Part 4

TABLE OF CONVERSION FACTORS. ─────────────────────────────────────────────────────────────── GEOMETRICAL EQUIVALENTS. ──────────────────────────┬──────────────────────────┬───────── SCALE C. │ SCALE D. │If C = 1, │ │ D = ──────────────────────────┼──────────────────────────┼───────── Diameter of circle │Circumference of circle │3·1416 „ „ │Side of inscribed square │0·707 „ „ │„ equal square │0·886 „ „ │„ „ equilateral │ │ triangle │1·346 Circum. of circle │„ inscribed square │0·225 „ „ │„ equal square │0·282 Side of square │Diagonal of square │1·414 Square inch │Circular inch │1·273 Area of circle │Area of inscribed square │0·636 ──────────────────────────┴──────────────────────────┴───────── MEASURES OF LENGTH. ──────────────────────────┬──────────────────────────┬───────── Inches │Millimetres │25·40 „ │Centimetres │2·54 8ths of an inch │Millimetres │3·175 16ths „ „ │„ │1·587 32nds „ „ │„ │0·794 64ths „ „ │„ │0·397 Feet │Metres │0·3048 Yards │„ │0·9144 Chains │„ │20·116 Miles │Kilometres │1·609 ──────────────────────────┴──────────────────────────┴───────── MEASURES OF AREA. ──────────────────────────┬──────────────────────────┬───────── Square inches │Square centimetres │6·46 Circular „ │„ „ │5·067 Square feet │„ metres │0·0929 „ yards │„ „ │0·836 „ miles │„ kilometres │2·59 „ „ │Hectares │259·00 Acres │„ │0·4046 ──────────────────────────┴──────────────────────────┴───────── MEASURES OF CAPACITY. ──────────────────────────┬──────────────────────────┬───────── Cubic inches │Cubic centimetres │16·38 „ „ │Imperial gallons │0·00360 „ „ │U.S. gallons │0·00432 „ „ │Litres │0·01638 Cubic feet │Cubic metres │0·0283 „ „ │Imperial gallons │6·23 „ „ │U.S. gallons │7·48 „ „ │Litres │28·37 „ yards │Cubic metres │0·764 Imperial gallons │Litres │4·54 „ „ │U.S. gallons │1·200 Bushels │Cubic metres │0·0363 „ │„ feet │1·283 ──────────────────────────┴──────────────────────────┴───────── MEASURES OF WEIGHT. ──────────────────────────┬──────────────────────────┬───────── Grains │Grammes │0·0648 Ounces (Troy) │„ │31·103 „ (Avoird.) │„ │28·35 „ „ │Kilogrammes │0·02835 Pounds (Troy) │„ │0·3732 „ (Avoird.) │„ │0·4536 Hundredweights │„ │50·802 Tons │„ │1016·4 „ │Metric tonnes │1·016 ──────────────────────────┴──────────────────────────┴───────── COMPOUND FACTORS—VELOCITIES. ──────────────────────────┬──────────────────────────┬───────── Feet per second │Metres per second │0·3048 „ „ │„ minute │18·288 „ „ │Miles per hour │0.682 „ minute │Meters per second │0·00508 „ „ │„ minute │0·3048 „ „ │Miles per hour │0·01136 Yards per „ │„ „ │0·0341 Miles per hour │Metres per minute │26·82 Knots │„ „ │30·88 „ │Miles per hour │1·151 ──────────────────────────┴──────────────────────────┴───────── COMPOUND FACTORS—PRESSURES. ──────────────────────────┬──────────────────────────┬───────── Pounds per sq. inch │Grammes per sq. mm. │0·7031 „ „ │Kilos. per sq. centimetre │0·0703 „ „ │Atmospheres │0·068 „ „ │Head of water in inches │27·71 „ „ │„ „ feet │2·309 „ „ │„ „ metres │0·757 „ „ │Inches of Mercury │2·04 Inches of water │Pounds per square inch │0·0361 „ „ │Inches of mercury │0·0714 „ „ │Pounds per square foot │5·20 Inches of mercury │Atmospheres │0·0333 Atmospheres │Metres of water │10·34 „ │Kilos. per sq. cm. │1·033 Feet of water │Pounds per square foot │62·35 „ „ │Atmospheres │0·0294 „ „ │Inches of mercury │0·883 Pounds per sq. foot │„ „ │0·01417 „ „ │Kilos. per square metre │4·883 „ „ │Atmospheres │0·000472 Pounds per sq. yard │Kilos. per square metre │0·5425 Tons per sq. inch │„ square mm. │1·575 „ sq. foot │Tonnes per square metre │10·936 ──────────────────────────┴──────────────────────────┴───────── COMPOUND FACTORS—WEIGHTS, CAPACITIES, ETC. ──────────────────────────┬──────────────────────────┬───────── Pounds per lineal ft. │Kilos. per lineal metre │1·488 „ per lineal yd. │„ „ „ │0·496 „ per lineal mile │Kilos. per kilometre │0·2818 Tons „ „ │Tonnes „ │0·6313 Feet „ „ │Metres „ │1·894 Pounds per cubic in. │Grammes per cubic cm. │27·68 „ per cubic ft. │Kilos. per cubic metre │16·02 „ per cubic yd. │„ „ „ │0·593 Tons per cubic yard │Tonnes „ „ │1·329 Cubic yds. per pound │Cubic metres per kilo. │1·685 „ per ton │„ „ per tonne │0·7525 Cubic inch of water │Weight in pounds │0·03608 Cubic feet of water │„ „ │62·35 „ „ │„ kilos │28·23 „ „ │Imperial gallons │6·235 „ „ │U.S. gallons │7·48 Litre of water │Cubic inches │61·025 Gallons of water │Weight in kilos │4·54 Pounds of fresh water │Pounds of sea water │1·026 Grains per gallon │Grammes per litre │0·01426 Pounds per gallon │Kilos. per litre │0·0998 „ per U.S. gal. │„ „ │0·115 ──────────────────────────┴──────────────────────────┴───────── COMPOUND FACTORS—POWER UNITS, ETC. ──────────────────────────┬──────────────────────────┬───────── British Ther. Units. │Kilogrammetres. │108 „ „ │Joules │1058 „ „ │Calories (Fr. Ther. units)│0·252 „ „ per sq. ft. │„ per square metre │2·713 „ „ per pound │„ per kilogramme │0·555 Pounds per sq. ft. │Dynes, per sq. cm. │479 Foot-pounds │Kilogrammetres │0·1382 „ „ │Joules │1·356 „ „ │Thermal Units │0·00129 „ „ │Calorie │0·000324 Foot-tons │Tonne-metres │0·333 Horse-power │Force decheval (Fr.H.P.) │1·014 „ „ │Kilowatts │0·746 Pounds per H.P. │Kilos. per cheval │0·447 Square feet per H. P. │Square metres per cheval │0·0196 Cubic „ „ │Cubic „ „ │0·0279 Watts │Ther. Units per hour │3·44 „ │Foot-pounds per second │0·73 „ │„ per minute │44·24 Watt-hours │Kilogrammetres │367 „ „ │Joules │3600 Kilogrammetres │„ │9·806 ──────────────────────────┴──────────────────────────┴─────────

_Inverse Proportion._—If “more” requires “less,” or “less” requires “more,” the case is one of _inverse_ proportion, and although it will be seen that this form of proportion is quite readily dealt with by the preceding method, the working is simplified to some extent by inverting the slide so that the C scale is adjacent to the A scale. By the aid of the cursor, the values on the inverted C (or Ɔ) scale, and on the D scale, can be then read off. These will now constitute a series of inverse ratios. For example, in the proportion

─────────── Ɔ 8 4 ─────────── D 1·5 3

the 4 on the Ɔ scale is brought opposite 3 on D, when under 8 on Ɔ is found 1·5 on D.[5]

GENERAL HINTS ON THE ELEMENTARY USES OF THE SLIDE RULE.

Before the more complex operations of involution, evolution, etc., are considered, a few general hints on the use of the slide rule for elementary operations may be of service, especially as these will serve to enforce some of the more important points brought out in the preceding sections.

Always use the slide rule in as _direct_ a light as possible.

Study the manner in which the scales are divided. Follow the graduations of the C and D scales from 1 to 10, noting the values given by each successive graduation and how these values change as we follow along to the right. Do the same with the two halves of the A and B scales and note the difference in the value of the subdivisions, due to the shorter scale-lengths.

Practise reading values by setting 1 on C to some value on D and reading under 2, 3, 4, etc., on C, checking the readings by mental arithmetic. To the same end, find squares, square roots, etc., comparing the results with the actual values as given in tables. Practise setting both slide and cursor to values taken at random. Aim at accuracy; speed will come with practice.

When in doubt as to any method of working, verify by making a simple calculation of the same form.

Follow the orthodox methods of working until entirely confident in the use of the instrument, and even then do not readily make a change. If any altered procedure is adopted, first work a simple case and guard carefully against unconsciously lapsing into the usual method during the operation.

Unless the calculation is of a straightforward character, time taken in considering how best to attack it (rearranging the expression if desirable) is generally time well spent.

In setting two values together, set the cursor to one of them on the rule, and bring the other, on the slide, to the cursor line.

In multiplying factors, as 57 × 0·1256, take the fractional value first. It is easier to set 1 on C to 1256 on D and read under 57 on C, than to reverse the procedure. When both values are eye-estimated, set the cursor to the second factor on C and read the result on D, under the cursor line.

In continuous operations avoid moving the slide further than necessary, by taking the factors in that order which will keep the scale readings as close together as possible.

SQUARES AND SQUARE ROOTS.

We have seen that the relation which the upper scales bear to the lower set is such that over any number on D is its square on A, and, conversely, under any number on A is its square root on D, the same remarks applying to the C and B scales on the slide. Taking the values engraved on the rule, we have on D, numbers lying between 1 and 10, and on A the corresponding squares extending from 1 to 100. Hence the squares of numbers between 1 and 10, or the roots of numbers between 1 and 100, can be read off on the rule by the aid of the cursor. All other cases are brought within these ranges of values by factorising with powers of 10, as before explained.

The more practical rule is the following:—

_To Find the Square of a Number_, set the cursor to the number on D and read the required square on A under the cursor. The rule for

_The Number of Digits in a Square_ is easily deducible from the rule for multiplication. If the square is read on the _left_ scale of A, it will contain _twice_ the number of digits in the original number _less_ 1; if it is read on the _right_ scale of A, it will contain _twice_ the number of digits in the original number.

EX.—Find the square of 114.

Placing the cursor to 114 on D, it is seen that the coinciding number on A is 13. As the result is read off on the _left_ scale of A, the number of digits will be (3 × 2) − 1 = 5, and the answer is read as 13,000. The true result is 12,996.

EX.—Find the square of 0·0093.

The cursor being placed to 93 on D, the number on A is found to be 865. The result is read on the _right_ scale of A, so the number of digits = −2 × 2 = −4, and the answer is read as 0·0000865 [0·00008649].

_Square Root._—The foregoing rules suggest the method of procedure in the inverse operation of extracting the square root of a given number, which will be found on the D scale opposite the number on the A scale. It is necessary to observe, however, that if the number consists of an _odd_ number of digits, it is to be taken on the _left-hand_ portion of the A scale, and the number of digits in the root = (N + 1)/(2), N being the number of digits in the original number. When there is an even number of digits in the number, it is to be taken on the _right-hand_ portion of the A scale, and the root contains _one-half_ the number of digits in the original number.

EX.—Find the square root of 36,500.

As there is an _odd_ number of digits, placing the cursor to 365 on the L.H. A scale gives 191 on D. By the rule there are (N + 1)/(2) = (5 + 1)/(2) = 3 digits in the required root, which is therefore read as 191 [191·05].

EX.—Find √(0·0098.)

Placing the cursor to 98 on the right-hand scale of A (since −2 is an _even_ number of digits), it is seen that the coinciding number on D is 99. As the number of digits in the number is −2, the number of digits in the root will be (−2)/(2) = −1. It will therefore be read as 0·099 [0·09899+].

EX.—Find √(0·098).

The number of digits is −1, so under 98 on the left scale of A, we find 313 on D. By the rule the number in the root will be (−1 +1)/(2) = 0, and the root is therefore read as 0·313 [0·313049+].

EX.—Find √(0·149.)

As the number of digits (0) is _even_, the cursor is set to 149 on the right-hand scale of A, giving 386 on D. By the rule, the number of digits in the root will be (0)/(2) = 0, and the root will be read as 0·386 [0·38605+].

Another method of extracting the square root, by which more accurate readings may generally be obtained, is by using the C and D scales only, with the slide inverted. If there is an _odd_ number of digits in the number, the _right_ index, or if an even number of digits the _left_ index, of the inverted scale Ɔ is placed so as to coincide with the number on D of which the root is sought. Then with the cursor, the number is found on D which coincides with the same number on Ɔ, which number is the root sought.

EX.—Find √(22·2.)

Placing the left index of Ɔ to 222 on D, the two equal coinciding numbers on Ɔ and D are found to be 4·71.

Note that under the cursor line we have the original number, 22·2, on A, and from this the number of digits in the root is determined as before.

The plan of finding the square of a number by ordinary multiplication is often very convenient. The inverse process of finding a square root by trial division is not to be recommended.

To obtain a close value of a root or to verify one found in the usual way, the author has, on occasion, adopted the following plan:—Set 1 (or 10) on B to the number on the A scale (L.H. or R.H. as the case may require), and bring the cursor to the number on D. If the root found is correct, the readings on C under the cursor and on D under the index of C, will be in exact agreement.

If 1 on B is placed to a number _n_ on the L.H. A scale, the student will note that while root _n_ is read on D under 1 on C, the root of 10 _n_ is read on D under 10 on B. Hence, if preferred, the number can be taken always on the first scale of A and the root read under 1 or 10 on B, according to whether there is an odd or even number of digits in the number. Obviously the second root is the first multiplied by √(10).

CUBES AND CUBE ROOTS.

In raising a number to the third power, a combination of the preceding method and ordinary multiplication is employed.

TO FIND THE CUBE OF A NUMBER.—_Set the_ L.H. _or_ R.H. _index of C to the number on D, and opposite the number_ ON THE LEFT-HAND _scale of B read the cube on the_ L.H. _or_ R.H. _scale of A_.

By this rule four scales are brought into requisition. Of these, the D scale and the L.H. B scale are _always_ employed, and are to be read as of equal denomination. The values assigned to the L.H. and R.H. scales of A will be apparent from the following considerations.

Commencing with the indices of C and D coinciding, and moving the slide to the right, it will be seen that, working in accordance with the above rule, the cubes of numbers from 1 to 2·154 (= ∛(10)) will be found on the first or L.H. scale of A. Moving the slide still farther to the right, we obtain _on the_ R.H. _A scale_ cubes of numbers from 2·154 to 4·641 (or ∛(10) to ∛(100)). Had we a _third_ repetition of the L.H. A scale, the L.H. index of C could be still further traversed to the right, and the cubes of numbers from 4·641 to 10 read off on this prolongation of A. But the same end can be attained by making use of the R.H. index of C, when, traversing the slide to the right as before, the cubes of numbers from 4·641 to 10 on D can be read off _on the_ L.H. _A scale_ over the corresponding numbers on the L.H. B scale. Hence, using the L.H. index of C, the readings on the L.H. A scale may be regarded comparatively as units, those on the R.H. A scale as tens; while for the hundreds we again make use of the L.H. A scale in conjunction with the _right-hand_ index of C.

By keeping these points in view, the number of digits in the cube (N) of a given number (_n_) are readily deduced. Thus, if the units scale is used, N = 3_n_ − 2; if the tens scale, N = 3_n_ − 1; while if the hundreds scale be used, N = 3_n_. Placed in the form of rules:—

N = 3_n_ − 2 when the product is read on the L.H. scale of A with the slide to the _right_ (units scale).

N = 3_n_ − 1 when the product is read on the R.H. scale of A; slide to the _right_ (tens scale).

N = 3_n_ when the product is read on the L.H. scale of A with the slide to the _left_ (hundreds scale).

With decimals the same rule applies, but, as before, the number of digits must be read as −1, −2, etc., when one, two, etc., cyphers follow immediately after the decimal point.

EX.—Find the value of 1·4^3.

Placing the L.H. index of C to 1·4 on D, the reading on A opposite 1·4 on the L.H. scale of B is found to be about 2·745 [2·744].

EX.—Find the value of 26·4^3.

Placing the L.H. index of C to 26·4 on D, the reading on A opposite 26·4 on the L.H. scale of B is found to be about 18,400 [18,399·744].

EX.—Find the value of 7·3^3.

In this case it becomes necessary to use the R.H. index of C, which is set to 7·3 on D, when opposite 7·3 on the L.H. scale of B is read 389 [389·017] on A.

EX.—Find the value of 0·073^3.

From the setting as before it is seen that the number of digits in the number must be multiplied by 3. Hence, as there is −1 digit in 0·073, there will be −3 in the cube, which is therefore read 0·000389.

The last two examples serve to illustrate the principle of factorising with powers of 10. Thus

0·073 = 7·3 × 10^{−2}; 0·073^3 = 7·3^3 × (10^{−2})^3 = 389 × 10^{−6} = 0·000389.

_Cube Root_ (_Direct Method_).—One method of extracting the cube root of a number is by an inversion of the foregoing operation. Using the same scales, _the slide is moved either to the right or left until under the given number on A is found a number on the_ L.H. _B scale, identical with the number simultaneously found on D under the right or left index of C_. This number is the required cube root.

From what has already been said regarding the combined use of these scales in cubing, it will be evident that in extracting the cube root of a number, it is necessary, in order to decide which scales are to be used, to know the number of figures to be dealt with. We therefore (as in the arithmetical method of extraction) point off the given number into sections of three figures each, commencing at the decimal point, and proceeding to the left for numbers greater than unity, and to the right for numbers less than unity. Then if the first section of figures on the left consists of—

1 figure, the number will evidently require to be taken on what we have called the “units” scale—_i.e._, on the L.H. scale of A, using the L.H. index of C.

If of 2 figures, the number will be taken on the “tens” scale—_i.e._, on the R.H. scale of A, using the L.H. index of C.

If of 3 figures, the number will be taken on the “hundreds” scale—_i.e._, on the L.H. scale of A, using the R.H. index of C.

To determine the number of digits in cube roots it is only necessary to note that when the number is pointed off into sections as directed, there will be one figure in the root for every section into which the number is so divided, whether the _first_ section consists of 1, 2, or 3 digits.

Of numbers wholly decimal, the cube roots will be decimal, and for every group of _three_ 0s immediately following the decimal point, _one_ 0 will follow the decimal point in the root. If necessary, 0s must be added so as to make up complete multiples of 3 figures before proceeding to extract the root. Thus 0·8 is to be regarded as 0·800, and 0·00008 as 0·000080 in extracting cube roots.

EX.—Find ∛(14,000.)