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.[4]
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.[5]