Part 7

For quantities less than 0·1 the digit place numbers will be negative. The troublesome addition of these may be avoided by transferring them to the opposite side and treating them as positive.

2 4 0·00356 × 27·1 × 0·08375 Thus:— ───────────────────────── = 288 0·1426 × 9·85 × 0·00002 2 1 1

The first numerator, 0·00356, has −2 digits. Note this by placing 2 below the lower line as shown. 27·1 has 2 digits; place 2 over it. 0·08375 has −1 digit; hence place 1 below the lower line. The first denominator has no digits; the second, 9·85, has 1 digit; hence place 1 under it. 0·00002 has −4 digits; place 4 above the upper line. The sum of the top series is 2 + 4 = 6; of the bottom series 2 + 1 + 1 = 4. Subtracting the bottom from the top, we have 6 − 4 = 2 digits, to which 1 has to be added for an uncancelled memo-mark, and the result is read as 288.

Moving the decimal point often facilitates matters. Thus, (32·4 × 0·98 × 432 × 0·0217)/(4·71 × 0·175 × 0·00000621 × 412000) is much more conveniently dealt with when re-arranged as (32·4 × 9·8 × 432 × 2·17)/(4·71 × 17·5 × 6·21 × 4·12) = 141.

To determine the number of figures in the result by rough cancelling and mental calculation, we note that 4·71 enters 432 about 100 times; 9·8 enters 17·5 about 2; 6·21 into 32·4 about 5; and 2·17 into 4·12 about 2. This gives (500)/(4) = 125, showing that the result contains 3 digits. From the slide rule we read 141, which is therefore the result sought.

The occasional traversing of the slide through the rule, to interchange the indices—a contingency which the use of the C and D scales always involves—may often be avoided by a very simple expedient. Such an example as (6·19 × 31·2 × 422)/(1120 × 8·86 × 2.09) = 3·93 is sometimes cited as a particularly difficult case. Working through the expression as given, two traversings of the slide are necessary; but by taking the factors in the slightly different order, (6·19 × 31·2 × 422)/(8·86 × 2·09 × 1120), so that the significant figures of each pair are more nearly alike, we not only avoid any traversing the slide, but we also reduce the extent to which the slide is moved to effect the several divisions.

Such cases as (a × b)/(c × d × e × f × g) or (a × b × c × d × e)/(f × g) really resolve themselves into (a × b × 1 × 1 × 1)/(c × d × e × f × g) and (a × b × c × d × e)/(f × g × 1 × 1 × 1), but, of course, if rules are used to locate the decimal point, the 1’s so (mentally) introduced are not to be counted as additional figures in the factors.

MULTIPLICATION AND DIVISION WITH THE SLIDE INVERTED.

If the slide be inverted in the rule but with the same face uppermost, so that the Ɔ scale lies adjacent to the A scale, and the right and left indices of the slide and rule are placed in coincidence, we find the product of any number on D by the coincident number on Ɔ (readily referred to each other by the cursor) is always 10. Hence, by reading the numbers on Ɔ as decimals, we have over any unit number on D, its reciprocal on Ɔ. Thus 2 on D is found opposite 0·5 on Ɔ; 3 on D opposite to 0·333; while opposite 8 on Ɔ is 0·125 on D, etc. The reason of this is that the sum of the lengths of the slide and rule corresponding to the factors, is always equal to the length corresponding to the product—in this case, 10.

It will be seen that if we attempt to apply the ordinary rule for multiplication, with the slide inverted, we shall actually be multiplying the one factor taken on D by the reciprocal of the other taken on Ɔ. But multiplying by the reciprocal of a number is equivalent to dividing by that number, and dividing a factor by the reciprocal of a number is equivalent to multiplying by that number. It follows that with the slide inverted the operations of multiplication and division are reversed, as are also the rules for the number of digits in the product and the position of the decimal point. Hence, in multiplying with the slide inverted, we place (by the aid of the cursor) one factor on Ɔ opposite the other factor on D, and read the result on D under either index of Ɔ. It follows that with the slide thus set, any pair of coinciding factors on Ɔ and D will give the same constant product found on D under the index of Ɔ. One useful application of this fact is found in selecting the scantlings of rectangular sections of given areas or in deciding upon the dimensions of rectangular sheets, plates, cisterns, etc. Thus by placing the index of Ɔ to 72 on D, it is readily seen that a plate having an area of 72 sq. ft. may have sides 8 by 9 ft., 6 by 12, 5 by 14·4, 4 by 18, 3 by 24, 2 by 36, with innumerable intermediate values. Many other useful applications of a similar character will suggest themselves.

PROPORTION.

With the slide in the ordinary position and with the indices of the C and D scales in exact agreement, the ratio of the corresponding divisions of these scales is 1. If the slide is moved so that 1 on C agrees with 2 on D, we know that under any number n on C is n × 2 on D, so that if we read numerators on C and denominators on D we have

C 1 1·5 2 3 4 ───────────────────────────────────────── D1 2 3 4 6 8.

In other words, the numbers on D bear to the coinciding numbers on C a ratio of 2 to 1. Obviously the same condition will obtain no matter in what position the slide may be placed. The rule for proportion, which is apparent from the foregoing, may be expressed as follows:—

RULE FOR PROPORTION.—Set the first term of a proportion on the C scale to the second term on the D scale, and opposite the third term on the C scale read the fourth term on the D scale.

EX.—Find the 4th term in the proportion of 20 ∶ 27 ∷ 70 ∶ x. Set 20 on C to 27 on D, and opposite 70 on C read 94·5 on D. Thus

C 20 70 ───────────────── D 27 94·5.

It will be evident that this is merely a case of combined multiplication and division of the form, (20 × 70)/(27) = 94·5. Hence, given any three terms of a proportion, we set the 1st to the 2nd, or the 3rd to the 4th, as the case may be, and opposite the other given term read the term required.

Thus, in reducing vulgar fractions to decimals, the decimal equivalent of (3)/(16) is determined by placing 3 on C to 16 on D, when over the index or 1 of D we read 0·1875 on C. In this case the terms are 3 ∶ 16 ∷ x ∶ 1. For the inverse operation—to find a vulgar fraction equivalent to a given decimal—the given decimal fraction on C is set to the index of D, and then opposite any denominator on D is the corresponding numerator of the fraction on C.

If the index of C be placed to agree with 3·1416 on D, it will be clear from what has been said that this ratio exists throughout between the numbers of the two scales. Therefore, against any diameter of a circle on C will be found the corresponding circumference on D. In the same way, by setting 1 on C to the appropriate conversion factor on D, we can convert a series of values in one denomination to their equivalents in another denomination. In this connection the following table of conversion factors will be found of service. If the A and B scales are used instead of the C and D scales, a complete set of conversions will be at once obtained. In this case, however, the left-hand A and B scales should be used for the initial setting, any values read on the right-hand A or B scales being read as of tenfold value. With the C and D scales a portion of the one scale will project beyond the other. To read this portion of the scale, the cursor or runner is brought to whichever index of the C scale falls within the rule, and the slide moved until the other index of the C scale coincides with the cursor, when the remainder of the equivalent values can then be read off. It must be remembered that if the slide is moved in the direction of notation (to the right), the values read thereon have a tenfold greater value; if the slide is moved to the left, the readings thereon are decreased in a tenfold degree. Although preferred by many, in the form given, the case is obviously one of multiplication, and is so treated in the Data Slips at the end of the book.

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.

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.