Part 12

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.

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.

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.