────────────── A,C,_a_ │ _c_ │ _c_ = _a_(Sin. C)/(Sin. A) ───────────┼─────────┼───────────────────────────────────────────────── A,_a_,_c_ │ C │ Sin. C = (_c_ Sin. A)/(_a_) ───────────┼─────────┼───────────────────────────────────────────────── _a_,_c_,B │ A │ Tan. A = (_a_ Sin. B)/(_c_ − _a_ Cos. B) ───────────┼─────────┼───────────────────────────────────────────────── _a_,_b_,_c_│ A │Sin. ½A = √(((_s_ − _b_)(_s_ − _c_))/(_b_ × _c_)) „ │ „ │ Cos. ½A = √((_s_(_s_ − _a_))/(_b_ × _c_)); „ │ „ │ Tan. ½A = √(((_s_ − _b_)(_s_ − _c_))/(_s_(_s_ − │ │ _a_))) ───────────┴─────────┴─────────────────────────────────────────────────

COMPOUND ANGLES.

Sin. (A + B) = Sin. A Cos. B + Cos. A Sin. B. Sin. (A − B) = Sin. A Cos. B − Cos. A Sin. B. Cos. (A + B) = Cos. A Cos. B − Sin. A Sin. B. Cos. (A − B) = Cos. A Cos. B + Sin. A Sin. B.

Tan. (A + B) = (Tan. A + Tan. B)/(1 − Tan. A Tan. B).

Tan. (A − B) = (Tan. A − Tan. B)/(1 + Tan. A Tan. B).

SLIDE RULE DATA SLIPS, COMPILED BY C. N. PICKWORTH, WH.SC.

(_It is suggested that this page be removed by cutting through the above line, and selected portions of the Sectional Data Slips attached to the back of the Slide Rule._)

¹⁄₃₂ │0·03125 ¹⁄₁₆ │0·0625 ³⁄₃₂ │0·09375 ⅛ │0·125 ⁵⁄₃₂ │0·15625 ³⁄₁₆ │0·1875 ⁷⁄₃₂ │0·21875 ¼ │0·25 ⁹⁄₃₂ │0·28125 ⁵⁄₁₆ │0·3125 ¹¹⁄₃₂ │0·34375 ⅜ │0·375 ¹³⁄₃₂ │0·40625 ⁷⁄₁₆ │0·4375 ¹⁵⁄₃₂ │0·46875 ¹⁷⁄₃₂ │0·53125 ⁹⁄₁₆ │0·5625 ¹⁹⁄₃₂ │0·59375 ⅝ │0·625 ²¹⁄₃₂ │0·65625 ¹¹⁄₁₆ │0·6875 ²³⁄₃₂ │0·71875 ¾ │0·75 ²⁵⁄₃₂ │0·78125 ¹³⁄₁₆ │0·8125 ²⁷⁄₃₂ │0·84375 ⅞ │0·875 ²⁹⁄₃₂ │0·90625 ¹⁵⁄₁₆ │0·9375 ³¹⁄₃₂ │0·96875

Circ. of circle = 3·1416 _d_.

Area „ „ = 0·7854 _d_^2.

Sq. eq. area to cir., _s_ = 0·886 _d_.

Circle eq. to sq., _d_ = 1·128 _s_.

Sq. inscbd. in circ., _s_ = 0·707 _d_.

Circsb. circ. of sq., _d_ = 1·414 _s_.

Area of ellipse = 0.7854 _a_ × _b_.

Surface of sphere = 3·1416 _d_^2.

Volume „ „ = 0·5236 _d_^3.

„ „ cone = 0·2618 _d_^2 _h_.

Radian = (180°)/(π) = 57·29 deg.

Base of nat. or hyp. log. = e = 2·7183.

Nat. or hyp. log. = com. log. × 2·3026.

g (at London) 32·18 ft. per sec., per sec.

Abs. temp. = deg. F. + 461° = deg. C. + 274°.

C.° = (5)/(9)(F.° − 32°); F.° = (9)/(5)C.° + 32°.

Cal. pr.—Ther. units per lb.: Coal, 14,300;

petrol’m, 20,000; coal gas per cu. ft., 700.

Sp. heat:—Wt. iron, 0·1138; C.I., 0·1298;

copper, brass, 0·095; lead, 0·0314.

Inch = 25·4 mil’metres; mil’metre = 0·03937 in.

Foot = 0·3048 metres; metre = 3·2809 feet.

Yard = 0·91438 metre; metre = 1·0936 yards.

Mile = 1·6093 kilomtrs.; kilomtr. = 0·6213 mile.

Sq. in. = 6·4513 sq. cm.; sq. cm. = 0·155 sq. in.

Sq. ft. = 9·29 sq. decmtr.; sq. decmtr. = 0·1076 sq. ft.

Sq. yd. = 0·836 sq. metre; sq. metre = 1·196 sq. yds.

Sq. ml. = 258·9 hectares; hectare = 0·00386 sq. ml.

Cu. in. = 16·386 c. cm.; c. cm. = 0·06102 cu. in.

Cu. ft. = 0·0283 c. metre; c. metre = 35·316 cu. ft.

Grain = 0·0648 gramme; gram. = 15·43 grs.

Ounce = 28·35 grams.; „ = 0·03527 oz.

Pound = 0·4536 kilogm.; kilogm. = 2·204 lb.

Ton = 1·016 tonnes; tonne = 0·9842 ton.

Mile per hr. = 1·466 ft., or 44·7 cm., per sec.

Lb. per cu. in. = 0·0276 kilogram per cu. cm.

Kilogram per cu. cm. = 36·125 lb. per cu. in.

Lb. per cu. ft. = 16·019 kilogm. per cu. mtre.

Grain per gall. = 0·01426 gramme per litre.

Gramme per litre = 70·116 grains per gall.

Ultimate Strength│Lb. per Sq. in. „ │Tens’n.│Comp’n. ─────────────────┼───────┼─────── Wt. iron │ 50,000│ 50,000 Cast „ │ 16,000│ 95,000 Steel │ 80,000│ 70,000 Copper │ 21,000│ 50,000 Brass │ 18,000│ 10,500 Lead │ 2,500│ 7,000 Pine │ 11,000│ 6,000 Oak │ 15,000│ 10,000

Weight of Metals.│ Cub. In. │ Cub. Ft. │12 Cu. In. ─────────────────┼──────────┼──────────┼────────── Wt. iron │ 0·277│ 480│ 3·33 Cast „ │ 0·260│ 450│ 3·12 Steel │ 0·283│ 490│ 3·40 Copper │ 0·318│ 550│ 3·82 Brass │ 0·300│ 520│ 3·61 Zinc │ 0·248│ 430│ 2·98 Alumin’m │ 0.096│ 168│ 1·16 Lead │ 0.411│ 710│ 4·93

Lb. per sq. in. = 2·31 ft. water = 2·04 in. mercury = 0·0703 kilo. per sq. cm. Atmosphere = 14·7 lb. per sq. in. = 33·94 ft. water = 1·0335 „ „ Ft. hd. water = 0·433 lb. per sq. in. = 62·35 lb. per sq. ft. = 0·0304 „ „ Cub. ft. of water = 62·35 lb. = 0·0278 ton = 28·315 litres = 7·48 U.S. galls. Gall. (Imp.) = 277·27 cu. in. = 0·1604 cu. ft. = 10 lb. water = 4·544 litres. Litre = 1·76 pints = 0·22 gall. = 61 cu. in. = 0·0353 cu. ft. = 0·264 U.S. gall. Horse-power = 33,000 ft.-lb. per min. = 0·746 kilowatt = 42·4 heat units per min. Heat unit = 778 ft.-lb. = 1055 watt-sec. = 107·5 kilogrammetres = 0·252 calorie. Foot-pound = 0·00129 heat unit = 1·36 joules = 0·1383 kilogrammetres. Kilowatt = 1·34 H.P. = 44,240 ft.-lb. per min. = 3412 heat units per hour.

-----

Footnote 1:

It will be recognised that n is the characteristic of the logarithm of the original number.

Footnote 2:

The special case in which the numerator is 1, 10, or any power of 10 must be treated by the rule for reciprocals (page 27).

Footnote 3:

The possible need for traversing the slide, to change the indices, when using the C and D scales, is not considered as a setting.

Footnote 4:

The reader may be reminded that cross-multiplication of the factors in any such slide rule setting will give a constant product, _e.g._, 20 × 94·5 = 27 × 70.

Footnote 5:

In this case cross _dividing_ gives a constant quotient, _e.g._, 8 ÷ 3 = 4 ÷ 1·5. Since the upper scale is now a scale of reciprocals, the ratio is really

O ⅛ ¼ ─────────── D 1·5 3

Footnote 6:

These lines should not be brought to the working edge of the scale but should terminate in the horizontal line which forms the border of the finer graduations, their value being read into the calculation by means of the cursor (see page 55).

Footnote 7:

The same principle may be applied to the cursor.

Footnote 8:

Philosophical Transactions of the Royal Society, 1815.

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[Illustration: [Slide Rule]]

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[Illustration: [Slide Rule]]

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[Illustration: [Centre Screw Spring Bow Half Set]]

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[Illustration: [Halden Calculex]]

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TRANSCRIBER’S NOTES

Page Changed from Changed to

24 the right, so the number of the right, so the number of digits in the answer = 3 − 2 × 1 digits in the answer = 3 − 2 + 1 = 2 = 2

116 grammes, we have the equation, grammes, we have the equation, _x_ × (Cl.)/(Ag.Cl.) × _x_ = (Cl.)/(Ag.Cl.) × (_a_)/(_s_). Hence, the mark (_a_)/(_s_). Hence, the mark

● Typos fixed; non-standard spelling and dialect retained. ● Used numbers for footnotes, placing them all at the end of the last chapter. ● Enclosed italics font in _underscores_. ● Enclosed bold font in =equals=. ● The caret (^) serves as a superscript indicator, applicable to individual characters (like 2^d) and even entire phrases (like 1^{st}). ● Subscripts are shown using an underscore (_) with curly braces { }, as in H_{2}O.