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CONDENSED LIST.
Ap. means within 2%.
phyiloal Quantities, Relations,
Dimensional Formulas,
etc., p. 18-^6
I«eneths: complete tables, p. 30-34
1 mil"' 0.025 40 mm. Ap. %o
1 mm. » 30.37 mils. Ap 40
1 cm.»0.393 7 inch. Ap. Mo
1 inch =2.540 centimeters. Ap. ^%
1 foot<=0.304 8 meter. Ap. ^o
1 yard —0.914 4 meter. Ap. %o or i%i
1 meter=30.37 inches. Ap. 40
" -3.281 feet. Ap. i%
" =1.094 yards. Ap. »Ho
1 kilometer-3 281. feet. Ap. HX
10 000
'* =1 094. yards. Ap. 1 100
" -0.621 4 mile. Ap ^
1 mile -5 280. feet. Ap. 5 300
•* -1760. yards. Ap.%X1000
" -1.609 km. Ap. add^o
1 knot — 1.853 kilometers. Ap. ^^
" -1.152 miles. Ap. add Vr .
Inches in fractions, decimals, milli-
meters and feet, p. 35-38
Digit conv. tables (1 to 100), p. 39-40
Surfaces: complete tables, p. 41-44
1 circular mU = 0.000 5067 sq, mm.
Ap. 1^+1000
1 sq. mm. — 1 974. circ. mils. Ap.
2000
1 sq. cm. —0.155 sq. inch. Ap. ^3
1 sq. inch-1 273 240. circ. mils
-6.452 sq. cm. Ap. 6H
1 sq. ft. -0.092 90 sq.m. Ap.i%2+ 10
1 sq. yd.— 0.836 1 sq. meter. Ap. %
1 sq. m. -10.76 sq. ft. Ap. ^4i X 10
** -1.196 sq. yds. Ap. add%
1 acre— 43 560. sq. feet
** -0.404 7 hectare. Ap. Mo
1 hectare — 2.471 acres. Ap. *%
-0.003 861 aq. mile
Isq.km. -247.1 acres. Ap.^XlOOO
" =0.386 1 sq.mUe. Ap.»%6
l8q.mile=640. acres
-2.590 sq.km. Ap.26 + 10
Digit conversion tables, p. 44
Tol umes : complete tables p . 46-55
1 cb. cm. -0.061 02 cb. in. Ap. %oo
1 cb. in. -16.39 cb. cm. Ap. 10%
1 pint -28.875 cb. in. Ap.^XlOO
" -0.473 2 liter. Ap. %i X 10
1 quart— 0.946 4 liter. Ap. sub. %o
lhter-61.02 cb. inches. Ap. 60
" —1.057 quarts. Ap. add %o
*• -0.035 31 cb. ft. Ap.T/a + lOO
1 gal. (U. S.) -231. cb. inches
-3.785 liters. Ap. V&XliL
-0.133 7cb.ft. Ap.*^'^^*^
" (Brit.) -4.646 liters. .
1 cb. ft. -28.32 liters. Ap. JJ
•* -7.481 gal. (U.S.).
1 bushel (U.S.)-1.244cb.ft..
" =»0.352 4 hectoliter. Ap
1 hectoliter -26.42 gab. (U(.S
%X10
\
1 cb. yd. —0.764 6 cb. meter. Ap. H
1 cb. m. -35.31 cb. ft. Ap. T/^X 10
-1.308 cb. yds. Ap. IH
Digit conversion tables, p. 51
Weights: complete table, p. 57-61
1 gram— 64.80 mg. Ap. 65
1 gram — 15.43 grains. Ap. 15H
= 0.035 27 OS. Ap. % + 1 00
1 ounce - 28 .35 grams. Ap. 44 X 1 00
1 pound (av.) =0.453 6 klg. Ap. %o
1 kilogram - 35 27 oz. Ap. T^ X 1
-2.205 pounds. Ap. 2^
1 short ton— 0.907 2 met. ton. Ap.
subtr. Mo
do.— 0.892 9 long ton. Ap. subtr. Mo
1 metric ton — 1.102 short tons.
Ap. add Mo
do. -0.984 2 long ton. Ap. 1
1 long ton = 1.12 short tons. Ap. 1%
— 1.016 met. tons. Ap.jl
Digit conversion tables, p. 59
Weigh tM and liengths; Wt. of
Bars: complete table , p . 62-63
1 lb. per mile— 0.281 8 kg. per km.
Ap. %
1 kg. per kilometer— 3.548 lbs. per
mile. Ap. %
1 lb. per yard— 0.496 1 kg. per
meter. Ap. j/^
1 kg. per meter— 2.016 lbs. per
yard. Ap. 2
do. -0.672 lb. per ft. Ap. %
lib. per ft. — 1.488 kg. perm. Ap. %
Pressures: complete table, p. 64-67
1 lb. per sq. ft.— 4.882 kg. per sq.
meter. Ap. *%
1 ft. water column— 62.43 lbs. per
sq. foot. Ap. 509^
do. -0.029 50 atm. Ap. Mno
1 lb. per sq. in. —0.070 31 kg. per sq.
cm. Ap. Moo
do.— 0.068 04 atmosphere. Ap.%Q
1 kg. per sq. cm. — 14.22 lbs. per
sq. in. Ap. *o%
do. —0.967 8 atmosphere. Ap.
subtr. Ho
1 atm. == 14.70 lbs. per sq. in. Ap. <•%
" —1.033 kg. per sq. cm. Ap.
add Ho
Digit conversion tables, p. 57
Weights and Voluines: complete
table, p. 68-69
1 lb. per cb. yd. -0.593 3 kg. per
cubic meter. Ap. <Mo
1 kg. per cb. meter — 1.686 lbs n«f
cb.yd. Ap.io/6 ^^
,riniurnnfi8>.^1h ,iyrrfc-ft. Ap.%o
1^^- per cb.
complete
P- %0Q
Ap.io%
t* ^^-^
p. add ^^Q
\
f
r'.c.
CONDENSED LIST.
91-
' hi -r ■.
<i
1 liter « 2.205 pounds. Ap. *%o
1 cb. ft. "62.43 pounds. Ap. % X 100
" =28.32 kilograms. Ap. »%
1 cb. yd. = 1 686. lbs. Ap H X 10 000
=764.6 kg. Ap. kx 1 000
= 0.752 5 ton (long). Ap.H
lcb.meter=2 205. pounds
= 0.984 2 ton ( long). Ap 1
Tolames of Water: complete
table, p. 71
1 gram =0.061 02 cb. in. Ap. %no
lounce=28.35cb. cm. Ap. 4^X100
llb.=27.68cb.in. Ap. ifexiO
" -0.453 6 Uter. Ap. %i
" -0.016 02 cb. foot. Ap.%-4-100
1 kilogram = 61.02 cb. in. Ap. 60
" =0.035 31 cb.ft. Ap.% + 100
Bnerffjr; Work; Heat: complete
table, p. 74-77
1 joule =0.737 6 ft.-lb. Ap. H
" -0.238 9 small calorie.
" =0.102 kg.-met. Ap. Mo
1 ft.-lb. = 1.356 joules. Ap. %
=0.3239 small cal. Ap. i%o
=0.138 3 kg.-met. Ap. %o
=0.001 285 thermal umt. Ap.
^^-1-1000
1 kg.-m. =9.806 joules. Ap. 10
^* =7.233 ft.-lbs. Ap. 89ii
" =2.342 g.-calories. Ap. %
1 ther. unit = 1 055. joules. Ap. 210%
*• =778.1 ft.-lb. Ap. 700%
= 107.6 kg.-met. Ap. 108
=0.252 kg.-cal. Ap. H
I watt-hour = 2 655 .ft.-lbs. Ap. «oo%
=367.1 kg.-met. Ap."o%
=0.8600 kg.-gal. Ap.%
1 cal. (kg.) =4 186. joules. Ap. 4 200
-3 088. ft.-lbs. Ap. 3 100
=426.9 kg.-met. Ap.a»%
—3.968 ther. units. Ap.4
- 1.163 watt-hrs. Ap. %
1 met. hp.-hr. = l 952 910. ft.-lbs.
= 270 000. kg.-met.
1 hp.-hr. = 1 980 000. foot-pounds
" =273 745. kilogram-meters
1 kw.-hr. = 2 655 403. foot-pounds
" -367 123. kilogram-meters
Digit conversion tables, p. 77
Relations betireen Torque and
Knergry, p. 78
Traction Knergy, p. 78
1 ton (met.)-km. =0.684 9 ton ^h.)-
mile. Ap. %3
1 ton (8h.)-mile = 1.460 ton (mct.)-
kilometer. Ap. ^%
Tractive Force, p. 78
1 lb. per ton (sh.)— 0.500 kilogram
per ton (met.)
1 kg. per ton (met.) — 2.000 pounds
per ton (short)
Power: complete table, p. 80-82
1 watt— 44.26 ft.-lbs. per minute.
Ap. HXlOO
do. — 14.33 gram-cal. per minute.
Ap.MXlOO
do. —6.119 kg. -met. per minute.
Ap. 6
tt
1 met. hp.— 75.00 kilogram-meters
per sec. OrJiXlOO
do. -0.986 3 hp. Ap. 1
do. -0.735 4 kw. Ap. 2^i-s- 10
1 hp. — 33 000. ft.-lbs. per min.
Ap. MX 100 000
" —1.014 metric hp. Ap. 1
** -0.7457 kilowatt. Ap. ^
Ikw. — 1.360 m. hp. Ap. add M
" -1.341 hp. Ap. addM
Digit conversion tables, p. 82
Forces: complete table, p. 83
1 d3aie — 1.020 milligrams. Ap. 1
1 gram -980.6 dsmes. Ap. 1000
1 pound— 444 791. dynes
Moments of Inertia and of
Momentum, p. 84
Itinear Velocities: complete
table, p. 85-86
1 km. per hr. — 16.67 met. per min.
Ap. MX 100
1 ft. per sec. —0.681 8 mile per hour.
Ap. 68 + 100
1 mile per hour = 1 .467 ft. per second.
Ap. >H
1 meter per sec. —3.6 km. i^r hour
Angular Velocities, p. 86
Frequency, p. 86
Linear Accelerations: complete
table, p. 88
Gravity =32. 17 ft. per sec. per sec.
Ap. 32
Gravity —9.806 m. per s. per s. Ap.lO
Angular Accelerations, p. 88
Angles: complete table, p. 89
1 deg. -0.017 45 radian. Ap. % + 100
1 radian — 57.30 degrees
1 right angle = 1.571 radians. Ap. ^^
Solid angles, p. 89
Grades: complete table, p. 90-91
1 foot per mile =0.018 94%
1% —52.80 feet per mile
Conversion table 1 to 15%, p. 92
Time, p. 94
Discliarges; Flow of Water;
Irrigation Units^ p. 95
1 miner's mch — 1.5 cb. ft. per min.
1 acre-foot ? 325 851 . gallons
Klectric and Magnetic Units:
Mean, efifec. and max. values, p. 97
Resistance, Impedance, React-
ance, p. 99
1 Brit. Assn. unit — 0.986 7 ohm.
Ap. subtr. 1%
1 ohm— 1.014 Brit. Assn. units. Ap.
add 1%
Resistance and Iicngth, p. 101
Resistance and Cross-section,
p. 101
Resistivity; Specific Resist-
ance, complete table, p. 103-4
1 ohm, circ. mil, ft., unit =0.001 662
ohm, Bq.mm.,m.,unit. Ap. V^oo
1 ohm, sq. mm., m., unit — 601.5
ohm, circ. mil, foot units
Resistivity of pure copper:
= 10.03 ohm, circ. mil, foot units
= 0.016 67 ohm, sq. mm., m., unit
CONDENSED LIST.
Resistivity of mercury:
=565.9 omn, circ. mil, foot units
=0.940 7 ohm, sq. mm., meter unit
Conductance, Admittance, Sim-
ceptanee* p. 1*05
Conductivity, Specific Conduc-
tance, complste table, p. 107
1 mercury umt=0.017 72 copper unit
1 copper unit » 55.76 mercury units
Electromotive Force, complete
table. p. 109-111
1 volt -0.981 Weston cell at 20«C.
Ap. subtr. 2%
" - 0.697 4 Clark cell 15^. Ap. %o
1 Weston cell at 20* C. :
= 1.019 4 volts Ap. add 2%
=0.710 9 CUrk cell at 15° C. Ap. %
1 Clark cell 16° -1.434 volts. Ap. i%
-1.407 Weston cells
at 20°. Ap. %
E.M.F. of Clark and Weston cells at
different temperatures, p. Ill
Current, p. 113
Currcmt Density, complete table,
p. 114-115
1 ampere per square inch:
—0.155 amp. per sq. cm. Ap. %3
1 ampere per square centimeter:
— 6.452 amp. per sq. in. Ap. 6H
1 ampere per square millimeter:
—0.000 5067 ampere per circular
mU. Ap. H-i- 100
1 ampere per circular mil:
— 197 4. amp. per sq. mm. Ap. 2 000
Quantity; Cliarge, p. 116
Capacity, p. 117
Inductance, p. 119
Time Constant, p. 120
Frequency: complete table, p. 121
Frequency — 0. 1 59 2 X angular veloc .
Angular velocity — 6 J283 X frequency
Kinetic lEn^rgy of a Current,
p. 122
iriectrical Energry, p. 123
(see also under Energy , p. 74)
IClectrical Fower, p. 125
(aee also under Power, p. 80)
Electrochemical EquW., p. 126
Ionic charge =96 539, coulombs
Electrolytic Deposits, p. 126-7
1 pound per day — 0.182 6 ton (short)
per year. Ap. iV&-*-10
1 kg. per day —0.365 3 met. ton per
year. Ap. ii^+lO
Electrochemical Energy: com^
plete table, p. 129
Volts — kg. -calories per gram-mole-
cule X 0.043 36. Ap.^^o
Magnetic Reluctance, p. 129
Oersteds — gilberts -i- maxwells
" —ampere-turns X 1.257 -»-
maxwells
AEagrnetic Reluctivity, p. 130
Sfagaetio Permeance, p. 131
Mag^netic Permeability, p. 132
—gausses in iron -^ gausses in air
Magnetic SuHceptibility, p. 132
Magnetomotive Force: complete
table, p. 133-4
1 gilbert — 0.795 8 ampere-turns.
Ap. subtr. %
1 amp.-tum = 1.257 gilb't. Ap. add 14
Gilberts — maxwells X oersteids
Magnetizing Force, p. 136-7
1 amp.-tum per inch— 0.494 7 gil-
, bert per centimeter. Ap. ^
1 gilbert per cm. —2.021 amp.-turn
per inch. Ap. 2
Magnetic Flux, p. 138-9
Maxwells— gausses Xsq. centimeters
' ' — fldlberts -?- oersteds
Magnetic Flux Density: com-
plete table, p. 141-2
1 inch-gauss — 0.1550 gauss. Ap.^n
1 gauss — 6.452 inch-gausses. Ap. 6>^
Gausses— maxwells-i-sq. centimeters
Magnetic Moment, p. 142
Intensity of Magnetization, 142
Magnetic Energy, p. 143
Magnetic Power, p. 144
Photometric Units:
Intensity of Ldglit; Candle
Power: complete table, p. 146
1 hefner— 0.88 English candle
1 English candle — 1.14 hefners
Flux of liight; Spherical or
Hemispli. Candle Power, 147
Illumination: compl. table, p. 148
1 met.-cp.— 0.081 8 it.-cp. Ap. Ma
1 ft.-cp. — 12.2 met.-cp (hefners)
Brightness of Source, p. 148
Quantity of Light, p. 149
Light Efficiency, p. 149
Thermometer Scales:
Reduction factors, p. 150
Readings in:
C.° = (F.°--32)X%
F.°-(C.°X%)+32
Scales: Centigrade, Fahrenheit;
Reaumur, Absolute and Con-
crete, from absolute zero to
6000° C. p. 151-163
Money, Foreign, p. 164
Money and Length, p. 166
Money and Weight, p. 167
Scales of Maps and Drawings,
p. 168
Functions of ir, p. 169-170
Useful Numbers, p. 170
Systems of Logarithms, p. 171
Acceleration of Gravity, p. 171
Mechanical Equivalent of
Heat, p. 171
Specific Heat of Water, p. 171
Miscellaneous Foreign Meas-
ures, p. 172
Index, p. 175-196
REAPY-REFERENCE TABLES
Vol. I
READY
REFERENCE TABLES.
Volume I.
CONVERSION FACTORS
OF EVERY UNIT OE MEASURE IX USE, INCLUDINa THOSE OP
LENGTH, SURFACE, VOLUME, CAPACITT, WEIGHT, WEIGHT AND LENGTH,
PRESSURE, WEIGHT AND VOLUME, WEIGHT OF WATER, ENERGY, HEAT,
POWER, FORCE, INERTIA, MOMENTS, VELOCITT, ACCELERATION,
ANGLES, GRADES, TIME, ELECTRICITT, MAGNETISM, ELBO-
TROCHEMISTRT, LIGHT, TEMPERATURE, MONET, MONET
AND LENGTH, MONET AND WEIGHT, NUMEROUS COM-
POUND UNITS, USEFUL FUNCTIONS AND NUMBERS,
ETC., ETC. WITH THEIR ACCURATE AND
THEIR APPROXIMATE VALUES, THEIR
LOGARITHMS, RELATIONS, DIGIT
CONVERSION TABLES, EXPLA-
NATIONS OF CALCULA-
TIONS, ETC., ETC.
BASED ON THE ACCURATE LEGAL STANDARD VALUES OF THE
UNITED STATES.
CONVENIENTLY ARRANGED FOR
ENGINEERS, PHYSICISTS, STUDENTS, MERCHANTS, ETC.
BY
. GAEL HEEING, M.E.,
Past President Americcm Institute of Electrical Engineers;
President Engineers* Club of Philadelphia;
Delegate to International Congresses ; etc.
I^IRST BCDITION.
FIRST THOUSAND.
NEW YORK:
JOHN WILEY & SONS.
London : CHAPMAN & HALL, Limited.
1904.
Copyright, 1904,
BY
CARL HERma.
Entered at Stationers' Hall.
PRESS OF
BRAUNWORTH & CO.
BOOKBINDERS AND PRINTERS
BROOKLYN, N. Y.
I
J
**/ look upon our English system (%s a wickedly brain-des'roying piece of
bondage under which we suffer. I say this seriously. I do not think any
one knows how seriously I speak of it. '* — Sir William Thomson (now Lord
Kelvin), Philadelphia. Sept. 24, 1884.
t«..
PREFACE.
The present is the first of several volumes in preparation by the author,
which are intended to contain collections of data conveniently arranged
for ready reference. In this first volume, all the various measures used
in practice, more especially by engineers and physicists, are given with
their values in terms of as many of the others as they are likely to be con-
verted into in practice; the reciprocals of these are also given, thus enabling
every calculation involving the conversion of one measure into another
to be reduced to a single, simple multiplication. Moreover, they are stated
in such a form that errors due to dividing instead of multiplying are entirely
avoided. It has been the intention to include every unit or measure used
in practice, besides mamr that are now obsolete but are occasionally met
witn. The more usual foreign units or measures have also been adaed.
These conversion factors are not compiled, but have all been especially
recalculated for this volume, under the direction of the author, from the
exact legal values as far as such values existed, and from the very best,
most standard, and most authoritative values obtainable, when no legal
values existed. The greatest possible care was taken in selecting these
fundamental values. They were obtained from the National Bureau of
Standards, the Director of the Nautical Almanac, the International ~Geo-
vi detic Association, the U. S. Coast and Geodetic Survey, the U. S Treasury
( Department, the adoptions and recommendations of international con-
\^ grosses and national societies, standard works of reference, and personal
^^ authorities, preference in the few cases of disagreement being generally
I given in the order named. The authority for each fundamental value is
^ given. As all the conversion factors were calculated from the same set of
'^ fundament£d values, they are all consistent with each other, forming a
L^ single, interconvertible, uniform system. It is believed that tins is the
^ first time tSuch a complete set of conversion factors of all measures has been
3 published, based on the accurate legal standard values. It is also bdieved
to be the first complete collection of all the electric, magnetic, and photo-
metric units, together with their interrelations, that has been published
in convenient tabular form for ready reference.
Accuracy, completeness, and convenience for ready reference were the
chief aims in the preparation of these tables. The calculations were very
carefully made, in many cases by two entirely different methods, and the
resulting values were checked and cross-checked, often several times. For
most of the values a final comparison was made between the electrotyped
plates and the original calculation sheets. ^ All values marked with aster-
isks were checked by the original authorities after the pages were electro-
typed It is therefore believed that there are no errors, but should any
errors or omissions be foimd, the author would greatly appreciate being
notified of them, as it is the intention to have the values correct and the
^ collection complete, so as to serve as a reliable standard of reference.
Cr A new feature has been added in the form of convenient approximate
; values consisting often of only one and generally of only two digits. These
'^^ digits have been so chosen that they reduce the calculation to the simplest
possible; they will be found to sum.ce for most of the .ordinary computa-
iJc tions, being always correct within 2%. The correct logarithms have also
_L been given for nearly all of the conversion factors.
y Another new feature not generally found in tables of conversion factors,
^ is that the values of most of the compound units are given. This saves
double, triple, or even more lengthy calculations, in which errors are likely
^ to be made, as they often involve both multiplications and divisions.
., A table of physical quantities has been added giving the derivation,
Bjrmbols, dimensional formula, etc., of each; the table is similar to the
y one approved by the International Congress at Chicago, but includes many
^ more quantities.
\
«
»" N ^
PREFACE.
The usual method of giving tables of "Weights and Measures'' has been
entirely abandoned here as too cumbersome, inadequate, and entirelv
impracticable for giving numerous values each with its reciprocals; such
tables are moreover quite unsuited for ready reference The author has,
instead, adopted a system in which all interconvertible units (like the
mechanical, neat, and electrical units of energy, for instance) are given
together in one table, and are there placed in the order of their size. This
system is not only a more condensed and practical one, but is also far more
convenient for ready reference.
The present tables are an extension and entire recalculation of the very
much smaller ones published b^ the author about twenty years ago under
the title of "Equivalents of Units of Measurement."
Unusual, foreign, and obsolete units, or those used for special trades,
have been addecT as far as they were obtainable, and in every case their
values are given in terms of the usual, legal, or modem units.
For some units, moro particularly the electric, magnetic, photometric,
etc., explanations have been added of the meaning, derivation, and proper
use of tne units, which are often incorrectly understood or applied. Some
explanatory notes on the usual methods and accuracies of calculations
are also given in the introduction.
The index has been made very complete, and in addition to this, atten-
tion is called to the Condensed List beginning on the inside cover-page,
which contains the most frequently used values with page references to
the others.
A comparison with other tables will show that many of the latter are
not based on the legal standards of the country. It is not generally known,
for instance, that the legal foot is derived from the meter and not from a
standard vard. Msuiy of these other tables are based on an unauthoriaed
value of tne meter in inches.
Although the accuracy adopted throughout, namely, six places of figures
and seven of logarithms, may not always be warranted by the accuracy of
some of the fundamental figures, yet it has been maintained uniform
throughout these tables in order to enable changes to be made by mere
proportion, when more accurate fundamental values become obtainable in
the future.
The author takes pleasure in expressing his appreciation of the valuable
assistance contributed by others, which has aidded greatly to the com-
pleteness and reliability of the information and makes much of^it authori-
tative. 'He desires to express his obligations to the following: To the
National Bureau of Standards, and in particular to its Director, Prof S. W.
Stratton, for very full and complete published and unpublished data con-
cerning the values of the legal standards of this and other countries; also
for his consent to allow the name of that Bureau to be added to these data,
and for much other valuable information and assistance. To L. A. Fischer,
Assistant Physicist of that Bureau for the laborious work of checking the
correctness of many values of the conversion factors of the fundamental
mechanical units. To Dr. F. A. Wolfif, Assistant Physicist of that Bureau,
for similar service in connection with the standard values of some of the
electrical units. To Prof. Walter S. Harshman, Director of the Nautical
Almanac, for his revision of the table of un-ts of time. To Prof. Thomas
Gray, author of the Smithsonian Physical Tables, for suggestions and
recommendations concerning some ofi the derivational and dimensional
formulas in the table of Physical Quantities. To Prof. W. S. Franklin
for recommendations, suggestions, and a revision of those relations between
the electrical and magnetic units which apply to varying currents. To Dr.
Ed. L. Nichols for information concerning some of the standards of candle-
power. To Dr. A. E. Kennelly for a revision of parts of the table of physi-
cal quantities. To the author's former assistant, Dr. E. F. Roeber, for his
able work in mathematical physics and his reliable calculations of many of
the values in the tables.
CARL HERINQ.
Philadelphia, April, 1904.
TABLE OF CONTENTS.
Condensed last inside cover page
Symbols and Abbreviations xvj-xvMi
Introduction. . . 1-26
Inter-relation of Units 1-4
Compound Names of Units 4
Distmction between Units and Quantities Measured in them . . 4-5
Reducing Formulas from one Kmd of Units to Another 6-7
Ratios 7
Percentage 7-8
Condensed Numbers. The Use of 10» 8-9
Prefixes Used in the Metric System 9
Accuracy of Approximate or Abbreviated Numbers 9-10
Accuracy of Logarithms , 10
Absolute System of Units. The C. G. S. System 11-12
Dimensional Formulas. 12-14
Decisions of International Electrical Congresses concerning
Electric, Magnetic, and Photometric Units and Definitions. 14-16
Tables of Physical Quantities and Relations 17-26
Fimdamental 18
Gteometric 18
Mechanical 18
Magnetic, electromagnetic system • 20
' ' electrostatic system 21
Electric, electromagnetic system 22-23
•* electrostatic system 24-26
Electrochemical, electromagnetic system 23
' ' electrostatic system 25
Photometric 25
Thermal 26
TABLES OF CONVERSION FACTORS 27-173
General Remarks 27-29
Lengths 29-40
'j'gjjij og
Table oif Usual Measures 30-31
Table of Unusual, Special Trade or Obsolete Measures 31-32
Table of Foreign Measures 33-34
Tables of Inches in Fractions, Decimals, Millimeters, and Feet. 35-38
Digit Conversion Tables, 1 to 100 39-40
Surfaces 41-44
Text 41
Table of Usual Measures 41-43
Table of Unusual, Special Trade, or Obsolete Measures 43-44
Table of Foreign Measures 44
Digit Conversion Tables 44
Volumes; Cubic and Capacity Measures 45-65
Text 45-46
Table of Usual Measures 46-51
Digit Conversion Tables 51
Table of Unusual, Special Trade, or Obsolete Measures 52-54
Table of Foreign Measures 64-55
Xn TABLE OF CONTENTS.
PAGES
Weights or Masses 66-61
Text 66-57
Table of Usual Measures ; 67-58
Digit Conversion Tables 59
Table of Unusual, Special Trade, or Obsolete Measures 59-60
Table of Relative Weights (used in Chemistry) 60
Table of Foreign Measures 60-61
Weights or Masses and Lengths; Weight of Wires, Rails, Bars;
Forces and Lengths; Film or Surface Tension; Capillarity. . 62-63
Pressures; Pressures of Water, Mercury, and Atmosphere; Stress
or Force per Unit Area; Weights or Forces and Surfaces;
Weights 01 Sheets, Deposits, Coatings, etc 63-67
Digit Conversion Tables 67
Weights or Masses and Volumes; Densities; Weights of Materi-
als; Masses per Unit of Volume 67-69
Weights and Volumes of Water 69-71
Table of Weights of Water 70
Table of Volumes of Water 71
Energy; Work; Heat; Vis-Viva; Torque 72-77
Text 72-74
Tables 74-77
Digit Conversion Tables 77
Relations between Torque and Energy 78
Traction Energy 78
Tractive Force; Tractive Effort; Traction Resistance; Traction
Coefficient 78
Power; Rate of Energy; Rate of Doing Work ; Momentum 79-82
Text 79
Tables 80-82
Digit Conversion Tables 82
Forces; Weights Considered as Forces S3
Moments of Inertia. Text. '. 84
Moments of Inertia in Terms of the Mass 84
Moments of Inertia in Terms of the Surface 84
Moments of Momentum* Angular Momentum 84
Linear Velocities; Speeds 85-86
Angular Velocities, Rotary Speeds 86
Frequency; Periodicity; Period; Alternations 86-87
Linear Accelerations; Rate of Increase in Velocities ; Gravity.... 87-88
Angular Accelerations; Rate of Increase in Angular Velocities. ... 88
Angles (plane); Circular Measures 89
Solid Angles. 89
Grades; Slopes; Inclines 90-92
Ck)nversion Table, 1 to 15% 92
Time 93-95
Text 93
Tables 94-95
Discharges; Flow of Water; Irrigation Units; Volume and Time. 95
Electric and Magnetic Units 96-144
General Remarks 96
Mean, Effective, and Maximum Values 97
Resistance; Impedance; Reactance 97-100
Resistance and Length for the Same Cross-section 101
Resistance and Cross-section for the Same Length 101
Resistivity; Specific Resistance 101-104
Conductance; Admittance; Susceptance 104-105
Conductivity; Specific Conductance 106-107
Electromotive Force ; Potential ; DiflFerence or Fall of Poten-
tial; Stress; Electrical Pressure; Voltage 108-111
E. M. F. of Clark and Weston Cells at Different Tempera-
tures Ill
Electric Current; Current Strength or Intensity 112-114
CJurrent Density 114-115
Electrical Quantity; Charge 115-116
Electrical Capacity 1 17
Inductance; Coefficient of Self- or Mutual Induction 118-120
TABLE OF CONTENTS. Xlll
PAGES
Time Constant (of Inductive Circuits) 120
Frequency; Periodicity; Period; Alternations 121
Kinetic Energy of a Current in a Circuit 122
Electrical Energy or Work 122-123
Electrical Power 124-125
Electrochemical Equivalents and Derivatives 125-126
Electrolytic Deposits 126-127
Electrochemical Energy 128-129
Magnetic Reluctance; Magnetic Resistance 129-130
Magnetic Reluctivity: Specific Magnetic Reluctance; Mag-
netic Resistivity; Specific Magnetic Resistance 130
Magnetic Permeance; Magnetic Conductance; Magnetic
Capacity 130-131
Magnetic Permeability; Specific Permeance; Magnetic Con-
ductivity 131-132
Magnetic Susceptibility 132
Magnetomotive Force; Ainpere-tums; Magnetic Potential;
Difference of Magnetic Potential; Magnetic Pressure. .. . 132-134
Magnetizing Force; Magnetomotive Force per Centimeter;
Magnetic Force; Field Intensity; Magnetic Calculations. 134-137
Magnetic Flux; Lines of Force; Flux of Force; Amount of
Magnetic Field; Pole Strength 137-139
Magnetic Flux Density ; Manietic Induction ; Lines of Force
per Unit Cross-section ; Earth's Field 140-142
Magnetic Moment 142
Intensity of Magnetization; Moment per Unit Volume; Pole
Strength per Unit Cross-section 142
Magnetic Work or Energy 143
Magnetic Power 144
Photometric Units 144-149
Intensity of Light; Candle Power 144-146
P'lux of Light ; Spherical or Hemispherical Candle Power 147
Illumination 148
Brightness of Source 148
Quantity of Light 149
Light Efficiency ; Power per Candle Power 149
Thermometer Scales 150-163
Reduction Factors for One Degree 150
'Reduction Factors for Readings of a Temperature in Degrees. 150
Tables of Values from Absolute Zero to 6000* C 151-163
Money. 164-166
Fluctuating Currencies 166
Money and Length 166
Money and Weight 167
Scales of Maps and Drawings 168
Paper Measure 168
Miscellaneous Measures 168
Useful Functions of TT 169-170
Useful Numbers 170
Systems of Logarithms 171
Acceleration of Gravity 17 1
Mechanical Equivalent of Heat ' 171
Specific Heat of Water 171
Miscellaneous Foreign Measures 172-173
Index 175
SYMBOLS AND ABBREVIATIONS
USED IN THE TABLES AND TEXT.
The page-numbers refer to the pages where the explanation or most impor-
tant use is given.
a acceleration, 19
A acre, 43
A, a ampere, 15, 113
a are, 43
ah ampere-hour, 116
ap ai>othecary measures^ 46, 57
Aprx approxmiate within 2%, 28.
96
atm atnK>sphere, 66
av avwrdupois weights, 57
a-t ampere-turns, 132
B magnetic induction, 140
B, b, B, h sufloeptanoe^ 22, 104
B.A.U. British Association unit,
99
bbl barrel, 53
Brit. British or English, 30
BTU Board of Trade unit (kilo-
watt-hour), 77
BTU British thermal unit ,74,75
bu bushel, 50
C, c , C , c capacity , electnc , 23, 117
C, c coulomb, 15, 116
C^ Centigrade degrees, 150
c cord, 54
Cal calorie, large, 76
cal calorie, small, 75
Cal/min calorie (large) per mm-
ute, 81
cal/min calorie (small) per min-
ute, 80
cal/s calorie (small) per second, 81
cb cubic. 46
Cksm circular centimeter 42
Cft circular foot, 42
eg centigram, 57
C.G.S. or CGS centimeter-gram-
second, 11
ch chain, 32
C^ circular inch, 42
circ. circular 41
cl centiliter, 52
cm centimeter, 30
cm' square centimeter, 42
cm' cubic centimeter, 46
cm/s centimeter per second, 85
cm/s' centimeter per second per
second, 88
CM circular mil, 41
Cmm circular millimet^, 41
cp candle pK>wer, 144, 146
cwt hundredweight, 58
d diameter, 41
dg decigram, 57
dkg deca^n^am or dekajgram , 59
dkl decaliter or dekahter, 52
dkm decameter or dekameter, 32
dks decastere or dekastere, 54
dl deciliter, 47
dm decimeter, 30
dm' square decimeter, 42
dm' cubic decimeter, 48
ds decistere, 53
dwt pennyweight, 59
dyne-cm dyne-centimeter (erg) , 7 4
dyne/cm dvne per centimeter , 62
dyne/cm' dyne per square centi-
meter, 64
e base of Naperian logarithms, 17 1
e, e brightness of source of light,
16, 148
E, e, E, e electromotive force, 22.
108
Et electrcnnotive force at t degrees,
111
E, e ell, 32
E, E illumination, 16, 148
eff. eflfective value
elmg electromagnetic, 96
elst electrostatic. 96
e.m.f . electromotive force, 108
F* Falu*enhett degrees, 150 .
F farad, 15, 117
F, f force, 18
F magnetomotive force, 132
fl fluid measures, 52
ft foot or feet, 30
ft' square foot, 42
ft' cubic foot, 49
ft-gr foot-grain^ 74
ft-gr/s foot-gram per second, 80
ft-ib " ~
foot-pound, 74
XV
XVI
SYMBOLS AND ABBREVIATIONS.
ft-lb/min foot-pound per minute,
80
ft-lb/s foot-pound per second, 80
ft/C foot per hundred, 90
ft/ft foot per foot, 90
ft/M foot per thousand, 90
ft/min foot per minute, 85
ft/ml foot per mile, 90
ft/s foot per second, 85
ft/s^ foot per second per second , 88
fur furlong, 32
G, s, G, a conductance, 22, 104
g acceleration of gravity, 171
g gram, 58
gal gallon, 49
gi 8^.52
gr grain, 57
g-C gram-Centi^frade heat unit, 75
g-om gram-oentuneter, 74
g-cm/s gram-centimeter per sec-
ond, 80
g/cm gram per centimeter, 62
g/cm' gram per square centimeter,
64
g/cm' gram per oubio centimeter,
69
g/dm^ gram per square decimeter,
64
g/h gram per hour, 127
g/m gram per meter, 62
g/min gram per minute, 127
gr/in grain per inch, 62
gr/in' grain per cubic inch, 68
H heat, 26
H henry, 119
H hydrogen. 126
H magnetic flux density, 140
H magnetising force, 134
h hour, duration, 94
" hour, time of day, 93
ha hectare, 43
hf hectogram, 60
Hg mercury
hhd hogshead, 53
hks hektostere, 54
hi hectoliter, 50
hp horse-power, 81
hp-h horse-power hour, 77
hp-m horse-power-minute, 76
hp-s horse-power-second, 75
I , i, /, i current, (elec.) intensity of,
22, 112
I, /^ intensity of light. 16, 144
I intensity of magnetisation, 142
in inch, 30
in^ square inch, 42
in* cubic inch, 47
in Hg inch of mercury column, 65
in/m inch i>er mile, 90
int. international
J, j joule, 15, 74, 123, 143
J mechanical equivalent of hoaX ,26
k electric inductive capacity, 14,
18, 23. 24
k dielectric constant, 23
K moment of inertia,: 19
kg kilogram, 58
kg cal kilogram calorie, 171
kg-0 kilogram Centigrade heat
unit, 76
kg-km kilogram -kilometer, 76
kg-km/min kilo^pun-kilinneterper
mmute, 81
kg-m kilogram -meter, 75
kg-m/min kilogram -meter per min-
ute, 80
kg-m/s kilogram -meter per second,
81
kg/cm' kilogram per pquare centi-
meter, 66
kg/cm' kilogram per cubic centi-
meter, 69
kg/day kilogram per day, 127
kg/h kilogram per hour, 127
kg/hi kilogram per hectoliter, 68
kg/km kilogram per kilometer, 62
kg/1 kilogram per liter, 69
kg/m kilogram per meter, 63
kg/m2 kilogram per square meter.
64
kg/m' kilogram per cubic meter, 68
kg/mm' kilogram per square milli-
meter, 66
kg/t kilogram per ton (metrie), 78
kg/3rr kilogram per year, 126
kl kiloliter, 54
km kilometer, 30
km' square kilometer. 43
kw-h kilowatt-hour. 77
km/hr kilometer per hour. 85
km/hr/min kilometer per hour per
minute, 88
km/hr/s kilometer per hour per
second. 88
km/min kilogram -meter per min«
ute, 86
kw kilowatt, 82
kw-m kilowatt-minute , 76
kw-s kilowatt-second, 75
L , 1 , L , I inductance , self-inductioni
23. 118
£, I length, 18
1 liter, 48
lb pound, 58
Ib-C pound Centigrade heat unit,
75
Ib-C/min pound-Oentigrade heat
unit per miute, 81
Ib-F pound-Fahrenheit heat unit,
75
Ib-F/min do. per minute, 81
Ib/bu pound per bushel, 68
lb/day pound per day, 127
lb/ft pound per foot, 63
lb/ft' pound per square foot, 64
lb/ft' pound per cubic foot, 68
lb/gal pound. per gallon, 68
Ib/h pound per hour, 127
lb/in pound per inch, 63
lb/in' pound per square inch, 65
lb/in' pound per cubic inch, 69
lb/ml pound per mile, 62
Ib/qt pound per qt, 68
Ib/tn pound per ton (av), 78
lb/yd pound per srard, 62
lb/yd' pound pereubio jrard, 68
SYMBOLS AND ABBREVIATIONS.
XVU
Vb/yr pound per year, 126
li Unk. 31
log logarithm c 171
logio common logarithm, 171
log; Naperian logarithm 171
m magnet pole, strength, 20
M mass, 18
M mechanical equivalent of heat,
171
m meter. 30
m minute* duration, 94
"» minute, time of day, 93
m^ square meter. 43
m^ cubic meter. 50
m/min mdter per minute, 85
m/s meter per second, 85
m/sec* meter per second per sec-
ond, 88
mg milligram, 57
mg/mm milligram per millimeter,
62
mg/s milligram per second, 127
mU^ square mil, 41
min mmute, 94
ml milliliter, 46
ml mile, 31
mP square mile, 43
ml-lb mile-pound, 76
ml-lb/min mile-pound per minute,
81
mlAr mile per hour, 85
ml/hr/min mile per hour per min-
ute, 88
ml/hr/sec mile per hour per sec-
ond, 88
ml/min mile per minute, 86
mm millimeter, 30
mm' square millimeter, 41
mm^ cubic millimeter, 52
mm Hg millimeter of mercury col-
umn. 65
mm/m millimeter per meter, 90
m.m.f. magnetomotive force, 132
mo month, 94
mol gram molecule, 60
N number of turns, 20
n,n any number
n frequency, 23, 87, 121
na nail, 31
O ohm, 15
O oxygen, 126
OS ounce, 58
oz/h ounce per hour, 127
oz/min ounce per minute, 127
p page
p pole (length), perch, 32
p pressure, 19
P, P power, activity (mechanical,
electric, magnetic, etc.), 19.
20, 23. 124, 144
p.d. di£ference of potential, 108
per" between two units means
first divided by second, 4'
phys. physical or physics
pk pecK, 49
pt pint, 47
Q« Q» Qi Q quantity of electricity.
22, 115
Q, Q quantity of light, 16, 149
qr quarter, 31, 53, 60
qt quart, 48
R, r, R, r resistance (electric), 22,
97
R Rdaumur degrees, 150
R reluctance, 129
R rood, 44
r rod, 32
rev revolution, 89
rev/h revolution per hour, 86
rev/min revolution per minute, 86
rev/min/min revolution per min-
ute per mmute, 88
rev/min/s revolution per minute
per second, 88
rev/s revolution per second, 86
rev/s/s revolution per second per
second, 88
rhp revolution per hour, 86
rpm revolution per minute, 86
rps revolution per second, 86
s second, duration, 94
" second, time of day, 93
s shilling, 165
s stere, 51
S, 8 surface, 18
sec second
sp specific
sp. gr. specific gravity, 18
sq square, 41
S.U. Siemens unit, 99
subtr. subtract
T, t time, 18
t temperature degrees, 111
t ton, metric, 58
t-km ton-kilometer, 78
t/cm' ton (metric) per square cen-
timeter, 67
t/m' ton (metric) per square
meter, 65
t/m' ton (metric) per cubic meter.
69
t/yr ton (metric) per year, 127
tn ton (avoirdupois), 58
tn -ml ton-m ile , 7 8
tn/ft' ton (av) per square foot, 66
iTi/iD* ton (av) per square inch, 66
tn/yd' ton (av) per cubic yard, 69
tn/yr ton (av) per year, 127
tJ, u, C7, M difference of potential,
22 108
U.S. United States
V velocity, 19
v. V velocity of light, 14, 21, 24, 96
V, V volume, 18, 69
V, V volt, 15, 110
vol. volume •
W, w watt, 15,80, 125
W weights, 69
W, W work, energy (mechanical,
electric, magnetic, etc.)i
vis-viva, impact, 19. 20,
23, 122, 143
w-h watt-hour, 76
w-s watt-second, 74
X, X, ^, a; reactance, 22, 97
arc capacity reactance, 22
XVIU
SYMBOLS AND ABBREVIATIONS.
Xa magnetic reactance, 22
x,y,Y,y admittance, 22, 104
yd yard, 30
yd2 square yard, 42
yd^ cubic yard, 50
r year, 94
. z, Z, X, impedance, 22, 97
S
a (alpha) angle, 18
ff (beta) angle. 18
r (gamma) conductivity (electric).
22, 105
r 0.001 milligram, 69
9 (delta) density, 18
(theta) temperature, 13, 18, 26
K (kappa) susceptibility, 20, 132
fi (mu) magnetic permeability, 14,
18, 20, 131
fi micron or micro-meter, 31
fif.1 milli-micron, 31
w (nu) reluctivity, 20, 130
n (pi), angle of 180°, 89
K ratio of circumference to diam-
eter, 169
P (rho) resistivity, 22, 101
(phi) angle of phase difference,
124
*, ^ (phi) magnetic flux, 20, 137
* flux of light, 16, 147
w (omega) angular velocity, 19,
23, 86, 121
(B flux density, magnetic induc-
tion, 20, 140
C magnetic capacity, 20
^ magnetomotive force, 20, 132
3C magnetizing force, field inten-
^sity, 20. 134
3C flux density, 140
3 intensity of magnetization, 20,
911 magnetic moment, 20, 142
(n reluctance, 20, 129
X multiplication
- (hyphen) between two units
means their product. 4
-t- division, first quantity divided
by the second
% per cent or per hundred, 7 . 90
"/oo per mil or per thousand, 8, 90
y/ square root
4/ cube root
■"* reciprocal
I square, 41
* square root
« cube or cubic, 46
3 cube root
low for condensed numbers, 8
** degree, 89
' foot
' minute, 89
" inch
" second, 89
'" line (1/12 inch)
£ pound sterling, 165
$ dollar in the United States, 166
3 dram, apothecary, 69
§ ounce, apothecary, 69
9 scruple, 69
CO frequency, 87, 121
INTRODUCTION.
inter-relation of units.
In order to establish a stable, systematic set of relations between the
various units in use, the first requisite is to find which the proper funda-
mental relations are, otherwise the secondary relations may have diflferent
values depending upon how they have been calculated, that is, they will
form an unstable system. Such simple relations as those between pounds,
ounces, grams, kilograms etc., are definitely established by law and are
well known. The relations between units of length and units of capacity
(volume) are sli^tly less simnle but are also, or at least should be, estab-
lished by law. The relation between what are commonly called weights
(more correctly masses) and lengths, becomes somewhat more complicated,
but by means of the mass of water these two kinds of units become defi-
nitely connected with each other, at least in the metric system, and from
that to all others; this relation is also defined by law.
But with the more widely dififering units the relations become more
complicated. It may even seem at first thought as though there was no
relation between such units as a foot and a horse-power or a watt; or
between a pound and a degree of a thermometer, or between a foot and an
ampere of electric current, etc. Yet all such units can be shown to be con-
nected with each other, often more simply than might at first appear. In
a set of values of different units in terms of the others it is therefore neces-
sary to find and establish all the necessary fundamental relations, and no
more, or else the system is not a stable one, and the derived values become
different, depending upon how they are calculated.
To find the connecting links between such widely different units, it is
necessary to reduce them to some unit or quantity common to aU, as a
direct numerical equivalent can be given only between units of the same
kind. This common unit, or quantity, and the only one, is energy. Energy
can be expressed in terms of combinations of all the different units com-
monly used in practice, and by expressing the same amount of energy in
terms of these different combinations of units their relations to each other
may be established.
An analysis shows that there are three typicallv different sets of well-
established units in common use, in terms of which energy is expressed or
measured, and these three sets or groups include all the units generally
used.
The first and most stable or fundamental are the absolute units and all
those having a known and invariable relation to them; these include the
electrical units, which are the newest, and were wisely based on the absolute
ones. They are the same throughout the universe; they are invariable
and are independent of any constant of nature, except perhaps the unit
of time, which depends on the revolution of the earth around the sun, but
as this has been so accurately determined it may safely be considered here
as an absolute quantity; it is at least invariable throughout the universe.
The next group of units is the one involving the constant of nature
called the attraction of gravitation of our earth, and they are therefore
often called gravitation imits; they are, therefore, purely terrestrial units,
and would be quite different on other celestial bodies, and, in fact, are,
strictly speaking, different even on different parts of this earth, although
for most purposes they may be considered to be the same. They includo
such \mits as the pound or kilogram considered as weights, the horsa-
INTRODUCTION.
power as usually defined, etc. It will be found upon investigation that this
whole group is linked with the first group, namely, the absolute and elec-
trical units, through the value of the acceleration of gravity, a terrestrial
constant of nature, and unfortunately one which is variable and has no
universally, accepted normal value. If some standard normal value of this
constant were universally established and were fixed definitely, this group
of units would be connected with the absolute units by a fixed relation, and
the former would then all have definite and invariable values in terms of
the absolute units. The acceleration of gravity is therefore the connect-
ing link, and the only one, between these two groups. Throughout the
tables in this book the standard value of the acceleration of gravity has
been taken as 980.596 6 cm (the authority is given under units of Accel-
eration).
The third and last group of units is the one involving a property of
water, which is also a constant of nature. This group includes such units
as the calorie, the thermal unit, the degrees of thermometer scales, etc. It
will be found upon investigation that this whole group of units is linked
with the first group, namely, the absolute and electrical units, through
the value of the specific heat of water; and moreover this is the only con-
necting link. This constant differs from the acceleration of gravity, which
is the connecting link between the first and second groups, in that it has
not a variable value like gravity; but, on the other hand, its value has not
yet been determined with such accuracy. It is like gravity in that it is a
constant of nature whose value must be determined by experiment and
definitely established before fixed and stable relations between this group
of units and the absolute units can be estabhshed.
The relations between the various units in any one group are, as a rule,
fixed by definition or by law; the horse-power, for instance, is defined in
terms of the f oot-poimd and the minute ; or the calorie is defined in terms
of the thermometer scale and a quantity of water. Such relations are there-
fore established and require no experimentally determined . constant in
order to express their relations to each other* no matter what the value
of the specific heat of water is, the relation between the calorie and the
thermometer scale is fixed. But when we wish to express the values of
any of the units of one group in terms of units of the other, then the numer-
ical values of these two connecting links, namely, the acceleration of gravity
and the specific heat of water, must be known. If, for instance, horse-
powers are to be expressed in absolute or electrical units like watts, the
value of gravity must be known ; or if calories are to be expressed in abso-
lute or electrical units like joules, the value of the specific heat of water
must be known; or if calories are to be expressed in foot-pounds, then
both these constants of nature must be known.
Moreover, the values of gravity and the specific heat of water are the only
two constants that must be known in order to reduce any of these different
units of energy to any of the others. By accepting or fixing a definite
value for each of these two constants the relations between all the units
of these three groups become linked together into one single stable system
in which all the relations between any of the units remain absolutely the
same no matter how they have been calculated. As no standard values
of either of these two constants have as yet been universally accepted or
agreed upon, which is very unfortunate, the writer has, in the preparation of
these tables, selected certain values as the ones which seem to be the best
that exist at the present time, and which at least have a semi-official en-
dorsement.
As the establishment of fixed values for these two constants forms an
inflexible stable system of units, there must exist some relation between
these two constants themselves. It can readily be shown that the specific
heat of water divided by the acceleration of gravity gives the mechanical
equivalent of heat, when all are reduced to the same terms. The value used
throughout this book is 426.9 kilogram-meters per kilogram-Centigrade
heat unit (see authority under units of Energy). When thus defined this
constant becomes a secondary or derived one.f which is as it should be,
* »■ ■ ■ ^■— I ■ I I ■■■■■» — — -■ - ..■■ ^
t While this is, strictly speaking, the more rational way of deducing those
constants, yet the author has preferred to accept the simpler value above
given for tne mechanical equivalent and has made the specific heat the in-
commensurable derived unit for reasons explained under units of Energy.
INTER-EELATION OF UNITS.
because it does not involve the absolute units, but is the relation between
the second and third groups, both of which are separated from the absolute
units by one of these two empirical and therefore less stable constants.
If the heat units and what are called the gravitational units, like the
pound (weight), the horse-power, etc., had originally been defined in terms
of the absolute units, as was wisely done with the electrical units, there
would have been no necessity of knowing the values of gravity, the specific
heat of water, or the mechanical equiv^ent of heat, in order to establish
fixed and invariable relations between all the units in use. Such fixed rela-
tions would then be established definitely by definition. Concrete stand-
ards of measurement complying as closely as possible with the defined
values would then require the experimental determinations of these con-
stants, but the exact theoretical relations would not. Such is the case with
the electrical units which are defined with absolute accuracy in terms of
the absolute units and require no experimentally determined constant to
connect them with the absolute units; in the electrical system of units the
problem therefore is not to find such an empirical relation, but to deter-
mine concrete standards which comply with the defined relations as closely
as possible.
if those on whom the duty will fall to establish a system of units of light
would follow the good example set by those who established the electrical
units, and base them on the absolute system of units, instead of on another
different constant of nature, like the candle or incandescent platinimi, they
would avoid creating a fourth group of units whose empirical relation to the
absolute, electrical, and to the other two groups of units would have to be
determined, and would be uncertain until it was. What is called radiated
light, usually measiu'ed in spherical candle-power, is a true power or a rate
of work, and the absolute unit would therefore be a dyne-centimeter per
second. If the practical unit could be defined as a decimal multiple or
fraction of this, like the watt is, it would at once become a definitely known
unit. It would then become necessary, as it was with the three funda-
mental electrical units, to establish a concrete standard which would com-
ply as definitely as possible with the defined unit. If, however, the prac-
tical unit is defined on some independent basis, like the present unsatis-
factory and uncertain temporary light units, it will become necessary to
determine experimentally the mechanical eauivalent of light, before any
of the much-needed relations could be established with the other existing
units, like the electrical, gravitational, or heat imits. In the case of light
units the matter becomes complicated on account of the different wave-
lengths and combinations of w&ve-lengths which have different effects on
the eye in perceiving light. Owing to the absence of any known relations
between lignt enerp^ and energy stated in other units, no relations^between
those groups of units could be given in this book.
The relations above described, which exist between the different groups
of units, may be represented graphically as shown in the following diagram:
GRAVITATIONAL.
UNITS and their
derivatives; gram
or pound as weights,
f oot-poands, horse-
powers, etc*
^Mechanical equiva^
HEAT UNITS and
their derivatives;
calorie, thermal
unit, thermometer
scale, etc.
LIGHT UNITS and
their derivatives;
candle power,
lux, etc.
ABSOLUTE or ELECTRICAL UNITS
and their derivatives; gram or pound
as masses, dyne, centimeter, foot,
second, watts, joules, etc.
INTRODTTCTION.
The most fundamental, namely, the absolute or electrical units, are shown
at the bottom. The gravitational units to the left are shown linked to
these through the value of the acceleration of gravity. The heat units
to the right are shown linked to the absolute and electrical units through
the value of the specific heat of water. Finally, the ^avitational units are
linked to the heat units through the mechanical equivalent of heat, which
is the quotient of the other two linking values. The relation of any one
of the three to either one of the others can be found either by means of the
direct link or indirectly by means of the two other links; the result must
be the same if the values expressed by these links are definitely established.
The group of units of hgnt at present in temporary use are shown as a
fourth p^oup, the connecting links of which are unfortunately not yet known,
for which reason no relations to other units can be given, although such
definite relations it seems ought of necessity to exist at least whenlight is
considered as a radiation of some definite wave-length.
COMPOUND NAMES OF UNITS.
When in compound names of imits of measurement the names of the sim-
ple units are joined by a hyphen, as in foot-pounds, horsepower-hours, etc. . it
signifies the prodtuA of the simple units; that is, an amount of energy stated
in foot-pounds means that the number of feet is multiplied by the number
of pounds, to give the foot-pounds; in other words, the compound quantity
varies directly with each of the others. If, however, the simple units are
joined by the word ' *per" as in feet per second^ pounds per mile, etc., it means
that the first is divided by the second; that is, a velocity in feet per second
means that the number of feet is divided by the number of seconds. In
other words, the compound quantity varies directly with the first and in-
versely with the second. An acceleration in * 'miles per second i>er second **
means that the velocity in miles per second is divided by the number of
seconds during which this velocity^ was acquired.
The above is the correct practice, but it is unfortunately not followed
by idl writers. For instance, "horsepower per hour" is wrong; it should
be horsepower-hour, as the number of hours is a multiplier and not a
divisor.
DISTINCTION BETWEEN UNITS AND QUANTITIES
MEASURED IN THOSE UNITS.
Serious errors are not infrequently made by failing to distinguish between
the calculations of the values of the units themselves and the calculation of
quantities measured in terms of those units. By using the reduction factors
in the way thejr are given in the tables in this book, no mistakes can be
made as the units have all been calculated, and the value of each unit is
given in terms of each of the others; they are just Uke the selling prices of
one apple in terms of different moneys, the only remaining calculation is
to mmtiply the quantity of these units by the reduction factor given in the
table' thus, if 1 foot-pound =0.14 kilogram-meter, then 3 of these foot-
pounds^ 3X0.14 kilogram-meters. But in determining the reduction fac-
tors themselves, or in similar calculations, which are not infrequent, errors
are very apt to be made by multiplying where one should divide. A fre-
quent case is that in which a formula reads in, say, distances in meters and
weights in kilograms and it is desired to change it to read in feet and pounds *
this will be described in the next section; or, in dealing with compound
units, such as foot-pounds (ft X lbs) and feet per pound (ft -^ lbs), which
were described above.
A clear distinction should be made between the units themselves and the
quantities measured or expressed in terms of those units. Thus, 1 foot «- 12
inches is the value of one unit in terms of another, but it is not correct to
interpret this by saying: a length in feet "« 12 X its length in inches, as that
would be 144 times too great. It should be interpreted as follows: If 1 foot
equals 12 inches, then a length of any other number of feet must be mul-
tiplied by 12 to reduce that length to inchee, or to express it in terms of
inches. Simple as this may seem in this almost self-evident illustration,
mistakes are easily and frequently made in more obscure relations, especially
DISTINCTION BETWEEN UNITS AND QUANTITIES. 5
in substituting one kind of a unit for another in a formula, which will be
described in the next section.
The larger the unit the smaller will be the number of those units which are
contained in a given quantity; that is, the smaller will be the number ex-
pressing the size of that quantity in terms of those units. For instance, a
meter is greater than a yard, hence the nimiber of meters in a given distance
is less than the nimiber of yards. This seU-evident rule is often of value
as a check to avoid confusion between the calculation of the units them-
selves and the quantities in terms of those units; errors of this kind are
especially liable to occur when the two units have nearly the same values.
This nile should not be confounded with another quite different case in
which the values of two' different units are given in terms of the same third
unit; for instance, 1 kilogram-meter = 2.3 heat units and 1 foot-pound = 0.32
of the same heat units; here the foot-pound is a smaller unit than the kilo-
gram-meter, hence its value in terms of a third unit will of course also be
smaller. But 1 heat unit — 3.1 foot-pounds and 1 heat unit —0.43 kilogram-
meter; here the same quantity, 1 heat unit, is expressed first in small units,
namely, foot-pounds, tnen in large units, kilogram-meters; hence 3.1, the
number of the smaller units, is of course greater than 0.43, which is the
number of the larger units.
In calculating one imit from another unit, great care must be taken to
avoid multiplying when one should divide, or the reverse. Thus if the
unit 1 kilogram-meter equals 2.3 heat units, it would not be correct to say
that because the unit 1 kilogram = 2.2 lbs and 1 meter =3.3 feet, therefore
the unit 1 foot-pound = 2.3 X (2.2 X 3.3) = 16.8 of those heat units; it should
>e 1 foot-pound = 2.3 -5- (2.2X3.3) = 0.31 heat units. The safest way to
avoid such errors is to remember that the statement 1 kilogram-meter = 2.3
heat units really is a true equation between units and means 1 kilogram X 1
meter = 2.3 heat units; one should then substitute for 1 kilogram its equal
in terms of the other unit, namely 2.2 X 1 pound, and for 1 meter its equal,
namely 3.3 X 1 foot ; then 2.2 X 1 poimd X 3.3 X 1 foot = 2.3 heat units, which
when reduced gives the imit 1 foot-poimd = 2.3 ^(2.2X3.3) or 0.31 heat
unit. These terms here always refer to units and not to quantities meas-
ured in terms of those units, which latter is the case in formulas.
Expressing this relation in algebraical terms, it means A XB=2.3C, in
which A is one kilogram and not a weight denoted in kilograms, B is one
meter and not a distance expressed in meters, and C is one heat unit ; they
should here be considered as concrete things like a certain piece of brass,
a certain stick, and a certain lump of coal, as distinguished from abstract
units or as distinguished from a mere number representing a measurement
in terms of those units. ^ Now if in the same way a is one pound or a differ-
ent piece of brass, and h is one foot or a different stick, then as 1 kilo^am^
2.2 pounds, it follows that A = 2.2a and similarly i} = 3.3&. Substituting
these in the first equation gives 2.2aX3.3& = 2.3C, which when reduced
gives aX6 = 2.3-i-(2.2X3.3)C = 0.31C.
In formulas, however, the letters do not stand for the units themselves,
but for quantities measured in terms of the units, and hence one must multi-
ply instead of divide, or the reverse, ns will be explained in the next section.
A similar error is very apt to arise in calculating units containing the
word "per," which word simiifies a division, as was explained above under
compound units. This is the case, for instance, in finding the value of 1
kilogram per kilometer from the value of 1 pound per foot, or a velocity
of 1 mile per minute from that of 1 foot i>er second. It must be remembered
that the unit following the word "per" is a divisor. The safest way always
is, as above described, to write out the whole expression carefully, then
substitute equivalent values and reduce. Thus: 1 mile per minute =26.8
meters per second, may be written 1 mile -5- 1 minute = 26.8 meters per second ;
then to find the value of 1 foot per second from it, substitute 5 280 feet for
1 mile, and 60 seconds for 1 minute, thus (5 280 X 1 foot) -4- (60 X 1 second) =
26.8 meters per second; then reduce by dividing both sides by 5 280 and
multiplying both by 60, giving 1 foot + 1 second, or 1 foot per second = 26.8
-i-5 280X60 = 0.30 meter per second.
When the unit following the word "per" is not to be changed, then the
division just referred to does not enter into the calculation. Thus to
reduce some value of 1 foot per minute to that of 1 mile per minute or to that
of 1 meter per minute^ is merely a reduction of feet to iniles, or to meters.
INTRODUCTION.
RBDUOING FORMULAS FROM ONE KIND OF
UNITS TO ANOTHER.
It often occurs in practice that a formula is given for one set of units,
such as meters, kilograms, seconds, etc., and it is desired to use it for other
units such as feet, pounds, minutes, etc. While this is generally a simple
calculation, it is very apt to be made incorrectly^, giving an entirely wrong
result. The error arises from the fact that one is apt to forget the distinc-
tion between a unit and a quantity measured in terms of that unit (see the
preceding section) which distinction involves the difference between whether
one should multiply or divide. For instance, a foot is larger than an inch,
but the number of feet expressing a certain distance is amaUer than the num^
her of inches expressing that same distance.
The best way to avoid mistakes is to remember that in the uetud formulaa
the letters represent quantities measured in terms of certain units; they do not
rei>resent the units themselves. To change a formula from one kind of
units to another, it is therefore necessary to replace each letter of the original
formula by another letter combined with such a reduction factor as will
reduce the measurement made in terms of the new unit, to that made in
terms of the original one. Thus if a formula contains the letter L as repre-
senting length expressed in meters, and it is desired to change this to I
expressed in feet, then substitute for L the equivalent Z + 3.28 or ZX 0.305,
because any length I measured in feet when divided by 3.?S (or multiplied
by 0.305) will be that same length expiessed in meters; and as the original
formula is correct for meters, it will be correct to substitute iX 0.305 for L
because the number thus obtained will be the same as the number which
expresses L in meters. Or. if the original formula contains the letter W^
representing weight expressed in kilo^ams, and it is desired to change it
to w expressed in pounds, then substitute for W the equivalent 10X0.454,
becaiLse any weight in poiwds when multiplied by 0.454 will give the same
number as when expressed in kilograms, which latter number is what the
original formula called for. After having thus replaced each letter by a
new one and a constant, all these constants may be combined into one if
desired.
It must not be forgotten that the quantitv which the whole formula rep-
resents, that is, the quantity which is to be calculated by means of the
formula, is generally also in terms of some unit (imless it is a mere ratio,
percentage, or number), and care must therefore be taken to also substi-
tute a new letter and reduction factor for it, if it is desired to change it also.
For instance, if a formula for determining the metric horse-power from data
given in meters and kilograms, is to be cnanged to read in feet and pounds,
as just described, the value obtained in applying the new formula woula
still be in metric horse-power, notwithstanding that the formula has been
reduced to feet and pounds; if the result is to be in English horse-powers,
then the letter representing the metric horse-power must likewise be changea
to a new one with its reduction factor, just as was done with the others.
This will be illustrated below by an example.
The safest rule of thumb for the changing of the units of a formula is
therefore as follows: Substitute for each letter of the old formula a new letter
multiplied by the value of the new unit in terms of the old one, as given in the
tables in this book. Then the new letters will represent quantities meas-
ured in terms of the new units. Thus for L (meaning meters) substitute I
([meaning feet) X 0.305, this number being the value of one foot (new unit)
in terms of meters (old unit), as obtained from the tables. For if 1 foot =
0.305 meters, any other number of feet (namely I) must be multiplied by
0.305 to reduce it to the number of meters (L) called for by the ori^nai
formula; or, in other words, L and ZX 0.305 arc equivalents, and either
may be substituted for the other.
(3r put into algebraical terms: if according to the table 1 foot »" 0.305
meter, then I feet are equal to ZX 0.305 meters; or a length represented by I
in feet is represented by ZX 0.305 in meters; but the latter quantitv is
nimierically equal to L, hence L — lX 0.305, which is a true equation in which
the letters represent the same length measured first in meters and then in
feet; and (ZX 0.305) may therefore be substituted for L in any formula.
RATIOS. — PERCENTAGE.
The above rules are illustrated in the following example: Let
in which P represents the power in metric horse-power which is required
to operate a windlass or elevator to raise W metric tons L meters in height
in T seconds. The constant 16 includes the friction loss, and the reduction
factors. It is required to change this to English horse-powers (p), short
tons (t&), feet (2), and minutes (/).
From the tables, one English horse-power (the new unit) is equal to 1.01
metric horse-powers (the old imit), hence P=pX1.01. One short ton of
2 000 lbs. is equal to 0.907 metric ton, hence Tr=«>X 0.907. One foot
equals 0.305 meter, hence L — 2X0.305, and one minute equals 60 seconds,
hence 7^=tX60. The constant 16 being in terms of no unit, as it is a mere
nimiber or ratio, remains unchanged. Substituting all these in the old
formula, gives
W1A1 .^ ti>X 0.907 X/X 0.305
PX 1.01 = 16 j^^ ,
which when all the constants are combined, gives the new formula
p=0.073-T-,
which will be correct for quantities expressed in terms of the new units.
When the orip^nal formula is to be applied only once for one set of values
in other units, it may sometimes be simpler to reduce the original data in
short tons, feet, and minutes to metric tons, meters, and seconds by means
of the tables; then substitute them in the original formula and calculate
the result, which will be in metric horse-power; then reduce this result back
to English horse-power by means of the tables.
RATIOS.
A ratio means the relation of one quantity to another, that is, one quan-
tity divided by the other. It is therefore always a mere number and is
independent of any units, except that the two quantities concerned must
always be in terms of the same imits. When the statement 's made that a
certain number is the ratio of one quantity to another, it invariably means
that the -first was divided by the second, and never the reverse. A con-
venient rule of thumb for calculating the ratio is to divide by the one that is
preceded by the word ''to."
Unless otherwise stated, a ratio is the figure thus obtained by division
and when it is reduced to a single number (as distinguished from a vulgar
fraction) it means so-and-so much per unity. Thus a ratio may be ex-
pressed as 3 to 4 (or ^) for instance, or as 0.75; in the latter case it means
0.75 per unity. Ratios are, however, more frequently stated as percent-
ages, that is, as ratios per idO, in which case the above defined value must
be multiplied by 100. Thus the ratio of 3 to 4 is then expressed as 75% ,
as described more fully below.
PERCENTAGE.
Percentage values represented by the sign % and meaning jyer hundred^
are often used as a convenient way of representing a ratio, fraction, or rela-
tion, so as to avoid the use of fractions, at least in most cases. ^ Thus a
recovery of 3 horse-power out of 4 may be expressed as the fraction % or
0.75, or it may more conveniently be expressed as 75% , which avoids frac-
tions and reduces the relation to one which is based on 100. The objection
to it is that such values of necessity introduce a multiplication or division
by 100, which sometimes gives rise to confusion. To have adopted the
system of stating such values per unity instead of per hundred, would have
avoided this confusion, but the values would then in most cases be frac-
tions, for which reason the percentage values are usually preferred.
Percentage values, like other ratios, are mere numbers and are not in
terms of any imits ; they are moreover the same for all units provided only
that the two quantities from which the percentage value is calculated are
always in terms of the same units.
JJ
INTRODUCTION.
While percmt«ces ocouning in practice are geaeislly between 1 and 1'
.... .1 . -urej be leastlian 1 or greater than 100. When leestl
,e BJid read them properLy; thui .25%, f
IS a of 1%, ia often incorrectly read as "" • -
rh n. riuiF i= (|> mv-iAnt 1 ■!. ''nr '
. .o the ton__
minus 6) o( 8"; then divide 2 by 8 and muit^
by loO?'^v^ng'33>|%' '
TheBii(n%oi3™nietiiL._. ^ .
Tlii:^ differs from percent only in that it means per 1 000 instead a\
OOHDENSSD NTTMBERS. The Use of 10".
orvery small numbers is often moved to the next to the to^ left-hand dlipt.
Hud the amount by which it has thereby been lowered or raised is written
in the form of 10 to the required power. Thus the large Dum-
"- ■ ■ ^.■^^VM^ (that is, 1.23X 1 000000); or
is, 1,23X0.000001). A pomtivo
' Rer than the one given,
gxp&nfni, whether ^ai-
live or uef^BiivH, always mruL-aLea mv Tzu'wrr cif piaceB that the deemial
point has been moved. When the exponent is negative it also means that
m the oriwnil number there ii one less zero between tho point and the first
diitit than the exponent indicates: thus 10-« means that there are 5 »eros
between the point and the first figure; this, however, applies only when
the decimal jioint in the condensed number ia between the first and second
left-hand digits, sa usual; sometimes, though not according to good pmo-
tice, i.23X10-» is written 12.3X10-', in which ease this rule of thumb
To multiply or divide such condensed numbers, perform that operation
with the nmnerical part, and then merely add (algebraically) the exponents
of the lO's in the case of a multiplication or subtract them (algebraically)
in the case of a division. Thus (3.4xlO")X(5.8XHH)-l9.04>! IQi" or
I.BOIXIO". Or(3.4X10-»)X(5.6X10->)-19.0*X10-'°,orbelter
3 oHeo 1
It of 10 indicates that'the real nun
1.904X10-«. Aeain, (l.ixl0")X(3.4Xl0-')-4.l
(3.4X10-')-1.2X10», as -4 aubtrsctei' ' ■ '
To square, cube, etc, such a condense
with the numerical part and tnulliply . .... . . . .,
respecUvely. For extracting the square or cube root, perform that oper^
reepeotive^; if this does not divide evenly, then first change the exponent
so that it will, by moving the decimal point in the pumencsl part; thtls, to
extract tho cube root rf6.4 X 10', write it 64. X 10", then the cube root ia
4. X lOa.
To add or subtract such condenned numbers they must first be reduced
to the same power of 10; then add or subtract the numerical paRs and
affix 10 to the same power. Thus (1.23X 10') + (4.5SXI0») = (I2.3X 10»)+
PREFIXES — ^APPROXIMATE NUMBERS.
(4.66X10«)=16.86X10», or better, 1.686X10^. The same rule applies
when all the exponents are negative, or when some are positive and some
negative, although in the latter case it will generally be simpler to reduce
all to the real numbers bv eliminating the factor 10.
10*^ is never used in such numbers, but it sometimes results from calcula-
tions. It b equal to 1, and is therefore always omitted.
TABLE OF CONDENSED NUMBERS.
Let 1.23 represent any number, of any number of places of figures; then:
1.23 X10« -1230 000.
1.23 X10» = 123 000.
1.23 X10< - 12 300.
1.23X108 - 1230.
1.23X102 - 123.
1.23X101 - 12.3
1.23X100 «=. 1.23
1.23X10-1= 0.123
1.23X10-2- 0.012 3
1.23X10-3- 0.00123
1.23X10-*- 0.000123
1.23X10-5- 0.000 012 3
1.23XlO-«- 0.000 00123
BFIXES used in the METRIC 1
3TS1
Micro- 0.000 001 or
10-8
Milli- 0.001 or
10-3
Centi- 0.01 or
10-2
Deci- 0.1 or
10-1
Deca- or I>eka- 10. or
101
Hecto- or Hekto- 100. or
102
Kilo- 1 000. or
103
Myria- 10 000. or
10«
Mega- 1 000 000. or
10*
Thus one mtZh'meter equals 0.001 meter, or one kilometer equals 1 000
meters.
AOOURAOT OF APPROXIMATE OR ABBREVIATBD
NUMBERS.
When a value is known to be correct only to a limited number of places
of figures, or when the figures are abbreviated to a few places, the possible
error in such approximate figures, or in other words their acciu'acy, varies
not only with the number of places of figures, but also with the value of the
left-hand digit, as will be explained below. It is here assumed that the
last right-hand figure has always been increased by one, when the next
figure would be 5 or over; thus an^hing between 12.85 and 12.89 (includ-
ing the former) would be abbreviated to 12.9, while anything between
12.80 and 12.85 (but not including the latter) would be abbreviated to 12.8.
The possible error is less the greater the number of places of figures. For
two places of figures (from 10 to 99) the greatest error (namely ± 5 in the 3d
place) is from 5% for the smaller numbers to ^% for the larger. For three
places of figures (100 to 999) the greatest errors are ^% to %o%. For
four places of figures, the greatest errors are Ho% to Hoo%> For five
places J^% to Hooo%. etc.
The possible error is also less as the left-hand digit is greater. Thus the
abbreviated number 10.1 may mean anything from 10.05 to nearly 10.15,
and the greatest error may therefore be ± 5 in the next (2d) place of deci-
mals, or numerically this would be 5 times the left-hand digit 1. But in
the abbreviated number 99.9, which may mean anything from 99.85 to
nearly 99.95, the greatest error, namely ±5, is only about %o times the
left-hand digit 9; that is, in percentage only about M.0 as great an error as
it was for 10.1. It follows therefore that values beginning (at the left hand)
with the low digits 1, 2, 3, etc.. should be stated to one place more than
those beginning with the high aigits 9, 8, 7, etc., if the accuracies of the
10 INTRODUCTION.
abbreviated values are to be more nearly the same for all. This can be
made use of to advantage in tables of such approximate or abbreviated
values. If, for instance, such a table is intended to be limited to three
places of figures, the greatest error will be 0.5^ for the low numbers (100)
and 0.05% for the high ones (999); but by giving four places of figures for
all nimibers between 100 and 499, and three places for those from 500 to
999, making an increase of only 1 figure for every 6, or about 16%, the
greatest error will be reduced fivefold, namely to 0.1%. The table will then
take a mean position (in accuracy) between a three-place and a four-plrx»
table; the variations of the greatest errors in different parts of the table
will, however, be the same in all three.
The location of the decimal point does not, of course, affect the percentage
accuracy; thus 101., when it represents an approximate or abbreviated
value, is just as accurately stated as 0.000 101 is.
A zero (0) at the right-hand end of a whole niunber may mean either a
definite known quantity like a digit or an indefinite unknown one, merely
filling a vacant place of figures; there is unfortunately no way of distin-
guis^n^ between them. Thus in the value 5280. feet to the nule,the zero
might either mean that the last place of figures is exactly correct (which is
the case in this particular value) or that only three places are correct and
'the fourth unknown. In a decimal fraction, however, a zero in the last
place is always, or should always be, understood to be a definite known
quantity; thus 0.150 is understood to be known or correct to three places,
while 0.15 is known or correct to only two places.
The following table gives the errors in abbreviated numbers in a more
concise form:
Abbreviated numbers. Greatest error. Mean probable error.
10 5.% 2.5%
60 l.% 0.5%
100 0.5% 0.25%
600 0.1% 0.05%
1 000 0.05% 25 parts in 100 000
6 000 0.01 % 5 parts in 100 000
10 000 6 parts in 100 000 25 parts in 1 000 000
50 000 1 part in 100 000 5 parts in 1 000 000
100 000 5 parts in 1 000 000 25 parts in 10 000 000
500 000 1 part in 1 000 000 .5 parts in 10 000 000
It will thus be seen that by using only three places of figures to represent
any data or results (provided the last right-hand figure has been raised by
unity when the next is 5 or over) the greatest possible error is only half a
percent, and the probable mean error is about ^o to %o%. Three places
of figures therefore suffice for most engineering data.
In the more usual simple calculations with three-place values, the mean
probable error in the result is in general no greater than this, but the greatest
§088{ble error becomes larger and may affect the third place by one unit,
uccessive multiplications like cubing increase the error. Hence if the
result is to be quite correct to three places, then four places must be used
to start with.
AOCURACT OF LOGARITHMS.
In general, the number of (decimal) places in the logarithms themselves
should be one greater than the number of places of figures, in order to be
as accurate (in percent) as the arithmetical calculations would be. The
errors are therefore easily obtained from the above table for numbers. Four-
place tables are therefore sufficiently accurate for three-place figures.
ABSOLUTE SYSTEM. — C. G. S. SYSTEM. 11
ABSOLUTE SYSTEM of UNTTa The O. G. S. SYSTEM.
Among the physical quantities there are some that are fixed, definite,
independent of each other, and invariable all over the universe, while others
are variable, indefinite, dependent, arbitrary, or involve empirical con-
stants. A length, for instance, belongs to the former class and is invariable
throughout the universe, while a weight considered as a force, is different
on different parts of the earth or on different planets. The former are
for the sake of a distinction, called absoltUe quantities.
Most physical quantities are dependent on others or are derived from
others; for instance, a surface is the product of two lengths, and a velocity
is a length passed through in a certain time. But there are a few that are
independent of any others and may, therefore, be considered to be funda-
mental; a length, for instance cannot be derived from anything else. By
selecting three of the latter from among the absolute quantities most and
perhaps all of the other physical quantities may be derived from them, or
defined in terms of them, thus making a single uniform system in which
all quantities can be expressed in terms of some function or combination
of these three fundamental quantities. When a definite amount of each
of these fundamental quantities is taken as a unit, such a system is called
an absoltUe system of units.
Various systems of this kind have been devised differing in the three
quantities which have been selected as the fundamental ones. The most
important one and the only one which is in general use, is based on the
three quantities, length, mass* and time, usually represented by the letters
L, M, and T respectively, or I, m, and t, and when the unit amounts are
taken as the centimeter, the gram (mass), and the second (mean solar),
the system is called the centimeter-gram-second system, usually denoted
as the C. O. S. system.
In this system the units of each of the derived quantities are the amounts
which correspond to the unit amounts of those of the fundamental quanti-
ties which are involved. Thus the unit of velocity in this system is one
centimeter per second; the unit of force, called a dyne, is that whish act-
ing on a mass of one gram at rest produces in one second a velocity of one
centimeter per second; the imit of energy, called an erg, is one dyne of
force acting through one centimeter; the unit of electricity or magnetic
pole is that which attracts another equal amount at one centimeter dis-
tance, with a force of one dyne; etc., etc.
Sometimes the units of the derived quantities depend upon how they are
defined, in which case there may be several different units. The most
important case of this kind is that of the electrical units; in the one ^stem,
called the electrostatic system, a whole set of units is based on the dennition
that a unit of electricity is that which attracts an equal amount one centi-
meter distant, with a force of one dyne, while in the other, called the electro-
magnetic system, the unit current is defined as that which, flowing through
an arc of one centimeter, curved to one centimeter radius, generates a unit
magnetic pole at the center; this definition thus connects the unit of elec-
tricity with the unit of magnetism ; on it the electrical units in common use
are based. The relation between the two units of each electric quantity
thus defined is found to be the velocity of light in the C. G. S. system,
or some power of it.
* Mass represents amounts of matter and must be clearly distinguished
from weight. A given mass of iron, for instance, is the same whether it
be solid, liquid, volatilized, oxidized, dissolved, etc., and is the same all
over the universe, while its weight is enormously greater on the sun than
on the earth. Two given masses always have the same attraction of gravi-
tation to each other at the same distance.
12 INTRODUCTION.
Sometimes the units thus defined are inconveniently large or small for
practical purposes, and therefore certain practical units have been chosen
which are made some multiple of 10 times as small or as large as the C. G. S.
unit. Thus the ampere is 1/10 and the volt is one hundred million times
the C. G. S. unit (electromagnetic) of current and electromotive force
respectively. The definitions of all these units and the relations between
them are given in their respective places in the table of conversion factors.
While most ph^^sical quantities can thus be deduced from, or defined in
terms of, the centimeter, the gram, and the second, yet it has not yet been
possible to do this with all. Among the important exceptions are tem-
perature, magnetic permeability, and electric inductive capacity; they
should therefore be added to the fundamental quantities in defining or
deducing some of the quantities. For most purposes the two latter may
be, and in fact are, eliminated from consideration by making them equcd
to the niuneral 1 or imity in the definitions. They are therefore called
"suppressed" fundamental quantities.
Tliere are in general two ways of establishing units of various quantities.
One way is to define an absolute unit in accordance with the C. G. S. system,
and then establish a concrete unit in terms of the absolute one, which will
represent the latter or some simple decimal multiple of it, as closely aa
possible; this seems to be the more rational way, as it maintains the whole
system uniform, although it is sometimes difficult to establish the concrete
unit correctly; this is the method adopted in establishing the electrical
and magnetic units. The other way is to arbitrarily establish the concrete
unit first, by selecting some convenient standard, and then determining
experimentally what its relation is to the natural, absolute unit; this is
the method which was adopted in establishing the units of weight (con-
sidered as a force and not as a mass), heat, temperature, light, etc.; it has
the disadvantage that the relations to other units then always involve an
empirical constant which is necessarily incommensurat-e, like the accelera-
tion of gravity, the mechanical equivalents of heat and of light ; but it has
the advantage that the concrete unit is established definitely at the start,
and is not subject to occasional readjustment like the concrete electrical
units.
DIMENSIONAL FORMUIiAS.
When any physical quantity is represented algebraically in terms of
the fundamental quantities, the expression is called its dimensional formula.
Thus, in the length, mass, and time or L, M, and T system, a surface
which is the product of two lengths is represented by L XL or L^, a volume
by Z/3; these are the dimensional formulas of a surface and a volume, and
tne exponent of the letter is called its dimension.
It frequently happens in the^ more complex formulas, that some of these
letters occur as divisors, that is, as the denominator of a fraction, and to
avoid stating the formula in terms of a fraction, the exponents are made
negative in such cases. Thus a velocity which is length divided by time,
or L/T, is generally written LT—^ ; an acceleration is a velocity divided by
time or LT—^/T, which is written LT^^; a force is this multiplied by mass
or L Af 2^2 and energy is a force multiplied by a length or L* M I*-', etc.
Fractional exponents denote roots, thus Li means the square root of L\
or Z/~f denotes the cube of the square root of L in the denominator, etc.
In some cases the letters cancel each other, leaving the exponent 0; the
dimensional formula is then simply 1, sometimes called a niunber; an angle,
for instance, is defined as an arc divided by the radius, that is, L-^L*"LO«» 1;
DIMENSIONAL FORMULAS. 13
the same is true of an efficiency which is energy divided by energy, or power
dirided by power, etc.
When intelligently interpreted and applied, such dimensional formulas
are often very useful. They frequently give an idea of the physical nature
of a quantity, and more particularly of its relation to other quantities;
they sometimes show that several quantities which have been defined in
entirely different ways, and which originated dififerently, are really the
same; they sometimes point out the existence of some law; they aid one
in determining what the rational unit is, of a quantity for which no unit has
been chosen; they are useful for finding whether a physical formula is
eocreet and complete; these are only a few of the uses of such formulas.
The quotient of two such formulas often shows that the relation between
the quantities is a third simple, well-known quantity. Thus the relation
between energy and pressure is a volume, or the relations between the
formulas for the electrical units defined electrostatically, and those defined
eleetromaspetioally, is in this way shown to be the velocity of light or its
square, fdiich is one of the foundations of Maxwell's electromagnetic theory
of lii^t.
It must not be forgotten that the dimensional formulas usually used are
based on the fundamental units, length, mass, and time, and that they
would be quite different if other fundamental quantities are used ; they are
thnefore only relative uid not absolute, and their chief use is therefore
to show the relaHon» between different quantities rather than the physicsd
nature of any one considered bv itself.
Althoufi^ dimensional formulas are very useful, great caution should be
exercised in applying them, as inconsistencies and absurdities may resudt
if they are treated as mere algebraic formulas. Energy, for instance, is
force X length, which gives the formula L^ M T—^, but torque is also force X
length, and therefore has the same formula, although it is phjrsically an
entire^ different quantity. When torque acts through an angle it becomes
energy, hence torque Xan|^e» energy, but as the dimensional formula of an
angle is 1 the formula for torque is not changed by multiplying it by an
angle. The length involved in torque is perpendicular to the force, while
in energ3r it is in the same direction, therefore the two lengths have a differ-
ent physical meaning in the two cases, as they are at right angles to each
other; as this cannot be indicated in the formula, it shows that the system
is defective. The "angle" in such a case has been termed a "suppressed
quantity" in the formula, and it is due to such quantities that errors may
arise in using dimensional formulas. Its real existence in the formula for
torque should at least be indicated by some auxiliary letter like a, for in-
stance, calling attention to its suppression in the rest of the formula.
It seems to have been impossible so far to find the true dimensional formula
of temperature, or even to determine whether one exists; it is therefore
safest at present to consider it an auxiliary fundamental quantity, usually
represented by B, and add it to the formula. Further remarks on this sub-
ject will be foimd in a footnote imder the thermal units in the table of
physical quantities given below.
In the formulas for the electrical and magnetic units there are also some
"suppressed" quantities similar in some respects to the angle above men-
tioned; like in the case of temperatiue, their dimensional formulas are
not known, they are generally omitted in the formulas for other derived
quantities, but without them these formulas axe not complete and may
mislead. Ordinarily, they are considered to be unity and therefore are
algebraically eliminated from the formulas, but as was shown above con-
cerning the quantity "angle," this may lead to misconceptions. Until
their dimensional formulas are known it is recommended to consider them
as auxiliary fundamental quantities and to add to the formulas a symbol
which represents them, in order to call attention to the fact that they really
exist but are suppressed in the particular system used.* In the dimensional
formulas given below in the table of physical quantities, they have been
* Prof. Ruecker, in the paper referred to below, says: "I think the ssrm-
bols are thus made to express the limits of our knowledge and ignorance
on the subject more exactly than if we arbitrarily assume that some one of
the quantities is an abstract number."
r
14 INTRODUCTION.
added in parentheaes? for ordinary purposes they ma^ be oonaidered as
being unity. These quantities are the electric inductive capacity repre-
sented by k, and the magnetic permeability represented by fi. In the
definitions of the units their values for air are assimied to be unity, which
eliminates them quantitatively, but this is no reason why their dimensional
formulas should be unity; a definition, for instance, might be based on a
unit distance, but it would be quite wrong to therefore leave out of the
dimensional fonnula the L which represents it.
Energy being always the same quantity in all systems, never has any
of these suppressed factors in its formula; the same is true of power.
Although the dimensional formulas of these two suppressed quantities
individually are unknown, that of both combined is known in tne follow-
ing form:
in which v is the velocity of light in the C. G. S. system; hence when the
formula of one of them is known, that of the other can be determined.
It was thou^t by Williams (see reference below) that jt mav be a density.
The dimensional formulas of the photometric quantities nave, it seems,
never been given. Those in the following table are suggested by tne author.
A concise discussion of the deduction of the dimensiomd formulas of
many of the usual physical quantities will be foimd in the Smithsonian
Physical Tables, edited by Prof. Thomas Gray, 2d edit. p. xv to 4. The
*' suppressed'' quantities are discussed in a paper by Ruecker in the Phil.
Mag., Feb. 1889, vol. 27, p. 104, supplemented by atfother by Williams,
Phil. Mag., 1892, p. 234. The table of physical quantities given below con-
tains all the quantities whose formulas are given in these references besides
numerous others.
DBOISIONS OF INTERNATIONAIi BLSOTRIOAIj
OONGRSSSBS
Concerning Bleotric, Magnetic, and Photometric Units and
Definitions.
The following is a brief summary of these decisions, adoptions, and
recommendations.*
The official congress of 1881 in Paris adopted the fundamental units
centimeter, gram Tmass), and second, for the electric measures; the ohm
as equal to 10* C. O. S. units t * the volt as equal to 10^ C. G. S. units t ; the
ampere as the current produced by a volt through an ohm : the ooulomb
as the quantity corresponding to an ampere for one second; the fiarad as
the capacity corresponding to a charge of a coulomb by a volt. It in-
structed an international commission to determine what the length of a
column of mercury of one square millimeter cross-section must be, at 0^ C,
in order to have a resistance of one ohm as above defined; the commission
reported in 1884 that this length was 106 centimeters and called thi^ ohm
the leg^ ohm; it also recommended that this be adopted internationally;
it also recommended that the ampere be made equal to 10~i C. G. S.
(electromagnetic) units, and that the volt be that electromotive force
which maintains a current of one ampere (presumabty as just defined)
through a resistance of one legal ohm; (this volt has since often been re-
ferred to as the leiral volt, although there seems to be no legal sanction
for this name). Tne congress of 1881 also recommended that an inter-
national commission be appointed to define a standard of light ; this com-
mission reported in 1884 that the unit of each kind of simple light be
* For a more detailed simimary up to 1900 see Recapitulation dea DM-
aions des Congrba Antirieura, by Hospitalier, in the report entitled "CJon-
gr^s International d'^lectricit^," 1900, pages 11-22; for the adoptions of
the congress of 1900 see pages 369 and 370 of that report.
t The electromagnetic system was unquestionably meant, and is undec^
stood to be meant in all that follows here.
1
DECISIONS OF CONGRESSES. 15
the quantity of U^t of the same kind emitted perpendiculaxiy from a square
centimeter of surface of melted platinum at the tonperature of it« solidi-
fication; and that the practical unit of ^irhite li|^t be the total licht
thus eniitted. (The name Tiolle has since come into u^ie for this unit,
although apparently without official adoption.^
The official eongrees of 1889 in Paris adopted the Joule as equal to 1(F
C. G. S. units of work (ergs) and defined it also as the eneiiy represented
per second by one ampere through one ohm; uhe ohm here referred to is
presumably that defined in terms of the absolute system, and not the **lecal '*
ohm); the ^ratt as equal to Kfi C. G. S. units of power (ergs per second).
and therefore equal to a joule per second; the kilowatt as the industrial
unit of power in place of the horse-power; the boayle dtelniale (decimal
candle) for the practical unit of light as equal to the twentieth part of the
absolute standard of light defined by the commission in 1884 (namely, the
platinum standard described above and called the vioUe); the quadrant
for the practical imit of self-induction (now called inductance) as eoual
to 10* centimeters; the period of an idtemating current was defined to
be the duration of one complete oscillation, and the freqaoncy the num-
ber of periods per second; the mean intensity (of a current) was defined
by an algebraic expression which signifies the arithmetical aA*erage of all
the instantaneous values; the effeetlTe intensity (of a current) was
defined to be the square root of the mean square oi the intensity of the
current, and the efltectiTe electroniotiTe force, the square root of
the mean square of the electromoti^'e force ; t he apparent resistance (now
called impedance) was defined to be the factor by which the effective in-
tensity of the current must be multiplied to give the effective electromo-
tive force; the positive plate of an accumulator was defined to be that
which is the positive pole during discharge.
The unofficial cons^«ss of 1891 at Frankfort (Germany) decided '^ that
all units be expressed in Roman type, all physical quantities in italics, and
all physical constants and angles in Gredc type; also that the quantities
ampere, coulomb, farad, joule, ohm, volt, and watt be expresseil by their
initial letters. A, C, F, J, O, V, and W. (These decisions were not con-
firmed by the subsequent official congress, and have not come into general
use.)
The official congress of 1893 in Cliicago t adopted the international
olini, based on the ohm equal to 10^ C. G. S. units and represented by the
resistance of a column of mercury at 0** C, 106.3 centimeters long, weigh-
ing 14.452 1 grams, and having a uniform cross-section; (this cross-sec-
tion, although not so stated officially, is practically one square millimeter);
the international ampere as equal to 10^^ C. G. S. units, represented
for practical purposes by the current which will deposit 0.001 118 gram of
silver per second; the international volt as that electromotive force
which will maintain one international ampere through one international
ohm, represented for practical purposes by 1 -5-1.434 of that of a Clark cell
at 15° C. ; the international coulomb as the quantity corresponding to
one international ampere in one second; the international farad as
corresponding to a charge of one international coulomb by one interna-
tional volt; the joule as equal to 10^ C. G. S. units, and represented in
practice by the energy in one second of an international ampere passing
through an international ohm; the iratt as equal to 10^ C. G. S. units and
represented in practice by one joule per second; the henry as **the induc-
tion in a circuit when the electromotive force induced in this circuit is one
international volt, while the inducing current varies at the rate of one
ampere per second" (presumably meaning the international ampere).
These eight units, substantially as defined by this international congress,
were made legal in the United States by Act of 0>ngress in 1894. For
practical magnetic units, the C. G. S. units were commended by the Chicago
congress, but no names were given them; a report of a committee on
notation and nomenclature was received and ordered printed as an appen-
* See Electrical World, vol. 18, 1891, page 248.
t For further details see page 20 of the Proceedings of the International
Electrical Congress held at Chicago, 1893, published by the Amer. Inst.
Elect. Engineers.
16 INTRODUCTION.
dix, no further action being taken on it; (see the table of physical quanti-
ties below, in which that report is included).
The unofficial congress of 1896 in Geneva adopted the bougie d^ciuiale
as the unit of intensity of light [/] equal to Ho of a violle, but provisionally
considered it to be represented m practice by the Hefner lamp ; the lumeu
as the unit of flux of li^ht If] equal to one bougie d^imale for one solid
angle; the lux as the uiut of illumination [E] equal to one lumen per square
meter; the bougrie per square centimeter as the unit of brightness [e];
the lumen-hour as the unit of guantity of Ught [O].
The official congress of 1900 in Paris adopted the name gauss for the
C. G. S. unit of intensity of magnetic field or flux density, and the name
maxirell for the C. G. S. unit of magnetic flux.
For the values of these official units in terms of each other and of other
units, see the respective tables of measures.
PHYSICAL QUANTITIES AND RELATIONS. 17
TABLES of PHYSICAL QUANTTTIBS and RELATIONS.
The following table gives the physical quantities and relations in use,
with their names, symbols, derivation, dimensional formulas in the C. G. S.
system, and the units whenever such units have been generally adopted.
A similar though much smaller table, limited chiefly to the more important
electrical and magnetic quantities, was recommended by the Committee
on Notation of the Int-ernational Electrical Congress of 1893 in Chicago;*
this has been included here after revision, correction, and some rearrange-
ment. The "suppressed" factors k and pi have been added, which neces-
sitated some changes in the derivational formulas of the Congress table,
notably that of magnetic flux which now becomes BS instead of H8. The
dimensional formulas then become consistent throughout; moreover, they
then represent the conditions of practice better, as iron is used in nearly
all forms of magnets.
The magnetic quantities based on the electrostatic system, which are
not generally given in text-books, have here been added for the sake of
completeness.
For the definitions of the units and the quantitative relations between
them, see the respective tables of measures.
* Reprinted with some additions in the "Electrical World and Engineer,"
vol. 37, January 5, 1901, p. 501.
18
INTRODUCTION.
PHYSICAL QUANTITIES AND RELATIONS.
Name.
Fundamental.
Length
Mass
Time
Axtxiliary funda-
mental quantities
Temperatm'e
Electric inductive
capacity
Magnetic permea-
bility
Geometrio.
Surface
Volume. . . .
Angle, plane.
Angle, solid.
Curvature or Tor-
tuosity
Specific curvature
of a surface. . . .
Mechanical.
Weight: see Mass
and Force.
Density
Specific gravity. . ,
Force
Intensity of attrac-
tion or force at a
point
Gravitation con-
stant
Force of a center
of attraction or
strength of a cen-
ter
Surface tension.
Sym-
bol.
M
T,t
d
k
S,8
V
sp. gr
F,f
\
Derivation.
LL
LLL
arc
radius
spherical area
radius^
angle
length
solid angle
surface
M
V
density
density
Ma
F
M
M^
M
JL
L
Dimensional
Formula.
L
M
T
6
k
L2
L3
number
number
L-2
L-3M
number
LMT-^
MT-^
C.G.S.Unit,*
centimeter
gram (mass)
second (mean
solar)
1 sq. centimeter
cubic centimeter
radian
1 sq. cm at 1 cm
radius
gram (mass) per
cb. cm
1
dyne
dyne per gram
dyne per centi-
meter
* For the names and abbreviations of the numerous practical units and
for their values in the C. G. S. imits, see the various tables of measures.
PHYSICAL QUANTITIES AND RELATIONS.
19
PHT8ICAI. QUANTITIES AND RBLATIONS {Continued).
Name.
Pressure or inten-
sity of stress. . . .
Modulus of elas-
ticity
Resilience.
Torque , moment ,
couple
Directive force (as
in suspensions). .
Moment of inertia.
Inertia
Velocity, linear. . .
Velocity, angular
Acceleration, lin-
ear
Acceleration, an-
gular
Momentum or
cj^uantity of mo-
tion.
Moment of mo
mentum or angu-
lar momentum .
Energy, work. . .
Vis-viva
Impact
Power or activity.
Efficiency.
Sym-
bol.
K
ta
a
W
W
W
P
Derivation.
F
S
IF
LS
W_
V
FL or
W
angle
torque
angle
MXradius^
LW
L
Z/
T
angle
T
V
T
to
T
mass X veloc-
ity
momentum X
length
FL
W_
T
power
power
Dimensional
Formula.
L-^MT-^
L^MT-2
LM
LT-^
T-i
LT-2
rp—2
LMT-^
number
C. G. S. Unit.
baric*
djme-centimeter
gm (mass) cm sq.
centimeter per
second; kine
radian per second
centimeter per
sec. per sec.
radian per sec.
per sec.
erg
erg
erg
erg per second
1%
* It equals a dyne per sq. centimeter. This name has not yet been defi-
nitely aaopted; some use it for the pressure of one atmosphere.
t More correctly this should also contain the reciprocal of an angle.
20
INTRODUCTION.
PHYSICAL QUANTITIES AND RBI^TIONS (Coniinued)
Name.
Sym-
bol.
Derivation.
Dimensional
Formula.
C. G. S. and
Practical Units.
Maf^etic.
(El'mag. system.)
Reluctance ormag-
netic resistance. .
(R
L
L-Ku-')
oersted *
Reluctivity; spe-
cific reluctance. .
V
1
/A
number X(/i~^)
Permeance
1
L(ti)
Magnetic capacity
(£
(R
1
(R
L(m)
Permeability or
specific induc-
tive capacity . . .
M
(B
5C
number X(/*)
Susceptibility. . . .
K
a
JC
number X(^)
Magnetomotive
force
^
4;riV/t
W
in
Lt
gilberts
Magnetic poten-
tial
Magnetizing force.
3C
gilbert 1^ per cen-
timeter; gauss
Field intensity. . .
JC
F
m
L-iMir-K/r-h
gauss
Flux density
(B
9
S
L-iMiT-\(tii)
gauss
Magnetic induc-
tion
(B
3
an
V
gauss
Intensity of mag-
netization
Flux or magnetic
lines of force. . . .
«
(RS
L^MiT-^ifii)
maxwell
Strength of pole or
quantity of mag-
netism
\/L2F^
ml
Magnetic moment.
Magnetic energy...
W
^
L^MT-^
erg
Magnetic power. .
P
*(Fn§
L^MT-3
erg per second
♦ Provisionally adopted by the Amer. Inst, of Electrical Engineers,
t iV = number of turns. J L— length, of coil. § n=» frequency.
^ Provisionally adopted by the Amer. Inst. Elec. Engineers. The usual
unit is the ampere-turn. 1 gilbert =» 0.795 8 ampere-turn.
PHYSICAL QUANTITIES AND RELATIONS.
21
PHY8ICAI. QUANnTmS AND RELATIONS {.Continued).
Name.
Magrnetio.
(Electrostatic system.)
Reluctance or magnetic
resistance
Reluctivity; specific re-
luctance
Sym-
bol.
Permeance.
Magnetic capacity
Permeability or specific
inductive capacity. . . .
Susceptibility
Magnetomotive force. . .
Magnetic potential. . . .
Magnetizing force
Field intensity.
Flux density.
Magnetic induction
Intensity of magnetiza-
tion
Flux (or magnetic lines
of force)
Strength of pole or quan-
tity of magnetism
Magnetic moment.
Magnetic energy. .
Magnetic power.
(R
K
JC
3C
CB
3
m
m
w
Derivation.
_1^
fi
\_
(R
J_
(R
®.
5C
n_
3C
AnNI
m.
JL
L
Z.
m
S
s
m
V
ET
W
magn.poten.
ml
*JF
T
Dimensional
Formula.
L^T-^k)
L-ir2(fc-»)
L-iy2(Jfc-»)
L-2r2(jfc-i)
L-2r2(jt-i)
L^M^T-^kh
L^M^T-^ik^)
L^M^T-Kkh
L^M^T-^(,k^)
L-^M^^k-h
L-iM*(Jfc-i)
L-iM^{k-^)
J>M*(Jfc-*)
L^Mhk-h
/JM*(fc-*)
Ratio of
Electro-
static to
Elec'mag-
n'tic Units*
»2
«-2
«-2
V
V
V
1
* wis the velocity of light in the C. G. S. system, namely 3X10^0 centi-
meters per second. The factors fi and k are here neglected, as the square
root of their product is the reciprocal of the velocity of light, and these
ratios would therefore all become unity if these factors were included.
22
INTRODUCTION.
PHTSICAL QUANTITIES AND RELATIONS {CmUinued).
Name.
JBlectric.
(El' mag. system.)
Resistance
Resistivity or spe-
cific resistance. .
Reactance
Magnetic react-
ance
Capacity react-
ance or condens-
ance
Impedance. . . .
Conductance. . .
Conductivity or
specific conduct-
ance. ,
Sym-
bol.
fl.rt
P
X
\-
Derivation.
_P
RS
L
2imL —
1
Admittance.
Susceptance.
Electromotive
force
Potential
Diflference of po-
tential
Electromotive
force at a point .
Intensity of elec-
tric field
}
r
B,6t
E,e
U,u
Vector potential
Current
Current density. . .
Quantity of elec-
tricity; charge. .
Surface density or
electric displace-
ment
1,1
Q,Q
2itnC
2;mL
1
2jmC
1 r
— or -5
p
1 1
T
W_
Q
RI
L
Z.
Q
ET
L
E_
R
I_
S
IT
Q_
S
Dimensional
Formula.
LT-Hpi)
L^T-Hfi)
LT-Kp)
LT-Kp)
LT-Hp)
LT-Kp)
L-^Tifi-^)
L^M^T-K/ih
lAM^T-Hfih
L^M^T-Kp-h
L-hM^T-^p-h
Practical Units.*
ohm
ohm-centimeter
ohm
ohm
ohm
ohm
mho
mho per centi-
meter
mho
mho
volt
volt
volt
volt
ampere
ampere per sq. cm
coulomb; am-
pere-hour
coulomb per sq.
cm
* The C. G. S. units have no names. For the relations between the values
of these units and the C. G. S. units, see the various tables of measures,
t Vector quantities when used should be denoted by capital italics.
PHYSICAL QUANTITIES AND RELATIONS. 23
PHTSIOAL QUANTTTIBS AND RELATIONS {C<mtinwd).
Name.
Capacity.
Electric inductive
capacity, or Di-
electric constant,
or Specific induc-
tive capacity. . . .
Inductance or co-
efficient of self-
induction or elec-
tro-kinetic iner-
tia
Mutual inductance
Inductance factor:
see below.
Time constant. . . .
Period.
Frequency
Angular velocity..
Electro-kinetic
momentum
Thermoelectric
height or specific
heat of electricity
Coefficient of Pel-
tier effect
Ditto.
Electric energy . . .
Kinetic energy.. . .
Electric power. . . .
Power factor.
Inductance factor.
Bleotroclieinl-
cal.
(El' mag. system.)
Ionic charge
Electrochemical
equivalent
Electric deposition
Sym-
bol.
C,c
^ r
I
J
I.
n
at
i.
W
W
Derivation.
E
jC
L
C C
I
L_
R
\^
n
2ir7l
/Xinductance
heat
IT
energy
IT
EQ
nL
EQ
T
realP
apparent P
wattless P
apparent P
Q_
M
K
Q
M
T
Dimensional
Formula.
L-^TKit-^)
L-^TKpr^)
number
T-i
L^M^T-H/ih
L^M^T-^ie-Hfih
L^M^T-Knh
L^MT-»
L^MT-2
L^MT-»
number
number
MT-i
Practical Units.
micro-farad
henry
henry
second; henry
per ohm
second
periods per sec.
radians per sec.
joule; watt-hour
joule
watt; kilowatt
coulomb per uni-
valent gram ion
gram per cou-
lomb
gram per second
* JV =» number of turns.
t 0= temperature.
24
INTRODUCTION.
PHYSICAL QUANTITIES AND RELATIONS (Continued).
Name.
Bleotrlo.
(Electrostatic system.)
Resistance
Resistivity or specific
resistance
Reactance
Magnetic reactance. . .
Capacity reactance or
condensance
Impedance. .
Conductance.
Conductivity or specific
conductance
Admittance.
Susceptance
Electromotive force or
potential
Difference of potential
Electromotive force at
a point
Intensity of electric field
Vector potential.
Current.
Current density
Quantity of electricity,
charge
Surface density or elec-
tric displacement. . . .
Capacity.
Electric induction ca-
pacity or Dielectric
constant or specific in-
ductive capacity. . . .
Ssnn-
bol.
R,r
o
X, X
X
m
he
G.g
B,b
E,e
U,u
Li
Q,Q
C,c
Derivation.
E
I
r
2imL
1
2nnC
\/r2 + a:2
7
E
Q
STE/L
1
g
W
Q
L
F
Q
ET
L
T
J_
S
S
E
L
L ' L
Dimensional
Formula.
L-ir(ifc-i)
Tik-^)
L-^nk-^)
L-Jr(Jfc-»)
L-^T(k-^)
LT-Kk)
T-Hk)
LT-Hk)
LT-Hk)
L^M^T-Kk-^)
rJM^T-Hk-h
L-^M^T-Hk-h
L-^M^T-Hk-h
L-^M^(k-^)
L^M^T-^k^)
L-^M^T-Kk^)
L^M^T-Hk^)
L-iM^T-Kkh
Uk)
number (k)
number
Ratio of
Electro-
static to
Elec'mag-
netic Units.
«-2
t>-2
«-2
tr-2
v-2
t>2
v-i
V
V
V
va
PHYSICAL QUANTITIES AND RELATIONS.
25
PHYSICAL QUANTmES AND REUITIONS (Con/mi«ed).
Name.
Inductance or coefficient
of self' -induction or
electrokinetic inertia. .
Mutual inductance
Time constant.
Period.
Frequency
Electrokinetic momen-
tum
Thermo-electric height
or specific heat of elec
tricity
Coefficient of Peltier ef-
fect
Ditto
Electric energy.
Electric power.
Electrochemical.
(Electrostatic system.)
Ionic charge
Electrochemical equiva-
lent
Electric deposition. .
Photometrfc*
Quantity of light. . . .
Flux of light
Intensity of light
Brightness
Illumination.
Sym-
bol.
\
L,l
n
£ m • • •
w
p
Q
9
E
Derivation.
2m
R
J_
n
/Xinductance
E
e
heat
IT
energy
IT
EQ
W
T
_Q,
M
M
Q
M_
T
*r
Q_
T
9
solid angle
J_
S
JL
s
Dimensional
Formula.
jr^_,j,2(jfc-l)
T
T
TSikh
L-^M^Te(k-h
L^M^T-^k-i)
LKMT-^
L^MT-^
L-^M^T{k-h
MT-i
L^MT-^
LniT-^
L^MT-^
MT-3
MT-s
Ratio of
Electro-
static to
Elec 'mag-
netic Units.
«-«
1
1
1
1
1
Unit.
lumen hour
lumen
candle ;hef-
ner
candle per
sq. cm
lux
* The dimensional formulas for the photometric quantities here given
have been deduced by the author on the basis that radiated light is power,
or that quantity of light is energy; according to this, the formula for can-
dle-power is that for power divided by that for a solid angle; the latter,
having the dimensional formula 1 in the C. G. S. system, is a "suppressed"
factor which does not appear in the formula.
J
26
INTRODUCTION.
PHYSICAL QUANTITIES AND RELATIONS {Concluded^
■
•
1
H
■
6
J
Derivation.
Dimensional Formulas.!
Name.
Dynamical
Thermal
Therm o-
metric.
O^H-^M.
Thermal.*
Heat
energy
H
T
X,2Mr-2
ex
L2T-2
LMT-^d-^
number
M
number
Md
MT-^6
ex
M
e
L-^MT-^
number
M ■•
ex
LT-i
number
M
M
LT^d
L^MT-^
Rate of heat
production...
Temperature. .
Coefficient of
expansion . . .
Entropy
Latent heat. . .
Conductivity. .
Emissivity or
immissivity. .
Specific heat. ..
Capacity
Mechanical
equivalent. . .
iV-V)
H
H
M
HL
TLH
H
TL^e
H
H
MXsp. heat
mec. energy
L-^T*
M
number
M
number
heat-energy
* For the names and definitions of heat-units see tables of units of Energy.
t The dimensional formulas of the various thermal quantities or relations
are still a matter of some conjecture, excepting only that of quantity of
heat, which is simply energy, and that of the rate of production or trans-
mission of heat or radiant heat, which is simply power; that of tempera-
ture is the uncertain factor. For this reason four different systems are
here given, based on four different fundamental conceptions. The first is
based on the dynamical units, that is, on the formula of energv combined
with a fourth fundamental unit representing temperature. Tne second is
based on the thermal units ; quantity of heat then is mass X temperature ;
in this system the specific heat of water is unity by definition and is
therefore suppressed in the formula. In the third system volume is sub-
stituted for mass in the second. In the fourth system the author has
eliminated by defining temperature as energy per unit of mass, that is, as
specific mass energy. Most of the data in columns 3, 4, 5, and 6 have been
taken from the Smithsonian Physical Tables prepared by Prof, Thomas
Gray ; the present author has extended them and supplied some omissions ;
the new matter has been approved by Prof. Gray.
t By giving 6 the dimensional formula L^T'^, those based on the dynam-
ical units and on the thermal units become identical; they are given in
the last column. Or by making it L~^MT~^^ those based on the dynam-
ical and the thermometric units become identical. L^T~2 is the dimen-
sional formula of energy per unit of mass, and L~'^MT~2 is that of pres-
sure. Ruecker suggests giving 6 the formula L^MT~^, which is energy.
No great importance should be attached to attempts to give temperature a
formula, as they are all merely speculative.
TABLES
OP
CONVERSION FACTORS.
GBNERAL RBMARKS.
In the following set of tables every unit which is used to measure quan-
tities is given with its value in terms of the other units of its kind. Such
numbers are variously termed values, reduction or conversion factors,
equivalents, relations, ratios, constants, etc.; they give the relations be-
tween the different units and enable one to reduce each one (or a quantity
nkeasured by it) to the other (or a quantity measured by it) by means of a
single multiplication. The table includes all the values and relations
usually given in books under the title of ''Weights and Measures, " although
these form only a very small part, as by far the greater number have never
before been published together as a complete set of values.
The term Weights and Measures" has not been used in connection with
these tables, partly because it is not apparent why weights are not also
measures, and partly because the present tables consist largely of various
measures ru«ly if ever given in books under the title of weights and meas-
ures.
Owing to the very large number of units belonging in such a table,
their classification and arrangement becomes important in order to enable
one to find any particular unit readily. All units for measuring the same
quantity, that is, all units of length for instance, or all units of volume, etc.,
have here been brought together in one group, and in each group they are
arranged in the order of their size ; all the different values of each unit are
also arranged in order of size.
To give the value of each unit in terms of each of the others would have
made the tables many times as lon^, and they would have become unnec-
essarily cumbersome, as the majority of the values are never used; the
number of values or reduction factors have therefore been limited to those
likely to occur in practice ; if the others are ever needed they can readily be
found from these.
To avoid unnecessary repetition of values, all units capable of being
reduced from one to the other have here been put in the same group. Thus
the group of units of volumes includes both cubical units such as cubic feet,
etc., as well as capacity measures, such as gallons, liters, etc. All energy
units are similarly grouped together, whether they be mechanical, thermal,
or electrical. In a few cases, however, such units have been separated into
different groups* forces, for instance, have not been given together with
weights, althougn they are interconvertible, and electrical resistances have
been given separately and not with velocities, although in the C. G. S.
electromagnetic system of units they are velocities.
Sometimes some of the units in one group are for measuring entirely dif-
ferent kinds of quantities, and even have entirely different dimensions, yet
their equivalents are the same, and they have therefore been brought
together to save repetition; for instance, pounds per sauare foot may de-
note either a pressure or the weight of sheet metal ; in either case the equiv-
alents in other units, such as kilograms per square meter, will be the same.
Similarly, grams per centimeter may denote either the weights of a wire or
27
28 TABLES OF CONVERSION FACTORS.
a surface tension, yet its equivalents in other units are the same. Power
and momentum, or energy and torque, are other illustrations.
In the case of lengths, surfaces, volumes, and weights (masses) there are
so many unusual, special trade, obsolete, or foreign imits the values or
reduction factors of which are not often needed, that they have been grouped
separately, so as to make the main table less cumbersome to use.
The reciprocal values have been given in all cases in which it was thought
thev were likely to occur in practice, thus reducing all calculation to a mere
multiplication, as distinguished from a long division. They are ^ven in
their proper places, and one should therefore first look for the units that
one wants. Thus to convert meters into feet, see value under Meters, and
to convert feet into meters, see value under Feet.
Special attention is here called to the approximate values which have
been given in nearly every case. These have been carefully chosen with a
view to reduce the calculation to the smallest possible, generally to a multi-
plication by one digit and a division by another single digit, followed by
pointing on the decimal. They are believed to be the simplest values that
exist. The accuracy of all these approximate values is within 2% and
often within 1 % ; they are therefore sufficient for most calculations.
The symbols or abbreviations which are given are either those in common
use or those which have been recommended by societies, journals, or indi-
viduals, and in a few unimportant cases where none existed they have been
supplied by the author in conformity with the others. The advantages
and desirability of a uniform and universal system of abbreviations or sym-
bols are too evident to need further comment here.
This whole system of tables, which is not a compilation but a complete
recalculation, has been based on the very best fundamental values that
are obtainable. Their numerical values are given in their respective places
or in the introductory notes; .they are printed in bold-faced type. When-
ever legally adopted values existed, as for instance for the relations between
the metric and the older units, they have been used. In other cases the
fundamental values have been obtained from the best existing sources, and
wherever possible those values have been chosen which have been deter-
mined upon by the best authorities and are used by them. Tlie chief of
these authorities was the National Bureau of Standards, from which the
author has obtained the legal values and all the other standard values used
or recommended by it, besides much valuable assistance; among the others
were the International Congresses, the Director of the Nautical Almanac,
the Coast Survey Department, etc. The greatest care was taken in obtain-
ing all these fundamental values, and it is believed that they are the best
which exist at the present time. They are limited to only those which are
absolutely necessary, as was explained above under the inter-relations of
units, and none of tnem is therefore inconsistent with any other, nor can any
derived value then have two values depending upon which way it has been
calculated from the fundamental values; all the values together form a
single, uniform, stable system.
Tiie fundamental values have here been given to as many places of figures
as in the original source. The derived values have been given throughout
to six significant figures. In some cases the fundamental value itself may
not have six figures, or its possible error may not warrant so many places
of figures in the derived units; but in such a completely recalculated set
of values it was thought best to retain six places of significant figures
throughout in all the derived values, as the con*ection due to any subse-
quently adopted more accuratie fundamental value can then be made by
mere proportion instead of by a complete recalculation. The derived values
are exactly correct for the particular fundamental values used here, except-
ing of course the usual slight inaccuracy (i:l) of the last right-hand digit
in the numbers and in the logarithms.
The calculations have all been made with the greatest care; each was
checked by calculating it in another way and in many cases the values were
thus checked twice; it is therefore believed that there are no errors. There
is, of course, the unavoidable uncertainty of one unit in the sixth place of
figures or in the seventh place of the logarithms, although in most of these
cases the next place was also calculated in order to insure the accuracy of
those that were retained. Many of the values in the first few groups have
been carefully checked by L. A. Fischer, Assistant Physicist of the National
Bureau of Standards, and may therefore be assumea to be those used by
LENGTHS. 29
that Bureau. The author takes thu opportunity to acknoiriadge hia appro-
ciatioD of Mr. Fischer's very valuable reviaion of those values.
J^ obsolete and foreign units were compiled from varioua sources. . They
toSid which Value waa'the co™i''one"''^ "" ^"^ ^^ ' '^*" **" ™
otuly. they have here also been placed where they might be looketl (or. and
are there aocompanied by a refBrence to the place where they properly be-
S comparison of the values in these tables with those in other books will
show that few of the latter have been based on tbe lettsl standards of this
country. Moreover, the older published values will often be found U '
being cecalcu
rith eai^b other,
Isted througho
as they seem to have been oompilac
1 instead of
!iS
,n is called to the co
und in other books; BB
(blld
PHS.
id uuita. moat of whieh are
)r tj«» save from two to four
LENGK
„2:as
"a*
tJ'^fi
ftX".
of the United States is
of 00 percent platinum a
tbe Inter-
JidlOper-
A the International Bureau of Weights ai__
Ueaaurea, near Paris. Copiesof this bar are possessed by each of the twenty
countries contributory to the support of Ihe International Bureau of Wndits
and Measures, and these copies a» known Ai National Prototypes. Tnie
United States owqb two of ttese bars, whose values in lerme of the Inter-
Meter, thus insuring the use of
gen«'^ le^at'ion in the Unite!
ards of weights and measures,
legaliied. and it is the only leg
nF AT.n.ni)RrHq in Wnnliinfftnn unii iq in ntnftrni use in this country. 3ince
iUT«H has been authorised to
with this relation; the legal
an fixed and Te&nite'uniU.' The ralation' between the (j. 8. yard and the
■upposmi, but is Ibied definitely by precise definition. There is no legal
authority in this oountry for the old Kater relation of 1818, namely.
39,37078. which is still in use. notably by one well-known maier of gauges.
In Great Britain the relation, legalised in 1806, between this same Inter-
national Meter and the British Imperial yard of 36 inches is 1 meter-
3S,37D 113 inches. The Kater value. 30.370 79. was used there for trade
purposes until 1896. In 1867 the Clark value, 3B.370 432, was recom-
mended by the Warden of Standards, London, to supersede the Kater
value, and was generally used in scientiGc work in Ureat Britain until 1S96.
There exists therefore a very slight diflarence between the present U. B. and
parts in one million, the U. 3. yard being the longer; it is therefore abeo-
lutely negligible eicept in the most refined physical measurements. Unless
otherwise stated, the values in the following tables are based on the U. S.
legal relation.
Tbe value of the nautical mile in meters |1 853.25) is the one adopted
many years ago by the U. S. Coast and Geodetin Survey, and has not been
changed since its first adoption. It is tbe length of one minute of arcof
a great circle of a true sphere whose area is equal to that of the earth. The
"Committee Meter" used by the U, S. Coast and Geodetic Survey prior
to 1889 is equal to the international meter of 39,370 inches (U. S.).
30 LENGTHS.
LENGTHS. Usual.
** Accepted by the National Bureau of Standards.
* Checked by L. A. Fischer, Asst. Phys. National Bureau of Standards.
Aprx. means within 2%. \
Logarithm
I mil « 0.025 400 05"^ millimeter. Aprx. Mo 3-404 8346
•• = 0.001 inch 5.000 0000
I millimeter [mm]» 39.370 0* mils. Aprx. 40 1.595 1654
= 0.039 370 0* inch. Aprx. %5 2.595 1654
= 0.001 meter 5.000 0000
I centimeter [cm] » 0.393 700* inch. Aprx. ^o 1595 1654
= 0.032 808 3* foot. Aprx. ^ 5.515 9842
= 0.01 meter 2000 0000
1 inch [in]= 1 000. mils 3.000 0000
= 25.400 05* millimeters. Aprx. H X 100 1.404 8346
= 2.540 005* centimeters. Aprx. i% 0404 8346
= 0.083 333 3* foot or Vi2 2.920 8188
= 0.027 777 8* yard or Mie. Aprx. ^M-s- 100 3-443 6975
= 0.025 400 05* meter. Aprx. %o 3-404 8346
1 decimeter [dm]=> 10. centimeters 1.000 0000
= 3.937 00* inches. Aprx. 4 Q.595 1654
= 0.328 083* foot. Aprx. %
-515 9842
=-0.109 361* yard. Aprx. % .038 8629
= 0.1 meter .000 0000
1 foot [ft] (Brit.)= 0.999 997 1* foot (U. S.). Aprx. 1 .999 9988
= 0.304 800* meter. Aprx. %o 1-484 0146
1 foot [ft] (U. S.)= 304.801* miUimeters. Aprx. 300 2-484 0158
= 30.480 1* centimeters. Aprx. 30 1-484 0158
12. inches 1-079 1812
= 3.048 01* decimeters. Aprx. 3-. 0.484 0158
= 1.000 002 9* feet (Brit.). Aprx. 1 0.000 0012
= 0.333 333 yard or % J.522 8787
= 0.304 801* meter. Aprx. %o 1.484 0158
= 0.000 304 801* kilometer. Aprx. 3 -t- 10 000. 4.484 0158
= 0.000 189 394* mile. Aprx. 19-5- 100 000 |.277 3661
1 yard [yd] (Brit.) = 0.999 997 1* yard (U. S.). Aprx. 1 1.999 9988
= 0.914 399 2* meter. Aprx. %o or i<Hi 1-961 1359
1 yard [yd] (U.S.)= 91.440 2* centimeters. Aprx. 90 1.961 1371
= 36. inches 1-556 3025
•« _. 3^ feet 0-477 1213
=» 1.000 002 9* yards(Bnt.). "Aprx. l.V.'. 0.000 0012
= 0.914 402** meter. Aprx. %o or i%i. . 1.961 1371
-0.000 914 402* kilomt'r. Aprx. Vii"*- 100. i.961 1371
= 0.000 568 182* mile. Aprx. ^ -4-1 000 3.754 4873
1 meter [m]= 1 000. millimeters 3-000 0000
= 100. centimeters 2-000 0000
= 39.370 1 13* inches (Brit.). Aprx. 40 1.595 1666
= 39.370 000** inches(U.S.). Aprx. 40 1.595 1654
•* = 10. decimeters 1.000 0000
= 3.280 83** feet. Aprx. i% 0.515 9842
= 1.093 61** yards. Aprx. i^o 6-038 8629
« 0.546 806* fathom. Aprx. i^o 1-737 8329
= 0.001 kilometer 3.000 0000
= 0.000 621 370* mile. Aprx. %-5- 1 000 4.793 3503
1 kilometer [km] = 3 280.83* feet. Aprx. Vs X 10 000 3-515 9842
= 1 093.61* yards. Aprx. 1100 3038 8629
=-1 000. meters 8000 0000
= 0.621 370* mile. Aprx. % 1.793 3503
=0.539 61 1 knot (Brit.). Aprx. %i 1.732 0806
=0.639 593 nautical mile ( U. S.) Aprx. %i . . 1.732 0660
1
14
<«
14
It
11
11
LENGTHS. 31
1 mile [ml] » same as statute mile or land mile.
- 6 280 * feet. Aprx. 5 300 3722 6339
« 1 760.* yards. Aprx. % X 1 000 3245 5127
= 1 609.35* meters. Aprx. 1 600 3206 6497
« 1.609 35* kilometers. Aprx. add %o 0206 6497
»0.868 421 knot (Brit.). Aprx. subtract % 1.938 7303
=»0.868 392 nautical mile (U. S.). Aprx. subt. ^. . 1.938 7157
1 knot or nautical mile (Brit.):
— 6 080. feet. Aprx. 6 000 3-783 9038
= 2 026.67 yards. Aprx. 2 000 3306 7823
= 1.853 19 kilometers. Aprx. i% 0267 9194
= 1.151 52 miles. Aprx. add ^ 0.061 2697
»0.999 966 nautical mile or knot (U.S.). Aprx. 1 1.999 9854
1 nanticflJ mile or knot (U.S.) same as geographical or sea mile :
= 6 080.20 feet. Aprx. 6 000 3783 9182
=■ 2 026.73 yards. Aprx. 2 000 3-306 7969
= 1 853.^5 meters. Aprx. ^VoX 1 000 3267 9340
« 1.853 25 kilometers. Aprx. i^ 0-267 9340
— 1.151 55 miles. Aprx. add Vt 0-061 2843
= 1.000 034i knots (Brit.). Aprx. 1 0000 0146
a 1 minute of earth's circumference 0-000 0000
LENGTHS (continued). Unusual, Special Trade, or Obsolete.
1 Ang^stroem unit (spectroscopy) = 0.1 milli-micron, or micro-milli-
meter = 0.000 1 micron = 0.000 003 937 00 mil =» 0.000 000 1 millimeter.
1 mil = 254 000.5 Angstroem units.
1 milli-micron [mii (spectroscopy) or micro-millimeter (microscopy)
= 10 Angstroem units = 0.001 micron = 0.000 039 370 mil = 0.000 001 milli-
meter. 1 mil = 25 400.1 milli-microns. The term micro-millimeter is also
used, though incorrectly, in biology for 0.001 millimeter, which length is
more properly called a micron or micro-meter.
1 wave len^k of blue light is of the order of about 5000 Angstroem
units or 500 miUi-microns or 0.5 micron or 0.02 mil. For accurate values
see below under meter.
1 micron or microne or micro-meter [ft\ (spectroscopy and micros-
copy) =10 000 Angstroem units =1000 milli-microns or micro-milli-
meters =0.039 3700 mil (aprx. %5) = 0.001 millimeter. 1 mil = 25.400 1
microns.
1 terze (Brit.) = Via second =M44 line = ViT28 inch.
1 second (Brit.) = 12. terzes = Ha line = M.44 inch.
1 point = 0.008 inch.
1 point (typography) = % line = 1^2 inch.
lllne (U. S.) = H2inch.
1 line (Brit.) = 144. terzes = 12. seconds = Via inch. Also pven as Vio inch.
1 liairsbreadth = % line = Ms inch.
1 barleycorn = \i inch.
1 nail [na] (cloth) = 2^ inches = >^ span.
1 palm = 3. inches.
1 hand = 4. inches.
1 link ni] (surveyor's) = 7.920 inches =0.201 17 meter.
1 link [lij (engineer's) = 12. inches=l. foot = 0.304 80 meter.
1 span = 9. inches = 4. nails =1. quarter = ^4 yard.
1 quarter [qrl (cloth) = 9. inches = 4. nails = l. 8pan = V4 yard.
1 cubit =18. inches = 1V& feet.
1 cnbit (in Bible) = 21.8 inches.
1 Tara (California; legal) = 33.372 inches.
1 pace = 3. feet.
1 military pace = 3. feet.
1 common pace = 2V^ feet.
1 meter used by Pratt & Whitney Co. = 39.370 79 inches » 1.000 020
meters (int.).
32 LENGTHS.
1 meters 1 553 163.5 wave lengths of red light. t
= 1966 249.7 " " " green " t
= 2 083 372.1 •• *• •• blue " t
1 Committee M^ter (of U. S. Coast Survey) » 39.370 inches- 1
meter (int.j.
1 ell [E, e] (cloth; Brit.) = 45. inches^ 1.143 meters.
1 fathom (U. S.)=-6. feet = 1.828 8 (aprx. %) meters.
1 fathom (Brit.) = 6.080 feet = 1.853 2 (aprx. i^^) meters = ^ooo nauti-
cal mile (Brit.).
1 rod [r], pole [p], or perch [p] (surveyor's) = 6% yards =5.029 2 meters.
1 decameter or dekameter [dkm]= 10. meters.
1 chain [ch] (Gunter's or surveyor's) = 100. links (surveyor's) = 66. feet —
20.117 meters = 4. rods, poles, or perches=Ho furlong = %o mile.
1 chain, [ch] (engineer's) = 100. links (engineer's) = 100. feet = 30.480
meters.
1 chain [ch] (Philadelphia standard) = 100% feet = 30.556 meters.
1 bolt (cloth) = 40. yards.
1 hectometer =100. meters.
1 furlong [fur] = 660. feet = 201.17 meters=40. rods, poles, or perches=
10. chains (Gunter's or surveyor's) =% mile.
1 cable or cable's leng^th (Bnt. navy) =608. feet (sometimes stated as
608.6 feet) = Mo nautical mile (Brit.).
1 cable's leng^th (U. S. Navy) = 720. feet = 219.457 meters= 120. fathoms
(U. S.).
1 car-mile, see under units of Energy.
1 knot (telegraph; Brit.) = 2 029. yards =1 855.32 meters.
1 g:eo|$raphlcal mile, sometimes used for nautical mile; see also under
international geographical mile.
1 International nautical or sea mile = 6 076.10 feet = 1 852. meters =
^ oLl** of meridian.
1** of latitude at equator = 60. (aprx.) nautical miles.
•• =68.70 miles (statute),
"lat. 20° = 68.78
•• •• 40* = 69.00
.. .. 600 = 69 23
.. .. 800 = 69.39
.. .. 900 = 69.41
1** of longitude at equator = 60. (aprx.) nautical miles.
= 69.16 miles (statute).
•• lat. 20° = 65.02 "
♦• .. 400 = 53.05 •*
.. .. 60° = 34.67 "
.. .. 800 = 12.05 •*
1 legaa (California; legal) = 2.633 5 miles = 5 000. varas.
1 league (U. S.) =4.828 05 kilometers = 3. miles (statute); also given as
3. nautical miles.
1 international geographical mile = 24 350.3 feet = 7 422. meters =
4.611 80 miles (statute) = 4. (aprx.) nautical miles =^5 of 1° at equator.
The term geographical mile is sometimes used also for nautical mile.
1 myriameter or mirlameter = 10. kilometers.
1 mean diameter of the earthy (astronomy) = 12 742.0 kilometers =
7 917.5 miles.
1 mean diameter of earth's orbit§ (astronomy) = 149 340 870 ±
96 101 kilometers = 92 796 950 ± 59 715 miles.
t Michelson; cadmiimi light waves for air at 15° C. and a pressiire of
760 mm. mercury.
t Based on the accepted value of a nautical mile as defined above.
S Harkness.
LENGTHS. 33
LENGTHS (concluded). Foreign.
These are mostly obsolete, as the metric system is now used in most
foreign countries. The British measures are included among the U. S.
measures, being very nearly, and sometimes quite the same. The trans-
lated terms are merely synonymous, and not the exact equivalents.
Germany. Pruatia. Legal May 16, 1816. 1 Fuss H (foot) «= 12 ZollH
(inches) of 12 Linien ['"] (hnes). 1 Fuss (also called " rheinlaendiscner
Fuss," that is, Rhineland foot) = 0.313 853 5 meter. Road measiu*e: 1
Meile (mile) » 2 000 Ruthen (rods) of 12 Fuss (feet); 1 MeUe»7.532 5
kilometers; 1 Ruthe» 3.766 25 meters. From Jan. 1, 1872, to Jan. 1, 1874:
1 deutsche Meile (German mile) =° 7.500 kilometers. Trade measure: 1
EUe(yard)'»2>^ Fuss (feet)»25H ZoU (inch) « 0.666 939 meter. 1 Lach-
ter«80 Zoll (inches) » 2.092 36 meters.
Bavaria. 1 Fuss (foot) » 12 Zoll (inches) of 12 Linien (lines). More
rarely 1 Fu8s-> 10 Zoll of 10 Linien. 1 Fuss = 0.291 859 meter.
Saxony. 1 Fuss = 0.283 19 meter; subdivisions like in Bavaria.
Wvertemberg. 1 Fuss = 0.286 49 meter; subdivisions like in Baden.
Baden. 1 Fuss (foot) — 10 Zoll (inches) of 10 Linien (lines). 1 Fus8=>
0.3 meter.
Hanover. 1 Fuss =0.292 1 meter.
The following values of various German feet in inches are given in Nys-
trom's Mechanics: Bavaria, 11.42; Berlin, 12.19: Bremen, 11.38; Dres-
den, 11.14; Hamburg, 11.29; Hanover, 11.45; Leipsic, 11.11; Prussia,
12.36; Rhineland, 12.35; Strasburg, 11.39. Also the following road meas-
ures: Germany, mile, long, 10 126- yards; Hamburg mile, 8 244 yards;
Hanover mile, 11 559 yards; Prussia mile, 8 468 yards.
France. "Old measures" (systdme ancien) used prior to 1812: 1 toise
(fathom) = 6 pieds (du roi) (feet) of 12 pouces (inches) of 12 lignes (lines)
of 12 points. In geodesy: 1 pied = 10 pouces of 10 lignes of 10 points.
1 toise = 1.949 037 meters. Road measure: 1 lieue (league) = 2 283 toises =
4 449.65 meters; 1 lieue marine 7=2 854 toises = 5 562.55 meters; 1 lieue
moyenne = 2 534 toises = 4 938.86 meters. Field measure: 1 perche
(perch) = 18 or 22 pieds. For depths of the sea: 1 brasse = 5 pieds. Trade:
1 aune (yard) de Paris= 1.188 45 meters. According to Nystrom's Me-
chanics: 1 pied du roi = 12.79 inches; 1 league (lieue) marine = 6 075 yards;
1 league, common =4 861 yards; 1 league, post, = 4 264 yards.
"Usual" measures (systdme usuel) used from 1812 to 1840: 1 toise
(fathom) = 2 meters; 1 pied (foot) = >i meter; 1 aune (yard) = 1.2 meters.
For subdivisions into other units see above under "old measures." 1 noeud
— 1 knot or nautical mile.
Austria. 1 Ruthe (rod) = 10 Fuss (feet) of 12 Zoll (inches) of 12 Linien
(lines). 1 Fuss =0.316 10 meter. 1 Meile (mile) = 7.586 kilometers =
4p00Klafter of 6 Fuss. 1 Elle (yard) = 2.46 Fuss. According to Nys-
trom's Mechanics: 1 Vienna foot = 12.45 inches.
Sweden. 1 meile (mile) = 6 000 famn of 3 alnar of 2 fot (foot) of 10 tum
(inches) (or 12 vorktum) of 10 linier (lines). 1 meile = 10.688 4 kilometers.
1 fot = 0.296 901 meter. 1 ruthe (rod) = 16 fot. 1 corde = 10 stangen
of 10 fot. According to Nystrom's Mechanics: 1 Swedish foot = 11.69
inches; 1 Swedish mile = 11 700. yards-
Ruttia. 1 werst = 5(X) saschenn of 3 arschin of 4 tschetwert of 4 wer-
schock. 1 werst = 1.066 8 kilometers =3 500 feet =0.662 88 mile. 1 sa-
8ohehn = 2.133 6 meters = 7.0 feet (U.S.). 1 arschin = 71.12 centimeters =
28.0 inches. 1 tschetwert = 17.780 centimeters = 7.0 inches. 1 werschock =
4.445 centimeters =1.75 inches.^ The British foot is also used; 1 foot =
12 inches of 12 lines. According to Nystrom's Mechanics: 1 Russian
foot = 13.75 inches; 1 Moscow foot = 13.17 inches; 1 Riga foot = 10.79
inches; 1 Warsaw foot = 14.03 inches; 1 verst = l 167. yards.
Switzerland. 1 Fuss (foot) = 10 Zoll (inches) of 10 Linien (lines).
1 Fuss = 0.3 meter. According to Nystrom's Mechanics: 1 Geneva foot =
19.20 inches; 1 Zurich foot — 11.81 inches; 1 Swiss mile = 9 153 yards.
Holland. 1 Ruthe (rod) =12 Fuss (feet). 1 Ruthe = 3.767 36 meters]
1 Fuss =0.31 3 947 meter. According to Nystrom's Mechanics: 1 Amster-
dam foot = 11.14 inches; 1 Utrecht foot = 10.74 inches; 1 Flanders mile*
34 LENGTHS.
6 869 yards; 1 Holland mae = 6 395 yards; 1 Netherlands mile = 1093
yards.
Spain. 1 vara (yard) = 32.874 8 inches » 0.835 022 meter. According
to Nystrom's Mechanics: 1 Spanish foot = 11.03 inches; 1 toesas — 66.72
inches; 1 palmo=8.64 inches; 1 Spanish common league = 7 416 yards.
Italy. According to Nystrom's Mechanics: 1 Florence braccio = 21.69
inches; 1 Genoa palmo = 9.72 inches; 1 Malta foot = 11. 17 inches; 1 Naples
palmo = 10.38 inches; 1 Rome foot = 11.60 inches; 1 Sardinia palmo =
9.78 inches; 1 Sicily palmo = 9.53 inches; 1 Turin foot = 12.72 inches;
1 Venice foot = 13.40 inches; 1 Rome mile = 2 025. yards.
Japan. Long measure: 1 ri = 36 cho of 60 ken of 6 shaku of 10 sun of
10 bu. 1 jo = 10 shaku. 1 ri = 3.93 kilometers = 2.44 miles. 1 kilometer =
0.255 ri. 1 mile = 0.410 ri. For cloth measure: 1 jo = 10 shaku of 10 sun
of 10 bu ; the unit is the jo ; in this measure the units are 3^ longer than those
of the same name in the long measure.
Miscellaneous (from Nystrom's Mechanics): Antwerp foot =11.24
inches; Brussels foot = 11.45 mches. Denmark mile = 8 244 yards; Copen-
hagen foot = 12.35 inches. Portugal league =6 760 yards; Lisbon foot =
12.96 inches; Lisbon palmo=8.64 inches. Ireland mile = 3 038 yards;
Scotland mile = 1984 yards. Hungary mile = 9 113 yards. Bohemia
mile =10 137 yards. Poland mile, long, =8 101 yards. Turkey berri =
1826 yards. Persia parasang = 6 086 yards; Persia arish = 38.27 inches.
Arabia mile =2 148 yards. China li = 629 yards; mathematic foot = 13.12
inches; builder's foot = 12.71 inches; tradesman's foot = 13.32 inches;
surveyor's foot = 12.58 inches. .
Ancient. Biblical: 1 digit = 0.912 inch; 1 pahn = 4 digits = 3.648
inches; 1 span = 3 palms = 10.94 inches; 1 cubit = 2 spans=21.888 inches;
1 fathom = 3.46 cubits = 7.296 feet. Egyptian: 1 finger =0.737 4 inch;
1 nahud cubit = 1.476 feet; 1 royal cubit = 1.722 feet. Grecian: 1 digit =
0.754 inch; 1 pous = 16 digits=1.007 3 feet; 1 cubit = 1.133 feet; 1
stadium = 604.375 feet; 1 mile = 8 stadiums =4 835. feet. Hebrew: 1
cubit = 1.822 feet; 1 Sabbath day's journey = 3 648. feet ; 1 mile = 4 000
cubits = 7 296 feet; 1 day's journey = 33.164 miles; 1 sacred cubit = 2.02
feet. Roman: 1 digit = 0.725 7 inch; 1 uncia (inch) = 0.967 inch; 1 pes
(foot) = 12 uncias= 11.60 inches; 1 cubit = 24 digits = 1.45 feet; 1 passuss
3.33 cubits = 4.835 feet; 1 millarium (mile) = 4 842 feet. Arabian: 1 foot»a
1.095 feet. Babylonian: 1 foot = 1.14 feet.
Leng^ths in -which OTerhead Telegraph Ijines Are or Were Ez-
gressed in Diflferent Countries. (From Munro and Jamieson's Pocket
ook.) The lengths here given are in English or statute miles of 5 280 feet.
Arabia, mile 1.220 4. Austria, mile 5.753 4. Bohemia, mile 6.759 6.
Brabant, league 3.452 2. Burgundy, league 3.516 6. China, li 0.359 1.
Denmark, mile 4.684 1. Flanders, league 3.90. Hamburg, mile 4.684 1.
Hanover, mile 6.567 6. Hesse, mile 5.992 6. Holland, mile 4.602 8. Hun-
gary, mile 5.177 8. Italy, mile 1.150 5. Lithuania, mile 5.557 3. Nor-
way, mile 7.018 3. Oldenburg, mile 6.147 7. Poland, long mile 4.602 8:
short mile 3.4517. Portugal, league 3.840 9. Prussia, mile 4.80Q8;
Rome, mile 0.925 0. Russia, verst 0.663 0. Saxony, mile 5.627 8. Silesia,
mile 4.024 4. Spain, common legua of 8 000. varas, 4.213 6; legal legua
of 5 000 veras, 2.753 4. Swabia, mile 5.633 5. Sweden, mile 6.647 7.
Switzerland, mile 5.200 5. Turkey, berri 1.037 5. Tuscany, mile 1.027 2.
Westphalia, mile 6.903 9.
LENGTHS.
35
Inches in Fractions, DecimalH, Millimeters, and Feet.
For every 64th of an inch up to 1 inch; for every 32d of an inch up to 6
inches; for every 16th of an inch up to 12 inches; for every inch up to 10
feet. The equivalents of other intermediate values, or of values beyond
the table, may be found by adding together two or more values from the
table; thus for ll%a inches add that for 11 inches to that for %2.
Milli-
meters.
Inches.
Feet.
0.001 302
0.002 604
0.003 906
0.005 208
MiUi-
meters.
Inches.
Feet.
0.396 876
0.793 762
1.190 63
1.587 50
H4
H2
Vie
0.016 625
0.031 250
0.046 875
0.062 500
19.446 9
19.843 8
20.240 7
20.637 5
*%4
2%2
«H4
'%6
0.765 625
0.781 250
0.796 875
0.812 500
0.063 80
0.065 10
0.066 41
0.067 71
1.984 38
2.381 25
2.778 13
3.175 01
%2
0.078 125
0.093 750
0.109 375
0.125 000
0.006 510
0.007 813
0.009 115
0.010 42
21.034 4
21.431 3
21.828 2
22.225
»%4
•2%2
8%4
'A
0.828 125
0.843 750
0.859 375
0.875 000
0.069 01
0.070 31
0.071 61
0.072 92
3.571 88
3.968 76
4.365 63
4.762 51
%4
%2
»H4
%6
0.140 625
0.156 250
0.171 875
0.187 500
0.011 72
0.013 02
0.014 32
0.015 63
22.621 9
23.018 8
23.415 7
23.812 5
B%4
2%2
5»/64
1%6
0.890 625
0.906 250
0.921 875
0.937 500
0.074 22
0.075 52
0.076 82
0.078 13
5.159 39
5.556 26
5.953 14
6.350 01
1%4
%2
1%4
0^03 125
o!218 750
0.234 375
0.250 000
0.016 93
0.018 23
0.019 53
0.020 83
24.209 4
24.606 3
25.003 2
25.400 1
«H4
«%4
1
0.953 125
0.968 750
0.984 375
1.000 000
0.079 43
0.080 73
0.082 03
0.083 33
6.746 89
7.143 76
7.540 64
7.937 52
1%4
%2
1%4
•He
0.266 625
0.281 250
0.296 875
0.312 500
0.022 14
0.023 44
0.024 74
0.026 04
26.193 8
26.987 6
27.781 3
28.575 1
IH2
IVie
1%2
1^8
1.031 250
1.062 500
1.093 750
1.125 000
0.085 94
0.088 54
0.091 15
0.093 75
8.334 39
8.731 27
9.128 14
9.525 02
2^4
0.328 125
0.343 750
0.359 375
0.375 000
0.027 34
0.028 65
0.029 95
0.031 25
29.368 8
30.162 6
30.956 3
31.750 1
1%2
1%6
1%2
IK
1.156 250
1.187 500
1.218 750
1.250 000
0.096 35
0.098 96
0.101 6
0.104 2
9.921 89
10.318 8
10.715 6
11.112 5
2%4
1%2
2%4
0.390 625
0.406 250
6.421 875
0.437 500
0.032 55
0.033 85
0.035 16
0.036 46
32.543 8
33.337 6
34.131 3
34.925 1
1%2
l«^i6
1»M2
1.281 250
1.312 500
1.343 750
1.375 000
0.106 8
0.109 4
0.112
0.114 6
11.509 4
11.906 3
12.303 1
12.700
2%4
1%2
33^4
y2
0.453 125
0.468 750
0.484 375
0.500 000
0.037 76
0.039 06
0.040 36
0.041 67
35.718 8
36.512 6
37.306 3
38.100 1
11%2
l^ie
ll'y82
1.406 250
1.437 500
1.468 760
1.500 000
0.117 2
0.119 8
0.122 4
0.125
13.096 9
13.493 8
13.890 7
14.287 5
«%4
1%2
«%4
^i'6
0.515 625
0.531 250
0.546 875
0.662 500
0.042 97
0.044 27
0.045 57
0.046 88
38.893 8
39.687 6
40.481 3
41.275 1
11%2
l»/io
11%2
1.531 250
1.562 500
1.693 750
1.625 000
0.127 6
0.130 2
0.132 8
0.135 4
14.684 4
15.081 3
15.478 2
15.875
«%4
1%2
«%4
0.578 125
0.593 750
0.609 375
0.625 000
0.048 18
0.049 48
0.050 78
0.052 08
42.068 8
42.862 6
43.656 3
44.450 1
VH2
12%2
1.656 250
1.687 500
1.718 750
1.750 000
0.138
0.140 6
0.143 2
0.145 8
16.271 9
16.668 8
17.065 7
17.462 5
*H4
*%4
0.640 625
0.656 250
0.671 875
0.687 500
0.053 39
0.054 69
0.055 99
0.057 29
45.243 8
46.037 6
46.831 3
47.625 1
12%2
I'^ie
13%2
m
1.781 250
1.812 500
1.843 750
1 .875 000
0.148 4
0.151
0.153 6
0.156 3
17.859 4
18.256 3
18.653 2
19.050
*%4
2%2
H
0.703 125
0.718 750
0.734 375
0.750 000
0.058 59
0.069 90
0.061 20
0.062 50
48.418 8
49.212 6
50.006 3
60.800 1
12%2
l^'A2
2
1.906 250
1.937 500
1.968 750
2.000 000
0.158 9
0.161 5
0.164 1
0.166 7
36
LENGTHS.
Milli-
meters.
Inches.
Feet.
Milli-
meters.
Inches.
Feet.
51.693 9
52.387 6
53.181 4
63.976 1
2^2
2^6
2%2
2H
2.031 250
2.062 500
2.093 750
2.125 000
0.169 3
0.171 9
0.174 6
0.177 1
96.043 9
96.837 7
97.631 4
98.426 2
32%2
327^2
3.781 260
3.812 600
3.843 760
3.875 000
0.315 1
0.317 7
0.320 3
0.322 9
64.768 9
65.662 6
66.366 4
67.150 1
2%2
2^6
2%a
2M
2.166 260
2.187 600
2.218 760
2.260 000
0.179 7
0.182 3
0.184 9
0.187 6
99.218 9
100.013
100.806
101.600
32%2
31«^l6
33^2
4
3.906 260
3.937 600
3.968 760
4.000 000
0.325 5
0.328 1
0.330 7
0.333 3
67.943 9
68.737 6
69.631 4
60.326 1
2«/62
2«Ho
21^2
21^
2.281 260
2.312 600
2.343 750
2.375 000
0.190 1
0.192 7
0.196 3
0.197 9
102.394
103.188
103.981
104.776
4^2
4%a
4>^
4.031 260
4.062 600
4.093 750
4.125 000
0.335 9
0.338 6
0.341 1
0.343 8
61.118 9
61.912 6
62.706 4
63.600 1
21%2
27,16
21%2
2J^
2.406 260
2.437 60Q
2.468 760
2.600 000
0.200 5
0.203 1
0.205 7
0.208 3
105.569
106.363
107.166
107.960
4%2
4JH6
4%2
4H
4.166 250
4.187 500
4.218 750
4.250 000
0.346 4
0.349
0.351 6
0.354 2
64.293 9
65.087 6
65.881 4
66.675 1
21 %2
2«ji6
21%2
2^
2.531 250
2.662 600
2.693 760
2.626 000
0.210 9
0.213 6
0.216 1
0.218 8
108.744
1 109.638
1 110.331
111.126
4%2
4%e
41H2
4H
4.281 260
4.312 600
4.343 760
4.376 000
0.366 8
0.369 4
0.362
0.364 6
67.468 9
68.262 6
69.056 4
69.860 1
22>i2
2Hi6
22%2
234
2.666 260
2.687 500
2.718 760
2.760 000
0.221 4
0.224
0.226 6
0.229 2
111.919
112.713
113.506
114.300
41%2
4%6
41%2
4>^
4.406 250
4.437 600
4.468 750
4.500 000
0.367 2
0.369 8
0.372 4
0.375
70.643 9
71.437 6
72.231 4
73.026 1
22%2
2i'H6
22%2
2K
2.781 260
2.812 500
2.843 760
2.875 000
0.231 8
0.234 4
0.237
0.239 6
116.094
116.888
116.681
117.475
41%2
4»/i6
41%2
4.631 260
4.562 600
4.593 760
4.626 000
0.377 6
0.380 2
0.382 8
0.386 4
73.818 9
74.612 7
76.406 4
76.200 2
22%a
21^6
23^2
3
2.906 260
2.937 600
2.968 760
3.000 000
0.242 2
0.244 8
0.247 4
0.260
118.269
119.063
119.856
120.650
42,^2
4Hi6
42%2
4M
4^656 260
4.687 600
4.718 760
4.760 000
0.388
0.390 6
0.393 2
0.396 8
76.993 9
77.787 7
78.681 4
79.376 2
3^2
3H6
3%2
3H
3.031 260
3.062 500
3.093 760
3.126 000
0.262 6
0.266 2
0.267 8
0.260 4
121.444
122.238
123.031
123.826
42%2
4i%e
42%2
4.781 250
4.812 500
4.843 750
4.875 000
0.398 4
0.401
0.403 6
0.406 3
80.168 9
80.962 7
81.766 4
82.560 2
3%2
3%6
3%2
3.166 260
3.187 600
3.218 750
3.260 000
0.263
0.265 6
0.268 2
0.270 8
124.619
125.413
126.207
127.000
42%2
4i%6
43>^2
6
4.906 250
4.937 500
4.968 750
5.000 000
0.408 9
0.411 5
0.414 1
0.416 7
83.343 9
84.137 7
84.931 4
85.725 2
3%2
3%6
31M2
3.281 260
3.312 600
3.343 750
3.376 000
0.273 4
0.276
0.278 6
0.281 3
127.794
128.588
129.382
130.176
5?^2
5Vi6
5JJ^2
53 8
5.031 260
5.062 500
5.093 750
5.125 000
0.419 3
0.4210
0.424 5
0.427 1
$6,518 9
87.312 7
88.106 4
88.9d0 2
31%2
37^6
31%2
3J^
3.406 260
3.437 600
3.468 750
3.600 000
0.283 9
0.286 5
0.289 1
0.291 7
130.969
131.763
132.567
133.360
5%2
5%2
5H
5.156 260
5.187 600
5.218 750
5.250 000
0.429 7
0.432 3
0.434 9
0.437 5
89.693 9
90.487 7
91.281 4
92.076 2
3i%3
3^6
31%2
3^^
3.631 260
3.662 600
3.693 750
3.625 000
0.294 3
0.296 9
0.299 6
0.302 1
134.144
134.938
136.732
136.525
5%2
5%e
5»H2
5.281 260
5.312 500
5.343 760
5.375 000
0.440 1
0.442 7
0.445 3
0.447 9
92.868 9
93.662 7
94.466 4
96.250 2
32,^2
31M6
32%2
3M
3.666 260
3.687 500
3.718 750
3.760 000
0.304 7
0.307 3
0.309 9
0.312 5
137.319
138.113
138.907
139.700
51%2
57,i6
51%2
5H
5.406 250
5.437 500
5.468 750
5.500 000
0.450 6
0.463 1
0.455 7
0.468 3
LENGTHS.
37
Milli-
meters.
Inches.
Feet.
MilU-
meters.
Inches. '
Feet.
140.494
141.288
142.082
142.875
61%2
5%6
51%2
5H
5.531 250
5.562 500
5.593 750
5.625 000
0.460 9
0.463 6
0.466 1
0.468 8
217.488
219.075
220.663
222.250
8%e
8.562 500
8.626 000
8.687 500
8.750 000
0.713 6
0.718 8
0.724
0.729 2
143.669
144.463
145.257
146.050
52^2
51H0
52%a
5H
5.656 250
5.687 500
5.718 750
5.750 000
0.471 4
0.474
0.476 6
0.479 2
223.838
225.425
227.013
228.600
8i%e
81%6
9
8.812 600
8.876 000
8.937 500
9.000 000
0.734 4
0.739 6
0.744 8
0.750
146.844
147.638
148.432
149.225
52%a
51%6
52%2
5Vh
5.781 250
5.812 500
5.843 750
5.875 000
0.481 8
0.484 4
0.487
0.489 6
230.188
231.776
233.363
234.960
9Me
9H
9%6
9H
9.062 500
9.125 000
9.187 500
9.250 000
0.766 2
0.760 4
0.765 6
0.770 8
150.019
150.813
151.607
152.400
52%2
51%6
6
5.906 250
5.937 500
5.968 750
6.000 000
0.492 2
0.494 8
0.497 4
0.500
236.538
238.125
239.713
241.300
9«H6
9%e
9H
9.312 600
9.375 000
9.437 600
9.500 000
0.776
0.781 3
0.786 5
0.791 7
153.988
155.575
157.163
158.750
6^6
6^
6%6
6>i
6.062 500
6.125 000
6.187 500
6.250 000
0.505 2
0.510 4
0.515 6
0.520 8
242.888
244.475
246.063
247.650
9%6
91V46
9.562 500
9.626 000
9.687 600
9.750 000
0.796 9
0.802 1
0.807 3
0.812 5
160.338
161.925
163.513
165.100
me
6«"8
6vie
6.312 500
6.375 000
6.437 500
6.500 000
0.526
0.531 3
0.536 5
0.641 7
249.238
260.825
252.413
254.001
91%6
9i«He
10
9.812 500
9.875 000
9.937 500
10.000 000
0.817 7
0.822 9
0.828 1
0.833 3
166.688
168.275
169.863
171.450
6%8
m
61^6
m
6.562 500
6.625 000
6.687 500
6.750 000
0.546 9
0.552 1
0.557 3
0.562 5
255.588
257.176
258.763
260.351
lOMe
lOH
lO^ie
10.062 500
10.125 000
10.187 500
10.250 000
0.838 5
0.843 8
0.849
0.854 2
173.038
174.625
176.213
177.800
6IH0
QVh
6i«Ho
7
6.812 500
6.875 000
6.937 500
7.000 000
0.667 7
0.672 9
0.578 1
0.583 3
261.938
263.526
265.113
266.701
10«K6
10%6
lOV^
10.312 500
10.375 000
10.437 500
10.500 000
0.869 4
0.864 6
0.869 8
0.875
179.388
180.975
182.563
184.150
7^6
7»h6
7H
7.062 600
7.125 000
7.187 500
7.250 000
0.588 5
0.593 8
0.599
0.604 2
268.288
269.876
271.463
273.061
10%6
10^
1011^6
lOM
10.562 600
10.625 000
10.687 600
10.760 000
0.880 2
0.886 4
0.890 6
0.895 8
185.738
187.325
188.913
190.500
7^6
7H
7.312 500
7.375 000
7.437 500
7.500 000
0.609 4
0.614 6
0.619 8
0.625
274.638
276.226
277.813
279.401
10i%e
103^^
10i«H6
11
10.812 600
10.875 000
10.937 500
11.000 OUO
0.9010
0.906 3
0.911 5
0.916 7
192.088
193.675
195.263
196.850
7%6
7iJ^6
7«i
7.562 500
7.625 000
7.687 500
7.750 000
0.630 2
0.635 4
0.640 6
0.645 8
280.988
282.576
284.163
285.751
llMe
IIH
ll%e
UK
11.062 500
11.125 000
11.187 600
11.250 000
0.921 9
0.927 1
0.932 3
0.937 6
198.438
200.025
201.613
203.200
7HH6
7^
71^16
8
7.812 500
7.875 000
7.937 500
8.000 000
0.651
0.656 3
0.661 5
0.666 7
287.338
288.926
290.513
292.101
ll'He
11^/^ ■
llTie
IIH
11.312 500
11.375 000
11.437 500
11.600 000
0.942 7
0.947 9
0.953 1
0.958 3
204.788
206.376
207.963
209.550
8%6
8li
8.062 500
8.125 OOC
8.187 50C
8.250 000
0.671 9
0.677 1
0.682 3
0.687 5
293.688
' 296.276
296.863
298.451
11%6
im
113^
11.662 600
11.626 000
11.687 600
11.750 000
0.963 6
0.968 8
0.974
0.979 2
211.138
212.726
214.313
216.900
8Vie
8%6
8H
8.312 500
8.375 000
8.437 500
8.500 000
0.692 7
0.697 9
0.703 1
0.708 3
300.038
301.626
303.213
304.801
lli%e
IVA
1U%6
12
11.812 500
11.875 000
11.937 500
12.000 000
0.984 4
0.989 6
0.994 8
1.000
38
LENGTHS.
Millimeters
Feet and Inches.
Millimeters.
Feet and Inches.
304.801
1 foot inches
1 676.40
5 feet 6 inches
330.201
1 1
1 701.80
5 7
355.601
1 2
1 727.20
5 8
381.001
1 3
1 752.60
5 9
406.401
1 4
1 778.00
5 10
431.801
1 6
1 803.40
5 11
457.201
1 6
1 828.80
6
482.601
1 7
1 854.20
6 1
508.001
1 8
1 879.60
6 2
533.401
1 9
1 905.00
6 3
558.801
1 10
1 930.40
6 4
584.201
1 11
1 955.80
6 5
609.601
2 feet
1 981.20
6 6
635.001
2 1
2 006.60
6 7
660.401
2 2
2 032.00
6 8
685.801
2 3
2 057.40
6 9
711.201
2 4
2 082.80
6 10
736.601
2 5
2 108.20
6 11
762.002
2 6
2 133.60
7
787.402
2 7
2 159.00
7 1
812.802
2 8
2 184.40
7 2
838.202
2 9
2 209.80
7 3
863.602
2 10
2 235.20
7 4
889.002
2 11
2 260.60
7 5
914.402
3
2 286.00
7 6
939.802
3 1
2 311.40
7 7
965.202
3 2
2 336.80
7 8
990.602
3 3
2 362.20
7 9
1 016.00
3 4
2 387.60
7 10
1 041.40
3 5
2 413.00
7 11
1 066.80
3 6
2 438.40
8
1 092.20
3 7
2 463.80
8 1
1 117.60
3 8
2 489.20
8 2
1 143.00
3 9
2 514.61
8 3
1 168.40
3 10
2 540.01
8 4
1 193.80
3 11
2 565.41
8 5
1 219.20
4
2 590.81
8 6
1 244.60
4 1
2 616.21
8 7
1 270.00
4 2
2 641.61
8 8
1 295.40
4 3
2 667.01
8 9
1 320.80
4 4
2 692.41
8 10
1 346.20
4 5
2 717.81
8 11
1 371.60
4 6
2 743.21
9
1 397.00
4 7
2 768.61
9 1
1 422.40
4 8
2 794.01
9 2
1 447.80
4 9
2 819.41
9 3
1 473.20
4 10
2 844.81
9 4
1 498.60
4 11
2 870.21
9 5
1 524.00
5
2 895.61
9 6
1 549.40
5 1
2 921.01
9 7
1 574.80
5 2
2 946.41
9 8
1 600.20
5 3
2 971.81
9 9
1 625.60
5 4
2 997.21
9 10
1 651.00
5 5
3 022.61
9 11
3 048.01
10 feet
LENGTHS.
39
CToBTersion Tables for Liens^bs-
Ins. =
milixu't 1
Bims =
tnehenj.
Feet =
meters.
feet
yards.
klmtrs.
Mtrs =
Yds.=
aieters.
Miles =
Klms^
miles.
1
25.4001
0.039370
0.304801
3.28083
0.914 402
1.093 61
1.609 35
.621 370
2
50.8001
0.078 740
0.609 GOl
6.561 67
1.828 80
2.187 22
3.218 69
1.242 74
3
76.200 2
0.118 110
0.914 402
9.842 50
2.743 21
3.28083
4.828 04
1.86411
4
101.600
0.157 480
1.219 20
13.123 3
).*>57 61
4.374 44
6.437 39
2.485 48
5
127.000
0.196850
1.52400
16.404 2
1.57201
5.468 06
iMQ 73
3.10685
6
152.400
0.236220
1.828 80
19.6850
5.486 41
6.561 67
9.656 08
3.728 22
7
177.800
0.275 590
2.13360
22.9658
6.40081
7.655 28
11.265 4
4.34959
8
203.200
0.314960
2.438 40
26.2467
7.315 21
8.748 89
12.874 8
4.97096
9
228.600
0.354 330
2.743 21
29.527 5
S.229 62
9.84250
14.484 1
5.592 33
10
254.001
0.393 700
3.04801
32.808 3
D.14402
10.936 1
16.093 5
6.213 70
11
279.401
0.433 070
3.35281
36.089 2
10.058 4
12.029 7
17.702 8
6.835 07
12
304.801
0.472 440
3.657 61
39.3700
10.972 8
13.123 3
10.312 2
7.456 44
13
330.201
0.511 810
3.962 41
42.650 8
11.887 2
14.216 9
20.921 5
8.077 81
14
355.601
0.551 180
4.267 21
45.931 7
12.801 6
15.310 6
22.5309
8.699 18
15
381.001
0.590 550
4.572 01
49.212 5
13.716
16.404 2
24.1402
9.320 55
16
406.401
0.629 920
4.87681
52.493 3
14.630 4
17.497 8
25.740 6
9.941 92
17
431.801
0.669 290
5.181 61
55.774 2
15.544 8
18.591 4
27.358 9
10.563 3
18
457.201
0.708 600
5.486 41
59.055
16.459 2
19.685
28.968 2
11.184 7
19
482.601
0.748 030
5.791 21
62.335 8
17.373 6
20.778
30.577 6
11.8060
20
508.001
0.787 400
6.096 01
65.616 7
18.288
21.872 2
32.186 9
12.427 4
21
533.401
0.826 770
6.400 81
68.897 5
19.202 4
•22.965 8
W.706 3
13.0488
22
558.801
0.866 140
6.705 61
72.178 3
20.1168
24.059 4
15.405 6
13.670 1
23
584.201
0.905 510
7.010 41
75.459 2
21.0312
25.153 1
^7.015
14.291 5
24
609.601
0.944880
7.315 21
78.740
21.945 6
26.246 7
]8.G24 3
14.912 9
25
635.001
0.984 250
7.620 02
82.020 8
22.860
27.340 3
10.233 7
15.534 2
26
360.401
1.023 62
7.924 82
85.301 7
23.774 4
28.433 9
11.843
16.155 6
27
685.801
1.062 99
8.229 62
88.582 5
24.688 9
29.527 5
13.452 4
16.777
28
711.201
1.102 36
8.534 42
91.863 3
25.603 3
30.621 1
15.061 7
17.398 4
29
736.601
1.14173
S.839 22
95.144 2
26.517 7
31.714 7
16.671 1
18.019 7
30
762.002
1.181 10
9.144 02
98.425
27.432 1
32.808 3
48.280 4
18.641 1
31
787.402
1.220 47
9.448 82
101.706
28.346 5
33.901 9
49.889 8
19.262 5
32
SI 2. 802
1.259 84
9.753 62
104.987
29.2609
34.995 6
51.499 1
19.883 8
33
838.202
1.299 21
10.058 4
108.268
30.175 3
36.089 2
53.108 5
20.505 2
34
S63.602
1.338 58
10.363 2
111.548
31.089 7
37.182 8
54.717 8
21.126 6
35
389.002
1.377 95
10.668
114.829
32.004 1
38.276 4
56.327 2
21.747 9
36
914.402
1.417 32
10.972 8
118.110
32.918 5
39.3700
57.936 5
22.369 3
37
939.802
1.456 69
11.277 6
121.391
33.832 9
40.463 6
59.545 8
22.990 7
38
365.202
1.496 06
11.582 4
124.672
34.747 3
41.557 2
61.155 2
23.612 1
39
990.602
1.535 43
11.887 2
127.953
35.661 7
42.650 8
62.764 5
24.233 4
40
1 016.00
1.574 80
12.192
131.233
36.576 1
43.744 4
64.373 9
24.854 8
41
1 041.40
1.614 17
12.496 8
134.514
37.490 5
44.838 1
65.983 2
25.476 2
42
1 066.80
1.653 54
12.801 6
137.795
38.404 9
45.931 7
37.592 6
26.097 5
43
1 092.20
1.692 91
13.106 4
141.076
39.319 3
47.025 3
39.201 9
26.718 9
44
1 117.60
1.732 28
13.411 2
144.357
40.233 7
48.118 9
70.811 3
27.340 3
45
1 143.00
1.771 65
13.716
147.638
41.1481
49.212 5
72.420 6
27.961 6
46
1 168.40
1.81102
14.020 8
150.918
42.062 5
50.306 1
74.030
28.583
47
1 193.80
1.850 39
14.325 6
154.199
42.976 9
51.399 7
75.639 3
29.204 4
48
1 219.20
1.889 76
14.630 4
157.480
43.891 3
52.493 3
77.248 7
29.825 8
49
1 244.60
1.929 13
14.935 2
160.761
44.805 7
53.586 9
78.858
30.447 1
50
1 270.00
1.968 50
15.240
164.042
45.720 1
54.680 6
80.467 4
31.068 5
40
LENGTHS.
Conversion Tables for I«eng:ths (concluded).
Ins. =
Mms=
Feet =
Mtrs«
Yds.=
Miless
Klins»
milim't
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
1 295.40
1 320.80
1 346.20
1371.60
1 397.00
1 422.40
1 447.80
1 473.20
1 498.60
1 524.00
1 549.40
1 674.80
1600.20
1 625.60
1651.00
1 676.40
1 701.80
1 727.20
1 752.60
1 778.00
1 803.40
1 828.80
1 854.20
1 879.60
1 905.00
1 930.40
1 955.80
1 981.20
2 006.60
2 032.00
2 057.40
2 082.80
2 108.20
2 133.60
2 159.00
2 184.40
2 209.80
2 235.20
2 260.60
2 286.00
2 311.40
2 336.80
2 362.20
2 387.60
2 413.00
2 438.40
2 463.80
2 489.20
2 514.60
2 540.01
inches.
2.007 87
2.047 24
2.086 61
2.125 98
2.165 35
2.204 72
2.244 09
2.283 46
2.322 83
2.362 20
2.401 57
2.44094
2.480 31
2.519 68
2.559 05
2.598 42
2.637 79
2.677 16
2.716 63
2.755 90
2.795 27
2.834 64
2.87401
2.913 38
2.952 75
2.992 12
3.031 49
3.070 86
3.110 23
3.149 60
3.188 97
3.228 34
3.267 71
3.30708
3.346 45
3.385 82
3.425 19
3.464 56
3.503 93
3.543 30
3.582 67
3.622 04
3.661 41
3.700 78
3.740 15
3.779 52
3.818 89
3.858 26
3.897 63
3.937 00
meters.
15.544 8
15.849 6
16.154 4
16.459 2
16.764
17.068 8
17.373 6
17.678 4
17.983 2
18.2880
18.592 8
18.897 6
19.202 4
19.507 2
19.812
20.116 8
20.421 6
20.726 4
21.031 2
21.3360
21.640 8
21.945 6
22.250 4
22.565 2
22.860
23.1648
23.469 6
23.774 4
24.079 2
24.3840
24.688 8
24.993 6
25.298 4
25.603 3
25.908 1
26.212 9
26.517 7
26.822 5
27.127 3
27.432 1
27.736 9
28.041 7
28.3465
28.651 3
28.956 1
29.260 9
29.565 7
29.870 5
30.176 3
30.480 1
feet.
167.323
170.603
173.884
177.165
180.446
183.727
187.008
190.288
193.569
196.860
200.131
203.412
206.693
209.973
213.264
216.635
219.816
223.097
226.378
229.658
232.939
236.220
239.601
242.782
246.063
249.343
262.624
265.906
259.186
262.467
266.748
269.028
272.309
276.690
278.871
282.162
286.433
288.713
291.994
296.276
298.666
301.837
306.118
308.398
311.679
314.960
318.241
.321.522
324.803
328.083
meters.
46.634 6
47.548 9
48.463 3
49.377 7
60.292 1
61.206 6
62.120 9
53.035 3
53.949 7
64.864 1
66.778 5
66.692 9
57.607 3
68.621 7
69.436 1
60.360 5
61.264 9
62.179 3
63.093 7
64.008 1
64.922 6
66.836 9
66.761 3
67.666 7
68.580 1
69.494 5
70.408 9
71.323 3
72.237 7
73.162 1
74.066 6
74.981
76.896 4
76.809 8
77.7242
78.638 6
79.663
80.467 4
81.3818
82.296 2
83.2106
84.126
86.039 4
85.963 8
86.868 2
87.782 6
88.697
89.6114
90.625 8
91.440 2
yards.
66.774 2
66.867 8
67.961 4
69.066
60.148 6
61.242 2
62.336 8
63.429 4
64.623 1
65.616 7
66.710 3
67.803 9
68.897 6
69.991 1
71.084 7
72.178 3
73.271 9
74.366 6
76.469 2
76.652 8
77.646 4
78.740
79.833 6
80.927 2
82.020 8
83.114 4
84.208 1
86.301 7
86.396 3
87.488 9
88.682 6
89.676 1
90.769 7
91.863-3
92.9669
94.060 6
96.144 2
96.237 8
97.331 4
98.426
99.618 6
100.612
101.706
102.799
103.893
104.987
106.080
107.174
108.268
109.361
klmtrs.
82.076 7
83.686 1
86.296 4
86.904 7
88.514 1
90.123 4
91.732 8
93.342 1
94.951 6
96.560 8
98.170 2
99.779 5
101.389
102.998
104.608
106.217
107.826
109.436
111.045
112.654
114.264
116.873
117.482
119.092
120.701
122.310
123.920
125.529
127.138
128.748
130.357
131.966
133.676
136.185
136.795
138.404
140.013
141.623
143.232
144.841
146.461
148.060
149.669
161.279
162.888
164.497
156.107
167.716
169.326
160.936
miles.
31.689 9
32.311 2
32.932 6
33.5540
34.175 3
34.7967
36.418 1
36.039 5
36.660 8
37.282 2
37.903 6
38.524 9
39.146 3
39.767 7
40.389
41.010 4
41.6318
42.263 2
42.874 5
43.496 9
44.117 3
44.738 6
46.360
46.981 4
46.602 7
47.224 1
47.845 5
48.466 9
49.088 2
49.709 6
50.331
60.952 3
51.573 7
62.195 1
52.816 4
53.4378
54.059 2
54.680 6
55.301 9
65.923 3
66.544 7
57.166
57.787 4
58.408 8
59.030 1
69.661 5
60.272 9
60.894 3
61.615 6
62.137
SURFACES. 41
SURFACES.
There are no fundamental standards of surfaces or areas, as these meas-
ures are all based on the linear measures, which see for their fimdamental
values. Tlie legal U. S. yard, foot, inch, mile, etc., being about 3 parts
in one million larger than the legal values in Great Britain, the U. S. square
yard, square foot, etc., will be about 6 parts in one million larger than
the British, a difference which is absolutely negligible in all but tne most
refined physical measurements. The values given in the following tables
are based on the U. S. legal relation. To reduce the values in these tables
to British square miles, square yards, square feet, and similar units, when
very great accuracy is required subtract 6 parts for every million from
all those values of a sq. mile, sq. yard, etc., which are in terms ef other
units than miles, yards, etc.; lor instance, add this correction in the value
of one sq. mile in sq. meters, but not in sq. feet, as the latter is the same
for both; or add 6 parts for every million to all the values of other units
which are given in terms of sq. miles, sq. yards, etc. This will never affect
the 5th place of figures by more than one ubit and generally less.
A circular nnit or measure (such as a circular mil) is the area of a
circle whose diameter is one unit or measure (as one mil). It is used for
cross-sections of round wires, pipes, rods, etc., and avoids the necessitv
of using the value of n ( = 3.141 592 65). If the diameter of a circle is a,
its area in circular units is simply cP. Some of the specific relations given
in the table may also be expre^d as follows:
If d is the diameter of a circle expressed in one kind of a unit, then the
area in the other unit will be:
if d is in mils the area in sq. millimeters ^d^X 0.000 506 709*
if d is in millimeters the area in sq. mils —d^Xl 217.36*
if d is in centimeters the area in sq. inches =d2x 0.121 736*
if d is in centimeters the area in sq. feet —d^X 0.000 845 391*
if d is in inches the area in sq. centimeters --d^X 5.067 09*
if d is in inches the area in sq. feet '^d^XO.OOS 454 15*
if d is in feet the area in sq. centimeters «= d^ x 729.662*
if d is in feet the area in sq. inches -^d^X 113.097*
SURFACES; Circular or Cross-section Measures. Usual.
Aprx. means within 2% ; "sq." means square and is often used as a suffix
instead of the exponent (*), being simpler for printing and type-writing;
thus sq. cm for cm^; sq. ft for ft^.
* Checked by L. A. Fischer, Asst. Phys. National Bureau of Standards.
Logarithm
1 circular mil [CM]= 0.785 398* sq. mil. Aprx. Ho 1895 0899
= 0.000 645 163* Cmm. Aprx. %i -4- 1 000 .. |.809 6692
=0.000 506 709* SCI. mm. Aprx. M + 1 000. 1.704 7591
= 0.000 001 circular inch g.OOO 0000
1 sq. mil [mil2]= 1.273 24* circular mils. Aprx. i% 0.104 9101
=0.000 821 447* circ. mm. Aprx. Ms-*- 100 i.914 5793
=0.000 645 163* sq. millimeter. Aprx. ^^i -5- 1 000 |.809 6692
= 0.000 001 sq. inch g.OOO 0000
1 circular millimeter [Cmm]:
= 1 550.00* circular mils. Aprx. ^ViXl 000 3.190 3308
= 1 217.36* sq. mils. Aprx.%X 1 000 8.085 4207
=0.785 398* S9. millimeter. Aprx. %o 1895 0899
= 0.01 circular centimeter 3-000 0000
1 sq. millimeter [mm2]= l 973.52* cir. mils. Aprx. 2 000.. 3.295 2409
- 1 550.00* s<i. mils. Ap. i^ X 1 000 3.190 3308
= 1.273 24* circ. mm. Aprx. 1%. . . 0'.104 9101
= 0.012 732 4* circ. cm. Aprx. Ho 2.104 9101
= 0.01 sq. centimeter 3.000 0000
^0.001 550 00* sq. in. Apneas + 100 . g.l90 8808
(C
it
it
It
tt
It
tl
tt
tl
tl
42 SURFACES.
1 circular centimeter [Ccm]:
= 155 000.* circular mils. Aprx. ^Vi X 100 000 5190 3308
= 121 736.* sq. mils. Aprx. 12 X 10 000 5085 4207
= 100. circular millimeters 2-000 0000
= 78.539 8 * sq. millimeters. Aprx. 80 1.895 0899
= 0.785 398 * sq. centimeter. Aprx. %o 1895 0899
= 0.155 000 * circular inch. Aprx. ^a 1.190 3308
= 0.121 736 * sq. inch. Aprx. 12 -i- 100 1.085 4207
= 0.001 076 39 * circular foot. Aprx. 108 -^ 100 000 g.031 9683
= 0.000 845 391 * sq. foot. Aprx. Vl2 -J* 100 4.927 0582
1 gq. centimeter [cm^]:
= 197 352.* circular mils. Aprx. 2 X 100 000 5.295 2409
= 155 000.* sq. mils. Aprx. i^ X 100 000 5-190 3308
= 127.324 * circular millimeters. Aprx. J^ X 1 000 2104 9101
= ^ 100. sq. millimeters 2.000 0000
= 1.273 24 * circular centimeters. Aprx. i% 0104 9101
= 0.197 352 * circular inch. Aprx. %o 1-295 2409
= 0.155 000 * sq. inch. Aprx. %s 1.190 3308
= 0.01 * sq. decimeter 2-000 0000
= 0.001 370 50 * circular foot. " Aprx. ^}4^l 000 §.136 8784
= 0.001 076 387 * sq. foot. Aprx. 108 -h 100 000 §.031 9683
1 circular incli [Cin]:
= 1 000 000. circular mils 6-000 0000
= 785 398.* sq. mils. Aprx. % x 1 000 000 5-895 0899
= 645.163 * circular millimeters. Aprx. i% X 100 2-809 6692
= 506.709 * sq. millimeters. Aprx. HX 1 000 2-704 7591
= 6.451 63 * circular centimeters. Aprx. i% 0-809 6692
= 5.067 09 * sq. centimeters. Aprx. 5 0-704 7591
= 0.785 398 * sq. inch. Aprx. %o 1-895 0899
=0.006 944 44 * circular foot. Aprx. 7 h- 1 000 §.841 6375
= 0.005 454 15 * sq. foot. Aprx. %i -^ 100 §.736 7274
1 sq. inck [in2]= 1 273 240.* cir. mils. Aprx. >^ X 10 000 000 6-104 9101
= 1 000 000. sq. mils 6-000 0000
= 821.447 * cir. mm. Aprx. %X1 000 2-914 5793
= 645.163 * sq. mm. Aprx. i% X 100 2-809 6692
= 8.214 47 * cir. centimeters. Aprx. % X 10. . 0.914 5793
= 6.451 63 * sq. cm. Aprx. i% or 6M 0-809 6692
= 1.273 24 * circular inches. Aprx. i% 0104 9101
= 0.064 516 3 * sq. decimeter. Aprx. T4i -^ 10. . . 2-809 6692
= 0.008 841 94 * circular foot. Aprx. 9-1-1 000. . . §.946 5476
= 0.006 944 44 * sq. foot. Aprx. Tiooo S-841 6375
1 sq. decimeter [dm2]= 100. sq. centimeters 2.000 0000
= 15.500 0* sq. inches. Aprx. 1^X10. 1.190 3308
= 0.107 638 7* sq. foot. Aprx. 11 -J- 100.. LOSl 9683
= 0.01 sq. meter 2-000 000(
= 0.011 959 ;9* sq. yard. Aprx. 12-^1 000 2-077 7258
1 circular foot [Cft]= 929.034* cir. cm. Aprx. 1M2X 1 000.. . 2-968 0317
= 729.662* sq. cm. Aprx. ^ii X 1 000 2-863 1216
= 144. circular ins. Aprx. ^^X 1 000. 2-158 3625
= 113.097* sq. inches. Aprx. % X 1 000. . 2-053 4524
= 0.785 398* sq. foot. Aprx. %o 1.895 0899
1 sq. foot [ft2] (Brit.) = 0.999 994 2 sq. foot (U. S.). Aprx. 1 T.999 9975
= 0.092 902 9 sq. meter. Aprx. 1V42 -5-10. . 2-968 0292
1 sq. foot [ft 2] (U. S.)= 1 182.88* cir. cm. Aprx. Ve X 1 000. . 3-072 9418
= 929.034* sq. cm. Aprx. 31,12X1 000. 2-968 0317
= 183.346* cir. in. Aprx. ^H X 100 2-263 2726
= 144. sq. ins. Aprx. ^Xl 000. .. 2-158 3625
= 9.290 34* sq. dm. Aprx. 1^2 X 10. . . 0.968 0317
= 1.273 24* circular feet. Aprx. 1% 0104 9101
= 1.000 005 7 sq. feet (Brit.). Aprx. 1. . . 0000 0025
= 0.111 111* sq. yard, or% 1-045 7576
= 0.092 903 4* sq. meter. Aprx. 11^2-5-10. 2-968 0317
1 sq. yard [yda] (Brit.) = 0.999 994 3 sq. yard (U. S.). Aprx. 1... T.999 9975
•' = 0.836 126 sq. meter. Aprx.% 1-922 2717
it
it
it
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It
It
It
It
It
II
It
It
It
II
it
it
11
It
n
II
It
SURFACES. 43
1 sq. yard ryd2](U. S.)- 8 361.31 * sq. cm. Ap.%X10000 3.922 2742
= 1 296.* sq. ins. Aprx. 13 X 100 8.112 6050
= 83.613 1 * sq. dm. Aprx. % X 100 1.922 2742
=• 9. sq. feet 0954 2425
•* = 1.000 005 7 sq. yds. (Brit.). Ap. 1. 0000 0025
= 0.836 131 * sq. meter. Aprx.% . . I.922 2742
- 0.008 361 31 * are. Aprx. %-*-100. . .. f .922 2742
= 0.000 206 612* acre. Ap. 21-J-lOOOOO. i.315 1545
1 sq. meter [m2]=i 10 000. sq. centimeters 4-000 0000
= 1 550.00 * sq. ins. Aprx. 11^ X 1 000 3. 190 8308
" =• 100. sq. decimeters 2-000 OOOO
= 10.763 93 sq.ft. (Brit.). Aprx. 1^1 X 10 1.081 9708
= 10.763 87 * sq. ft. (U. S.). Aprx. i%i X 10 1.031 9683
= 1.195 99 sq. yds. (Brit.). Aprx. add ^^ 0077 7283
= 1.195 99 *sq. yds. (U.S.). Aprx. add % 0-077 7258
" = 0.01 are 3-000 0000
= 0.000 247 104 * acre. Aprx. H-^l 000 1.392 8804
1 are or ar [a]- 1 076.387* sq. feet. Aprx. i%i X 1 000 8.031 9688
" = 119.599* sq. yards. Aprx. 120 2-077 7258
" « 100. sq. meters 2000 0000
'* = 10. meters square 1-000 0000
*' = 1 . sq. decameter 0-000 0000
•• - 0.024 710 4* acre. Aprx. M -^ 10 2.392 8804
— 0.01 hectare 2.000 0000
1 acre [A]- 43 560.* sq. feet. Aprx. 3^3 X 1 000 000 4-639 0879
■ « 4 840.* sq. yards. Aprx. 4 800 8-684 8454
« 4 046.87 * sq. meters. Aprx. 4X1 000 8607 1196
« 208.710 * feet square. Aprx. 210 2-319 5440
= 40.468 7 * ares. Aprx. 40 1-607 1196
= 0.404 687 * hectare. Aprx. ^0 1-607 H96
« 0.004 046 87 * sq. kilometer. Anrx. 4-5- 1 000 3-607 1196
=0.001 562 50 * sq. mile. Aprx. ^Vr -J- 1 000 g.193 8200
1 hectare [ha]= 107 638.7* sq. feet. Aprx. i%i X 100 000. . .. 5031 9683
"11 959.9* sq. yards. Aprx. 12 X 1 000 4077 7258
" « 10 000. sq. meters 4-000 0000
** -= 100. ares 2000 0000
= 2.471 04* acres. Aprx. 1% 0-392 8804
" ■= 0.01 sq. kilometer 2.000 0000
= 0.003 861 01* sq. mile. Aprx. He-?- 10 g.586 7004
l8q.kiloiiaeter[km2]= 10 763 867.* sq.ft. Ap. i%iX 10000000 7-0319683
= 1 195 985.* sq. yds. Aprx. 12 X 100 000 6-077 7258
" =1 000 000. sq. meters 6-000 0000
— 10 000. ares 4.OOO 0000
= 247.104 * acres. Aprx. J4Xl 000 2392 8804
" = 100. hectares 2000 0000
- 0.386 101 * sq. mile. Aprx. He X 10. . . 1.586 7004
1 sq. mile [ml2] = 27 878 400.* sq. feet. Aprx. ^}4 X 10 000 000 . 7.445 2678
«= 3 097 600.* sq. yards. Aprx. 31 X 100 000. . . 6-49102*^3
•= 2 589 999.* sq. meters. Aprx. 26 X 100 000. . 6-413 2996
= 25 900.0 * ares. Aprx. 26 X 1 OOO 4-413 2996
= 640. acres 2-806 1800
= 259.000 * hectares. Aprx. 260 2-413 2996
«= 2.590 00 * sq. kilometers. Aprx. 26 -i- 10 0.413 2996
•
SURFACES (continued). Unusual, Special Trade, or
Obsolete.
1 inilliare = 0.1 sq. meter.
1 centare, centar, or centaire = 10.763 87 sq. feet = l. sq. meter.
1 square (building) = 100. sq. feet.
1 deciare = 10. sq. meters = 0.1 are.
1 sq. rod, or sq. pule, or sq. perch = 625. sq. links (surveyor's) = 272^
sq. feet = 30*'^ sq. yards = 25.293 sq. meters = ^60 acre.
1 sq- chain (Gunter's or surveyor's) = 4 356. sq. feet = 484. sq. yards=
404.687 sq. meters = 16. sq. rods, poles, or perches = 4.046 87 ares = ^0 acre =
0.040 468 7 hectare = 0.000 404 687 sq. kilometer = 0.000 156 250 sq. mile.
44
SURFACES.
1 sq. meter «=0.002 471 04 sq. chain. 1 aTe»0.247 104 sq. chain. 1 acre —
10. sq. chains. 1 hectare » 24.710 4 sq. chains.
1 decare (not used) = l 000. sq. meters »= 10. ares.
1 rood rR]«10 890. sq. feet=-l 210. sq. yards =1 011.72 sq. meters —
40. sq. rods, poles, or perches— M acre.
1 farding^deal (Brit.)>ol. rood.
1 circular acre ^^ 235.504 feet diameter.
1 section (of land) » 1. mile square.
1 township =» 36. sq. miles.
1 sq. myriaineter or miriameter^KX). sq. kilometers = 38.610 1 sq.
miles.
SURFACES (concluded). Foreign.
These are mostly obsolete as the metric system is now used in most
foreign countries. The British measures are included among the U. S.
measures, being very nearly, and sometimes quite, the same. The trans-
lated terms are merely synonymous, and not the exact equivalents.
Germany. Prussia 1 Morgen«180 sq. Ruthen (rods). 1 Morgen —
25.532 2$ ares » 2 553.225 sq. meters =» 0.630 912 acres. According to
Nystrom's Mechanics, 1 Berlin Morgen, great, — 6 786 sq. yards; ditto,
small, « 3 054 sq. yards. 1 Hamburg Morgen = 11 545 sq. yards. 1 Han-
over Morgen = 3 100 sq. yards. 1 Prussian Morgen = 3 053 sq. yards.
France. 1 arpent (acre)=sl(X) sq. perches (French, nearly same as
U. S.) = 34.19 are, or 51.07 are =0.844 849 acres, or 1.261 96 acres.
Austria. 1 Joch=16(X) sq. Klafter»5 755 sq. meters » 1.422 08
acres. 1 Vienna Joch = 6 889 sq. yards.
Sweden. 1 Tunnland = 56 000 sq. fot»° 49.364 1 ares — 1.219 81 acres.
According to Nystrom it is equal to 5 900 sq. yards.
Russia. 1 Dessaetine = 2 400 sq. saschehn =» 10 925 sq. meters-"
2.699 61 acres. 13 066.2 sq. yards (Nystrom).
Switzerland. 1 Creneva arpent = 6 179 sq. yards; 1 faux =» 7 856 sq.
yards; 1 Zurich acre = 3 875.0 sq. yards.
Spain. 1 fanegada (since 1801) = 1.587 1 acres » 69 134.08 sq. feet;
according to Nystrom it is equal to 5 500 sq. yards.
Japan. 1 cho (apparently not 1 cho length squared) = 10 tan of
10 se of 30 tsubo. 1 tsubo»l ken square = 3.31 square meters = 35.6
square feet. 1 square meter = 0.302 tsubo. 1 square foot = 0.028 1 tsubo.
1 cho = 2.45 acres.
Miscellaneous. (From Nystrom's Mechanics.) 1 fanegada, Canary
Isles, = 2 422 sq. yards. 1 acre, Ireland, = 7 840 sq. yards. 1 acre, Scot-
land, =6 150 sq. yards. 1 mo^gia, Naples, = 3 998 sq. yards. 1 Pezza,
Rome, = 3 158 sq. yards. 1 Geira, Portugal, = 6 970 sq. yards.
Conversion Tables for Surfaces.
Sq. in.=
Sq.cin. =
Sq. ft.=
Sq. m.=
Sq.yd.=
Acres ='
sq. cm.
sq. ins.
sq. met.
sq. ft. .
sq. yds.
hect'res.
sq.met.
Hect'r.=
acres.
1
2
3
4
5
6
7
8
9
10
6.451 63
12.903 3
19.354 9
25.806 5
32.258 1
38.709 8
45.161 4
51.613
58.064 6
64.516 3
0.155 000
0.309 999
0.464 999
0.619 999
0.774 998
0.929 998
1.085 00
1.239 99
1.395 00
1.550 00
0.092 903
0.185 807
0.278 710
0.371 614
0.464 517
0.557-420
0.650 324
0.743 227
0.836 131
0.929 034
10.763 9
21.527 7
32.291 6
43.055 5
53.819 3
64.583 2
75.347 1
86.1110
96.874 8
107.639
0.836 13
1672 26
2.508 39
3.344 52
4.180 65
5.016 78
5.852 91
6.689 05
7.525 18
8.361 31
1.195 99
2.391 97
3.587 96
4.783 94
5.979 93
7.17591
8.371 90
9.567 88
10.7639
11.959 9
0.404 687
0.809 375
1.21406
1.618 75
2.023 44
2.428 12
2.83281
3.237 50
3.64219
4.04687
2.471 04
4.94209
7.413 13
9.884 18
12.355 2
14.826 3
17.297 3
19.768 4
22.239 4
24.7104
VOLUMES. 45
VOLUMES. Cubic and Capacity Measures.
The fundamental standard measures of volumes in the United States
are; (a) the cubes of linear dimensions, in terms of units baaed on the
international meter; (6) the liter which is the volume of the mass of one
international kilogram of pure water at its maximum density, and at
760 mm. barometric pressure; (c) the gallon, which is equal to exactly
231 cubic inches; (d) the bushel (the old Winchester bushel of England)
which is equal to exactly 2 150.42 cubic inches. The inch here referred
to is the one derived from this meter. The liter, being based on this kilo-
gram, is the same in all the countries participating in the international con-
vention. These four measures of volumes and capacities are those now used
by the National Bureau of Standards in Washington and are in general use
in this country. They are definite and accurate, except that the pre-
cise volume of the liter expressed in cubic centimeter is still slightly un-
certain as explained below.
The liter is, according to the decision some years ago of the Interna-
tional Committee of Weights and Measures, defined as "the volume of the
mass of one kilogram of pure water at its maximum density, and under
normal atmospheric pressure." The accepted temperature, at present,
at which the density of water is a maximum, is 4** C.» but this may be
altered slightly by subsequent determinations. The normal atmospheric
pressure referred to is that exerted by a vertical column of mercury
760 mm. high, at latitude 45** and at sea level, the mercury having a den-
sity of 13.595 93. This definition of the liter has been adopted by the
National Bureau of Standards. It was originally intended that the kilo-
gram should be the weight of a cubic decimeter of water, thus making
the liter exactly a cubic decimeter. The kilogram, however, being now
fixed definitely as the weight of a certain piece of metal, and as this estab-
lishes the liter, it still remains to determine by measurement the precise
volume of the liter in cubic decimeters. The International Bureau began
this determination some five years ago, and the results thus far obtaine^d
indicate that the liter is about 25 parts in 1 000000. greater than the cubio
decimeter, t As this relation will always be subject to slight correctioncr,
and as the difference is of no consequence in practice, the National Bureau
of Standards has for the present assumed that the liter and the cubic
decimeter are equivalent; this identity is assumed also in all the tables
in this book. Tiie difference above mentioned would at most affect only
the fifth place of figures slightly* and generally only the sixth. The capac-
ities of vessels are determined by weighing the water necessary to fill
them, and not by measuring their dimensions (see the table of the weights
and volumes of water).
The Customs Service and the Internal Revenue Bureau use the gallon
of 231 cubic inches above referred to; it is the old British wine gallon.
In these tables this gallon and the measures based on it are indicated as
•'liquid; U. S." to distinguish them from the "dry; U. S." measures and
the Imperial or "Brit." measures.
The gallon and liter are the units for measuring liquids, while the bushel
and liter are the units for dry measures, as for grain, iruits, vegetables,
etc. The "diy" measures are about one sixth greater than the corre-
sponding "liquid" measures; accurately:
dry measures X 1.163 65 (aprx. add H) = liquid meas. (log Q065 8213)
liquid measures X 0.859 367 (aprx. subtr. ^) = dry meas. (log 1.934 1787)
The fluid ounce is in use chiefly by apothecaries. The barrel is no fixed
unit, and no value of a barrel has ever been adopted by Congress; in the
Customs Service and Internal Revenue Bureau every barrel is gauged.
Concerning the bushel the Revised Statutes of the United States, under
the ascertainment of duties on grain, chapter 6, sec. 2919, says: "For
the purpose of estimating the duties on importations of grain, the num-
t Guillaume in a recent book on the International Bureau of Weights
and Measures states that this Bureau has found the mass of a cubic oeci-
meter of water at 4*» C. to be 0.999 955 kg. This makes a liter about 45
parts in 1 000 000. greater than a cubic decimeter. But it seems that no
formal adoption of any value has yet been made.
1
46 VOLUMES.
ber of bushels shall be ascertained by weieht, instead of by measuring:
and sixty pounds of wheat, fifty-six pounds of corn, fifty-six pounds of
rye, forty-eight pounds of barley, thirty-two pounds of oats, sixty pounds
of pease, and forty-two pounds of buckwheat, avoirdupois weight, shall
respectively be estimated as a bushel."
In Great Britain the liter is exactly the same as in the United States.
The gallon, which in this country is often called the Imperial or British
gallon, was originally defined as the volume of '*t€n imperial pounds of
distilled water, weighed in air against brass weights, with the water and
the air at the temperature of 62° Fahrenheit, the barometer being at 30
inches." According to the latest computations of the Standards 0£Sce
in London, the British or Imperial gallon is equal to 4.545 963 1 liters;
this value was made legal in 1896 1 in Great Britain, and is the one used
as a basis throughout these tables, even for computing such values as
the gallon in cubic inches ; the figures are therefore quite consistent through-
out, except that the U. S. yard, foot, inch are used, which are about 3
parts in one million larger than the British, a difference too small to be
considered in any but the most refined physical measurements. Owing
to this difference, the U. S. cubic yard, cubic foot, etc., are about 9 parts
in one million greater than the British. Hence to reduce these values
in these tables to British yards, feet, and inches, when very great accuracy
is required, subtract 9 parts for every million from all the values of a cubio
yard, cubic foot, and cubic inch given in terms of other units than yards,
feet, and inches; for instance, add this correction in the value of a cb.
foot in cb. meters, but not in cb. inches, as the latter is the same for both;
or add 9 parts for every million to all the values of other units given in
terms of cubic yards, cubic feet, and cubic inches. This will never affect
the fifth place of figures by more than one unit, and generally not that
much. I
In Great Britain the measures of capacities (gallons, quarts, etc.) are
the same for liquid as for dry materials. The British measures of capacity
differ but little from the corresponding dry measures in the United States ;
accurately :
Brit. meas.X 1.032 02 (aprx. add >io) = dry U. S. meas. (log 0-013 6888)
dry U.S. meas. X 0.968 972 (aprx. sub. Ho) = Brit. meas. (log 1.986 3112)
The British measures of capacity are about 20% greater than the corre-
sponding liquid measures in the United States; accurately:
Brit. meas.X 1.200 91 (aprx. add %) = liquid U.S. meas. (log 0-079 5101)
liquid U.S. meas. X 0.832 602 4 (aprx. i^a) = Brit. meas. (log T .920 4899 )
The U. S. and the British apothecary fluid measures, smaller than the
pint, differ very little from each other, almost exactly 4%, the former
being the greater; for conversion add 4% to the quantity expressed in
the U. S. measures, or subtract 4% when expressed in British measures.
VOLUMES. Cubic and Capacity Measures. Usual.
** Accepted by the National Bureau of Standards.
* Checked by L. A. Fischer. Asst. Phys. National Bureau of Standards.
Aprx. means within 2%.
ap. means apothecary measures: cb. means cubic, and is often used as a
suffix instead cf the exponent (S), being simpler for printing and type
writing; thus cb. cm. for cm. 3, or cb. ft. for ft.3
1 cb. centimeter [cm^ or 1 milliliter [ml]: Logarithm
= 1 000. cb. millimeters S-OOO 0000
= 0.061 023 4* cb. inch. Aprx.6-i-100 2-785 4962
= 0.01 deciliter 5.000 0000
= 0.002 113 36 pint* (liquid; U.S.). Aprx. 21 -s- 10 000. . 3-324 9742
= 0.001 816 16* pint (dry; U. S.). Aprx. s^i ^ 100 §.259 1529
= 0.001 759 80 pint (Brit.). Aprx. '>A -«- 1 000 |.245 4641
= 0.001 056 68* quart (liquid; U. S.). Aprx. aVi-t-lO 000. §.023 9442
= 0.001 liter or cb. decimeter §-000 0000
= 0.000 908 078* quart (dry; U. S.). Aprx. 9^ 10 000 |.958 1229
= 0.000 879 902 quart (Brit.). Aprx. J^-^ 1 000 4944 4341
t Formerly the legal value is said to have been 4.543 46 liters.
VOLTJMES.
47
1 cb. ifich[in3] =
16 387.16* cb. mm. Aprx. J^X 100 000. . .
16.387 16* cb. cm. or ml. Aprx. HX 100..
= 0.163 871 6* deciliters. Aprx. M
= 0.034 632 0* pt. (liq. ; U. S.). Aprx. % -^ 100.
=. 0.029 761 6* pt. (dry; U. S.). Aprx. 3 -MOO.
- 0.028 838 2 pt. (Brit.). Aprx. % -s- 10
« 0.017 316 0* qt. (liq.; U.S.). Aprx.%-f-100.
« 0.016 387 16 liter or cb. dm. Aprx. ^ -s- 10 . .
« 0.014 880 8* qt. (dry; U.S.). Aprx. %-»- 100
- 0.014 419 1 qt. (Bnt.). Aprx. ^^h- 10
« 0.004 329 00* gal. (liq. ; U. S.). Aprx. % -s- 100
^ 0.003 604 77 gal. (Brit.). Aprx. ^i h- 100. . .
= 0.000 578 704* cb. foot. Aprx. ^ -i- 1 000
21 + 100.
?ii
10
2)^ + 100.
I deciliter [dl]= 100. cb. centimeters
« 6.102 34* cb. inches. Aprx.6^o
=• 0.211 336* pt. (liq.; U. S.). Aprx
- 0.181 616* pt. (dry; U.S.). Aprx
== 0.175 980 pint (Brit.). Aprx.%H
« 0.105 668* qt. (liq.; U. S.). Aprx.
" = 0.1 liter or cb. decimeter ,
- 0.090 807 8* quart (dry; U.S.). Aprx.Vii..
« 0.087 990 2 quart (Bnt.). Aprx. J^-:- 10. . .
« 0.026 417 0* gal. (liq. U.S.). Aprx. %-5- 100
= 0.021997 5 gal. (Brit.). Aprx. 22 -s-1 000.
= 0.003 531 45* cb. foot. Aprx. %-5- 1 000. . . .
I pint [pt] (liquid; U.S.):
= 473.179* cb. centimeters. Aprx. Hi X 10 000
= 28.875* cb. inches. Aprx. % X 100
= 16. fluid ounces (ap. U. S.)
=» 4.731 79* deciliters. Aprx. 4H
4. gills (liquid : U. S.)
= 0.859 367* pint (dry; U. S.). Aprx. %
= 0.832 702 4 pint (Bnt.). Aprx. %
= 0.5 quart (liquids U.S.); or V^
= 0.473 179* liter or cb. decimeter. Aprx. Hi X 10. . .
=. 0.125 gallon (liquid; U. S.): or H
= 0.016 710 1* cb. foot. Aprx.H-5-10
«= 0.004 731 79* hectoliter. Aprx. Hi -*- 10
«0.000 618 891* cb. yard. Aprx. 5^+1 000
= 0.000 473 179* cb. meter or stere. Aprx. Hi -^ 100
1 pint [pt] (dry; U.S.):
=- 550.614* cb. centimeters. Aprx. 550
= 33.600 312* cb. inches. Aprx. H X 100
— 5.506 14* deciliters. Aprx. bHor^H
= 1.163 65* pints (liquid; U.S.). Aprx. lKor%. ..
«= 0.968 972 pint (Brit.). Aprx. subtract Ho
=• 0.550 614* liter or cb. decimeter. Aprx. i>^ ■*- 10. . .
=. 0.5 quart (dry; U. S.); or ^
= 0.121 122 gallon (Brit.). Aprx. % -s- 10
= 0.062 500* peck (U. S.); orM-s- 10
= 0.019 444 6* cb. foot. Aprx. 2^1 -s- 100
= 0.015 625* bushel (U. S.). Aprx. 11^ -s- 100
= 0.005 506 14* hectoliter. Aprx. 13^ + 1 000
=0.000 720 171* cb. yard. Aprx. ^ + 1 000
=0.000 550 614* cb. meter or stere. Aprx. ^H-^ 10 000. . .
1 pint [pt] (Brit.):
cb. centimeters. Aprx. ^ X 1 000. . . .
cb. inches. Aprx. % X 10
fluid ounces (ap. Brit.)
deciliters. Aprx. 5%
gills (Brit.)
pints (liquid; U. S.). Aprx. 1%
pints (dry; U. S.). Aprx. l^o
liter or cb. decimeter. Aprx. ^
quart (Brit.); otH
gallon (Brit.); or V^
peck (Brit.); or^ + lO
4.214 5088
1214 5088
I-214 5088
3.539 4780
2.473 6567
459 9679
238 4480
_ 214 5088
3.172 6267
3158 9879
3636 8880
§.556 8779
4762 4563
2000 0000
0-785 4962
f-324 9742
1259 1529
245 4641
023 9442
000 0000
_ 958 1229
2.944 4841
3.421 8842
3.342 8741
3.547 9525
568.245 39
34.676 2
20.
5.682 453 9
4.
1.200 91
1.032 02
0.568 245 39
0.5
0.125
0.062 5
2675
1460
1204
0675
0602
!;.984
920
.608
; 675
.096
3.222
3675
J.791
i.675
2740
1526
0.740
1.065
:.986
740
698
.083
3.795
3-288
3198
3.740
4857
i-740
2.754
1.540
1-301
0-754
0.602
0079
0013
-754
-698
-096
1795
0258
5220
1200
0258
0600
1787
4899
9700
0258
9100
9788
0258
6145
0258
8471
3433
8471
8218
3112
8471
9700
2215
8800
7996
8200
8471
4358
8471
5359
0321
0300
5359
0600
5101
6888
5359
9700
9100
8800
48 VOLUMES.
1 pint [pt] (Brit.);
= 0.020 067 3 cb. foot. Aprx. 2-*- 100 5-302 4884
= 0.015 626 bushel (Brit.). Aprx. i^^-s- 100 2193 8200
=. 0.005 682 453 9 hectoliter. Aprx. Vi -s- 100 3.754 5359
« 0.000 743 232 cb. yard. Aprx. %^\ 000 i.871 1246
=0.000 568 245 39 cb. meter or stere. Aprx. ^ -*- 1 000 J.764 5359
1 quart [qt] (liquid; U. S.).
= 946.359* cb. centimeters. Aprx. %i X 10 000 2976 0558
» 57.750 0* cb. inches. Aprx. ^ X 100 1.761 5520
= 9.463 59* deciliters. Aprx. %i X 100 0.976 0558
= 8. gills (liquid: U. S.) 0-903 0900
=. 2. pints (liquid; U. S.) 0301 0300
— 0.946 359* liter or CD. decimeter. Aprx. subtr. Ho. • • • 1.976 0558
= 0.859 367* quart (dry; U. S.). Aprx. % T.934 1787
= 0.832 702 4 quart (Brit.). Aprx. % 1.920 4899
= 0.25* gallon (liquid ; U. S. ) ; or K 1-397 9400
= 0.208 170 gallon (Brit.). Aprx. 21 + 100 1.318 4299
= 0.033 420 1* cb. foot. Aprx. M-*- 10 2-524 0083
= 0.009 463 59* hectoliter. Aprx. %i -^ 10 §.976 0558
= 0.001 237 78* cb. yard. Aprx. V^^ 100 §.092 6445
= 0.000 946 359* cb. meter or stere. Aprx. %i -*- 100 |.976 0558
1 cb. decimeter [dm^] = 1 . liter which see for values 0-000 0000
1 liter [1]= 1 000. cb. centimeters 3-000 0000
= 61.023 4* cb. inches. Aprx. 60 1.785 4962
-= 10. deciliters 1-000 0000
= 2.113 36* pints (liquid; U. S.). Aprx. ^Mo 0-324 9742
- 1.816 16* pints (dry; U. S.). Aprx. 2%i 0-259 1529
= 1.759 80 pints (Brit.). Aprx. lHor% 0-245 4641
=- 1.056 68* quarts (liquid; U. S.). Aprx. add Ho-. 0-023 9442
= 1 . CD. decimeter 0-000 0000
= 0.908 078* quart (dry; U. S.). Aprx. »Ao 1-958 1229
= 0.879 902 quart (Brit.). Aprx. "H 1-944 4341
= 0.264 170* gallon (liquid; U.S.). Aprx.%o 1-421 8842
= 0.219 975 gallon (Brit.). Aprx. 22 + 100 1.342 3741
= 0.113 510 peck (U.S.). Aprx. 9^ -4-10 1.055 0329
= 0.109 988 peck (Brit.). Aprx. 11 ■<- 100 1-041 3441
= 0.035 314 5 cb.foot. Aprx.%-HlOO 2547 9525
= 0.028 377 4 bushel (U. S.). Aprx. J^ + 10 2-452 9729
<«
11
It
ti
«i
41
(I
t<
• «
(<
t(
=. 0.027 496 9 bushel (Brit.). Aprx. i^ + 100 2-439 2841
= 0.01 hectoliter 2-000 0000
=0.001 307 94 cb. yard. Aprx. 13-i- 10 000 5.116 5887
" = 0.001 cb. meter or stere §-000 0000
1 quart [qt] (dry; U. S.):
= 1 101.23 cb. centimeters. Aprx. 1 100 3-041 8771
= 67.200 625 cb. inches. Aprx. ^ X 100 1.827 3733
= 11.012 3 deciliters. Aprx. 11 1.041 8771
= 2. pints (dry; U. S.) 0-301 0300
= 1.163 65 quarts (liquid; U. S.). Aprx. add H 0-065 8213
«= 1.101 23 liters or cb. decimeters. Aprx. IHo 0-041 8771
= 0.968 972 quart (Brit.). Aprx. subtract 3^o 1-986 3112
= 0.242 243 gallon (Brit.). Aprx. 24-^100 1.384 2512
= 0.125* peck (U.S.); orH 1-096 9100
= 0.038 889 3* cb. foot. Aprx. He 2-589 8296
= 0.031 250* bushel (U. S.). Aprx. 31-1-1 000 2-494 8500
=- 0.011 012 3 hectoliter. Aprx. 11 -*- 1 000 2041 8771
= 0.001 440 34* cb. yard. Aprx. ^ + 100 3-158 4658
= 0.001 101 23 cb. meter or stere. Aprx. 11 -j- 10 000 3-041 8771
1 quart [qtl (Brit.):
= 1 136.490 8 cb. centimeters. Aprx. ^Xl 000 3.055 5659
= 69.352 5 cb. inches. Aprx. 70 1.841 0621
= 11.364 908 deciliters. Aprx. % X 10 1-055 5659
8. gills (Brit.) 0908 0900
= 2. pints (Brit.) 0-301 0300
= 1.200 91 quarts (liquid; U.S.). Aprx. 1% 0079 5101
= 1.136 490 8 liters or cb. decimeters. Aprx. add^ 0-055 50G9
= 1.032 02 quarts (dry; U. S.). Aprx. IHo Q-018 G008
- 0.300 227 gallon (liquid; U. S.). Aprx. %o 1.477 4501
VOLUMES. 49
I quart [qt] (Brit.)'
» 0.25 gallon (Brit.); otH I.397 9400
— 0.126 peck (Brit.); otH 1.096 9100
■» 0.040 134 6 cb. foot. Aprx. 4-*- 100 2-603 5184
c- 0.031 250 bushel (Brit.). Apnc. 31 + 1 000 2494 8500
= 0.011 364 908 hectoliter. Aprx. %-h100 2065 5659
— 0.001 486 46 cb. yard. Apnc. %-h1 000 f .172 1546
=> 0.001 136 490 8 cb. meter or store. Aprx. %-*-l 000 3-055 5659
1 grallou [gal] (liquid; U. S.):
=- 3 785.43* cb. centimeters. Aprx. ^X 10 000 3.573 1158
— 8S1.** cb. inches. Aprx. % X 100 2-363 6120
= 37.854 3* deciliters. Aprx. % X 100 1.578 1158
= 32. gills (liquid: U. S.) 1.505 1500
« 8. pints (hquid; U. S.) 0-903 0900
»» 4. quarts (hquid; U. S.) 0-602 0600
=- 3.785 43* hters or cb. de<»meters. Aprx. 5^ X 10. ... 0-578 1158
= 0.832 702 4 gallon (Brit.). Aprx. % 1.920 4899
= 0.133 681* cb. foot. Aprx.%-*-10 T.126 0683
= 0.037 854 3* hectoliter. Aprx. ^ X 10 §.578 1158
»0.004 95113* cb.yard. Aprx. )4 + 100 3-694 7045
«=0.003 785 43* cb. meter or stere. Aprx. ^g -j- 100 3.578 1158
1 Imperial gallon [see gallon (Brit.)].
1 grallon [gal] (Brit.)-
= 4 545.963 1 cb. centimeters. Aprx. %X1 000 8.657 6259
=» 277.410 cb. inches. Aprx. ^i X 100 2-443 1221
= 32. gills (Brit.) 1.505 1500
-= 8. pints (Brit.) 0-903 0900
>- 4.545 063 1 liters or cb. decimeters. Aprx. 4H 0-657 6259
— 4. quarts (Brit.) 0-602 0600
= 1.200 91 gallons (liquid; U.S.). Aprx. 1% 0-079 5101
=« 0.160 538 cb.foot. Aprx.%-J-10 T.205 5784
— 0.125 bushel (Brit.); or^ 1.096 9100
= 0.045 459 631 hectoliter. Aprx. %-5- 100 2-657 6259
« 0.005 945 86 cb. yard. Aprx. 6-i- 1 000 §.774 2148
=-0.004 545 963 1 cb. meter or stere. Aprx. % -s- 1 000 3-657 6259
1 peck [pk] (U. S.) = 8 809.82* cb. cms. Aprx. ^ X 10 000. . 3-944 9671
=■ 537.605 0* cb. ins. Aprx. %i X 1 000. . . 2-730 4633
*• — 16. pints (dry; U. S.) 1.204 1200
-= 8.809 82* lit's or cb. dms. Aprx. HXlO 0-944 9671
— 8. quarts (dry; U.S.) 0-903 0900
= 1.937 94 gals. (Brit.). Aprx. 2^1. ... 0-287 3412
= 0.968 972 pk. (Brit.). Aprx. subt. Ho. . 1-986 3112
= 0.311 114* cb. foot. Aprx. 31-*- 100. . . . 1.492 9196
-= 0.25 bushel (U. S.) or >i 1-397 9400
— 0.088 098 2* hectoliter. Aprx. J^ + 10 §.944 9371
= 0.011 522 7* cb.yard. Aprx.%-J-100 2-0615558
" «0.008 809 82* cb. meter. Aprx. J^-i-100... . 5-944 9671
I i;eck[pk](Brit.)=- 9 091.926 2 cb.cms. Aprx. 9 000 3958 6559
=» 554.820 cb. ins. Aprx. i>4X 100. .. 2744 1521
— 16. pints (Brit.) 1.204 1200
" = 9.091 926 2 liters or cb. dms. Aprx. 9.. 0958 6559
'* = 8. quarts (Brit.) 0-903 0900
*• = 2. gallons (Brit.) 0-301 0300
•• = 1.032 02 pecks (U.S.). Aprx. l?^o. 0013 6888
•• = 0.321076 cb.foot. Aprx. 32 -s- 100 . . 1.506 6084
•* « 0.25 bushel (Brit.) or M 1-897 9400
•• - 0.090 919 262 hectoliter. Aprx. 9-*- 100.. 2-958 6559
= 0.011 891 7 cb. yard. Aprx. 12-1-1 000 3.075 2446
" =0.009 091926 2 cb. meter. Aprx. 9-»-l 000. 3-958 6559
I cb. foot [ft3]- 28 317.0* cb. cms. Aprx. ^^ X 100 000 4-452 0475
-=■ 1 728.* cb. inches. Aprx.%X1000 3-237 5437
■= 59.844 2 pints (liq.; U.S.). Aprx. 60 1-777 0217
— 51.428 09 pints (dry; U.S.). Aprx. 51 1.7112004
= 49.832 4 pints (Brit.). Aprx. 50 1-697 5116
« 29.922 1 quarts (liq.; U.S.). Aprx. 30 1.475 9917
« 28.317 0* liters or cb. dms. Aprx.^XlOO... 1-452 0475
- 25.714 05* quarts (dry; U.S.). Aprx. 26 X.410 1704
<«
<«
0.803 564*
0.778 630
0.283 170*
50 VOLUMES.
1 cb. foot [ft3] = 24.916 2 quarts (Brit.). Aprx. 25 1.396 4816
7.480 52* gals. (liq. ; U. S.). Aprx. 5iX 10. .. 0-873 9317
6.428 51* gallons (dry; U.S.). Aprx.6^.. . 0-808 1104
6.229 05 gallons (Brit.). Aprx. 6>i 0-794 4216
3.214 26* pecks (U. S.). Aprx. 3H 0-507 0804
3.114 52 pecks (Brit.). Aprx. 3^^ 493 3916
jushel (U. S.). Aprx. % 1-905 O'>04
>ushel (Brit.). Aprx. % J 891 3316
lectoliter. Aprx. ^ 1-452 0475
«0.037 037 0* cb. yard. Aprx. ^-s- 10 3-568 6362
=0.028 317 0* cb. meter or stere. Aprx. % -i- 10 . . 2-452 0475
1 bushel [bu] (U. S.)-
= 35 239.28* cb. centimeters. Aprx. ■% X 10 000 4-547 0271
= » 150.480 0** cb. inches. Aprx. i% XI 000 3.332 5233
« 64. pints (dry; U.S.) 1.806 1800
« 35.230 28* liters or cb. decimeters. Aprx. 35 1-547 0271
= 32. quarts (dry; U. S.) 1-505 1500
= 7.751 78 gallons (Brit.). Aprx. 7H 0-889 4012
= 4. pecks f U. S-) 0-602 0600
= 1.244 46* cb. feet. Aprx. IH 0-094 9796
= 0.968 972 bushel (Brit.). Aprx. subtr. Ho 1-986 3112
= 0.352 392 8* hectoliter. Aprx. % -*- 10 1-547 0271
= 0.125 quarter (U. S.) or i^ 1-096 9100
= 0.046 091 0* cb. yard. Aprx. %i -s- 10 3.663 6158
= 0.035 239 28* cb. meter or stere. Aprx. % -«- 100 3-547 0271
1 bushel [bul (Brit.):
= 36 367.704 8 cb. centimeters. Aprx. ^i X 100 000 4-560 7159
= 2 219.28 cb. inches. Aprx. % X 10 000 8 346 2121
= 64. pints (Brit.) 1.806 1800
= 36.367 704 8 liters or cb. decimeters. Aprx. ^iX 100. 1-560 7159
= 32. quarts (Brit.) 1-505 1500
= 8. gallons (Brit.) 0-903 0900
= 4. pecks (Brit.) 602 0600
= 1.284 31 cb. feet. Aprx. % 0-108 6684
« 1.032 02 bushels (U. S.)- Aprx. add ^o 0013 6888
= 0.125 quarter (Brit.) or H 1-096 9100
= 0-047 566 9 cb. yard. Aprx. 48-5-1 000 3-677 3046
= 0.036 367 704 8 cb. meter or stere. Aprx. "Hi -s- 10 3-560 7159
1 hectoliter [hl]= 6 102.34* cb. inches. Aprx. 6 000 3-785 4962
- 211.336 pints (liq.; U.S.). Aprx. 210 2324 9742
= 181.616*pts. (dry; U.S.). Aprx. 1^X100.. 2-259 1529
= 175.980 pints (Brit.). Aprx. %X 100 2-245 4641
=- 105.668 quarts (liquid; U.S.) Aprx. 106. 2023 9442
•• =» 100. liters or cb. decimeter 2-000 0000
= 90.807 8 quarts (dry; U.S.). Aprx. 90.... 1-958 1229
= 87.990 2 quarts (Brit.). Aprx. ^X 100 .... 1 944 4341
26.417 gals, (liq.; U. S.). Aprx.%XlO.. 1-4218842
= 21.997 5 gallons (Brit.). Aprx. 22 1.342 3741
= 11.3510 pecks(U.S.). Aprx.^XlO 1055 0329
= 10.998 8 pecks (Brit.). Aprx. 11 1.041 3441
= 3-531 45 cb. feet. Aprx. 3Hor% 0-547 9525
= 2.837 74 bushels (U. S.). Aprx. JJ^ X 10 0-452 9729
= 2.749 69 bushels (Brit.). Aprx. ^H 0-439 2841
= 0.130 794 cb.yard. Aprx. 13-i- 100 1-116 5887
*' = 0.1 cb. meter or stere I-OOO 0000
1 cb. yard [yd^]- 46 656. cb. inches. Aprx. i% X 10 000 4-668 9075
= 1615.79 pints (liq.; U.S.). Aprx.%XlOOO. 3-208 3855
= 1 388.56 pints (dry; U. S.). Aprx. % X 1 000 3142 5642
= 1 345.47 pints (Brit.). Aprx.%X 1 000 3-128 8754
= 807.896 quarts (liquid; U.S.). Aprx. 800.. 2-907 3555
= 764.559 liters or cb. dms. Aprx. ^i X 1 000 . 2883 4113
= 694.279 quarts (dry; U.S.). Aprx. 700 28415342
= 672.737 quarts (Brit.). Aprx. 2^X 1 000... . 2-827 8454
= 201.974 gallons (liq.; U.S.). Aprx. 200. -. 2-305 2955
= 168.184 gallons (Brit.). Aprx. V^ X 1 000 .. 2-225 7854
= 86.784 9 pecks (U. S.). Aprx. Vh X 100 1.938 4442
= 84.092 1 pecks (Brit.). Aprx. %X 100 1-924 7554
VOLUMES.
I ob. yard (yds)- 2T cb feet Apni^iXIO
- 21696 2 bU3helB(U S) Apnt H
- 21023 busbela (But) Aprx 21
-0764559 cb meterorstcra Apnt
I cb. meter [m'] or atere [a]
-61 023 4 cb luches Apri 00 000
-2 113.36 pinlsdiquid TJ S) Apnt 2100
■ 816.15 pints(dry U8"
759.80 "WH ''^''— * <--iuuu
^""*^" -'^ ^XlOO
Volnioes. Cablcal; Cspacltf.
Cb.ius.-
b.cmo
^•f,-
b. met's
cb..e«
cb-mec-
Ob. yd,
SK
!?«-
1
1
.387
i
lilO
S8f
Si
.366 4
:|i
if
176:S7
il
iJIS
lii
i
Si
ill?
3:079
:.8B«7
4:787
29,574
6:967
:54lb2
fflf
.8231
.893 8
F].o>. -
*.~
fl. ounce
liters
liters ■
eallons
liedfll't
sss-
!
6
■8
29.57^
18.2E
207 JK
233.59
0.033 81;
III
0:236 70
>.946 36
3:785 4
b:624 5
b;463G
si
3.7854
|i
22.713
30^283
as
1.264 17
i!792 51
.056 7
.5860
1:377 6
i:7620
2:819 I
5:675 6
14:i89
22:7M
52 VOLUMES.
VOLUMBS. Cubic and Capacity Measures (continued).
Unusual, Special Trade, or Obsolete.
ap. means apothecarv measures; av. means avoirdupois weights'; aprx.
means approximately.
1 luolecale. There are about a million, million, million molecules in
a cb. millimeter of air (Woodward).
1 cubic inilllmeter [mm3]»0.()01 milliliter or cb. centimeter.
1 initiim or drop (ap. Brit.) = 59.192 2 cb. millimeters = H9 fluid scruple
(ai>. Brit.) = 0.059 192 2 milliliter or cb. centimeter = Ko fluid dram (ap.
Brit.). 1 milliliter or cb. centimeter = 16.894 1 minims or drops (ap. Bnt.}.
1 minim or drop (ap. U. S.) = 61.612 cb. millimeters = 0.061 612 milli-
liter or cb. centimeter = Ko fluid dram (ap. U. S.). 1 milliliter or cb. centi-
meter =16.230 6 minims or drops (ap. U. S.).
1 milliliter [ml] = 1 cb. centimet^er, which see above in other table;
it equals 0.001 liter.
1 fluid scruple (ap. Brit.) = 20. minims (ap. Brit.) = 1.183 845 milliliters
or cb. centimeters = >^ fluid dram (ap. Bnt.) = H4 fluid ounce (ap. Brit).
1 milliliter or cb. centimeter =0.844 705 fluid scruple (ap. Brit.).
1 fluid drachm same as fluid dram.
1 fluid dram (ap. Brit.) = 60. minims (ap. Brit'.) = 3.551 534 milliliters
or cb. centimeters = 3. fluid scruples (ap. Brit.) = 0.216 73 cb. inch = H fluid
ounce (ap. Brit.)=H6o pint (Brit.). 1 cb. centimeter =0.281 67 fluid
dram (ap. Brit.).
1 fluid dram (ap. U. S.)=60. minims (ap. U.S.) =0.225 59 (aprx. %)
cb. inch = H fluid ounce (ap. U. S.)=i428 pint (liquid; U. S.) = 3.696 71
milliliter or cb. centimeter. 1 cb. centimeter = 0.270 51 fluid dram (ap.
U. S.).
1 centiliter [cl] = 10. cb. centimeters = 0.610 234 cb. inch = 0.01 liter.
1 fluid ounce [fl ozl (ap. Brit.)=480. minims (ap. Brit.) = 28.412 27
cb. centimeters or milliliters =24. fluid scruples (ap. Brit.) = 8. fluid drams
(ap. Brit.) = 1.733 81 cb. inches = 0.960 73 fluid ounce (ap. U. S.) = >^o pint
(Brit.) = 0.028 412 27 liter. 1 liter = 35.196 fluid ounces (ap. Brit.).
1 fluid ounce [fl oz] (ap. U. S.) = 480. minims (ap. U. S.)»= 29.573 7 cb.
centimeters = 8. fluid drams (ap. U. S.)= 1.804 69 (aprx. %) cb. inches =
1.040 88 fluid ounces (ap. Brit.) = H6 pint (liquid; U. 8.)=0.029 573 7
liter. 1 liter = 33.813 8 fluid ounces (ap. U. S.).
1 gill [gi] (liquid; U. S.) = 118.295 cb. centimeters = 7.218 760 cb. inches
= 1.182 95 deciliters = 0.832 702 4 gill (Brit.)=)^ pint (liquid; U. S.) =
\i quart (liquid ; U. S.) = 0.1 18 295 liter. 1 cb. inch = 0.138 528 gill (liquid ;
or. S.). 1 liter = 8.453 44 giUs (liquid; U. S.).
1 gill [gi] (Brit.) = 142.061 35 cb. centimeters =8.669 05 cb. inches =
1.420 613 5 deciliters^ 1.200 91 gills (liquid; U. 8.) = M pint (Brit.) = H
quart (Brit.) = 0.142 061 35 liter. 1 cb. inch=0.115 352 8 gill (Brit.).
1 liter = 7.039 20 giUs (Brit.).
1 pint[pt] (ap. U. S.) = 128. fluid drams (ap. U. S.) = 16. fluid ounces
(ap. U. S.) =0.473 179 liter = same as liquid U. S. pint.
1 pint [pt] (ap. Brit.) = 480. fluid scruples (ap. Brit.) = 160. fluid drams
(ap. Brit.) = 20. flmd ounces (ap. Brit.) =0.568 245 4 liter = same as ordi-
nary Brit. pint.
1 millistere = 1 cb. decimeter =1 liter.
1 pottle (Brit.) = H gallon Brit.
1 board foot =144. cb. inches = ^2 cb. foot.
1 gallon [gal] (ap. U. S.), same as ordinary liquid U. S. gallon of 231.
cb. inches.
1 gallon [gal.] (wine; old Brit.), same as ordinary liquid, U. S. gallon
of 231. cb. inches.
1 gallon (dry; U. S.) (obsolete; term *'>^ peck" used instead) =
268.802 5 cb. inches=8. pints (dry; U. S.) = 4.404 91 liters = 4. quarts
(dry; U. S.) = H peek (U. S.).
1 gallon [gal] (ap. Brit.), same as ordinary Brit, or Imperial gallon.
1 beer gallon (obsolete) = 282. cb. inches = 8. beer pints = 4. beer quarts.
1 decaliter or dekaliter [dkl] = 10. liters = 1 centistere.
1 centistere = 10. cb. decimeters or liters =1 decaliter =V4oo stere or
cb. meter.
VOLUMES. 53
1 sfram-molecule of any gas at 0** C. and 760 mm. pressure has a vol-
ume of 22 380. cb. centimeters.
1 foot (solid, timber) — 1 cb. ft.
1 solid foot »1 cb. ft.
1 hektollter Thl], same as hectoliter.
1 deoistere [ds]=alOO. cb. decimeters or liters »1 hectoliter »Vio stere
or cb. meter.
1 firkin « 9. gaUons (Uquid; U. S.) = 7.494 3 gallons (Brit.) -34.068 9
liters -"^ barrel. Of butter -=66. pounds (av.).
1 Winchester bushel [bu], same as ordinary bushel. in U. S.«2 150.42
cb. inches.
1 struck bushel, same as ordinary bushel in U. S. — 2 150.42 cb. inches.
«= l>i struck bushels or IH ordinary bushels (U. S.).
1 bushel « 60 pounds of wheat = 56 pounds of com or rye =-48 poun<
of barley » 32 pounds of oats — 60 pounds of peas « 42 pounds of buck-
wheat. (U. S. Customs; 1^^.)
1 coomb (Brit.) = 16. pecks (Brit.) =4. bushels (Brit.) = 1.454 708 hecto-
liters =J4 qiiarter (Brit.).
1 barrel {bbl], has no legal or fixed value ; it varies between 30 and 43
gallons (liquid; U. S.). Barrels should therefore always be gauged. The
following are the most usual values, about in the probable order of im-
portance:
1 barrel (Uquid; U. S.)-31H gallons (Uquid; U. S.)- 4.211 cb. feet =
1.192 4 hectoliters =H hogsheaa.
1 barrel (wine and brandy; Brit.)-=31V^ gallons (Brit.) = 1.431 98
hectoUters. Also given as 36 gallons (Brit.).
1 barrel (Uquid; U. S.) = 31. gaUons (liquid; U. S.) = 1.173 5 hectoliters.
1 barrel (refined oil) = 42. gaUons (liquid; U. S.; used by Standard
Oil Co.).
1 barrel (flour; U. S.) = 3. bushels (U. S.); also given as 3.75 cb. feet;
•* legal" (t) weight given as 196 pounds (av.).
1 barrel (dry; U. S.) = 21H bushels (U. S.) = 26.756 cb. feet = 7.676 45
1 barrel (beer; U. S.) = 36. beer gaUons of 282 cb. inches = 1.663 6
hectoUters.
1 barrel (Uquid; Penna.) = 32. gaUons (liquid; U. S.).
1 sack (coal; Brit.) = 3 bushels (Brit.)=^a chaldron.
1 deci8tere = l hectoUter = 0.1 stere or cb. meter.
1 tierce (Uquid; U. S.) = 42. gaUons (Uquid; U. S.) = 1.589 9 hecto-
Uters =% hogshead (U. S.).
1 tierce (Brit.) = 42. gaUons (Brit.) = 1.909 30 hectoUters = % hogs-
head (Brit.).
1 hoipBhead [hhd] (beer; U. S. obsolete) = 54 beer gallons of 282 cb.
inches.
1 hogshead [hhd] (liquid- U. S.) = 63. gaUons (liquid; U. S.) = 2. bar-
rels of 31H gaUons (liquid; U. S.) = 2.384 8 hectoliters = 1J^ tierces (liquid;
U. 8.).
1 hogshead [hhd] (Brit.) = 63. gaUons (Brit.) = 2.863 96 hectoliters.
1 quarter [qf] (dry; U. S.) = 8. bushels (U. S.) = 2.819 1 hectoUters=M
(aprx.) tun (Brit.).
1 quarter [qr] (Brit.) = 8. bushels (Brit.) = 2.909 416 hectoUter8=2.
coombs (Brit.) = J^ (aprxO tun (Brit.)==Ho last (Brit.).
1 puncheon (Uquid- U. S.)=84. gaUons (Uquid; U. S.) = 3.179 76 hec-
toUters =2. tierces (U. B.).
1 puncheon (Brit.)=84. gallons (Brit.) = 3.818 61 hectoliters =2.
tierces (Brit.).
Ipipe or butt (liquid; U. S.) = 126. gaUons (liquid; U. S.) = 4.769 6
hectoUters =2. hogsheads (U. S.).
Ipipe or butt (Brit.) = 126. gallons (Brit.) = 5.727 91 hectoliters- 2.
hogsheads (Brit.).
1 cord foot (wood) = 4X4 XI foot = 16. cb. feet = 0.453 07 cb. meter or
stere ^^ V^ cord
1 tun (Uquid; U. S.) = 252. gallons (liquid; U. S.) =9.539 3 hectoliters-
6. tierces (liquid; U. S.)=4. hogsheads (U. S.) = 3. puncheons (U. S.) = 2.
pipes or butts.
1 tun (Brit.) = 252. gallons (Brit.) = 11.455 8 hectoUters = 6. tierces
(Brit.) = 4. hogsheads (Brit.) = 3. puncheons (Brit.) = 2. pipes or butts
(Brit.).
54 VOLUMES.
1 perch (masonry) -16HX1H XI foot=24Ji cb, feet (generally 26 cb.
feet, sometimes 22 cb. feet) = 0.916 67 cb. yard «= 0.700 85 stere or cb. meter.
1 solid yard = 1 cb. yard.
1 Htere [s]= 1 cb. meter, which see above in other table.
1 kilollter [kl] =» 1 000. liters =10. hectoliters =» 1 cb. meter or stere.
1 g^rou ton (2240 pounds av.) displacement of w<Uer = 35. SSI 3 cb. feet =
1.328 93 cb. yards = 1.016 05 cb. meters.
1 shipping ton (for cargo; U. S.) = 40. cb. feet » 32.142 6 bushels
(U. S.) = 31.145 2 bushels (Brit.) « 11.326 8 hectoliters « 1.132 68 cb. meters.
1 shipping: ton (for cargo; Brit.) =42. cb. feet =« 33.749 7 bushels
(U. S.) = 32.702 5 bushels (Brit.) = 11.893 1 hectoliters = 1.189 31 cb. meters.
Iclialdron (dry; U. S.) = 44.800 6 cb. feet « 36. bushels (U. S.) =
12.686 1 hectoliters.
1 chaldron (coal; Brit. ) = 12. sacks»36. bushels (Brit.) -=68.658 cb.
feet; weight 3 136. pounds (av.).
1 chaldron (Canada) = 58.64 cb. feet or about 45 bushels (Brit.); stated
also as 25.64 cb. feet or about 20 bushels (Brit.).
1 chaldron (Newcastle) weight 5 936. pounds (presumably coal).
1 "wey (Brit.) = 51.372 4 cb. feet=40. busheb (Brit.)«6. quarters
(Brit.) = ^ last (Brit.V
1 reg^ister ton (snipping; for whole vessels) = 100. cb. feet — 2.831 7
i last (Brit.) = 102.745 cb. feet = 80. bushels (Brit.) = 10. quarters (Brit.)
= 2. weys (Brit.). ^
Icord [c] (wood) = 4X4X8 feet = 128. cb. feet = 8. cord feet = 3.624 58
cb. meters or stere. 1 stere or cb. meter = 0.275 894 cord.
1 toise (Canada) = 261^ cb. ft. = 9.685 18 cb. yds. = 7.404 90 cb. meters.
1 rod (brickwork; Brit.) = 16H feet square X 14. inches = 272Ji sq. feet
of 14. inch wall; conventionally 272. sq. feet of 14. inch wall=317>^ cb. feet
= 8.985 9 cb. meters.
1 rod (engineering works; Brit.) = 306. cb. ft. = llM cb. yards = 8.666
cb. meters.
1 myrialiter or niyrioliter = 10 000. liters = 100. hectoliters = 10. cb.
meters or steres = 1 decastere.
1 decastere or dekastere [dks]=100. hectoliters = 10. cb. meters or
steres = 1 myrialiter.
1 hectostere or hektostere [hks]=100. steres or cb. meters.
1 acre-foot (irrigation) = 325 851. gallons =43 560. cb. feet = l 613.33
cb. yards = 1 233.49 cb. meters.
VOLUMES. Cubic and Capacity Measxures (concluded).
Foreign,
These are mostly obsolete, as the metric system is now used in most
foreign countries. The British measures are included among the U. S.
measures, being very nearly, and sometimes quite, the same. The trans-
lated terms are merely synonymous, and not the exact equivalents.
Germany. Prussian. 1 Fuder = 4 Oxhoft of IH Ohm of 2 Eimer
(bucket) of 2 Anker of 30 Quart (Prussian). 1 Wispel = 6 Tonne (tun) of
4 Scheflfel of 16 Metzen of 3 Quart (Prussian). 1 Quart (Prussian) = 64
cub. Zoll = >^ cub. Fuss = 1.145 03 liters; 1 Wispel= 13.191 hectobters ;
1 Tonne = 2. 198 46 hectoliters; 1 Scheflel = 0.549 61 hectoliter. 1 Schacht-
ruthe = 144. cub. Fuss = 4.451 9 cub. meters. 1 Klafter = 108 cub. Fuss =
3.338 9 cub. meters.
The following values are given by Nystrom; they are in cubic inches.
For liquid measures: 1 Stubgen in Bremen = 194.5, in Hamburg 221, in
Hanover 231. For dry measures: 1 Scheffel in Berlin 3 180, in Bremen
4 339, in Hamburg 6 426. 1 Hanover Malter = 6 868.
France. "Old Measures" (systdme ancien) used prior to 1812. 1 muid
(hogshead) f= 2 feuillettes of 2 quartants of 9 setiers or veltes of 4 pots of
2 pintes (pint, though more nearly equal to a quart) of 2 chopines 9f 2
demi-setiers of 2 possons of 2 demi-possons of 2 roquilles (gill). 1 pinte
(ancienne) = 0.931 32 liter.
VOLUMES. 55
**New measures" (systdme usuelle) used from 1812 to 1840. 1 boisseau «
2.751 2 gallons (Brit.); 1 litron (old liter) -1.760 8 pints (Brit.); hence it
seems that 1 boisseau »12H litrons; 1 pinte»sl liter.
The following values are given bv Nystrom; they are in cubic inches.
For liquid measures: 1 Bordeaux barnque » 14 033. For dry measures:
1 Marseilles charge «» 9 411.
Austria. Liquid measures: 1 Eimer — 40 Maass of 4 Seidel of 2 Piff.
1 Eimer = 66.689 liters; 1 Maass -1.414 724 liters -0.044 8 cub. Fuss.
Dry measures: 1 Mut or Muth — 30 Metzen of 16 Maassel of 4 Futter-
maassel of 2 Becher. 1 Metze— 61.486 82 liters » 1.947 1 cub. Fuss.
The following values are given by Nystrom; they are in cubic inches.
For liquid measures: 1 Hungarian Eimer 4 474; 1 Trieste Orne»=4 007;
1 Vienna Eimer— 3 462; 1 Vienna Maass 86.33. For dry measures: 1
Trieste Stari 4 621 ; 1 Vienna Metzen 3 763.
Sweden. 1 am = 4 anker of 16 kannen of 2 stop. 1 am — 167.030 liters;
1 kanne — 2.617 liters— 100 cub. decimal turn. For grain: 1 tonne — 2
spon of 16 koppen. 1 tonne — 56 kannen — 146.566 liters. According to
Nystrom 1 Swedish eimer — 4 794 cub. inches; 1 kanna— 159.67 cub.
inches; 1 tunna = 8 940 cub. inches.
Russia. 1 tschetwert — 2 osmini of 2 pajok of 2 tschetwerik of 4 tschet-
werka of 2 gamez. 1 tschetwert = 209.9 liters; 1 tschetwerik = 26.209
liters; 1 tschet werka— 6.662 2 liters; 1 gamez = 3.276 1 liters. 1 botschka
(barrel) — 40 wcdro of 10 kruschky or stoof of 10 tscharky. 1 kruschky —
1.228 5 liters. According to Nystrom 1 Russian weddras — 752 cub. incheF;
1 kunkas— 94 cub. inches; 1 Riga loop — 3 978 cub. inches; 1 chetwert —
12 448. cubic inches.
Spain. 1 cantara of wine (castile) — 4.263 gallons (liquid; U.S.); in
Havana 4.1 gallons. 1 fanega of corn — 1.699 14 bushels (U. S.). Accord-
ing to Nystrom 1 azumbras — 22.6 cub. inches; 1 quartellos — 30.6 cub.
inches; 1 catrize — 41 269. cub. inches; 1 Malaga fanaga — 3 783. cub. inches.
Japan. 1 koku — 10 to ; 1 to - 10 sho of 10 go of 10 seki. 1 koku — 180.39
liters = 47.66 gallons (U. S.). 1 liter = 0.006 54 koku. 1 gallon (U. S.)-
0.021 koku.
Miscellaneous. The following are given in Nystrom's Mechanics in
cubic inches. For liquid measures: Amsterdam Anker — 2 331; stoop 146;
Copenhagen Anker 2 356; Antwerp stoop 194. Florence oil barille 1 946,
wine barille 2 427; Genoa wine barille 4 530, pinte 90.5; Leghorn oil
barille 1 942; Naples wine bariUe 2 644, oil staio 1 133; Rome wine barille
2 560, oil barille 2 240, boccali 80; Sicily oil caffiri 662; Malta caffiri 1 270;
Venice Secchio 628. Lisbon almude 1040; Oporto almude 1655; Con-
stantinople almude 319. Greneva setier 2 760. Canaries arrobas 949.
Scotland pint — 103.5. Tripoli mattari 1 376. Tunis oil mattari 1 167.
For dry measures: Amsterdam mudde 6 596, sack 4 947; Rotterdam
sach 6 361; Copenhagen toende 8489; Antwerp viertel 4705. Florence
stari 1 449; Genoa mina 7 382; Leghorn stajo 1 601, sacco 4 603: Milan
mog^ 8 444; Naples temoli 3 122; Rome rubbio 16 904, quarti 4 226;
Sardmia starelli 2 988; Sicily salme gros 21014, salme generale 16 886;
Malta salme 16 930; Corsica stajo 6 014; Venice stajo 4 945. Lisbon
alquiere 817, fanega 3 268; Madeira alquiere 684; Oporto alquiere 1 051.
Alexandria rebele 9 687, kislos 10 418 ; Constantinople kislos 2 023 ; Smyrna
kislos 2 141. Algiers tarrie 1219. Tripoli caffiri 19 780; Tunis caffiri
21 855. Candia charge 9 288. Greece medimni 2 390. Persia artaba
4 013. Pobud zonEcc 3 120. Geneva coupes 4 739. Scotland firlot 2 197.
56 WEIGHTS OR MASSES.
WEIGHTS or MASSES.
The fundamental standard of mass (commonly called weight) of the
United States is the International Kilogram, a cylindrical mass of metal
made of 90% platinum and 10% iridium, and preserved at the Interna-
tional Bureau of Weights and Measures, near Paris. Copies^ of the In-
ternational Kilogram are possessed by each of the 20 countries contrib-
uting to the support of the International Bureau of Weights and Measures,
and these copies are known as National Protot3n>es. The United States
possesses two of these standards, whose values in terms of the International
Kilogram are known with the greatest accuracy. One of the kilograms
known as No. 4 is used as a working standard and the other, No. 20, is
kept under seal and only used to check No. 4. One of the objects for tho
maintenance of the International Bureau of Weights and Measures is to
provide for the recomparison at regular intervals of the various National
Protot3n;>es with the International Kilogram, thus insuring the use of the
same standard throughout the world.
According to the Act of Congress of July 28, 1866, which was the first
general legislation upon the subject of fixing the standard of weights and
measures, the pound is to be derived from the kilogram. This act defined
the relation 1 kilogram » 2.204 6 avoirdupois pounds; but this has since
been changed to the more accurate value, 1 kilogram =» 15 432.356 39 grains,
which corresponds to 2.204 622 34 avoirdupois pounds, or 1 avoirdupois
Soimd = 453.592 427 7 grains. This value is the one now used by the
rational Bureau of Standards in Washington and the avoirdupois pounds,
ounces, grains, etc., in common use now in this country are derived from
the kilogram according to this relation, and are consequently fixed and
definite units. In this country the relation between the pound and the
kilogram is therefore no longer to be determined by^ measurement, as is
often supposed, but is fixed definitely by precise definition.
The troy pound was definitely adopted by Congress in 1828 for coinaffe
purposes and was supposed to be an exact copy of the British troy poimd ;
formerly the avoirdupois pound was derived from this according to the
relation 1 avoirdupois poimd ''00%76o troy pound; this relation is £*ill
correct, but in this country both the troy and the avoirdupois pounds as
now standardized by the Government are derived from the kilogram as
stated above, and hence both values are fixed definitely. The old troy
pound, "although totally unfit for such purpose, ''is the legal standard for
coinage purposes in this country; according to the National Bureau of
Standaras, tne troy pound of the Mint, and the troy pound of that Bureau
(based on the kilogram) are the same.
The kilogram was originally intended to be the mass of a cubic deci-
meter or liter of pure^ water at the temperature of its maximiun density.
The present International Prototype Kilogram is an exact copy of the
original kilogram of the archives, which when made was supposed to be
equal to. the weight of one cubic decimeter of water. It is a certain mass
of platinum-indium. The determination of the precise relation of this
adopted kilogram to the mass of a cubic decimeter of water is now in
progress, and it may be several years before the finsd results lire announced.
The results thus far indicate that the kilogram is heavier than it would be
according to the original definition by about 25 milligrams, or about 25
parts in 1 000 000. The present kilogram, however, is definite, and the
result of such a discrepancy would make the liter, which is the volume of
a kilogram of water, very slightly larger than the cubic decimeter by about
25 parts in 1 000 000. The question of correcting the kilogram to agree
with its original theoretical definition may be considered when the deter-
mination now being made at the International Bureau has been com-
pleted. For all but the most refined measurements, however, this slight
discrepancy is absolutely negligible. The National Bureau of Standards
has for the present assumed that the liter and the cubic decimeter are
equivalent;^ this identity is assumed also in all the tables in this book.
The relation, legalized in Great Britain in 1898, between the avoirdupois
pound and the kilogram, is precisely the same as that in this country, hence
the pounds are exactly the same in both countries. All the avoirdupois,
troy and apothecary weights are therefore also the same in the United
States and in Great Britain.
WEIGHTS OR MASSES. 67
Tlie metric weights are now in use everywhere for all accurate scientific
measurements. In this country they are coming into more general use;
chemists use them entirely. The avoirdapois weig^hts (iU[>breyiation av.)
are used for most purposes, including merchandise in general. The tr*^
welghto are used for weighing gold, alver, etc. ; they are used by the U. S.
Mint; quantities, even much larger than an ounce, are usually stated in
ounces and not in pounds. In the apothecary weights (abbreviation ap.)
only the grain, scruple, and dram are in general use in this country* the
ounce is used only when called for in prescriptions; apothecaries almost
always use the avoirdupois pound and ounce. The apothecary ounce and
pound are the same as the troy. The grain is the same in all three systems.
No fixed rules can be given concerning the distinction between the use
of the short or net ton of 2 000 lbs. and the long or gross ton of 2 240
lbs., but the following general rules ma/ serve as a gniide. The long
ton seems to be the only official one; section 2951 of the Revised Statutes
of the U. S., 2d Ed., 1878, CoUectioli of Duties upon Imports, Chapter 6,
sasfs that by the word "ton " in that chapter is meant 2 240 pounds. With
fr»^t on railroads, a ton of 2 240 pounds seems to be generi^y used. In
the iron and steel trades, pig iron, steel rails, and iron ore are bought and
sold by the ton of 2 240 pounds. Coal seems to be weighed in long tons
also; coke, however, is weighed in short tons of 2 000 pounds. The short
ton of 2 000 pounds seems to be in general use for weighing chemical prod-
ucts, or in general for the more expensive products ; but in shipping them
on railroads the ton of 2 240 pounds is often used. The short ton seems
to be in general use also in traction engineering calculations.
What are popularly termed weights should more correctly be called
masses, but for all practical purposes the two terms are the same. The
mass of a body is the same anywhere in the universe, but its weight depends
on the attraction of gravitation. The mass of a body is most convemently
measured by its weight, and if the attraction of gravitation is the same
(and it varies but slightly at different parts of eeuth) the weight will be a
correct measure of the mass. With the usual beam balance, the com-
parison of two masses by means of their weights is absolutely exact and
quite independent of the value of gravity, which acts equally on both;
but with spring scales the same mass may have slightly different weights,
depending on the force of gravity.
When weights are considered as such, and not as masses, they are
forces and have the dimensions of forces. The reduction factors in
the following table are correct whether the units are considered as masses
or as forces. The reduction factors between them and the true units of
force (such as dyne and poundal) are given in a separate table of forces so
as to avoid confusion, bee also note under Forces and under Acceleration.
WEIQHTS or MASSES. (See also FORCES.) UsuaL
** Accepted by the National Bureau of Standards.
* Checked by L. A. Fischer, Asst. Phys. National Bureau of Standards.
av. means avoirdupois. Aprx. means within 2%.
Logarithm
1 milligram [mg]«i 0.1 centigram I-OOO 0000
= 0.015 432 4* grain. Aprx.%s-*-10 3.188 4322
" = 0.001 gram 3000 0000
1 centigram [cg]=» 10. milligrams 1-000 0000
^' =0.154 324* grain. Aprx.%s 1188 4822
•• = 0.01 gram 2-000 0000
1 grain [grj^'same in avoirdupois, troy, or apothecary weights.
=• 64.798 9* milligrams. Aprx. 65 1811 6678
= 6.479 89* centigrams. Aprx. QH 0811 5678
= 0.064 798 9* gram. Aprx. i%-i- 100 2-811 5678
" =0.002 285 71* ounce (av.). Aprx. %-»- 1 000 5-359 0219
1 decigram [dg]» 1.543 24* grains. Aprx. ^^ 0-188 4322
= 0.1 gram I-OOO 0000
*•
58 WEIGHTS OR MASSES.
1 gram [g]= 1 000. milligrams 3.000 0000
= 100. centigrams 2-000 0000
= 15.432 356 39** grains. Aprx. 15M 1.188 4822
= 0.035 274 0* ounce (av.). Aprx. % -h 100 2547 4541
= 0.032 150 7* ounce (troy). Aprx. 32-5-1 000 2.507 1910
= 0.002 204 62* pound (av.). Aprx. 22-4-10 000. . . 3.843 8842
= 0.001 kilogram §.000 OOOO
1 ounce [oz] (av.) = 2 834.95* centigrams. Aprx. % X 10 000 . 8.452 5459
= 437.500* grains. Aprx. i% X 100 2640 9781
= 28.349 5* grams. Aprx. ^ X 100 1.452 5459
= 16. drams (av.) 1.204 1200
= 0.911 458* ounce (troy). Aprx. i%i f .959 7869
= 0.062 500* pound (av.) or Ho 2.795 8800
= 0.028 349 5* kilogram. Aprx. ^-^ 10 2.452 5469
1 pound [lb] (av.) = 7 000.** grains 3-845 0980
=453.592 427 7 ** grams. Aprx. % X 100 2656 6658
= 256. drams (av.) 2408 2400
= 16. oimces (av.) 1-204 1200
= 14.583 3* oz. (troy). Aprx. 3^X100. 1.163 8569
= 0.453 592 4* kilogram. Aprx. %-*- 10 .. 1.656 6668
1 kilogram or kilo [kg]:
= 16 43^.366 39** grains. Aprx. ^J^X 1 000 4-188 4322
= 1 000. grams 3.000 0000
= 35.274 0* ounces (av.). Aprx. % X 10 1-547 4541
= 32.150 7* ounces (troy). Aprx. 32 1.507 1910
= 2.204 62* poimds (av.). Aprx. 2^^ 0343 3342
= 0.022 046 2* hundredweight (sh.). Aprx. 22-*-l 000. 2-343 3342
= 0.019 684 1* hundredweight (long). Aprx. 2 -J- 100. . 2-294 1162
= 0.001 102 31* short or net ton. Aprx. % -i- 100 §.042 3042
= 0.001 metric ton 3-000 OOOO
= 0.000 984 206 long or gross ton. Aprx. 1 -f- 1 000 i.993 0862
1 hundredweight [cwt] (short):
= 100. pounds (av.) 2-000 OOOO
= 45.359 24* kilograms. Aprx. % X 10 1.656 6658
= 0.892 857* hundredweight (long). Aprx. %o 1-950 7820
= 0.05 short or net ton 2-698 9700
=0.045 359 24* metric ton. Aprx. H2 2.656 6658
= 0.044 642 9* long or gross ton. Aprx. % -5- 100 2-649 7520
1 hundredweight [cwt] (long):
= 112. pounds (av.). Aprx. MtXl 000 2-049 2180
= 50.802 4* kilograms. Aprx. 50 1.705 8838
= 1.120 00* hundredweights (short). Aprx. 1% 0-049 2180
= 0.056 000* short or net ton. Aprx. % -5- 10 2748 1880
= 0.050 802 4* metric ton. Aprx. Ho 2-705 8838
= 0.05 long or gross ton or Ho 2-698 9700
1 short or net ton [tn]:
= 2 000. pounds (av.) 3-301 0800
= 907.185* kilograms. Aprx. 900 2957 6958
= 20. hundredweights (short) 1-301 0300
= 17.857 1* hundredweights (long). Aprx. % X 10 1.251 8120
=0.907 185* metric ton. Aprx. subtract Vio 1-957 6958
= 0.892 857* long or gross ton. Aprx. subtract ^0 1-950 7820
1 metric ton, tonne, tonneau, millier, or bar [t]:
= 2 204.62* pounds (av.). Aprx. 22 X 100 3-343 3342
= 1 000. kilograms 3000 OOOO
= 22.046 2* hundredweights (short). Aprx. 22 1-343 3342
= 19.684 1* hundredweights (long). Aprx. 20 1-294 1182
= 1.102 31* short or net tons. Aprx. add^o 0042 3042
= 0.984 206* long or gross ton. Aprx. 1 1-993 0862
1 long or gross ton [tn]:
= 2 240 pounds (av.). Aprx. 22 X 100 3.35O 2480
= 1 016.05* kilograms. Aprx. 1 000 3006 9138
= 22.4(X) 0* hundredweights (short). Aprx. 22 1-350 2480
= 20. hundredweights (long) , 1-301 0300
= 1.12 short or net tons. Aprx. 1% 0-049 2180
= 1.016 05* metric tons. Aprx. 1 0-006 9138
WEIGHTS OS MASSES.
59
Conversion Tables for Weights.
Note. — By pounds and ounces are meant avoirdupois pounds and ounces.
Grains =
MU'grs»
Ounces —
migrs
grains
gram
ounces
klgrms
pound
met.
met.
tons
Grams =
Pounds =
KIgms ae
Sh. ton SB
Lff.ton«=
lit.ton »
tons
shrt
tons
long
tons
1
2
3
4
5
6
7
8
10
64.799
129.60
194.40
259.20
323.99
388.79
453.59
518.39
583 19
647.99
0.015 432
0.030 865
0.046297
0.061 730
0.077 162
0.092 594
0.108 03
0.123 46
0.138 89
0.154 32
28.350
56.699
35.049
113.40
141.75
170.10
198.45
226.80
255.15
283.501
0.035 274
0.070.548
0.105 82
0.141 10
0.176 37
0.211 64
0.24692
0.282 19
0.317 47
0.352 74
0.45359
0.907 18
1.360 8
1.814 4
2.268
2.721 6
3.175 1
3.628 7
4.082 3
4.535 9
2.204 6
4.409 2
6.613 9
8.818 5
11.023
13.228
15.432
17.637
19.842
22.046
0.907 1.102 1.016
1.814 2.205 2.032
2.722 3.307 3.048
3.629 4.409 4.064
4.536 5.512 5.080
5.443 6.614 6.096
6.350 7.716 7.112
7.257 8.818 8.128
8.165 9.921 9.144
9.07211.02 10.16
0.984
1.968
2.953
3.937
4.921
5.905
6.889
7.874
8.857
9.842
WEIGHTS or MASSES (continned). Unusual, Special
Trade, or Obsolete.
ay. means avoirdupois weight; ap. means apothecary weight; aprx.
means approximately.
0.001 milligram [^J (has no name, symbol used instead )=» 0.000 015 433 4
grains.
1 Jeweller's grralii=)^ carat (diamond) of various weights.
1 carat (diamond) = 4. jeweller's grains — (according to Streeter) in
U. S. 205.500 milligrams or 3.171 4 grains, and in England 205.409 milli-
grams or 3.170 grains: other authorities give 3.168, 3.18, and 3.2 grains
in U. S., and 3.17 in England. For values in other countries see below
under Foreign Weights.
1 scruple [9] (ap.) = 20. grains » 1 .295 978 (aprx. %) grams^H dram
(ap.) = V^4 ounce (troy or ap.). 1 gram = 0.771 618 (aprx. %) scruple.
1 pennyweight [dwt] (troy)=»24. grains^ 1.555 17 (aprx. ^^) grams»
yio ounce (troy or ap.). 1 gram— 0.643 015 (aprx. %i) pennyweight.
1 drachm, same as dram.
Idram (av.) = 271^2 or 27.343 75 (aprx. 27^) grains = 1.771 85 (aprx.
%) grams =» 0.455 729 (aprx. %i) drams (ap.)=V46 ounce (av.). 1 gram =
0.564 383 ^aprx. ^) dram (av.).
1 dram [ 3 ] (ap.) = 60. grains =•3.887 934 grams = 3. scruDles=» 2.194 29
(aprx. 1%) drams (av.) = M ounce (ap.). 1 gram =0.257 206 arams (ap.).
1 decagram or dekagram [dkgj= 154.323 6 grains = 10. grams =0.352 74
ounce (av.).
1 ounce (troy, silk) = 360. grains = 23.327 6 grams.
1 ounce [oz] (troy) (used chiefly for gold and silver) = 480. grains »»
31.103 5 grams = 20. pennyweights = 1.097 14 (aprx. ^V4o) ounces (av.) = l
ounce (ap.) = M3 or 0.083 333 3 pound (troy or ap.) = 0.068 571 4 pound
(av.). 1 gram = 0.032 150 7 ounce (troy). 1 ounce (av.) = 0.911 458 (aprx.
i%i) ounce (troy). 1 poimd (av.) = 14.583 3 (aprx. *<>%) ounces (troy).
1 kilogram = 32.150 7 ounces (troy).
1 ounce [ 5 ] (ap.) (used only in prescriptions) = 480. grains = 31.103 5
grams = 24. scruples = 8. drams (ap.) = 1.097 14 (aprx. ^Vio) ounces (av.) =
1 ounce (troy)=M2 or 0.083 333 3 pound (ap. or troy) =0.068 571 4 pound
(av.). 1 gram=0.032 150 7 ounce (ap.). 1 ounce (av.) = 0.911 458 (aprx.
i%i) ounce (ap.). 1 pound (av.) = 14.583 3 ounces (ap.). 1 kilogram =
32.150 7 ounces (ap.).
60 WEIGHTS OR MASSES.
1 hecto£:rani [hg] = 100. grains » 3.527 40 ounces (av.).
1 pound (troy) (seldom used ; troy ounces used instead) = 6 760. grains =»
240. pennyweights = 12. ounces (troy or ap.) = l pound iB.p.)=^'^^%ooo or
0.822 857 (aprx. %) pound (av.) =0.373 242 (aprx. %) kilogram. 1 pound
(av.)=70o%7eo or 1.215 28 (aprx.%) poimds (troy). 1 kilogram = 2.679 23
pounds (troy).
1 mint pound (U. S.), same as troy pound.
1 pound (troy, silk) = 16. ounces (troy, silk) = 6 760. grains = 1 pound
(troy).
1 pound (ap.) (obsolete) = 5 760. grains = 288. scruples = 96. drams
(ap.) = 12. ounces (ap. or troy) = 1 pound (troy) ^^Te^^^^^ or 0.822 857 (aprx.
%) pound (av.) = 0.373 242 (aprx. %) kilogram. 1 pound (av.)= "^^^rw
or 1.215 28 (aprx. %) pounds (ap.). 1 kilogram = 2.679 23 pounds (ap.).
1 stone (Brit.) = 14. pounds (av.) =6.350 29 kilograms.
1 myriagram = 10 000. grams = 22.046 2 pounds (av.) = 10. kilograms.
1 quarter [qrl (short) = 25. pounds (av.) = 11.339 8 kilograms = ^
hundredweight (snort).
1 quarter [qrj (long) =28. pounds (av.) = 12.700 6 kilograms =>^ hun-
dredweight (long).
1 firkin (butter) = 56. pounds (av.) (really a capacity measiu^).
1 bushel (salt) = 70. pounds (av.) (really a capacity measure).
1 quintal (av.) = 100. pounds (av.) = 45.359 24 kilograms»l hundred-
weight (short).
1 barrel of flour (' 'legal "?) = 196. pounds (av.).
1 barrel of beef or pork = 2(X). pounds (av.).
1 quintal (metric) = 100. kilograms = 220.462 pounds (av.).
1 pig (metal) = 301. pounds (av.) = 21>^ stones.
1 lother (iron, lead, etc.) = 2 408. pounds (av.) = 172. stones — 8. piga.
1 bloom ton = l^o long tons = 2 464. lbs.
Relative Weights (used in chemistry).
1 millimol = 0.001 mol or gram molecule.
1 mol or mole = 1 gram molecule, which see below.
1 gram molecule = as many grams of a substance as is represented
numerically by its molecular weight. A gram molecule of any gas at 0® C.
and 760 mm pressure occupies a volume of 22 380. cubic centimeters.
1 kilogram molecule = 1 000. gram molecules.
1 gramatom=as many grams of an elemental substance as is repre-
sented numerically by its atomic weight.
WmOHTS or MASSES (concluded). Foreign.
These are mostly obsolete, as the metric system is now used in most
foreign countries. The British measures are included among the U. S.
measures, being very nearly, and sometimes quite, the same. The trans-
lated terms are merely synonymous, and not the exact equivalents.
Germany. Pnissia. To 1839 inclusive: 1 Centner (hundredweight) =
110 Pfund (pound) of 32 Loth of 4 Quentchen. 1 Pfund =0.467 711 kilo-
gram. From 1840 for customs and from 1858 for trade: 1 Centner or
Z.C. = 100 Pfund (pound) of 30 Loth of 10 Quentchen of 10 Cent or zent of
10 Kern or Korn (grain); 1 Pfund = 0.5 kilogram. Apothecaries weight:
1 Pfund = 12 Unzen (ounces) of 8 Drachmen of 3 Skrupel of 20 Gran (grain)
(signs the same as in U. S. apoth. measure; values slightly smaller);
1 Pfund = 0.350 783 kilogram (see also under Baden); 1 Schiffslast (shipping
weight) = 40 Centner = 2 000 kilograms. 1 carat (diamond) in Berlin
205.440, in Frankfort o. M. 205.770, and in Leipsic 205.000 milligrams
(Streeter).
Bavaria. 1 Pfund (pound) = 32 Loth of 4 Quentchen; 1 Pfund =
0.560 kilogram.
Saxony. 1 Pfund (pound) = 4 Pfenniggewicht (pennyweight) of 2
Hellergewicht ; 1 Pfund = 0.467 6 kilogram.
Wiirtemberg. To 1850: 1 Pfund (pound) = 32 Loth of 4 Quentchen of
4 Richtpfennig: 1 Pfund = 0.467 7 kilogram. Since 1850 like in Baden.
Baden. 1 Pfund (pound) = 2 Mark of 2 Vierlingen of 4 Unzen (oimces);
or 1 Pfund = 10 Zehnlingen of 10 Centas of 10 Dekas of 10 As; or 1 Pfund-
WEIGHTS OR MASSES. 61
«ram,^ (A4iolher authority
I Pfund-O.MOS kiloiTTaiii.
-lOOkoFDof lOOar
0.425 1 o
und) of
ling pouad)-20 lle9pund-40O skalpund. Nyatr
golJ graim, 1_ miirco =0.S07 8 pounds av. in Spaii
SwediBh poun3-0.a375 pound av,; 1 miner'a pound-0.S28 6
RnaalBl 1 ptund (pound)-32 loth ol 3 aolotnick of 06 doli; 1 pfund =
0. 409 512 or 0.409 531 kilogram. Shipping weights; 1 tierkawitl - 10 puil
or pood of 40 pfund (pounda): 1 berkowiti- 163.S1 kiloKrami, Nynrom
gives 1 Russian pouDd-0.902 pounds av.; 1 Warsaw paund-O.SQl pound
Bwtticrland. Like Baden eieept that 1 Ptund-32 Loth of Ifl Uugon.
Nystrom gives 1 Geneva pound, heavy,- 1.214 pounds av.
carat (diamond) in Amsterdam = 205.700 milligram-.
Spain. I marcr or uiark-SO castellanos of 8
gold grains. 1 marco =0."
pounds av. in South Ameri
71.07 to 71.04 irMns (V. t., ,.__, __ ,
weigbt)ot 4 arroba (quarters) of 25 libra (pounda)i 1 tonelaiia of Castile -
2032.2 pounds avoirdupois; 1 libra - 0.46 1 kilogram-l.Oie 1 pounds
av.; 1 arroba of Castile or Msdrid = 25.402 5 pDunds av.; it has various
values in different parts of Spain. Nystrom eives 1 Barcelona pound -
0.888 1 pound avoirdupoie, Streeter gives 1 carat (diamond) -205.303
milligram.
Ital]'. Nystrom gives the following values in pounds avoirdupois:
Bologna pound 0.798; Corsica pound 0.750; Florence pound 0.749: Genoa
poimd 1.077; Leghorn pound 0.740; Naples Rottoli 1.964: Rome pound
0.748: Sicily pound 0.700: Venice pound, heavy, 1.056, light, 0.667. Slreeter
gives I carat (diamond) in Florence-lOS.200. in Leghorn 215.900 milli-
^jSpan. 1 kwan-1000 momme of 10 fun. 1 kwan-6H <aprit.) kin
of 160 momme. 1 liwan-3.76 kilograms -8.29 pounds (av.). 1 kilo-
gram-0,266 kwan, 1 pound-0.121 Ttwan,
Mlacellsneana. Nystrom gives the following values which have here
been reduced to pounds avoirdupois; Antwerp pound 1.034; Copenhagen
pound l.lOl; Madara pound 0.608; Tangiers pound 1.061; Cairo rottoli
0,952; Alexandria rottoli 0.035: Algiers rottoli 1.190; Damascus rotloli
3.96; Tunis rottoli I.IIO; Tripoli rottoli 1.120; Cyprus rottoli 5.24; Can-
dia rottoli 1.164: Aleppo rottoli 4.89; Aleppo oke 2.79; Const aniinople
0ke2.Sl; Smyrna oke 2.74 : Mocha maund 3.00; Morea pound l.IOl : R>>n.
gal seer 1.867; Batavia catty 1.302; China cattf 1.326^ Js^aa^ci
in Borneo 105.01
62 WEIGHTS AND LENGTHS.
WmOHTS or MASSES and LENGTHS; WEIGHTS of
WIRES, RAILS, BARS; FORCES and LENGTHS;
FILM or SURFACE TENSION; CAPILLARITT.
(Mass -s- length ; force -4- length.)
In this group of units the masses, forces, and the weights considered as
both masses and forces have all been combined to avoid repetition of their
relations to each other. When the units involve masses or weights con-
sidered as masses, as in pounds per foot, they are used to measure such
quantities as the weights of rails, bars, wires, etc. ; while when the units
Involve forces or weights considered as forces, they are used to measure
such guantities as surface tension, capillapty, etc. The dimensions of the
units in the two cases are different. Weights considered as forces involve
the value of gravity, but masses and forces do not.
Aprx. means within 2%.
Logarithm
1 dyne per centimeter [dyne/cm]:
= 0.039 973 9 grain per inch. Aprx. 4 -i- 100 2.601 7764
= 0.001 019 79 gram per centimeter. Aprx. 1-5-1 000 5.008 5096
=0.000 183 719 poundal per inch. Aprx. i%-i- 10 000 1.264 1530
1 pound (av.) per mile [lb/ml]:
= 0.281 849 kilogram per kilometer. Aprx. ^ 1450 0161
= 0.110 480 grain per mch. Aprx. % f .043 2829
= 0.002 818 49 gram per centimeter. Aorx. ^ -i- 100 3-450 0161
=0.000 568 182 pound per yard. Aprx. ^ -;- 1 000 4754 4873
=0.000 281 849 kilogram per meter. Aprx. ^ -s- 1 000 4450 0161
=0.000 189 394 pound per foot. Aprx. 19-5-100 000 |.277 3661
1 kilogfram per kilometer [kg/km] or gram per meter
[g/m] or milligrram per millimeter [mg/mm]:
= 9.805 97 dynes per centimeter. A^rx. 10 0-991 4904
= 3.548 00 pounds per mile. Aprx. % 0-549 9839
= 0.391 983 grain per inch. Aprx. Vio 1-598 2668
= 0.01 gram per centimeter 2-000 0000
= 0.002 015 91 pound per yard. Aprx. 2-5-1 000 g.804 4713
= 0.001 801 54 poundal per inch. Aprx. %-i-l 000 §.255 6434
= 0.001 kilogram per meter 3-000 0000
=0.000 671 970 pound per foot. Aprx. M ■*- 1 000 4-827 3500
1 grain per incli [gr/in]:
= 25.016 3 dynes per centimeter. Aprx. KX 100 1.398 2236
= 9.051 4 pounds per mile. Aprx. 9 0-956 7171
= 2.551 14 kilograms per kilometer. Aprx. ^XIO.. . . 0-406 7332
= 0.025 511 4 gram per centimeter. Aprx. M-?- 10 2-406 7332
= 0.005 142 86 pound per yard. Aprx. 51 -^ 10 000 |.7ll 2045
= 0.004 595 96 poundal per inch. Aprx. ^s -5- 100 1-662 3766
= 0.002 551 14 kilogram per meter. Aprx. K-^ 100 5-406 7332
= 0.001 714 3 pound per foot. Aprx. 17-5-10 000 5-234 0832
= 0.000 142 857 pound per inch. Aprx. Vi-i-1 000 4-154 9020
1 g:rain per centimeter [g/cm];
= 980.597 dynes per centimeter. Aprx. 1 000 2-991 4904
= 354.800 pounds per mile. Aprx. % X 100 2-549 9839
= 100. kg per km or g per m or mg per mm 2-000 0000
= 39.198 3 grams per inch. Aprx. 39 1-593 2668
= 0.201 591 pound per yard. Aprx. % - 1.304 4713
= 0.180 154 poundal per inch. Aprx. %-^ 10 1-255 6434
= 0.1 kilogram per meter 1-000 0000
= 0.067 197 pound per foot. Aprx. H-i-10 2.827 3500
=0.005 599 75 pound per inch. Aprx. % -4- 100 §.748 1688
1 pound per yard [lb/yd]:
= 1 760. pounds per mile. Aprx. % X 1 000 3.245 5127
= 496.054 kilograms per kilometer. Aprx. MXl 000.. . 2-695 5287
= 194.444 grains per inch. Aprx. ^^i X 10 000 2288 7965
= 4.960 54 grams per centimeter. Aprx. 5 0-695 5287
= 0.496 054 kilogram per meter. Aprx. J^ 1-695 5287
= 0.333 333 pound per foot or ^ T.522 8787
=0.027 777 8 pound per inch. Aprx. ^M"*- 100 2-443 6975
PRESSURES. 63
1 poondal per inch =5 443.11 dsrnes per cm. Aprx. 'H X 1 000 S.7S5 8470
=:: 217.582 grains per inch. Aprx. ^V^ X 100 2-887 6284
" >» 5.550 81 grams per centimeter. Aprx. i>^ 0-744 8566
1 kllogrram per meter [kg/m]:
« 3 548.00 poimds per mile. Aprx. %X1 000 8-649 9889
» 1 000. kilograms per kilometer 3-000 0000
— 391.983 grains per inch. Aprx. 400 2-598 2668
« 10, grams per centimeter 1-000 0000
« 2.015 91 pounds per yard. Aprx. 2 Q-804 4718
— 0.671 970 pomid per foot. Aprx. H 1-827 8500
' -0.055 997 5 pound per inch. Aprx. i>^-s- 100 2-748 1688
1 pound per foot [lb/ft1:
— 5 280. pounds per mile. Aprx. %© X 100 000 8-722 6840
— 1 488.16 kilograms per kilometer. Aprx. % X 1 000 ... 8-172 6500
— 583.333 grains per mch. Aprx. Ht X 10 000 2-765 9168
->■ 14.881 6 grams per centimeter. Aprx. % X 10 1-172 6500
« 3. pounds per yard 0-477 1218
» 1.488 16 kilograms per meter. Aprx. % Q-172 6500
=^0.083 333 3 pound per inch. Aprx. %-i- 10 2-920 8188
1 pound per inch [lb/in] =« 7 000. grains per inch 8-845 0980
= 178.579 gram/cm. Aprx. % X 100. 2251 8812
= 36. poimds per yard 1-556 3025
•= 17.857 9 kg per metr. Aprx. % X 10 1.251 8312
"» 12. poimds per foot 1-079 J812
Ton per mile. A term popularly though incorrectly used for "ton-muej"
which see under units of Ener^. It is never used in
the sense that a mile of something weighs a ton, except
possibly in referring to submarine cables, rails, etc.
«t
41
PRESSURSS; PRESSTTRBS of WATER, MEROURT,
and ATMOSPHERE ; STRESS or FORCE per UNIT
AREA. WEIGHTS or FORCES and SURFACES;
WEIGHTS of SHEETS, DEPOSITS, COATINGS,
etc. (Force -f- surface; mass -^sarface.)
The pressures of water columns have all (including those involving only
non-metric units) been calculated on the uniform basis that a cubic deci-
meter of water weighs one kilogram (see notes under Volumes and Weights).
The pressures of mercury columns have all been calculated on the oasis
that the specific gravity of mercury is 13.595 93, which is the value accepted
by the International Bureau of Weights and Measures, and by the (U. S.)
National Bureau of Standards. This is the value used in the legal defini-
tion of the liter in reference to the atmospheric pressure.
The pressure of the atmosphere here used as a standard is that equal to
760 miUimeters of mercury of the specific gravity given above.
To convert barometric pressures from millimeters to inches or the reverse,
use the reduction factors for one millimeter of mercury in inches of mer-
cury or the reverse.
In this group of units, the forces, masses^ and the weights, considered as
both forces and masses, have all been combined to avoid repetition of their
relations to each other. When the units involve forces, or weights con-
sidered as forces, they are used to measure pressures or stresses per unit
area; while when the units involve masses, or weights considered as
masses, they are used to measure the weights of sheets, as those of metals.
For instance, pounds ^r square foot may represent a pressure or the weight
of a sheet of metal; in the former case the pound is a force and in the
latter a mass. The dimensions of the units in the two cases are dififerent.
Weights considered as forces involve the value of gravity, but masses and
true units of force do not.
Aprx. means within 2%. Hg means mercury.
64 PRESSURES.
Logarithm
1 dyne per square centimeter [dsme/cm*]:
= 1 . bariet 0000 OOOO
= 0.067 197 poundal per square foot. Aprx.%-J-10 . . . 3-827 8501
= 0.010 197 9 kilogram per square meter. Aprx. Moo ... 2.008 5096
= 0.01 megadyne per square meter 2.000 OOOO
= 0.002 088 70 pound per square foot. Aprx. 21 -!- 10 000. §.319 8754
=0.000 750 068 millimeter of mercury. Aprx. % -s- 1 000. . . 1.875 1006
=0.000 466 646 poimdal per square inch. Aprx. ^% ■*- 10 000 |.668 9876
1 barie f (Fr. barye) — 1. dsme per square centimeter.
1 gn^am per square decimeter [g/dm^:
= 0.1 kilogram per sq. m, which see for other values. T.QOO OOOO
=0.020 481 7 pound per square foot. Aprx. 205-^ 10 000. . . 2.311 8659
1 poundal per square foot:
= 14.881 6 dynes per square centimeter. Aprx. %X 10.. 1.172 6499
= 0.151 761 kilogram per square meter. Aprx. % t- 10. . . . 1.181 1595
=0.031 083 2 pound per square foot. Aprx. J^2 2.492 5258
=0.011 162 2 millimeter of mercury. Aprx. H-s- 10 2047 7504
1 kilogram per square meter [kg/m^:
= 98.059 66 dynes per square centimeter. Aprx. 100. 1.9914904
«= 10. grams per square decimeter 1.000 OOOO
= 6.589 32 poundals per square foot. Aprx. ^ X 10. 0.818 8405
= 0.204 817 pound per sq. foot. Aprx. 205-4-1 000.. . f .311 3659
«" 0.1 gram per square centimeter 1.000 OOOO
= 0.073 551 4 millimeter of mercury. Aprx. ^-^10 . . . 2.866 5910
= 0.045 759 2 poundal per square inch. Aprx. % -J- 100. 2.660 4780
= 0.003 280 83 foot of water. Aprx. >^ -*- 100 5.515 9842
= 0.002 895 72 inch of mercury. Aprx. %-^ 100 |.461 7564
« 0.001 422 34 poimd per square inch. Aprx. H -s- 100 . . 3.153 0034
= 0.001 meter of water 3.000 OOOO
= 0.000 102 408 ton (short )/sq. ft. Aprx. 102 -5- 1 000 000 j.OlO 8359
= 0.000 1 kilogram per square centimeter 4.000 OOOO
=0.000 096 778 2 atmosphere. Aprx. a%o -J- 100 000 §.985 7774
=0.000 091 436 1 ton (long) per sq. ft. Aprx. Mi -*-l 000. . 5.96I 1179
1 megadsme per square meter:
= 100. djoies per sq. cm, which see for other values 2*000 OOOO
1 pound per square foot [Ib/ft^]:
= 478.767 dynes per square centimeter. Aprx. 480. . . 2.680 1246
= 48.824 1 grams per square decimeter. Aprx. 49. . . . 1.688 6341
= 32.171 7 poundals per square foot. Aprx. ^1 X 1 000. 1-507 4746
= 4.882 41 kilop-ams per square meter. Aprx.- *% .... 0.688 6341
= 0.359 108 millimeter of mercury. Aprx. Mi 1555 2251
= 0.016 018 4 foot of water. Aprx. %-J- 100 2.204 6188
— 0.014 138 1 inch of mercury. Aprx. %-5- 100 2.150 3905
= 0.006 944 44 pound per square inch. Aprx. 7-^1 000. . . 3.841 6375
— 0.004 882 41 meter of water. Aprx. 49-5-10 000 3.688 6341
= 0.000 5 ton (short) per square fopt or J^-^ 1 000 1.698 9700
=0.000 488 241 kilogram per sq. cm. Aprx. 49-4-100 000. . . 4.688 6841
=0.000 472 511 atmosphere. Aprx. 47-5-100 000 |.674 4115
=0.000 446 429 ton (long) per sq. foot. Aprx. % -5- 1 000 . . . 4649 7520
1 gram per square centimeter [g/cm^J:
= 10. kilograms per sq. m, which see for other values 1.000 OOOO
t Recommended by a Committee of the International Physical Congress
of 1900, in Paris, for the absolute unit of pressure, that is, for one dyne per
sq. centimeter. It seems it was not officially adopted by that Congress.
The recommendation includes that the megabarie (or megabarye) is repre-
sented with sufficient accuracy for practical purposes by the pressure of
75 cm of mercury at 0° C. ; this latter is nearly what is usually accepted as
the pressure of one atmosphere, namely 76 cm of mercury. It was origi-
nally proposed to the Congress to adopt this name barie for the atmos-
pheric pressure, making it equal to a megadyne per sg. centimeter, but
this was changed by the Committee of that Congress; it is, however, some-
times used in this sense.
PRESSURES. 66
1 millimeter of mercury column [mm Hg]:
= 1 333.21 dynes per sq. cm. Aprx. ^^ X 1 000 8124 8994
— 89.587 9 poimdals per square foot. Aprx. 90 1-952 2495
= 13.595 93t kilograms per sq. meter. Aprx. % X 10. . . . 1.183 4090
= 2.784 68 pounds per square foot. Aprx. ^H 0.444 7749
— 0.622 138 poundal per square inch. Aprx. fs 1-798 8870
= 0.044 606 foot of water. Aprx. %-*-10 2.649 8982
= 0.039 370 inch of mercury. Aprx. 4-^-100 2595 1854
«= 0.019 338 pound per square inch. Aprx. 19^1 000. . 2-288 4124
=0.013 695 93 meter of water. Aprx. % -^ 100 2-138 4090
=0.001 392 34 ton (short) per square foot- Aprx. % ^ 1 000. 3-143 7449
=0.001 359 59 kilogram per sq. cm. Aprx. Va^l 000 3-133 4090
—0.001 315 79 atmosphere. Aprx. ^i -^ 1 000 3-119 1864
=0.001243 16 ton (long) per sq. foot. Aprx. 3^-5-100 §-094 5269
1 poundal per square inch:
= 2 142.95 dynes per sq. cm. Aprx. %4 X 10 000 8331 0124
= 21.853 6 kilograms per square meter. Aprx. i% X 10. . 1.339 5220
= 1.607 36 millimeters of mercury. Aprx. % 0-208 1130
=0.031 083 2 pound per square inch. Aprx. ^2 2-492 5253
1 foot of water column = 304.801 kg per sq. m. Aprx. 300- . 2-484 0158
= 62.428 3 Ibs/sq. ft. Aprx. 5^X100 1.795 8817
** = 22.418 5mm Hg. Aprx. %X 10... 1.350 6068
" -= 0.882 617 inch Hg. Ap. subtr. %. 1.945 7722
" = 0.433 530 lb per sq. inch. Aprx.4^. 1-637 0192
*• = 0-304 801 meter of water.. Aprx. %o 1-484 0158
" =0.031 214 2 sh. ton/ft2. Aprx. ^2- - . 2-494 3517
" =0.030 480 1 kg/cm2. Aprx. 3 -J- 100. . 2-484 0158
" =0.029 498 atm. Aprx. 3-1-100 2.469 7982
= 0.027 869 8 1. ton/ft^. Ap. 1^-5-100.. 2-445 1387
1 incli of mercurjr column [in Hg]:
= 345.337 kilograms per sq. meter. Aprx. % X 100 2-538 2488
= 70.731 pounds per square foot. Aprx. 70 1-849 6095
= 25.400 05 millimeters of mercury. Aprx. MX 100 1-404 8846
= 1.132 99 feet of water. Aprx. i% 0-054 2278
= 0.491 187 pound per square inch. Aprx. }4 1-691 2470
= 0.345 337 meter of water. Aprx. %o 1-538 2438
=0.035 365 5 ton (short) per sq. foot. Aprx. 7^^100 2-548 5795
=0.034 533 7 kilogram per sq. centimeter. Aprx. % -*- 100. . 2-538 2486
=0.033 421 1 atmosphere. Aprx. Ho 2524 0210
=0.031 576 3 ton (long) per sq. foot. Aprx. H2 2-499 8615
1 pound per square inch [Ib/in^]:
= 703.067 kilograms per sq. meter. Aprx. 700 2-846 9966
= 144. pounds per square foot. Aprx. ^ X 1 000. . . . 2-158 3825
= 51.711 6 millimeters of mercury. Aprx. 8%X 10 1.713 5876
= 32.171 7 poundals per sq. inch. Aprx. HiXl 000 1.507 4746
= 2.306 65 feet of water. Aprx. % 0362 9808
= 2.035 88 inches of mercury. Aprx. 2 0808 7580
= 0.703 067 meter of water. Aprx. %o 1-846 9968
= 0.072 ton (short) per square foot. Aprx. ^4 2-857 3325
=0.070 306 7 kilogram per sq. centimeter. Aprx. 7-*- 100. . 2-846 9966
=0-068 041 5 atmosphere. Aprx. Ho 2-832 7740
=0.064 285 7 ton (long) per sq- foot. Aprx. %i n- 10 2-808 1145
1 meter of water column or
1 metric ton per square meter [t/m^:
= 1 000. kilograms per square meter 3-000 0000
= 204.817 pounds per square foot. Aprx. 205 2-3118659
= 73.551 4 millimeters of mercury. Aprx. MX 100 1-866 5910
= 3.280 83 feet of water. Aprx. MX 10 0515 9842
= 2.895 72 inches of mercury. Aprx. a<)^ 0461 7564
= 1.422 34 pounds per sq. inch. Aprx. ^X 10 Q153 0034
= 0.102 408 ton (short) per sq. foot. Aprx. ^M"*- 100 J.OIO 8359
= 0.1 kilogram per square centimeter l-OOO 0000
=0.096 778 2 atmosphere. Aprx. 97 -^ 1 000 2-985 7774
=0-091 436 1 ton (long) per square foot. Aprx. Hi 2.981 1179
t This is the specific gravity of mercury used throughout in these tables.
r
66 PRESSURES.
1 ton (short) per square foot [tn/ft^]:
= 9 764.82 kilograms per square meter. Aprx. 9 800. . 3-989 6641
= 2 000. pounds per square foot 3.301 0800
= 718.216 millimeters of mercury. Aprx. ^ X 1 000 . . 2.856 2551
= 32.036 7 feet of water. Aprx. 32 1.505 6483
= 28.276 2 inches of mercury. Aprx. % X 100 1.451 4206
= 13.888 9 pounds per square inch. Aprx. % X 10 1-142 6676
= 9.764 82 meters of water. Aprx. 98-5-10 0-989 6641
== 0.976 482 kilogram per sq. cm. Aprx. subtract Ko • • • 1989 6641
= 0.945 021 atmosphere. Aprx. subtract Ho 1-976 4415
= 0.892 857 ton (long) per sq. ft. Aprx. subtract J^^ 1-950 7820
= 0.006 944 44 ton (Short) per sq. inch. Aprx. 7-^ 1 000.. . . §.841 6375
= 0.006 200 40 ton (long) per sq. inch. Aprx. ^-i-100 3.792 4195
1 kilog^ram per square centimeter [kg/cm^:
= 10 000. kilograms per square meter 4-000 0000
= 2 048.17 pounds per square foot. Aprx. 2 050 3.311 3669
= 735.614 millimeters or mercury. Aprx. HXl 000 2-866 5910
= 32.808 3 feet of water. Aprx. MX 100 I.515 9842
= 28.957 2 inches of mercury. Aprx. % X 100 1-461 7564
— 14.223 4 pounds per sq. inch. Aprx. V^X 100 1-153 0034
= 10. meters of water 1-000 0000
= 1.024 08 tons (short) per sq. foot. Aprx. add )io 0010 3359
=0.967 782 atmosphere. Aprx. subtract Ho • ■ ■ ■ 1-985 7774
=0.914 361 ton (long) per square foot. Aprx. i%i 1961 1179
= 0.001 metric ton per square centimeter \ . . §.000 0000
1 barie t = 75 centimeters of Hg (aprx.). Accurately 75.0068.
*' =1 megadyne per square centimeter.
1 uie^abariet =1 megadyne per square centimeter 0-000 0000
1 nieg^adynepersq. cm. <=750.068mm.ofHg. Aprx. ?4^X 1000 2-875 1006
-=0.986931 atmosphere (stand.) Ap-1 1-994 2870
1 atmosphere [atm] (standard):
== 10 332.9 kilograms per square meter. Aprx. 10 300 4-014 2226
= 2 116-35 pounds per square foot- Aprx- 2 100 3-325 5885
= 760. millimeters of mercury. Aprx. ^X 1 000 2880 8136
= 33.900 6 feet of water. Aprx. >iX 100 I.53O 2068
= 29.921 2 inches of mercury. Aprx. 30 1-475 9790
= 14.696 9 pounds per square inch- Aprx- *% 1-167 2260
= 10.332 9 meters of water. Aprx. 10)^^ 1-014 2226
= 1-058 18 tons (short) per sq- foot. Aprx. add Ho 0-024 5585
— 1.033 29 kilograms per sq. cm. Aprx. add Ho 0-014 2226
= 1.013 24 megadynes per sq. centimeter. Aprx.l 0-005 7130
== 1.013 24 megabaries.t Aprx. 1 0-005 7130
= 0.944 801 ton (long) per sq. foot. Aprx. subtract ^0 •-- - 1-975 3405
1 ton (long) per square foot [tn/ft^]:
= 10 936-6 kilograms per sq. meter. Aprx. 1 1 000 4-038 8821
= 2 240- pounds per square foot. Aprx. % X 1 000 . . 3. 3 50 2480
= 804.402 millimeters of mercury. Aprx. 800 2.905 4731
= 35.881 1 feet of water. Aprx. ^i X 100 1.554 8663
== 31.669 3 inches of mercury. Aprx. 32 1-500 6385
= 15.555 6 poimds per square inch. Aprx. ^H 1-191 8855
= 10.936 6 meters of water. Aprx. 11 1-038 8821
= 1.12 tons (short) per sq- foot. Aprx. add H . . . . 0-049 2180
= 1.093 66 kilograms per square cm. Aprx. add Vio - . - 0-038 8821
= 1.058 42 atmospheres. Aprx. add Ho 0-024 6595
= 0.007 777 78 ton (short) per sq. inch. Aprx. % -5- 100 3-890 8555
= 0-006 944 44 ton (long) per sq. inch. Aprx. 7-^ 1 000 . . . §.841 6375
1 kilogram per square millimeter [kg/mm^l:
= 100. kilograms per sq. cm, which see for other values 2000 OOOO
1 ton (short) per square incli [tn/in^]:
= 144. tons (short) per sq. foot. Aprx. ^^ X 1 000. . . . 2158 3625
= 140.613 kilograms per sq. cm. Aprx. % X 100 2148 0266
= 136.083 atmospheres. Aprx. %X 100 2133 8040
=0.892 857 ton (long) per sq. inch. Aprx. subtr. Mo 1-950 7820
=0.140 613 metric ton per sq. centimeter. Aprx. ^ 1-148 0266
t Not authoritative; see foot-note on page 64.
X Authoritative ; see foot-note on page 64.
L. . _
WEIGHTS AND VOLUMES.
67
1 ton (long) per square Inch [tn/in^):
= 157.487 kilograms per sq. centimeter. Aprx.^HXlOO. 2-197 2446
= 152.413 atmospheres. Aprx. % X 100 2188 0220
= 144. tons (long) per sq. foot. Aprx. ^ X 1 000 2158 8825
» 1.12 tons (short) per square inch. Aprx. add ^i; . . . 0-049 2180
=0.157 487 metric ton per sq. cm. Aprx. l^ -*• 10 1-197 2448
1 metric ton per square centimeter [t/cm^j:
» 1 000. kilograms per sq. cm, which see for other values.. 8-000 0000
=967.782 atmospheres. Aprx. 970 2-985 7774
— 7.111 70 tons (short) per sq. inch. Aprx.%XlU 0-851 9734
— 6.349 73 tons (long) per sq. inch. Aprx. % X 10 0-802 7564
Conversion Tables for Pressures.
Pounds per
sq. inch »
kilogram
persq.cm.
atmos-
pheres.
Kilograms
per sq. cm =*
lbs. per
sq. m.
atmos-
Atmosph's »
lbs. per
sq. m.
kg. per
sq. cm.
pheres.
1
2
3
4
5
6
7
8
9
10
0.070 307
0.140 61
0.210 92
0.281 23
0.351 53
0.421 84
0.492 15
0.562 45
0.632 76
0.70307
14.223
28.447
42.670
66.894
71.117
85.340
99.564
113.79
128.01
142.23
14.697
29.394
44.091
58.788
73.485
88.181
102.88
117.58
132.27
146.97
0.068042
0.13608
0.204 12
0.272 17
0.340 21
0.408 25
0.476 29
0.544 33
0.612 37
0.680 42
1.033 3
2.066 6
3.099 9
4.133 2
5.166 5
6.199 7
7.233
8.266 3
9.299 6
10.333
0.967 78
1 .935 6
2.903 3
3.871 1
4.838 9
5.806 7
6.774 5
7.742 3
8.7100
9.677 8
WEIGHTS or MASSES and VOLUMES; DENSITIES;
WEIGHTS of MATERIALS; MASSES per unit of
VOLUME. (Weight -h volume.)
Only the more usual units are given here, as the table would otherwise
have become very long and cumbersome. The relations between such
compound units as these are the same as those between their individual
units whenever one of the latter is the same in both; for instance, the rela-
tion between pounds per cubic yard and kilograms per cubic yard is the same
as between pounds and kilograms, and as these are given in the tables of
weights they are not repeated here. In such a reduction multiply the
pounds per cubic yard by the value of 1 pound in kilograms. Similarly,
the relation between pounds per cubic yard and pounds per cubic meter is
the same as that between a cubic meter and a cubic yaru, but in this case
care must be taken in the reduction on account of the word ' ' per," not to
multiply the former by the value of one cubic yard in cubic meters, but to
divide instead, as a cubic yard is smaller than a cubic meter, hence the
weight per cubic meter is larger. To avoid such a long division use instead
the reciprocal relation, namely, the value of one cubic meter in cubic yards
and then multiply. The general rule for all compound units is that if the
individual unit to be changed is preceded by the word " per," then divide
by the value of the old unit in terms of the new one (or multiply by its re-
ciprocal); in all other cases multiply, even when the unit follows a hyphen,
as, for instance, in the case of pounds in foot-pounds.
In this group of units the weights are always ma.sses and never forces;
no unit exists having the dimensions of force divided by volume. The
value of ^gravity is therefore not involved in these values.
'Weights of Bf aterials. In' the metric system the number represent-
ing the density or specific gravity also represents the actual weight in grams
of a cubic centimeter of the material. Hence the actual weight of any other
68 WEIGHTS AND VOLUMES.
unit of volume of that material in terms of any other unit of weight is deter-
mined by merely multiplying the specific gravity or density (when based
on water, as is usual for all materials except gases) by the value of 1 gram
per cubic centimeter in terms of those units as ^ven in the table below.
Thus the weight of any material in pounds per cubic foot is 62.43 multiplied
by its specific gravity, this figure 62.43 being the value of 1 gram per cubic
centimeter in terms of pounds per cubic foot. Similarly, the specific
gravity or density is easily calculated by means of the figures in this table,
when the weight of a unit of volume is given in terms of any of the usual
non-metric units. Thus if the weight of any material is, say, 100 pounds
per cubic foot, its specific gravity is 0.016 02 (which is the value of 1 pound
per cubic foot in terms of grams per cubic centimeter in the table) multi-
plied by 100, that is 1.602.
Aprx. means within 2%.
Logarithm
1 pound per cubic yard [Ib/yd^]:
» 0.593 273 kilogram per cubic meter. Aprx. %o I -778 2645
= 0.037 037 or V^ pound per cb. ft. Aprx. 5^ + 10 2-668 6862
=0.000 593 273 gram per cb. cm orkg per lit. Ap. .0006. . . |.773 2545
-0.000 593 273 ton (met.) per cb. meter. Aprx. 6-i- 10 000. 2.778 2545
= 0.000 5 ton (short) per cubic yard or H -«- 1 000 |.698 9700
» 0.000 446 429 ton (long) per cubic yard. Aprx. %-*-! 000. 4.649 7520
1 kllog^ram per cubic meter [kg/m^]:
» 1.685 56 pounds per cubic yard. Aprx. ^% 0-226 7455
— 0.062 428 3 poimd per cubic foot. Aprs. 5i + 10 5-795 8817
B 0.001 gram per cb. cm or kilogram per liter S-OOO 0000
» 0.001 ton (met.) per cubic meter 8-000 0000
=0.000 842 782 ton (short) per cb. yd. Aprx. % + 1 000. . . 4.925 7155
=0.000 752 484 ton (long) per cb. yd. Aprx. M-*- 1 000 i.876 4976
1 gn^ain per cubic inch [gr/in^]:
= 0.246 857 pound per cubic foot. Aprx. }i 1.392 4457
=0.003 954 25 gram per cb. cm or kg per lit. Aprx. ^ ooo... 3-597 0640
1 pound per bushel [Ib/bu] (U. S.):
» 12.871 8 kilograms per cubic meter. Aprx. % X 10 1-109 6887
= 1.287 18 kilograms per hectoliter. Aprx. ^ 0-109 6887
» 1.032 02 poimds per bushel (Brit.). Aprx. add Ko Q-013 6888
=0.803 564 pound per cubic foot. Aprx. % 1.905 0204
1 pound per bushel [Ib/bu] (Brit.):
= 12.472 4 kilograms per cubic meter. Aprx. ^X 100 1.095 9499
= 1.247 24 kilograms per hectoliter. Aprx. i% 0-095 9499
= 0.968 972 pound per bushel ( U. S.). Aprx. subtr. ^io 1.986 8112
=0.778 630 pound per cubic foot. Aprx. % 1-891 8816
1 iLilog^am per hectoliter [kg/hi]:
= 10. kilograms per cubic meter, which see for other values. 1-000 0000
1 pound per cubic foot [Ib/ft^l:
= 27. pounds per cubic yard. Aprx. % X 10 1.481 8688
« 16.018 4 kilograms per cubic meter. Aprx. % X 10. . . 1.204 6183
= 4.050 93 grains per cubic inch. Aprx. 4 0-607 5543
= 1.601 84 kilograms per hectoliter. Aprx. % 0.204 6183
» 1.284 31 pounds per bushel (Brit.). Aprx. ^ 0-108 6684
= 1.244 46 pounds per bushel (U. S.). Aprx. 1^ 0-094 9796
= 0.160 538 pound per gallon (Brit.). Aprx. %-5-10 1.205 5784
= 0.133 681 poimd per gallon (liquid; U.S.). Aprx. H -5- 10 1.126 0688
=0.016 018 4 gram per cb. cm or kg per lit. Aprx. %+ 100.. 2204 6188
= 0.016 018 4 ton (met.) per cubic meter. Aprx. %-»• 100. . 2204 6188
= 0.013 5 ton (short) per cubic yard. Aprx. H -J- 100. . 2-130 3888
=0.012 053 6 ton (long) per cubic yard. Aprx. %-i-lOO . . 2-081 1158
1 pound per g^allon [lb/gal] (liquid; U. S.):
= 7.480 52 pouncis per cubic foot. Aprx. ^^X 10 0-873 9817
= 1.200 91 pounds per gallon (Brit.). Aprx. add % 6-079 5101
=0.119 826 gram per cb. cm or kg per lit. Aprx. 12 -i- 100... 1.078 5600
1 pound per g^allon [lb/gal] (Brit.}:
= 6.229 05 pounds per cubic foot. Aprx. 6H 0.794 4216
=0.832 702 4 pound per gallon (liquid ; U. S.). Aprx. % . . . 1-920 4899
=0.099 779 2 gram per cb. cm or kg per liter. Aprx. Ho- . . 2.999 0399
1 pound per quart [lb/qt]»4. pounds per gallon 0-602 0600
WEIGHTS AND VOLUMES OF WATER. 69
1 eram per cubic centimeter [g/cm^] or
1 kilofpram per liter [kg/11 or
1 ton (met.) per cubic meter [t/mfl:
= 1 685.57 pounds per cubic yard. Aprx. HXIO 000.. . 3226 7465
— 1 000. kilograms per cubic meter S.QOO 0000
» 252.893 grains per cubic inch. Apnc. hi XI 000 2-402 9860
"> 100. kilograms per hectoliter 2-000 0000
= 80.177 1 pounds per bushel (Brit.). Aprx. 80 1.904 0501
— 77.68 93 pounds per bushel (U. S.). Aprx. % X 100 . . 1.890 3618
— 62.428 3 pounds per cubic foot. Aprx. HX 100 1.705 3817
— 10.022 1 pounds per gallon (Brit.). Aprx. 10 1.000 9601
— 8.345 45 pounds per gal (liquid ; U.S.). Aprx.M2 X 100 0-921 4500
=■ 0.842 783 ton (short) per cubic yard. Aprx. subtr. H • 1-925 7155
=- 0.752 484 ton (long) per cubic yard. Aprx. ^ 1-876 4975
—0.036 127 5 poimd per cubic inch. Aprx. %i -i- 10 2-657 8380
— 0.(X)1 kilogram per cubic centimeter 3-000 0000
1 ton (short) per cubic yard [tn/yd^:
— 1 186.55 kilograms per cubic meter. Aprx. % X 1 000. . . 3-074 2845
— 118.655 kilograms per hectoliter. Aprx. % X 100 2-074 2845
» 95.133 7 pounds per bushel (Brit.). Aprx. 95 1-978 3346
— 02.181 9 pounds per bushel (U. S.). Aprx. Mi X 1 000. . . 1.964 6458
=• 74.074 1 pounds per cubic foot. Aprx. H X 100 1-869 6662
» 1.186 55 tons (met.) per cb. m or kg per lit. Ap. add ^... Q-074 2845
">0.892 857 ton (long) per cubic yard. Aprx. %o 1-950 7820
1 ton (long) per cubic yard [tn/yd^]:
» 1 328.93 kilograms per cubic meter. Aprx. % X 1 (XX) .... 3.123 5025
a* 132.893 kilograms per hectoliter. Aprx. % X 100 2-123 5025
= 106.550 pounds per bushel (Brit.). Aprx. 107 2027 5526
— 103.244 pounds per bushel (U. S.). Aprx. 103 2013 8638
■"82.963 pounds per cubic foot. Aprx. % X 100 I.gi8 8842
<- 1.328 03 tons (met.) per cb. m or kg per lit. Ap. add H-- 0123 5025
= 1.12 tons (short) per cubic yard. Aprx. aad % 0-049 2180
1 pound per cubic inch [Ib/inS]:
= 27.679 7 grams per cb. centimeter. Aprx. i^^X 10. . . . 1.442 1620
=0.027 679 7 kilogram per cubic centimeter. Aprx. »>i-*- 100 2-442 1620
1 kilogram per cubic centimeter [kg/cm^j:
— 1 (XX). grams per cb. cm or tons (met.) per cb. m 8.000 0000
»36.127 5 pounds per cubic inch. Aprx. 36 1*557 8380
WEIGHTS and VOLUMBS of WATBR.
Factors for calculating weights or volumes of materials from
their specific gravity.
The following two groups of numbers give the weights (W) of all the
different units of volume of water occurring in practice; also the volumes
(V) of all the different units of weight of water occurring in practice; the
latter are, of course, the reciprocals of the former and are given so as to
avoid the lon^ divisions by the former.
Besides their direct application to hydraulics and to the calibration of
vessels and for measuring, or for the indirect determinations of irreg^ular
volumes by means of weights, they are also of use for determining the
weights of materials, as the weight of a unit of volume of any material,
whether solid or liquid, is its specific gravity or density multiplied by one
of these factors, W; or the volume of a unit of weight of any material,
whether solid or liquid, is one of these factors, V, divided by its specific
gravity or density. They are applicable also to gases provided the value
of the specific gravity or density which is used is based on water and not
on air or hydrogen.
For the weights of columns of water, mercury, or the air, see under
Pressures.
All these values, even those given entirely in English units, have been
calculated from the uniform bases that one liter of water weigh.<( one kilo-
gram, and that a liter is equal to a cubic decimeter. (See notes on the. lltor
m the introductory remarks on xmits of Volume and Weight.)
it
it
<«
70 WEIGHTS AND VOLUMES OF WATER.
WEIGHTS of WATER, W.
Aprx, means within 2%.
Logarithm
1 cubic centimeter => 15.423 4 grains. Aprx. 3i^or 15J^. . 1.188 4322
= 1. gram 0000 0000
= 0.035 274 oz (av.). Aprx. %-5.100 . . §.547 454]
=0.002 204 62 lb (av.). Aprx. % -5- 100 . . . §.343 3342
1 cubic inch = 252.893 grains. Aprx. HXl 000 2-402 9360
« 16.387 2 grams. Aprx. M X 100 or »% 1.214 5038
= .578 040 ounce (av.). Aprx. t^ 1.761 9579
=0.036 127 5 pound (av.). Aprx. ^10 2-557 8380
1 pint (liquid; U. S.)= 1.043 18 pounds (av.). Aprx. add Mo. . 0-018 3600
=0.473 179 kilogram. Aprx. Mi X 10 1.675 0258
1 pint (dry; U. S.)= 1.213 90 pounds (av.). Aprx. % 0-084 1818
=0.550 614 kilogram. Aprx. i^-^-lO 1.740 8471
1 pint (Brit.)= 1.252 77 pounds (av.). Aprx. % 0097 8701
=0.568 245 39 kilogram. Aprx. ^ I.754 5359
1 quart (liquid ; U. S.) = 2.086 36 lb (av.). Aprx. 2Mo or 2M0 . . 0-319 3900
=0.946 359 kilogram. Aprx. subt. Mo. . - X.976 0558
1 liter = 2.204 62 pounds (av.). Aprx. 22JI0 0-343 3342
" = 1. kilogram 0-000 0000
1 quart (dry; U. S.) =2.427 79 pounds (av.). Aprx. 2^0 or 1%. . 0-385 2113
" =1.101 23 kilograms. Aprx. add^o 0041 8771
1 quart (Brit.)= 2.505 53 pounds (av.). Aprx. 1% 0-398 9001
= 1.136 490 8 kilograms. Aprx. add^ 0055 5659
1 gallon (liquid; U. S.) = 8.345 45 pounds (av.). Aprx. b% 0-921 4500
= 3.785 43 kilograms. Aprx. H X 10 0.578 1158
1 gallon (Brit.)= 10.022 1 pounds (av.). Aprx. 10 LOOO 9601
= 4.545 963 1 kilograms. Aprx. % or 4M 0-657 6269
1 peck (U. S.) = 19.422 3 pounds (av.). Aprx. %i X 1 000 1.288 3013
= 8.809 82 kilograms. Aprx. J^ X 10 0-944 9671
1 peck (Brit.)'= 20.044 3 pounds (av.). Aprx. 20 I.30I 9901
= 9.091 926 2 kilograms. Aprx. 9 or .io%i 0-958 6559
1 cubic foot= 62.428 3 pounds (av.). Aprx. ^XlOO 1.795 3817
= 28.3170 kilograms. Aprx. %X 100 1-452 0475
= 0.031 214 2 ton (short). Aprx. Ms: 2-494 3517
= 0.028 317 ton (met.). Aprx. ^ho 2.452 0475
=0.027 869 8 ton (long). Aprx. 1^00 2-445 1337
1 busliel (U. S.) = 77.689 3 pounds (av.). Aprx. % X 100 1.890 3613
= 35.239 28 kilograms. Aprx. 35 1.547 0271
1 busliel (Brit.)= 80.177 1 pounds (av.). Aprx. 80 1-904 0501
= 36.367 704 8 kilograms. Aprx. Hi X 100 1.560 7159
1 hectoliter = 220.462 pounds (av.). Aprx. 220 2-343 3342
= 100. kilograms 2-000 0000
= 0.110 231 ton (short). Aprx. % T.042 3042
= 0.1 ton (met.) T.OOO 0000
=0.098 420 6 ton (long). Aprx. Vio 2-993 0862
1 cubic yard= 1 685.56 pounds (av.). Aprx. MX 10 000 3-226 7455
= 764.559 kilograms. Aprx. ?^ X 1 000 2-883 4113
=0.842 782 ton (short). Aprx. % T.925 7155
=0.764 559 ton (met.). Aprx. H 1.883 4113
= 0.752 484 ton (long). Aprx. H 1-876 4975
1 cubic meter = 2 204.62 pounds (av.). Aprx. 2 200 3-343 3342
= 1 000. kilograms 3.OOO 0000
= 1.102 31 tons (short). Aprx. add ^0 0-042 3042
= 1. ton (met.) 0-000 0000
=0.984 206 ton (long). Aprx. 1 1.993 0862
it
tt
4<
tt
tt
il
«t
WEIGHTS AND VOLUMES OP WATER.
71
VOLUMES of WATER, V.
1 f^raiii =» 0.0fi4 798 9 cubic centimeter. Aprx. i% -?- 100 3-811 5678
=0.003 954 25 cubic inch. Aprx. 4 -5- 1 000 §.597 0640
1 g^ram ^ 1. cubic centimeter Q-000 0000
=0.061 023 4 cubic inch. Aprx. 6^ 100 2.785 4952
1 ounce (av.) = 28.349 5 cubic centimeters. Aprx. % X 100 1-452 5459
= 1.729 98 cubic inches. Aprx. % 0-238 0421
1 pound (av.)— 453.592 cb. centimeters. Aprx. %X 100.. 2-656 6658
=» 27.679 7 cubic inches. Aprx. ^K X 10 1-442 1620
- 0.958 606 pt (liquid; U.S.). Aprx. subtr.^. 1.981 6400
<= 0.823 794 pint (dry; U. S.). Aprx. % 1-915 8187
- 0.798 233 pint (Brit.). Aprx. % or ^io 1-902 1299
- 0.479 303 qt (liquid; U. 8.). Aprx. 10-4-21.. 1.680 6100
= 0.453 592 hter. Aprx. lO-f-22 or %u f .656 6658
- 0.411 897 quart (dry; U. S.). Aprx. i%4 .. 1.614 7887
- 0.399 117 quart (Brit.). Aprx. Mo 1-601 0999
-= 0.119 826 gal (liquid; U.S.). Aprx. 12 -i- 100. T.078 5500
- 0.099 779 2 gallon (Brit.). Aprx. Mo 2-999 0399
= 0.051 487 1 peck (U. S.). Aprx. 51 h- 1 000. . . 2-711 6987
= 0.049 889 6 peck (Brit.). Aprx. ^ 3-698 0099
- 0.016 018 4 cubic foot. Aprx. % -f- 100 3-204 6188
- 0.012 871 8 bushel (U.S.). Aprx. %-*- 100 .. . 3-109 6387
- 0.012 472 4 bushel (Brit.). Aprx. Ho 3-095 9499
'• = 0.004 535 92 hectoliter. Aprx. %-s-l 000 §.656 6658
«=0.000 693 273 cubic yard. Aprx. 6-5-10 000 4.773 2545
=0.000 453 592 cubic meter. Aprx. % -^ 10 000 . . |.656 6658
I kilogram = 1 000. cubic centimeters 3-000 0000
= 61.023 4 cubic inches. Aprx. 60 1-785 4962
= 2.113 36 pints (liquid; U. S.). Aprx. 2^0 . . - 0-324 9742
- 1.816 15 pints (dry; U.S.). Aprx. 2%i 0259 1529
= 1.759 80 pints (Brit.). Aprx. % 0-245 4641
= 1.056 68 quarts (liquid; U.S.). Aprx. add >^. 0.023 9442
** = 1. liter 0-000 0000
- 0.908 078 quart (dry; U. S.). Aprx. »Ao 1.958 1229
- 0.879 902 quart (Brit.). Aprx. J4 1-944 4341
= 0.264 170 gallon (hquid; U. S.). Aprx. %o 1-421 8842
= 0.219 975 gallon (Brit.). Aprx. 22-»- 100 1.842 3741
= 0.113 510 peck (U.S.). Aprx.^-s-lO 1.055 0329
= 0.109 988 peck (Brit.). Aprx. 11 -h 100 1.041 3441
-= 0.035 314 6 cubic foot. Aprx. % + 100 3-547 9525
- 0.028 377 4 bushel (U. S.). Aprx. 9^ -h 10 3-452 9729
- 0.027 496 9 bushel (Brit.). Aprx. »M-^ 100 3-439 2841
" = 0.01 hectoliter 3-000 0000
-0.001 307 94 cubic yard. Aprx. -^-i-l 000 |.ii6 5887
" — 0.001 cubic meter 3-000 0000
1 ton (short) = 32.036 7 cubic feet. Aprx. 32 1-505 6483
= 9.071 85 hectohters. Aprx. 9 0957 6958
*• = 1.186 55 cubic yards. Aprx. % / 0074 2845
" =0.907 185 cubic meter. Aprx. %o 1-957 6958
1 ton (metric) = 35.314 5 cubic feet. Aprx. %X 10 1547 9525
= 10. hectoliters 1-000 0000
*' —1.307 94 cubic yards. Aprx. % 0-116 5887
" — 1. cubic meter 0000 0000
1 ton riong) — 35.881 1 cubic feet. Aprx.'Hi X 100 I.554 8663
" = 10.160 5 hectoliters. Aprx. 10 1-006 9138
" = 1.328 93 cubic yards. Aprx. ^ 0-123 5025
•* — 1.016 05 cubic meters. Aprx. 1 0-006 9188
72 energy; work; heat.
ENSROT; WORE; HEAT; VIS-VIVA; TORQUE.
(ForceX length ; mass X temperature; elec. quant. Xe. m. f.)
All energy units or measures are here grouped together, be they mechan-
ical, electrical, thermal, chemical, kinetic, potential, etc. Units of power,
not being energy but rcUe of energy, are not included in this group (see note
under units of Fower). For the relations between energies stated in terms
of power units, as horse-power-hours, and kilowatt-hours, see the relations
between horse-powers and kilowatts under Power.
The relations between the mechanical and thermal units of energy are
based on the mechanical equivalent of heat ; those between the mechanical
and electric or absolute units are based on the value of gravity; those be-
tween the thermal and the electrical or absolute units are baaed on the
specific heat of water in absolute imits. These three bases are the three
connecting links between these three classes of energy units. The three
Unks are again interlinked bjr the relation that the mechanical equivalent
of heat multiplied by gravity is equal to the specific heat of water, when all
three are reduced to grams, centimeters, and seconds. This will be found
explained more fully under Inter-relations of Units in the Introduction.
Torque is a force multipUed by a lever-arm; it is a "moment" and is
therefore measured in imits like foot-pounds or kilograms-meters; the
relation between units of torque are therefore the same as those in the
table between the same named imits of energy. In order to show whether
torque or energy is meant by such similarly named units, it has become
customary to reverse the accepted term whenever torque is meant ; that is,
to use the term pound-feet for torque and foot-paunda for enerfy^ also tneter-
kilograina for torque and kilogram-metera for energy; this distinction is to
be recommended even though the length factor unfortunately comes first
in foot-pounds of energy and last in kilogram-meters of energy. For the
relations between imits of torque and units of energy, see the end of the
following table. Torque multiphed by an angle is energy, and as an angle
has no dimensions, it follows that the dimensions of torque and of energy
are the same, notwithstanding the fact that there is no direct equivalent
between a foot-pound of torque, for instance, and a calorie of energy. If
the torque is measured statically it is performing no work, yet it is a true
torque; it is then like a suspended weight which is capable of doing work
when released; but this weight (force) must be multiplied by a length or
distance through which it acts in order to give energy, while torque a&eady
includes this factor length; this apparent discrepancy arises from the fact
that an angle, by which torque must be multiplied to reduce it to ener^,
has no dimensions. In units of energy the length factor is in the direction
of the force, while in units of torque it is peipendicular to the force. The
two lengths are, therefore, at relatively dififerent angles with each other,
which explains how the angle enters into the difiference in the nature of the
two units; torque differs from energy somewhat like the so-called "wattless
component" differs from the true energy in electrical quantities. See also
the relations given at the end of the following table.
Mechanical Equivalent of Heat, and the Heat Unit. The me-
chanical equivalent of heat, sometimes called Joule's equivalent, is an
empirical number which gives the relation that heat units bear to mechani-
cal units, thus enabling one to calculate how much mechanical energy stated
in foot-pounds, kilogram-meters, horse-power-hours, etc., is equal to any
given amount of heat energy stated in calories or thermal units, or the
reverse.
The mechanical units are all based on the C. G. S. or absolute svstem of
units and on the acceleration of gravity, while the heat units are based on
an inherent property of water, and on the thermometer scale and kind of
thermometer. The two bases are therefore entirely different, and the re-
lation between them, namely, the mechanical equivalent of heat, therefore
is, and must ^ways remain, one that has to be determined by experiment.
Moreover, these experiments are intricate and the heat units theinselves
are not yet definitely and accurately established, owing to the variations
in the specific heats of water between 0° and 1(X)° C. and to the differences
in the kinds of thermometers; therefore the mechanical equivalent of heat
energy; work; heat. 73
is not 3ret known to very great accuracy, although the accuracy is quite
sufficient for all purposes except perhaps for very refined physical research.
Quite a numper of researches have been made, among which are a few very
accurate ones, but the most authoritative value is unquestionably the one
recommended by Griffiths and by Ames in their reports to the International
Physical Ck>ngres8 of 1900, which met in Paris (Ri4>ports, Congr^s Inter-
national de Physique, 1900, tome 1, pp. 226 and 204). Griffiths there
concludes, after a careful comparison and discussion of the best determi-
nations, to recommend the number 4.187 joules for the calorific capacity
(more generally called the specific heat) of 1 gram of water raised from 15°
to 16** C, measured on the hydrogen scale of the International Bureau.
The probable error, he says, is less than 1 in 2000. He also recommends
that this be considered the same as the mean value per decree between 0°
and 100° C, and believes it to be very improbable that the error in this
assumption attains 2 in 1 000. This therefore also defines the unit of heat
which he recommends as the intermediate thermic standard. The value
which he recommends, 4.187, agrees with that given as the most probable
in the report of Ames, after reduction to the same temperature interval.
In Ames' report all the important determinations of the mechanical equiva-
lent of heat are discussed and compared. This value is the mean of the
determinations of Rowland, Griffiths, Schuster, and Gannon, and CaUender
and Barnes, when all their results are reduced to the hydrogen scale of
temperature and when the electrical methods are corrected for the proba-
ble error of the present international volt as now legally fixed in terms of
the Clark cell.
Taking for the value of gravity at sea-level and at 45° latitude 9.805 966
in meters (Helmert. Die math. u. phys. Theorien der hoehem Geodaesie,
II, p. 241, 1884X this specific heat reduces to 426.985 kilogram-meters
as the value of the mechanical eauivalent of heat. According to Griffiths'
Erobable error, the true value therefore lies between 427.20 and 426.77
ilogram-meters, showing that the fourth figure is still uncertain. Some
recent, very carefully made researches by Barnes, which were not finished
in time to be included in Griffiths' report, give the value 426.6, and it is
probable, therefore, that the true value is lower than Griffiths' probable
mean, rather than higher, and may be even nearer to 426.6 than to 426.985.
But as the final value in Griffiths' report may be considered as semi-official,
the author has adopted that value throughout this book, except that in
order to make it approach rather than depart from Barnes' value, it has
been abbreviated to 426.9 instead of 427.0. In view of the fact that the
fourth place is still imcertain it would not be rational to retain more than
four places of figures.
This naturally also establishes the absolute value of the heat units and
of the mean specific heat of water to be used in this book. The value of the
specific heat of water in absolute imits is the same thing as the mechanical
eguivident of heat stated in ergs or joules. The specific heat of water is
different at different temperatures between 0° and 100° C, but the value
between 15° and 16° C. (according to Barnes at 16° C.) is as nearlv as has
been determined equal to the mean per degree for the whole value between
0° and 100° C, and a heat unit based on this mean is therefore much more
definitely defined than if based merely on a rise of temperature of one degree
without stating which degree.
Although it would perhaps be more rational to consider the specific heat
of water to be the fundamental quantity, the mechanical equivalent of heat
being then derived from it, yet the author has reversed this by adopting; a
more simple abbreviated number for the mechanical equivalent, and letting
the specific heat be the incommensurable, derived quantity. This was
done because the former is used very frequently^, while the latter is not.
The uncertainty of the fourth place of figures in either of these values,
which maybe ±2, does not warrant any fine distinction between the funda-
mental and the derived value. The resulting value of the specific heat of
water then becomes 4.186 17 instead of 4.187. In view of the researches
of Barnes above mentioned, the former value is probably even more nearly
correct than the latter.
The value of the mechanical equivalent of heat and that of the calorie
or heat unit adopted in this book, are therefore ^ven bv the following
statements: 426.9 kilogram-meters of energy will raise 1 kilogram of water
from 15° to 16° C, hydrogen scale, at sea-level, latitude 45°, and are there-
74 energy; work; heat.
fore equivalent to one large calorie or kilogram calorie. The small calorie
or gnxn calorie, equal to one thousandth of the large calorie, is the amount
of heat that will raise the temperature of 1 gram of water from 15^ to 16** C,
hydrogen scale ; this is taken as equal to one hundredth of the amount of
heat that will raise the temperature of 1 gram of water from 0° to 100*^ C.
Unfortunately writers generally do not state which of the two calories they
mean. The thermal unit, or British thermal unit, or BTU, is the amount
of heat which will raise one pound (av.) of water one degree Fahrenheit; in
these tables it is a derived unit whose value is determined from that of
either of the calories. The hybrid unit based on the pound and the Centi-
grade scale has no name.
This mechanical equivalent, namely 426.9 kilogram-meters per kilo^am
Centigrade heat unit , corresponds to 778. 104 foot-pounds per pound Fah-
renheit heat unit or per thermal unit. The probable error in them is
thought to be within 1 in 2 000. For further equivalents or converions
factors based on this, see the table.
ENERGY; WORE; HEAT; VIS-VIVA; TORQUE.
Aprx. means within 2%.
Logarithm
1 erg or dyne-centimeter [dsme-cm]:
— 0.001 019 79 gram-centimeter. Aprx. M ooo 3-008 5096
-= 0.000 516 328 foot-grain. Aprx. ^H -^ 10 000 |.712 9260
= 0.000 000 1 joule 7.000 0000
«0.000 000.073 761 2 foot-pound. Aprx. ^ -<- 10 000 000. . . 8867 8279
1 g^am-ceiitimeter [g-cm]:
= 980.596 6 ergs. Apr. 1 000 2-991 4904
=0.000 098 059 66 joule. Aprx. Mo ooo 5991 4904
= 0.000 072 330 foot-pound. Aprx. % -J- 10 000 5-859 3184
= 0.000 01 kilogram-meter 5.000 0000
1 foot-grain [ft-gr]: = 1 936.75 ergs. Aprx. 1900 3.287 0740
= 1.975 08 gram-centimeters. Aprx. 2. 0-295 5886
■=0.000 193 675 joule. Aprx. 19 -^ 100 000. 1-287 0740
-0.000 142 857 ft-lb. Aprx. M -*- 1 000 1-154 9020
I Joule [j] or volt-coulomb or watt-second [w-s]:
= 10 000 000. ergs 7.OOO 0000
= 10 197.9 gram-centimeters. Aprx. 10 000 4-008 5096
= 0.737 612 foot-pound. Aprx. H 1-867 8279
« 0.238 882 small calorie. Aprx. 24-*- 100 f.378 1834
== 0.101 979 kilogram-meter. Aprx. Mo 1-008 5096
« 0.001 359 72 metric hp-second. Aprx. ^3^-f-l 000 5.133 4433
= 0.001 341 11 horse-power-second. Aprx. %-t-l 000 3-127 4653
=» 0.001 kilowatt-second. Aprx. M 000 5-000 0000
=0 000 947 960 thermal unit. Aprx. 95 -s- 100 000 I.976 7901
= 0.000 526 645 pound-Centgr. heat unit. Aprx. Viq -s- 100. 1.721 5176
=0.000 277 778 watt-hour. Aprx. %i -f- 1 000 I.443 6975
=0.000 238 882 large calorie. Aprx. 24 -^ 100 000 1.373 1334
1 f«»ot-pound [ft-lb]:
= 13 557 300. ergs. Aprx. 27^ x 1 000 000 7.132 1721
= 13 825.5 gram-centimeters. Aprx. %X 10 000.. 4.140 6817
= 1.355 73 joules. Aprx.% 0-132 1721
= 0.323 859 small calorie. Aprx. 1^0 1-510 3555
= 0.138 255 kilogram-meter. Aprx. %o 1-140 6817
« 0.001 843 40 metric hp-second. Aprx. i>i-*-l 000.. . §.265 6204
^ 0.001 818 18 horse-power-second. Aprx. ^i-f- 100.. 3.259 6378
=. 0.001 355 73 kilowatt-second. Aprx. % -hi 000 §.132 1721
= 0.001 285 17 thermal unit. Aprx. % -h 1 000 |.108 9622
= 0.000 713 986 Ib-Centgr. heat unit. Aprx. ^^ -i- 1 000 . 4.353 6897
« 0.000 376 591 watt-hour. Aprx. H + l 000 J.575 8696
= 0.000 323 859 large calorie. Aprx. ^H^iO 000 I.510 3555
=«0.000 000 505 051 horse-power-hour. Aprx. J^-j- 1 000 000 f .703 3348
-0.000 000 376 591 kilowatt-hour. Aprx. ^g -*- 1 000 000. . . 7.575 8696
energy; work; heat. 75
1 meter-kilogram: see kilogram-meter; also under /orgtic, below.
1 fcilogrram-meter [kg-m]:
= 98 059 660. ergs. Aprx. 100 000 000 79914904
=. 100 000. gram-centimeters 5000 GOOD
-= 9.805 966 joules. Aprx. 10 0991 4904
— 7.233 00 foot-pounds. Aprx. »%i 0859 3184
=- 2.342 47 small calories. Aprx. % 0.369 6738
« 0.013 333 3 metric hp-second. Aprx. % -^ 100 2124 9387
« 0.013 150 9 horse-power-second. Aprx. ^i -*- 100 3-118 9557
= 0.009 805 966 kilowatt-second. Aprx. ^oo 3-9914904
= 0.009 295 67 thermal unit. Aprx. 2% -;- 1 000 3-968 2805
= 0.005 164 26 Ib-Centgr. heat unit. Aprx. 52-^10 000. 3-713 0080
«= 0.002 723 88 watt-hour. Aprx. %i -j- 100 5-435 1879
= 0.002 342 47 large calorie. Aprx. % -^ 1 000 5-369 6738
=0.000 003 703 70 metric hp-hour. Aprx. H -5- 100 000 6-568 6362
=0.000 002 723 88 kilowatt-hour. Aprx. %i -^ 100 000 g.435 1879
1 metric liorse-power-second [hp-s]:
= 735.447 joules. Aprx. MX 1 000 2-866 5517
« 542.475 foot-pounds. Aprx. »1^ X 100 2-734 3797
«= 175.685 small calories. Aprx. % X 100 2-244 7351
= 75. kilogram-meters. Aprx. ^X 100 1-875 0613
=0.204 291 watt-hour. Aprx. 204 -^ 1 000 1310 2492
1 horse-power-second [hp-s]:
= 745.650 joules. Aprx. »i^X 1 000 2-872 5348
= 550. foot-pounds or i^X 100 2740 3627
= 178.122 small calories. Aprx. %X 100 2-250 7182
= 76.040 4 kilogram meters. Aprx. ^X 100 1.881 0444
=0.207 125 watt-hour. Aprx. 207-^1 000 1.316 2323
1 kilowatt-second [kw-s]:
= 1 000. joules 3-000 0000
= 737.612 foot-pounds. Aprx. MX 1 000 2-867 8279
= 238.882 small calories. Aprx. 240 2-378 1834
= 101.979 kilogram-meters. Aprx. 100 2-008 5096
=0.277 778 watt-hour. Aprx. ^Ho 1-443 6975
1 calorie (small) [cal] or gram-Centigrade heat unit [g-C]:
=0.001 large calorie or kg-Centigr. heat unit, which see. . . . 5-000 0000
1 thermal unit [BTU] or pound -Fahrenheit heat nnitt [Ib-F]:
= 1 054.90 joules. Aprx. si^X 100 3023 2099
= 778.104 foot-pounds. Aprx. % X 1 000 2-891 0379
— 251.996 small calories. Aprx. MX 1 000 2-401 3933
— 107.577 kilogram-meters. Aprx. 108 2-031 7195
= 1.434 36 metric horse-power-seconds. Aprx. i%.... 0-156 6582
= 1.414 74 horse-power-seconds. Aprx. i% 0-150 6752
= 1.054 90 kilowatt-seconds. Aprx. add ^o 0023 2099
« 0.555 556 pound-Centgr. heat unit. Aprx. % 1744 7275
= 0.293 027 watt -hour. Aprx. %7 1-466 9074
= 0.251 995 8 large calorie. Aprx. M 1-401 3933
=0.000 398 433 metric hp-hour. Aprx. 4-*- 10 000 J.600 3557
=0.000 392 982 horse-power-hour. Aprx. 4-5-10 000 1.594 3727
=0.000 293 027 kilowatt-hour. Aprx. s^i-*- 100 000 |.466 9074
1 pound-Ceutigrade heat unit [Ib-C]:
= 1 898.81 joules. Aprx. 1 900 8-278 4824
= 1 400.59 foot-pounds. Aprx. 1 400 3146 3104
= 453.592 4 small calories. Aprx. % X 100 2-656 6658
= 193.639 kilogram-meters. Aprx. 194 2-286 9920
= 2-581 85 metric horse-power-seconds. Aprx. ^Vis-- - - 0-411 9307
= 2.546 52 horse-power-seconds. Aprx. i% 0-405 9477
= 1.898 81 kilowatt-seconds. Aprx. i%o 0-278 4824
— 1.800 00 thermal units. Aprx.% 0255 2725
— 0.527 448 watt-hour. Aprx. ^%9 J. 722 1799
= 0.453 592 4 large calorie. Aprx. i%2 1-656 6658
=0.000 717 180 metric hp-hour. Aprx. % -f- 1 000 4-855 6282
=0.000 707 368 horse-power-hour. Aprx. % ^ 1 000 i.849 6452
=0.000 527 448 kilowatt-hour. Aprx. Vl9 -^ 100 1-722 1799
t Often called a British Thermal Unit. BTU also means kilowatt-hour.
76 energy; work; heat.
1 watt-hour [w-h]:
= 3 600. joules 8.556 8025
= 2 655.40 foot-poundsi. Aprx. % X 1 000 3-424 1305
= 859.975 small calories. Aprx. % X 1 000 2-934 4859
= 367.123 kilogram-meters. Aprx. i^X 100 2564 8121
=« 4.894 98 metric horse-power-seconds. Aprx. *%o... 0-689 7508
= 4.828 01 horse-power-seconds. Aprx. *^io 0-683 7678
=. 3.6 kilowatt-seconds. Aprx. »%o or *%i 0-556 3025
= 3.412 66 thermal units. Aprx. 8^o or i% 0-533 0926
== 1.895 92 pound-Centigr. heat units. Aprx. i%o 0-277 8201
= 0.859 975 large calorie. Aprx. % 1934 4859
=0.001 359 72 metric horse-power-hour. Aprx.%-Hl 000.. 3.133 4483
=0.001 341 11 horse-power-hour. Aprx. %-^l 000 5-127 4652
= 0.001 kilowatt-hour S-000 0000
1 calorie (large) [Cal] or kilogram- Centigrade lieat unit fkg-C]:
=. 4 186.17 joules. Aprx. 4200 3621 8166
=- 3 087.77 foot-pounds. Aprx. 3 100 3489 6446
a 1 000. small calories 3-000 0000
a 496.900 kilogram-meters. Aprx. ^ X 1 000 2-630 3262
« 5.692 metric horse-power seconds. Aprx.*% 0-755 2649
— 6.614 12 horse-power-seconds. Aprx.*% 0-749 2819
= 4.186 17 kilowatt-seconds. Aprx. *%o 0-621 8166
— 3.968 32 thermal units. Aprx. 4 0-598 6067
= 2.204 62 pound-Centgr. heat units. Aprx. 25Jio 0-343 3342
= 1.162 82 watt-hours. Aprx. % 0065 5141
=0.001 581 11 metric horse-power-hour. Aprx. %-f-l 000. 5.198 9624
=0.001 559 48 horse-power-hour. Aprx. ^Vt-^l 000 5.192 9794
= 0.001 162 82 kilowatt-hour. Aprx. % -«- 1 000 5-065 5141
1 mile-pound [ml-lb]:
= 5 280. foot-pounds. Aprx. 5 300 3-722 6339
=0.002 703 66 metric horse-power-hour. Aprx. %^ 1 000 . 5-431 9518
= 0.002 666 67 horse-power-hour. Aprx. %^ 1 000 5.426 9687
= 0.001 988 40 kilowatt-hour. Aprx. % ooo 5-298 5035
1 kilogram-kilometer [kg-km]:
= 7 233.00 foot-pounds. Aprx. "^ X 10 000 3-859 3184
= 1 000. kilogram-meters 3-000 0000
= 1.369 89 mile-pounds. Aprx. ^H 0.136 6845
=0.003 703 70 metric horse-power-hour. Aprx. ^'s-s-lOO. . 5-568 6362
=0.003 653 03 horse-power-hour. Aprx. i^-s-l 000 5-562 6532
= 0.002 723 88 kilowatt-hour. Aprx. i^ + l 000 5-435 1879
1 metric horse-power-minute [hp-m]:
= 44 126-8 joules. Aprx. % X 100 000 4-644 7029
= 32 548.5 foot-pounds. Aprx. i5:t X 10 000 4-512 5309
= 4 500. kilogram-meters. Aprx. % X 1 000 3-653 2125
= 12.257 5 watt-hours. Aprx. "% 1.088 4004
= 10.541 1 large calories. Aprx. ^H 1022 8868
1 horse-power-minute [hp-m]:
= 44 739.0 joules. Aprx. % X 100 000 4-650 6860
= 33 000. foot-pounds. Aprx. H X 100 000 '4-518 5139
= 4 562.42 kilogram-meters. Aprx. % X 1 000 8-659 1956
= 12.427 5 watt-hours. Aprx. B0^ 1.094 3885
= 10.687 3 large calories. Aprx. lO'Ho 1028 8694
1 kilowatt-minute [kw-m]:
= 60 000. joules 4-778 1513
= 44 256.7 foot-pounds. Aprx. % X 100 000 4-645 9798
=6 118.72 kilogram-meters. Aprx. 6 000 3786 6609
= 16.666 7 watt-hours. Aprx. H X 100 1-221 8488
=. 14.332 9 large calories. Aprx. Vr X 100 1-156 8347
energy; work; heat.
77
1 ittetric horse-power-hour [hp-h]:
•=2 647 610. joules. Aprx. %X 1 000 000 8-422 8542
= 1 952 910. foot-pounds. Aprx. %i X 100 000 000 6-290 6822
— 270 000. kilograra-meters. Aprx. 27 X 10 000 6-431 3638
= 3 600. metric horse-power-seconds 3-556 3025
«= 2 509.83 thermal units. Aprx. MX 10 000 S-399 6443
= 1 394.35 pound-Centgr. heat units. Aprx. 1 400 3-144 8718
=. 736.447 watt-hours. Aprx. 740 2-866 5517
= 632.467 large calories. Aprx. 630 2-801 0376
» 270. kilogram-kilometers 2-431 3688
— 60. metric horse-power-minutes 1-778 1513
=0.986 318 horse-power-hour. Aprx. 1 1-994 0170
= 0.735 447 kilowatt-hour. Aprx. H or 2%o 1-866 5517
1 liorse-power-hour [hp-h]:
= 2 684 340. joules. Aprx. % X 1 000 000 6-428 8873
= 1 980 000. foot-pounds. Aprx. 2 000 000 6-296 6652
« 273 745. kilogram-meters. Aprx. i>i X 100 000 5-437 3469
» 3 600. horse-power-seconds 3-556 8025
= 2544.65 thermal units. Aprx. ^X 10 000 8-405 6274
» 1 413.69 pound-Centgr. heat units. Aprx. 1 400 3150 3549
« 745.650 watt-hours. Aprx. fix 1 000 2-872 5348
= 641.240 large calories. Aprx. 640 2807 0207
-= 375.000 mile-pounds or H X 1 000 2574 0313
■» 60. horse-power-minutes 1-778 1613
= 1.013 87 metric horse-power-hours. Aprx. 1 0-005 9830
= 0.745 650 kilowatt-hour. Aprx. H 1-872 6848
1 kUowatt-honr [kw-h] [BTU]t:
— 3 600 000. joules 6-556 3025
=2 655 403. foot-pounds. Aprx. % X 1 000 000 6-424 1806
= 367 123. kilogram-meters. Aprx. »>iX 100 000 5-564 8121
»■ 4 828.01 horse-power-seconds. Aprx. 4 800 3-683 7678
= 3 412.66 thermal units. Aprx. 3 400 • 3-533 0926
= 1 895.92 pound-Centgr. heat units. Aprx. 1 900 3277 8201
= 1 000. watt-hours 3-000 0000
= 859.975 large calories. Aprx. % X 1 000 2934 4859
= 502.917 mile-pounds. Aprx. K X 1 000 2-701 4966
= 367.123 kilogram-kilometers. Aprx. ^^X 100 2564 8121
= 60. kilowatt-minutes 1778 1513
= 1.359 72 metric horse-power-hours. Aprx. add )^ 0-133 4488
= 1.341 11 horse-powcr-hours. Aprx. add H 0-127 4652
Conversion Tables for Energry, Work, Heat.
Foot-lbs. =
Klgr-met' s
kg-mets
ft-ibs.
thermal u
calories
(large)
kg-mt
Thermal u.
ft-lbs.
Calories (1)
Calories ( 8^
joules
Joules ^
calories
(smaU)
1
2
3
4
5
6
7
8
9
10
0.138 26
0.276 51
0.414 77
0.553 02
0.69128
0.829 53
0.967 79
1.106
1.244 3
1.382 6
7.2330
14.466
21.699
28.932
36.165
43.398
50.631
57.864
65.097
72.330
0.001 285 2
0.002 570 3
0.003 855 5
0.005 140 7
0.006 425 9
0.007 7110
0.008 996 2
0.010 281
0.011567
0.012 852
778.10
1 556.2
2 334.3
3112.4
3 890.5
4 668.6
5 446.7
6 224.8
7 002.9
7 781.0
0,002 342 5
0.004 684 9
0.007 027 4
0.009 369 9
0.011712
0.014 055
0.016 397
0.018 740
0.021 082
0.023 425
426.90
853.80
1 280.7
1 707.6
2 134.5
2 561.4
2 988.3
3 415.2
3842.1
4 269.0
0.238 88
0.477 76
0.716 65
0.955 53
1.194 4
1.433 3
1.672 2
1.9111
2.149 9
2.388 8
4.1862
8.3723
12.559
16.745
20.931
25.117
29.303
33.489
37.676
41.862
t In Great Britain this is often called a Board of Trade Unit, or simply
a Unit, and is abbreviated to BTU; these letters also stand for a British
Thermal Unit, which has an entirely different value.
78
TORQUE. — TRACTIVE FORCE.
RELATIONS BETWEEN TORQUE AND ENERQT.
Let foot-pounds, kllogrram -meters, etc., represent units of energ^y,
and let pound-feet, meter-kilogrrams, etc., represent units of torqne;
a radian is the angle whose arc is equal to its radius (about 57^°); torque
acting through an angle gives energy; the general relations between the
units then are:
units of enerjfy = units of torque X radians ;
units of torque = units of energy -r- radians.
The numerical relations between the units bearing similar names are:
1 foot-pound [ft-lbl « 1 p<iund-foot-radian \
1 pound-foot [Ib-ft J = 1 foot-pound per radian;
1 foot-pound =0.159 155 pound -foot-re volution;
1 pound -foot — 6.283 19 foot-pounds per revolution;
1 foot-pound per revolution =0.159 155 pound-foot;
1 pound-foot-revolution » 6.283 19 foot-pounds.
The same relations are true between kilogram-meters of energy and
meter-kllog^ams of torque, or between any other pairs of units having
similar names.
TRACTION ENERGY.
Ton-mile. A unit used in traction calculations representing the energy
(work, not power} which is required to draw one (short) ton of 2 000 lbs.
over a distance oi one mile. It has no fixed value, being dependent upon
the nature of the track, the grade, and the speed. It is never used simi-
larly to the term ' ' foot-pound " as representing one ton raised one mile
vertically. See also note on "poimd per ton" under tractive effort,
below.
Ton-kilometer. A unit similar to "ton-mile," but meaning one metric
ton drawn over a distance of one kilometer.
Ton per mile. A term popularly (though not correctly) used for * 'ton-
mile" (see above). The term "per" is here used incorrectly, as in all
other cases it means that the first quantity is divided by the second, and
not multiplied, as in this case.
Car-mile. A term used in traction calculations representing the energy
required to draw a car one mile. It is analogous to ton-mile (see above),
but is even less fixed in value, as it also involves the weight of the car.
The only fixed relations which these units have are the following:
Logarithm
1 ton-kilometer [t-km] = 0.684 943 ton-mile. Aprx. %8 1.836 6545
1 ton-mile [tn-mlj = 1.459 98 ton-kilometer. Aprx. i%. . . 0.164 8455
(Aprx. means within 2%.)
TRACTIVE FORCE; TRACTIVE EFFORT; TRAC-
TION RESISTANCE; TRACTION COEFFICIENT.
(Force -4- weight.)
The following units, although really of the nature of forces, have been
placed here in order to accompany traction energy.
Pound per ton. A unit used in traction calculations, representing the
force in pounds required to move one (short) ton of 2 000 pounds hori-
zontally against the friction of the rails, wheels, roads, etc. It has no fixed
value, as it varies with this friction. It is really a mere coefficient, rela-
tion, or mere number, and has no dimensional formula. See also note on
"ton-mile" under Energy units.
Kilogrram per ton. A unit similar to "pound per ton," but meaning
the force in kilograms required to draw one metric ton.
The only fixed relations which these units have are the following:
Logarithm
1 pound per (short) ton [Ib/tn]: = 0.500 000 kg per (met.) tn. 1.698 9700
1 kilogram per (met.) ton [kg/t]=>2.000 0(X) lb per (short) ton. Q.SOl 0800
POWER. 79
POWER; RATE of ENERGY; RATE of DOING
WORK; MOMENTUM. (Energy -^ time ; mass X
velocity.)
velocity.)
Units of power are for measuring the rate of doing work, and should
therefore be clearly distinguished from the units of work or heat, which
are energy and not power. Much confusion is often caused by confound-
ing these two terms with each other. Power bears the same relation to
enercy (work, heat, etc.) as a velocity does to length; power is energy
divided by time, just as velocity is length divided by time. A reduction
from power to energy or the reverse therefore always involves the factor
of time.
Tliere are really only two true power units in common use — ^the horse-
power and the watt (or kilowatt) — but powers are also often expressed in
energy per unit of time, as in foot-pounds i>er minute. This table of re-
duction factors is confined in general to the true power imits. A large
number of reduction factors in terms of units of energy have, however,
also been included, but these are in general given here only per mintUe and
not also per second and per hour, as this would have made tne table many
times as long and very cumbersome to use ; a mere multiplication or divi-
sion by 60 will then reduce the energy per minute to its equivalent in energy
per hour or per second, respectively.
In such cases confusion may arise as to whether one should multiply
or divide by 60, owing to the difference between a unit and a quantity
measured in terms of that unit ; the following general reduction factors will
avoid all such confusion:
If W is any imit of energy (such as work or heat) like a foot-pound, heat
unit, etc., then
1 W per hour =%o TT per minute;
" = H 6op W^ per second ;
> = 60 Ir per hour;
1 W per minute «» 60 W per hour;
" ""^o W per second ;
1 W per second = 3 600 W per hour;
= 60 TT per minute.
Thus 120. ft-lbs per min - (120 X 60) =- 7 200. ft-lbs per hour or (120 + 60)
—2. ft-lbs per sec.
For reducing powers which are expressed in energy units per minute (like
ft-lbs per min) to powers expressed m other energy units, but also per min-
tUe (like heat units per min) use the table for the energy units (ft-lbs into
heat units in this case); the element of time then does not enter, as it is the
same in both. If one is per minute and the other per second, they must,
of course, both be first reduced to either minutes or seconds.
Some of these same units also measure momentum, only that they
then mean masses multiplied by velocities. In the units of power the
pounds, kilograms, etc., represent forces, while in the units of momentum
they represent masses.
Power factor is a term used to show the amount of true power con-
tained in a given amount of apparent power. It is the ratio of the true
power to the apparent power. Its use is limited chiefly to electric power
^nerated by alternating currents. With direct electric currents the power
IS equal to the product of the volts and the amperes, and is called watts;
with alternating currents, however, this is true only when the volts and
amperes are exactly in phase with each other, which often is not the case.
When there is such a aifference in phase, that is, when the current lags
behind or precedes the voltage, their product is only apparent power and
is usually measured in volt-amperes. If the true power in such a case is
measured in watts, then the power factor will be the number of watts
divided by the number of volt-amperes, and it will always be less than
imity, in practice usually between about 0.7 and 0.95. For true sine waves
the real power in watts is equal to the voltage X current X cos 6, in which
^ is the angular phase difference; hence it follows that in such cases the
power factor is numerically equal to cos ^, which is found directly from a
table of cosines. Sometimes the power factor is stated in percent, m which
case it is equal to the above figure multiplied by 100.
80 POWER.
Load factor is a term commonly applied to electric, steam, or hvdratilie
power stations to show how much of the total i>os8ible amount ox power
has actually been generated or used during a limited time. It is the ratio
of the mean power used during a limited time (generallv 1 day) divided by
the total power that the station could have generated during that time:
as it is usually stated in percent, this ratio must be multiplied by 100. If
the average power generated during a day, is H of that which the station
is capable of generating, the, load factor is 25%. A 100% load factor
means that the, station js running at its full output all the time. In water-
power installations or in stations having storage batteries, this quantity
is of use in determining the amount of storage capacity desired.
POWER; RATE of ENERGY; RATE of DOING
WORE; MOMENTUM.
Aprx. means within 2%.
Logarithm
1 ergf per second or 1 dyne-centimeter per seoond:
-0.000 000 1 watt 7.000 0000
1 g^am-centimeter p«r seoond [g-cm/sj:
<»0.000 098 059 7 watt. Aprx. ^oooo 5.991 4904
1 foot-g^ratn per seoond [ft-gr/sl:
«0.000 193 675 watt. Aprx. %i + 100 J.287 0740
1 foot-pound per n&lnute [ft-lb/min]:
=- 0.022 595 4 watt. Aprx.%-J-100 3.354 0208
•» 0.011 363 6 mile-pound per hour. Aprx. §^ -♦- 100 2.055 51 74
« 0.000 030 723 4 metric horse-power. Aprx. Ms + 10 000. . 5.487 4691
« 0.000 030 303 horse-power. Aprx. 3-*- 100 000 5.48I 4860
= 0.000 022 595 4 kilowatt. Aprx. % + 100 000 5.354 0208
1 calorie (small) per minute [cal/min]:
=0.069 769 5 watt. Aprx. %oo 2.843 6668
1 kilognntm-meter per minute [kg-m/min]:
0.163 433 watt. Aprx.H.
' -213 3391
=- 0.082 193^2 mile-pound per hour. Aprx. % + 10.! '.!!.! j i.914 3357
=0.000 222 222 metric horse-power. Aprx. % + 1 000 - .346 7876
= 0.000 219 182 horse-power. Aprx. % -h 1 000 i .340 8044
=0.000 163 433 kilowatt. Aprx. H + 1 000 |.213 3891
1 watt [w] or 1 Joule per second:
= 10 000 000« ergs per second 7.000 0000
= 10 197.9 gram-centimeters per second. Aprx. 10 000. 4.008 5096
= 5 163.28 foot-grains per second. Aprx. 5 200 3712 9260
— 44.256 7 foot-pounds per minute. Aprx. % X 100 1.645 9798
— 14.332 9 small calories per minute. Aprx. Vt X 100. . . 1.156 3847
— 6.118 72 kilogram-meters per'minute. Aprx. 6 Q.786 6609
— 0.737 612 foot-pound per second. Aprx. % 1.867 8279
— 0.502 917 mile-pound per hour. Aprx. ^i 1.701 4965
= 0.238 882 small calorie per second. Aprx. 24-{- 100. . . 1.378 1884
— 0.101979 kilogram-meter per second. Aprx. Mo 1008 5096
= 0.056 877 6 thermal unit per minute. Aprx. ^ -»- 10 2-754 9414
= 0.031598 7 Ib-Centgr. heat unit per min. Aprx. ^^ + 100 2.499 6689
= 0.014 332 9 large calorie per minute. Aprx. V^o 2-156 8347
=0.001 359 72 metric horse-power. Aprx. % + 1 000 §.138 4488
=0.001 341 11 horse-power. Aprx. % + l 000 5.127 4658
= 0.001 kilowatt 3.OOO 0000
watts = volt-amperes X cos angle of lag.
▼olt-amperes = volts X amperes.
** = watts + cos angle of lag.
1 foot-pound per second [ft-ib/s]:
= 60. foot-pounds per minute 1.778 1518
= 8.295 32 kilogram-meters per minute. Aprx. % X 10. 0-918 8330
= 1.355 73 watts. Aprx. ^ 0.182 1721
=0.001 843 40 metric horse-power. Aprx. i^ + l 000 §-265 6204
—0.001 818 18 horse-power. Aprx. ^4i -f- 100 |,259 6378
-0.001 355 73 kilowatt. Aprx. % -5- 1 000 §.182 1721
POWER. 81
1 mil«-poand per hour
•=• 88. foot-pounds per minute. Aprx. JiXlOO. . . . 1.044 4827
— 12.166 5 kilogram-meters per minute. Aprx. 12 1.085 1644
— 1.988 40 watts. Aprx. 2 Q-298 5035
-0.002 666 67 horse-power. Aprx. %-»- 1 000 5.425 9687
1 calorie (small) per second [cal/s]— 4.186 17 watts. Ap. 6%a. 0.621 8166
1 kllognram-meter per second Hcg-m/s]:
— 433.980 foot-pounds per minute. Aprx. % XI 000. 2.637 4696
— 60. kilogram-meters per minute 1.778 1513
— 9.805 97 watts. Aprx. 10 Q.991 4904
— 0.013 333 3 metric horse-power. Aprx. %-s- 100 2.124 9387
-• 0.013150 9 horse-power. Aprx. % + 100 2-118 9557
-0.009 805 97 kilowatt. Aprx. 1 -*- 100 §.991 4904
1 thermal unit per n&lnute pb-F/min]:
— 17.581 6 watts. Aprx. % X 10 1.245 0586
—0.023 906 metric horse-power. Aprx. 1%+ 100 3.878 5069
—0.023 578 9 horse-power. Aprx. % -s- 100 2.372 S239
-0,017 581 6 kilowatt. Aprx. %-5- 100 2-245 0586
1 pound-Centlg^ade heat unit per minute pb-C/min}:
— 31.646 9 watts. Aprx. 32 1.600 3811
-0.043 030 8 metric horse-power. Aprx. % -i- 10 5.683 7794
—0.042 442 1 horse-power. Aprx. %-4- 10 2-627 7964
-0.031 646 9 kilowatt. Aprx. 32 -*■ 1 000 2-500 3811
1 watt-hour per minute — 60. watts 1-778 1518
1 calorie (large) per minute [Cal/min]:
— 69.769 5 watts. Aprx. 70 1.843 6658
—0.094 866 7 metric horse-power. Aprx. %i 2.977 1186
—0.093 568 7 horse-power. Aprx. %a 2-971 1806
-0.069 769 5 kilowatt. Aprx. %oo 2-843 6658
1 mile-pound per minute [ml-lb/min]:
— 119.304 watts. Aprx. 120 2076 6548
— 0.162 220 metric horse-power. Aprx. %-*-10 1-210 1081
—0.160 000 horse-power. Aprx. %-^ 10 1.204 1200
--0.119 304 kilowatt. Aprx. %-i-lO 1.076 6548
1 kilogram-lcilometer per minute [kg-km/min]:
— 163.433 watts. Aprx. HXl 000 2.218 3891
—0.222 222 metric horse-power. Aprx. % 1.346 7875
—0.219 182 horse-power. Aprx. %. 1.840 8044
—0.163 433 kilowatt. Aprx. ^ 1.218 8391
1 metric horse-power [hpl or French horse-power or
ehevalvapeur or force de cheval or Pferde-lcraf t :
— 7.354 48X 10» ergs per second. Aprx. ^X 10» 9.866 5517
— 32 548.5 foot-pounds per minute. Aprx. 33 000 4-512 5309
— 4 500. kilogram-meters per minute. Aprx. % X 1000 3-653 2125
— 735.448 watts. Aprx. ^ X 100 2-866 5517
— 642.475 foot-pounds per second. Aprx. %i X 1 000 . . 2-784 3797
— 76. kilogram-meters per second or H X 100 1-875 0613
— 41.830 5 thermal units per minute. Aprx. 42 1.621 4931
— 23.239 2 lb-C!tg. heat units per minute. Aprx. %X 10. 1.866 2206
— 10.541 1 large calories per minute. Aprx. *H 1-022 8864
— 0.986 318 horae-power. Aprx. 1 1-994 0170
— 0750 000 poncelet 1.875 0618
— 0.735 448 kilowatt. Aprx. ^-^10 1.866 5517
1 horse-power [hpl:
— 7.456 50 X 10^ ergs per second. Aprx. H X lO^o 9-872 5348
— 33 000. foot-pounds per min. Aprx. H X 100 000. . . 4518 5189
— 4 562.42 kg-meters per minute. Aprx. %Xl 000 3-659 1956
— 745.650 watts. Aprx. ^X 1 000 2872 5848
— 550. foot-pounds per second. Aprx. *>^ X 1(X) . , . 2-740 3627
— 375.000 mile-pounds per hour. Aprx. HXl 000.. . . 2. 574 0313
— 76.040 4 kg-meters per second. Aprx. ^X 100 1.881 0444
— 42.410 8 thermal units per min. Aprx. % X 100 1.627 4762
— 23.561 5 Ib-Ctg. heat units per min. Aprx. % X 10 . . 1.372 2037
— 10.687 3 large calories per min. Aprx. ^ 1028 8695
— 1.013 87 metric horse-powers. Aprx. 1 0.005 9830
— 0.760 404 poncelet. Aprx. H 1881 0444
» 0J45 650 kilowatt. Aprx. H 1-872 5348
82
POWER.
poncelet=» 100. kilogram -meters per second 2*000 0000
" — 1.333 33 metric horse-powers. Aprx. % 0>124 9887
" = 1.315 09 horse-powers. Aprx. % Q.118 9557
= 0.980 597 kilowatt. Aprx. 1 1.091 4904
kilowatt [kw]= 1 X 10^ ergs per second 10-000 0000
= 44 256.7 ft-lbs per min. Aprx. %X 100 000.. 4.845 9798
•* =6 118.72 kg-met. per min. Aprx. 6X1 000. . 8-786 6609
" =1 000. watts 8000 0000
*' = 737.612 ft-lbs. per sec. Aprx. MX 1 000... . 2-867 8279
" =101.979 kilogram-metr. per sec. Aprx. 100. 2-008 5096
" =56.877 6 thermal u. per min. Aprx. ^ X 100. 1.764 9414
*' =31.598 7 Ib-Ctg. heat u. per min. Aprx. «%.. 1.499 6689
= 14.3329 large cal. per min. Aprx.^XlOO.. 1.156 3347
*' =1.359 72 metric horse-powers. Aprx. add ^. 0-183 4483
'* =1.341 11 horse-powers. Aprx. add H 0-127 4652
= 1.019 79 poncelets. Aprx. 1 0008 6096
watt-honr per second = 3 600. watts. Aprx. ^HXl 000. . . 8-556 3025
= 3.600 kilowatts. Aprx. ^M 0-556 3025
metric liorse-power-hoar per minute:
=60. metric horse-powers 1-778 1513
horse-power-hour per minute =60. horse-powers 1-778 1513
kilowatt-hour per minute = 60. kilowatts 1-778 1513
metric horse-power-liour per second:
= 3 600. metric horse-powers. Aprx. ^M X 1 000 8-556 3025
horse-power-hour per second:
= 3 600. horse-powers. Aprx. ^MX 1 000 8-556 3025
kilowatt hour per second;
= 3 600. kilowatts. Aprx. ^MX 1 000 8-556 8025
Conversion Tahles for Power.
Horse-powers =
Metric hp's =
kilowatts .
metr hp's
kilowatts
horse-
Kilowatts =
horse-
powers
metr hp's
powers
1
2
3
4
5
6
7
8
9
10
0.745 65
1.491 3
2.237
2.982 6
3.728 3
4.473 9
5.2196
5.965 2
6.710 9
7.456 5
1.341 1
2.682 2
4.023 3
5.364 4
6.7056
8.046 7
9.387 8
10.729
12.070
13.411
0.735 45
1.470 9
2.206 3
2.941 8
3.677 2
4.412 7
5.1481
5.8836
6.6190
7.354 5
1.359 7
2.719 4
4.079 2
5.438 9
6.798 6
8.158 3
9.518
10.878
12.237
13.597
1.013 9
2.027 7
3.041 6
4.055 5
5.069 4
6.0832
7.097 1
8.1110
9.124 8
10.139
0.986 32
1.9726
2.9590
3.945 3
4.931 6
5.917 9
6.904 2
7-890 5
8.876 9
9.863 2
FORCES. 83
FORCES; WEIGHTS Considered as Forces. (See also
Weights.)
Only true units of force (dsmes and poundals) are given here, together
with tneir values in terms of weights, and the reciprocals of these values.
The relations between two weights when considered as forces are, of course,
the same as when they are considered as masses; these have been given
under Weights and are therefore not repeated here.
A true unit of force is independent of the value of gravity and is the
same throughout the universe. But a weight considered as a force in-
cludes the value of gravity and is therefore different for different values
of gravitv. The value of gravity used in these tables is 980.596 6 (see
note on this value imder the units of Acceleration).
A dyne is that force which, acting on a mass of one gram for one second,
produces a velocity of one centimeter per second; this definition refers
to a space which is free from the attraction of other bodies. A poundal
is similarly that force which, acting on a mass of one pound for one second,
produces a velocity of one foot per second.
The attraction of gravity of the earth is really a force, but it cannot be
used as a unit of force, as the amount of this force which acts on anybody
depends on the mass on which it acts. When reduced to the force per
gram mass, it becomes the same thing as the force represented by one gram
considered as a weight, and this together with all similar values is given
in the table. The attraction of gravitation becomes a constant quantity
when it is stated as an acceleration, as in this form it is independent of
the mass on which it acts; it can then be'used as a unit and is included
as such in the table of accelerations, which see for its reduction factors.
For tractive forces or tractive efforts, see under this title at the end of
units of Energy.
Aprx. means within 2%.
Logarithm
1 microdyne^ 0.000 001 dyne B-OGG 0000
1 mtlllgn^am » 0.980 596 6 dyne. Aprx. 98 -^ 100 f.ggi 49G4
= 0.000 070 926 5 poundals. Aprx. 7 + 100 000. . . 5.85G 8088
1 djme— 1.019 79 milligrams. Aprx. 1 0008 5096
" =» 0.015 737 7 grain. Aprx. i^ + 100 5.196 9418
" = 0.001 019 79 gram. Aprx. 1 -«- 1 000 j.008 5096
•• =. 0.000 035 971 9 ounce (av.). Aprx. 4 -s- 110 000 5555 9637
•* =» 0.000 072 330 poimdal. Aprx. % -J- 10 000 5-859 3184
" =0 000 002 248 25 pound (av.). Aprx. % -s- 1 000 000 6351 8487
1 grrain = 63.541 6 dynes. Aprx. Til X 100 1.803 0582
=•0.004 595 96 poundal. Aprx. 46 + 10 000 §.662 8766
1 8^am»980.596 6 dynes. Aprx. 1 000 2991 4904
= 0.070 926 5 poundal. Aprx. 7 -s- 100 2.850 8088
1 kilodyne = 1 000. dynes 3000 0000
1 my I'iadyne = 10 000. dynes 4-000 0000
1 poundal =■ 14 099.1 milligrams. Aprx. V^ X 100 000 4.149 1912
«- 13 825.5 dynes. Aprx. ^HXIO 000 4.I4O 6816
— 217.582 grains. Aprx. 220 2337 6284
— 14.099 1 grams. Aprx. Vj X 100. 1.149 1912
— 0.497 331 ounce (av.). Aprx. H- ■ ' 1-696 6453
=0.031 083 2 pound. Aprx. 31-1-1 000 2492 5253
= 0.014 099 1 kilogram. Aprx.V^o 2-149 1912
1 ounce (av.) = 27 799.5 dynes. Aprx. i,^ X 10 (X)0 4-444 0368
" =2.010 73 poundals. Aprx. 2 0-303 3547
I pound (av.)= 444 791. dynes. Aprx. % X I 000 000 5.648 1568
= 32.1717 poundals. Aprx. 32 1-507 4746
1 kilogram =980 596.6 dynes. Aprx. 1 000 000 5991 4904
= 70.926 5 poimdals. Aprx. 70 1.850 8088
'• =0.980 597 megadyne. Aprx. 1 I.99I 4904
1 megadjme = 1 000 000. dynes 6000 0000
= 72.330 poundals. Aprx. %X 100 1-859 3184
= 2.248 25 pounds. Aprx. % 0351 8437
= 1.019 79 kilograms. Aprx. 1 0008 5096
1 IXg?.?;^ P^r°on} «* underr^active Force-, p. 78.
4t
• 4
• *
• t
4t
44
tt
44
•4
84 MOMENTS OF INERTIA.
MOMENTS of INERTIA. (Mass X square of length.)
The moment of inertia of a body is its mass multiplied by the square of
the radius of gyration ; it must therefore always refer to some axis of rota^
tion. It represents that weight which when concentrated at a unit distance
from the axis of rotation would require the same energy to cause a eiven
increase in its angular velocity, that the body itself requires. It bears
the same relation to angular acceleration as weight bears to linear acceler-
ation.
Moments of inertia (so-called) are frequently used in calculating the
strength of beams of various cross-sections. In such cases the formulas
usually represent the moments of inertia of each of differently shaped cross-
sections, and the mass is then usually represented by its cross-sectional
area, as the formulas are then the same for all materials provided only that
the shape of the cross-section is the same; the numbers thus obtained are
often (though not correctly) called the moments of inertia; they might
better be called the specific moment of inertia of the respective cross-section.
The figure obtained from such a formula must then be multiplied by the
mass (weight) of a cube (of unit side) of the material to give the true mo-
ment of inertia, if that is required, of a slab or section having a thickness
of one unit of length. In calculations involving the ratio of two differ-
ent moments of inertia, the weight or mass need not be introduced pro-
vided the material is the same in both ; in all such cases the figures obtained
from the formulas just mentioned can be used directly. In such cases the
relation between the units in which the moments of inertia are expressed
is as the fourth power of the respective linear imits.
MOMENTS of INERTIA in terms of the mass.
Logarithm
1 unit in pounds and inches:
= 2.926 41 units in kilograms and centimeters 0*466 8860
1 unit in kilograms and centimeters:
»0.341 716 unit in pounds and inches , I'688 6650
MOMENTS of INERTIA In terms of the surface.
1 unit in inches = 41 .623 5 units in centimeters 1-619 8384
1 unit in centimeters = 0.024 024 9 unit in inches 2880 6616
Thus if the moment of inertia (so-called) of the cross-section of a betun,
for instance, has been calculated in inches and square inches and is then
multiplied by 41.62, the result would be the same as if the moment of
inertia of the same cross-section had been calculated in centimeters and
square centimeters.
MOMENTS of MOMENTUM ; ANGULAR MOMENTUM.
(Momentum X length.)
/Hiese units are simply those of momentum multiplied by a length, and
as they are seldom used it is not necessary to give them in a separate table.
The units of momentum are given above (see under Power); tne length by
which they are to be multiplied must of course be in terms of the same unit
as the one already included in the respective unit of momentum. The
moment of momentiim is also equal to the moipent of inertia divided by
LINEAR velocities; SPEEDS. 85
UNBAR VELOCITIES; SPEEDS. (Length -^ time.)
For simple reductions or relations between lengths, see units of Length.
Aprx. means within 2%.
Logarithm
I foot per minute [ft/min]:
» 0.304 801 meter per minute. Aprx. Ho I-484 0158
=0.018 288 kilometer per hour. Aprx. %i + 10 2-2e2 1671
=0.016 666 7 foot per second. Aprx. H -i- 10 5.221 8487
=0.011 363 6 mile per hour. Aprx. 9^-<-100 2055 5174
1 kine » 1 centimeter per second 0-000 0000
I centimeter per second [cm/s]:
"- 1 kine 0-000 0000
=0.01 meter per second, which see for other values 2-000 0000
1 meter per minute [m/min]:
= 3.280 83 feet per minute. Aprx. i% 0-515 9842
= 0.06 kilometer per hour 3-778 1518
=-0.054 680 6 foot per second. Aprx. ^^-i-lOO 5-737 8329
=0.037 282 2 mile per hour. Aprx. ^-i- 10 3.571 5016
1 kilometer per hour [km/hr]:
=- 54.680 6 feet per minute. Aprx. ^HX 10 1.787 8829
= 16.666 7 meters per minute. Aprx. ^ X 100. 1-221 8487
=0.911 343 foot per second. Aprx. i%i 1.959 6816
—0.621 370 mile per hour. Aprx. ^. . 1.793 8508
=0.539 611 knot (Brit.) per hour. Aprx. %i 1.782 0806
=0.639 693 knot (U. S.) per hour. Aprx. ^i 1.782 0660
1 foot per second [ft/s]:
— 60. feet per minute 1-778 1518
= 18.288 meters per minute. Aprx. ^HX 10 1.262 1671
= 1.097 28 kilometers per hour. Aprx. add ^io 0040 3184
= 0.681 818 mile per hour. Aprx. 68^100 1.833 6687
= 0.592 105 knot (Brit.) per hour. Aprx. %o 1-772 8990
=. 0.592 085 knot (U. S.) per hour. Aprx. %a 1.772 3844
= 0.304 801 meter per second. Aprx. %o 1-484 0158
= 0.018 288 kilometer per minute. Aprx. 'H-*- 100 5.262 1671
=0.011 363 6 mile per minute. Aprx. %-i- 100 3-055 5174
1 mile per kour [ml/hr]:
= 88. feet per minute 1-944 4827
= 26.822 4 meters per minute. Aprx. % X 10 1.428 4984
= 1.609 35 kilometers per hour. Aprx. % 0-206 6497
= 1.466 67 feet per second. Aprx. i% 0-166 3318
= 0.868 421 knot (Brit.) per hour. Aprx. Vs T.988 7808
= 0.868 392 knot (U. S.) per hour. Aprx. J^ 1.938 7157
= 0.447 041 meter per second. Aprx. % -s- 10 1.650 3471
= 0.026 822 4 kilometer per minute. Aprx. % -!- 100 3-428 4984
=0.016 666 7 mile per minute. Aprx. M + 10 3-221 8487
1 knot (Brit.) per hour :
= 1.853 19 kilometers per hour. Aprx. ^H 0-267 9194
— 1.688 89 feet per second. Aprx. i% 0-227 6011
= 1.151 62 miles per hour. Aprx. add ^ 6-061 2697
=0.999 966 knot ( U. S.) per hour. Aprx. 1 1.999 9854
1 knot (U. S.) per hour :
= 1.853 26 kilometers per hour. Aprx. ^H 0.267 9340
— 1.688 94 feet per second. Aprx. i% 0-227 6157
= 1.161 65 miles per hour. Aprx. add Vr 0-061 2848
= 1.000 034 knots (Brit.) per hour. Aprx. 1 0000 0146
1 meter per second [m/s]:
= 196.860 feet per minute. Aprx. 200 2-294 1355
— 100. centimeters per second 2-000 0000
= 60. meters per minute 1-778 1518
= 3.6 kilometers per hour 0-556 3025
= 3.280 83 feet per second. Aprx. i% 0-515 9842
= 2.236 93 miles per hour. Aprx. % 0-849 6529
= 06 kilometer per minute 3-778 1518
=0.037 282 2 mile per minute. Aprx. K'7 3-571 5016
86 ANGULAR velocities; rotary speeds.
1 kilometer per minute [km/min]:
= 54.680 6 feet per second. Aprx. 55 1.737 8329
= 37.282 2 miles per hour. Aprx. 37 1.571 6016
= 16.666 7 meters per second. Aprx. H X 100 1-221 8487
= 0.621 370 mile per minute. Aprx. ^ 1.793 3503
1 mile per minute [ml/mini:
=B 88. feet per second 1.944 4827
= 60. miles per hour 1.778 1618
»26.822 4 meters per second. Aprx. 27 1-428 4984
— 1.609 35 kilometers per minute. Aprx. % 0.206 6497
Miscellaneous concrete units: Average velocity of molecules
about 500. meters per second (Woodward). Velocity of
light about 300 000. kilometers per second.
ANQULAR VELOCITIES; ROTARY SPEEDS.
(Angle -7- time.)
For simple reductions or relations between angles, see values under Angles.
Aprx. means within 2%.
Angular velocity s angle moved through divided by time.
* * * * in degrees per second = angle in degrees -f- time in seconds.
= revolutions per second X 360.
'* "in revolutions per minute = revolutions-*- time in minutes.
" '* in radians per 8econd»2;rX revolutions per second.
Logarithm
1 revolution per hour [rev/h or rph]:
=0.1 degree per second I-OOO 0000
1 radian per minute:
=« 9.549 30 revolutions per hour. Aprx. *% 0.979 9714
= 0.954 930 degree per second. Aprx. subtract J^ f.979 9714
= 0.002 652 58 revolution per second. Aprx. % -J- 1 000 3.423 6689
1 degree per second:
=> 10. revolutions per hour 1 QOO 0000
= 1.047 20 radians per minute. Aprx. add >^.. 0.020 0287
=« 0.166 667 revolution per minute or H 1.221 8487
=• Hw or 0.002 777 78 rev per second. Aprx. %i -<- 100 g.443 6975
1 revolution per minute [rev/min or rpm]:
— 6. degrees per second 0.778 1513
= 0.104 720 radian per second. Aprx. 2^^-5-100 1-020 0287
= 0.016 666 7 rev per second or Ho 3.221 8487
1 radian per second [oi]:
= 57. 296 8 degrees per second. Aprx. 57 1.768 1226
= 0.159 155 revolutions per second Aprx. i%oo 1-201 8201
1 revolution per second [rev/s or rps]:
= 60. revolutions per minute 1-778 1513
= 6.283 185 radians per second. Aprx. ^X 10 0-798 1799
FREQUENCY; PERIODICITY; PERIOD; ALTERNA.
TIONS. (l-^time; time.)
Frequency or periodicity is the number of recurrences of some periodic
or wave motion during a given time; this time is always understood to
be a second unless otherwise stated; the frequency always refers to the
number of complete waves. The '* number of alternations,*' however,
refers to the number of changes of the direction of the motion or to the
reversals, and therefore refers to half waves, and is always equal to double
the frequency, if the time is the same. The period is the time of one com-
plete wave or oscillation and is therefore the reciprocal of the frequency.
The term frequency is the one most generally used, and always refers to a
second; the term number of alternations is unfortunately preferred by
some and when it refers to electric currents the time is usually a minute;
the term period is used comparatively rarely as a measure, its use being
generally limited to scientific discussions.
FREQUENCY. — LINEAR ACCELERATIONS. 87
Tn mathematical discussions of electric alternating-current problems the
frequency is often replaced by an angular velocity, generally represented
by at and measured in radians per second (see under Angular Velocities
above). Then a> =>2 Tin, in which a; is in radians per second and n is the true
frequency in cycles per second, a cycle being here considered the same
thing as a complete revolution.
The freauencv is also equal to the velocity of propagation divided by
the wave lengtn. The wave lengths are therefore measured in units of
length, but when the velocity^ for a class of waves is a constant (as those
of light or the electromagnetic waves), the wave lengths may also be in-
dicated in units of time, in which case a wave length becomes equal to the
period of the wave. Wave length should not be confounded with the
amplitude, which latter measures the intensity of the wave and has noth-
ing to do with the frequency, period, or wave length.
If n is the frequency per second [f^], then:
the period in seconds = l/n;
the number of alternations per minute = 120n.
If n is the number of alternations per minute, then:
the frequency per second = n/120 ;
the period in seconds = 120/n.
If n is the period in seconds, then:
the frequency per second = l/n;
the number of alternations per minute = 120n.
If n is the frequency in cycles per second, to the angfular velocity in
radians per second, and if a cycle is represented by one revolution, then:
oi = 2itn; or
n=0.159 165 w:
6> « 6.283 19 n
An electrical degree is the 360th part of one complete cycle.
IiINEAR ACCELERATIONS; RATE of INCREASE in
VELOCITIES; QRAVITT. (Velocity -^ time.)
The mean value for "gravity'' which has been taken as a basis in all
these tables is an acceleration of 9.S05 966 meters per second per second,
at sea-level and in latitude 45°. This is probably the best available mean
value and is known as Helmert's value (Die math. u. phys. Theorien der
hoehern Geodaesie; II, p. 241 ; 1884). It is used up to date by the Inter-
national Geodetic Association, according to its latest publisned report.
No value has been adopted by the National Bureau of Standards. At the
International Bureau of Weights and Measures the value 9.809 91 is taken,
from which the normal value at sea-level and in latitude 45** would be
9.806 65; the difference is only about 7 in one hundred thousand.
This value enters into many figures involving relations between forces
and masses, as, for instance, in the relation between watts and horse-power,
because a watt is defined in terms of a dyne (that is, a true force, which is
independent of gravity), while a horse-power is defined in terms of pounds
or kilograms (that is, masses which must be multiplied by the value of
gravity to be reduced to true forces comparable with dynes). The value
of gravity, however, falls out in all figures involving the relation between
two masses when both are considered as forces, as, for instance, in the rela-
tion between foot-pounds and kilogram-meters, which involve pounds and
kilograms considered as forces. It does not enter at all in the inter-rela-
tions between watts, joules, ergs, and dynes, as these are the same through-
out the universe. Pounds, kilograms, etc.. are really masses, and as such
are the same throughout the universe, but when considered as weights
or forces their values depend on gravity and are slightly different on differ-
ent parts of the earth ; when measured on a beam balance by comparison
with other weights, their values would remain the same, but when meas-
ured on a spring balance their values would change slightly, as the spring
measures the force of gravity acting on the masses.
The "gravity" here referred to, which is an acceleration and p>ertains
only to our earth, should not be confounded with what is called the "gravi-
tation constant" or "Newton's constant," which pertains to the whole
universe, and is the constant by which one must multiply the product of
88 LINEAR AND ANGULAR ACCELERATIONS.
the masses of two bodies divided by the square of their distance apart, in
order to get the force of attraction between them.
Aprx. means within 2%.
Logarithm
1 kilometer per hoar per min. [km/hr/min](or per mln* per hoar):
= 0.016 666 7 or ^0 kilometer per hour per second (or per
second per hour) which see for other values. 5.221 8487
1 mile per hoar per minute [ml/hr/min] (or per minate per hoar):
=0.016 666 7 or ^o niile per hour per second (or per sec. per
hr.), which see for other values 5.221 8487
1 centimeter per second per second [cm/s^:
= 0.036 kilometer per hour per second (or per second
per hour) 3556 8025
= 0.032 808 3 foot per second per second. Aprx. M + 10. . . 2515 8842
=» 0.022 369 3 mile per hour per second (or per second per
hour). Aprx. % -5- 100 2.849 6529
= 0.001 019 79 gravity. Aprx. % oqo 5-008 5096
1 Icilometer per hoar per second [km/hr/s] or per second per hoar):
== 27.777 8 centimeters per second per second. Aprx. 28 1.448 6975
= 0.911 343 foot per second per second. Aprx. subtr. ^i . I.959 6816
= 0.621 370 mile per hour per second (or per second per
hour). Aprx. ^ 1.793 3503
=» 0.277 778 meter per second per second. Aprx. ^11 1443 6975
= 0.028 327 4 gravity. Aprx. % -5- 10 2.452 2071
1 foot per second per second [ft/s^:
= 30.480 1 centimeters per second per second. Aprx. 30. . 1.484 0158
= 1.097 28 kilometers per hour per second (or per second
per hour). Aprx. add Mo 0040 3184
= 0.681 818 mile per hour per second (or per second per
hour). Aprx. *Ho 1.833 6687
= 0.304 801 meter per second per second. Aprx. %o 1.484 0158
= 0.031 083 2 gravity. Aprx. 31 -J- 1 000 §.492 5254
1 mile per liour per second [ml/hr/sec] or per second per hoar:
= 44.704 1 centimeters per sec per sec. Aprx. % X 10. . . . 1.650 3471
= 1.609 35 kilometers per hour per second (or per second
per hour). Aprx. % ^ 0206 6497
= 1.466 67 feet per second per second. Aprx. *%o 0166 3313
= 0.447 041 meter per second per second. Aprx. %-*-10. . 1.850 3471
= 0.045 588 6 gravity. Aprx. %-i-lOO 5.658 8567
1 meter per second per second [m/sec^]:
= 3.6 kilometers per hr. per sec. (or per sec. per hr.) . . 0.556 3025
= 3.280 83 feet per second per second. Aprx. |^X 10 0.515 9842
= 2.236 93 miles per hr per sec (or per s per hr). Aprx. %.. Q-849 6529
= 0.101 979 gravity. Aprx. Vio 1.008 5096
Gravity = 980.596 6 centimeters per sec per sec. Aprx. 1 000. . 2991 4904
= 35.301 5 kilometers per hour per second (or per
second per hour). Aprx. %X 10 1-547 7929
= 32.171 7 feet per second per second. Aprx. 32. . . 1-507 4746
= 21.935 3 miles per hr per sec (or /sec/hr). Aprx. 22 I.341 1433
= 9.805 966 meters per second per second. Aprx. 10. 0-991 4904
«<
n
ANGULAR ACCELERATIONS; RATE of INCREASE
in ANGULAR VELOCITIES. (Angular velocity -^ time.)
Logarithm
1 revolation per minate per minate [rev/min/min]:
= 0.016 666 7 or J^o rev per min per sec (or per sperm).. 3.2218487
= 0.000 277 778 or H 600 rev per s per s. Aprx. %i -5- 1 000. . I.443 6975
1 rcTolution per minute per second [rev/min/s] or per
second per minute:
= 60. revolutions per minute per minute 1.778 1513
=0.016 666 7 or }4o revolution per second per second 2.221 8487
1 reTolation per second per second [rev/s/s]:
=3 600. revolutions per minute per minute 8.556 8025
— 60. rev per minute per second (or per sec per min.) .... 1.778 1513
1 1
« t
ANGLES. 89
ANGLES (plane); dROULAR MEASURIS.
Aprs, means within 2%. Logarithm
1 second ("1=0.016 666 7 minute, or %o 2221 8487
1 minatet'j s 60. seconds 1-778 1513
. = 0.016 666 7 degree.or %o 3221 8487
= 0.000 290 888 radian i.463 7261
•= 0.000 185 185 quadrant 1.267 6062
=0.000 046 296 3 circumference $.665 5462
1 grade =0.01 right angle S-OOO 0000
1 degree i**l:
— - 3 600. seconds 3.556 8025
— 60. minutes 1.778 1513
— 0.017 453 3 radian. Aprx. %-5- 100 2-241 8774
-« 0.011 111 1 quadrant, or Vw 5-045 7575
«=0.005 555 56 x's (considered as an angle of 180°). or Mso • . 3-744 7275
—0.002 777 78 circumference, or H«o 3-443 6975
1 radian:
spangle whose arc equals radius.
— 206 265. seconds 5.314 4252
— 3 437.75 minutes. Aprx. 3400 3-536 2739
— 57-295 8 degrees. Aprx. 57 1-758 1228
-=0.636 620 quadrant. Aprx. Hi 1-803 8801
—0.318 310 jr's (considered as an angle of 180*). Ap. »^ioo . . 1-502 8501
—0.159 155 circumference. Aprx. i%oo-" 1-201 8201
1 quadrant or rlg^ht angle:
— 100. grades 2-000 0000
— 90. degrees 1-954 2425
— 1.570 80 radians. Aprx. ^^ 0196 1199
— 0.5 KB (considered as an angle of 180°), or H 1-698 9700
— 0.25 circumference, or M 1-397 9400
ir (as an angle) or senii-circumfereuee:
— 180. degrees 2-255 2725
— 3-141 59 radians. Aprx. 2% 0-497 1499
— 2. quadrants 0-301 0300
— 0.5 circumference, or H I-698 9700
1 clrcamference or reToIntion [rev],
— 21 600. minutes 4334 4538
— 360. degrees 2-556 3025
—6.283 185 radians. Aprx. %X 10 0798 1799
— 4. quadrants 0-602 0600
— 2. ic'b (considered as an angle of 180°) 0-301 0300
1 electrical degree— the 360th part of a cycle; see p. 121.
SOLID ANGLES. (Surface -h radius.)
A solid angle is an angle, like that at the point of a cone, which is sub-
tended by a spherical surface. The unit solid angle is that angle which,
at the center of a sphere of unit radius, subtends a imit area on the surface
of the sphere; this unit is sometimes called a steradian. A spherical
right an^le is assumed to be an angle, like at the comer of a rectangular
block, which is bounded by three planes perpendicular to each other.
Logarithm
1 unit— 0.636 620 spherical right angle, or 2/ff 1-803 8801
" — 0.159 155 hemisphere, or 1 -5-2»r 1.201 8201
*• —0.079 577 5 sphere, or 1 -^4;: 2.900 7901
1 steradian — 1 unit solid angle; (see above) 0-000 0000
1 spherioal right angle — 1.570 80 units, or jc/2 0-196 1199
'* — 0.25 hemisphere, or 14 1-397 9400
•* = 0.125 sphere, or >s 1-096 9100
1 hemisphere— 6.283 19 units, or 2;r 0-798 1799
** — 4. spherical right angles 0-602 0600
•• = 0.5 sphere, or M 1-898 9700
1 sphere — 12.566 4 units, or 4ic 1099 2099
— 8. spherical right angles 0-903 0900
B 2. hemi^heres 0-301 0300
90 grades; slopes; inclines.
QRADES ; SLOPES ; INCLINES. (Angle ; length -^length.)
Grades are iudicated in terms of so many different kinds of units, some
of which involve trigonometric relations, that the relations between some of
them become complicated. Those given in the following table are mathe-
matically correct and are strictly proportional; they may therefore be
used like those in the other tables; for instance, a 1% grade from the
table is 62.8 feet rise per mile, hence a 5% grade will be 5 times this; or
from the table, 1 foot p>er mile is a 0.018 9% grade, hence 100 feet p>er mile
will be a 1.89% grade. •
Much unnecessary labor and confusion would be avoided if grades were
uniformly represented in percent, that is, in the rise per himdred. There
seems to be a tendency to adopt this unit generally. In the following
table all the relations are given in terms of the percent unit as a basis; the
relation between any two others is readily found by reducing both to the
same percentage value.
In using percentage values it should be remembered that they mean
the rise per liiiiidred, and it may therefore be necessary sometimes to
multiply or divide by 100 when the actual or total distances or rises are
involved. Thus a rise of 12 feet in 600 feet is a 2% grade, as the 600 feet
must first be divided by 100 to reduce it to hundreds, before dividing it
into 12. Or if the total rise on a 2% grade is given as 12 feet, it means
12-4-2 = 6 feet per hundred, which must therefore be multiplied by 100
to get the total distance.
Much confusion arises from the incorrect way in which percentage values
are not infrequently written. Thus fifty percent should be written 50%
and not .50%, which latter means half of one percent, or fifty hundredths
of one percent.
Much confusion also arisies from the fact that sometimes the sloping or
inclined distance is meant instead of the horizontal distance, and gener-
ally it is not stated which one is understood. In referring to profiles or to
distances on a map, the horizontal distance is always understood, thus
involving the tangent of the angle; while in formulas for the traction on
grades, or when the distances are the actual lengths of track or road, the
sloping or inclined distance is generally implied, as the traction formulas
then become simpler; they then involve the sine of the angle. The pres-
ent table gives the correct reduction factors for both. Some of these are
the same in both systems; but it should be understood that a grade of a
given percent based on horizontal distances is slightly different from a
grade of the same percent based on sloping distances ; the latter is alwavs
the larger angle or steeper grade. The difference is however ^nerally
negligibly small for all but exceptionally steep grades; up to a 14% grade,
which is about the limit for traction on rails and for the usual roads, the
difference is less than 1%.
The following reduction factors are the same whether the units are
based on the horizontal or on the sloping distances.
Logarithm
1 inch permilerin/ml] =0.001 578 28% 3.198 1849
1 foot per mile [ft/ml] = 0.018 939 4% 2.277 3661
1^00 = 0.1% I-OOOOOOO
1 per mil [0/(»o] = 0.1% I.QOO 0000
1 millimeter per meter [mm/m] — ^1% lOOO 0000
1 foot per thousand feet [ft/M] = 01% lOOO 0000
1 foot per 100 feet [ft/C] = 1.% 0000 0000
1 foot rise per foot [ft/ft] = 100.% 2 000 0000
1% =633.6 inches per mile 2-801 8152
52.8 feet per mile 1.722 6839
10. %o, or lOper mil 1.000 0000
10. millimeters per meter 1-000 0000
10. feet per thousand feet 1.000 0000
1. foot per hundred feet 0-000 0000
0.01 foot rise per foot 2-000 0000
it
<(
It
<i
It
It
grades; slopes; inclines. 91
n miles per foot rise — 1/n feet rise per mile.
= (0.018 939 4 -*-n)%.
n% « (0.018 939 4 +n) miles per foot rise,
n feet (rise) per mile — 1/n miles per foot rise.
n feet per foot rise = 1/n feet (ri.se) per foot.
= (100/n)%.
n% =(100/n) feet per foot rise,
n foot (rise) per foot = 1/n foot per foot rise.
The following reduction factors are only for units based on the hori-
zontal distances.
n%» % based on sloping distances.
V10024-n2
n%-100Xsin.
n degrees rise » (100 tan n)%.
n% •-' number of degrees whose tangent is O.Oln.
If n is the tangent of angle, then the rise in % a lOOn.
If n is the rise in %, then the tan » O.Oln.
1 AfV|«
If n is the sine of angle, then the rise in % = — , Or find from a
« VI -n2
table the tangent corresponding to this sine, then the rise in % »■ 100 X tan.
n
If n is the rise in %,then the sine of tlie angle » . Or
Vi002+n2
the sine may be found from a table as that corresponding to tan = 0.0 In.
The following reduction factors are only for units based on the sloping
distances.
n% — — % based on horizontal distances.
V1002-n2
n%-100Xtan.
n degrees rise = (100 sin n)%.
n% = number of degrees whose sine is O.Oln.
If n is the sine of angle, then the rise in % a>100 n.
If n is the rise in %, then the sine =0.01 n.
If n fs the tangent of angle, then the rise in %^—jr==. Or find
vi+n2
from a table the sine corresponding to this tangent, then the rise in % =
100 X sin.
n
If n is the rise in % , then the tangent of the angle =• . — . Or
V1002-n2
the tangent may be found from a table as that corresponding to sine = O.Oln.
Approximate: for small angles and for most engineering calculations
concerning grades, the sine and the tangent are very nearly equal, hence
the simpler of the formulas given above can in most cases be used for both,
and no tables are then necessary. Up to a 14% grade (about 8 degrees)
the error made thereby is less than 1%.
The actual values given below in the fifteen-colimin table avoid the
calculations with the above reduction factors. Intermediate values suffi-
ciently accurate for most purposes may be found from this table by
ordinary interpolation.
i
1
.1""
t
i
grades; slope
3; INCLINES.
§||8| slip ^1=3=
1
88SSS =l:SS. „— »„
slililSi§§S§S=3
0660a ddood dodod
l1ll
"s^'isbs "a'-SS'j Sia'-ls
fe°-2.M. %,%%Z,i, l=i?-So&
I-
Equivalent Percent
IliSllipllfl;
S SSggS =2=35
= ooodd ddc'dd
ft. - ssaae
8 Kssss isassis
liilissssss
= ooood 66666
ls3lgiSiiS|
isllsSSilcS
SSSSS2=223S
d 00000 ddddd
PerMLl(%,)ot
or Feet per two Feet.
SgS5SgSSgS2§gSg
Feet per lf& Feet.
-""'"•—-2=2233
TIME. 93
TIMB.
There are in use two di£Ferent eorstems of units of time, the mean solar
time and the sidereal time. The mean solar time is based on the appar-
ent motion of the sun with respect to the earth, that is, on the motion of the
earth with respect to the sun. The sidereal tune is based on the appar-
ent motion of the stars with respect to the earth, that is, on the motion of
the earth with respect to the stars.
The mean solar time is that indicated by the clocks in common use;
it ifl that which is furnished by the JJ. S. Naval Observatory and is used
in all physical research. In all derived imits in use, such as velocities,
forces, power, electrical quantities, etc., which involve the element of time,
this mean solar time is understood to be meant. Accordingto the National
Bureau of Standards, Lord Kelvin (formerly Sir William Thomson), Prof.
Walter S. Harshman the Director of the Nautical Almanac, I^of. R. S.
Woodward, and other authorities, it is the second of mean solar time which
is the imit of time in the centimeter-gram-second system of units. Mean
solar time is always understood to be meant in all designations of time un-
less otherwise specifically stated. In all the units in this book which involve
time, the mean solar time is understood.
Sidereal time is used only for astronomical purposes. It is considered
as possessing more nearly the essential qualification of a standard unit,
namely invariability, but the mean solar time has been adopted instead.
However, the relation between the two is known to such a oegree of pre-
cision, that mean solar time is also perfectly uniform. Tables for inter-
changing sidereal and mean solar time are given in the American Ephemeris.
Besides these two sets of units of time, there are probably hundreds of
other terms used to designate various periods of time, chiefly different
kinds of years, months, cvcles, etc. These are generally used only in
astronomy and history, and many of them are obsolete; they have there-
fore not been included in the following table.
All the impK>rtant values in the table have been checked through the
kindness of Frof. Walter S. Harshman. Director of the Nautical Almanac;
many of these have been accepted as the best obtainable values, and as
such are used in the American Ephemeris and Nautical Almanac.
The International Bureau of Weights and Measures has established an
important distinction in the notation of time. When it refers to the epoch,
that is, the date or time of day,, the reference letters are used as indices;
and when it refers to the duration of a phenomenon, they are on the same
line with the niunbers. For instance, an experiment began at 2*^ IS*" 46"
lasted 2h 15m 46s, and ended at 4>' 31» 32*.
Standard Railway Time in the United States and Canada. On
November 18, 1883, a new system of railway time called "Standard Time "
went into effect on most of the railroads of the United States and Canada,
and lias since been adopted by^ most of the principal cities. According to
this system the coimtry is divided into five strips or zones running north
and south, each 15° in width. Throughout each strip the time of the clock
is the same, and it differs from that in the two neighboring strips by pre-
cisely one hour; for instance, when it is 4 o'clock in one strip it is 5 o clock
in the next one east, and 3 o'clock in the next one west.
The following table ^ves the longitude of the middle line of each strip*
the time in that strip is the correct time for that longitude. The actual
lines midway between these, where the time changes by one hour, do not
always correspond exactly with the theoretical ones, for obvious reasons.
The table also gives the name by which that time is designated in each
strip and the conventional color by which it is indicated on maps. Eastern
time is exactly 5 hours later than Greenwich time.
^^^Greenl^*^ ^*^® °^ Standard Time. Conventional Color.
60** Intercolonial time Brown
75° Eastern time Red
00° Central time Blue
105° Mountain time Green
120° Pacific time Yellow
J
94 TIME.
TIMB.
Mean solar time is the time in universal use except in astronomy. Un-
less otherwise stated, mean solar time is understood in this table.
* Accepted by Prof. Walter S. Harshman, Director of the Nautical Almanac.
Logarithm
1 sidereal second » 0.997 269 57* second (mean solar) I.998 8128*
1 second [s] (mean solar) => 1.002 737 91* sidereal seconds 0001 1874*
1 sidereal niinate=»60.* sidereal seconds.
1 minate [min or m] (mean solar) = 60.* seconds (mean solar).
1 Aidereal hour = 60.* sidereal minutes, or 3 600.* sidereal seconds.
1 hour [h] (mean solar) :
= 60.* minutes (mean solar), or 3 600.* seconds (mean solar).
1 sidereal day :
=» 86 164.1* seconds (mean solar).
« 86 400.* sidereal seconds.
•= 1 440.* sidereal minutes.
«= 24.* sidereal hours.
«» 23. h, 56. m , 4.091* s (mean solar).
=0.997 269 57* dav (mean solar) 1.998 8128*
== 1 mean solar day less 3 m and 55.909* s (mean solar).
1 day (mean solar) :
= 86 400.* seconds (mean solar).
= 86 636.555* sidereal seconds.
=« 1 440.* minutes (mean solar).
— 24.* hours (mean solar).
= 24. sidereal hours, 3 m, 56.555* s sidereal time.
•= 1.002 737 91* sidereal days 0-001 1874*
= 1 sidereal day plus 3. minutes 56.555* seconds sideread.
365.242 20* mean solar days = 366.242 20* sidereal days.
1 civil or calendar day :
— 1 day (mean solar) ; is reckoned from mean midnight to mean midnight.
An apparent solar or a natural <lay is variable.
An astronomical or naatical day is reckoned from mean noon to mean
noon.
1 -week =7. days (mean solar).
1 anomalistic month;
»27. days, 13. hours, 18. minutes, 37.4 seconds (mean solar 7).
1 civil or calendar month [mo]:
= 28, 29, 30, and 31 days (mean solar).
— aprx. M2 year (mean solar).
1 average Innar or synodic month:
= 29. days, 12. hours, 44. minutes, 2.8* seconds (mean solar).
=29.630 59* days (mean solar).
1 average sidereal month = 27. days. 7. hours. 4.3. minutes, 11.5* seconds.
1 lunar year = 354. days, 8. hours, 48. minutes, 34. seconds (mean solar).
*• = 12. lunar or synodic months.
1 common lunar year = 354. days.
1 year [yr] (mean solar) :
=366.242 20* sidereal days.
= 365.242 20* days (mean solar).
= 365. days, 5. hours, 48. minutes, 46.* seconds (mean solar).
«« 1. sidereal year less 20. minutes, 26.9* seconds (sidereal).
1 sidereal year:
= 366.256 399 2* sidereal days.
=365.256 360 4* days (mean solar).
■» 366. days, 6. hours, 9. minutes, 9.5* seconds (mean solar).
= 1. moan solar year plus 20. min, 23.6* sec (mean solar).
1 civil or calendar year, ordinary = 365.* days (mean solar).
** leapt =366.* days (mean solar).
1 common year = 1. ordinary civil or calendar year.
1 Julian year = 365.25* days (mean solar).
t A leap year is one whose number is divisible by 4, except when the
number ends in two ciphers, then it must be divisible by 400.
discharges; irrigation. 95
1 Gregorian year » 365. days, 5. hours, 49. minutes, 12. sec (mean solar).
1 tropical or natural year = 1. year (mean solar).
1 anomalistic year » 365. d, 6. h. 13. m, 53.* s (mean solar).
A legal year is obsolete.
1 solar cycle = 28.* Julian years.
1 centniyt = 100.* civil or calendar years.
1 Innisolar cycle » 532. years.
1 millenlaiu^ 1 OCX), calendar years.
DISOHAROES; FLOW of WATER; IRRIGATION
UNITS; VOIiUMB and TIME. (Volume -^ time.)
(See also Volumes.)
IMscharges (as of water) are generally measured in terms of some volume
per second as cubic feet per second, gallons per second, cubic meters per
second, etc. The relations oet ween them are therefore the same as between
those respective volumes, which see under the imits of Volumes, l^e
same is true if they are given per minute. When one is per second and
the other per minute, reduce either to the same time as the other and then
use the taole of volumes.
The only unit differing from these is the miner's inch which is sometimes
used in the western United States for measoiring the flow of water in streams,
particularly ,for mining. It is a somewhat vague unit, generally insuffi-
ciently defined, and has so-called "legal" values in different States, which
values differ. Its value varies from about 1.20 to about 1.76 cubic leet per
minute; the mean is ^nerally taken as about 1.5; for this value the reduc-
tion factors are given m the following table. Aprx. means within 2%.
Logarithm
1 miner's Inch - 1.5 cubic feet per minute 0176 0918
** ^ 0.187 013 gallon per second. Aprx. Me •.. 1-271 8717
** a 0.025 cubic foot per second, or^io .... 2-897 9400
** »0.000707 925 cb. meters per sec. Ap. ^ooo. . 4-849 9875
The acre-foot is sometimes used as a unit for measuring irrigation: it
means a body of water 1 acre in area and 1 foot in depth. It is therefore
really a true unit of volume. Its chief equivalents are:
1 acre-foot » 325 851. gallons.
- 43 560. cubic feet.
•* - 1 613.33 cubic yards.
" -> 1233.49 cubic meters.
t The twentieth century is generally assumed to have begnn with Janu-
ary 1, 1901.
96 ELECTEIC AND MAGNETIC UNITS.
ELECTRIC and MAGNETIC UNITS.
General Bemarks. —In the following tables C.G.S. refers to the ceaii-
meter-gram-second system of units; elmg means electromagnetic; elst
means electrostatic; v means the velocity of light in air, which is here
taken as equal to about SXIO^^ centimeters per second, which value has
been included in the logarithms of those relationa which involve this v ; the
word ** about,** used with such derivatives of this velocity, means that they
include whatever inaccuracy there is in this velocity. Api*x. means that
the simple fractions given are correct within 2%. The values of the derived
figures in these tables are generallv given to six significant figures and
seven-place logarithms, even though the original fundamental data may
sometimes not warrant such accuracy: the object is to enable the correc-
tions due to any subsequently adopted more accurate fundamental values
to be made by mere proportion instead of by complete recalculations.
The fundamental values of the electrical units used in these tables
are those adopted by the International Electrical Congress at Chicago.
Besides the exact values in terms of the C.G.S. units, that Congress also
defined certain concrete units as the closest approximations which existed
at that time; these concrete units are the ones in use in practice at pres-
ent. In calculating the relations between the electrical and the mechan-
ical and thermal units like foot-pounds, horse-powers, heat-units, etc., for
these tables, it had to be assumed that these concrete units are exactly
equal to those defined in terms of the C.G.S. units which they represent, as
it would otherwise be impossible to calculate those relations until the dif-
ferences which may exist between the concrete and the exact values are
known; such relations therefore must always involve these differences if
they exist ; whatever they may be they are absolutely negligible in all but
the most refined physical research.
The absolute viklnes of the electric units are given for both the
usual electromagnetic system and the less usual electrostatic system, but
the magnetic units have been confined to those in the electromagnetic
system, as the others are rarely if ever used; the dimensional formulas and
interrelations of tha latter are given in the table of Physical Quantities
in the Introduction. In a paper read before the American Institute of
Electrical Engineers in July, 1903, Dr. A. E. KennSUy suggests the prefixes
ab- or abii- to the names volt, ohm, etc., to designate the corresponding
absolute elestromagnetis units; thus abvolt, absohm, etc., mean the
absolute or C.G.S. electromagnetic units. Similarly the prefix abstat-
designates tha corresponding absolute electrostatic units. The suggestion
seems a good one, as it is often very convenient to have easily, remembered
specific names for the absolute electrical units.
The American Institute of Electrical Engineers has adopted the rule that
vector quantities when used should be denoted hy capital italics.- This
applies chiefly to electromotive force, current, and impedance.
Owing to the numerous important and very useful interrelations
between the various electric and magnetic quantities, many of
which are simple unit relations like Ohm's law or Joule's law, there have
been added to the tables of these units numerous formulas giving the rela-
tions between quantities measured in terms of various different electric and
magnetic units, many of which will frequently be found useful; they are
correct numerically also, and may be used like any other formulas. As
there is a very large number of such relations, only those likely to be used
are given; the others can be readily derived from them. Such relations
are usually given in the form of algebraic formulas, but the method adopted
here is preferred because it shows directly for what particular units the
relations are numerically correct. Treatises on alternating currents should
be consulted for the limiting conditions under which these relations apply
to alternating or other periodically varying quantities.
The author is indebted to Prof. W. S. Franklin for important suggestions
concerning the quantitative interrelations of electric and magnetic units
when their intensities are varying or alternating. Also to Dr. Frank A.
Wolff, Jr., Assistant Physicist of the National Bureau of Standards, for his
kindness in endorsing the correctness of a number of the more important
derived values of the electrical units.
resistance; impedance; reactance. 97
Mean, effective and maximum values in periodically Taryine
functions. — With alternating currents the instantaneous values of both
the electromotive force and the current vary continually. When the
arithmetical mean or average of all these momentary values of the electro-
motive force or current is taken, it is called the mean value; this value is
seldom if ever used; for a true alternating current the algebraic mean is
always zero for one whole period, hence the arithmetic mean of the whole
period or the algebraic or arithmetic mean of half a period is used instead.
When the square root of the mean square is used it is called the effective
value; this is the value almost always used and is the one meant when not
otherwise specified ; it is this value which corresponds to the electromotive
force or current of a direct current circuit in calculations of the energy.
Similarly, there is a mean and an effective magnetomotive force, mag-
netizing force, flux, flux density, etc., but as the arithmetic mean values
of these are of less importance the effective values are the ones gen-
erally understood unless otherwise specified. By the maximum value
of any of these is meant the greatest value reached in one period; this value
is sometimes the important one, notably in the strain on the insulation,
which depends on the maximum electromotive force, or in the calculation
of the hysteresis loss, which depends on the maximum flux density. Hiese
terms, mean, effective, and maximum, are not used in practice in connec-
tion with the power in watts; the mean watts are always understood and
are equal to tne product of the effective volts, the effective amperes, and
the cosine of the angle of phase difference.
The following table gives the relations between the maximum, effective,
and mean values when the variations follow the sine law. By mean value
is here meant that for a half period, as the algebraic mean for a whole period
is always zero.
Aprx. means within 2%,
Mean value ( half period ) : Logarithm
—effective value X 0.900 316 (or(2-i- t)V2). Aprx. subt. 10%.. T.964 3951
—maximum value X 0.636 620 (or 2/r). Aprx. ^ 1.808 8801
XSffectlve value:
-mean valueX 1.11072 (or (fr-!-4)\/2). Aprx. 1% Q-045 6048
—maximum value X 0.707 107 (or iV2). Aprx. %© 1.849 4850
Maximum value:
-mean value X 1.57P 80 (or «/2). Aprx. ^Vi 0196 1199
—effective value X 1.414 21 (or \/2). Aprx. 1% 0-150 Q150
RESXSTANOX! [R, r]; IMPBDANOE [Z, z]; RSAOT-
ANCB [Z, z]. (Bleotromotive force 4- current j length X
resistivity -^ cross-section.)
These imits are used to measure the opposition offered to the passage of
a current through a circuit or part of a circuit. The greater the resistance
the greater this opposition. It is similar to the mechanical friction of a
moving body, like that of water in a pipe, although the analogy does not
extend to the numerical laws, nor are there any specific units For mechan-
ical frictional resistance, as there are for electrical resistance. According
to Ohm's law the resistance in ohms is equal to the electromotive force in
volts divided by the current in amperes; or according to Joule's law it is
equal in ohms to the power in watts divided by the square of the current
in amperes; or to the square of the voltage divided by the watts. These
relations apply to direct currents, and refer to the true or "ohmic" resist-
ance of the conductor itself, which is dependent only on the material, size,
and temperature of the conductor. They also apply to alternating cur-
rents when there is no reactance in circuit (caused by inductance or capacity)
in which case the effective values of the electromotive force and current,
and the true watts, are naeant. Resistance refers to a given circuit or part
of a circuit, while resistivity (which see) refers to the specific resistance ox the
material irrespective of the size or shape of the circuit.
98 resistance; impedance; reactance.
Reactance, although not a resistance, and Impedance, which is often
called apparent resistance, and is a resistance combined with a react-
ance, are both correctly measured and expressed in the same units as resist-
ance, namely, ohms, although the reactance depends on the inductance and
capacity of the circuit and on the frequencv of the alternating current, and
is therefore not true resistance. Both of these terms are limited chiefly to
alternating current circuits. The impedance in ohms is equal to the effec-
tive electromotive force in volts, divided by the effective current in amperes,
regardless of what the phase difference may be (that being embraced by the
vector character of the impedance}. It is also equal to the square root of
the sum of the squares of the resistance and the reactance, all being ex-
pressed in ohms. For further explanations concerning the calculations of
alternating current circuits, reference should be made to treatises on this
subject.
In direct current circuits, resistance (true or "ohmic^') in ohms is the
reciprocal of conductance in mhos; similarly, in alternating current cir-
cuits impedance in ohms is the reciprocal of admittance in mhos. When
various parts of a circuit are connected in series their total resistance or
impedance in ohms is simply the sum of all the individual resistances or
impedances in ohms. If they are in parallel or multiple, however, it is
best to add their conductances or admittances in mhos and then take the
reciprocal of this sum, which will then be the total joint resistance or im-
pedance in ohms.
The unit now universally used is the ohm, by which is here meant the
international ohm of the International Congress of 1893 at Chicago, based
on the value 10^ C.Q.S. units, and represented by the resistance of a column
of mercury at 0° C, 106.3 centimeters long, weighing 14.452 1 grams, and
having a uniform cross-section. It was made legal in this country by Con-
gress in 1894 and is adopted by the National Bureau of Standards. The
Reichsanstalt has also adopted this value, and there is therefore uniformity
in the resistance standards used in those two institutions. The National
Bureau of Standards at present uses 1-ohm manganin resistance standards,
verified from time to time at the Reichsanstalt, so that the results of the
Bureau are at present expressed in terms of the particular mercurial resist-
ance standards of the Reichsanstalt. The construction of primary mercurial
standards is about to be undertaken by the Bureau. The British National
Physical Laboratory is also undertaking the construction of such standards,
and the present definition of the unit of resistance in that country in terms of
the Board of Trade ohm and the B. A. units, may be replaced by one in
terms of the primary mercurial standard.
In some of the relations with other units, such as the absolute or those
of energy, the value of the international ohm as above defined is in the
following tables assumed to be equal to the theoretical value, namely,
10^ electromagnetic C.G.S. units, which value is sometimes called the tme
ohm. In the relations between the international ohm and the other mer-
cury units given in the following table, it is assumed that the uniform cross-
section referred to in the definition of the international ohm is one square
millimeter.
The legal ohm was a mercury standard in use for a number of years
prior to the adoption of the international ohm ; it differs from the latter only
in that the length is 106 centimeters and that its cross-section is defined to
be 1 sq. millimeter, while with the international ohm it is the weight which
is defined.
The Siemens unit formerly used is the resistance of a column of mer-
cury, at 0° C, having a cross-section of one square millimeter and a length
of one meter. It was never legalized, but has often been used as a well-
defined standard of reference.
The British Association unit, or B. A. unit, was formerly the standard
in Great Britain ; the legal standard now used there is a Board of Trade coil
or ohm equal to the international ohm, on the assumed relation that 1 inter*
national ohm = 1.013 68 B. A. units. The relation accepted by the National
Bureau of Standards is the mean of the relations determined by Glazebrook
and by Lindeck in 1892, namely, 1 international (Reichsanstalt) ohm —
1.013 48 B. A. units. . The primary standards of the Reichsanstalt, in terms
of which this value is given, are themselves subject to various sources of
error involved in the construction of such standards.
< <
t <
1 1
resistance; impedance; reactance. 99
The electromagrnetic C.G.S. unit (or absolute unit) is the resistance
through which 1 C.G.S. unit of electromotive force will cause 1 C.G.S. unit
of current to flow.
The electrostatic C.G.S. unit (or absolute unit) is similarly defined
with respect to the electrostatic units of e.m.f. and current.
RBSISTANOB ; IMPEDANCE; REAOTANOB.
** Accepted by the National Bureau of Standards.
* Checked by Dr. Frank A. Wolff, Jr., Asst. Phys. National Bureau of
Standards.
Aprx. means within 2%. By "ohm" is here meant the international
ohm, unless otherwise stated, v is the velocity of light.
Logarithm
1 CGS unit [elmg]= 1 absohm 0-000 0000
= 0.001 microhm §.000 0000
= 10-» ohm 9.000 0000
= l/v2 CGS unit (elst). About % X 10-». . . 21045 7575
1 Hbsohm = 1 CGS unit (elmg) 0-000 0000
1 inicrolini-= 1 000. CGS units (elmg) 3-000 0000
= 0.000 001 ohm f.QOO 0000
1 Siemens unit (S.U.) = 0.940 734* ohm. Aprx. subtract 6%. 1.973 4667
I Uritish Association unit [B.A.U.]:
= 0.986 699** ohm.t Aprx. subtract 1% 1-994 1848
= 0.986 602* ohm.t Aprx. subtract 1% 1-994 1420
1 egal ohm = 0.997 178* ohm. Aprx. 1 1-998 7726
1 olim = 10» (X3S imits (elmg) 9-000 0000
= 10^ microhms 6.000 0000
= 1.063 00* Siemens units. Aprx. add 6% 0-026 5333
= 1.013 58* British Association units.t Aprx. add 1%. 0-005 8580
= 1.013 48** British Ass'n units.t Aprx. add 1% 0-005 8152
= 1.002 83* legal ohms. Aprx. 1 0-001 2274
= 10~* megohm B-000 0000
- 10»/t>2 CJGS unit (elst). About V9X 10"" 12-045 7575
1 international ohni=»l ohm, which see above.
1 true ohm = 10» (XJS units (elmg) 9.OOO 0000
1 wlim of Beiclisanstalt => 1 ohm 0-000 0000
1 Hoard of Trade (Brit.) ohui:Y
= 1.013 58 Brit. Ass'n unit. Aprx. add 1% 0005 8580
>» 1 ohm 0-000 0000
1 megohm =» 10 '• ohms 8-000 0000
= 1015/^2 CGS units (elst). About V9 X lO"* g.045 7575
I CGS unit (elst)= v^ CGS units (elmg) About 9X 10»J . 20954 2425
«t>2x 10-9 ohms. About 9 X 10" 11-954 2425
" = 1 abstatohm 0-000 0000
1 abstatolim = 1 CGS unit (elst) 0-000 0000
1 absolute unit = 1 CGS unit either elst or elmg 0-000 0000
The relations to other measures are as follows: §
Ohms = volts + amperes.
•• >= volts X seconds H- coulombs.
" =volts2-i- watts.
• ' -= watts T- amperes^.
= watts X seconds^ -i- coulombs^.
' volts* X seconds -!- joules.
• I . r Ovy
4 i
« 4
4 (
> 4
I (
1 <
4 4
• 4
4 4
4 4
i I
= joules -i- (amperes^ X seconds).
= joules X seconds -J- coulombs^.
t Mean of Glazebrook's and Lindeck's values of B. A. units in terms of
Keichsanstalt primary mercurial standards, accepted by the National
Bureau of Standards.
X Legal relation in Great Britain.
§ Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other period-
ically varying quantities.
^Latest value, 1903: 1 Keichsanstalt ohm = 1.000 165 Board of Trade ohms.
100 resistance; impedance; reactance.
Ohms resistance:
— 1 ^mhos conductance. For direct currents only.
— henrys-5-time constant in seconds.
•=» induced volts-*- (rate of change of amperes per second X time con-
stant in seconds).
=henrysX final amperes X applied volts -J- (joules of kinetic energ^r of
the current X 2).
= applied volts XVEhenrys-t- (joules of kinetic energy of the current
X2)]-
= applied volts^X time constant in seconds -*- (joules of kinetic energy of
the current X 2).
« joules of kinetic energy of the current X 2 -5- (time constant in seconds
X final amperes^).
•=» (maxwells X number of turns) -^ (final amperes X time constant in
seconds X 10^). When the flux is due only to the current, as in
self-induction.
Micrblims resistance:
■»l-^megamhos conductance. For direct currents only.
For alternating current circuits: f
Ohmg resistance :
= volts energy component of e.m.f.-*- total amperes.
= watts -T- amperes^.
=»v^(ohms impedance^ — ohms reactance^).
=■ v [(1 "*• mhos admittance^) — ohms reactance^].
<=-mhos conductance X ohms impedance^.
=»mhos conductance-^ mhos admittance^.
a«v'(applied volts^— induced volts^)-*- amperes.
— applied volts -J- (amperes X ^/[(time constant in seconds X frequency X
6.283 19t)2-t-l ).
— \/r(apphed volts -5- amperes)2—(henrysX frequency X 6.283 19t)*].
— y rohms impedance^ - (henrys X frequency X 6.283 19* )2].
—induced volts -^ (time constant in seconds X amperes X frequency X
6.283 19t).
Ohms reactance :
B volts wattless component of e.m.f.-f- total amperes.
«=\/(ohms impedance* — ohms resistance*).
•= V [( 1 -*-mhos admittance*) — ohms resistance*].
«=ohms impedance* X mhos susceptance.
«=mhos susceptance -t- mhos admittance*.
«ohms magnetic reactance— ohms capacity reactance.
= (henrys X frequency X 6.283 19t) -[0.159 155 §-^ (farads X frequency)].
Ohms magnetic reactance = henrys X frequency X 6.283 19.t
Ohms capacity reactance «= — 0.159 155§ -s- (farads X frequency).
'* => —159 155. ^(microfarads X frequency).
Ohms impedance— total effective volts -^ total effective amperes.
' ' — '\/(ohms resistance* 4- ohms reactance*).
• • «= 1 -s- mhos admittance.
*' " =l-t-\/(nihos conductance* -h mhos susceptance*).
• * a \/(ohms resistance -s- mhos conductance).
' * •=\/(ohms reactance -^ mhos susceptance).
•• =- V[ohms resistance* + (henrys X freq. X 6.283 19t)^-
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other period-
ically varying quantities.
X Or 2ff. Aprx. % X 10. Log 0-798 1799.
§ Or 1 -*-2>r. Aprx. %-i- 10. Log 1.201 8201.
RESISTANCE AND LENGTH. 101
RBSXSTANOX! and LENGTH, for the SAME CROSS-
SBOTION. (Resistance -s- length.)
For wires of the same cross-section and material the following relations
exist. Aprx. means within 2%.
Logarithm
1 ohm per mile:
=« 0.621 370 ohm per kilometer. Aprx. H I.79S 8503
=0.000 621 370 ohm per meter. Aprx. ^ -*- 1 000.. . . 1.793 3503
=0.000 189 394 ohm per foot. Aprx. 19-»- 100 000 . . J.277 3661
1 ohm per kilometer :
= 1.609 35 ohms per mile. Aprx. add %o Q-20e 6497
=» 0.001 ohm per meter 3.000 0000
= 0.000 304 801 ohm per foot. Aprx. 3 -^ 10 000 4.484 01 58
1 ohm per meters 1 609.35 ohms per mile. Aprx. 1 600 3.206 6497
" *» 1 000. ohms per kilometer 3-000 0000
• ' =0.304 801 ohm per foot. Aprx. %o 1.484 0158
1 ohm per foot « 5 280. ohms per mile. Aprx. 6 300 3-722 6339
»3 280.83 ohms per khn. Aprx. M X 10 000 . . 8-515 9842
» 3.280 83 ohms per meter. Aprx. 1% 0.515 9842
< (
RBSISTANOB and OROSS-SBOTION, for the SAMB
LENGTH. (Resistance X cross-section.)
When the lengths of wires of the same material are the same and the
cross-sections are different, the relations between the respective compound
units representing the product of the resistance and the cross-section are
the same as the relations between the different cross-section units, which
see under the units of Surface. Thus if the compound unit ohm-centl-
meters^ jg the product of the resistance in ohms and the cross-section in
sauare centimeters of any wire, and if similarly ohm-inches^ is the product
01 the ohms and the square inches cross-section of another wire of the same
length, then 1 ohm-centimeter2>*>0.155 000 ohm-inch^, in which 0.155 000
is the value of 1 sq. centimeter in sq. inches.
These units are used for converting the values of resistivities from one
unit to another (see also under units of Resistivity, below); thus if n is the
resistivity of a material in ohm, circular mil, foot, units, and it is required
to change this into ohm, sq. mil, foot, units, midtii^ly n by the value of
1 circTilar mil in square mils as given in the table of units of Surface, namely.
0.785 398.
RESISTryiTT [p] ; SPEOIFIO RESISTANOB.
(Resistance X cross-section -i- length.)
These units are used to measure the inherent quality of a material to
resist the passage of an electric current. Resistivity diners frpm resistance
in that the latter refers to the number of ohms of any given circuit or part
of a circuit, and depends on the length, cross-section, and quality of the
material; while resistivity refers only to the nature of the material itself
and.is always the same for the same material ; it is the resistance of a unit
amount of the material, like a cube of one centimeter, or a mil-foot, or a
meter of one sq. millimeter section. It bears somewhat the same relation
to resistance as the density of a material does to the weight of any given
102 ' resistivity; specific resistance.
amount of it; the density is always the same for that material, being the
weight of a unit of volume, while the total weight of an actual piece depends
upon the size of that piece. As the resistivity or specific resistance is a
quality of a material, its values are usually given in tables of physical con-
stants; the resistivity may also be calculated from the resistance of any
given piece by multiplying this resistance in ohms by the cross-section and
dividing by the length- the result will of course be different, depending
upon the units used. The resistivity, being a property of a material, is
the same for direct as for alternating currents.
There are several units in use. The most rational one is the resistance
in ohms between two parallel sides of a cube of one centimeter of the ma-
terial. This imit is sometimes called the ohin-centimeter unit, or 1 ohm
per cubic centimeter, or more correctly, 1 oliin-square-centimeter
per centimeter; it will here be called the olim, cubic centimeter,
unit. For most conducting materials in use, except electrolytes, the
resistivity when stated in these units is a very small number, hence it is
often stated in microhms (millionths of an ohm) instead of ohms.
The electromagnetic and the electrostatic C.G.S. units are the
same as the above except that the resistance is stated in the respective
C.G.S. units instead of in ohms.
Another unit in common use is the meter-millimeter anit, that is, the
resistance in ohms of a wire one meter long and one square millimeter in
cross-section; it also has the name 1 ohm per meter per square milli-
meter, or more correctly 1 ohm-square-millimeter per meter; it will
here be called the ohm, square millimeter, meter, unit. ' This unit has
the advantage that it generally involves less calculation than the cubic-
centimeter unit, as the lengths and cross-sections are in practice usually
stated in meters and square millimeters when the metric system is used. A
more rational imit for circular wires, although apparently not in general
use. is similar to this one except that the cross-section is a circle of one
millimeter diameter, and is therefore equal to a "circular millimeter" as
distinguished from a "square millimeter" (see explanation of circular units
under units of Surface). This has the advantage of eliminating from the
calculation for the usual round wires the troublesome factor it (or 3.141 59),
because when the cross-sections are stated in circular units they are directly
equal to the squares of the diameters.
The imit usually used when the lengths are in feet and the diameters in
mils (that is, thousandths of an inch), is the mil-foot or circular mil-foot,
which is the resistance in ohms of a round wire one foot long and one mil
in diameter. The cross-section then is one circular rail, and the cross-
sections of other round wires are then equal to the squares of their diam-
eters in mils (see explanation in preceding paragraph). This unit also has
the name of 1 ohm per foot per circular mil or per mil diameter, or more
correctly, 1 ohm-circular-mil per foot; it will nere be called the ohm,
circular mil, foot, unit. A similar unit, though less frequently used, is
the square mil-foot; it differs from the other only in that the cross-section
is a square mil instead of a circular mil, and it is more convenient to use
with bars of rectangular cross-section. It also has the name of 1 ohm per
foot per square mil, or more correctly, 1 olim-square-mil per foot;
it will here be called the ohm, square mil, foot, unit.
Sometimes resistivities are denoted in terms of that of mercury or pure
copper as a basis. They are then mere relative resistivities or ratios of two
resistivities, that of the material divided by that of mercury or copper, and
are therefore not in terms of any real units, although they might oe called
mercury or copper units. In the following table the resistivities of mercunr
and of copper have been added to facilitate making calculations with such
relative resistivities. The resistivity of mercury here used is that
deduced from the definition of the concrete international ohm, namely, ihat
a column of uniform cross-section, 106.3 centimeters long, weighing 14.4521
grams, has a resistance of one ohm at 0® C. The cross-section is for this
purpose assumed to be one sauare millimeter. For the resistivity of pure
copper the Matthiessen value is still in use, namely 1.687 microhms for
one centimeter length and one square centimeter cross-section, at 15** C,
according to Prof. Lindeck of the Reichsanstalt. Pure copper as now
made has a lower resistivity than this; according to Prof, Lindeck the
value used in GJermany (presumably under the authority of the Reichsan-
stalt) is 1.667 in the same units.
resistivity; specific resistance. 103
Aprx. means within 2%. v is the velocity of light.
Logarithm
1 CGS unit (elmg) — 10~^ ohm, cubic centimeter, unit 9'000 0000
= 1/1)2 CGS unit (elst). AboutHXlO-^".. . 21.045 7575
1 ohm, circular mil, foot, unit:
= 0.785 398 ohm, sq. mil, foot, unit. Aprx. 9io 1-895 0899
■= 0.166 243 microhin, cb. cm, unit. Aprx. K 1-220 7433
—0.002 116 67 ohm, circ. mm, meter, unit. Aprx. mo ooo- 3-325 6534
=0.001 662 43 ohm. sq. mm, meter, unit. Aprx. H -^ 100. . 3220 7433
1 mil-foot unit. See 1 ohm, circular mil, foot, unit.
1 ohm-circular-mil per foot. See 1 ohm, circular mil. foot, imit.
1 ohm per foot per circular mil. See 1 ohm, circular mil, foot, imit.
1 ohm per foot per mil diameter. See 1 ohm, circular mil, foot, unit.
1 ohm, sq. mil, foot,- unit:
= 1.273 24 ohm, circular mil, foot, units. Aprx. ^% 0-104 9101
= 0.211 667 microhm, cb. cm, unit. Aprx. 21 -i- 100.. . 1.325 6534
=0.002 69503 ohm, circ. mm, met. unit. Aprx. % -^ 1 000. 3-430 5635
= 0.00211667 ohm, sq. mm, met. unit. Aprx. 21-5-10 000. 3.325 6534
1 ohm-sq. mil per foot. See 1 ohm, sq. mil, foot, lulit.
1 ohm per foot per sq.-mll. See 1 ohm, sq. mil, foot, imit.
1 microhm, cb. centimeter, unit:
= 1 000. CGS units (ebng) 3000 0000
= 6.015 29 ohm, circular mil, foot, units. Aprx. 6. . . . 0-779 2567
= 4.724 40 ohm, sq. mil, foot, units. Aprx. Hi X 100. . 0674 3466
=0.012732 4 ohm, circ. mm, met. unit. Aprx. H-^-lO... . 3-104 9101
= 0.01 ohm, sq. mm, meter, unit 2-000 0000
= 10~* ohm, CD. cm, unit S-000 0000
1 microhm-sq. centimeter per centimeter. See 1 microhm, cb. cm,
unit.
1 microhm per cubic centimeter. See 1 microhm, cb. cm, unit.
1 ohm, circular mm, meter, unit:
- 472.440 ohm, circ. mil, ft, units. Aprx. lo o<>%i. 2-674 3466
- 371.064 ohm, sq. mil, ft, units. Aprx. H X 1 000 2-569 4365
= 78.539 8 microhm, cb. cm, units. Aprx. 80 1-895 0899
>a 0.785 398 ohm, sq. mm, meter, unit. Aprx. ^io • . 1-895 0899
=0.000 078 639 8 ohm, cb. cm. unit. Aprx. 80^ 1 000 000. 5-895 0899
1 ohm-clrcular-mm per meter. See 1 ohm, circular mm, meter, unit.
1 ohm per meter per circular mm. Seel ohm, circular mm, meter,
imit.
1 ohm, sq. uim, meter, unit:
= 601.529 ohm, circular mil, foot, units. Aprx. 600. . . . 2-779 2567
= 472.440 ohm, sq. mil. foot, units. Aprx. H\ X 10 000. . 2674 3466
= 100. microhm, cb. cm, units 2-000 0000
= 1.273 24 ohm, circular mm, meter, units. Aprx. i%. .. 0104 9101
= 0.000 1 ohm, cb. centimeter, unit 1-000 0000
1 ohm-sq. mm per meter. See 1 ohm, sq. mm, meter, unit.
1 ohm per meter per sq. mm. See 1 ohin, sq. mm, meter, unit.
1 ohm, cb. centimeter, unit:
= 109 CGS units (elmg) 9000 0000
= 1 000 000. microhms, cb. cm, units 6000 0000
= 12 732.4 ohm.circ. mm, met- units. Aprx. ir^X 100 000. 4104 9101
= 10 000. ohm, sq. mm, meter, units 4000 0000
= lOVt?2 CGS unit (elst). About Vo X 10"" 12-045 7575
1 ohni-sq. cm per cm. See 1 ohm, cb. centimeter, unit.
1 ohm per cubic centimeter. See 1 ohm, cb. cm, unit.
1 meg^ohm, cb. centimeter, unit:
= 1 000 000. ohm. cb. cm. units 6-000 0000
1 COS unit (elst)= v^ CGS units (elmg). About 9 X lO* 20-954 2425
=v2Xl0-9 ohm, cb. cm, units. Abt. 9X10" 11-954 2425
=v^X 10~^5 megohm, cb. cm, units. About
9X 105 5-954 2425
it
104 resistivity; conductance.
When resistivities, or specific resistances, are given in terms of eopper
or mercury, then miiltiply them by the reduction factors in the following
table to reduce them to the corresponding values in the respective units.
Logarithm
KesistiTity of copper: f
«= 10.027 5 ohm, circular mil, foot, units 1-001 1923
=- 7.875 57 ohm, sq. mil, foot, units 0-896 2822
»■ 1.667t microhm, cb. cm, imits 0-221 9356
— 0.021224 9 ohm, circular mm, meter, unit 3-326 8457
-= 0.017 720 2 times that of mercury 2-248 4689
>» 0.016 67 ohm, sq. mm, meter, unit 2-221 9356
= 0.000 001 667 ohm, cb. cm, unit 6-221 9356
Resistivity of copper (Matthiessen): }
= 10.147 8 ohm, circular mil, foot, units 1>006 3718
= 7.970 06 ohm, sq. mil, foot, units 0901 4617
=» 1.687^ microhm, cb. cm, units 0-227 1151
>» 0.021 479 5 ohm, circular mm, meter, unit 2-332 0252
=» 0.017 932 8 times that of mercury 5.253 6484
»■ 0.016 87 ohm, sq. mm, meter, unit 2-227 1151
«0.000 001 687 ohm, cb. cm, unit g.227 1151
Resistivity of mercury : §
= 565.879 ohm, circular mil, foot, units. 2-752 7234
» 444.40 ohm, sq. mil, foot, units 2-647 8133
» 94.073 4 microhm, cb. cm, units 1-973 4667
= 56.432 7 times that of copper 1.751 5311
B 55.7637 times that of copper (Matthiessen). . .' . 1-746 3516
a 1.197 78 ohm, circular mm, meter, units Q-078 3768
«=■ 0.940 734§ ohm, sq. mm, meter, imit ,. . . . 1.973 4667
-0.000 094 073 4 ohm. cb. cm, unit 5973 4667
The relations of resistivity to other measures are as follows:
Resistivity (in ohm, cb. cm, units) =• 1 -i- conductivity (in mho, cb. cm tmits).
^ohmsXsq. cm section -i- cm length.
4< ii
CONDUCTANCE [G, g]; ADMITTANCE [T, y]; SUS-
CEPTANCE [B, b]. (Current -i- electromotive force ;
1 -7- resistance ; cross-section X conductivity -s- leng^)
^ These units are used to measure the quality of a circuit or part of a
circuit to conduct a current: the greater the conductance the better does
the circuit conduct. For direct current circuits it is the reciprocal or
opposite of resistance, which see. There appears to be no mechanical
analogy, as in mechanics the opposite quality, namely, the mechanical
resistance, is used. Conductances are not often used in calculations, as
the resistances are generally used by preference. When, however, there
are several circuits in parallel or multiple arc, the calculation of their joint
action is simpler if made with conductances, as their joint conductance
is then merely the sum of the individual conductances, while when resist-
ances are used the joint resistance is equal to the reciprocal of the sum of
the reciprocals of the individual resistances.
For direct current circuits, or when there is no reactance in alternating
current circuits, the conductance in mhos is equal to the reciprocal of
the resistance in ohms. It follows from Ohm's law that for direct current
circuits, and for alternating current circuits without reactance, the con-
ductance in mhos is equal to the current in amperes divided by the elec-
tromotive force in volts. The conductance in mhos is also equal to the
resistance in ohms divided by the sum of the squares of the resistance in
ohms and the reactance in ohms. The relations to joules and watts are
rarely if ever used.
t Pure copper at 15® C; according to Prof. Lindeck.
" Matthiessen's value for pure copper at 15° C. ; according to Prof. Lindeck.
Pure mercury at 0° C, oased on definition of international ohm.
\
conductance; admittance; susceptance. 105
Adnalttanoe, which is the reciprocal of impedance, and susceptance,
which together with conductance make admittance, are both correctly
expressed and measured in the same units as conductances, namely, mhos,
although they depend on the inductance and capacity of the circuit and
on the irequency of the alternating current and are therefore not true con-
ductances. Both of these terms are limited chiefly to alternating current
circuits. The admittance in mhos is equal to the effective current in
amperes divided by the effective electromotive force in volts, regardless
of what the phase difference may be; its value in mhos is equal to the
reciprocal of the impedance in ohms. It is also equal to the square root
of tne sum of the squares of the conductance and the susceptance, all
values being in mhos. The susceptance in mhos is equal to the wattless
current in amperes divided by the electromotive force in volts. It is
also equal in mhos to the reactance in ohms divided by the sum of the
squares of the resistance in ohms and the reactance in ohms. For further
explanations concerning the calculation of alternating current circuits
reference should be made to treatises on that subject.
The onl3r unit used in practice is the mho, which is the reciprocal of
the ohm; it is the word onm written reversed to indicate the reciprocal.
There^ is no official sanction for its use, but as there is no other practical
unit, it has come iato use.
The electromagnetic and electrostatic C.G.S. units are the re-
ciprocals of the corresponding units of resistance, v is the velocity of light.
Logarithm
1 CGS unit (elst) «= 10V»' mho. About H X 10-" 12045 7576
- l/r2 CGS unit (elmg). About HX10-». 21045 7575
1 mho-v^X 10-0 CGS units (elst). About 9X 10" 11.954 2425
•• . io-« CGS unit (ehng) 8.000 0000
1 megamho « 1 000 000. mhos B-OOO 0000
- 0.001 CGS unit (elmg) §.000 0000
1 CGS unit (ehng)- v^ CGS units (elst). About 9X 10». . 20-954 2425
= 10» mhos 9000 0000
«" 1 000. megamhos 3.000 0000
4*
The relations to 9ther measures are as follows, t (See also the reciprooab
of those under resistance.)
Mlios= 1-i-ohmfl.
** =- amperes -I- volts.
" =» watts -i- volts*.
•* = amperes* -I- watts.
Megamhos = 1 -(-microhms.
Mhos conductance:
« 1 -s- ohms resistance. For direct currents onlv.
=» amperes energy component of current -»- volts total e.m.f.
= watts -s- volts*.
^VXnahos admittance*— mhos susceptance*).
■=vT(l"*-ohms impedance*) — mhos susceptance*].
«obms resistance -5- ohms impedance*.
«ohms resistance^ (ohms resistance* -f ohms reactance*).
«ohms resistance X mhos admittance*.
Mhos susceptance:
«= amperes wattless component of current -s- volts total e.m.f.
^^/(mhos admittance* — mhos conductance*).
«=V[(l"*"ohms impedance*) — mhos conductance*].
= ohms reactance -s- ohmsimpedance*.
sohms reactance -5- (ohms resistance* -f ohms reactance*).
«ohms reactance X mhos admittance*.
Mhos admittance <= total effective amperes -t- total effective volts.
" •• ^^(mhos conductance*-!- mhos susceptance*).
" *• =1-*- ohms impedance.
" " =l-s-\/(ohms resistance* -f ohms reactance*).
" " ^^(mhos conductance -5- ohms resistance).
*• •* ^vCnahos susceptance -s- ohms reactance).
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to fUt^mating or other peri-
Q4ically varying qu^ntitie?,
106 conductivity; specific conductance.
OONDUOTIVITY [r]\ SPBOIFIO CONDUCTANCE.
(Conductance X length -^ cross-section ; 1 -^ resistivity.)
These units are used to measure the inherent quality of a material to
conduct an electric current. Conductivity, which is the reciprocal of resis-
tivity, differs from conductance in that the latter applies to a given circuit
or part of a circuit, and its amount depends on the length, cross section,
and quality of the material; while conductivity refers only to the nature
of the material and is always the same for the same material; it is the
conductance of a unit amount of the material like a cube of one centimeter.
It bears somewhat the same relation to conductance as the density of a
material does to the weight of a given amount of it ; the density is alwavs
the same for that material, being the weight of a unit of volume, while the
total weight of an actual piece depends on the size of that piece. As the
conductivity is a quality of a material, its values are usually given in
tables of physical constants. The conductivity may also be calculated from
the conductance of any piece or part of a circuit by multiplying the con-
ductance in mhos (or 1 -5- resistance in ohms) by the length and dividing by
the cross-section ; the result will of course be different depending upon the
units used. The conductivity, being a property of a material, is the same
for direct and for alternating current.
The values of the conductivities of materials are not in general use, the
reciprocal quantity, namely resistivity, being generally preferred, as it is
simpler to use in most calculations. The use of conductivities in practice
is generally limited to electrolytes and for making comparisons of other
conducting materials with copper, or for comparing different qualities of
copper with each other.
The most rational nnit and the one which seems to be coming into more
general use, is the conductivity of a material of which a cube of one centi-
meter has a conductance of one mho between two parallel sides. It will
here be called the mho, cubic centimeter, unit. This means that the
resistance of a column of that material one centimeter long and one square
centimeter cross-section, is one ohm, that is, it corresponds to the ohm,
cubic centimeter, unit of resistivity; conductivities stated in the mho,
cubic centimeter, unit are the numerical reciprocals of the resistivities
stated in the ohm, cubic centimeter, unit. .This unit of conductivity is
therefore the best one to use when conductivities are to be converted into
resistivities or the reverse. This unit is used chiefly for electrolj^s; the
best conducting aqueous solutions of acids at 40° C. have a conductivity of
about one, in terms of this unit.
The electromasrnetic C.G.S. unit is the same, except that the con-
ductance is stated in C.G.S. units instead of mhos. Similarly with the
electrostatic C.G.S. unit.
The conductances in mhos of a wire one meter long and one square milli-
meter or one circular millimeter cross-section, or one foot long and one
square mil or one circular mil cross-section, may also be used as units of
conductivity, each being the reciprocal of the corresponding unit of resis-
tivity, which see.
The most usual way of stating the conductivity of a solid material is to
give the ratio of its conductivity to that of pure copper as a standard; this
avoids the use of the little-known unit of conductance (mho). This ratio
is usually ^ven as a percent, but may also be stated as so-and-so many
''copper units" of conductivity. The Matthiessen value for the resistivity
of pure copper at 15° C. is, according to Prof. Lindeck of the Reichsanstalt,
1 687 in microhm, cubic centimeter, units. The conductivity correspond-
ing to this is 502 768. in mho, cubic centimeter, units. In the following
table this is called the copper unit (Matthiessen). Better qualities of
copper are now made, which explains the apparent anomaly of conductiW-
ties greater than 100% when based on the Matthiessen standard. A better
value for pure copper is the one used as standard in Grermany, presumably
by authority of the Reichsanstalt, which according to Prof Lindeck ia a
resistivity of 1 667 in microhm, cubic centimeter, units, at 15° C. The
conductivity; specific conductance. 107
conductivity corresponding to this is 599 880. in mho, cubic centimeter,
units. In the following table this is called the copper unit. Mercury
was formerly often used as a standard of comparison, particiilarly in stat-
ing the conductivities of electrolytes. The resistivity determined from the
definition of the international ohm is 94.073 4 in microhm, cubic centimeter,
units. The conductivity corresponding to this is 10 630. in mho, cubic
centimeter, units. In the following table this is called the luerciiry unit.
Logarithm
ICGS unit (elst):
= 10 Vv* mho, cb. cm, imit. About H X 10"" 13.045 7575
-94 073.4/»2 mercury unit. About 1.045 26 X lO-io 11.019 2242
— l/t»2 COS unit (ehng). About % X lO"*^ 31.045 7575
1 mho, cb. centimeter, unit:
-= t;2x 10-9 CGS units (elst). About 9X 10" 11.954 2425
= 0.000 094 073 4 mercury unit 5.973 4667
-= 0.000 001 687 copper unit (Matthiessen) |.227 1151
-= 0.000 001 667 copper unit 6.221 9356
« 10-» CGS unit (elmg) §.000 0000
1 mercury unit:
=i;2X 1.063 OX 10-5 CGS units (elst). Ab. 9.567 000 X 10" 15.980 7758
=• 10 630. mho, cb. cm, units 4-026 5333
— 0.017 932 8 copper unit (Matthiessen) 2253 6484
— 0.017 720 2 copper unit 2.248 4689
= 0.000 010 630 CGS unit (elmg) 5026 5333
1 copper unit (Matthiessen) :
= 592 768. mho, cb. cm, units 5-772 8849
-= 55.763 7 mercury units 1.746 3519
= 0.000 592 768 CGS unit (ehng) |.772 8849
1 copper unit = 599 880. mho, cb. cm, units 5.778 0644
* • = 56.432 7 mercury units 1.751 5311
=0.003 599 880 CGS unit (elmg) i.778 0644
1 CGS unit (ehng) = t>2 CGS units (elst). About 9 X 10^^ 20954 2425
» 10® mho, cb. cm, units 9.000 0000
= 94 073.4 mercury units 4973 4667
= 1 687. copper units (Matthiessen) 3.227 1151
«" 1 667. copper units 3.221 9856
< i
• •
« t
< 4
Conductivity of mercury : f
= 10 630. mho. cb. centimeter, units 4.026 5333
= 0.017 932 8 times that of copper (Matthiessen) 2253 6484
= 0.017 720 2 times that of copper 2-248 4689
Conductivity of copper (Matthiessen): X
— 592 768. mho, cb. centimeter, units 5-772 8849
— 55.763 7 times that of mercury 1.746 3516
Conductivity of copper: §
= 590 880. mho, cb. centimeter, units 5.778 0844
= 56.432 7 times tliat of mercury 1-751 5311
The relations of conductivity to other measures are as follows:
Conductivity (in mho, cb. cm, units) :
= l-i- resistivity (in oljm, cb. cm, units).
=mhosXcm length -t-sq. cm section.
=cm length -s- (qhms X sq. cm section).
t Pure mercury at 0* C. ; based on the definition of the international ohm.
t Matthiessen's value for pure copper at 15** C. ; according to Prof. Lin-
deck.
S Pure copper at 15^ C. ; according to Prof. Lindeck.
108 ELECTROMOTIVE FORCE; POTENTIAL.
EUESCTROMOTIVE FORCE [e.iii.f., E, e]; POTENTIAI.;
DIFFERENCE OR FALL OF POTENTIAL [p. d.,
n, u]; STRESS; ELECTRICAL PRESSURE ; VOLT-
AGE. (Magnetic flux -^ time; current X resistance.)
These units are used to measure the electrical pressure, stress, or motive
force which produces or tends to produce a current, just as pounds measure
the pressure of air either in the form of compressed air or wind, or as the
differences of level of water measure the force which causes the water to
flow and which might similarly be called the. hydraulic motive force. Ac-
cording to Ohm's law the electromotive force in volts is equal to the current
in ampNeres multiplied by the resistance in ohms; or according to Joule's
law it is equal to the power in watts divided by the amperes; or to the
square root of the product of the watts and the ohms; these apply to the
electromotive forces or differences of potential of direct currents; they
apply to alternating current electromotive forces also when there is no
reactance in the circuit and therefore no phase shifting (caused by induct-
ance or capacity), in which case they refer to the effective electromotive
force; when there is reactance in such circuits the electromotive force la
volts equals the current in amperes multiplied by the impedance in ohms.
The terms electromotive force and potential are used synonymously
ia practice, although the use of the latter term is not to be commended
owing to its more general meaning in physics. Diflference of potential
means in general the difference between two absolute potentials, e.m.f.'s,
or potentials in general, the actual values of which need not be known;
this term is often distinguished from electromotive force, in that the latter
applies to the total which is generated in a battery or dynamo while the
difference of potential applies only to a portion of it, like that available at
the terminals, or that oetween any two points on a circuit. Voltage
means any e.m.f. or difference of potential when expressed in volts. Ab-
solute potential is sometimes used to denote tne potential above or
below some assumed zero, which is usually taken as that of the earth.
The unit universally used is the volt, by which is here meant the inter-
national volt of the International Congress of 1893 in Chicago, defined as
that electromotive force which will maintain one international ampere
through one international ohm, represented for practical purposes by
1 4-1.434 of that of a Clark cell at 15° C.f It was made legal in this country
by Congress in 1894, and is adopted by the National Bureau of Standards.
The ** saturated" Weston or cadmium standard cell (with excess of
crystals) may eventually be substituted for the Clark cell as the official
standard because it has a much smaller temperature coefficient (see table
below) • the relation between the Clark and this Weston cell being known
quite definitely (see table below), either may be used as the standard; in
tnis table this ratio and the value of the Clark cell are used as the funda-
mental values. There is another type of cadmium cell called the '* unsatu-
rated" Weston cell, in which there is no exeess of crystals at ordinary
temperatures, as the solution is saturated at 4° C. ; this has the advantage
of having a still lower temperature coefficient, which can be neglected en-
tirely at ordinary temperatures; the National Bureau of Standards does
not regard it safe to assign a definite value to this unsaturated Weston cell
owing to the possibility of the seal being imperfect and the consequent
change in the concentration of the solution, and also the impossibility
of ascertaining the exact temperature at which the solution was saturated.
The e.m.f. of this cell, with a solution saturated at 4° C, is, however, 1.019 8
international volts, this value being the same as that of the saturated cell
at the same temperature. A number of such cells belonging to the Bureau
have been intercompared, and were found to differ by a number of units
t This is the value of the volt used in calibrating the Weston voltmetera.
ELECTROMOTIVE FORCE; POTENTIAL. 109
in the last decimal place ; hence the cell should not be employed as a stand-
ard of reference, although as a working standard it can hardly be improved
upon.
The value adopted at the Reichsanstalt for the electromotive force of the
Clark cell is based upon a determination of its e.m.f. in terms of the elec-
trochemical equivalent of silver and the unit of resistance, and also upon a
similar detemunation of the e.m.f. of the Weston or cadmium cell, together
with a determination of the ratio of the values of these two cells. As the
values thus obtained for the Clark and Weston or cadmium cells by the
silver voltameter did not agree with the directly determined ratio, each of
the silver voltameter determinations was given eciual weight and the two
separate values adjusted so as to give the ratio directly determined. The
Reichsanstalt's value thus obtained for the Clark cell is 1.432 85 instead of
1.434 as defined by the International Congress and legal in this country, f
This Reichsanstalt value may be more accurate, but is not legaUated here.
As the cells are the same, this makes a very slight difiference oetween the
volt used by the Reichsanstalt and that legal ana used in this country (the
international volt). The National Bureau of Standards uses as the funda-
mental units those of resistance and electromotive force, obtaining the
ampere from them, thus bringing all three into agreement with each other.
According to Weston the international concrete volt; ampere, and ohm,
as definea by the Chicago Congress, agree with each other.
In some of the relations with other imits, such as the absolute, the mag-
netic, the energy units, etc., the value of the international volt as above
defined in terms of the Clark cell is in the following tables assumed to be
equal to the theoretical value, namely 10^ electromagnetic C.G.S. units,
wnich value is sometimes called the true volt. According to the theoret-
ical definition of a volt, it is the difference of potential generated in a con-
ductor which cuts 10^ C.G.S. units of magnetic flux (or 10* maxwells or
lines of force) per second; or it is the difference of potential generated per
centimeter len^h of a conductor moving transversely throu^ a ma^etic
field of a density of 1 C.G.S. unit of density (or 1 gauss) at a velocity of
10^ centimeters (or 1 (XK) kilometers) per second. A difference of potential
thus generated is moreover directly proportional to the amount of magnetic
flux traversed per second. In the older literature the e.m.f. of a Daniell
cell (about 1.1 volt) was often used as a unit.
The electromagnetic C.G.S. unit (or absolute unit) is the difference
of potential generated at the ends of a conductor 1 centimeter long moving
through a magnetic field of unit density (one gauss) at a speed of 1 centi-
meter per second perpendicularly to tne direction of the field; or more
briefly, it is the difference of potential induced in a conductor which cuts
one unit of magnetic flux (one maxwell or one line of force) per second.
The electrostatic C.G.S. unit (or absolute unit) is that difference of
potential through which one electrostatic unit of quantity falls when the
work done by it is one erg.
EliBCTROMOTlVE FORCB.
** Accepted by the National Bureau of Standards.
* Checked by Dr. Frank A. Wolff, Jr., Asst. Pfays. Nfttional Bureau of
Standards.
Aprx. means within 2%. By "volf is meant international volt unless
otherwise stated, v is the velocity of light.
Logarithm
1 CGS unit (elmg) » 1 abvolt 0-000 0000
** =»0.01 microvolt 5-000 0000
" B 10-8 volt g.OOO 0000
= l/i> CGS unit (elst). AboutHXlO-w. .. xf.522 8787
1 abvolt » 1 CGS unit (elmg) 0000 0000
1 microvolt » 100. CGS units (elmg) 2000 0000
" -=0.000 001 volt g.OOO 0000
1 milliTolt*- 0.001 volt S-000 0000
1 legal volt = 0.997 178* volt. Aprx. 1 1.998 7726
** ■- 1 ampere X 1 legal ohm.
t The difference corresponds almost exactly to that due to one Centigrade
difference of temperature ; it is about 8 hundredths of one percent.
no ELECTROMOTIVE FORCE; POTENTIAL,
1 volt [V, v] : Logarithm
=- IQs COS units (elmg) 8000 0000
= 1 000 000. microvolts 6000 0000
= 1 000. millivolts 8000 0000
= 1.002 83* legal volts. Aprx. 1 001 2274
"-0.999 198** volt of Reichsanstaltf Aprx. subtr. ^ioo%.. 1.999 6516
S:
= 0.980 962** Weston (satur. ) cell at 20° C. Ap. sub. 2%. . . J .991 6523
= 0.980 567 Weston (unsat.) cell at any temp.f Aprx.
subtr. 2% T.991 4774
= 0.697 350* aark cell at 15° C. Aprx. T4o 1848 4508
= 108/t> COS unit (elst). About Mco 5.522 8787
=• 0.001 kilovolt 3 000 0000
1 international volt = 1 volt, which see above 0000 0000
1 trne volt = 10^ CGS units (elmg) 8000 0000
1 volt of KeichsanBtalt = 1 .000 803** volts, t Ap. add %oo% . 0000 3484
1 Weston (cadmium; saturated) cell at 90° C. -with excess of crystals:
= 1.019 4** volts. Aprx. add 2% 0008 8477
= 1.018 6** volts of Reichsanstalt. Aprx. add 2% 0007 9993
= 0.710 88* Clark cell at 15° C. Aprx. % 1.851 7985
1 Weston (cadmium; unsaturated! ) cell at any ordinary temperature:
= 1 .019 8* volts. Aprx. add 2% 0008 5226
= 1.019 0* volts of Reichsanstalt. Aprx. add 2% 0008 1742
1 Daniell cell = 1.1 volts approximately (unreliable) 0041 3927
1 Clark cell at 16° C:
= 1.434** volts. Aprx. 1% 0156 5492
= 1.482 86** volts of Reichsanstalt. Aprx. 1% 0156 2008
= 1.406 7** Weston cells at 20° C Aprx. % 0148 2015
1 CGS unit (elst)= v CGS units (elmg). Ab. 3X10*0. 10477 1213
= i>X 10-» volts. About 300 2477 1213
= 1 abstavolt 0000 0000
= t) X 1Q~" kilovolt. About Mo I 477 1213
1 abstavolt = 1 (XrS unit (elst) 0000 0000
1 kilovolt = 1000. volts 3.OOO 0000
=, 10" /t> CGS units, (elst). About lo^ 0522 8787
1 megravolt = 1 (XK) 000. volts 6000 0000
1 absolute unit = 1 CGS unit, either elmg or elst 0.000 0000
The relations of volts to other measures are as follows: t
Volts = amperes X ohms.
= ohms X coulombs + seconds.
= watts + amperes.
= kilowatts X 1 000. -J- amperes.
= \/( watts X ohms).
= watts X seconds -s- coulombs.
= joules -J- coulombs.
= joules -»- (amperes X seconds).
= \/( joules X ohms + seconds).
' coulombs -»- farads.
it
ti
(1
11
11
It
><
II
<i
«4
41
t<
' * = coulombs X 1 000 000. + microfarads.
** ->\/( joules of stored energy X 2-*- farads).
tt
tt
1 000. X \/( joules of stored energy X 2 -1- microfarads).
= maxwells X number of turns -s-( seconds XIO^).
=gausses X sq. centimeters -i- (seconds X 10^).
Induced volts:
= henrys X rate of change of amperes per second.
= time constant in seconds X ohms X rate of change of amperes per sec.
Applied volts:
= henrys X final amperes-*- time constant in seconds.
= joules of kinetic energy of the current X ohms X 2 -^ (henrys X final
amperes).
= ohmsXyTjoules of kinetic energy of the current X 2-*- henrys J.
= joules of kinetic energy of the current X 2-:- (time constant in sec-
onds X final amperes).
■"\/[( joules of kinetic energy of the current X ohms X 2) -1- time con-
stant in seconds .
1 See explanatory note above.
t (Consult treatises on alternating currents for the limiting conditions.
ELGCTROMOTIYB FORCE; POTENTIAL.
Ill
For alternating current circuits: f
Volts = watts -f- (amperes X cos ^J)
' * -■ \/[( watts X ohms) + cos ^J]
'• — amperes-!- (farads X frequency X 6.283 19 J ).
*• - amperes X 1 (XX) (XX). + (microfarads X frequency X 6 .283 19J ).
Induced Tolts:
— \/[( applied volts)* — (amperes X ohms resistance)*].
= henrysX amperes X frequency X 6.283 19§.
a time constant in seconds X ohms resistance X amperes X frequency X
6.283 19§.
Applied voltfi:
■»\/[(ampere8Xohms resistance)* -h (induced volts)*].
— amperes X \/[(henry8 X frequency X 6.283 19§ )* -I- (ohms resistance)*] .
— amperes X ohms resistance X v[(time constant in seconds X fre-
quency X 6.283 19§ )* + !].
Formulas for the temperature oorrections in Centigrade de-
grees, between 0° and 30°, determined by the Reichsanstalt and accepted
by the National Bureau of Standards. They apply equally well to the
international values and to the Reichsanstalt values, t is the temperature
in Centigrade degrees; Et is the electromotive force at that temperature;
while Eis and E^a are the standard values at 15° and 20° respectively
as given in the above table.
For the Clark cell:
Et'^Eis-O.OOl 19 «- 15) -0.000 007 a-15)*.**
For the Weston (saturated) cell:
Et-'Em- 0.000 038 (t - 20) - 0.000 000 65 « - 20)*. **
E.M.F. of Clark and Weston Cells at Different Temperatures
Calculated from these Formulas.
F°.
Clark.
Weston (saturated).
c°
International
Reichsanstalt
International
Reichsanstalt
Volts.
Volts.
Volts.
Volts.
32.0
1.4503
1.44912
1.0199
1.0191
10
50.0
1.4398
1.4386 2
1.0197
1.0189
13
55.4
1.4364
1.4.35'?
1.0196
1.0188
14
57.2
1.4352
1.4340 3
1.0196
1.0188
16
59.0
1.4340
1.4328 5
1.0196
1.0188
16
60.8
1.4328
1.4316 5
1.0195
1.0187
17
62.6
1.4316
1.4304 4
1.0195
1.0187
18
64.4
1.4304
1.4292 2
1.0195
1.0187
19
66 2
1.4291
1.4279 8
1.0194
1.0186
20
68.0
1.4279
1.4267 3
1.0194
1.0186
21
69.8
1.4266
1.4254 6
1.0194
1.0186
22
71.6
1 .4253
1.42418
1.0193
1.0185
23
73.4
1.4240
1.4228 8
1.0193
1.0185
24
75.2
1.4227
1.4215 7
1.0192
1.0184
30
86.0
1.4146
1.41.34 3
1.0190
1.0182
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other peri-
odically varying quantities.
1 ^ is the phase difference in degrees.
} Or 2»r. Aprx. HX 10. Log. 0.798 1799.
112 ELECTRICAL CURRENT.
EIiEOTRIOAL CURRENT [1,1]; CURRENT STRBNQTH
or INTENSITY. (Electromotive force -^ resistance;
quantity -i- time.)
These units are used to measure the rate of flow or passa^ of units of
electricity per second, just as the rate of flow of water or air is measured in
units like cubic feet per second. The current in amperes is, according to
Ohm's law, equal to the electromotive force in volts divided by the resistance
in ohms; or, according to Joule's law, to the power in watts divided by the
volts; or to the square root of the quotient of the watts divided by the
ohms. These apply to direct currents without coimter-electromotive forces ;
they apply to alternating currents also when there is no reactance in the
circuit (caused by inductance or capacity), in which case they refer to the
effective current. In any alternating current circuit with reactance the
current in amperes is equal to the electromotive force in volts divided by
the impedance in ohms. An ampere is also equal to a passage of one
coulomb per second.
The unit imiversally used is the ampere, by which is here meant the
international ampere of the International Congress of 1893 at Chicago,
defined as e<iual to Ho of the C.G.S. electromagnetic unit of current; it was
made legal in this country by Congress in 1894. For practical purposes it
is defined by that congress as the current which (under specified conditions)
deposits 0.001 118 gram of silver per second. t The ampere is, however,
preferably determined from the ohm and the voltage of a standard cell.
Owing to the slight discrepancy in the units of current, electromotive force,
resistance, and Ohm's law, the National Bureau of Standards has selected
two of these as the fundamental units, namely those of resistance and elec-
tromotive force established by the International Congress and legalized in
this country, and from these the ampere is derived, thus bringing all three
into agreement with Ohm's law. Individual Clark standard cells agree
with each other to within at least 2 parts in 10 000. , and by the use of care-
fully purified materials this agreement can be still closer, while different
determinations of the electro-chemical equivalent of silver differ by con-
siderably larger amounts, unless repeated under perfectly definite condi-
tions and made with great care. The Reichsanstsut measured the electro-
motive force of the Clark cell in terms of the ohm and the ampere based
on the silver voltameter, but obtained a slightly different value for this cell
from that defined by the International Congress; hence the ampere of the
Reichsanstalt, which is in agreement with the volt and ohm there used, is
slightly different from the ampere used in this country, which is based on
the international volt and ohm, although both these amperes were originally
intended to be the same. But this discrepancy, which is only about 8 hun-
dredths of one percent, is auite negUgible in ordinary practice. According
to Weston the concrete volt, ampere, and ohm as, denned by the Chicago
Congress are in agreement with each other to within one parr in 1000.
In some of the relations with other units, such as the absolute, the mag-
netic, the energy units, etc., the value of the international ampere as above
defined is in the following tables assumed to be equal to the theoretical
value, namely Ho of the electromagnetic C.G.S. unit, which value is some-
times called the true ampere.
The electromag:netic C.G.S. unit (or absolute unit) is that current
which, flowing in the circumference of. a circle of one centimeter radius,
will, for every centimeter length of circumference, exert in air a force of one
dyne on a unit magnetic pole placed at the center* one whole circumfer-
ence therefore exerts a force of 2;r dynes on that pole.^ Or under the same
conditions, every centimeter of the circumference will in air i)roduce at the
center a magnetic field of one unit density (one gauss), that is, one unit of
magnetic flux (one maxwell) per square centimeter.
The electrostatic C.G.S. unit (or absolute unit) is the current which
flows when one electrostatic C.G.S. unit of quantity passes per second.
t This is the value used in caUbrating the Weston amperemeters.
< 4
I t
4 I
ELECTRICAL CURRENT. 113
BLKCTRICAIi CURRKNT.
** Accepted by the National Bureau of Standards.
* Checked by Dr. Frank A. Wolff, Jr., Asst. Phys. National Bureau of
Standards.
Aprx. means within 2%. Bv "ampere" is meant the international am*
pere, assumed to be equal to the international volt divided by the interna-
tional ohm, unless otherwise stated, v is the velocity of light.
Logarithm
1 CGS unit (elst) — 1 abstatampere 0000 0000
•• =10V» microampere. About >i-*-l 000 1.522 8787
' • = lO/r ampere. About M X 10-» IQ.522 8787
= l/t> CGS unit (ehng). About H X 10-»o.. . 11-622 8787
1 abstatampere — 1 CGS unit (elst) 0000 0000
1 microampere » v/W CGS units (elst). About 3 000. . 8.477 1213
'* «0.000 001 ampere g.OOO 0000
1 milllampere -v/lO 000 CGS units (elst). About 3 000 000. 6-477 1213
=» 0.001 ampere 3-000 0000
" - 0.0001 CGS unit (ebng) 4-000 0000
1 ampere [A, a]=- v/lO CGS units (elst). About 3X lO^. 9477 1213
— 1000. milliamperes 8-000 0000
—0.999 198** ampere of Reichsanstaltf 1-999 6516
-= 0.1 CGS unit (elmg) I-OOO 0000
1 International ampere = 1 ampere, which see above Q-000 0000
1 true ampere — 0.1 CGS unit (elmg) 1000 0000
1 ampere of Reichsantitalt = 1.000 803** amperesf 0-000 8484
1 ampere of Nat. Bureau of Standards = 1 volt + 1 ohm.
1 CGS unit (ehng) =• v CGS units (elst). About 3 X 10 o 10-477 1213
* • =10. amperes l-OOO 0000
" =1 absampere 0-000 0000
1 absampere = 1 CGS unit (elmg) 0-000 0000
1 kiloampere = 1 000. amperes 3-000 0000
• • = 100. CGS units (ehng) 2-000 0000
1 absolute unit — 1 CGS unit, either elst or elmg 0*000 0000
The relations to other measures are as follows: t
Amperes —volts -^ohms.
'* — coulombs + seconds.
* ' — watts ■*- volts.
* ' — 1 000 X kilowatts -J- volts.
** — ^( watts -s- ohms).
'* — joules -»-( volts X second ).
*' —V[jo\ile8-s- (ohms X seconds)].
Kate of change of amperes per second:
— induced volts -i-henrys.
—induced volts -s- (ohms X time constant in seconds).
Final amperes:
— applied volts X time constant in seconds -f-henrys.
— joules of kinetic energy of the current X ohms X 2 -{-(henrysX applied
volts).
— ^^(ioules of kinetic energy of the current X 2 -s- henr^rs).
—joules of kinetic energy of the current X 2 + (applied volts X time con-
stant in seconds).
— VTJoules of kinetic energy of the current X 2 -»- (ohms X time constant
in seconds) .
t See explanatory notes above.
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other periQO^
iQf41y v{^rying quantitiea-
114 ELECTRICAL CURRENT; CURRENT DENSITY.
When the flux is due only to the current, as in self-induction, and when
there is no magnetic leakage:
Final amperes:
= (maxwells X number of turns) -f- (henrys X 10^).
«= (maxwells X number of turns) + (time constant in seconds X ohms X
lOS).
—ergs of kinetic energy of the current X 20 -i- (maxwells X no. of turns).
«= joules of kinetic energy of the current X 2 X 10^ ■*- (maxwells X number
of turns).
When there is magnetic leakage, substitute in the above for the quantity
*' maxwells X number of turns," the mean flux turns, that is the "mean
maxwells X number of turns." Thus:
Final amperes:
=(mean maxwells X number of turns)-*- (henrys of self-induction X 10**).
When the flux is from an external source, and independent of the current
as in mutual induction, and when there is no magnetic leakage:
Final amperes:
—ergs of kinetic energy of the current X 10 h- (maxwells X no. of turns).
= joules ofkinetic energy of the current X 10^ -s- (maxwells X no. of turns)
When there is magnetic leakage, make the same substitution as described
above.
Final amperes in primary:
=mean maxwells through secondary X secondary turns -»- (henrys of
mutual induction X 10*).
For alternating-current circuits f:
Amperes:
«= watts -!- (volts X cos ^ J ).
= VT watts H- (ohms X cos <f> t )].
•^Vtapplied vol ts^— induced volts^ + ohms resistance.
= farads X volts X frequency X 6.283 19 § .
= microfarads X volts X frequency X 6.283 19 § -s- 1 000 000.
= induced volts + (henrys X frequency X 6. 283 19 § ).
—induced volts ^ (time constant in seconds X ohms resistance X fre-
quency X 6.283 19§).
—applied volts +\/[(henrysX frequency X 6.283 19 §)*+(ohms res.)*].
CURRENT DENSITY. (Current -t- surface.)
These units are used to measure the current flowing through a unit cross-
section of a wire ojr other conductor; or the amount of current flowing into
or out of a unit surface of an electrode in an electrolyte.
Aprx. means within 2%.
1 ampere per sq. meter: Logarithm
= 0.092 903 4 ampere per sq. foot. Aprx. ^Via -•-10 J 968 0317
= 0.01 ampere per sq. decimeter 2-000 0000
= 0.000 645 163 ampere per sq. inch. Aprx. Vn •*■ I 000 1-809 6602
1 ampere per sq. foot:
= 10.763 87 amperes per sq. meter. Aprx i%i X 10 1 031 9683
— 0.107 638 7 ampere per sq. decimeter. Aprx.i%i-»-100... 10819683
— 0.(X)6 944 44 ampere per sq. inch. Aprx. %ooo 5-841 6875
1 ampere per sq. decimeter:
= 100 amperes per sq. meter 2-000 0000
= 9.290 34 amperes per sq. foot. Aprx. iVi2 X 10 968 0817
= 0.064 5163 ampere per sq. inch. Aprx."^! -*-10 3-809 6692
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other peri-
odically varying quantities.
t ^ is the phase difference in degrees.
} Or 2«. Aprx. HXIO. Log 0-798 1799.
CURRENT density; ELECTRICAL QUANTITY. 115
Logarithm
1 ampere pejr gq. inch:
*» 1 550.00 amperes per sq. meter. Aprx. ^^ X 1 000 .... 8190 3308
"" 144. amperes per sq. foot. Aprx. ^ X 1 000 2-158 3625
"■ 15.500 amperes per sq. decimeter. Aprx. ^V^ X 10. . . 1.190 3308
•"0.785 398 ampere per circular inch. Aprx. %o 1-895 0899
">0.155 000 ampere per sq. centimeter. Aprx. ^s • ....... 1-190 3308
="0.121 736 ampere per circular cm. Aprx. 12 + 100 1-085 4207
1 ampere per circular incli:
= 1.273 24 amperes per sq. inch. Aprx. i% Q-104 9101
—0.197 352 ampere per sq. centimeter. Aprx. %o J-295 2409
»0.155 000 ampere per circular cm. Aprx. %8 1-190 3308
1 ampere per sq. centimeter:
= 929.034 amperes per sq. foot. Aprx. 1^2 X 1 000 2-968 0817
a 100. amperes per sq. decimeter 2 000 0000
— 6.45^ 63 amperes per sg. inch. Aprx. 6H 0-809 6692
" 5.067 09 amperes per circular inch. Aprx. 5 0-704 7591
"-0.785 398. ampere per circular centimeter. Aprx.%o* •- . 1-895 0899
1 ampere per circular centimeter:
=8.214 47 .amperes per sq inch. Aprx. % X 10 0-914 5798
=6.451 63 amperes per circular inch. Aprx. 1% 0-809 6692
— 1.273.34 amperes per sq. centimeter. Aprx. 1% 0104 9101
1 ampere p^r sq. millimeter:^
— 0,785 398 ampere p^r circular mm. Aprx. Mo 1-895 0899
=0.000 645 163 ampere per sq. mil. Aprx. Tii -»- 1 000. . . 4. 809 6692
=0.000 506 709 ampere per circular mil. Aprx. }4-i-l 000 . i 704 7591
1 ampere per circular millimeter:
= .1,273 24 amperes per sq. millimeter. Aprx. '%.... 0-i04 9101
= 0.000.821 447 ampere per sq. mil. Aprx. M« -^ 100 1-914 5793
-0.000.645 163 ampere per circular mil.. Aprx. ^Ai + 1 000 1809 6692
1 ampere pep sq. ntil:
= 1 55P,00 amperes, per sq. millimeter. Aprx. ^h X 1 000.. 3-190 3308
= 1 217.36 amperes per circular mm. Aprx. % X 1 000. . . 3 085 4207
=0.785 398 ampere per circular mil. Aprx. %o 1-895 0899
1 ampere per circular mil:
= 1 973..5^.^peres per sq. millimeter. Aprx. 2 000 3-295 2409
= 1 550,00 ajD9Peres per circular mm. Aprx. ^V^ X 1 000 . . . 3-190 3308
= 1.273.^4 .amperes per sq. mil. Aprx. ^% 0-104 9101
ELECTRICAL QUANTITT [Q, q]; CHARGE. (Current
■ Xtime; capacity X electromotive force.)
These units are used to measure the amount of electricity as such, just
as a quantity of matter might be measured in the number of units called
molecules, which it contains. The number of units of electricity in a
given quantity of electricity is the same whether or not it is flowing in
the form of a current, or whether or not it is subjected to an electrical
pressure, just as the number of molecules in a given weight of air is the
same whether or not it is in motion, as in a wind, or whether or not it is
under pressure, as in compressed air. The quantity of electricity in
coulombs is equal to the current in amperes multiplied by the time in
seconds; or to the energy in joules divided by the voltage; or to voltage
multiplied by the capacity of a condenser in farads.
The unit generally used is the coulomb, by which is here meant the
international conlomb of the International Congress of 1893 at Chicago,
defined as "the quantity of electricity transferred by a current of one
international ampere in one second." It was made legal in this country by
Congress in 1894, and is accepted by the National Bureau of Standards.
For practical purposes it is that quantity which will deposit 0.001 118
gram of silver in a silver voltameter. In some of the relations with
other units, such as the absolute, the energy units, etc., the above value
is in the following tables assumed to be equal to the theoretical value,
namely Mo the electromagnetic C.G.S. unit. Another unit frequently
used, especially with batteries and in electrochemistry, is the ampere-
116 ELECTRICAL QUANTITY; CHAHQE.
hoar; it is the quantity of electricity transferred by a current x>f one
ampere in one hour.
The electromagnetic C.G.S. unit (or absolute unit) is the quantity
of electricity transferred by one electromagnetic C.G.S. unit of current
in one second. The electrostatic C.G.S. unit (or absolute unit) is
that quantity of electricity which in air exerts a force of one dyne on an
equal quantity one centimeter distant.
EIiECTRICAIi QUANTITY; CHARGE.
Apnc. means within 2%. By "coulomb" is meant the international
coiilomb. V is the velocity of light.
Logarithm
1 CGS unit [elst]:
— 1 abstatcoulomb 000 0000
— W/v microcoulomb. About H-i-l 000 j 522 8787
— 10/t> coulomb. About H X 10-» lQ-622 8787
— l/v CGS unit (elmg). About H X 10" lo n-522 8787
= 1 -»-(360 v) ampere-hour. About 9.259X 10"" ll-96e 6762
1 abstatcoalomb =« 1 (XjtS unit (elst) ; . . 0-000 0000
1 microcoulomb:
— vX 0.000 000 1 CGS units (elst). About 3 000 8477 1218
— 0.000 001 coulomb B-OOO 0000
— 0.000 000 1 CGS imit (elmg) f.QOO 0000
« 2.777 78 X 10-10 ampere-hour. Aprx. ^H X lO-io 10448 6976
1 coulomb [C, c] :
— t>/10 CGS units (elst). About 3 000 000 000... . 9477 1218
— ■ 1 000 000. microcoulombs 6-000 0000
— 0.1 CGS unit (elmg) T.OOO 0000
-0.000 277 778 ampere-hour. Aprx. i)^-s- 10 000 4448 6975
1 international coulomb — l coulomb, which see above.
1 true coulomb = 0.1 CGS unit (elmg) I.OOO 0000
1 ampere-second »1 coulomb, whicn see above.
1 CGS unit (elmg):
*- 1 abscoulomb OOO 0000
— V CJGS units (elst). About 3 X 10^ 10-447 1218
— 10 coulombs 1-000 0000
=0.002 777 78 ampere-hour. Aprx ^Ji+l 000 §.443 6975
1 abscoulomb » 1 CGS unit (elmg) 0-000 0000
1 ampere-hour [ahl:
-r X 360. CGS units (elst). About 1.08 X 10i» 18083 4288
— 3600. coulombs 8-556 8028
= 360. CGS imits (ekng) 2-556 8025
1 absolute unit =» 1 (XjS unit, either elst. or ehng 0<000 0000
The relations to other measures are as follows :t
Coulombs » amperes X seconds.
*• -= amperes X hours X 3 600.
• * » volts X seconds + ohms.
•• "» watts X seconds + volts.
*• =-'y/(watt8X8econds2-t-ohms).
' * «■ joules ■*- volts.
•* «» \/(ioules X seconds -*- ohms).
" = farads X volts.
* • =- microfarads X volts -s- 1 000 000.
Ampere-ho urs — coulombs -J- 3 600.
" » amperes X hours.
■= volts X hours -*- ohms.
» watts X hours -*- volts.
«=■ \/( watts X hours* -*- ohms).
"=» joules -s- (volts X 3 600).
- VTJoules X hours + (ohms X 3 600).
Microcoulombs » microfarads X volts.
t Treatises on alternating currents should be consulted for the limitinji
conditions under which these relations apply to alternating or Qther peri"
odicaUy varying quantities.
«*
•I
ti
4«
ELECTRICAL CAPACITY. 117
ELBOTRIOAL CAPACITT [O, c]. (Quantity -h electro-
motive force.)
These units are used to measure the ability of a body (like a condenser)
to hold charges of electricity (measured in coulombs) under electrical
stress, pressure, or potential (measured in volts). The capacity of a con-
denser is greater the greater the charge in coulombs that it will hold for the
same pressure in volts, or the less the pressure in volts required for the
same charge. The capacity in farads is equal to the charge in coulombs
divided by the electromotive force in volts. In some respects an electrical
capacity is analogous to the capacity of a closed vessel to hold air under
pressure, the greater the pressure the greater the quantity of air, yet the
capacity of the vessel remains the same and could be measured in terms
of the air and its pressure.
The unit imiversally used is the microfarad or millionth of a farad;
by farad is here meant the International farad of the International
Congress of 1893 at Chicago, defined as equal to one international coulomb
divided by one international volt. It was made legal in this country by
Congress in 1894, and is accepted by the National Bureau of Standards.
As the farad is an inconveniently large unit, never occurring in practice,
the microfarad is generally used. The electromagnetic C.O.8. nnit (or
absolute unit) is the capacity of a condenser which when charged at one
C!.G.S. imit of potential will hold one C.G.S. unit of quantity. The elec-
trostatic C.G.S. unit (or absolute unit) is similarly defined with respect
to the electrostatic units of potential and quantity.
By "farad" is meant the international larad. v is the velocity of li^ht.
Logarithm
1 COS unit [elst]« 1 abstafarad 0-000 0000
= 10i«/»2 microfarad. About % X 10-5 g.045 7575
= 109/t;2 farad. About % X 10"" 13046 7575
= IM COS unit (ehng). About % X 10-» . 21.045 7575
1 ab8tafara<l »1 CGrS unit (elst) 000 0000
1 microfarad = v'X 10" »« (XIS units (elst). About 9 X lO^ 5-954 2425
" = 1 microcoulomb -?- 1 volt 0-000 0000
* • =0.000 001 farad gOOO 0000
•• « lO-w COS unit (elmg) 15-000 0000
1 farad [F]- v^X 10-» CXJS units (elst). About 9X 10" 11954 2425
** -1 000 000. microfarads 6 000 0000
•« =, 10-9 COS unit (elmg) 5-000 0000
1 international farad = 1 farad, which see above. . 0-000 0000
1 COS unit (ehng)= v^ CGS units (elst). About 9X1020.. . 20954 2425
= 101* microfarads 15000 0000
= 10* farads 9-000 0000
ss 1 abfarad 0-000 0000
1 abfarad = 1 CGS unit (elmg) 0000 0000
1 absolute unit — 1 CGS unit, either elmg or elst 0-000 0000
The relations to other measures are as follows: t
Farads » coulombs -^ volts.
* * «= joules of stored energy X 2 + volts^.
Microfarads^ coulombs X 1 000 000. -s- volts.
• * = microcoulombs -s- volts.
* * = joules of stored energy X 2 000 000. -h volts*.
For alternating-current circuits t
Farads — amperes -*- (volts X frequency X 6.283 19 t )•
• • = _ 1 ^. (ohms reactance X frequency X 6 .283 1 9 J ).
Microfarads = amperes X 1 000 000 -*- (volts X frequency X 6.283 19 t \
• ' =-1 000 000. -5- (ohms reactance X frequency X 6.283 19 T ).
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other period-
ically varjring quantities.
t Or 2ar. Aprx. H X 10. Log 0-798 1799.
i 4
* I
< I
14
4 (
118 INDUCTANCE.
INDUCTANCE [L, 1]; COEFFICIENT of SELF- or
MUTUAL INDUCTION. (E.m.f. -^ (current -^ time) ; re-
sistance X time; number of turns X flux •^ current; kinetic
energy 4- square of current.)
These units are used to measure the intensity of that property by virtue
of which an electromotive force is produced by changes of current in a
neighboring circuit (as in a transformer) or by changes of current in the
circuit itself; in the former case the phenomenon is called mutual induc-
tion, and in the latter self-induction. An electric current has a property
analogous in some respects to the inertia of a heav^^ moving body; a cur-
rent resists momentarily any chajige in its strength, just as a heavy moving
body resists anv change in its velocity. When a current is started, it en-
counters for a snort time (see time-constant below) a counter-electrontotive
force in its pwn circuit due either to itself (self-induction) or to a current
in the opposite direction which it is inducing in a neighboring circuit (mutual
induction). Similarly, if stopped it tends to prolong itself either in its own
circuit or (by induction) in a neighboring circuit; the quicker the change
the greater this tendency. This property is due to the inductance.
The term induction in electrodynamics applies broadly to the general
phenomenon of the generation of an electromotive force (which may or may
not produce a current) by magnetic flux, whether it be that of a magnet or
that surrounding a current. The terms self and mutual induction apply
to the special cases of this phenomenon when the induction is produced by
a current in its own circuit or in a neighboring circuit respectively, there
being no mechanical motion. The terms coefficient of sell or coefficient
of mutual induction apjjly to the numerical value of the intensity of this
phenomenon, by which it is measured; the terms ftelf-iiiductance and
mutual iiidiictHnce are now more generally used instead. The term
inductance applies to both of these coefficients, and is therefore the
measure of self or mutual induction. The inductance factor is the ratio
of the wattless volt amperes to the total volt amperes; or the ratio of the
wattless component of the current or e.m.f . to the total current or e.m.f .
The inductance depends for its value on the geometric conditions of the
circuit, and varies with the size and shape of the circuit or of a coil, with
the square of the number of turns of a coil, with the distance between the
wires, etc. ; it also varies according to an irregular law with the presence of
iron or other magnetic material. An inductance, in henrys, is measured
by and is equal to the electromotive force induced, in volts, divided by the
rate of change of current in amperes per second, which causes it; it is also
equal in henrys to twice the kinetic (magnetic) energy of the circuit in
joules divided by the square of the final current in amperes; a coefficient
of self-induction in henrys is also equal to the resistance of the circuit in
ohms multiplied by its time-constant (see below) in seconds. The induct-
ance of any particular circuit is also the constant relation of the product
of the magnetic flux and the turns, to the current producing; the flux.
In the C.G.S. system of units, on which the practical unit is based, this
measure happens to be of the same kind as a length; this is a consequence
of what is called a "suppressed factor" in the dimensional formula, and as
it is misleading and answers no useful pun)ose, the coincidence should not
be given any importance. While it is not incorrect to express inductances
in centimeters, it misleads and is not good practice. The electromotive
force produced by inductance is generated precisely as in dynamos by the
cutting of magnetic flux or lines of force and at the rate of 10^ such lines
(maxwells) per second for each volt, the magnetic flux in inductance being
that produced by the current, and is proportional in amount to the change
in the current stren^h. In the case of djrnamos the wire moves and the
flux is at rest, while m inductance the wire is at rest and the flux moves.
The unit now most generally used is the lienry. By henryis here meant
that of the International Congress of 1893 at Chicago, defined as the induc-
tion in a circuit when the electromotive force induced in this circuit is one
< 4
« <
INDUCTANCE. 119
international volt, while the inducing current varies at the rate of one inter-,
national ampere per second. It was made legal in this country bv Con-
gress in 1894, and is accepted bv the National Bureau of Standaras. In
some of the relations in these tables, this value is assumed to be equal to
10^ C.G.S. electromagnetic units of inductance. This unit was formerly, and
for some years officially, called a **quadrHnt." If the self-inductance of
a given circuit (usually a coil) is one henry, it means that the magnetic lines,
of force corresponding in that circuit to one ampere will form 10^ linkages
of unit magnetic lines of force (maxwells) witn that circuit. The- unit
called sec-ohm was at one time used, as in self-induction the inductance
is equal to the product of the resistance in ohms, and the time-constant of
the circuit in seconds.
The electromaf^netic C.G.S. unit (or absolute unit) is that induction
which will induce one C.G.S. unit of electromotive force by a change of cur-
rent at the rate of one C.G.S. unit of current per second. It happens to be
numerically equal to 1 centimeter of length and is sometimes so represented.
The electrostatic C.G.S. unit (or absolute unit) is similarly defined with
respect to the electrostatic units of electromotive force and current.
INDUCTANCE.
V is the velocity of light. Logarithm
1 CGS unit (elmg) » 1 centimeter Q 000 0000
• • —0.001 microhenry S-OOO 0000
' • — 10-» henry §.000 0000
- l/t)2 CGS unit (elst). About H X 10-» . . 51045 7575
1 centimeter inductance — 1 CGS imit (elmg) 0-000 0000
1 microhenry — 1 000. CHjS units (elmg) 8. 000 0000
-0.000 001 henry 6000 0000
1 millihenry— O.OOl henry §000 0000
1 henry [H]- 10» CGS units (elmg) 9000 0000
— 10 000. kilometers, or 1 earth's quadrant 4-000 0000
- 109/i>2 CGS unit (elst). About Vn X 10"" 12045 7575
1 quadrant = 1 henry 0000 0000
1 quad — 1 quadrant or henry 0-000 0000
1 ft«c-ohm — 1 henry 0000 0000
1 CGS unit (elst) - v^ CGS units (ehng). About 9 X 10* 20-954 2425
•• - r2 centimeters. About9Xl0» 20-954 2425
** -«2x 10-9 henrys. About 9 X lO^* 11-954 2426
. The relations to other measures are as follows :t
Henrys — induced volts -i- rate of change of amperes per second.
' ' — time constant in seconds X ohms.
' ' — time constant in seconds X applied volts -t- final amperes.
' ' — joules of kinetic energy of the current X 2 -*- final amperes^.
* * — joules of kinetic energy of the current X ohms^ X 2 -r- applied volts*.
When the flux is due only to the current, as in self-induction, and when
there is no magnetic leakage:
Henrys of self-induction:
— (maxwells X number of turns*) -f- (ampere-turns X 10®).
— (maxwells X number of turns) -f-( final amperes X 10®).
— (number of turns X 0.000 1)2X 1.256 64 % -J- oersteds.
When there is magnetic leakage, substitute in the above for the quantity
" maxwells X number of turns," the mean flux turns, that is, the " mean
maxwells X number of turns." Thus:
Henrys of self-induction!
— (mean maxwells X number of turns) -t- (final amperes X 10®).
When the flux is from an external source and independent of the current,
as in mutual induction , and when there is magnetic leakage :
Henrys of mutual induction:
— (mean maxwells through the second ary X secondary turns) + (final
amperes in primary X 10®).
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other period-
ically varying quantities.
X Or 4ir+10. Aprx. i%. Log 0-099 2099.
120 inductance; time-constant.
For altomating current circuits :t
Henrys'- induced volts + (amperes X frequency X 6.283 19 1)*
'* —ohms reactances (frequency X 6.283 19t).
* * — >/(ohms impedance' — ohms resistance') -i- frequency X 6.283 10 1 •
•• — vlCapplied volts + amperes)' — ohms resistance'] + frequency X
6.283 19 1
Indnetance factor:
—wattless component of current or e.m.f.-i- total current or ejn J.
■- y/i 1 — power factor*).
TIME-OONSTANT (of inductive circuita). (Inductance -^
resistance; time.)
When a current is started in a circuit containing inductance of the kind
called self-induction, as is the case for instance in coils, particularly if they
have many turns and iron cores, the current will not reach its full value
instantly, but owing to the self-induction it will at first be ojpposed by a
counter-electromotive force due to the inductance : this opposition will grow
less and less until the current has reached its full strength. It is similar to
what takes place when a heavy weight like a street car is started to move ;
its inertia will at first oppose the moving force, but this opposition will |^ow
less and less as the speed increases until the full, constant speed is attained,
and the inertia will then offer no further opposition.
It is sometimes of importance to know how loxig it takes before the cur-
rent has reached its ultimate value, but theoretically it takes an infinite
time, and therefore it is usual to state the time that it takes the current to
rise to a certain definite fractional part of its fuU value, namely nearly ^
(the exact figure is given below), and this time is called the "time-constant"
of that circuit. This time in seconds (often a very small fraction of a
second) is equal to the self-induction in henrys divided by the resistance
in ohms; or instead of the ohms one may of course use the applied volts
divided by the final steady current in amperes. This time-constant is
therefore greater the greater the self-induction and the less the resistance.
It ^ves more information about a circuit than the mere inductance does,
as it includes the resistance; the self -inductance of a coil, for instance, is
the same whether the wire is made of copper or of a high resisting metal,
but the time-constant is less in the latter case.
The exact fractional part of the fuU value of the current, above referred
to, is (« — 1 ) -t-«, in which e is the base of the Naperian logarithms. Numer-
ic^ly this is equal to 63.212%, or nearly %.
The unit in which time-constants are always given is the second, hence
there are no reduction factors. The most important relations to other
measures are as follows:
Time-constant in seconds » henrys -f- ohms resistance.
' ' — henrys X final amperes + applied volts.
For further relations see those for henrys under inductance, and divide
them by the ohms resistance.
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other period-
ically varying quantities.
t Or 2k. Aprx. % X 10. Log 0.798 1799*
FREQUENCY. 121
FRfiQUENOTj PERIODIOITT; PERIOD; ALTERNA-
TIONS, (l-s-time; time.)
Freqaencj or periodicity is the number of recurrences or cycles of
some periodic or wave phenomenon or oscillation during a given time which
is always understood to be a second unless otherwise stated; the frequency
always refers to the number of complete cycles. The number of alterna-
tions, however, refers to the number of changes of the direction or to the
reversals, and therefore refers to half-waves, and is always equal to double
the frequency, if the time is the same. The period is the time of one com-
plete wave or oscillation^ and is therefore the reciprocal of the frequency.
The term " frequency " is the one most generally used, and always refers
to a second; the term " number of alterations" is preferred by some, and
when it refers to electric currents the time is usually a minute; the term
" period" is used comparatively rarely as a measure, its use being gen-
erally limited to scientific discussions; the unit is generally the second.
In mathematical discussions of electric alternating-current problems the
frequency is often replaced by an angular velocity, generally represented
by at and measured in terms of radians per second (see under Angular Veloc-
ities above). Then a; = 2;m, in which to is in radians per second and n is the
true frequency in cycles per second, a cycle being here considered the
same thing as a complete revolution >
The frequency is also equal to the velocity of propagation divided by
the wave-length. The wave-lengths are therefore measured in tmits of
length, but when the velocity for a class of waves is a constant (as those
of ught or the electromapietic waves), the wave-lengths may also be in-
dicated in units of time, in which case a wave-length oecomes equal to the
period of the wave. Wave-length should not be confounded with the
amplitade, which measures the intensitv of the wave and has nothing
to do with the frequency, period, or wave-length.
If n is the frequency per second [CO], then:
the period in seconds = 1/n;
the number of alternations per minute = 120n.
If n is the number of alternations per minute, then:
the frequency per second = n/l 20;
the period in seconds » 120/n.
tf n is the period in seconds, then:
the frequency per second = 1/n ;
the number of alternations per minute » 120n.
If n is the frequency in cycles per second, and (a the angrnlar Telocity
in radians per second, and if a cycle is represented by one complete revolu-
tion, then:
w=»2ffn; or
n» 0.159 155a>; and
«i» 6.283 197k
An electrical degree is the 360th part of one complete cycle.
For the relations of frequency to other measures see those relations be-
tween other measures which involve the frequency, chiefly under farads
and henrys.
122 ELECTRICAL ENERGY.
EINETIO ENERGY of a CURRENT in a CIRCUIT [W].
(Inductance X current^ ; power X time-constant ; energy.)
' As was explained above under "'Time-constant/' it takes an appreciable
time to start a current in a circuit having inductance. During this interval
of time energy is being stored up in the circuit, which is given back again,
usually in the form of a spark, when the current is stoppled; this is called
the kinetic energy of the current in the circuit. It has an analogy in the
storing of energy in a heavy body like a street car when it is started to move,-
which enerigy is given back again when the body is stopped.
This kinetic energy is greater the greater the seli-induction and the
greater the current. It is equal in joules to half the product of the self-
inductance in henrys and the square of the current in amperes at any
instant, and therefore also of the amperes of the final steady current. Sim-
ilarly in mechanics, this kinetic energy is equal to half the product of the
mass and the square of the velocity at any instant, and therefore also of
the final steady velocity. This energy is stored in the form of magnetic
energy, and during that time it is potential energy; it remains stored as long
as the current continues, and is given out again when the current ceases.
Ir a transformer it is the kinetic energy of theprimary circuit which is trans-
mitted to and is led out by the secondary. The practical unit is the joule,
and the C.G.S. unit is the erg,
IThe chief relations to other measures are as follows *
Joules of kinetic energ^y of the current = henrys X final amperes^ + 2.
£rgs of kinetic energy of the current = henrys X final ami>eres2 X 5 X 10^.
;For further relations see under units of Electrical Energy, below.
EXiECTRICAL ENERGY OR WORE [W]. (QuanUtyX
electromotive force; current^ XresistanceX time; power
X time J power -^ frequency.)
The units of electrical energy are all convertible directly into units of
other kinds of energy, as energy is the one quantity common to all the
systems of units.-' The electrical and absolute units have therefore been
included with the mechanical, thermal, and other units in the general table
of all the units of Energy (p. 74); the present table is limited to a few
specific values and to some relations between the electrical unit of energy
and other electrical units.
The energy in joules delivered to a circuit is equal to the electromotive
force in volts multiplied either by the quantity of electricity in coulombs,
or by the product of the current in amperes and the time in seconds. These
apply to direct Currents; in alternating-current calculations involving
energy, the power and not .the work done is generally the important- con-
sideration; the energy of alternating currents of any wave form delivered
per cycle, in joules, is equal to the power in mean watts divided by the
irequency. For the energy stored in a current see the preceding section
on Kinetic Energy.
The unit universally used is the Joule, by which is here meant the joule
of the International Congress of 1893 in Chicago, defined as equal to 10^
C.G.S. units of work (ergs) and represented sufficiently well for practical
use by the energy expended in one second by an international ampere in
an international ohm ; in the relations in these tables this defined value is
used. It was niade legal in this country by Congress in 1894 and is accepted
by the National Bureau of Standards. , Sometimes the ampere-hour is used
as the unit of electrical quantity, in which case the corresponding unit of
energy becomes the volt-ampere-hour, usually called the watt -lion r; the
kilowatt-hour ( = 1000. watt-hours) is also common. The electro-
magnetic C.G.S. unit (or absolute unit) is the erg:, defined as the work
of one dyne acting through one centimeter. The electrostatic C.G.S. unit
(or absolute unit) is this same erg.
r
I
ELECTRICAL ENERGY, 123
ELECT KICAIi BNBKOT.
Aprx. means within 2%.
Logarithm
1 CGS unit (ehng) — 1 erg O-OOO 0000
1-CGS unit (elst) = l erg 0000 0000
1 absolute unit — 1 erg. 000 0000
^ ??^" } SS§ "^?* J®te?) • 0000 0000
.< ",^i P^? ^*^* ^®^^^ 0000 0000
-10-7 joule 7.000 0000
1 microjoule - 10. ergs ....;..-.. l;000 0000
1 Joule [J]- 10 000 000. ergs 7.000 0000
-0.000 277 778 watt-hour. Aprx. %i-M 000 4443 6975
1 kilojonle — 1 000. joules 8000 0000
1 watt-hour»3 600. joules 8-556 3025
" — 3.6 kilojoules 0-556 3025
1 kilowatt-hour « 3 600 000. joules 6-556 3025
" -» 1 000. watt-hours 8-000 0000
For further conversion factors see table of units of Energy, page 74.
The relations to other measures are as follows: f
tFonles = volts X coulombs.
* * — volts X amperes X seconds.
•• — volts* X seconds -»- ohms.
•* — amperes* X ohms X seconds .
** — ohms X coulombs* + seconds.
* * — watts X seconds.
Joules of stored energy = farads X volts* +2.
* * " - microfarads X volts* -f- 2 000 000.
KrgH of stored energy— microfarads X volts* X 5.
Joules of kinetic energy of the current:
—henrys X final amperes* + 2.
« henrys X applied volts* -*- (ohms* X 2) .
-e time-constant in seconds X ohms X final amperes* -t- 2.
—time-constant in seconds X final amperes X applied volts -i- 2.
—tune-constant in seconds X applied volts* -4- (ohms X 2).
Ergs of kinetic energy of the current = henrys X final amperes*X5XlO».
When the flux is due only to the current, as in self-induction, and when
there is no magnetic leakage:
Joules of stored energy = maxwells Xampere-tumsn- (2X10**).
Krg^ of stored energy— maxwells X ampere-turns + 2().
When there is magnetic leakage:
Joules of stored energy = mean maxwells X ampere-turns -«- 2X10*.
£rg^s of stored energy = mean maxwells X ampere-turns 4- 20.
When the flux is from an external source, and independent of the cur-
rent, as in mutual induction, and when there is no magnetic leakage:
Joules of stored energy — maxwells X ampere-turns -t- 10^.
JSrg^s of stored energy — maxwells X ampere-turns -s- 10.
When there is magnetic leakage substitute "mean maxwells" for
'• maxwells."
For alternating current circuits: t
HOT aiiernaung currcni^ circuits: t
Joules per cycle — watts + frequency.
—effective am E>eres X effective volts X cos ^-(-frequency.
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other peri-
odically varying quantities.
124
ELECTRICAL POWER.
BLEOTRtOAIi POWSR [P]. (Current X electromotive
force; energy 4- time; energy X frequency.)
The units of electrical power are all convertible directly^ into units of
other kinds of power, and the electrical and absolute units have there-
fore been included with the mechanical, thermal, and other units in the
general table of all the units of Power (p. 80). The present table is lim-
ited to a few specific values and to some relations between the electrical
unit of power and other electrical units.
The power in watts is equal to the energy in joules divided by the time
in seconds; this applies to both direct and alternating currents whether
there is phase shiftmg in the latter case or not. According to Joule's
law, the power in watts is also equal to the square of the current in am-
peres multiplied by the resistance in ohms, or to the current in amperes
multiplied by the electromotive force in volts, or to the square 01 the
electromotive force in volts divided by the resistance in ohms. These
apply to direct currents; they apply to alternating currents also, but
only under certain conditions, the cnief one of which is that there is no
reactance in the circuit and therefore no phase shifting (caused by in-
ductance or capacity), in which case the relations refer to the effective
values of the current and electromotive force. For further information
treatises on alternating currents should be consulted. In any case, these
relations give the true power when the result is multiplied by the power
factor (see below).
The unit universally used is the watt, by which is here meant the
watt of the International Congress of 1893, m Chicago, defined as equal
to 10^ C.G.S. units of power (erg per second) and represented sufficiently
well for practical use by the work done at the rate of one joule per second;
in the relations in these tables this defined value is used. It was made
legal in this country by Congress in 1894, and is accepted, by the National
Bureau of Standards. The kilowatt ( = 1 000 watts) is also common.
The electromagrnetic C.G.S. unit (or absolute unit) is an erg; per
second. The electrostatic C.G.S. unit (or absolute unit) is the same
eris per second.
Power Factor is a term used to show the amount of true power con-
tained in a given amount of apparent power. It is the ratio of the true
power to the apparent power. Its use is limited chiefly to electric power
generated by alternating currents. With direct electric currents, the
power is equal to the product of the volts and the amperes, and is called
watts; with alternating currents, however, this is true only when the
volts and amperes are exactly in phase with each other, which often is not
the case. When there is such a diflFerence in phase, that is when the current
lags behind or precedes the voltage, their product is only apparent
power and is usually measured in terms of the product of the volts and
the amperes and called volt-amperes. If the true power in such a case
is measured in watts, then the power factor will be the number of watts
divided by the number of volt-amperes, and it will always be less than
unity, in practice usually between 0.7 and 0.95. For true sine waves,
the real power in watts is equal to the voltage X current X cos ^, in which
4» is the angular phase diflFerence; hence it follows that in such cases the
power factor is numerically egual to cos <f>, whose numerical value is found
directly from a table of cosines. Sometimes the power factor is stated
in percent, in which case it is equal to the above figure multiplied by 100.
The inductance factor is the ratio of the wattless volt-amperes to
the total volt-amperes; or the ratio of the wattless component of the
current or the e.m.f. to the total current or e.m.f. The sum of the squares
of the inductance factor and the power factor is equal to unity.
4t
«t
<(
«1
• t
ELECTROCHEMICAL EQUIVALENTS. 125
jSIiECTBICAIi POWEB.
Logarithm
1 CGS unit (elmg) >- 1 erg per second 0-000 0000
1 CGS unit (elst) = 1 erg per second 0-000 0000
1 absolute unit = 1 erg per second 0-000 0000
1 erg per second » 1 CGS unit Celmg) 0-000 0000
=■1 CGS unit (elst) 0-000 0000
" =• 10-7 watt 7.000 0000
1 microwatt « 10. ergs per second 1-000 0000
1 watt [W, w] = 10 000 000. ergs per second 7000 0000
" . » 1 joule per second 0000 0000
1 kilowatt « 1 000 watts 3-000 0000
For further conversion factors, see table of units of Power, page 80.
The relations to other measures are as follows: t
lYatts * volts X amperes.
= amperes^ X ohms.
= volts*-*- ohms.
» coulombs X volts -i- seconds.
-■ coulombs* X ohms + seconds.*
-■ joules + seconds.
For alternating-current circuits: f
Watts -a volts X amperes X cos ^.
** « amperes* X ohms X cos ^.
•• ■» volts* X cos ^-f- ohms.
'* aeffec. volts Xefifec. amp. X ohms resistance -t- ohms impedance.
** = joules per cycle X f requeue jr.
Mean "watts »> effective volts X effective amperes X cos 4*,-
Poorer factor a- true power in watts + apparent power in volt-amperes.
** ' <->energy component of current or e.m.f.-i- total current or
e.m.f.
" —v'Cl — inductance factor*).
Inductance factor:
—wattless component of current or e.m.f. -»- total current or e-m.f.
— \/(l — power factor*).
BLEOTROOHEMIOAIi EQX7IVAI.IiNTS and DERIVA.
TIVES. (Weight -^ quantity of electricity; quantity of
electricity -4- weight; weight -i- energy ; energy -^ weight.)
The electrochemical equivalent of any chemical element or ion is the
amount by weight which changes its chemical combinajbion fter coulomb
of electricity during electrolysis. For these and their derivatives various
compound units are used such as millisrrams per coulomb, grams per
ampere-hour, pounds per ampere-hour, etc. The relations between
most of these are simply the relations between the respective units of
wei^t, which see under Weights. The reciprocals of these are also used
frequently.
Tue electrochemical equivalents of any elements or ions are, according
to Faraday's law, proportional to their atomic weights and inversely
proportional to their changes of valency. For determining the actual
values, that of some one element must be determined exi)erimentally,
after which all the others can be calculated. In the following relations
the value taken as a basis is the electrochemical equivalent of silver adopted
by the International Electrical Congress of Chicago in 1893, in the defi-
mtion of the ampere, and le^l in this country, namely 0.001 118 gram
per coulomb. The atomic weight of silver used in these relations is 107.93,
t Treatises on alternating currents should be consulted for the limiting
conditions under which these relations apply to alternating or other peri-
odically varsring quantities.
126 ELECTROCHEMICAL EQUIVALENTS. — ^DEPOSITS.
which is the usually accepted value on the basis of = 16. These funda-
mental values correspond with the usuaUy accepted value of the ionic
charge, 96 540. coulombs per monovalent gram ion, within the limits of
accuracy of the data. For a complete table of the equivalents and their
derivatives, of all the various elements and for various changes of valency
accompamed by descriptions of how to use them, see the author's Table
of ElectrochenucalEquvmlerUsand their Denvatives, in Electrochemical In-
dustry, Jan. 1903, p. 169. The following relations apply to all elements
or compounds. (The atomic weights are all based on oxygen = 16. but
the electrochemical equivalents are independent of whether the atomic
weights are based on 0"»16 or on H = l.)
Milligrams per coulomb = 0.010 359X1 «* • • u*
GramH per ampere-hour = 0.037 291 X ]■ "■^op^^c weight
Pounds per amp«re-hour» 0.000 082 21 X J c^^^i^Be of valency *
Grams per watt-hour =0.037 291 X 1
Kilosrrams per kilowatt hour =0.037 291 X «*r. * • k*
Kilog^rams per horse-power hour =0.027 806 X )■ atomic weight
Pounds per Isiiowatt-hour » 0.082 21 X I change of val. X volts.
Pounds per horse-power hour = 0.061 30 X J
Coulombs per miUigrram = 96.54 X ") chamre of valencv
Ampere hours per g:raiii =26.816 X ^ * ^ - vaiency
)
Ampere-hours per pound = 12 164. X J atomic weight.
change ofval.
atomic weight
X volts.
Watt-hours per g^ram =26.816 X
Kilowatt-hours per kilogrram = 26.816 X
Kilowatt-hours per pound = 12.164 X
Horse-power liours per kilogram = 35.964 X
Horse-power hours per pound = 16.31 3 X
Ionic charge for a monovalent gram ion = 96 539. coulombs.
For the amount of j^as in cubic centimeters at 0° C. and 760 mm mer-
cupr pressure, developed at one electrode, on the basis that one gram mole-
cule of a gas has a volume of 22.38 liters, the following relations exist, in
which n is the number of atoms per molecule :
Cb. centimeters per ampere-hour =834.6 -s-(nX change of valency).
Ampere-hours per cb. centimeter = 0.001 198 X n X change of valency.
ELECTROLYTIO DEPOSITS. (Mass -^ time; massH-
surface.)
Two kinds of units are used for measuring deposits, such as those in elec-
trolysis. One is for measuring the weight of the deposit on a limited sur-
face, and includes such units as grx'&ms per square decimeter, ounces
per square foot, etc., these are all given above in the same table as that
for Pressures (page 63) and are therefore not repeated here
The other kind of unit is for measuring the rate of deposition, and in-
cludes such units as millip^rams per second, pounds per day, tons
per year, etc.; the reduction factors for these are given in the following
table. The year is taken as equal to 365>^ days and the ton as equal to
the short ton of 2000. pounds.
Aprx. means within 2%.
Logarithm
1 pound (av) per year [Ib/yrJ:
= 0.002 737 85 pound per day. Aprx. ^^-^ 1 000 3437 4098
= 0.000 862 409 gram per minute. Aprx. % -M 000 I.936 7131
1 kilog^rain per year [kg/yr]:
=0.006 035 93 pound per day. Aprx. 6 •+- 1 000 5.780 7440
=0.002 737 85 kilogram per day. Aprx. ^^-hI 000 3.437 4098
=0.001 901 29 gram per minute. Aprx. 19 •*■ 10 000 3.279 0478
•
823 9087
656 6658
498 3033
261 5602
219 2660
ELECTROLYTIC DEPOSITS. 127
Logarithm
1 isrram per hour [g/h :
= 0.016 666 7 gram per minute, or %o. which see 2-221 8487
1 milligram per second [mg/s]:
=0.06 gram per minute, which see for other values 5-778 1518
1 pound (av) per day pb/day]
— 365.25 pounds per year. Aprx. ^HX 100 2-562 5902
=» 165.675 Kilograms per year. Aprx. % X 1 000 2219 2560
= 0.666 667 ounce per hour, orH * ^'^—
= 0.453 592 kilogram.per day. Aprx. % ■*• 10
— 0.314 995 gram per minute. Aprx. 31 -h 100
— 0. 182 625 ton (short) per year. Aprx. i V« -+-10
-■ 0.165 675 metric ton per year. Aprx. %
■=0.041 666 7 poimd per hour, or%4 5-619 7888
=0.018 899 7 kilogram per hour. Aprx. 19 -i- 1 000 2-276 4546
I ounce (av) per hour [oz/h]:
= 1.5 pounds per day, or% 0176 0918
-=0.680 389 kilogram per day. Aprx. 68 + 100 1.832 7571
=0.472 492 gram per minute. Aprx. 1% + 10 1.674 3946
I kilogram per day [kg/day :
— 805.238 pounds per year. Aprx. 800 2-905 9244
= 365.250 kilograms per year. Aprx. »>^X 100 2-562 5902
= 0.694 444 gram per minute. Aprx. "Ho 1-841 6376
= 0.402 619 ton (short) per year. Aprx. Vio 1-604 8944
= 365 250 metric ton per year. Aprx. ^H-hIO 1562 5902
—0.091 859 3 pound per hour. Aprx. Vii 2 968 1280
= 0.041 666 7 kilogram per hour, or^* 2-619 7888
1 gram per minute [g/min]C
= 1159.54 pounds per year. Aprx. % X 1 000 3 064 2869
=- 525.96 kilograms per year. Aprx. ^{9 X 10 000 2720 9527
= 60 grams per hour 1-778 1518
= 16.6667 milligrams per second, or 10% 1221 8487
= 3.174 66 pounds per day. Aprx. 8V10 0-501 6967
= 1.440 00 kilograms per day. Aprx. 1% Q-158 8625
=0.579 772 ton (short) per year. Aprx. ^ ' |.763 2569
= 0.525 96 metric ton per jrear. Aprx. ^%9 , -720 9527
= 0.132 277 pound per hour. Aprx. ^i -i- 10 T.12I 4856
= 0.06 kilogram per hour 2-778 1513
1 ton (short) per year [tn/yr]:
= 5.475 70 poimds per day. Aprx. *H 0-738 4398
= 2.483 74 kilograms per day. Aprx. 1% 0-395 1056
= 1.724 82 grams per minute. Aorx. % Q-236 7431
= 0.228 154 pound per hour. Aprx. % -h 10 T.358 2288
=0 103 489 kilogram per hour. Aprx. 3>^-5- 10 I.014 8944
I metric ton per year [t/jr];
— 6.035 93 pounds per day. Aprx. 6 0-780 7440
= 2.737 85 kilograms per day. Aprx. ^ii X 10 0437 4098
= 1.901 29 grams per minute. Aprx ^%o 0-279 0473
=0.251 497 poimd per hour. Aprx. M 1-400 5328
=0.114 077 kilogram per hour. Aprx. ^ 1-067 1986
1 pound (av) per hour [IbA]:
= 8 766. pounds per year. Aprx. ^ X 10 000 3-942 8015
= 3 976.19 kilograms per year. Aprx. 4 000 3-599 4672
= 10.886 2 kilograms per day. Aprx. 11 1.036 8770
=7.559 87 grams per minute. Aprx. MX 10 0-878 5146
= 4.383 tons (short) per year. Aprx. % X 10 0-641 7715
= 3.976 19 metric tons per year. Aprx. 4 0-599 4672
1 kilogram per hour [kg/h]:
= 19 325.7 poimds per year. Aprx. 19 000 4-286 1856
= 8 766. kilograms per year. Aprx. ^ X 10 000 3-942 8015
= 52.910 9 pounds per day. Aprx. 53 1-723 5454
= 16.666 7 grams per minute, or i*>% 1-221 8487
= 9.662 86 tons (short) per year. Aprx. 2% or %i X 100 985 1056
= 8.766 metric tons per year. Aprx. JiXlO 0942 8016
1 ounce (av) per minute [oz/min]t
= SH pounds per hour, which see for other values 0-574 0313
128
SLECTBOCHEMICAL ENERGY.
ELEOTROOHEBnOAL ENERGY.
The term electrochemical energy is used to refer to electrical energy
when it performs chemical work, or to chemical energy when it is set free as
electrical energy. The most convenient unit for expressing electrochemical
energy is the Joule, as the calculations are then the simplest; the calorie
is, however, the one generally used in tables which give the energy of com-
bination of chemical compoimds; values in calories or any other units of
energy are readily reduced to joules with the aid of the reduction factors
given under Energy, page 74. Care must be taken to distinguish between
the large and the small calorie, a distinction which is often not made in
text-books and tables.
There is a very simple and direct way of calculating how many volts
will be required to decompose a clieinical compoand electrolytically.
or how many volts will be generated in a battery in which chemical
compounds are formed, when the heats or enemas of combination of the
compounds are known. It is sometimes called Thomson's law, and can be
directly deduced from Faraday's law and the principle of the conservation
of energy. It is imjjortant to remember, however, that the number of
volts thus calculated is limited to that required to supply the necessary
energy of decomposition, or that generated in a battery by the formation
of compounds; it includes nothing more. The actual voltage involves
some further correction factors, which, though generally small as com-
pared with the voltage of decomposition, ar3 sometimes of importance.
Among these correction factors is the voltage required to overcome the
resistance of the electrolsrte ; this depends on the current flowing and on
the resistance of the electroljrte. Another correction factor is the Gibbs
or Helmholtz temp>erature coefficient, namely the rate of change of the
voltage with the temperature; this falls out when there is no such change,
or in practice when this change is inappreciable. Another correction fac-
tor is what is called the " over- voltage'' ; this depends on the fact that it
takes different voltages to set free the same gas at electrodes of different
metals. For these correction factors treatises on electrochemistry should
be consulted.
In this rule for calculating the voltage due to the heat of combination,
which is given below, the constant is oased on the relation between the
ioule and the calorie given in the table of units of Energy, and on the same
fundamental electrochemical and chemical constants for silver which
were used to calculate the reduction factors for Electrochemical Equiva-
lents above, namely 0.001 118 gram per coulomb as the electrochemical
equivalent of silver, and 107.93 as the atomic weight of silver, on the basis
of = 16. These fundamental values correspond with the usually
accepted value of the ionic charge, 96 540. coulombs per monovalent gram,
ion, within the limits of accuracy of the data. For the heats of com-
bination of compounds, see reference-books on this subject; the data are
usually given in calories per gram molecule, but care must be taken to find
out whether the large calorie or the small calorie is the one meant; also
whether it is a gram molecule or a kilogram molecule that is meant ; care
must also be taken to see that the given number of calories apply to the
exact decomposition or combination under consideration, whether gases
are set free and escap>e as such, or whether they recombine, and if the
latter, whether this recombination enters into the electrochemical reaction
or whether it is pureljf chemical, as by local action, also whether water or
some other accompanying product is formed or decomposed, etc.
ELECTROCHEMICAL ENERGY.— RELUCTANCE. 129
Rule for calculating the voltage of decomposition or composi-
tion, from the heat of combination.
Logarithm
For monovalent ions, or for one equivalent weight:
The number of volts:
»the number of kilogram calories per gram molecule X
0.043 363. Aprx. % -i- 10 5.687 1162
■"the number of kilogram calories per kilogram molecule,
or gram calories per gram molecule, X 0.000 643 363,
Aprx. ^ -5- 10 000 5.037 iie2
>-the number of kilogram calories per gram molecule +
23.061. Aprx. % X 10 1.362 8888
-a the number of kilogram calories per kilogram molecule,
or gram calories per gram molecule, -i-23 061. Aprx.
%X10 000 4.362 8838
For multivalent ions divide the volts thus obtained by the valency.
That is, for bivalent ions calculate the voltage from the above and then
divide by 2; for trivalent ions divide by 3, etc. Or the niunber of calories
may be reduced to that for one equivalent weight (by dividing the calories
by the valency), and the above rules will then give the volts directly.
MAGNETIC R£LtrOTANOE [(R, R] ; MAGNETIC RESIS-
TANCE. (Magnetomotive force -^ flux; length -i- (surface
X permeability) .)
Reluctance measures the amount by which the material in a magnetic
circuit or part of a circuit resists or opposes the flux; the more it resists,
the greater the reluctance; it is the opposite to permeance and numericall;/
it is equal to the reciprocal of permeance. As the reluctance of a centi'
meter cub6 of air (or more correctly, of a vacuum) in the C. G.S system is, by
definition, equal to unity (that is, to one oersted), it follows that the amount
of reluctance of any magnetic circuit or any part of it also represents the
number of times that its reluctance is greater than that of the air of the same
volume and shape.. Reluctance is analogous to resistance in an electric
circuit, but differs in these three important features: (1) in circuits con-
taining magnetic materials it generally varies very greatly with the flux
density, being constant only for air or other diamagnetic materials ; (2) there
is no such a thing as a magnetic insulator, that is, an infinitely great reluct-
ance; (3) maintenance of a flux through a reluctance does not necessarily
require the continuous expenditure of energy. ^ A reluctance in oersteds
is equal to the magnetomotive force in gilberts divided by the flux in max-
wells The reluctance of a circuit in oersteds is equal to the length of the
circuit in centimeters divided by its cross-section in square centimeters and
then either multiplied by the reluctivity or divided oy the permeability ;
but in magnetic materials the reluctivity and the permeability vary with
the density of the flux in the circuit, hence this method of calculation is
practicable only for non-magnetic materials like air, the permeability of
which is constant and is equal to unity; it may be applied to air-gaps, for
instance. The reluctance is not used very often in magnetic calculations,
these being usually based on the property called permeability, as is ex-
plained below under the units of magnetizing force.
The only unit used is the C.G.S. unit called an oersted, which is the
reluctance through which a CG.S. unit of magnetomotive force (called a
gilbert) will produce a CG.S. unit of flux (calledfa maxwell).
1 CGS nnlt (elmg) = l oersted.
1 oersted » 1 CGS unit (elmg).
130 MAGNETIC RELUCTIVITY.
The relations to other measures are as follows:
Oersteds:
= gilberts -Sr maxwells.
•= gilberts -5- (gausses Xsq. centimeters section).
= ampere-turns X 1.256 64t -^maxwells.
= ampere-turns X 1 .256 64t ^ (gausses X sq. centimeters section).
.=CGS unit of current-turns X 12.566 4t n- maxwells
=CGS unit current-turns X 12.566 4 1 -^- (gausses X sq. cm. section).
= 1 -5- permeance in COS units.
= centimeters length -s- (sq. centimeters section X permeability).
= centimeters length X reluctivity + sq. centimeters section.
=» inches length X 0.393 700-5- (sq inches section X permeability).
= inches length X reluctivity X 0.393 700-5- sq. inches section.
= (number of turns X 0.000 1 )2 X 1 .256 64t -5- henrys.
Oersteds in iron = oersteds in air -s- permeability.
" in air —oersteds in iron X permeability
MAGNETIC RELtrOTiyiTY[v]; SPEOIFIO MAGNET-
IC RELUCTANCE; MAGNETIC RESISTIVITY;
SPECIFIC MAGNETIC RESISTANCE. (Imperme-
ability; magnetizing force h- magnetic induction; reluc-
tance X surface -^length ; reluctance -^ reluctance.)
Reluctivity measures the number of times that a material resists or
opposes magnetic flux, more than air (or vacuum) does. It is a property
of a material and numerically it is equal to the reciprocal of the permea-
bility, which see. It corresponds in some respects to resistivity or specific
resistance in electrical circuits. It is specific reluctance based on air; but
as the reluctivity of air is unity, that of any other material is numericallv
equal to the reluctance in oersteds of a centimeter cube of that material.
Reluctivity is seldom used in magnetic calculation, its reciprocal, the
permeability, being used instead. Like permeability, it is a property of
the material in a magnetic circuit, and its values are different with different
flux densities.
■ There are no units, as it is a mere ratio. The relations of reluctivity to
other magnetic measures are the reciprocals of those for permeability.
The chief relations are :
Keluctl vity = 1-5- permeability.
' * = sq. centimeters section X oersteds -5- centimeters length.
* • = sq. inches section X oersteds X 2.540 01 -^ inches length.
MAGNETIC PERMEANCE; MAGNETIC CONDUCT-
ANCE; MAGNETIC CAPACITY. (1 -^ reluctance;
flux H- magnetomotive force ; surface -i- (length X permea-
biHty).)
Permeance measures the amount by which a given magnetic circuit con-
ducts the flux, the better it conducts, the greater the permeance; it is the
opposite to reluctance, and numerically it is equal to the reciprocal of reluct-
ance. It is analogous to conductance in an electrical circuit Just as the
reluctance of a given m£tgnetic circuit refers to that particular circuit or
its parts, so permeance refers to a particular magnetic circuit or its parts.
In referring to the property of the material itself, independently of its
t Or 4jr/10. Aprx. H X 10. Log 099 2099-
t Or 47r. Aprx. >iX 100. Log 1.099 2099-
PERMEANCE. -PERMEABILITY. 131
nse and shape, the term permeability or specific permeance is used, which
see below. When a given magnetic circuit contains a magnetic material
like iron, the permeance does not remain constant like the conductance of
an electric circuit, but it varies very greatly with the density of the flux;
it is constant only for air. Permeance is seldom used in magnetic calcu-
lations (see under magnetizing force). It can always be avoided by using
the reciprocal of the reluctance instead ; both may be avoided by basing the
calculations on the permeability.
The unit is the C.G.S. uult, which is the reciprocal of the absolute or
C.G.S. unit of reluctance, that is, the reciprocal of an oersted; it has
no name. The permeance of a centimeter cube of air (or vacuum) between
two parallel sides is unitv in the C.G.S. system. Hence the permeance of
a magnetic circuit or of apart of it, also represents ths number of times
that its permeance is greater than that of an equal volume of air of the same
size and- shape. It is equal, in C.G.S. imits, to the permeability multiplied
by the cross-section in square centimeters and divided by the length in
centimeters.
The relations to other measures are as follows:
Permeance «= maxwells -i- gilberts.
* * — 1 -*• oersteds.
" «^p>ermeabilityXsq. centimeters section -^centimeters length.
' ' •= permeability X sq. inches section X 2.540 01-4- inches length.
Permeaiice of iron =;= permeance of air X permeability.
*' of air = permeance of iron + permeability.
- See also the reciprocals of the relations given under Reluctance.
MAGNBTIO PERMEABILITY [//]; SPEOIFIO PERME-
ANOE; MAGNETIO OONBTTOTIVITY. (Magnetic
induction -Mnagnetizing force ; flux density -r- flux density;
permeance -7- permeance; 1-^ reluctivity.)
Permeability is a very important quantity in most magnetic calculations ;
it measures the number of times that a material conducts or aids magnetic
flux better than air does; if, for instance, the permeability of a certain
kind of iron imder certain conditions is 300., it means that it conducts
magnetic flux or lines of force 300. times as well as an equal amount of air
would imder the same conditions, that is, it is 300. times as permeable to
the flux It may be said to be maf^etic conductivity compared to air as a
standard.. If a ^ven inagnetomotive force produces a certain amount of
flux in a given circuit of air, it will produce 300. times this flux if the air
were replaced by this kind of ir^n. It follows from this that the magnet-
izing force (usually represented by H) multiplied by the permeability (;i)
gives the induction (B) in the iron, that is, the flux density in the iron*
or permeability » induction -i- magnetizing force; this is further explained
under the units of magnetizing force. Permeability is an inherent property
of materials and its values are usually given in tables or curves. It ifi also
the same as specific permeance, which is the permeance of a material as
compared with that. of air; as the permeance of a centimeter cube of air
is unity, the law above given follows. Permeability is the reciprocal of
reluctivity. It is analogous to conductivity or specific conductance in
electrical calculations, with a very important difference, however, namely
that while the electrical conductivity of a material is constant and does not
change with the current which flows, the permeability of the chief magnetic
materials varies very greatly with the density of the flux, and its values
must therefore be given for each flux den.«ity, for which reason they are
usually given in the form of curves called permeability curves or magneti-
zation curves. For air, however, the permeability is always constant and
is numerically equal to unity. Paramagnetic budieH are those whose
permeability is greater than unity, and dianiagnetlc bodies those whose
permeability is less than unity.
132 PERMEABILITY. — MAGNETOMOTIVE FORCE.
There fire no units of permeability, as it is a mere relation, nmnber, or
ratio between two quantities of the same kind. In this respect it is like
specific gravity.
For a Drief description of calculations involving permeabilities see under
the units of Magnetizing Force.
The relations to other measures are as follows:
Permeability:
— i^usses in iron -^ gausses in air.
omch gausses in iron + inch gausses in air.
» permeance of iron •*■ permeance of air.
—oersteds of air -J- oersteds of iron.
■■1-s- reluctivity.
- 1 + [susceptibility X 12.566 4 (or 47r)].
•- gausses -t- gilberts per centimeter.
=- gausses X 0.795 775 (or 10 ■*• 4 ;r)+ ampere-turns per centimeter.
-= gausses X 2.540 01 -f- gilberts per inch .
■«|{aussesX 2.021 27 -5- ampere-turns per inch.
—mch gausses X 0.393 700-^ gilberts per inch.
—inch gausses X 0.3 13 297 -4- ami>ere-turns per inch.
— gausses X 0.079 577 5 (or 1 -i-47r)-HCGS unit current-turns per cm.
—centimeters length ^(sq. centimeters section X oersteds).
—inches length X 0.393 700 + (8q. inches section X oersteds).
— permeance X centimeters length -^sq. centimeters.
For the relations with maxwells substitute for "gausses," in any of the
above, "maxwells -i-sq. centimeters section"; and for "inch gausses," sub>
stitute "maxwells -i-sq. inches section."
BfAGNETIO SUSOEPTIBILIT7 [k\ (Intensity of mag-
netization-^ magnetizing force.)
^ This quantity is used chiefly in physical conceptions; it is somewhat
similar to permeability in that it expresses the magnetisability of a ^b-
stance. It is equal to the intensity of magnetization (see below) divided
by the ma^etising force which produces it. There are no units, as it is
a mere ratio or number. Its relation to permeability is as follows:
Susceptibility - (permeabiUty- 1) X 0.079 577 5 (or 1 -h4»c). ,
MAGNETOMOTIVE FORCE [m.m.f. (F,F]; AMPERE.
TURNS [a-t]; MAGNETIC POTENTIAL; DIFFER-
ENCE OF MAGNETIC POTENTIAL; BfAGNETIC
PRESSURE. (Current X turns ; flux X reluctance ; energy -f-
pole strength.)
These units are used to measure the magnetic pressure or "motive force"
which produces or ends to produce a magnetic flux in a magnetic circuit,
just as an electromotive force tends to produce a current of electricity, or
a pressure of water tends to produce a flow of water and mi^ht similarly
be called the hydraulic motive force. In practice magnetomotive forces are
generally produced by, and are often measured in terms of, what are called
ampere-turns; this term means the product of an electric current in am-
Seres and the number of turns or windings of the coil through which it
ows; such a current-carrying coil produces a definite magnetomotive
force, which in turn produces an amount of flux dependent on the amount
of reluctance in the whole magnetic circuit.^ According to the laws of
electromagnetism, the magnetomotive force in C.G.S. units (gilberts) is
always numericsdly equal to the ampere-turns multiplied by 4ir-t- 10, whether
MAGNETOMOTIVE FORCE. 133
there is iron in the magnetic circuit or not. The magrnetomotive force in
gilberts in any closed loop encircling a long straight wire through which a
current passes ia 4x times the C.G.S. unit of current, or 4»r-*-10 times the
number of amperes. It is always directly proportional to the current pro-
ducing it. The law of the magnetic circuit is similar to Ohm's law for the
dectnc circuit, namely that the flux (corresponding to the current) is equal
to the magnetomotive force (corresponding to tne electromotive force),
divided by the reluctance (corresponding to the resistance); hence the
magnetomotive force in C.G.S. units (gilbenrts) is equal to the flux in C.G.S.
units (maxwells) multiplied by the reluctance in C.G.S. units (oersteds).
This is true whether there is iron in the magnetic circuit or not.
Most calculations occurring in practice are simplified by using the quan-
tity called the magnetizing force instead of the magnetomotive force, as is
explained below under the units of Magnetizing Force.
There are three units of magnetomotive force in use; the more general,
and often the more convenient one. is the ampere-turn, which is equal to
the ma^etomotive force produced by one ampere flowing once around a
magnetic circuit; this is irrespective of the shape or size of the electric or
magnetic circuit; the latter affect only the reluctance of the magnetic cir-
cuit and thereby the resulting flux. This unit bears an incommensurate
relation to the absolute magnetic units owing to the factor 4x.
The second usual unit is the electromg^anetic C.G.S. unit (or absolute
unit) , called a g:ilbert. Its definition is based on the fact that the magneto-
motive force produced in any one closed loop around a long straight wire
through which a C.G.S. unit of current (or 10 amperes) flows, is 4»r C.G.S.
units of magnetomotive force; hence one such unit is equal to 14-4^ of
this magnetomotive force. It mtryr also be defined as that magnetomotive
force which will produce a flux of one C.G.S. unit (maxwell) through one
C.G.S. unit of reluctance (oersted).
The third unit is like the ampere-turn except that the current is the
C.G.S. unit of current (10 amperes) instead of the ampere. This unit is
therefore equal to 10 times the ampere-tum unit. It has no name other
than the C.O.S. unit current-turn. It is never used in practice.
Logarithm
1 CGS unit (elmg) — 1 gilbert, which see for other values.
1 g^ilbert:
<« 1 CGS imit (elmg) of magnetomotive force.
■=» 0.795 775 (or 10/4»r) ampere-tum. Aprx. subtr. % . . . . T.900 7901
«0.079 577 5 (or 1/4 Jt) <UGS unit of current-timi. Aprx.%00 2-900 7901
1 ampere-turn » 1.256 637 (or 47r/10) CGS units. Aprx. add H 0099 2099
* • =1 .256 637 (or 47r/10) gilberts. Aprx. add >i. . Q099 2099
' * = 0.1 CGS unit of current-turn LOOO 0000
1 COS unit of current- turn: «
= 12.566 37 (or 4n) C(3S units (ehng). Aprx. io^& 1.099 2099
= 12.566 37 (or 4»r) gilberts. Aprx. 10% 1.099 2099
» 10. ampere-turns 1-000 0000
The relations to other measures are as follows:
Gilberts:
= ampere-turns X 1.256 64.t
= CGS unit current-turns X 12.566 4.$
= maxwells X oersteds.
*= maxwells -f- permeance (in CGS (elmg) units).
= maxwells X cm length -^(sq. cm section X permeability).
«= maxwells X inches length X 0,393 700 -f- (sq. inches section X permea-
bility).
= maxwells X centimeters length -^sq. centimeters section. For air.
— maxwells X ins. length X 0393 700 -*- sq . ins. section. For air.
"-gausses X centimeters length -h permeability.
«=» gausses X inches length X 2.540 01 -f- permeability.
«= inch gausses X inches length X 0.393 700-4- permeability.
= gausses X centimieters length. For air.
= gausses X inches length X 2.540 01 . For air.
= mch gausses X inches length X 0.393 700. For air.
=gaussesX oersteds Xsq. centimeters section.
■k
t Or 4)r/10. Aprx. add H- Log 099 2099-
t Or 4ff. Aprx. H X 100. Log 1.099 2099-
134 MAGNETOMOTIVE FORCE. — MAGNETIZING FORCE.
Gilberts for iron =eilberts for air+penneability.
Gilberts for air »= Alberts for iron X permeability.
Ampere-tarns:
=gilbert8X 0.795 775. t
= UGS unit current-turns X 10,
— max welUX oersteds X 0.795 775. t
= maxwells X 0.795 775t-«- permeance (in CGS (elmg) units).
«= maxwells X cm length X 0.795 775t -s- (sq. cm section X permeability).
«= maxwells X ins. length X 0.313 296 -t- (sq. ins. section X permeability).
= maxwells X cm length X 0.795 775t ■*■ sq. cm section. For air.
= maxwells X ins. length X 0.313 296 -*- sq. ins. section. For air.
= gausses X centimeters length X 0.7 95 77 5 1 -«- permeability.
= gausses X inches length X 2.021 27 ^- permeability.
=inch gausses X inches length X 0.313 296 -J- permeability.
= gausses X centimeters length X0.795 775.t For air.
= gausses X inches length X 2.02 1 27 . For air.
= inch gausses X inches length X 0.313 296. For air.
= gausses X oersteds X sq. centimeters section X 0.795 775. t
Ampere-turns for iron = ampere-turns for air -s- permeability.
Ampere-turns for air — ampere-turns for iron X permeability.
CGS unit of current-turns = ampere-turns X 0.1.
= gilberts X 0.079 577 5. J
(For further relations divide those for ampere-turns by 10.)
When the flux is due only to the current, as in self-induction, and when
there is no magnetic leakage:
Ampere-turns = ergs of kinetic energy of the current X 20 -J- maxwells.
* ' == joules of kinetic energy of the current X 2 X 10* -s- maxwells.
* ' = maxwells X number of turns^ -j- (henrys X 10*).
When the flux is from an external source, and independent of the current
as in mutual induction, and when there is no magnetic leakage:
Ampere-turns =ergs of kinetic energy of the current X 10 -s- maxwells.
' * =» joules of kinetic energy of the current X 10* -*- maxwells.
MAGNETIZING FORCE [JC, H]; MAGNETOMOTIVE
FORGE per CENTIMETER; MAGNETIC FORCE;
FIELD INTENSITY, (Turns Xcnrrent-r- length; magneto-
motive force ^ length ; induction -^ permeability; flux den-
sity -^ permeability ; force -^ pole strength.)
This quantity, which is one of the most important in the more usual mag-
netic calculations, is used to measure the magnetomotive force produced
per unit len^h of a coil or solenoid carrying an electric current; or the
magnetomotive force required per unit length of any part of a magnetic
circuit to produce the desired flux density in that part. The usual calcu>
lations of magnetic circuits then often become simpler than they would be
if the whole magnetomotive force itself is used. In electric circuits it has
its analogy in the electromotive fbrce produced per centimeter length of
active wire in an armature of a dynamo or in a transformer, or the difference
of potential required per centimeter length of a conductor in order to pro-
duce the desired current density in that conductor.
There are no specifically named units of magnetizing force. The one
most frequently used when the metric system is employed, is an ampere-
turn per centimeter length. When inches are used the unit is an am-
pere-turn per incli. The absolute or C.G.S. unit is one gilbert per
centimeter. When the magnetic circuit consists of air, the magnetizing
tOrlO-t-47r. Aprx. 9io. Log 1.900 7901-
i Or 1 +4«. Aprx. 8 + 100. Log 3.9OO 7901.
MAGNETIZING FORCE. 135
force can also be expressed and measured in terms of units of flux density,
namel^f gausses, for, although they mean something different, they are
numerically the same as gilberts per centimeter for air, as is shown below.
If a current flows through a imiform coil of wire which is very long as
compared with its diameter, the flux density produced in its interior will
be practically uniform except near its ends; it will be nearly uniform
throughout if the two ends are brought together to form a ring coil. More-
over, this flux density is independent of the diameter of the coil or the shape
of its cross-section, which anect only the reluctance and the total flux.
The magnetizing force produced by such a coil, whether it contains iron
or not, is numerically equal in gilberts per centimeter to 4innc, in which n
is the number of turns or windings per centimete length of coil, and c is the
current in absolute units; nc is therefore the number of current-turns per
centimeter, corresponding to (but not numerically equal to) the ampere-
turns per centimeter. Imagine a series of planes perpendicular to the axis
and one centimeter apart; then the magnetizing force given by this formula
wiU be the magnetomotive force in gilberts produced between each plane
and the next. As an analogy, suppose the interior were replaced by a con-
ductor carrying an electric current, and the coil itself were replaced by a
device which induces an electromotive force in that conductor, then the
volts induced per centimeter length of this device will evidently be the
volts of electromotive force which exist between each of these parallel planss
and the next.
It is also true for iron as well as for air that the magnetizing force equals
the flux density divided bj^ the permeability, but as the permeability of
air is unity by definition, it follows that for air (or more exactly for a
vacuimi) the magnetizing force of such a coil when expressed in gilberts
per centimeter, is numerically equal to the flux density in its interior in
gausses The above general formula therefore also gives, as a special case,
the flux density in air in gausses. This has given rise to much confusion, as
it makes it appear at first sight as though, a magnetomotive force was of
the same nature as a flux density, which with the electric units would be
like saying that an electromotive force was of the same nature as a current
density. The explanation is that the reluctance of a centimeter cube of
air is unity, hence the flux density in gausses through each centimeter cube
of air will be numerically the same as the magnetomotive force in gilberts
acting between its two opposite faces; or in the above illustration with
the imaginary parallel planes one centimeter apart, the flux density in
gausses in each space between two such planes will for air be numerically
equal to the magnetomotive force (in gilberts) between each plane and the
next. Analogously, if a wire happens to have a resistance of one ohm per
foot, the number of volts acting at the ends of each foot will be numerically
the same as the nimiber of amperes flowing. The identity is in the num-
bers and not necessarily in the nature of the units; moreover, the numerical
identity exists only between the units gilberts per centimeter and gauss,
and not between the other units like ampere-turns.
The formula 4jmc therefore always gives the magnetizing force produced
in gilberts per centimeter length of coil .whether there is iron in the coil or
not. It also gives the flux density in gausses in the interior of the coil,
but for air only, c is the current in absolute units and must be replaced
in the formula by C-^10, if C is to be in amperes. When the result given
by this formula is multiplied by the entire length of the coil in centi-
nieters, it gives the total magnetomotive force of the whole coil, in gilberts.
The same magnetizing force will' produce entirely different flux densities
in materials of different permeabilities, just as the same voltage will pro-
duce entirely different currents in materials of different conductivities.
The magnetizing force in gilberts per centimeter multiplied by the permea-
bility of any material gives the flux density in gausses, or the induction in
gausses, produced in that material by that magnetizing force. Or in dif-
ferent terms, if a coil produces in its interior a certain flux density in gausses
in air, then that flux density must be multiplied by the permeability of the
material to gfet the fhix density or induction in that material in gausses,
when the air circuit is completely replaced by that material, as in trans-
formers; for the case in which the circuit is partly air and partly iron, see
the next paragraphs. The more usual calculations, such as those for dy-
namos and transformers, start with the desired induction, and in order to
avoid the calculation of the reluctance, the troublesome factor P,4;r, and
136
MAGNETIZING FORCE.
the various other reduction factors when inch units are used, such calcula*
tions are generally made as described in the following paragraphs.
Maguetic calculations, like those for dynamos and transformers. The
values of the permeabilities of the particular iron or steel which is to be
used, are usually given in the form of a curve called a permeability curve
or a magnetization curve, whose horizontal distances are the magnetiising
forces in am[>ere-tums per centimeter (H), and whose vertical ones are the
corresponding inductions (B) or flux densities, in gausses or lines of force
per square centimeter in that quality of iron or steel. It must be decided
at the start what the inductions are to be in the various parts of the circuit :
then from the curve for the kind of iron in each part, find what the requires
magnetizing force is in ampere-turns per centimeter for that part; then
multiply this by the axial or center-line length in centimeters, of this par-
ticular part of the iron under consideration, be it the cores or the yoke^
pieces, or the armature, and the result will be the total number of ampere-
turns or the magnetomotive force, which is required to magnetize that part
to that particular induction. Notice that the length here used is that of
the path of the flux through that part of the iron, and not necessarily the
length of the coil. Having done this for each of the iron parts making up
the whole circuit, add them all together and the result will be the total
ampere-turns required for the total iron part of the circuit. The ampere-
turns required for producing the flux in the air-gap are calculated from the
desired flux density in the gap, as follows: amF>ere-tums — flux density in
gausses X length of air-path of flux (that is, twice the length of the gap) in
centimeters X 0.7 95 775. These latter ampere-turns (which genenuly are
by far the larger part of the whole) are then added to those required for the
iron part, thus giving the total required for the complete' magnetic circuit.
The coils for producing these ampere-turns may have any length and may
be woxmd around any convenient part of the circuit, for when the mag-
netic circuit is chiefly of iron, the magnetizing force of the coils will act m
that circuit very nearly the same way no matter how they are distributed
over the iron.
When the dimensions are all in inches and the flux densities in maxwells
per square inch (inch-gausses), then the magnetization curve should be
plotted for those units and the calculations are then precisely the same except
for the air-gap, for which the formula then becomes: ampere-turns — flux
density in maxwells per sq. inch X length of air-path of flux in inches X
0.313 296.
. When the linear dimensions are in inches, and the flux densities in gausses,
the curve should be plotted accordingly and the calculations are again the
same except for the air-gap, for which the formula then becomes: ampere-
turns— gausses X length of air-path of flux in inches X 2.021 27.
It will be noticed that in this method of calculation the reluctance need
not be known. The total required cross-section of the iron is determined
by dividing the given total flux by the given flux density. The reverse
calculation to the above is much more difficult; in that case the ampere-
turns of such a composite magnetic circuit together with all the dimen-
sions are given, and the flux produced by them is to be determined. In
such a case perform the calculation backwards, by making trial calcula-
tions as just described, using different assumed total fluxes, until one is
found which will require the given number of ampere-turns. ^ For a simple
magnetic circuit like that in a transformer, in which there is no air-gap,
either calculati9n amounts to little more than reading off the results from
the magnetization curve.
Logarithm
1 g^ilbert per inch:
=0.795 775 ampere-turn per inch. Aprx. subt. ^i 1.900 7901
— 0.393 700 gilbert per centimeter. Aprx. ^o 1.595 1654
—0.313 296 ampere-turn ];)er centimeter. Aprx. %6 1«495 9555
1 ampere-turu per inch:
=> 1.256 64 gilberts per inch. Aprx. add H 0099 2099
«0.494 738 gilbert per centimeter. Aprx. H L694 8763
=0.393 700 ampere-turn per centimeter. Aprx- ^o 1.595 1654
1 fl^ilbert per cm = 2.540 01 gilberts per inch. Aprx. i% 0-404 8346
*' = 2.02127 ampere-turns per inch. Aprx. 2. 0*305 6247
*• = 1. CGS unit (ehng) 0-000 0000
•* =0.795776 amp.-turn per cm. Ap. subt.%. 1.900 7901
MAGNETIZING FORCE. — MAGNETIC FLUX. 137
Logarithm
1 COS unit (elmg) = 1. gilbert per centimeter 0-000 0000
1 ampere-turn per cm:
■=3.191 86 gilberts per inch. Aprx. 3^^ 0504 0445
■- 2.640 01 ampere-turns per inch. Aprx. i% 0-404 8346
— 1.256 64 gilberts per cm. Aprx. add M 0.099 2099
1 CCKS unit of current-turn per centimeter:
■-31.918 6 gilberts per inch. Aprx. 32 1504 0445
—25.400 1 ampere-turns per inch. Aprx. 25 1-404 8346
— 12.566 4 gilberts per centimeter. Aprx. H X 100 1.099 2099
>» 10. ampere-turns per centimeter 1 .000 0000
For air (or vacuum) only :
1 gauss = 1 gilbert per centimeter.
The relations to other measures are as follows :
Ampere-turns per centimeter:
■- gausses X 0.795 77 5t + permeability.
—maxwells X 0.795 77 5t •♦■ (sq. cm section X permeability).
— gausses X 0.795 77 5t. For air.
■" maxwells X 0.795 77 5t + sq. cm section. For air.
Ampere-tUrns per inch:
-■ gausses X 2.021 27 -«- permeability.
—maxwells X 0.313 297 ■*- (sq. inches section X permeability).
—inch gausses X 0.313 296 -f- permeability.
— gausses X 2.021 27 . For air.
—maxwells X 0.313 297 -*- sq. inches section. For air.
—inch gausses X 0.313 296. For air.
Gilberts per centimeter » gausses -^permeability.
** «= maxwells -i-(sq. cm section X permeability).
** « gausses. For air.
** = maxwells -5- sq. centimeters. For air.
Gilberts per inch:
— gausses X 2.540 01 ■*- permeability.
— maxwells X 0.393 700 ->-(sq. inches section X permeability).
—inch gausses X 0.393 700 -*■ permeability.
—gausses X 2.540 01 . For air.
—maxwells X 0.393 700 -i- sq. inches section. For air.
— inch gausses X 0.393 700. For air.
CGS unit of cnrreiit-turns per centimeter:
= gausses X 0.079 577 5% -s- permeability.
- gausses X 0.079 677 5.% For air.
MAGNETIO FLUX [#, <f>]i LINBS OF FOROB; FLUX
OF FORCB; AMOUNT OF MAGNBTIO FIELD;
POLE STRENGTH [m]. (Magnetomotive force -^ reluct-
ance ; magnetic induction (or flux density) X surface ; elec-
tromotive force X time; length X V^*^''^®-)
These units are used to measure the total quantity or amount or number
of magnetic lines of force or the amount of flow or flux of magnetism, just
as amperes are used to measure the quantity or amount of electric current,
or as cubic feet per second measure the amount of a flow of water. This
magnetic flux or flow is in some respects analogous to a current of electricity,
as it must always form a closed circuit upon itself, and be the same in
amount in every cross-section of the circuit; it follows a law analogous to
Ohm's law, as the flux is equal to the magnetomotive force divided by the
t Or 10-^4 ». Aprx. 8/10. Log T.900 7901.
j Or I -^4^r, Apn?. 8/100. Log J.^OO 790X.
138 MAGNETIC FLUX.
reluctance. It differs, however, in that no work is being done continuously
in the circuit of a magnetic flux, and it can therefore continue to exist in-
definitely as in a permanent magnet, without consuming or producing
energy. Energy is stored in the flux when it is produced, and it is given out
again whenever the flux ceases to exist, but no energy is necessarily required
to maintain it ; it is therefore in this respect more like a mechanical pressure
or stress, as that of compressed air. The kinetic energy (which see abovis)
required to start an electric current is stored in the system as magnetic
energy, and given out again as electrical energy when the current stops.
From the energy standpoint, magnetic flux is more analogous to coiilombs
of electricity. A magnetic circuit also differs from an electric circuit in
that it can never be opened, as there is no such a thing as a magnetic insu**
lator; magnetic flux can cease only by contracting to a point somewhat
like an extremely small rubber band which is allowed to contract after hav-
ing been stretched.
The space surrounding a magnet or an electric current or that between
two magnetic poles, is called a magnetic field or mag^netic field of force,
as it contains magnetic flux or magnetic lines of force ; units of flux measure
the total amount of this field (but not its intensity or density) or the total
number of such lines of force ; this flux alsa continues through the magnet
itself.
Flux is also equal to the flux density (sometimes called induction), in
lines of force per square centimeter (or per square inch), multipUed by the
number of square centimeters (or square incnes) cross-section of its path,
and is then often called the total flux to distinguish it from the flux density.
An electric current is always encircled by such flux, and the two circuits,
namely the electric and the magnetic, are always linked together' like the
two links of a chain. When the magnetic flux enclosed in a coil or loop of
wire is increased or diminished, an electromotive force is produced in that
wire; this is the fundamental principle of a dynamo; or stated in different
terms, when a wire cuts through magnetic flux, an electromotive force is
produced in the wire; or the linking and unlinking of circuits of flux and
electric circuits produces an electromotive force; it is this that produces
self- and mutual induction.
The term fliix-turus or mean fiux-turns is sometimes used for denot-
ing the product of the number of turns and the mean flux (in maxwells) in
one turn; the magnetic leakage is thereby eliminated. The mean flux-
turns in maxwell-turns are equal to the self-inductance in henrys multiplied
by 10^ times the final current in amperes.
The unit universally used is the absolute or electromag^netic C.G.S.
unit or single line of force, and is defined as that amount of flux which
acting on a unit magnetic pole will propel it with a force of one djrne. It
can also be defined as the amount of flux passing through one souare centi-
meter cross-section of a field having a flux density of one C.G.S. unit. A
unit magnetic pole (imaginary) is one which will exert a force of one
dyne on another unit pole one centimeter distant. From each such pole
there radiates a flux equal to 47r (or 12.566 4) of these units or lines of force.
A single or unit line of force in a magnetic field may be said to stand for
or represent a tube of such a cross-section that it always embraces a unit
of flux ; a definite amount of flux may have widely different lengths of cir-
cuit or cross-sections without changing its amount.
This C.G.S. unit is called a maxwell, according to the International
Congress of 1900. A maxwell is therefore the same thing as a single or
unit line of force as above defined.
Logarithm
1 CGS unit (elmg) = 1 maxwell 0000 0000
1 maxwell:
« 1 CGS unit (elmg) 0000 0000
=» 1 line of force 0000 0000
= 1 gauss-cent imeter^ 0-000 0000
= 0.155 000 gauss-inch^. Aprx. ^is 1190 3308
= 0.079 577 5 (or ^;r) of the flux from a unit pole. Aprx. %oo 2-900 7901
1 weber (obsolete) = 1 maxwell 0-000 0000
1 unit pole (flux from) = 12.566 37 (or 4jr) maxwells 1.099 2099
1 Kapp llne<obsolete)»6 000. maxwells 3.778 1513
MAGNETIC FLUX. 139
The relations to other measures are as follows:
Maxwells:
«= gausses X sq. centimeters.
«m3h-gaus9esXsq. inches.
■= gilberts ■*■ oersteds.
«= gilberts X permeance (in CGS units).
«= gilberts X permeability Xsq. centimeters section -s- centimeters length.
= gilberts X permeability Xsq. inches section X 2.540 01 -5- inches length.
= gilberts Xsq. centimeters section ^centimeters length. For air.
« gilberts X sq. inches section X 2.540 01 -s- inches length. For air.
— gi berts per centimeter X permeability X sq. ceritimeters section.
=» gilberts per inch X permeability X sq, inches section X 2.540 01.
= gilberts per centimeter X sq. centimeters section. For air.
= gilberts per inch X sq. inch section X 2.540 01 . For air.
= ampere-turns X 1.256 64t ■*- oersteds.
=ampere-tumsX 1.256 64t X permeance (in CGS units).
=ampere-tumsX permeability Xsq. centimeters section X 1.256 64t-^
centimeters length.
«= ampere-turns X permeability Xsq. inches section X 3.191 86 -«- inches
length.
= ampere-turns X sq. centimeters section X 1.256 64 1 -^ centimeters
length. For air.
■=ampere-tums X sq. inches section X 3.191 86 -finches length. For air,
= ampere-turns per centimeter X permeability X sq. centimeters sec-
tion X 1.256 64. t
«=ampere-tums per inch Xp>ermeability Xsq. inches section X 3.191 86.
«= ampere-turns per cm X sq. centimeters section X 1 .256 64. t For air.
■= ampere-turns per inch Xsq. inches section X 3.191 86. For air.
= CGS unit current-turns X 12.566 4t -*- oersteds.
(For further relations with CGS unit current-turns, multiply those in
terms of ampere-turns by 10; that is, substitute for "ampere-turns" in
any of the above, the quantity "CGS unit current-turns X 10.")
Maxwells «* volts X seconds X 10^ ^ number of turns.
" = volts X seconds X 10®. For a single conductor.
When the flux is due only to the current, as in self-induction, and when
there is no magnetic leakage :
Maxwells = joules of stored energy X 2 X 10® -J- ampere-turns.
* ' = joules of stored energy X 2 X 10® ^ amperes. For a single wire.
* * = ergs of stored energy X 20 -«- ampere-turns.
' • = henrys X ampere-turns X 10® -f- number of tums^.
' ' = henrys X final amperes X 10® -*- number of turns.
When there is magnetic leakage, substitute for "maxwells" in the above?
"mean maxwells." The mean maxwells are the mean flux-turns divided
by the total number of turns.
When the flux is from an external source, and independent of the current,
as in mutual induction, and when there is no magnetic leakage:
Maxwells = joules of stored energy X 10® -s- ampere-turns.
* * = joules of stored energy X 10® -j- amperes. For a single conductor.
* ' = ergs of stored energy X 1 -t- ampere-t urns .
When there is magnetic leakage, make the same substitution as above
described.
Mean maxwells (through the secondary):
= henrys (of mutual induction) X final amperes (of primary) X 10® +
number of turns (of secondary).
t Or 4;r/10. Aprx. add H- Log 0099 2099.
t Or 4;r. Aprx. HX 100. Log 1.099 2099.
140 MAGNETIC FLUX DENSITY.
MAGNISTIO FLUX DENSITY [5C, H]; BIAGNBTIO
INDUCTION [(B, B] ; LINBS OF FORCE PER UNIT
CROSS-SECTION; EARTH'S FIELD. (FluxH-sur.
face ; magnetizing force X permeability.)
This quantity, which is one of the most important in magnetic calcula-
tions, measures the extent to which a body is magnetized as expressed by
the amount of flux which exists per square centimeter (or square inch) of
cross-section of the circuit; it gives the density of the flux in C.G.S. units
(maxwells or lines of force) per square centimeter or square inch cross-sec-
tion. It corresponds to current density in electrical calculations. As it
specifies or determines the strength of a magnetic field, it is often called
the field strength or field intensity;! the earth's magnetic field, for instance,
is expressed in these units. When it refers to air it is generally represented
by 3C or siniply H. When it refers to the flux density "induced" in a mag^-
netic material, such as iron, by an outside source, such as a current in a coil
of wire, it is often called the Indactioii, generally expressed by (R or sim-
ply by B, which is one of the most important quantities in magnetic cal-
culations. In the calculation and design of dynamos and transformers
this induction or flux density in the iron is of prime importance. From it
as a starting-point, the size of the core and the required ampere-turns are
calculated, as was explained briefly under the units of magnetizing force.
The saturation-point of magnetic material like iron is expressed in terms
of these units of flux density. The' total flux in maxwells is equal to the
flux density in gausses multiplied by the total cross^section in square centi-
meters. As the permeability or reluctivity of air is unity, it follows that
the magnetizing force in gilberts per centimeter of a long coil is numerically
the same as the flux density in gausses produced by it in the interior of the
coil, when there is no magnetic material in it. For this reason the magne-
tizing force is often confused with flux density or its equivalent the intensity
of field, as it is then often called. This applies only to the absolute units;
when ampere-turns or when inch units are used, a numerical factor must be
introduced.
The unit universally used when the dimension's are in the metric eysteBO.
is the electromagnetic C.G.S. unit, which, according to the International
Congress of 1900, is called a gauss. It is defined as that field intensity
which is produced at the center of a circle of one centimeter radius by 1
C.G.S. unit of current (or 10 amperes) flowing through an arc of this
circle one centimeter long. This is one of the relations which connect
the electric with the magnetic units. It can also be defined as the inten-
sity of the field at one centimeter distance from a unit pole, that is, at the
surface of a sphere of one centimeter radius, havibg an imaginary C.G.S.
unit pole at its center; from such a pole ^n unit lines of force (niaxweUs)
emanate, and as the area of the sphere is 47r square centimeters, it follows
that the flux density will be one line of force or maxwell per square centi-
meter of the spherical surface. It may also be defined as that field inten-
sity which will exert a pull of one djoie on an (imaginary) isolated unit
magnetic pole placed in it. All three of these definitions refer to the same
unit. The formulas giving the field intensity in coils, like those for mag-
nets or for galvanometers, all give the result in terms of this unit, provided
the current is stated in terms of the absolute unit of current which is equal
to 10 amperes; great care must be taken in such formulas to use the proper
unit of current; it should always be stated whether the formula has been
reduced to amperes or not.
The practical unit is therefore the same as the C.G.S. unit, namel:^
the RTBuss, which means one maxwell (or line of force) per square centi-
meter; and a flux density or induction stated in a number of gausses
means that number of maxwells (or lines of force) per square centimeter,
t It should be distinguished, however, from the term intensity of mag-
netization (see below), which is a term soQietime? used in physics and has a
different meaning.
( (
MAGNETIC FLUX DENSITY. 141
When the dimensions are in inches the flux density and induction are
often for convenience stated in lines of force (or maxwells) per square inch;
this inch unit has no generally accepted name; the name Incb gaass is
here proposed and is used in these tables.
Logarithm
1 maxwell per sq. inch = l. inch gauss 0000 0000
'* =0.155 000 gauss. Aprx. ?i« 1.190 8808
1 inch gAUBB^ 1. maxwell per sq. inch 0000 0000
•• "=0.155 000 gauss. Aprx. %a Il90 3308
1 COS unit (elmg) =• 1. gauss 0000 0000
1 grauss >s 6.451 63 maxwells per sq. inch. Aprx. ^% or G^i. . . . 0-809 6692
=6.451 63 inch gausses. Aprx. i% or 61^ 0-809 6692
-» 1. CGS unit (elmg) of flux density 0-000 0000
a 1. CXjrS unit (elmg) of flux per sq. centimeter. . . 0-000 0000
** "» 1. maxwell per sq. centimeter 0-000 0000
• • = 1 . magnetic ' ' line of force" per sq. centimeter.. . 0-000 0000
1 mazTirell per sq. centimeter => 1. gauss 0-000 0000
1 kilog:au8s « 1 000. gausses 3-000 0000
The relations to other measures are as follows:
Gausses >» maxwells -t-sq. centimeters.-
=inch gausses X 0.155 000.
= gilberts per centimeter X permeability.
= gilberts per inch X permeabilitjr X 0.393 700.
= gilberts per centimeter. For air.
=» gilberts per inch X 0.393 700. For air.
= gilberts X permeability + centimeters.
• • = gUberts X permeability X 0.393 700 -*- inches.
= gilberts-*- centimeters. For air.
= gilbert^ X 0.393 700 -i- inches. For air.
— gilberts -s- (oersteds X sq. centimeters) .
= ampere-turns per centimeter X permeability X 1.256 64. f
1 1
i <
< I
« I
< <
< I
1 1
« <
« «
* * — ampere-turns per inch X permeability X .494 7 38 .
* * = ampere-turns per centimeter X 1 .256 64. f For air.
« •
««
* * «= ampere-turns per inch X 0.494 738. For air.
* * •= ampere-turns X permeability X 1 .256 64t -*• centimeters.
* • =ampere-tums X permeability X 0.494 738 -f- inches.
* * = ampere-tums X 1 .256 64t -*- centimeters. For air.
= ampere-turns X 0.494 738-*- inches. For air.
»= ampere-tums X 1 .256 64t ■*- (oersteds X sq. centimeters).
= CXjIS unit current-turns per centimeter X permeability X 12.566 4.%
(For further relations with CGS unit current-turns, multiply those in
terms of ampere-tums by 10; that is, substitute for "ampere-tums" in
any of the above, the quantity "CGS unit current-turns X 10.")
Gausses = volts X seconds X 10® -*- (number of turns X sq. centimeters).
* * = CGS units of intensity of magnetization X 12.566 4.t
Inch g^ausses :
«= maxwells -f- sq . inches.
— gausses X 6.451 63.
— gilberts per centimeter X permeability X 6.451 63.
= gilberts per inch X permeability X 2.540 01 .
»" gilberts p>er centimeter X 6.451 63. For air.
—gilberts per inch X 2.540 01. For air.
— gilberts X permeability X 2.540 01 -*- inches.
— gilberts X 2.540 01 -?- inches. For air.
— gilberts -J- (oersteds X sq . inches) .
— ampere-tums per centimeter X permeability X 8 . 107 36
—ampere-tums per inch XpermeaoilityX 3.191 86.
— ampere-turns per centimeter X 8 . 107 35 . For air.
—ampere-turns per inch X 3.191 86. For air.
—ampere-tums X permeability X 3.191 86 -*- inches.
— ampere-tums X 3.191 86 -finches. For air. .
—ampere-tums X 1 .256 64t -*- (oersteds X sq. inches;.
— CGS unit current-turns per centimeter X permeability X 81.073 5.
, ■ ■ ■ 1 T
t Or 4;r/10. Aprx. add H- Log 099 2099 1
t Or 4;r. Aprx. >^X 100. Log 1-099 2099-
142 MAGNETIC MOMENT. — ^INTENSITY.
(For further relations with CGS unit current-turns, multiply those in
terms of ampere-turns by 10; that is, substitute for "ampere-turns" in
any of the above the quantity *'CGS unit current-turns X 10.")
Inch g^auMses = volts X seconds X 10*-*- (number of turns Xsq. inches).
Gausses in iron = gausses in air X permeability.
• ' in air =- gausses in iron -*- permeability.
Inch g^ausses in iron == inch gausses in air X permeability.
' * in air = inch gausses in iron + permeability.
MAGNETIO MOMENT [^]. (Pole strength X length.)
This quantity, used chiefly in magnetometry, is the product of the pole
strength of a magnet multiplied by its theoretical length, that is, by the
distance between the two centers at which the poles may be considered to
be condensed. As a pole (see under flux) is not measured in units like
force, such a moment is not directly comparable with a mechanical momer.t
called torque, but as the force existing between two unit poles one centi-
meter apart is one dsnie, a magnetic moment may be converted into a
mechanical moment. As a single pole has no real existence such a cal-
culation in practice always involves the action of two poles on two others.
lliere are no special units. The CO. 8. unit is a unit pole multiplied
by a centimeter, and would therefore be called a pole-centimeter. A max-
well-centimeter might also be used, as a unit pole has 4)r (=» about 12^)
maxwells or lines of force issuing from it; each line of force exerts a force
of one djoie on a unit pole. Two unit poles one centimeter apart attract
or repel each other with a force of one dyne; the force between any two
poles is proportional to the product of the two pole strengths in terms of
the above unit poles, and inversely proportional to the square of the dis-
tance between them in centimeters.
1 unit- pole-centimeter unit =1. unit pole X 1 . centimeter.
" — 1 . CGS unit of magnetic moment.
The relation to other measures are as follows:
Magnetic moments (in CGS units):
= CGS unit poles X centimeters.
= intensity of magnetization (in CGS units) X volume in cb. cm.
= gausses X 0.079 577 5 X volume in cb. cm.
INTENSITY OF MAGNETIZ^lTION [3, I\ j MOMENT
PER UNIT VOLUME; POLE STRENGTH PER
UNIT CROSS-SECTION. (Magnetic moment -^volume;
pole strength -^ surface.)
This quantitv, used chiefly in physical conceptions, measures the polar-
ized state of the interior of a magnet. If a magnet were cut into small
pieces (assuming that the magnetic state was not altered thereby) each
piece would be a separate magnet whose magnetic moment bears the same
proportion to its volume as the moment of the original magnet bears to
its volume, hence the magnetic state remains the same if stated in the mag-
netic moment per cubic centimeter, which quantity is called the intensity
of magnetization. It is also the pole strength per square centimeter cross-
section. As pole strength is convertible into flux (maxwells), it follows
that the intensity of magnetization is a unit of the same nature as flux den-
sity, that is, maxwells per square centimeter or gausses. They differ only
in the bases on which tney are defined.
The C.G.S. unit is one unit moment per cubic centimeter, that is, one
unit-pole-centimeter per cubic centimeter, or one unit pole per square
centimeter. This unit is numericallv equal to 4 5 gausses, as its relation to
gausses is the same as the relation of a unit pole is to a unit of flux.
1 CGS unit of intensity of mag^netization:
«= 1. CG^ unit of magnetic moment per cb. centimeter.
= 1. unit-pole-centimeter unit of magnetic moment per cb. cm.
= 12.566 4 (or 4;r) gausses.
The relations to other measures are as follows:
CGS unitA of intensity of magnetization:
»CGS units of magnetic moments -f-cb. centimeters.
- gausses X 0.079 577 5.
MAGNETIC ENERGY. 143
MAQNETIO WORK or BNBRQT [W]. (Magnetomotive
force X flux ; ampere-turns X flux.)
This quantity is seldom used in calculations. When a current is started
in a wire or in an electro-mafi^et, or when the armature of a steel magnet
is pulled off, masnetic energy is stored ; it is given out again in some other
form, often in the form of a spark, when the current is stopped, or as
mechanical eneror when a permanent magnet attracts its armature to
itself. In a transformer, the energy of the primary current is all converted
into magnetic energy which is reconverted into electrical energy in the
secondary circuit. The magnetic energy is equal to, and in fact is the same
thing as, the kinetic energy of a current (whiclr see above). It is equal to
the product of magnetomotive force and flux. It appears as heat in the
hysteresis loss. It should not be confused with the power used contin-
uously in exciting an electromagnet, as that power i.s all converted electric-
ally into heat ; it is only when the current is first started that any electric
energy is converted into magnetic energy. Magnetic flux itself is not
energy any more than coulombs of electricity or mechanical pressure;
energy is required to produce a pressure, but not necessarily to maintain
it, and so it is with magnetic flux (which see above).
There are no specific units of magnetic energy ; it is usually measured in
JouleH or oings* but may be measured in terms of anyoi the units of energy,
which see. The C.G.S. unit is the erg ; the practical unit is the Joule.
The relations to other measures are as follows:
Joules of matcnetic energ^y [J]r
= henrysX final amperes2-t-2.
«= henrys X applied volts* -t- (ohms* X 2) .
= time constant in seconds X ohms X final amperes* -4-2.
»time constant in seconds X final amperes X applied volts -4-2.
s^time constant in seconds X applied volts2-i-(onmsX2).
When the flux is due only to the current, as in self-induction, and when.'
there is no magnetic leakage:
Joules of magnetic energy :
= maxwells X ampere-turns + (2 X lO^).
— gausses X sq. centimeters X ampere-turns -i- (2 X 10^)>
= mch-gausses X sq. inches X ampere-turns -i- (2 X 10*).
= maxwells* X oersteds X 0.397 887 1 -^ lO*.
= maxwells X i^berts X 0.397 887 1 ^- lO^.
- gilberts* X 0.397 887 1 + (oersteds X 108).
» ampere-turns* X permeability X sq. centimeter section X 0.628 318
(or 2?r/10)-5-(centimeters lengtnX 10^).
« ampere-turns* X permeability X sq. inches section X 1 .595 93 ■*■ inches
length X 108.
(For further relations substitute for any of the above units their equiva-
lents in terms of, the desired units, as given in the other tables.)
When the flux is from an external source and independent of the current,
as in mutual induction, and when there is no magnetic leakage, the mag-
netic energy is twice as great as that given by the above relations; hence
aU the values above given must be multiplied by 2.
When there is magnetic leakage, use the "mean maxwells'' instead of
the "maxwells." T^e mean maxwells are the mean flux turns divided by
the totaJ niunber of turns.
!Ergs of magnetic energy » henrys X final amperes*X5X10°.
(For further relations of ergs to other units multiply those given above
for joules by 10^.)
t Or 10-5-8jr. Aprx. ^o- Log 1.599 7601.
144 MAGNETIC POWER.
MAQNBTIC POWER [P]. (Magnetomotive forceXflux-s-
time I ampere-turns X fltuc X frequency.)
This quantity is seldom used in calculations, and has little or no aigoiBr
cance in practice. It is the rate at which magnetic work or energy is per-
formed (see Magnetic Energy above); it is therefore equal to magnetic
energy divided by time. It is met with in practice in the alternating vaaLg"
netic fields of alternating electric currents, for in these the magnetic field
is continually being produced at a rate proportional to the frequency,
hence it is proportional to the product of the magnetomotive force, the flux,
and the frequency. The power of the primary current in a transformer is
transmitted to the secondary in the form of magnetic power, part of it
bein^ lost as magnetic power in the form of hysteresis.
With alternating magnetic fluxes the magnetic energy stored by tjie
current when flowmg in one direction, is in many cases returned to the
circuit again when the current is flowing in the reversed direction; it surges
to and rro in the iron, being alternately positive and negative, like in a
spring which is alternately compressed and released; hence calculations
oi the amount of the magnetic power involved are generally of no
importance in practice. Whatever energy leaves the circuit is generally
in the form of electrical energy (as in transformers) or heat (as in the
hysteresis loss); in such cases the power of this energy in watts is equal
to the amoimt in joules which leaves per cycle, multiplied by the frequency.
There are no specific units of magnetic power; such power is usually
measured in watts or in ergs per second, but it may be measured in terms
of any of the units of power. The C.G.S. unit is the erg per second;
the practical unit is tne watt.
The relations to other measures are as follows:
Watts o f m agrne t ic power [W , w] =« henrys X final amperes^ -¥■ seconds X 2.
(For further relations divide any of the values given for " jouJes," in the
preceding section, by *' seconds.")
PHOTOMETRIO UNITS.
The different kinds of units or measures given below and their relations
and sjrmbols (except *'cp'* for candle-power) are those adopted by the
unofficial International Cbngress at Geneva in 1806.
INTBNSITT OF LIGHT [I] ; OANDLB POWBR [op].
(Flux of light H- solid angle; power -^ solid angle.)
This quantity, which is the one most frequently used in photometnr.
measures the intensity of a source of li^ht in any one direction. For the
same total quantity or flux of light radiating from a source, the intensity
in any one direction becomes less as the solid angle through which it is
radiated becomes greater, hence the intensity is the total flux (in lumens)
divided by the number of units of solid angle (see under the table of units
of Solid Angles, p. 89).
There have been introduced from time to time various standards to be
used as reproducible units of the intensity of light, all of which are at best
only crude approximations to an exact standard. The most reliable of
these, and the one which is being generally accepted and is coming into
use internationally, is the hefner unit, an amyl acetate lamp of fixed dimen-
sions and height of flame. It has been carefully investigated by the
INTENSITY OF LIGHT. — CANDLE POWER. 145
Reichsanstalt and its coefficients have been determined; it was recom-
mended for international adoption to the International Electrical Ck)ngress
at Chicago in 1893, but was not adopted for reasons which probably do
not exist now; it has been adoi>ted by the Reichsanstalt; it has also been
accepted tentatively by the National Bureau of Standards through incan-
descent lamp secondary standards measured in terms of it at the, Reichs-
anstalt; it is endorsed by the American Institute of Electrical Engineers.
Tlie nefner standard lamp is very fully described with diagrams in an
article emanating from the Reichsanstalt, published in the Zeitachrift fiir
Inatrumentenkttnde, Vol. XlII, July, 1893, p. 257. In this article the
hefner unit is defined as the amount of light from a flame burning free in
stationary pure air, from a thick wick saturated with amvl acetate, the
wick completely filling a circular wick-tube of German silver, the inner
diameter of the tube being 8 mm, and the outer diameter 8.3 mm, the free
length of this tube being 25 mm ; the height of flame is 40 mm from the
edge of the wick-tube, measured at least 10 minutes after lighting.
The relation "1 hefner = 0.88 British standard candle" is the one gen-
erally used; it is the relation adopted by the Reichsanstalt and is the one
used by the National Bureau of Standards.
The next most important standard is the British standard candle or
Sngrlish spermaceti candle or simply £ng:lish candle, which is about
13^% greater than the hefner. The definition of the English candle used
by the National Bureau of Standards is the relation 1 hefner— 0.88 English
candles. This standard is the one universally referred to in this country by
the large incandescent electric lamp manufacturers and gas companies,
under the term ** candle power." The cand'e itself is not as easy to use
nor as reliable or constant as the hefner, for which reason the hefner is
generally used as the ultimate standard, the results being subsequently
reduced to such English candles by the relation 1 hefner =0.88 of these
candles. An official specification of the British standard candle is given
very fully in an article in the American^ Gas Light Journal, 1894, Jan. 8th,
p. 41. The candle has a. special wick, is made of spermaceti mixed with
3% to 4H% of bleached beeswax, weighs about a sixth of a pound, and
must bum at a rate not greater than 126 or less than 114 grains per hour.
In the comparisons made by the Reichsanstalt with the hefner, the height
of the flame of this English standard candle was 45 mm.
The unit called the platinum standard of lis^ht, sometimes im-
properly called an absolute unit, is the light emitted perpendicularly
from a square centimeter of surface of melted platinum at the tempera-
ture of its solidification. It is often called the vlollo. ^ This was virtually
adopted by an International Congress, but never came into use, and seems
to have oeen abandoned. Its value is not known definitely, but is ap-
proximately 20 candles. The bougie d^cimale, sometimes called a pyr, is
one twentieth of this, and at the unofficial Geneva Congress of 1896 its
value was {>rovisionally considered to be represented in practice by one
hefner. This is the value which will be used in the following tables.
The German parafiln candle has gone out of use, being replaced by
the hefner.
The carcel is an oil lamp formerly largely used as a standard in France ;
the oil is kept at a fixed level by means of a pump driven by clockwork.
The Harcourt pen tan e lamp is a flame using pentane gas; it is generally
made for 1 and for 10 candle-power ; it is used extensivelv as a secondary
standard and as such seems to be satisfactorv, but each lamp must be
calibrated 1^ comparison with some standard, as it seems they cannot
be made sufficiently uniform to have a definite value like the hefner.
Standard incandescent electric lamps which have been very care-
fully standardized by means of the hefner lampas the ultimate standard,
can now be purchased for use as standards. When their voltage is care-
fully adjusted, which can be done with great accuracy, they form very
satisfactory standards and are said to be very reliable. They are used
quite extensively and are probably the beat form of secondary standards
when a constant source of electric current is available.
Probably the best collection of values of the various standards is that
in a paper bjr Dr. Bunte in "The Technical Standards of Light," read before
the International Photometry Committee in June, 1903. It is translated
in the Journal of Gaa-lightipg, June 30, 1903, also in the Progressive Age,
Sept. 1 and 15, 1903. In the following table those values for which the
authority is given as Bxmte, have been taken from this paper.
146 INTENSITY OF LIGHT. — CANDLE POWER.
The following values, which are believed to be the best obtainable, are
mostly only approximate; different authorities do not agree.
** Means accepted by the Reichsanstalt and the National Bureau of
Standards. The names in parenthesis are the authorities.
1 liefner — 1. bougie decimale. (Geneva Congress.)
= 1.026 bougies d^imales. (Violle.)
»0.89 bougie d^imale. (Bunte.)
» 0.883 bougie decimale. (Laporte.)
»0.833 German candle. (Bunte.)
a 0.88** British or English standard candle (tmiversally accepted
value). Aprx. %.
=•0,092 carcel. (Bunte.)
1 boug^ie decimale :
*= 1 hefner. (Greneva Congress.)
— l.lShefners. (Violle.)
= 0.88 British or English standard candle. (1 hefner— 0.88 candle.)
=> 0.99 British or English standard candle. (Violle.)
— 0.94 German candle. (Violle.)
-0 05vioUe. (Official.)
1 py I* =*- 1 bougie d^imale.
1 British or JESngrlish standard candle [cp]:
«" 1.136 36 hefners (from the accepted value of the hefner). Aprx. %.
"» 1.14 hefners. (Bunte.)
=> 1.01 bougies d^cimales. (Bunte.)
— 0.950 (jennan candle. (Bunte.)
»- 0.105 carcel. (Bunte.)
1 candle or candle power [cpj :
= in this country and England 1 British or English standard candle.
1 German paraffin caudle (20 mm diam.):
*= 1.224 hefners. (Crerman Gas and Water Committee.)
= 1.20 hefners. (Bunte.)
= 1.16 hefners. (Lummer and Brodhun.)
=> 1.05 English candles. (Bunte.)
» 1.07 bougies d^cimales.
» 1.05 bougies dt^cimales. (Laporte.)
==0.110 carcel. (Bunte.)
1 c»ro«I = 10.87 hefners. (Bunte.)
'* =10.9 hefners.
" = 9.62 bougies d6cimales. (Violle.)
•• = 9.53 English candles. (Bunte.)
** = 9.05 German candles. (Bunte.)
= 0.481 violle. (Violle.)
1 violle « 22.6 hefners. ( Violle.)
" « 20. bougies d^cimales. (Official.)
'* '=' 20. hefners. (Geneva Congress.)
= 19.8 English candles. (Violle.)
** =18.8 German candles. (Violle.)
= 2.08carcels. (Violle.)
1 platinum standard = 1 violle. (Official.)
1 absolute unit = 1 violle.
1 Harcourt pentane lamp :
= secondary standard made for various candle-powers.
In the following relations a candle means a hefner unit.
Candles — lumens -^ units solid angle.
** = luxes X (distance in meters)*.
" = units of brightness Xsq. centimeters.
** = lumen-hours -^ (imits solid angle X hours).
FLUX OF LIGHT. 147
FLUX OF IiIQHT [#]; SPHERICAL OR HEMI-
SFHERIOAL CANDLE POWER. (Candle powerX
solid angle; power.)
This quantity measures the whole radiation or whole beam of light,
and is therefore equal to the intensity in candle powers multiplied by the
number of units of solid angle through which the conical beam radiates.
In practice the solid angles more usually used are either the hemiEq;>here
( « 6.283 19 units) or the whole sphere ( = 12.566 4 units). This quantity,
being a rad'ation of energy, is of the same nature as power, and a relation
between the two should exist and would be called the mechanical equiva-
lent of light but it is not yet known; it is believed to be of the order of
about 5.3 spherical candles per watt or 0.188 watts per spherical candle.t
The unit of flux is the amount of flux of li^ht in a beam of one unit
solid angle (one which subtends a square centimeter at a radius of one
centimeter) in which the intensity is one candle power. The name of
this unit adopted by the Geneva Congress is lumen. In practice the units
spherical candle power and hemispherical candle power are often
used instead, referring in this coimtry and England to the English candle.
In each of the following conversion factors the flux is the same, but
the solid angle within wmch it is confined is dififerent. Aprx. means
within 2%.
1 lumen =3 1. solid angle hefner.
** —0.159 155 hemispherical hefner. Aprx. 16+100.
" •=0.140 056 hemispherical (English) candle power. Aprx. ^.
•♦ « 0.079 577 spherical hefner. Aprx. 8 -«- 100.
" =0.070 028 spherical (English) candle power. Aprx. 7 + 100.
1 hemispherical hefner :
— 6.283 19 lumens. Aprx. 6H-
— 0.88 hemispherical (English) candle power. Aprx. Ji.
— 0.5(X) 000 spherical hefner.
•=0.440 000 spherical (English) candle power. Aprx. %.
1 hemispherical (English) candle power :
— 7.139 99 lumens. Aprx. »%,
■> 1.136 36 hemispherical hefners. Aprx. %.
=0.568 181 spherical hefner. Aprx. 4^.
*-> 0.500 000 spherical (English) candle power.
1 spherical hefner :
- 12.566 4 lumens. Aprx. H X 100.
» 1.76 hemispherical (English) candle powers. Aprx. T4.
— 2 hemispherical hefners.
» 0.88 spherical (English) candle power. Aprx. J4.
1 spherical (English) candle power :
- 14.280 lumens. Aprx. ^ X 100.
— 2 .27 2 7 3 hemispherical hefners. Aprx . % .
B 2. hemispherical (English) candle powers.
» 1.136 36 spherical hefners. Aprx. ^.
In the following relations one candle means a hefner unit
liumens »= candles X units solid angle.
** -=« candles Xsq. meters of illuminated surface -^ (distance in me-
ters)2.
*' B luxes X surface in sq. meters.
" —units of brightness X surf ace in sq. cm X units solid angle.
" «=■ lumen-hours + hours.
t See Mechanical Equivalent of Light. Elec. World and Eng., April 20,
1901, p 631.
148 ILLUMINATION. — BRIGHTNESS.
ILLUBAZMATZON [E]. (Candle power -^distance'; fins of
light -h surface.)
This quantity measures the amount of light falling on a surface; it
measures that which is received as distinguished from that which is given
out by the source. It is a very important quantity in photometry, as
illumination is that which light is intended to produce. Tne illumination
of a surface is proportional to the candle power of the source cf light and
inversely proportional to the square of the distance of the illuminated
surface from the source; if the intensity of the source is in hefners and the
distance in meters, then the illumination will be in luxes. The illumina-
tion in luxes is also equal to the total flux of light in lumens shining on
the surface, divided by the amount of the surface m square meters. 'These
relations give the amount of illumination which reaches that surface froni
the source, and not the amount of light reflected from the surface, as that
depends on the nature and color of the surface.
The unit adopted by the Geneva Ck>ngress is a lux, which is equal to
the illmnination produced by one hefner at a distance of one meter. When
the source is called a candle, then this unit is often called a meter-candle ;
but it is then improperly named, as it should be called a candle per meter
or still more correctly a candle per meter squared. The lux is also
equal to one lumen of flux per square meter. A unit called the foot-
candle (more properly candle per foot or candle per f«>ot squared) -
is also used ; it is equal to the illumination produced by an English candle
at a distance of one foot. With these units the amount of light required
to produce any desired illumination at a given distance can readily be
calculated.
1 lux:
» 1. lumen per sq. meter.
» 1. meter-candle (hefner) or 1 candle (hefner) per meter squared.
« 0.081 8 foot-candle (English) or candle (English) per foot squared.
Aprx. %2.
1 meter-eandle (hefner):
= l.lux.
B 1. lumen per sq. meter.
= 0.081 8 foot-candle (English). Aprx. Ma.
1 foot-candle ( English) » 12.2 luxes.
" = 12.2 meter-candles (hefner).
" b12.2 lumens per sq. meter.
In the following relations a candle means a hefner unit.
liuxes —candles -{-(distance in meters)^.
" =flux in lumens -i- surface in sq. meters.
" =flux in lumens -i- [(distance in meters)2X units solid angle].
'* = candles X units solid angle -*- surf ace in sq. meters.
" — imits of brightness X sunace of source in sq. cm . •«- (dist . in meters)'.
** « lumen-hours ■♦- (hours X surface in sq. meters).
BRIGHTNESS OF SOX7ROB [e]. (Candle power •^ surface
of source.)
This quantity measures the brightness of the source; it is the total
candle power of the source divided by its surface in sq. centimeters. It
is seldom used. The nnit is one Iiefner per sq. centimeter; no
name has been given to it. As there are no other units of this kind, there
are no conversion factors.
I unit of brightness <sl candle (hefner) per sq. centimeter.
In the following relations a candle means a hefner unit.
Units of brlg^htness :
« candles ■♦- centimeter^.
>»flux in lumens -{-(centimeter'Ximits solid angle).
QUANTITY OF LIGHT. — EFFICIENCY. 149
QUANTITY OP UGHT [Q]. (Flux of UghtX time j
energy.)
This quantity measures the total amount or volume of light together
with its duration. A rational payment for light, for instance, would
be made in terms of this unit. It is equal to the amoimt of flux in lumens
multiplied by the time in hours.
The unit is one lumen of flux for one hour, and is called a lumen-honr.
As there are no other units of the same kind, there are no conversion
factors.
1 lainen-honrssl lumen for one hour.
In the following relations a candle means a hefner unit.
liUmen-huurs :
=flux in lumens X hours.
«= candles X hours X units solid an^le.
«»candlesX illuminated surface in sq. meters X hours -•- (distance in
meters)*.
B*luxesX illuminated surface in sq. meters X hours.
•"Units of brightness Xsq. cm of source X units solid angle X hours,
■-units of brightness X sq. cm of source X illuminated surface in sq.
meters X hours-*- (distance in meters)*.
LIGHT EFFIOIENOT; POWER PER CANDLE
POWER. ( Candles -^ power; power -^ candles.)
The efliciencies of electric lights are frequently compared with each
other by comparing the watts required per candle. The number thus
obtained, by dividing the watts by the candles, is sometimes called the
efficiency, which term, however, is incorrectly used, because th: ^^reater
the watts per candle, the lower the efficiency; the term efficiency would be
more correctly applied to the number giving the candles per watt. Com-
parisons between such relative efficiencies are very useful, even though the
figures are not the absolute efficiencies; for the latter it would be necessary
to know the mechanical equivalent of light ,t and instead of the candle
power, the total flux in lumens or the spherical candle powers should be
used, so as to include the solid angle through which the light is radiated.
The following relations apply equally well to hefners as to English
candles.
n "watts per candle = 1/n candles per watt.
n candles per watt = l/n watts per candle. "^
n -watts per candle =7 35.448/n candles per metric horse^power.
*• =745.650/n candles per horse-power.
'* =a 1 000/n candles per kilowatt.
n candles per metric horse-power =735. 448/n watts per candle.
n candles per horse-power =745.650/n watts per candle.
n candles per kilowatt => 1 000/n watts per candle.
For the relation between watts, horse-powers, etc., see table of units of
Power, page 80.
t See notes on Flux of Light, p. 147, and the foot-note.
150 THERMOMETER SCALES.
THXSRMOMXSTBR SOALSa
Of the four scales in use, the Centifl^rade scale (also called Celsius) is the
most rational one and the one used in alL^ientific research and interna-
tional literature; it is also used exclusively in some of the European coun-
tries. The zero point is the melting point of ice, and the 100^ point is the
boilins point of water. The Fahrenheit scale is used in the United States
and England; on this scale the melting point of ice is exactly 32° and the
boiling point of water is 212^. The B^aumnr scale is in limited use in Ger-
many; it has the same zero point as the Centigrade scale, but the boiling
point of water on this scale is exactly 80°. The Absolute scale begins at a
theoretical, assumed point, supposed at present to be the lowest tempera-
ture which can exist ; this point is calculated from the expansion of gases
at ordinary temperatures and it is assmned that the same law holds good
down to an absolute zero ; it has never been reached, but has been approached
to within 17 degrees of the Centigrade scale.
The following table gives the corresponding values on these four different
scales for the complete range of all known temperatures, thus avoiding
most of the reductions. To these has been added a "concrete scale,"
which gives numerous temperatures at which certain materials change some
property, thereby enabling one to establish, maintain, or measure those
temperatures. But as most of these temperatures are not known defi-
nitely, they cannot be relied upon for more than approximate correctness.
They have been compiled from a large number of sources with due regard
for the authorities; many of them were taken from Camelley's excellent
table of melting and boiling points, and from Landolt and Boemstdn's
tables.
Reduction factors for one degree:
A Centigrade decree is % or 1.8 Fahrenheit degrees. It is % or 0.8
RtSaumur degree. It is the same as a degree of the Absolute scale.
A Fahrenheit degree is % or a little more than half of a Centigrade
degree or of a degree of the Absolute scale. It is % of a R^umur degree.
A B^anmnr de^ee is % or 1.25 Centigrade degrees, or degrees of the
Absolute scale. It is % or 2.25 Fahrenheit degrees.
A degree of the Absolute scale is the same as of the Centigrade soale.
Reduction factors for readings of a temperature in degrees;
To convert a reading in Centigrade degrees into the corresponding
one in Fahrenheit degrees, multiply by % and add 32. To convert it into
the one in Reaumur degrees multiply by %. To convert it into the one on
the Absolute scale, add 273.
To convert a reading in Fahrenheit degrees into the one in Centi-
grade degrees, subtract 32 and then multiply by ^, being careful about the
signs when the reading is below the melting point of ice. To convert it
into the one in Rdaumur degrees, subtract 32 and multiply by %. To con-
vert it into the one on the Absolute scale, subtract 32, then multiply by %
and add 273 ; or multiply by 5. add 2 297. and divide by 9.
To convert a reading in B^aumnr degrees into the one in Centigrade
degrees, multiply by %. To convert it into the one in Fahrenheit degrees,
multiply by % and add 32. To convert it into the one on the Absolute scale,
multiply by % and add 273.
To convert a reading on the Absolute scale to the one in Centigrade
degrees, subtract 273. To convert it into the one in Fahrenheit degrees sub-
tract 273, multiply by %, and add 32; or multiply by 9, subtract 2297. and
divide by 5. To convert it into the one in Reaumur degrees subtract 273
and multiply by %.
AU these reduction factors are strictly correct. Care must be taken to
add or subtract alffebraicalty when any of the readings are below the zero
of the respective scale, in which case they should be preceded by the nega-
tive sign. There can be no negative values on the Absolute scale (unless it
is shown in the future that the absolute zero has been fixed too high, a pos-
sibility which may not be remote).
THERMOMETER SCALES.
151
THERMOMETER SCALES.
Centi-
Fahren-
R^u-
Abso-
grade
Deg.
heit
mur
lute
Deg.
Deg.
Deg.
-273
-459.4
-218.4
-250
-418.0
-200.0
4- 23
-225
-373.0
-180.0
4- 48
-200
-328.0
-160.0
+ 73
-190
-310.0
-162.0
+ 83
-180
-292.0
-144.0
+ 93
-170
-274.0
-136.0
+ 103
-160
-256.0
-128.0
+ 113
-160
-238.0
-120.0
+ 123
-140
-220.0
-112.0
+ 133
-130
-202.0
-104.0
+ 143
-120
-184.0
- 96.0
+ 153
-110
-166.0
- 88.0
+ 163
-100
-148.0
- 80.0
+ 173
- 90
-130.0
- 72.0
+ 183
- 80
-112.0
- 64.0
+ 193
- 70
- 94.0
- 56.0
+203
- 60
- 76.0
- 48.0
+213
- 60
- 58.0
- 40.0
+223
- 45
- 49.0
- 36.0
+228
- 40
- 40.0
- 32.0
+233
- 35
- 31.0
- 28.0
+238
- 30
- 22.0
- 24.0
+ 243
- 28
- 18.4
- 22.4
+245
- 26
- 14.8
- 20.8
+ 247
- 24
- 11.2
- 19.2
+ 249
- 22
- 7.6
- 17.6
+251
- 20
- 4.0
- 16.0
+ 253
- 19
- 2.2
- 15.2
+254
- 18
- 0.4
- 14.4
+255
-17.8
- 14.2
+255.2
-17.2
+ 1.0
- 13.8
+255.8
-17.0
4- 1.4
- 13.6
+ 256.0
-16.7
4- 2.0
- 13.3
+256.3
-16.1
+ 3.0
- 12.9
+266.9
-16.0
4- 3.2
- 12.8
+ 257.0
-16.6
4- 4.0
- 12.4
+257.4
-16.0
4- 5.0
- 12.0
+258.0
-14.4
4- 6.0
- 11.6
+ 258.6
-14.0
4- 6.8
- 11.2
+259.0
-13.9
4- 7.0
- 11.1
+259.1
-13.3
4- 8.0
- 10.7
+269.7
-13.0
4- 8.6
- 10.4
+260.0
-12.8
4- 9.0
- 10.2
+260.2
-12.2
4- 10.0
- 9.8
+ 260.8
-12.0
4- 10.4
- 9.6
+261.0
-11.7
4- 11.0
- 9.3
+261.3
-11 1
4- 12.0
- 8.9
+261.9
-11.0
4- 12.2
- 8.8
+262.0
-10.6
4- 13.0
- 8.4
+262.4
Concrete Scale (mostly only approximate).
— 256° C. hydrogen freezes
-250° C. hydrogen boils
— 214° C. nitrogen freezes
— 200° C. temperature of liquid air
- 193.1° to - 194° C. (760 mm press.) nitro-
[gen boils
- 181.4° to - 184° C. (1 atm.) oxygen boils
— 167.0° C. nitric oxide solidifies
— 153.6° C. nitric oxide boils
- 130.5° C. pure ethyl alcohol freeze
— 1 12.5° C. hydrochloric acid melts
— 102° C. hydrochloric acid condenses
—85.5° C. hydrogen sulphide solidifies
— 76° C. sulphur dioxide solidifies
— 75° C. ammonia melts .
-61.8° C. hydrogen sulphide boils at 760
[mm
[sol.) solidifies
— 38° to —41° C. ammonium hydrate (sat.
— 39.5° C. mercury melts
— 33.7° C. ammonia boils
-33.6° C. chlorine boils
20.7° C.(CN)2 boils
18.0° C.SbHj boils
— 10.5° C. sulphur dioxide boils at 744 mm
162
THERMOMETER SCALES.
THERMOMETER SCALES— (Continued).
Centi-
grade
Deg.
-10.0
- 9.4
- 9.0
- 8.9
- 8.3
- 8.0
- 7.8
- 7.2
- 7.0
- 6.7
- 6.1
- 6.0
- 5.6
- 6.0
- 4.4
- 4.0
- 3.9
- 3.3
- 3.0
- 2.8
- 2.2
- 2.0
- 1.7
- 1.1
- 1.0
- 0.6
+ 0.6
1.0
1.1
1.7
2.0
2.2
2.8
3.0
3.3
3.9
4.0
4.4
6.0
6.6
6.0
6.1
6.7
7.0
7.2
7.8
8.0
8.3
8.9
Fahren-
R(?au-
heit
mur
Deg.
Deg.
+ 14.0
-8.0
+ 15.0
-7.6
+ 15.8
-7.2
+ 16.0
-7.1
+ 17.0
-6.7
+ 17.6
-6.4
+ 18.0
-6.2
+ 19.0
-5.8
+ 19.4
-6.6
+ 20.0
-5.3
+ 21.0
-4.9
+21.2
-4.8
+ 22.0
-4.4
+23.0
-4.0
+24.0
-3.6
+24.8
-3.2
+25.0
-3.1
+ 26.0
-2.7
+26.6
-2.4
+27.0
-2.2
+28.0
-1.8
+28.4
-1.6
+ 29.0
-1.3
+ 30.0
-0.9
+ 30.2
-0.8
+ 31.0
-0.4
+32.0
+ 33.0
+0.4
33.8
.0.8
34.0
0.9
35.0
1.3
35.6
1.6
36.0
1.8
37.0
2.2
37.4
2.4
38.0
2.7
39.0
3.1
39.2
3.2
40.0
3.6
41.0
4.0
42.0
4.4
42.8
4.8
43.0
4.9
44.0
5.3
44.6
6.6
45.0
6.8
46.0
6.2
46.4
6.4
47.0
6.7
48.0
7.1
Abso-
lute
Deg.
+263.0
+263.6
+264.0
+264.1
+264.7
Concrete Scale (mostly only approximate).
+265.0
+ 265.2
+ 265.8
+ 266.0
+ 266.3
+266.9
+ 267.0
+267.4
+ 268.0
+268.6
+ 269.0
+269.1
+269.7
+ 270.0
+ 270.2
+ 270.8
+ 271.0
+ 271.3
+ 271.9
+ 272.0
+ 272.4
+ 273.0
+ 273.6
274.0
274.1
274.7
275.0
275.2
275.8
276.0
276.3
276.9
277.0
277.4
278.0
278.6
279.0
279.1
279.7
280.0
280.2
280.8
281.0
281.3
281.9
—8.5° C. sulphuric acid melts, sp. gr. 1.732
■7.5*^ C. sulphuric acid melts, sp. gr. 1.727
7.2° C. bromine solidifies
0° C. freezing point of water
3° C. bensene (benxol) freeses
[molecular propirtion melts
4.5° C. alloy of potassium and sodium in
4.6* C. sulphuric acid melts, sp. gr. 1.749
6° C. alloy of 1 potassium and 1 sodium
[melts
THERMOMETER SCALES.
153
THERMOMETER SCALES— (Con/tnu«i).
Centi-
grade
Deg.
9.0
9.4
10.0
10.6
11.0
11.1
11.7
12.0
12.2
12.8
13.0
13.3
13.9
14.0
14.4
15.0
15.6
16.0
16.1
16.7
17.0
17.2
17.8
18.0
18.3
18.9
19.0
19.4
20.0
20.6
21.0
21.1
21.7
22.0
22.2
22.8
23.0
23.3
23.9
24.0
24.4
26.0
25.6
26.0
26.1
26.7
27.0
27.2
27.8
28.0
Fahren-
R<«au-
heit
mur
Deg.
Deg.
48.2
7.2
49.0
7.6
50.0
8.0
61.0
8.4
51.8
8.8
52.0
8.9
53.0
9.3
53.6
9.6
54.0
9.8
55.0
10.2
55.4
10.4
56.0
10.7
57.0
11.1
67.2
11.2
68.0
11.6
59.0
12.0
60.0
12.4
60.8
12.8
61.0
12.9
62.0
13.3
62.6
13.6
63.0
13.8
64.0
14.2
64.4
14.4
65.0
14.7
66.0
15.1
66.2
15.2
67.0
15.6
68.0
16.0
69.0
16.4
69.8
16.8
70.0
16.9
71.0
17.3
71.6
17.6
72.0
17.8
*
73.0
18.2
73.4
18.4
74.0
18.7
75.0
19.1
75.2
19.2
76
19.6
77.0
20.0
78.0
20.4
78.8
20.8
79.0
20.9
80.0
21.3
80.6
21.6
81.0
21.8
82.0
22.2
82.4
22.4
282.0
282.4
283.0
283.6
284.0
284.1
284.7
285.0
285.2
285.8
286.0
286.3
286.9
287.0
287.4
288.0
288.6
289.0
289.1
289.7
290.0
290.2
290.8
291.0
291.3
291.9
292.0
292.4
293.0
293.6
294.0
294.1
294.7
295.0
295.2
295.8
296.0
296.3
296.9
297.0
297.4
298.0
298.6
299.0
299.1
299.7
300.0
300.2
300.8
301.0
Concrete Scale (mostly only approximate).
10.5^ C. pure sulphuric acid freeses, sp. gr.
[1.854
17^ C. pure acetic acid freeses
19«-20* C. fresh butter solidifies
20**— 20.5* C. cocoanut oil solidifies
20.6** C. cocoa butter solidifies
21* C. fresh soft palm oil solidifies
24** C. fresh hard palm oil solidifies
24.5*' C. cocoanut oil melts
26.5* C. pure hydrocyanic acid boils
154
THERMOMETER SCALES.
THERMOMETER 8CALBQ— (Continued).
Centi-
Fahren-
Reau-
Abso-
grade
heit
mur
lute
Concrete Scale (mostly only approximate).
Deg.
Deg.
Deg.
Deg.
28.3
83.0
22.7
301.3
28.5° C. calcium chloride, (CaCl2+6H20),
28.9
84.0
23.1
301.9
[melts (see also under 719** C.)
29.0
84.2
23.2
302.0
29.4
85.0
23.6
302.4
30<» C. lard solidifies
30.0
86.0
24.0
303.0
30° C. fresh soft palm oil melts
30.6
87.0
24.4
303.6
30° C. gallium melts
31.0
87.8
24.8
304.0
31.1
88.0
24.9
304.1
31°-31.5° C. fresh butter mrfts
31.7
89.0
25.3
304.7
32.0
89.6
25.6
305.0
32.2
90.0
25.8
305.2
32.8
91.0
26.2
305.8
33.0
91.4
26.4
306.0
33° C. nutmeg butter solidifies
33.3
92.0
26.7
306.3
33° C. fresh beef tallow solidifies
33.9
93.0
27.1
306.9
33.5°-34° C. cocoa butter melts
34.0
93.2
27.2
307.0
34° C. old beef taUow solidifies
34.4
94.0
27.6
307.4
34.5° C. ether boils
35.0
95.0
28.0
308.0
35.6
96.0
28.4
308.6
36.0
96.8
28.8
309.0
36° C. fresh mutton taUow solidifies
36.1
97.0
28.9
309.1
36.7
98.0
29.3
309.7
37.0
98.6
29.6
310.0
37° C. blood heat of the human body
37.2
99.0
29.8
310.2
37.8
100.0
30.2
310.8
38° C. fresh hard palm oil melts
38.0
100.4
30.4
311.0
38° C. old palm oil soUdifies
38.3
101.0
30.7
311.3
38° C. pentane boilc
38.5° C. rubidium melt::
38.9
102.0
31.1
311.9
39.0
102.2
31.2
312.0
39.5° C. old mutton tallow solidifies
39.4
103.0
31.6
312.4
39.5° C. crystaUine phenol (carbolic acid)
[melts
40.0
104.0
32.0
313.0
40° C. magnesium chlorate melts
40.6
105.0
32.4
313.6
40.5°-41° C. Japanese wax solidifies
41.0
105.8
32.8
314.0
41.1
106.0
32.9
314.1
41.7
107.0
33.3
314.7
41.5°-42°C. lard melts
42.0
107.6
33.6
315.0
42° C. old palm oil melts
42.2
108.0
33.8
315.2
42.8
109.0
34.2
315.8
43.0
109.4
34.4
316.0
43° C. fresh beef tallow melts
43.3
110.0
34.7
316.3
43.5° C. old beef tallow melts
43.9
111.0
36.1
316.9
43.5°-44° C. nutmeg butter melts
44.0
111.2
35.2
317.0
44° C. spermaceti solidifies
44.4
112.0
35.6
317.4
44°-44.5° C. spermaceti melts
45.0
113.0
36.0
318.0
44.2°-44.5° C. yellow phosphorus melts
45.6
114.0
36.4
318.6
46.0
114.8
36.8
319.0
46.1
115.0
36.9
319.1
46.7
116.0
37.3
319.7
47.0
116.6
37.6
320.0
47° C. fresh mutton tallow melts
47.2
117.0
37.8
320.2
THERMOMETER SCALES.
156
THERMOMETER SCALES— (Conitntted).
Centi-
Fahren-
R<«au-
Abso-
fprade
Deg.
heit
mur
lute
Deg.
Deg.
Deg.
47.8
118.0
38.2
320.8
48.0
118.4
38.4
321.0
48.3
119.0
38.7
321.3
48.9
120.0
39.1
321.9
49.0
120.2
39.2
322.0
49.4
121.0
39.6
322.4
60.0
122.0
40.0
323.0
60.6
123.0
40.4
323.6
61.0
123.8
40.8
324.0
61.1
124.0
40.9
324.1
61.7
126.0
41.3
324.7
62.0
125.6
41.6
325.0
62.2
126.0
41.8
325.2
62.8
127.0
42.2
326.8
63.0
127.4
42.4
326.0
63.3
128.0
42.7
326.3
63.9
129.0
43.1
326.9
64.0
129.2
43.2
327.0
64.4
130.0
43.6
327.4
65.0
131.0
44.0
328.0
66.6
132.0
44.4
328.6
66.0
132.8
44.8
329.0
66.1
133.0
44.9
329.1
66.7
134.0
45.3
329.7
67.0
134.6
45.6
330.0
67.2
135.0
45.8
330.2
67.8
136.0
46.2
330.8
68.0
136.4
46.4
331.0
68.3
137.0
46.7
331.3
68.9
138.0
47.1
331.9
69.0
138.2
47.2
332.0
69.4
139.0
47.6
332.4
60.0
140.0
48.0
333.0
60.6
141.0
48.4
333.6
61.0
141.8
48.8
334.0
61.1
142.0
48.9
334.1
61.7
143.0
49.3
334.7
62.0
143.6
49.6
335.0
62.2
144.0
49.8
335.2
62.8
145.0
60.2
336.8
63.0
145.4
60.4
336.0
63.3
146.0
60.7
336.3
63.9
147.0
61.1
336.9
64.0
147.2
51.2
337.0
64.4
148.0
51.6
337.4
65.0
149.0
52.0
338.0
65.6
150.0
52.^
338.6
66.0
150.8
52.8
339.0
66.1
151.0
52.9
339.1
66.7
152.0
53.3
339.7
Concrete Scale (mostly only approximate).
50® C. hydrogen perchlorate melts
50.5** C. old mutton tallow melts
53.5°-54.5* C. Japanese wax melts
55° C. methyl alcohol boils
60.5 C. Wood's alloy (BiiCdPbaSn) melts
62° C. chloroform boils
62°-62.5° C. yellow bees' wax melts
62.1° C. potassium melts
63° C. bromine boils
63°-63.5° C. white bees' wax melts
65.5° C. alloy Cd4Sn5Pb6Biio melts
166
O'HERMOMfiTER SCALES.
THERMOMETER SCALES— (Con/tn«firf).
Centi-
Fahren-
Reau-
Abso-
•
grade
Deg.
heit
mur
lute
Concrete Scale (mostly only approximate).
Deg.
Deg.
Deg.
67.0
152.6
53.6
340.0
67.2
153.0
53.8
340.2
.
67.8
154.0
54.2
340.8
67.6'* C. alloy CdaRntPbiBig melts
68.0
154.4
54.4
341.0
68.3
166.0
54.7
341.3
68.9
156.0
65.1
341.9
68.6*» C. alloy CdSnPbBi, melts
69.0
156.2
65.2
342.0
69.4
157.0
65.6
342.4
70.0
158.0
56.0
343.0
70" C. hexane boila
70.6
159.0
56.4
343.6
71.0
159.8
56.8
344.0
71.1
160.0
56.9
344.1
71.7
161.0
57.3
344.7
72.0
161.6
67.6
345.0
72.2
162.0
67.8
345.2
72.8
163.0
68.2
345.8
73.0
163.4
58.4
346.0
73.3
164.0
68.7
346.3
73.9
165.0
59.1
346.9
74.0
165.2
59.2
347.0
74.4
166.0
69.6
347.4
76.0
167.0
60.0
348.0
75.6
168.0
60.4
348.6
76.0
168.8
60.8
349.0
76.1
169.0
60.9
349.1
76.7
170.0
61.3
349.7
77.0
170.6
61.6
350.0
77.2
171.0
61.8
350.2
77.8
172.0
62.2
350.8
78.0
172.4
62.4
351.0
78.3
173.0
62.7
351.3
78.4" C. pure ethyl alcohol boils
78.9
174.0
63.1
351.9
79.0
174.2
63.2
352.0
79.4
175.0
63.6
352.4
79.2" C. naphthalene melts
80.0
176.0
64.0
353.0
80.6
177.0
64.4
353.6
80.4" C. benzene, at 760 mm pressure, boils
81.0
177.8
64.8
354.0
81.1
178.0
64.9
354.1
81.7
179.0
65.3
354.7
82.0
179.6
65.6
355.0
82.2
180.0
65.8
355.2
82.8
181.0
66.2
355.8
•
83.0
181.4
66.4
356.0
83.3
182.0
66.7
356.3
83.9
183.0
67.1
356.9
84.0
183.2
67.2
357.0
84.4
184.0
67.6
357.4
«
85.0
185.0
68.0
358.0
85.6
186.0
68.4
358.6
86.0
186.8
68.8
359.0
THERMOMETER SCALES.
157
THERMOMETER SCALES (Continued).
Centi-
Fahren-
Reau-
Abso-
Srade
J>eg.
heit
Deg.
mur
Deg.
lute
Deg.
Concrete Scale (mostly only approximate).
86.1
187.0
68.9
359.1
86.7
188.0
69.3
359.7
87.0
188.6
69.6
360.0
87.2
189.0
69.8
360.2
87.8
190.0
70.2
360.8
88.0
190.4
70.4
361.0
88.3
191.0
70.7
361.3
88.9
192.0
71.1
361.9
89.0
192.2
71.2
362.0
89.4
193.0
71.6
362.4
89.5<* C. alloy CdPb8Bi4 melts
90.0
194.0
72.0
363.0
90.6
195.0
72.4
363.6
«
91.0
195.8
72.8
364.0
91.1
196.0
72.9
364.1
91.7
197.0
73.3
364.7
92.0
197.6
73.6
365.0
92** C. potash alum melts
92.2
198.0
73.8
365 2
92.8
199.0
74.2
365.8
93.0
199.4
74.4
366.0
93.3
200.0
74.7
366.3
93.9
201.0
75.1
366.9
93.7' C. Rose's alloy, Bi2PbSn, melts
94.0
201.2
75.2
367
94.4
202.0
75.6
367.4
96.0
203.0
76.0
368.0
95* C. alloy Cd2Pb7Bi8 melts
95.6
204.0
76.4
368.6
95.6° C. sodium melts
96.0
204.8
76.8
369.0
96.1
205.0
76.9
369.1
96.7
206.0
77.3
369.7
97.0
206.6
77.6
370.0
97.2
207.0
77.8
370.2
97.8
208.0
78.2
370.8
98.0
208.4
78.4
371.0
98.3
209.0
78.7
371.3
98.9
210.0
79.1
371.9
99.0
210.2
79.2
372.0
99.4
211.0
79.6
372.4
100.0
212.0
80.0
373.0
100<> C. water boils
100.6
213.0
80.4
373.6
101.0
213.8
80.8
374.0
101.1
214.0
80.9
374.1
101.7
215.0
81.3
374.7
102.0
215.6
81.6
375.0
•
102.2
216.0
81.8
375.2
102.8
217.0
82.2
375.8
103.0
217.4
82.4
376.0
103.3
218.0
82.7
376.3
103.9
219.0
83.1
376.9
104.0
219.2
83.2
377.0
104.4
220.0
83.6
377.4
104.4^ C. vitreous selenium melts
105.0
221.0
84.0
378.0
158
THERMOMETER SCALES.
THERMOMETER SCALES— (Coniinwed).
Centi-
Fahren-
Abso-
grade
heit
lute
Concrete Scale (mostly only approximate).
Deg.
Deg.
Deg.
105.6
222.0
378.6
'
106.0
222.8
379.0
106.1
223.0
379.1
106.7
224.0
379.7
•
107.0
224.6
380.0
107.2
225.0
380.2
107.8
226.0
380.8
108.0
226.4
381.0
108.3
227.0
381.3
108.9
228.0
381.9
109.0
228.2
382.0
109.4
229.0
382.4
110.0
230.0
383.0
110.6
231.0
383.6
111.0
231.8
384.0
111.1
232.0
384.1
111.7
233.0
384.7
112.0
233.6
385.0
112.2
234.0
385.2
112.8
235.0
385.8
110«>-115'> C. ferric sulphate melts
113.0
235.4
386.0
113.3
236.0
386.3
113.9
237.0
386.9
114.0
237.2
387.0
114.4
238.0
387.4
114.5** C. rhombic sulphur and copper nitrate,
[(Cu(N03)23H20). melt
115.0
239.0
388.0
115* C. iodine melts
115.6
240.0
388.6
116.0
240.8
389.0
116.1
241.0
389.1
116.7
242.0
389.7
117.0
242.6
390.0
117.2
243.0
390.2
117.8
244.0
390.8
118.0
244.4
391.0
118.3
245.0
391.3
--
118.9
246.0
391.9
119.0
246.2
392.0
119.4
247.0
392.4
120.0
248.0
393.0
120® C. prismatic sulphur melts
120.6
249.0
393.6
121.0
249.8
394.0
121.1
250.0
394.1
•
121.7
251.0
394.7
122.0
251.6
395.0
122.2
252.0
395.2
122.8
253.0
395.8
123.0
253.4
396.0
123.3
254.0
396.3
123.9
255.0
396.9
124.0
255.2
397.0
l
THERMOMETER SCALES.
159
THERMOMETER SCALES— <Con<tnii«d).
Centi-
Fahren-
grade
Deg.
heit
Deg.
124.4
256.0
125.0
257.0
125.6
258.0
126.0
258.8
126.1
259.0
126.7
260.0
127.0
260.6
127.2
261.0
127.8
262.0
128.0
262.4
128.3
263.0
128.9
264.0
129.0
264.2
129.4
265.0
130.0
266.0
130.6
267.0
131.0
267.8
131.1
268.0
131.7
269.0
132.0
269.6
132.2
270.0
132.8
271.0
133.0
271.4
133.3
272.0
133.9
273.0
134.0
273.2
134.4
274.0
135.0
275.0
135.6
276.0
136.0
276.8
136.1
277.0
136.7
278.0
137.0
278.6
137.2
279.0
137.8
280.0
138.0
280.4
138.3
281.0
138.9
282.0
139.0
282.2
139.4
283.0
140.0
284.0
140.6
285.0
141.0
285.8
141.1
286.0
141.7
287.0
142.0
287.6
142.2
288.0
142.8
289.0
143.0
289.4
143.3
290.0
Abso-
lute
Deg.
39; .4
398.0
398.6
399.0
399.1
399.7 1
400.0
400.2
400.8
401.0
401.3
401.9
402.0
402.4
403.0
403.6
404.0
404.1
404.7
405.0
405.2
405.8
406.0
406.3
406.9
407.0
407.4
408.0
408.6
409.0
409.1
409.7
410.0
410.2
410.8
411.0
411.3
411.9
412.0
412.4
413.0
413.6
414.0
414.1
414.7
415.0
415.2
415.8
416.0
416.3
Conciete Scale (mostly only approximate).
125.3*» C. alloy PbaBii melts
131** C. amyl alcohol boils
134» C. Al(N08)8-l-9HiO boilfl
136.4<» C. alloy SnsBU melts
138.12® C. sulphur monoohloride, at 760 mm pres-
[sure, boils
140° C. ferrous sulphate -1-6 aq. melts
160
THERMOMETER SCALES.
THERMOMETER SCALES— (Continued).
Centi-
grade
Deg.
143.9
144.0
144.4
145.0
145.6
146.0
146.1
146.7
147.0
147.2
147.8
148.0
148.3
148.9
149.0
149.4
150.0
162.0
154.0
156.0
158.0
160.0
162.0
164.0
166.0
168.0
170.0
172.0
174.0
176.0
178.0
180.0
182.0
184.0
186.0
188.0
190.0
192.0
194.0
196.0
198.0
200.0
205.0
210.0
215.0
220.0
225.0
230.0
235.0
240.0
Fahren-
Abso-
heit
lute
Deg.
Deg.
291.0
416.9
291.2
417.0
292.0
417.4
293.0
418.0
294.0
418.6
294.8
419.0
295.0
419.1
296.0
419.7
296.6
420.0
297.0
420.2
298.0
420.8
298.4
421.0
299.0
421.3
300.0
421.9
300.2
422.0
301.0
422.4
302.0
423.0
305.6
425.0
309.2
427.0
312.8
429.0
316.4
431.0
320.0
433.0
323.6
435.0
327.2
437.0
330.8
439.0
344.4
441.0
338.0
443.0
341.6
445.0
345.2
447.0
348.8
449.0
352.4
451.0
356.0
453.0
359.6
455.0
363.2
457.0
366.8
459.0
370.4
461.0
374.0
463.0
377.6
465.0
381.2
467.0
3S4.8
469.0
3S8.4
471.0
392.0
473.0
401.0
478.0
410.0
483.0
419.0
488.0
428.0
493.0
437.0
498.0
446.0
.503.0
455.0
508.0
464.0
513.0
Concrete Scale (mostly only approximate).
146° C.SnL melts
146.3<> C. ^oy CdBi4 melts
160*^ C. pure du^ar melts
161° C. turpentme boils
170° C. copper nitrate, (Cn(N08)23HsO), boils
173.8° C. alloy CdSna melts _ ^^
175° C. ordinary camphor, (CioHioO), melts
178° C. caffeine melts
180° C. aluminium chloride boils
180° C. lithium melts
181° C. aniline boils
181° C. alloy PbSna melts
182° C. phenol (carbolic acid) boils
187° C. alloy PbSni melts
197° C. alloy PbSn2 melts ^ . .,
204° C. ordina^ camphor, (CipHieO), boils
216.4°-216.8° C. naphthalene boils
217° C. crystalline selenium insol. in CS7 melts
218° C. silver nitrate melts
221° C. very faint yellow in tempering steel
230° C. silver chlorate melts .
232° C. pale straw yellow in tempenng ste^^l
235° C. tin melts
235° C. alloy PbSn melts
THERMOMETER SCALES.
161
THERMOMETER SCALES— (ConltniMrf).
Centi-
Fahren-
Abso-
grade
Ueg.
heit
lute
Deg.
473.0
Deg.
245.0
518.0
250.0
482.0
523.0
254.0
489.2
527.0
256.0
491.0
528.0
260.0
500.0
633.0
265.0
609.0
638.0
270.0
618.0
543.0
275.0
627.0
548.0
280.0
536.0
553.0
283.0
641.4
556.0
285.0
545.0
658.0
288.0
550.4
561.0
290.0
654.0
563.0
295.0
563.0
568.0
300.0
572.0
573.0
305.0
681.0
578.0
310.0
590.0
583.0
315.0
699.0
588.0
320.0
608.0
593.0
325.0
617.0
598.0
330.0
626.0
603.0
335.0
635.0
608.0
340.0
644.0
613.0
345.0
653.0
618.0
350.0
662.0
623.0
360.0
680.0
633.0
370.0
698.0
643.0
380.0
716.0
653.0
390.0
734.0
663.0
400.0
752.0
673.0
410.0
770.0
683.0
420.0
788.0
693.0
430.0
806.0
703.0
440.0
824.0
713.0
450.0
842.0
723.0
460.0
860.0
733.0
470.0
878.0
743.0
480.0
896.0
753.0
490.0
914.0
763.0
500.0
932.0
773.0
610.0
950.0
783.0
520.0
968.0
793.0
530.0
986.0
803.0
540.0
1004.0
813.0
660.0
1022.0
823.0
660.0
1040.0
833.0
570.0
1058.0
843.0
680.0
1076.0
853.0
690.0
1094.0
863.0
600.0
1112.0
873.0
Concrete Scale (mostly only approximate).
243** C. full yellow color in tempering steel
261^ C. silver chloride melts
254** C. brown color in tempering steel
255^ C. red phosphorus melts
262^ C. sine chloride melts
266** C. red color in tempering steel
269** C. bismuth melts
270*» C. alloy PbjSn melts
277** G. purple color in tempering steel
283** C. alloy PbaSn melts
276**-280*> C. glycerine distills
287.3** C. yellow phosphorus boils at 762 mm
288° C. mercuric chloride melts
288** C. bright-blue color in tempering steel
292"* C. alloy Pb4Sn melts
293^ C. full blue color in tempering steel
302** C. sodiimi chlorate melts
303** C. mercuric chloride boils
316** C. sodium nitrate melts
316** C. dark-blue color in tempering steel
320**-327** C. cadmium melts
327** C. lead melts
338** C. pure sulphuric acid, sp. gr. 1.864, boils
339** C. potassium nitrate melts
357.25** C. mercury boils
359** C. potassitmi chlorate melts
414** C. barium chlorate melts
419* C. zinc melts (Berthelot)
434** C. cuprous chloride melts
445** C. sulphur boils (Berthelot)
446** C. tellurium melts
448.4** C. sulphur at 760 mm boils (Camelley)
482** C. sodium perchlorate melts
486** C. silver perchlorate melts
498** C. lead chloride and cupric chloride melt
505** C. barium perchlorate melts
525** 0. first visible red of incandescent bodies
[(PouiUet)
541** C. cadmium chloride melts
561** C. calcium nitrate melts
561** C. borax melts
598** C. lithium chloride melts
162
THERMOMETER SCALES.
THERMOMETER QCALE&— (Continued).
Centi-
grade
Deg.
610
620
630
640
650
660
670
680
690
700
710
720
730
740
750
760
770
780
790
800
810
820
830
840
850
860
870
880
890
900
920
940
960
980
1000
1020
1 040
1060
1080
1 100
Fahr.
Deg.
120
140
160
180
200
220
240
1 260
1 280
1300
1 130
1 148
1 166
1184
1202
1220
1238
1
1
1
1
1
1
1
1
256
274
292
310
328
346
364
382
1400
1418
1436
1454
1472
1490
1508
1526
1544
1562
1580
1598
1616
1634'
1652
1688
1724
1760
1796
1832
868
904
940
976
2 012
2 048
2 084
2 120
2 156
2 192
2 228
2 264
2 300
2 336
2 372
933
943
953
963
973
983
993
1003
1013
1023
033
043
053
063
073
083
093
103
113
123
133
143
153
163
173
193
213
233
253
273
1293
1313
1333
1353
1373
1393
1413
1433
1453
1473
493
513
533
553
573
Concrete Scale (mostly only approzimAte).
610** C. potassium perchlorate melts
617'-628° C. stannous chloride boils
632^ C. antimony melts
650** C. NaCI andKCl in molecular proportions melt
667** C. alimiinium melts
664''-666° C. crystalline selenium boils at 760 mm
676*>-683° C. zinc chloride boUs
700° C. (about) dull«red incandescence (Pouillet)
708** C. magnesium chloride melts
719* C. calcium chloride, (CaClj), melts
719"»-731*> C. potassium boils
734** C. potassium chloride melts
750** C. magnesium melts
763*»-772** C. cadmium boils
772° C. sodium chloride (kitchen salt) melts
800° C. incipient cherry-red (Pouillet)
(see also
[28.5** C.)
814** C.
818** C
834** C.
847° C
850° C.
sodium carbonate melts
lithium sulphate melts
potassium carbonate melts
alloy 63% silver +37% copper melts
alloy 75% silver +25^ copper melts
861° C. sodium sulphate melts
870.5° C. alloy 71.9% silver+28.1% copper melts
886° C. " 82.1% " +17.9% '^
900° C. •• 57% ** +43%
900° C. cherry-red m candescence (Pouillet)
902° C. calcium fluoride melts
950° C. zmc boils
968° C. silver melts
975° C. alloy 60% silver + 40% gold melts
1 000° C. bright cherry-red (Pouillet)
1 015° C. potassium sulphate melts
1 050°-l 100° C. white pig iron melts
1 064° C. gold melts (Berthelot)
1 084° C. copper melts
1 100° C. dull-orange incandescence (Pouillet)
platinum melts
II II
1 100° C. alloy 95% gold + 5*
1 130°C. •• 90% •• +10^
1 150° C. limit of gas thermometers, bulb softens
1 190° C. alloy 80% gold + 20% platinum melts
1 200° C. bright-orange incandescence (Pouillet)
1 200° C. (about) gray pig iron melts
1 245° C. manganese 99% pure melts (Herseus)
1 255° C. alloy 70% gold + 30% platinum melts
1 285° C. *• 65% •• +35%
1 300° C. white incandescence (Pouillet)
THERMOMETER SCALES.
163
THERMOMETER SCAhES— (Concluded).
Centi-
{^rade
Fahr.
Deg.
Abso-
lute
Deg.
Deg.
1320
2 408
1593
1340
2444
1613
1360
2 480
1633
1380
2 516
1653
1400
2 552
1673
1420
2 588
1693
1440
2 624
1713
1460
2 660
1733
1480
2 696
1753
1500
2 732
1773
1520
2 768
1793
1540
2 804
1813
1560
2 840
1833
1580
2 876
1853
1600
2 912
1873
1620
2 948
1893
1640
2 984
1913
1660
3 020
1933
1680
3 056
1953
1700
3 092
1973
1720
3 128
1993
1740
3 164
2 013
1760
3 200
2 033
1780
3 236
2 053
1800
3 272
2 073
1825
3 317
2 098
1850
3 362
2 123
1875
3 407
2 148
1900
3 452
2 173
1925
3 497
2 198
1950
3 542
2 223
1975
3 587
2 248
2 000
3 632
2 273
2 100
3 812
2 373
2 500
4 532
2 773
3000
5 432
3 273
3 500
6 332
3 773
4000
7 232
4 273
5000
9 032
5 273
6 000
10 832
6 273
Concrete Scale (mostly only approximate).
1 320<* C. alloy 60% gold +40% platinum melts
1350*»C. ♦• 55% •• +45% " *; .
1 350** C. Bunsen flame ; mner blue flame (Kelvm)
I 370** C. furnace temperature for hard porcelain
1 375° C. glass furnace temperature
1 400° C. bright-white incandescence (Pouillet)
1 400° C. Bessemer steel melts
1 460° C. alloy 40% gold + 60% platinum melts
1 490° C. Siemens-Martin steel melts
1 500° C. dazzling-white heat (Pouillet)
1 500° C. palladium melts; tin boils
1 535° C. aUoy 30% gold +70% platinum melts
1 425°-l 625° G temp. Argand gas-burner (Lummer)
1 490°-I 580° C. temp, in Siemens-Martin furnace
1 475°-l 685° C. temp, candle-flame (Lummer)
1 610° C. alloy 20% gold +80% platinum melts
1650°C. ♦♦ 15% •• +85% , •• ,^
1 600°-1 825° C. temp. inc. elec. lamp (Lummer)
1 690° C. alloy 10% (afold+90% platinum melts
1 700° C. bismuth boils
1 730° C. alloy 5% gold +95% platinum melts
1 600°-l 800° C. thallium boils
1 779° C. platinimi melts
1 800° C. cobalt melts
1 820° C. Bunsen flame blast-lamp (Kelvin)
1 800°-2 000° C. pure wrought iron melts
1 900° C. manganese melts
1 950° C. iridium melts
-2 000° C. pure wrought iron melts
C. rhodium melts , -^ , i. .
-2 175° C. temp, of Nemst lamp and Welsbach
[mantel (Lummer)
C. osmium melts
C.'oxyhydrogen flame (Bunsen)
-3 927° C. temp, electric (carbon) arc light
[(Lummer)
6 000° C. temperature of the sun (Lummer)
1800°
2 000°
1 925°
2 500°
2 844°
3 475°
Exceedingly high temperatures of the order of milhons of degrees have
been calcinated to be produced, on certain theoretical assumptions, by the
iSct of some kinds of rays, like the X-rays; but owing to the exceed-
ing^r^nil quantity of the heat and the enormously rapid conduction they
'X^i ^ASy fncon^^^^^^^ that may appear in some of the high-temper-
ature values in the last column are no doubt due in part to the different
meSods by which they were determined; the values are generally only
approximations.
164 MONEr.
MONET.
The values of foreign moneys in the terms of the U. S. sold dollar, as
given below, are those used by the United States Treasury Department in
1903 for estimating the value of all foreign merchandise exported to the
United States expressed in the metallic currencies of those countries. (See
Department Circular No. 37 of 1903 of the Treasury Department.) The
values refer to the standard specie money of those countries and were
determined by the U. S. Mint in terms of the U. Si gold dollar; they are
therefore the true relative values, gold being the basis of comparison. The
reciprocal values were calculated from these. Hie coins of silver-standard
countries are valued by their pure silver contents at the average market price
of silver during the beginning of the year 1903. Much of the money in
foreign countries is paper money, the value of which is subject to many
fluctuations, and is generally not received at its par value.
Unless otherwise stated the standard of the countries in the foUowing
list is STold. France, Belgium, Italy, Switzerland, and Greece form the
Latin Union; their standard coins have the same value.
Argentine Republic. 1 peso— 0.965 U- S.gold dollar = 100 centavos.
1 argentine (gold) = 5 pesos =4.824 U. S. gold dollars. 1 U. S. gold dollar ««
1.036 pesos.
Austria-Hungary. 1 crown (Krone) » 0.203 U. S. gold dollar. 1
U. S. gold dollar = 4.923 crowns. The small unit is the heller. The former
imits were 1 florin =0.482 U. S. gold dollar = 100 kreutzer. (The usual
exchange value is much lower, being about 41 cts., and fluctuates consider-
ably.) Money orders issued in the United States are drawn on French
money (francs) ; 1 crown = about 1 .05 francs.
Belgium. 1 franc = 0.193 U.S. gold dollar = 100 centimes. 1 U. S.
gold dollar =5.18 francs
Bolivia (silver standard). 1 boliviano =0.352 U. S. gold dollar—
100 centavos. 1 U. S. gold dollar = 2.84 bolivianos.
Brazil. 1 milreis= 0.546 U. S. gold dollar = 1000 reis. 1 U. S. gold
dollar =1.83 milreis.
British Honduras. 1 dollar = 1 U. S. gold dollar = 100 cents.
British Possessionfl in North America (except Newfoundland).
1 dollar = 1 U. S. gold dollar.
Canada. See British Possessions.
Central American States. See individual States.
Ceylon. Rupee. See India.
Chile. 1 peso =0.365 U. S. gold dollar =10 dineros or decimos = 100
centavos. 1 U. S. gold dollar = 2.740 pesos. 1 condor (gold) =7.300 U. S.
gold dollars = 2 doubloons = 4 escudos = 20 pesos.
China (silver standard). 1 tael (average value) =0.549 U. S. gold dollar.
1 U. S. gold dollar = 1.82 taels (average value). The tael has different
values in different cities, varying from 0.520 to 0.580 dollar. The "British
dollar " has the same legal value as the Mexican dollar in Hong-kong, the
Straits Settlements, and Labuan.
Colombia (silver standard). 1 peso =0.352 U. S. gold dollar =10
diemos or decimos = 100 centavos. 1 U. S. gold dollar =2.84 pesos. 1
condor (gold) = 9.647 U. S. gold dollars.
Costa Rica. 1 colon =0.465 U. S. gold dollar = 100 centimos. 1 U. S.
gold dollar =2. 15 colons.
Cnba. 1 peso = 0.926 U. S. gold dollar. 1 U. S. gold dollar =1.080
pesos. 1 doubloon Isabella, centen = 5.017 U. S. gold dollars. 1 Alphonse —
4.823 U. S. gold dollars.
i>enmark. 1 crown (Krone) =0.268 U.S. gold dcllar= 100 oere. lU.S.
gold dollar = 3.73 crowns.
Ecuador. 1 sucre = 0.487 U. S. gold dollar. 1 U. S. gold dollar— 2.06
sucres.
Egypt. 1 pound = 4.943 U. S. gold dollars = 100 piasters— 4 000 paras.
1 U. S. gold dollar =0.202 pound.
Finland. 1 mark (markka) = 0.193 U. S- gold dollar — 100 penni.
1 U. S. gold dollar =5.18 marks.
MONEY. 165
France. 1 franc »0.193 U. S. gold dollar » 100 centimes. 1 Louis or
Napoleon » 20 francs. 1 sou = 5 centimes » about 1 cent (US). 1 U S
sold dollar » 5. 18 francs. 1 U. S. cent— 5 18 centimes => about 1 sou 20
francs » 3.86 U S gold dollars. 1 franc =0.811 mark (German) -0.793
shilling (English).
German Empire. 1 mark » 0.238 U. S gold dollar »1(X) pfennig 1
U S. gold dollar = 4.20 marks. 1 pfennig =0.238 cent; 1 cent— 4.20 pfen-
nigs. 1 mark — 1.233 francs (French])— U.978 shilling or 11 72 pence (English).
Great Britain. 1 pound sterling [£] or sovereign— 4.8665 U. S. gold
dollars-20 8hiUings-240 pence-960 farthings. 1 U S. dollar -0.205 49
pound sterling. 1 guinea — 21 shillings— 5.11 U. 8. gold dollars. 1 half
crown -2 6 shillings = .608 3 U. 8 gold dollar. 1 shilling [sl= 24.33 cents-
0.05 £ — 12 pence = 48 farthings— 1.022 marks (German) = 1.261 francs
(French). 1 U. 8. gold dollar — 4.109 8 shillings. 1 penny (plural pence)
[d}-2.03 cents -0.0833 shilling- 0.004167£- 10.51 centimes (French) -
8.52 pfennig (German). 1 cent— 0.493 penny.
Greece. 1 drachma -0.193 U. 8. gold dollar- 100 lepta. 1 U. 8. gold
dollar — 5.18 drachmas.
Guatemala (silver standard). 1 peso — 0.352 U. 8. gold dollar. 1 U. 8.
gold dollar — 2.84 pesos.
Haiti. 1 gourde = 0.965 U. 8. gold dollar- 100 cents (Haiti). 1 U. 8.
gold dollar— 1.036 gourdes.
Hawaii. 8ee United States.
Honduras (silver standard). 1 peso— 0.352 U. 8. gold dollar. 1 U. 8.
gold dollar— 2.84 pesos.
Honduras (British). 1 dollar -1 U. 8. gold dollar = 100 cents.
India. 1 ]>ound sterling [£] =4.866 5 U. S. gold dollars (same standard
as in Great Britain) = 15 rupees (money of account). 1 rupee — 0.324 4 U. 8.
gold dollar- 16 annas. 1 U. 8. gold dollar— 0.205 49 pound sterling -3.082
if^ly. 1 lira-0.193 U. 8. gold dollar- 100 centesimi. 1 U. 8. gold
dollar— 5.18 lire.
Japan. 1 yen = 0.498 U. 8. gold dollar -100 sen. 1 U. 8. gold dollar -
2.008 yen.
Liberia. Same as in the United States.
Mexico (silver standard). 1 dollar (silver), peso, or piastre — 0.383 U. 8.
gold dollar = 100 centavos. 1 dollar (gold) = 0.983 U. 8. gold dollar. 1 U. 8.
gold dollar — 2.61 1 dollars (silver) — 1 .017 dollars (gold). 1 once or doubloon
= 16 pesos.
Netherlands. 1 florin of 100 cents=0.402 U. 8. gold dollar. 1 U. 8.
gold dollar— 2.49 florins.
Newfoundland. 1 dollar = 1 .014 U. 8. gold dollars. 1 U. S. gold dollar
—0.986 dollar.
Nicaragua (silver standard). 1 peso — 0.352 U. S. gold dollar. 1 U. 8.
gold dollar— 2.84 pesos.
Norway. 1 crown = 0.268 U. S. gold dollar - 100 oere = 30 skillings. 1
U. 8. gold dollar — 3.73 crowns.
Persia (silver standard). 1 kran (silver)— 0.065 U. S. gold dollar. 1
tomans (gold) - 1 .704 5 U. 8. gold dollars. 1 U. 8. gold dollar- 15.4 krans
—0.5867 tomans (gold).
Peru. 1 sol = 0.487 U. 8. gold dollar— 10 dineros = 100 centavos. 1
libra (gold) -4.866 5 U. 8. gold doUars. 1 U. 8. gold dollar = 2.053 sols-
0.20549 Ubra (gold).
Portugal. 1 milreis = 1 .080 U. S. gold dollars - 10 testoons - 1 0(X) reis.
1 crown — 10 milreis. 1 U. 8. gold donar=0.925 9 milreis.
RuHsia. 1 ruble — 0.51 5 U. S. gold dollar = 2 poltinniks — 4 tchetvertaks
—5 abassis— 10 griviniks = 20 pietaks — 100 kopecks. 1 imperial (gold) —
16 rubles -7 .7 18 U. S. gold dollars. 1 U. S. gold dollar -1.942 rubles.
1 kopeck — 0.515 cent (U. 8.); 1 cent — 1.942 kopecks.
Sandwicli Islands. 83e United States.
Salvador (silver standard). 1 peso— 0.352 U. 8. gold dollar. 1 U. S.
gold dollar — 2.84 pesos.
Sicily. See Italy.
Spain. 1 peseta or pistareen — 0.193 U. 8. gold dollar — 100 centimes.
1 U. S. gold dollar— 5.18 pesetas.
Sweden. 1 crown = 0.268 U. 8. gold dollar - 100 oere. 1 U. 8. gold dol-
lar =3.73 crowns.
166 MONEY AND LENGTH.
Switaerland. 1 frano -0.193 U. S. gold dollar- 100 oentiiiiM 1 U. S.
gold dollar— 5.18 francs. ^ ^
Turkey. 1 piaster -0.044 U. S. gold dollar -40 paras. 1 V. S- gold
dollar — 22.73 piasters.
United States (and possessions). 1 dollar [$}— 1 U. S. gold dollar— 10
dimes— 100 cents 1 eagle — 20 dollars.
Urugaay. 1 peso — 1.034 U. S. gold dollars— 100 centavos or oentesi-
mos. 1 U. S. gold dollar— 0.967 peso.
Yenexaela. 1 bolivar — 0.193 U. S gold dollar— 2 decimos- 1 U- 8
gold dollar— 5. IS bolivars. 1 venesolaoo- 5 bolivars.
FLUCTUATING OURRENOIES.
The November, 1903, issue, No. 278, vol. 73, of the Monthly CoMtdar
Reports, published by the Department of Commerce and Labor, gives a
list of the fluctuating values of the silver imits of several countries. The
latest values, namely those for October 1, 1903, are as follows:
Bolivia, silver boliviano— $0,408.
Central America, silver peso =» $0,408.
China, tael, average value— $0,636.
Colombia, silver peso— $0,408.
Mexico, silver dollar— $0,443.
Persia, silver kraa— $0,076.
MONET and LENGTH.
The money values used in this table are those given in the above list.
1 shilling per mile a. 0.784 franc per kilometer.
" — 0.635 mark per kilometer.
•• -0.243 3 dollar per mile.
1 franc per kilometer » 1.276 shillings per mile.
" —0.811 mfcrk per kilometer.
" =0.311 dollar per mile.
1 marlc per kilometer « 1.575 shillings per mile.
" — 1 .233 francs per kilometer*
" -0.383 dollar per mile.
1 dollar per mile =4.11 shillings per mile.
** —3.22 francs per kilometer.
" —2.61 marks per kilometer.
1 '»^nt per foot — 0.493 penny per foot.
^' —0.170 franc per meter.
•* —0.138 mark per meter.
1 penny per foot= 2.03 cents per foot.
— 0.345 franc per meter.
•* —0.280 mark per meter.
1 franc per meter = 5. 88 cents per foot.
** — 2.90 pence per foot.
" —0.811 mark per meter.
1 mark per meter = 7.26 cents per foot.
•* — 3.67 pence per foot.
" — 1 .233 francs per meter.
MONEY AND WEIGHT. 167
MONET and WEIGHT.
The money values used in this table are those given in the above list.
Av. means avoirdupois weights.
1 franc per metric toii=0 811 mark per metric ton.
=0.806 shilling per long ton.
'* —0.720 shilling per short ton.
** =0.196 dollar per long ton.
" =0.175 dollar per short ton.
1 mark per metric ton = 1.233 francs per metric ton.
*' =0.994 shilling per long ton.
=0.888 shilling per short ton.
•* =0.242 dollar per long ton.
" =0.216 dollar per short ton.
1 shilling per long ton =1.24 francs per metric ton.
** =1.005 marks per metric ton.
=0.243 dollar per long ton.
*' =0.217 dollar per short ton.
1 shilling per short ton = 1.39 francs per metric ton.
** =1.13 marks per metric ton.
" =0.273 dollar per long ton.
" =0.243 dollar per short ton.
1 dollar per long ton = 5.10 francs per metric ton.
** =4.14 marks per metric ton.
'* =4.11 shillings per long ton.
*• =3.67 shillings per short ton.
1 dollar per short ton = 5.70 francs per metric ton.
** =4.63 marks per metric ton.
** =4.60 shillings per long ton.
" =4.11 shillings per short ton.
1 franc per kilogram = 0.811 mark per kilogram.
** = 0.547 cent per ounce (av.).
" = 0.360 shilling per pound (av.).
** = 0.270 penny per oimce (av.).
** =0.087 5 dollar per ounce (av.]|.
1 mark per kilogram = 1.233 francs per kilogram.
" =0.675 cent per oimce (av.).
** =0.444 shilling per pound (av.).
" =0.333 penny per ounce (av.^.
** =0.108 dollar per pound (av.).
1 cent per ounce [av.]= 1.83 francs per kilogram.
" = 1.48 marks per kilogram.
" =0.658 shilling per pound (av.).
" =0.493 penny per ounce (av.^.
** =0.160 dollar per pound (av.).
1 shilling per pound [av.]= 2.78 francs per kilogram.
** = 2.25 marks per kilogram.
" = 1.52 cents per oimce (av.).
** =0.750 penny per ounce (av.).
" =0.243 dollar per poimd (av.).
1 penny per ounce [av.]= 3.71 francs per kilogram.
*• = 3.00 marks per kilogram.
** — 2.03 cents per ounce (av.).
" =1.333 shillings per pound (av.).
•* =0.324 dollar per pound (av.).
L dollar per pound [av.]=11.4 francs per kilogram.
*• = 9.26 marks per kilogram.
** =6.25 cents per ounce (av.).
" =4.11 shillings per pound (av.).
•• = 3.08 pence per ounce (av.).
lt)8
SCALES OP MAPS AND DRAWINGS.
SCALES of MAPS and DRAWINGS.
The following table is limited to the more usual values. For a very
complete table of scales for maps, as distinguished from detail drawings,
see tiaupt, Scales of Maps, Proceedings Engineers' Club of Philadelphia,
Vol. 1, p. 47-59, 1879. The relation of the meter to the foot which is there
used is slightly different from the legal value in this coimtry; the follow**
ing are all based on the standard legal value.
When the scale of a map is given as — . then:
n
1 inch on the map »n X 2.540 005
=nX 1
-nX 0.083 333 3
" -nX 0.027 777 8
•• =nX 0.025 400 05
" -nX 0.005 050 51
-nX 0.001262 63
" =»nX 0.000 025 400 05
=nX 0.000 015 782 8
1 centimeter on the map »n X
-nX
-nX
-nX
=nX
-nX
-nX
-nX
•«
«•
•t
••
0.393 700
0.032 808 3
0.010 936 1
0.01
0.001 988 38
0.000 497 096
0.000 01
Logarithm
centimeters 0-404 8S46
inches 0000 0000
feet. 2.920 8188
yards 5.448 6976
meters S-404 8346
rods 3-708 8848
chains 3101 2749
kilometers 5-404 8346
miles 5 198 1849
1. centimeters . Q-000 0000
inches 1.595 1654
feet 5.515 9842
yards 3.088 8629
meters 3-000 0000
rods. . ■ 3.298 5002
chains 4-696 4408
kilometers . . a-OOO. 0000
miles f -793 8508
'nXO.000 006 213 70
Special Talnes of the above, frequently used. The first length
refers to the actual distance and the second to the map or drawing.
Scale of
•*
««
«•
1/12
1/120 =■
1/198 -
1/240 -
1/360 -
1/480
1/600 -
1/720
1/960 -
1/120 -
1/63 360 -
1/100 000
1 foot to the inch.
10 feet to the inch.
1 rod to the inch.
20 feet to the inch.
30 feet to the inch.
40 feet to the inch.
50 feet to the inch.
60 feet to the inch.
80 feet to the inch.
100 feet to the inch.
1 mile to the inch.
1 kilometer to the centuneter.
Scales for Detail Drawings. The first length refen to the drawing
and the second to the actual length.
Full size
Half size
Third size
Smarter size
ixth size
Eigrhth size
Twelfth size
Sixteenth size
Twenty-fonth size *
Thirty-second size »-
Forty-eiehth size »
Ninety-sixth size *>
12 inches to the foot.
f» (( (« (( («
4
3
2
mch
ft
*t
- 1
tt
it
if
tt
«•
it
t«
M
i(
•t
• •
t«
It
it
«t
PAPER MEASURE. MISCELLANEOUS MEASURES.
1 qnire »24 or 25 sheets.
1 ream «- 20 quires » 480 sheets.
1 ream « generally 500 sheets.
1 bundle Cobs.) »2 reams » 1 000 sheets.
1 bale (obsolete)*" 5 bundles.
1 dozen —12.
1 grross » 12 dozen — 144.
1 great jipross — 12 gross.
1 great gross » 144 dosen — 1 728.
1 score —20.
FUNCTIONS OF ;r. 16 Q
USEFUL FUNCTIONS OP 7C.
, . , . ^ Logarithm
Aprx. means within 2%.
K (called "pi") =»circuinference of a circle divided by the diameter and
is a constant.
ir— approximately 2^, which equals 3.142 86 or Moo% too much.
«» approximately s^lis, which equals 3.141 592 9.
X- 3.141 592 653 589 793 238 462 643 383 279 502 88. t • . . 0-497 1499
2»- 6.283 185 307 180 0798 1799
3jr- 9.424 777 960769 0974 2711
4;r=12.566 370 614 359. Aprx. HXlOO 1.099 2099
5;r=15.707 963 267 949 1.196 1199
6;r=18.849 555 921 539 1.275 8011
7;r=21.991 148 575 129 1.842 2479
8;r=25.132 741 228 718 1.400 2898
9«=' 28.274 333 882 308 I.45I 8924
lOjr-31.415 926 535 898 I.497 1499
4«/3- 4.188 790 204 786 0622 0886
jr/1 -3.141 592 7. Aprx. 2^ 0497 1499
jc/2-1.570 796 3. Aprx.i^ 0196 1199
w/3-1.047 197 6 Q 020 0286
jr/4-0.785 398 2. Aprx. %o T.895 0899
jt/5=-0.628 318 5 f.798 1799
;r/6=0.523 598 8 T.7I8 9986
jr/7 -0.448 799 1.652 0518
'/^-S 392 699 1 1.594 0599
jr/9 = 0.349 065 9 t.542 9074
«/12-0.261 799 4 f .417 9686
jt/16-0.196 349 5 T.298 0298
«/32=-0.098 174 8 2.991 9999
jr/64-0.049 087 4 3.690 9698
jr/108-0.029 088 8 3 463 7261
x/180-0.017 453 3 3 241 8774
jr/360-0.008 726 65 3940 8474
1 X ;r/4=0.785 398 2. Aprx. %o 1.895 0899
2X jr/4-1.570796 3 0196 1199
3X »r/4 = 2.356 194 5 372 2112
4X jc/4-3.141 592 7 0497 1499
5X jr/4 = 3.926 990 8 594 0599
6X jr/4-4.712 389 0678 2411
7 X jr/4-5.497 787 1 0-740 1879
8X jr/4-6.283 185 3 0-798 1799
9X jr/4-7.068 583 5 849 8324
lOX w/4-7.853 981 6 0-895 0899
l/jr-0.318 309 9. Aprx.%2 1502 8501
2/»r=0.636 619 8 T.803 8801
3/;r-0.954 929 7 1.979 9713
4/ »r- 1.273 239 5 104 9101
5/w- 1.591 549 4 201 8201
6/ JT- 1.909 859 3 0281 0014
7/;r-2.228 169 2 0-347 9482
8/jr-2.546 479 1 405 9401
9/ff=» 2.864 789 457 0926
lO/jr-3.183 098 9 0-502 8501
l2/;r-3.819 718 6 582 0314
t Ludolph's value. For Vega's value to 140 decimal places see Ifiatoire
dea Rech€rchea avr la Quadrature du CercUt by Montucla, 1831, p. 282.
170 FUNCTIONS OF n.
Xx>garithm
,rv/2 =. 4.442 882 9 0647 6649
;r-^x/2= 2.221 441 4 0-346 6349
4;r-*-10 = 1.256637061. Aprx.addM 099 2099
jr2 = 9.869 604 401 089. Aprx. 10 0-994 2997
4;r2 - 39.478 418. Aprx. 40 1.696 3597
jr2-5-4 = 2.467 40110. Aprx-MXlO 0.892a397
n^ = 31.006 276 680 300 1.491 4496
jrj = 97.409 091 1.988 5995
«f «306.019 69 2485 7494
= 961.389 19 2982 8992
r
.6
1-^v^ = 0.032 251 5 5.508 5504
9^" ^ lillkkH^^^^ 0.2485749
2\/jr = 3.544 907 6 0-549 6049
S?'' ^ ?-525?2S 0.3990899
-r Jr = 1.464 592 0165 7166
1-J- »r -= 0.318 309 866 T.502 8501
l-^2;r = 0.159 154 933. Aprx.%^-10 1.2018201
l-^4»r - 0.079577 47. Aprx.%00 2.9007901
10-^4,r - 0.7957747. Aprx.Ho 1-900 7901
10-i-8;r - 0.397 887 3. Aprx.^o 1-599 7601
1-5-Vff =. 0.564 189 6 T.75I 4251
l-^-^n = 0.6827841 1-8342834
Log JT = 0.497 149 872 694 133 854 35.
Log^ n = 1.144 729 885 849 400 174 14.
n = 180** considered as an angle.
«/180 = 0.017 453 29 5.041 8774
^/360 = 0.00872665 5-940 8474
180A = 57.295 779 5 1-758 1226
360/;r = 114.591 659 2-059 1626
USEFUIi NUMBERS.
Logarithm
V2 = 1.41421. Aprx.^^XlO 0150 5150
'^^ 1-732 05. Aprx. % 0-238 5606
^/2 = 1.259 92. Aprx. add M 0-100 3433
^3^= 1.442 25. Aprx. ^V^ X 10 0159 0404
V4-1.587 40. Aprx.% 0-200 6867
LOGARITHMS. — P^YS1CAL CONSTANTS. 171
SYSTEMS OF LOGARITHMS.
The logarithm [log] of a given number is the exponent which denoteB the
power to which a certain fixed numerical base is raised, in order to produce
this given number. Thus if the base is 10, then the log of 100 is 2, because
10 raised to the 2d power = 100. To multiply two numbers, add their
logs; to divide, subtract the log of the divisor from that of the dividend;
then from the resulting log find the corresponding number.
Log or logio means the conunon, usual, or Briggs' logarithm; the
base of this system is 10.
Loge or In means the Naperian, natural, or hyperbolic logarithm; the
base of this system is about 2.718 (see below), generally denoted by e.
Log of 1 =0 in any system.
Log of base = 1 in its own system.
e=base of Naperian, natural, or hyperbolic logarithms.
e=2 718 281 828.
logio of 6 = 0.434 294 481 903 252.
Log«ofe = l.
10 = base of usual, common, or Briggs' logarithms.
log« of 10 = 2.302 585 092 994 046.
Logio of 10 = 1.
log* 10 X logio « = 1.
The modalus of any system is the constant by which the Naperian
logarithm of a number must be multiplied to give the logarithm of the
number in that system. The modulus of any system is equal to the recip-
rocal of the Naperian log of the base of that system.
Modulus of Naperian system = 1 -s-log< of e = 1 .
Modulus of common system = 1 s- log* of 10 = 0.434 294 481 903 252.
The logarithm of a number (n) in any system is equal to the modulus
of that system multiplied by the Naperian logarithm of the number. Or:
Logio n= modulus (conmion system) Xlog^ n.
Log* n= modulus (Naperian system = 1) X log« n.
To change a logarithm from the common to the Naperian system, or
the reverse :
Common log X 2.302 585 092 994 046 = Naperian log.
Naperian log X 0.434 294 481 903 252= common log.
AOOJBLERATION OF GRAVITT [g].
(See also table, pp. 87, 88.)
Logarithm
17=9.805 966 meters per second per second. Aprx. 10 0-991 4904
v^ = 4.428 54 in meters per second per second. Aprx. % X 10. 0.646 2602
(7» 32.171 7 feet per second per second. Aprx. 32 1-507 4746
^'^» 8.021 44 in feet per second per second. Aprx. 8 0-904 2528
MEOHANIOAL EQUIVALENT OF HEAT [M].
(See also p. 72.)
ilf»- 426.900 kilogram-meters per kilogram calorie.
Aprx. %X1000 2-680 8262
1 + ilf =0.002 342 47 in same units. Aprx. % -s- 1 000 5-869 6788
M = 778. 104 foot-pounds per pound Fahrenheit heat-imit .
Aprx.%X1000 28910879
1 -i-ilf = 0.001 285 17 in same units. Aprx. »/r -*- 1 000 3-108 9621
(For further reduction factors see table of measures of Energy, pp. 74-77.)
SPECIFIO HEAT OF WATER.
(See also p. 72.)
Speelflc heat of water:
=4 186.17 jouJes per kilogram calorie. Aprx. 4 200 8-621 8166
Specific heat of water (in joules per kg oal)= mechanical
equivalent of heat (in kilogram-meters per kilogram calorie)
X acceleration of gravity (in meters per second per second).
172 MISCELLANEOUS FOKEIGN MEASURES.
MISCELLANBOUS FOREIGN MEASURES.
The following list of miscellaneous measures used in foreign countries
was received too late for classification with the others. The values are
taken from the November, 1903, issue (No. 278, vol. 73) of the MorUhly
Consular Reports published by the Department of Commerce and Labor
of the U. S. Government. They are the measures which are referred to in
the ConatUar Reports. Many of them are presumably onljr approximately
correct. They nave been rearranged here in the alpnabetical order of the
names of the countries. For an alphabetical list of the names of the meas-
ures see the index.
Arg^entine Republic. 1 pie«0.947 8 foot. 1 vara«"34.120 8 inches.
1 cuadra«= 4.2 acres. 1 fra8co=» 2.509 6 quarts. 1 baril— 20.078 7 ^fallens.
1 libra = 1.012 7 poimds. 1 arroba (dry) =25.317 5 pounds. 1 qumtal —
101.42 pounds.
Belgium. 1 last =85.134 bushels.
Bolivia. 1 marc— 0..S07 pound.
Borneo. 1 picul= 135.64 pounds.
Brazil. 1 arroba =32.38 pounds. 1 quintal = 130.06 poimds.
Celebes. 1 picul = 135.64 pounds.
Central America. 1 vara = 32.87 inches. 1 centaro= 4.263 1 gallons.
1 fanega (dry) = l 574 5 bushels. 1 libra = 1.043 poimds.
Chile. 1 vara = 33.367 inches. 1 fanega (dry) = 2.575 bushels. 1 libra
= 1 .014 pounds. 1 quintal = 101.41 pounds.
China. 1 tsim = 1.41 inches. 1 chih=14. inches. 1 li = 2115. feet.
1 catty = li pounds. 1 picul = 133i pounds.
Cochin-C h i na. 1 tael = 590 .7 5 grains.
Costa RIoa. 1 manzana = 1| acres.
Cuba. 1 vara = 33.384 inches. 1 arroba (liquid) =4.263 gallons. 1
fanega (dry) = 1.599 bushels. 1 libra = 1.016 1 pounds. 1 arroba (dry)«-
25.366 4 pounds.
Curacao. 1 vara = 33.375 inches.
Denmark. 1 mil (geographical) = 4.61 miles. 1 mil =4.68 miles. 1
tondeland = 1.36 acres. 1 tonde (cereals) =3.947 83 busheb. 1 centner »
110.11 pounds.
Bfirypt. 1 pic = 21i inches. 1 feddan = 1.03 acres. 1 ardeb=7.690 7
bushels. 1 oke = 2.722 5 pounds.
Greece. 1 drachme=a half oimce (presumably av.). 1 livre = l.l
pounds. 1 oke = 2.84 pounds. 1 quintal = 123.2 pounds.
Guiana. 1 livre = 1.079 1 pounds.
Holland. 1 last =85.134 bushels.
Honduras. 1 milla = 1 .149 3 miles.
Hungary. 1 oke = 3.08 17 pounds.
India. 1 bongkal=832. grains. 1 seer = 1 poimd 13 ounces. 1 maund
=82^ pounds. 1 candy (Madras) = 500. pounds. 1 candy (Bombay) =■
529. pounds.
Isle of Jersey. 1 verg(Bes=71.1 square rods.
Japan. 1 bu=0.1 inch. 1 sun = 1.193 inches. 1 shaku = 11.930 5
inches. 1 ken =6 feet. 1 tsubo = 6 feet square. 1 se= 0.024 51 acres. 1
tan =0.25 acre. 1 8ho = 1.6 quarts. 1 to =2 pecks. 1 koku=4.962 9
bushels. 1 catty = 1.31 pounds. 1 picul = 133i pounds.
Java. 1 catty = 1.35 pounds. 1 picul = 135.1 pounds.
Luxemburg. 1 fuder = 264.17 gallons.
Malta. 1 caffiso = 5.4 gallons. 1 barrel (customs) = 11.4 gallons. 1
cantaro (cantar) = 175. pounds. 1 salm = 490. pounds.
Mexico. 1 vara = 33. inches. 1 frasco = 2.5 quarts. 1 fanega (dry) =
1.547 28 bushels. 1 libra = 1.014 65 pounds. 1 quintal = 101.41 pounds.
1 carga = 300. pounds.
Morocco. 1 artel = 1.12 pounds. 1 fanega (dry) strike =70. pounds.
1 cantar = 113. poimds. 1 fanega (dry) full = 118. pounds.
Newfoundland. 1 quintal (fish) = 112. pounds.
Nicaragua. 1 manzana = 1} acres. 1 milla =1.149 3 miles.
Palestine. 1 rottle=6 poimds.
MISCELLANEOUS FOREIGN MEASURES. 173
Paragrnay. 1 vara = 34. inches. 1 cuadra=78-9 yards. 1 cuadra
square »8.077 square feet (7). 1 league (land) »4 633. acres. 1 arobe »25.
pounds. 1 quintal =» 100. pounda
Persia. 1 batman (tabriz) = 6.49 ppunds.
Peru. 1 vara » 33.367 inches. 1 libra — 1.014 3 pounds. 1 quintal «
101.41 pounds.
Philippine Islands. 1 picul — 137 .9 pounds.
Portug^al • 1 almuda — 4 .422 gallons. 1 arratel »- 1 .01 1 pounds. 1 libra
■"1.011 pounds. 1 arroba (dry) = 32.38 pounds.
BuMia. 1 arshine (square) = 5.44 square feet. 1 vedro = 2.707 gallons.
1 korree»3.5 bushels. 1 chetvert = 5.774 8 bushels. 1 klafter — 216. cubic
feet. 1 funt—0.902 8 pound. 1 pood = 36.1 12 pounds. 1 berkovets —
361.12 pounds.
RuBslau Poland. 1 vlocka —41.98 acres. 1 gamice—0.88 gallon. 1
last — 111 bushels.
SalTador. 1 mangana — 1| acres.
Sarawak. 1 coyan — 3 098. pounds.
Siam. 1 catty =1.35 pounds. 1 coyan — 2 667. pounds.
Spain. 1 pie =0.914 07 foot. 1 vara =0.914 117 yard. 1 arroba
(liquid) = 4.263 gallons. 1 fanega (liquid) = 16. ^lons. 1 butt (wine) =
140. gallons. 1 dessiatine = 1 .599 bushels. 1 hbra = 1.014 4 pounds. 1
arroba (dry) = 25.36 pounds. 1 frail (raisins) =50. poimds. 1 barrel
(raisins) = 100. pounds. 1 last (salt) = 4 760. pounds.
Sumatra. 1 bouw=7 096.5 square meters. 1 catty = 2.12 pounds.
Sweden. 1 tunna=4.5 bushels. 1 pund = 1.102 pounds. 1 centner —
93.7 pounds.
Syria. 1 rottle = 5|- pounds. 1 quintal = 125. pounds. 1 cantar (Da-
mascus) =575. pounds.
Turkey. 1 pik = 27.9 inches. 1 oke — 2.828 38 pounds. 1 cantar —
124.703 6 pounds.
Uruicuay. 1 cuadra —nearly 2 acres. 1 suerte — 2700. cuadras. 1
fanega (single) — 3.888 bushels. 1 fanega (double)— 7.776 bushels. 1 libra
— 1.014 3 pounds.
Venezuela. 1 vara — 33.384 inches. 1 fanega (dry) — 1.599 bushels.
1 libra = 1 .016 1 poimds. 1 arroba (dry) — 25.402 4 poimds.
Zanzibar. 1 frasila— 35. pounds.
i>
INDEX.
Ab-, prefix. 96
Abassis^ Russia, 165
Abbreviated numbers, 9
Abbreviations, text, 2S
do. table of, ix
Abfarad, 117
Abs-, prefix, 96
Absampere, 113
Abscoulomb, 116
Absohm, 99
Absolute:
potential, 108
temperature scale, 151-163
do. reduction factors, 150
system of units, text, 11
Absolute unit of:
candle power, 146
capacity, electric, 117
conductance, 105
conductivity, electric, 106, 107
current, electric, 112, 113
electromotive force, 109, 110
energy, electric, 122. 123
energy, magnetic, 143
flux, magnetic, 138
do., density, 140, 141
force, 83
inductance, 119
Ught, 3, 145, 146
magnetic moment, 142
magnetization intensity, 142
magnetizing force, 134, 137
magnetomotive force, 133
permeance, magnetic, 131
power, electric, 124, 125
power, magnetic, 144
quantity, electric, 116
reluctance, magnetic, 129
resistance, electric, 99
resistivity, 102, 103
time, 93
Absolute units :^
Congress decisions, 14
dimensional formulas, 18-26
generally called C.G.S. units, 11
Est of, 18-20
prefixes for, 96
relations to others, 3
Absolute units:
system of, text, 11
vs. concrete, 12
Absolute values, elec. imits, text, 90
Abstat-, prefix, 96
Abstafarad, 117
Abstatampere, 113
Abstatcoulomb, 116
Abstatohm, 99
Abstatvolt, 110
Abvolt, 109
Acceleration:
angular, table, 88
angular, physical, 19
gravity, 88, 171
do., mean value, 87
do., as a relation, 3
linear, table, 87
linear, physical, 19
Accuracy of logarithms, 10
Accuracy of numbers, 9
Acetate lamp, amyl, 144
Acre, 43
to hectares, digit table, 44
circular, 44
Ireland, Scotland, Switzerland, 44
Acre-foot, 95
Activity, 19
Admittance, table, 105
defined, 105
physical J 22, 24
Almude, Lisbon, Oporto, Constanti-
nople, 55
Almuda, Portugal, 173
Alnar, Sweden, 33
Alphonse, Cuba, 164
Alquiere , Lisbon , Madeira , Oporto , 55
Alternations, definition, 86, 121
number of, 87 , 121
Am, Sweden, 55
Ampere, table, 113
definition, 112
final, 113
international, 113
do., defined, 112
Nat. Bur. of Standards, 113
rate of change of. per second, 113
ReichsanstaTt, 113
true, 113
true, defined, 112
175
176
INDEX.
Ampere per circular centimeter, 115
per circular inch, 115
per circular mil, 115
per circular millimeter, 115
per square centimeter, 115
per square decimeter, 114
per square foot, 114
per square inch, 115
per square meter, 114
per square mil, 115
per square millimeter, 115
-hour, table. 116
do., defined, 115
do. per cubic centimeter, 126
do. per gram , 126
do. per poimd, 126
-second, 116
-turn, 133
do., defined, 132
do. per centimeter, 137
do. per inch, 136
Amplitude of waves, 121]
Amyl acetate lamp, 144
Ancient lengths, 34
Angles:
pUine, 89
do-, physical, 18
do. as a suppressed quantity, 13
solid, 89 .
do., physical, 18
spherical right, 89
grade, 91
Angular:
acceleration. 88
do., physical, 19
momentum, 84
do., physical, 19
velocity, 86
do., physical, 19
do. as a frequency, 121
do., rate of mcrease of, 88
Angstroem imit, 31
Anker, Amsterdam, Copenhagen,
Sweden, 55
Germany, 54
Annas, India, 165
Anomalistic month, 94
year, 96
Apothecary weights, table, 59
defined, 57
Apparent power, defined, 124
Apparent resistance, defined, 98
Apparent solar day, 94
Applied volts, 110
Approximate numbers, accuracy, 9
Approximate values, explained, 28
Ar, 43
Arabian lengths, 34
Ardeb, Egypt, 172
Are, 43
Argentine, Argentine, 164
Arish, Persia, 34
Arobe. Paraguay, 173
Arpent, France, Switezrland, 44
Arratel, Portugal, 173
Arroba, Argentine, Brazil, Cuba, 172
Spain, 61, 173
Portugal. Venezuela, 173
Arrobas, Canaries, 55
Arschine, Russia, 33 173
Art, Sweden, 61
Artaba, Persia, 55
Artel, Morocco, 172
As, Germany, 60
Ass, Sweden. 61
Astronomical dav, 94
Atmosphere, table, 66
defined, 63
digit converrion table, 67
Atom^ gram-, 60
Atomic weight of silver, 125
Aune, France, 33
Austrian lengths, 33
Avoirdupois weights, table, 59
defined, 57
Azumbras, Spain, 55
B
Babylonian lengths, 84
Bale, 168
Bar, 58
Bane. 64, 66
Baril, Argentine, 172
Barille, Italy, 55
Barleycorn, 31
Barrel, table, 53
no legal value, 45
fiour, 60
Sork or beef, 60
[alta, 172
Spain, 173
Barrique, France 55
Bars, weights of, 62
Barye (see Barie), 64, 66
Base of logarithms, 171
Batman, Persia, 173
Battery, voltage of, 128
B.A. unit, 99
do. defined, 98
Becher, Austria, 55
Berkovets, Russia, 173
Berkowitz, Russia, 61
Bern, Turkey, 34
Biblical lengths, 34
Bioom ton, 60
Board foot, 52
Board of Trade ohm, 99
do. defined, 98
Board of Trade unit (energy), 7
Boccali, Rome, 55
Boisseau, France, 55
Bolivar, Venezuela, 166
Boliviano, Bolivia, 164, 166
Bolt, 32
Bongkal, India, 172
Botschka, Russia, 55
Bougie decimale, 146
do. defined, 145
Bouw, Sumatra, 173
Braccio, Italy, 34
Brasse, France, 33
moEx.
177
Bricn^s' logarithms, 171
Brightness (light) table, 148
do. physical, 25
British Association unit, 90
do. defined, 98
British dollar, China, 164
British standard candle, 146
do. defined, 145
British thermal unit, 75
British to U. S., volumes, 46
BTU (kilowatt-hour), 77
BTU (thermal unit), 75
Bu, Japan, 34, 172
Building square, 43
Bundle, 168
Bushel, table, 50
legal, for grain, 45
U. S. standard, 45
to hectoliters, digit table, 51
salt (weight), 60
unusual values, 53
weight of water, 70
Butt, 53
Spain. 173
Cable, 32
Cable's length, 32
Cadmium cell, 108. 110, 111
Calendar day, 94
month, 94
year, 94
Calorie, defined, 73
larg^, table, 76
do. into k^-met., digit table, 77
do. per mmute, 81
small, table, 75
do. into joules, digit table, 77 .
do. per minute, 80
do. per second, 81
Calories into volts, 129
Candle, table. 146
defined, 144
British standard, table, 146
do. defined, 145
do. hemispherical, 147
do. i^herical, 147
English standard, table, 146
do. defined, 145
do. hemispherical, 147
do. sphencal, 147
German paraffine, table, 146
do. defined, 145
hefner, 146
do. defined, 144
do. hemispherical, 147
do. spherical, 147
hemispherical, 147
spermaceti, defined, 145
spherical, 147
relation to other units. 146
Candle power, see Candle
Candle per fopt, 148
per horse-power, 149
per horse-power, metric, 149
per kilowatt, 149
per meter, 148
per watt, 149
Candy, India, 172
Cantar, Malta, Morocco, 172
Syria, Turkey, 173
Cantara, Spain, 55
Cantaro, Malta, 172
Caffiri. Malta, Sicily. Tripoli. Tunis.
55
Caffiso, Malta. 172
Capacity, electrical, 117
cfo., C.G.S. units, 117
do. physical quantity. 23, 24
do. mductive, 23
do. reactance, 22, 24
do. specific inductive, 23, 24
Capacity, magnetic (permeance).130
do., physical, 20, 21
do., specific inductive. 20, 21
Capacity, heat, 26
Capacity, volumes, cubic, table, 46
do., digit conversion tables. 51
do., text; 45
do., foreign, 54. 172
Capillarity, 62
Carat, diamond, 59
Germany, 60
Amsterdam, Austria, Borneo.
France, Italy, Lisbon, Madras.
Spain, 61
Carcel, table, 146
defined, 145
Carga, Mexico, 172
Car-mile. 78
Castellanos, Spain, 61
Catrize, Spain, 55
(Datty, Batavia, 61
China, 61, 172
Japan. 61. 172
Java. 172
Siam, 173
Sumatra, 61, 173
Cent, Germany. 60
Haiti, Netherlands, 165
U. S., 166
per foot, 166
per oimce, 167
Centaire, 43
Ontar, 43
Centare. 43
Centaro, Central America, 172
Centas, (jiermanY. 60
Centavo, Ar^ntine, Bolivia, Cluli,
Colombia, 164
Mexico, Peru, 165
Uruguay, 166
Centen, Cuba, 164
Centesimo, Italy, 165
Uruguay, 166
Centi-, as prefix, 9
Centigrade :
heat unit, gram, 75
do. kilogram, 76
178
INDEX.
Centigrade:
degrees, reduction factors, 150
scale, to others, 151-163
Centigram, 57
Centiliter, 52
Centime, Belgium, 164
France, Spain, 165
Switzerland, 166
Centimeter, table, 30
circular, 42
inductance, 119
cubic, 46
do. to cb. inch, digit table, 51
do. to fluid drams, dijgit table, 51
do. to fluid ounces, digit table, 51
map scales, 168
per second, 85
per second per second, 88
square, 42
do. to sq. inch, digit table, 44
Centimeter-dyne, 74
do. per second, 80
-gram, 74
-gram per second, 80
Centimo, Costa Rica, 164
Centistere, 52 ,
Centner, Austria, Sweden, 61
Germany, 60
Denmark, 172
Sweden, 173
Century, 95
C.G.S.:
system of units, text, 11
unit current-turn, 133, 134
do. per centimeter, 137
units, see under respective names
Chain, 32
square, 43
Chaldron, 54
Charge electrical, 115
do., physical, 22, 24
ionic, 126
do. physical, 23, 25
Chargje, volume, Candia, France, 55
Chemical weights, 60
Chetvert, Russia, 55, 173
Cheval vapeur, 81
Chih. China, 172
Cho, Japan, 34, 44
Chopine, France, 54
Circuit, kinetic energy in, 122
Circular unit, defined, 41
acre J 44
centimeter, 42
foot, 42
inch, 42
mil, 41
mil-foot unit, 102
millimeter, 41
Circular measure (angles), 89
Circumference, 89
Civil day , month , year, 94
aark cell:
defined, 108
e.m.f. of, 110
temperature correction. 111
Clark meter, 29
Coatings, weights of, 63
Coefficient of:
expansion, 26
mutual induction, 118
Peltier effect 23, 25
self-induction, table, 119
do. defined, 118
do. physical, 23, 25
traction, 78
Colon, Costa Rica, 164
Committee meter, 29, 32
Common logarithms, 171
Common pace, 31
Compound names of unitf*, 4
Concrete temperature scale, 151-163
Concrete vs. absolute units, 12
Concrete electrical units, 14
Condensance, 22, 24
Condensed numoers, 8
Condor, Chile Colombia, 164
Conductance tables, 104
physical, 22 24
C.G.S. unit of. 105
magnetic, 130
specific, 106
do. physical, 22, 24
Conductivity, electric, tables, 107
text. 106
Shysical. 22, 24
.G.S. unit of, 106, 107
of copper, 106, 107
magnetic, 131
of mercury, 107
Conductivity, heat, 26
Congresses, dividons of electrical, 14
Conversion factors, tables, 30-
173
text, 27
digit tables:
do. capacities, 51
do. energy, 77
do. grades, 92
do. heat. 77
do. inches, fractions, mm., ft., 35
do. lengths, 39
do. power, 82
do. pressures, 67
do. surfaces, 44
do. volumes, 51
do. weights, 59
do. work. 77
Coomb, 53
Copper, conductivity, of 106, 107
resistivity of, 102, 104
unit of conductivity, 106, 107
Cord 54
Cord foot, 53
Corde, Sweden, 33
Coulomb, table, 116
defined, 115
international, 116
do. defined, 115
true, 116
per milligram. 126
Coupes, Geneva, 55
Couple, physical, 19
Coyan, Sarawak. Siam, 173
Cross-section and resistance, 101
Cross-section units, 41
INDEX.
179
Crown, Austria, Denmark, 164
Great Britain, Norway, Portugal,
Sweden, 165
Cuadra, Argentine, 172
Paraguay, Uruguay, 173
Cuadra sq., Paraguay, 173
Cubic centimeter, 46
to cb. inches, digit table, 51
to fluid drams, di^it table, 51
to fluid oimces, digit table, 51
weight of water, 70
per ampere-hour, 126
Cubic decimeter, 48
Cubic foot, 49
to cb. meters, digit table, 51
wei^t of water 70
Cubic inch, 47
to cb. centimeters, digit tables, 51
weight of water. 70
Cubic measures, 45-55
do. foreign, 54
Cubic meter 51
to cb. feet, digit table, 51
to cb. yards, digit table, 51
wei^t of water, 70
Cubic millimeter, 52
Cubic yard, 50
to cb. meters, digit table, 51
weight of water, 70
Cubit, 31
ancient, 34
Currencies, 164
fluctuating, 166
Current (elec), table, 113
text, 112
physical, 22, 24
C.G.S. unit, 112, 113
densitv, 114
do., physical. 22, 24
intensity, 112
kinetic energy of , 122
strength, 112
-tvan. C.G.S. imit of, 133
Ciu-vature, 18
Curvature, specific, of a surface, 18
Cycle, lunisolar, solar, 95
frequency, 86, 121
D
Daniell cell, 109. 110
Day, 94
apparent solar, astronomical, cal-
endar, civil, natiiral, nautical,
sidereal, solar, 94
Day's journey, ancient, 34
Deca-, as prefix, 9
Decagram, 59
Decaliter, 52
Decameter, 32
Decare, 44
Decastere, 54
Deci-, as prefix, 9
Declare, 43
Decigram, 57
Deciliter, 47
Decimeter, 30 «
cubic, 48
square, 42
Decuno, Chile, Colombia, 164
Venezuela, 166
Decisions of electrical congresses, 14
Decistere,53
Decomposition voltage, 128
Degree, angle, 89
Eade, 91
titude, longitude, 32
per second, 86
electrical, 87, 89, 121
Deka, Germany, 60
Deka-, as prefix, 9
Dekagram, 59
Dekaliter, 52
Dekameter, 32
Dekastere, 54
Demi-posson, France, 54
Demi-setier, France, 54
Denier, France, 61
Densities, mass, 67
weights and volumes from, 69
physical, 18
current, 114
do. physical, 24
flux (magnetic), 140
do. physical, 20, 21
surface, electric, 22, 24
Deposition, electric, 23, 25
electrolytic, 126
Deposits, electrolytic, 126
weights of , 63
Derivation of ph3/^sical quantities, 18
Dessaetine, Russia, 44
Spain, 173
Dimagnetic bodies, 131
Diameter, earth's, earth's orbit, 32
Diameter to cross-section, 41
Decistere, 53
Dielectric constant, 23, 24
Diemo, Colombia, 164
Difference of potential, electric, 108
physical, 22, 24
magnetic, 132
Digit « ancient, 34
Dig^lt conversion tables:
capacities, 51
energy, 77
ffrades, 92
heat, 77
inches, fractions, nun., ft., 35
lengths, 39
power, 82
pressures, 67
surfaces, 44
volumes, 51
weiehts, 59
work, 77
Dime, U. S., 166
180
INDEX,
Dimensional formulas, text, 12
tables, 18
Dinero, Chile, 164
Peru, 165
Directive force, suspensions, 19
Discharges (water), 95
Displacement, electric, 22, 24
Distinction between units and quan-
tities, 4
Doli, Russia, 61
Dollar, British Honduras, 164
British possessions, 164
China, 164
Mexico. 165, 166
Newfoimdland, 165
United States, 166
per mile, 166
per pound, 167
per ton, 167
Doubloon, Chile, Cuba, 164
Dozen, 168
Drachm, fluid, 52
Drachm, weight, 59
Drachma, Greece (money), 165
Drachme, Germany, 60
Greece (weight), 172
Dragme, France, 61
Dram, fluid, 52
weight, 59
Drop, 52
Dubloon, Mexico, 165
Dyne, 83
-centimeter, 74
do. per second, 80
per centimeter, 62
per sq. centimeter, 64
E
Eagle, U. S., 166
Ei^h, diameter, 32
Earth, diameter of orbit, 32
Earth's magnetic field, 140
Effective values, 97
Efficiency (power), 19
of light, 149
Effort, tractive, 78
Egyptian lengths, 34
Euner, Austria, Sweden, 55
Germany, 54
Kleotric:
and magnetic units, 96-144
do., interrelations of, 96
capacity, 117
degree, 87, 89, 121
deposition, 126
do., physical, 23, 25
displacement, 22, 24
energy, tables, 74, 123
do., text, 122
1 Electric :
energy, C.G.S. unit, 122, 123
do., physical, 23, 25
field, intensity, 22. 24
inductive capacity, 18, 23, 24
power, tables, 80, 125
do., text, 124
do., physical, 23, 25
pressure, 108
quantities, 22. 24
auantity, 115
o., physical, 22, 24
stress, 108
units, congress decisions, 14
units, relations to others, 3
Electrical, see 1. lectric
Electricity, quantity, 115
do., physical, 22, 24
Electrochemical :
equivalent, 125
do., physical, 23, 25
do. of silver, 125
energy, 128
quantities, 23, 25
Electro-kinetic inertia, 23, 25
Electro-kinetic momentum, 23, 25
Electrolytic deposits, 126
Electrolytic gas, 126
Electromagnetic system, 11
do., units and quantities, 20, 22
Electromagnetic units, see under re-
spective names
Electromotive force:
table, 109
text, 108
hysical, 22, 24
G.S. unit of, 109, 110
at a point, 22, 24
of Clark cell, 108, 110
do., temperature correction. 111
of Weston cells, 108. 110
do. temperature correction. 111
Electrostatic system, 11
do., units and quantities, 21, 24
Electrostatic units, see under re-
spective names.
Ell, 32
EUe, Austria, Germany, 33
Emissivity, heat, 26
ISneripy:
table, 74
digit tables, 77
text, 72
physical, 19
electric, 122 ■
do., physical, 23, 25
do.. C.G.S. unit, 122
do., stored, 123
kinetic (electro-), 23, 122
electrochemical, 128
magnetic, 20, 21, 143
rate of, 79
relations with torque, table, 78
do., text, 72
traction, 78
do. vs. torque, 13, 72, 78
English standard candle, 145, 146
Entropy, 26
8'
INDEX.
181
Ekiuivalent , mechanical :
heat, 3. 26.72, 171
l^ht. 147
Equivalents, electrochemical, 125
do., physical, 23, 25
Equivalents of imits, 30-173
Ei^, 74, 122
kinetic energy of current, 122, 123
magnetic energy, 143
stored energy, 122, 123
§er second, 80
o., electricj 124
do., magnetic, 144
Errors of abbreviated numbers, 10
EscudOj Chile. 164
Expansion, temperature, 26
Factor, inductance; 23, 118, 120.
124, 125
load, 80
power, 23.79, 124, 125
Factors, conversion, text, 27
do., tables, 30-173
reduction, text, 27
do., tables, 30-173
Fahrenheit degree red. factors, 150
scale, to others, 151-163
heat unit, 75
Fall of potential, 108
Famn, Sweden, 33
Fanega (dry). Central America,
Chile, Cuba, Mexico, Morocco,
172
Lisbon, 55
Spain 55, 173
Uruguay, Venezuela, 173
Fanegada, Canary Isles, Spain, 44
Farad, tables, 117
international, 117
Faraday's law, 125
Fardingdeal, 44
Farthing, Great Britain, 165
Fathom. 32
ancient, 34
Faux, Switzerland, 44
Feddan, Egypt, 172
Feet, see under Foot
Feuillette, France, 54
Field, earth's magnetic, 140
Field intensity, electric, 22, 24
do«« magnetic. 134
do., physical, 20,21
Film tension, 62
Final ampeiea. 113
Finger, ancient, 34
Firkin, 53
butter, 60
Firlou, Scotland, 55
Florin, Netherlands. 165
Flow of water, 95
Fluctuating currencies. 166
Fluid dram, 52
do. to cb. cm., digit table, 51
scruple, 52
oimce, 52
do. to cb. cm., digit table, 51
Flux:
light, table, 149
do., physical, 25
force, magnetic, 137
magnetic, tables, 138
do., text, 137
do., physical, 20, 21
do. density, 140
do. density, physical, 20, 21
do. density, C.G.S. unit, 140, 141
do. from unit pole, 138
-turns, 138
Florin, Austria, 164
Foot:
tables, 30
to inches and fractions, 35
to meters, digit table. 39
to millimeters, digit table, 35
board, 52
builder's, 34
circular, 42
cord, 53
cubic, table, 49
do. to cb. m., table, 51
do. weight of water, 70
foreign: Austria, Russia, 33
Spain, Italy, miscellaneous, 34
mathematic, 34
pressure of water, 65
rise per foot (grade), 90, 91, 92
solid (timber), 53
square, table, 42
do. to sq. meters, digit table, 44
surveyor's, 34
tradesman's, 34
water column, 65
Foot per:
foot (grade), 91, 92
100 feet (grade), 90, 92
1000 feet (grade), 90, 92
mile (grade), 90, 91, 92
minute, 85
second, 85
second per second, 88
Foot-:
-candle, 148
-grain, 74
-grain per second, 80
3)ound, 74
o. per minute, 80
do. per radian, 78
do. per revolution, 78
do. per second, 80
do. to kilogram -meters, table, 77
do. to thermal units, table, 77
do. to torque, 78
182
INDEX.
Force, table and text, 83
physical, 18
center of attraction, 18
directive, suspensions, 19
and length, 62
and surface, 63
per unit area, 63
de cheval, 81
flux of (naagnetic), 137
lines of (magnetic), 137
do., physical, 20, 21
magnetic, 134
magnetizing. 134
do., physical, 20, 21
magnetomotive, 132
do., physical, 20, 21
tractive, 78
Foreign measures, see under specific
names
do., miscellaneous, 172
Formulas, changing units of, 6
temperature standard cells. 111
Fot, Sweden, 33
Fot square, Sweden, 44
Fother, 60
Frail, Spain, 173
Franc, Belgium, 164
France, 165
Switzerland, 166
per kilogram, 167
per kilometer, 166
per meter, 166
per metric ton, 167
France, lengths, 33
Frasco, Argentine, Mexico, 172
Frasila, Zanzibar 173
French horse-power, 81
Frequency, general, 86
electric, 121
do., physical, 23, 25
Fuder, Germany, 54
Luxemburg, 172
Fun, Japan, 61
Functions, periodically vars'ing, 97
Fundamental, electric units, 96
quantities, 18
Funt, Russia, 173
Furlong, 32
Fuss, Austria, Germany, Holland,
Switzerland, 83
Futtermaassel, Austria, 55
a
Gallon, liquid, table, 49
standard vnJue, 45
to liters, digit table, 51
Gallon, apothecary, 52
beer, 52
British, 49
do., standard value, 46
dry, U. S., 52
imperial, 49
do., standard value, 46
weight of water, 70
wine, old British, 52
Gamez, Russia, 55
Gamice, Russian Poland, 173
Gas, electrolytic, 126
Gauss, 141
defined, 140
as magnetizing force, 137
do. defined, 135
inch-, 141
Geira, Portugal, 44
Geographical mile, 32
do. international, 32
Geometric quantities, 18
German lengths, 33
paraffine candle, 146
do. defined, 145
Gilbert, 133
defined, 133
per centimeter, 136
per inch, 136
Gill, 52
Go, Japan, 55
Gold grains, Spanish, 61
Gourde, Haiti, 165
Grade (angle), 89
Grade (incline), 90
conversion table, 92
Grain, mass, weight, 57
force, 83
French, 61
^old, Spanish, 61
jeweller's, 59
volume of water, 71
to milligrams digit table, 59
per cubic inch, 68
per inch, 62
Grain-foot, 74
per second, 80
Gram, mass, weight, 58
force, 83
volume of water, 71
to ounce, digit table, 59
per ampere-hour, 126
per centimeter, 62
per cubic centimeter, 69
per hour, 127
per meter, 62
per minute, 127
per sq. centimeter, 64
per sq. decimeter, 64
per watt-hour, 126
-atom , 60 .
-centigrade heat unit, 75
-centimeter, 74
do. per second, 80
-molecule, defined, 60
do. gas volume, 53
Gran^ Germany, 60
Gravitation constant, physical, 18
do. defined, 87
INDEX.
183
Gravitational units, relations, 3
Gravity, 88, 171
mean value, 87
as a relation, 3
Great gross, 168
Greciui lengths, 34
Gregorian year, 95
Grivinik, Russia, 165
Gros, France, 61
Gross, 168
Gross ton, 58
see also Ton, long
displacement of water, 54
Guinea, Great Britain, 165
H
Hairsbreadth, 31
Hand, 31
Harcourt pentane lamp, 146
do. defined, 145
Heaped bushel, 53
HeHt, tables, 74
digit tables, 77
text, 72
physical, 26
of combination into volts, 129
latent, 26
mechanical equivalent, 171
do. defined, 72
do., physical, 26
do. as a relation, 3
specific, capacity, 26
specific, of water, 171
fk>. as a relation, 3
unit, 75, 76
do. defined, 73
do. digit tables, 77
do. relations to others, 3
Hebrew lengths, 34
Recta-, as prefix, 9
Hectare, 43
to acres, digit table, 44
Hectogram, 60
Hectoliter, 50
to bushels, digit table, 51
weight of water, 70
Hectometer, 32
Hectostere, 54
Hefner or hefner unit, 146
defined, 144
hemispherical, 147
spherical, 147
Hekto-, as prefix, 9
Hektoliter, 50
Hektostere, 54
Heller, Austria, 164
HeUergewicht, Germany, 60
Hemisphere, 89
Hemispherical candle, 147
Hemispherical hefner, 147
Heniy, 119
defined, 118
relations to other units, 119
Hogshead, 53
Holland lengths. 33
Horse-power, table, 81,
metric, see \mder Metric
to metric hp., di^t table 82
to kilowatts, digit table, 82
-minute, 76
do., metric, 76
-second, 75
do., metric, 75
-hour, 77
do., metric, 77
do. per kilogram, 126
do. per minute, 82
do. per pound, 126
do. per second, 82
Hour (solar), 94
sidereal 94
Hyperbolic logarithms, 171
Hyphen, in names of units, 4
Hundredweight, 58
Illumination, 148
physical, 25
Immissivity, heat, 26
Impact, 19.
Impedance, 99
defined, 98
physical, 22, 24
Imperial, Russia, 165
Imperial gallon, 49
defined, 46
Incandescent lamp staj&daxds, 145
Inch, table, 30
fractions , mm., and feet , tables , 35
to millimeters, digit tables, 39
circular, 42
cubic, 47
do. to cb. cm., digit table, 51
do., weight of water, 70
square, 42
do. to sq. cm., digit table, 44
gauss, 141
on map, 168
mercury column, 65
per mile, 90, 92
Inclines. 90
Induced volts, 110
184
INDEX.
Inductance, table, 119
text, 118
C.G.S. unit, 119
physical, 23, 25
mutual, 25
Inductance factor, defined, 118, 124
do., physical, 23
do., values, 120, 125
Induction, 118
capacity, 24
magnetic f 140,
do., physical, 20, 21
mutual, 23. 118
self, 23, 118
Inductive capacity, electric, 23
do., relation to permeability, 14
do., suppressed factor, 12, 14
do., specific, 20. 21, 23
Inertia, physical, 19
electro-kmetic, 23, 25
moment of, 84
Intensity of:
attraction, 18
electric field, 22, 24
light, 144
do., physical, 25
masnetic field, 134*
do., physical, 20, 21
magnetisation. 142
do., C.G.S. unit of, 142
do., physical, 20, 21
stress, physical, 19
Intwnational:
ampttre, 113
do. defiofid, 112
couloftib, 116
do. defined, 115
farad, 117
meter, defined, 29
mile, 32
nautical mile, 32
ohm, 99
do. defined, 98
volt, 110
do. defined, 108
Inter-relation of imits, 1
Introduction. 1-26
Ion 125, 128, 129
energy of, 129
Ionic charge, 126
I>hysical, 23, 25
Irrigation units, 95
Italian lengths, 34
Japuiese lengths, 34
Jeweller's grain, 59
Jo, Japan, 34
Joch, Austria, 44
Joule:
table, 74, 123
defined, 122
to calories, digit table, 77
electro-chemical energy. 128
kinetic energy of current, 122, 123
magnetic energy, 143
per cycle, 123
per second, 80
Julian year, 94
K
Kanne, Sweden, 55
Kapp line, 138
Kater, value of meter, 29
Ken, Japan, 34, 172
square. Japan, 44
Kern, Germany, 60
Klafter, Austria. 33, 44
Germany, 54
Russia, 173
Kilo, 58
Kilo-, as prefix, 9
Kiloampere, 113
Kilodyne, 83
Kilogauss, 141
KiloKram:
table, 58
defined, 56
relation to liter, 56
force, 83
to pounds, digit table, 59
volume of water, 71
per cubic centimeter, 69
per cubic meter, 68
per day, 127
per hectoliter, 68
per horse-power-hour, 126
per hour, 127
per kilometer, 62
per kilowatt hour, 126
per liter, 69
per meter, 63
per sq. centimeter. 66
do. to atmospheres, digit table, 67
do. to lbs. per sq. in., table, 67
per square meter, 64
per square millimeter, 66
per ton, 78
per year, 126
-centigrade heat unit, 76
-kilometer, 76
do. per minute, 81
-meter, 75
do. to foot-pounds, digit table, 77
do. to large calories, digit table, 77
do. per minute, 80
do. per second, 81
-molecule, 60
Kiloioule, 123
Kiloliter, 54
INDEX.
185
Kilometer, table, 30
to miles, digit table, 39
square, 43
per hour, 85
do. per minute, 88
do. per second, 88
per minute, 86
Kilovolt, 110
Kilowatt, table, 82. 125
defined, 124
to hor8e-i>owers, digit table, 82
do., metric, digit table, 82
-hour, table, 77, 123
defined, 122
per kilogram, 126
per minute, 82
per pound, 126
per second, 82
-minute, 76
-second, 75
Kin, Japan, 61
Kine, 85
Kinetic energy of a current, 122
do.{ physical, 23
Kinetic inertia, electro-, 23
Kinetic momentum, electro-, 23
Kialos, Alexandria, Constuitinople,
Smyrna, 55
Knot, 31
telegraph, British. 32
perliour, 85
Koku, Japan, 55, 172
Kopek, Russia, 165
Koppa, Sweden, 55
Kom, Glermany, 60
Sweden, 61
Korree, Russia, 173
Kran, Persia, 165, 166
Kreutzer, Austria, 164
Kruschky, Russia, 55
Kunkas, Russia, 55
Kvintin, Sweden, 61
Kwan, Japan, 61
Lachter, Germany, 33
Lamps, standard, 145
Last, 54
Last, Belgium, Holland, 172
Russian Poland, Spain, 173
Latent heat, 26
Latitude, degree of, 32
League, 32
League, France, 33
Paraguay, 173
Spain, 34
miscellaneous foreign, 34
I Legal ohm, 99
do. defined, 98
volt. 109
year, 95
see idso Standards
Legua, 32
I«ensth8, table, 30
text, 29
physical, 18
British standards, 29
foreign, 33
fundamental standards, 29
syst^me ancien, France, 33
syst^me usuel, France, 33
U. S. standards, 29
and forces, 62
uid masses, 62
and money, 166
and weights, 62
Lepta, Greece, 165
Li, China, 34,172
Libra, Argentine, Central America,
Qiile, Cuba, Mexico, 172
Peru, 165, 173
Spam, 61, 173
Portugal, Uruguay, Venesuela,
173
Liespund, Sweden, 61
Lieue, France, 33
marine, France, 33
moyenne, France, 33
liight, 144-149
phsrsical, 25
brightness of source, 148
candle power, 144
dimensional formulas, 25
efficiency, 149
flux of, 147
illumination, 148
intensity of, 144
mechanical equivalent, 147
quantity of, 149
radiant, as power, 3, 25, 147
standards oi, 146
do. defined, 144
velocity of, 86
do. as a relation, 11, 14,21,24, 25u
• 96
units, 144-149
do., absolute, 3, 145, 146
do., relations to others, 3
Ligne, France, 33
Linear acceleration, 87
do., physical, 19
Linear velocity, 85
do., physicflil, 19
Line, 31
Lines of force (magnetic), 137
do. physical, 20, 21
dp. per unit cross-section, 140
Linie, Austria, Germany, Sweden,
Switzerland, 33
Link, 31
Lira, Italy, 165
lifter, table, 48
defined, 45
relation to kilogram, 45
standard, defined, 45
186
INDEX.
Liter, weight of water, 70
to gallons, difl^t table, 51
to quarts, digit table, 51
Litron, France, 55
livre, France, 61
Greece, Guiana, 172
Load factor, 80
Lod, Sweden, 61
Logarithms, accuracy of, 10
bases, 171
conversions of, 171
systems of, 171
Long ton, 58
do., see also Ton, long
Longitude, degree of, 32
Loop, Russia, 55
Loth, Germany J 60
Austria, Russia, Switzerland, 61
Louis, France, 165
Lumen, 147
Lumen-hour, 149
Lunar month, year, 04
Lunisolar cycle, 95
Lux, 148
M
Maass, Austria^ 55
Maassel, Austria, 55
Magnetio:
and electric units, 96-144
do., dimensional formulas, 20
do., introductory text, 96
do., interrelations, 96
calculations, 136
capacity, 130
do., physical, 20, 21
conductance, 130
conductivity, 131
energy, 143
do., physical, 20, 21
field, 137
do. defined, 138
force, 134
flux, tables, 138
do., text, 137
do., physical, 20, 21
do.. C.G.S. unit, 138
flux density, 140
do., physical, 20, 21
do., cAs. unit, 140, 141
flux from unit pole, 138
flux-turns, 138
induction, 140
do., physical, 20, 21
do., C.G.S. unit, 141
lines of force, 137
do. physical, 20, 21
do. per unit cross-section, 140
moment, 142
do., phyisical, 20, 21
do.. C.G.S. unit, 142
Magnetic:
permeability, 131
do., physical, 18, 20, 21
' do., suppressed factor, 12, 14
permeance, 130
do., C.G.S. unit, 131
pole, unit, 138
potential, 132
do., physical, 20, 21
power, 144
do., physical, 20, 21
pressure, 132
quantities, physical, 20, 21
reactance, 22, 24
reluctance, 129
do., physical, 20, 21
do.. C.G.S. unit, 129
reluctivity, 130
do., physical, 20, 21
resistance, 129
do., physical, 20, 21
resistivity, 130
susceptibility, 132
do., physiral, 20, 21
units, 129-144
do., congress decisions, 14
do., general, 96
work, 143
Magnetisation, see Majgnetisation
Magnetization, intensity, 142
do., physical, 20, 21
do., C.G.S. unit, 142
Magnetizing force, tables, 137
text, 134
Physical . 20, 21
.G.S. unit, 136
do. defined, 134
units, defined, 134
Magnetomotive force, 132
physical, 20, 21
C.G.S. unit, 133
per centimeter, 134
Malter, Germany, 54
Manzana, .Costa-Rica, Nicaragua,
172
Salvador, 173
Marc. France, 61
Bolivia, 172
Marco, Spain, 61
Mark, Finland, 164
Mark (money), Germany, 16$
per meter, 166
per metric ton, 167
per kilogram, 167
per kilometer, 166
(weight) Germany, 60
do., Sweden, 61
Masses:
tables, 57
digit tables, 59
text, 56
foreign, 60, 172
standards, defined, 66
physical, 18
water, 69
and lengths, 62
and surfaces, 63
and volumes, 67
INDEX.
187
lilaierials, weiffht of, 67
Mattari, Tripoli, Tunis, 55
Maund, India, 172
Mocha, 61
Maximum values, 97
Maxwell, 138
per square centimeter, 141
per square inch, 141
Mean solar time, 93
Mean values, 97
Mean watts, 125
Measures, see \mder respective names
apothecary, text, 45
dry, text, 45
liquid, text, 45
Mechanical equivalent of heat, 171
do. defined, 72
do. as a relation, 3
do., [physical, 26
Mechanical eqmvalent of light, 147
Mechanical quantities, 18
Medimni, Greece, 55
Meg:a-, as prefix, 9
Megabarie, 66
Megadyne, 83
per square centimeter, 66
per square meter, 64
Megamho, 105
Megavolt, 110
Megohm, 99
cubic centimeter, unit, 103
Meile, Austria, Germany, Sweden, 33
Mercury, conductivity, 107
density, 63
pressures, 63
do., inch of, 65
do., millimeter of, 65
specific gravity, 63
resist! vitv, 102, 104
imit, conductivity, 107
Meter, table, 30
standard, 29
to feet, digit table, 39
to yards, digit table, 39
Clark value, 29
committee, 29, 32
cubic, 51
do. to cb. ft., table, 51
do. to cb. yds., table, 51
international, 29
in wave lengths, 32
Kater value, 29
legal standard, 29
older values. 29
Pratt & Whitney Co., 31
square, 43
do. to sq. feet, table, 44
do. to sq. yards, table, 44
standard, 29
water column, 65
per minute, 85
per second 85
per second per second, 88
-candle, 148
-kilogram, energy, 75
do., torque, 78
millimeter unit, 102
Metercentner, Austria, 61
Metric horse-power, 81
to horse-power, digit table, 82
to kilowatt, digit table, 82
-hour, 77
do. per minute, 82
do. per second, 82
-minute, 76
-second, 75
Metric system, prefixes, 9
Metric ton, 58
per square meter, 65
per year, 127
Metze, Austria, 55
Germany, 54
Mho, 105
cubic centimeter unit, 107
do. defined, 106
Micro-, as prefix, 9.
Microampere, 113
Microcoiuomb, 116
Microdyne, 83
Microfarad, 117
Microhenry. 119
Microhm, 99
cubic centimeter unit, 103
per cubic centimeter, 103
square cm. per cm., 103
do. defined, 102
Microjoule, 123
Micro-meter. 31
Micro-millimeter, 31
Micron, 31
Microne, 31
Microvolt, 109
Microwatt, 125
Mil, 30
circular, 41
square, 41
Denmark, 172
Mil-foot unit, 102, 103
Mile, table, 31
to kilometers, digit table, 39
international geographical, 32
international nautical, 32
geographical, 32
nautical, 31
square, 43
statute, see Mile
telegraph, 34
German, Holland, Swiss, 33
Netherlands, Italy, miscellane-
ous, 34
per foot, 91, 92
per hour, 85
do. per minute, 88
do. per second, 88
per minute, 86
abound, 76
o. per hour, 81
do. per minute, 81
-ton, 78
Military pace, 31
Milla, Honduras, Nicaragua. 172
Millarium, ancient, 34
Millennium, 95
Milli-, as prefix, 9
Milliampere, 113
Milliare. 43
188
INDEX.
Millier, 58
Millier, France, 61
Milligram, mass, 57
force, 83
to grains, digit table, 59
per coulomb, 126
per millimeter, 62
pe- second, 127
Mimhenry, 119
Milliliter, 46
Millimeter, 30
to fractions of inch, digit table, 85
to inches, digit table, 39
circular, 41
cubic, 52
mercury column, 65
square, 41
per meter, 90, 92
Milli-micron, 31
Millimol. 60
Millistere, 52
MilUvolt, 109
Milreis, Brazil, 164
Portugal, 165
Mina, Genoa, 55
Miner's inch, 95
Miner's pound, Swedish, 61
Minim, 52
Mint pound, 60
Minute, time, 94
sidereal, 94
angle, 89
Miria-, see myria
Miriameter, 32
square, 44
Modulujs of elasticity, 19
Modulus of logarithms, 171
Moggi, Milan, 55
Moggia, Naples, 44
MoiTeo
Mole, 60
Molecule, 52
gram, kilosram, 60
average velocity of, 86
Moment, 72
torque, 72-78
physical, 19
. of mertia; 84
do., physical, 19
momentum, 84
do., physical, 19
magnetic, 14i2
do., physical, 20, 21
per unit volume, 142
Momentum 79
physical, 19
angular, 84
electro-kinetic, 23, 25
moments of, 84
Momme, Japan, 61
Money, 164
fluctuating, 166
and length, 166
and weight, 167
Monovalent ions, 129
Month, anomalistic, calendar, civil,
lunar, sidereal, synodic, 94
Morgen, Germany, 44
Motion, quantity of, 19
Mudde, Amsterdam, 65
Muid, France, 54
Mut or Muth, Austria, 55
Mutual inductance, 118
do., physical, 23, 25
induction, 118
do., physical, 23, 25
Myria-, as prefix, 9
Myriadyne, 83
Myria«ram, 60
Myri^ter, 54
Myriameter, 32
square, 44
MynoUter, 54
N
Nahud cubit, ancient, 34
NaU, 31
Names of units, compound, 4
Naperian logarithms, 171
Napoleon, France, 165
National Bur. Standards ampere ,113
National prototypes, 29
Natural day, 94
Natural logarithms, 171
Natural year, 95
Nautical dav, 94
Nautical mile, 31
defined 29
international, 32
Net ton, 58
do., see also Ton, short
Noeud, France, 33
Numbers, accuracy of, 9
Numbers, condensed, 8
Numbers, useful, 170
Oer, Denmark, 164
Norway, Sweden, 165
Oersted, 129
Ohm, table, 99
text, 98
Board of Trade, 98, 99
international, 98, 99
legal, 98. 99
INDBX.
189
Ohm, true, 98. 99
Reichsanstalt, 99
circular-mil, foot unit, 102, 103
circular-mil, per foot, 102, 103
circular-millimeter, meter unit,
103
circular millimeter fyer meter, 103
cubic centimeter unit, 102, 103
sq. cm. per cm., 103
sq. mil, foot imit, 102, 103
sq. mil per foot, 102, 103
sq. mm., met., imit, 102, 103
sq. millimeter per meter, 102, 103
per cubic centuneter, 103
per foot, 101
do. per circular mil, 102, 103
do. per mil diameter, 103
do. per square mil, 102, 103
per kilometer, 101
per meter, 101
do. per circular millimeter, 103
do. per sq. millimeter 102, 103
per mile, 101
-centimeter square, 101
-centimeter unit, 102
-inches square, 101
(volume), Grermany, 54
Ohmic resistance, 98.
Oke, oriental, 61
Egypt, Greece, Hungary, 172
Turkey, 173
Once, France, 61
Mexico, 165
Orbit, earth's, 32
Ome, Austria, 55
Osmini, Russia, 55
Ounce (av.), mass, 58
force, 83
to grams, digit table, 59
volume of water, 71
apothecary, 59
fluid, 52
Troy, 59
Troy, silk, 59
per hour, 127
per minute, 127
Oxhoft, Germany, 54
Pace. 31
common, 31
military, 31
Paiok, Russia, 55
Palm, 31
ancient, 34
Palmo, Italy, Spain, 34
Paper measure, 168
Para, Egypt, 164
Turkey, 166
Paraffin candle, 146
defined, 145
Paramagnetic bodies, 131
Parasang, Persia, 34
Passus, ancient, 34
Peck, 49
weight of water, 70
Penni, Finland, 164
Penny, Great Britain, 165
per foot, 166
per ounce, 167
Pennyweight, Troy, 69
Pentane lamp, 146
defined, 145
Peltier effect, 23, 25
Per, in names of units, 4
Percent (grade), 90-^2
Percentage, defined, 7
Per mil (grade), 90, 92
Perch, length, 32
masonry, 54
square, 43
Perche, France, 33, 44
Period, 86, 121
physical, 23, 25
Periodically-varying fimctions, 97
Periodicity, 86, 121
Permeability, magnetic, 131
physical, 2(), 21
relation to inductive capacity, 14
suppressed factor, 12, 14
Permeance, magnetic, 130
do., physical, 20, 21
specific, 131
Pes, ancient, 34
Peseta, Spain, 165
Peso, Argentine, Chile, Cuba, 164
Ckjlombia, 164, 166
Guatemala, Honduras, Nicaragua,
Mexico, Salvador, 165
Central Ainerica, Uruguay, 166
Pezza, Rome, 44
Pfennig (weight), Austria, 61
(money), Germany, 165
Pfenniggewicht, Germany, 60
Pferdekraft, 81
Pfund, Germany, 60
Austria, Russia, Switzerland, 61
Physical quantities, table, 17
derivation of, 18
dimensional formulas of, 18
symbols of, 18
Photometric quantities, 25
units, 144-149
do., congress decisions, 14
Pi (it), fimctions of, 169
do. as an angle, 89
Piaster, Egypt, 164
Turkey, 166
Piastre, Mexico, 165
Pic, Egypt, 172
Picul, Borneo, Celebes, Chins,
Japan, Java, 172
Phillippine Islands, 173
Pie, Argentine, 172
Spain, 173
Pied, France, 33
Pietak, Russia, 165
190
INDEX.
Piff, Austria, 55
Pig, metal, 60
Pik, Turkey, 173
Pint, 47
weight of water, 70
apothecary, 52
Scotland, 55
Pinte, France, 54
Genoa, 55
Pipe, 53
Pistareen, Spain, 165
Plane angle, 89
physical, 18
Platinum standard, 146
defined, 145
Poids de Marc, France, 61
Point, 31
France, 33
Pole, length, 32
square, 43
(magnetic) strength, 137
do., physical, 20, 21
do. per unit cross-section, 142
do. imit, 138
Poltinnik, Russia, 165
Poncelet, 82
Pood, Russia, 173
Posson, France, 54
Pot, France, 64
Potential, electric:
absolute, 108
electromotive force, 108
physical, 22, 24
vector, 22, 24
Potential, magnetic, 132
do., physical, 20, 21
Pottle, British, 52
Pouce, France, 33
Pound (mass), table, 58
defined. 56
force, 83
to kilograms, digit table, 59
volume of water, 71
ai)othecary, 60
(silk), Lyons, France, 61
miner's, Sweden, 61
mint, 60
Troy, 60
Troy, silk, 60
Amsterdam, Avistria, Italy, Rot-
terdam, Russia, Spain, miscel-
laneous, 61
sterling (money), Great Britain,
India. 165
Egypt. 164
Pound per:
ampere-hour, 126
bushel, 68
cubic foot, 68
cubic inch, 69
cubic yard, 68
day, 127
foot, 63
gallon, 68
horse-power hour, 126
hour, 127
inch, 63
kUowatt-hour, 126
Pound per:
mile, 62
quart, 68
square foot, 64
square inch, 65
do. to atmospheres, table, 67
do. to kg. per sq. cm. table, 67
ton, 78
yard, 62
year, 126
Pound-:
-Fahrenheit heat unit, 75
-foot revolution, 78
-centigrade heat unit, 75
do. per minute, 81
-foot, energy, 74
do., toraue, 78
-foot-raaian, 78
Poundal, 83
per inch, 63
per sq. foot, 64
per sq. inch, 65
Pous, ancient, 34
Power, tables, 80-82
digit tables, 82
text, 79
Sl^sical, 19
^.S. unit, 125
apparent, defined, 124
electric, 124
do., physical, 23, 25
magnetic, 144
do., physical, 20, 21
per candle-power, 149
radiant light, 3, 25. 147
Power factor, defined, 79, 124
do., physical, 23
do. values, 125
Practical units, 12
Prefixes in metric system, 9
Pressures, tables, 64
digit tables, 67
text, 63
physical, 19
water, mercury and atmosphere,
63
electrical, 108
magnetic, 132
Protot3i}es, national, 29
Pud, Russia, 61
Puncheon, 53
Pund, Sweden, 173
Pyr, 146
defined, 145
TT as an angle, 89
n, useful functions of, 169
Q
Quad. 119
Quadrant, angle, 89
inductance, 119
INDEX.
191
Quantities and units:
distinction between, 4
table of, physical, 17
Quantity :
elect ncal, 115
do., C.G.8. unit, 116
do., physical, 22, 24
light, 149
do., physical, 25
motion, 10
Quart, 48
to liters, digit table, 51
weight of water, 70
Germany, 54
Quartant, France, 54
Quartellos, Spain, 55
Quarter, length, 31
volume, 53
weight, 60
Quarti, Rome, 55
Quentchen, Austria, 61
Germany, 60
Quintal, 60
Argentine, Brazil, Chile, Greece,
Mexico, Newfoundland, Para-
guay, Peru, Syria, 173
Spain, 61
auinteau, France, 61
uire, 168
R
Radian, 89
per minute, 86
per second, 86
Radiant light as power, 3, 25, 147
Rails, weights of, 62
Railway time, 93
Rate of :
change of amperes per sec, 113
doing work, 79
energy, 79
heat production, 26
increase in velocity, 87-88
increase in angular velocity, 88
Ratios, described, 7
Reactance, 98, 100
physical, 22, 24
capacity, 22, 24
magnetic, 22, 24
imits and relations, 99-100
Ream, 168
R^umur degrees, reduction, 150
scale to others, 151-157
Rebele, Alexandria, 55
Reduction factors, tables, 30-173
do., text, 27
Reducing units in formulas, 6
Register ton, shipping, 54
Reichsanstalt, ampere of, 113
ohm of, 99
volt of, 110
Reis, Brazil, 164
Portugal, 165
Relations:
of units, tables, 30-173
do., text, 1
do., elec. and mag., 96
between quantities, 13
Shysical quantities, 17
uctance, magnetic, 129
do., physical, 20, 21
specific, 130
do., physical, 20, 21
Reluctivity, magnetic, 130
do., physical, 20, 21
Resilience, 19
Reals tance» electric:
table, 99
text, 97
physical, 22, 24
C.G.S. unit. 99
specific, table, 103
do., text, 101
do., physical. 22
and cross-section, 101
and length, 101
units and relations, 99
Resistance, magnetic, 129
do., physical, 20, 21
do., specific, 130
traction, 78
Resistivity, table, 103
text, 101
physical, 22, 24
C.G.S. unit, 102, 103
copper, 102, 104
mercury, 102, 104
magnetic, 130
Revolution, 89
per hour, 86
per minute, 86
per minute per minute, 88
per minute per second, 88
per second, 86
per pecond per second, 88
Rhineland foot, 33
Ri, Japan, 34
Richtpfennig, Germany, 60
Right angle. 89
do., spherical, 89
Rod, 32
Rod, square, 43
Rod, volume, 54
Roman lengths, 34
Rood, 44
Roquille, France, 54
Rotary speeds, 86
Rottle, Palestine, 172
Syria, 173
Rottoli, oriental, Naples, 61
Royal cubit, ancient, 34
Rubbio, Rome, 55
Ruble, Russia, 165
Rupee, Ceylon, 164
India. 165
192
index:
Russian lenfi:ths, 33
Ruthe, Austria, Germany, Holland,
Sweden, 33
Ruthe sq., Germany, 44
S
Sacco, leghorn, 55
Sach, Rotterdam, 55
Sack, 53
Amsterdam, 55
Sacred cubit, ancient, 34
Salm, Malta, 55, 172
Salme, Sicily, 55
Saschehn, Russia, 33
square, Russia, 44
Scale of maps and drawings, 168
Schachtruthe, Germany, 54
Schalpfund, Sweden, 61
Scheffel, Germany, 54
Schiffslast, Germany, 60
Schiffspund, Sweden, 61
SchifFstonne, Austria, 61
Score, 168
Scruple, apothecary, 59
fluid. 52
Scrupule, France, 61
Se, Japan, 44, 172
Sea mUe, 32,
Secchio, Venice, 55
Sec-ohm, 110
Second, angle, 89
length, 31
time, 94
sidereal, 94
Section, of land, 44
Seer, Bengal, 61
India, 172
Seidel, Austria, 55
Seki, Japan, 55
Self -inductance, 118
Self-induction, 118
physical, 23, 25
Semi-circumference, 89
Sen, Japan, 165
Setier, Geneva, 55
France, 54
Shaku, Japan, 34, 172
Sheets, weights of, 63
Shilling, Great Britain, 165
per mile, 166
per pound, 167
per ton, 167
Sho, Japan, 55, 172
Short ton, 58
do., see also Ton, short
Sidereal day, hour, minute, month,
second, year, 94
time, defined, 93
Siemens-unit, 99
defined, 98
Silver, atomic weight, 125
electrochemical equiv., 125
Sine, grades, 91, 92
Skalpund, Sweden. 61
SkiUing, Norway, 165
Skrupel, Germany, 60
Slopes, 90
Sol, Peru. 165
Solar c^cle, 95
Solar time, 93
Solid angle, 89
physical, 18
Solid yard, 54
Solotnick, Russia, 61
Sovereign, Great Britain, 165
Sou, France, 165
Span, 31
ancient, 34
Spanish lengths, 34
Specific:
conductance, 106
do., physical, 22, 24
gravity, 67
do., physical, 18
do., weights and volumes from, 69
heat, 26
do. of electricity, 23, 25
do. of water, 72, 171
do. as a relation, 3
inductive capacity, elec., 23, 24
do., magnetic, 20, 21
magnetic reluctance, 130
magnetic resistance, 130
permeance, 131
reluctance, 130
do., physical, 20, 21
resistance, table, 103
do., text, 101
do., physical, 22, 24
Speeds, 85
rotary, 86
Spermaceti candle, 145
Sphere, 89
Spherical candle, 147
hefner, 147
right angle, 89
Spon, Sweden, 55
Square:
Duilding, 43
centimeter, 42
do. to sq. inches, digit table, 44
chain, 43
decimeter, 42
foot. 42
do. to sq. meters, digit table, 44
inch, 42
do. to sq. cm., digit table, 44
ken, Japan, 44
kilometer, 43
meter, 43
do. to sq. feet, digit table, 44
do. to sq. yds., digit table, 44
mil, 41
mile, 43
millimeter, 41
miriameter, 44
INDEX.
193
Square :
myriameter, 44
perch, 43
pole, 43
rod, 43
yard, 42
do. to sq. meters, digit table, 44
Stadium, ancient, 34
Stajo, Corsica, Leghorn, Naples,
Venice, 65
Standard candles, table, 146
do. defined, 145
Standard cells, 108
do., temperatiire corrections, 111
Standard electric lamp, 145
Standard time, 93
Standards, length, 29
volume, 45
weight, 56
see also U. S. standards
Stange, Sweden, 33
Starelli, Sardinia, 55
Stari, Austria, Florence, 55
Statute mile (see Mile), 31
Steradian, 89
Stere, 54 ^
Stone, British, 60
Stoof . Russia, 55
Stoop, Amsterdam, Antwerp, Swe-
den, 55
Stored energy, electrical, 123
Strength of pole, 137
do., physical, 20, 21
Stress, electric, 108
Stress per unit area, 63
Struck bushel, 53
Stubgen, Germany, 54
Sucre. Ecuador, 164
Suerte, Uruguay, 173
Sun, Japan, 34, 172
Suppressed factor, 12, 13, 14, 25, 26
Surfaces, 41
digit tables, 44
physical, 18
British to U. S., 41
foreign, 44
and forces, 63
and weights, 63
density, 22, 24
tension, 62
physical, 18
Susceptance, 105
physical, 22, 24
Susceptibility, 132
physical, 20. 21
Swedish lengths, 33
Swiss lengths, 33
Symbols, text, 28
tables, ix
physical quantities, 18
Synodic month, 94
System, absolute, 11
C.G.S., 11
Systfemei ancien, French. 33, 54 61
Ssnstfeme, usuel, French, 33, 55, 61
Tables:
conversion factors, tables, 30-173
do., digit tables, see Digit
do., text, 27
physical quantities, 17
Tabriz, Persia, 173
Tael, China, 164, 166
Cochin-China, 172
Tan, Japan, 44, 172
Tangent, grades, 91, 92
Tame, Algiers, 55
Telegraph line measures, 34
Temoli, Naples, 55
Temperature, 2Q
physical, 18
corrections, standard cells. 111
dimensions of, 13, 26
scales, 150-163
Ten to the nth power, 8
Tension, film, 62
Terze, 31
Testoon, Portugal, 165
Thermal quantities (physical), 26
Thermal unit, 75
do. defined, 74
do. to foot-pounds, dig. tab., 77
do. per minute, 81
Thermoelectric height, 23, 25
Thermometer scales, 160-163
Thomson's law, 128
Tierce, 53
Time:
tables, 94
text, 93
physical, 18
mean solar, defined, 93
sidereal, defined, 93
standard, railway, 93
and volume (discharges), 96
constant (electric), 120
do., physical, 23, 25
To, Japan, 55, 172
Toende, Copenhagen, 55
Toesa, Spain, 34
Toise, 54
France, 33
Tomans, Persia, 165
Tomines, Spain, 61
Ton:
tables, 58
text, 57
bloom, 60
fross, see Ton, long
o., displacement of water, 54
long or gross, 58
do. to metric tons, digit table, 59
do. vs. short, 57
do., volume of water, 71
do., per cubic yard, 69
do., per square foot, 66
do., per square inch, 67
metric, 58
do. to long tons, digit table, 59
do. to short tons, digit table, 69
do., volume of water, 71
do , per cubic meter. 69
194
INDEX.
Ton. metric, per sq. centimeter, 67
do., per year, 127
do., kilometer, 78
net, see Ton, short
register, 54
shipping, 54
short or net, 58
do. to metric tons, digit table, 59
do., volume of water, 71
do., per cubic yard, 60
do., per mile, 78
do., per square foot, 66
do., per square inch, 66
do., per year, 127
do., -mile, 78
Ton-kilometer, 78
Ton-mile, 78
Tonde. Denmark, 172
Tondeland, Denmark, 172
Tonelada, Spain. 61
Tonne, 58
Tonneau, 58
Torque, tables, 74, 78
defined, 72
physical, 19
units, 78
vs. energy, 13, 72, 78
Tortuosity, 18
Township, 44
Traction coefficient, 78
Traction energy, 78
Traction resistance, 78
Tractive effort, 78
Tractive force, 78
Tropical year, 95
Troy weights, 57, 59
do. defined, 57
True ampere, 113
defined, 112
True coulomb, 116
True ohm, 99
defined, 98
True volt, 110
defined, 109
Tscharky, Russia, 55
Tschetvertaks, Russia, 165
Tschetwerik, Russia, 55
Tschetwerka, Russia, 55
Tschetwert, Russia, 33, 55
Tsubo, Japan, 44, 172
Tsun, China, 172
Turn, Sweden, 33
Tum, cubic, Sweden, 55
Tun, 53
Tunna, Sweden, 55, 173
Tunne, Germany, 54
Sweden, 55
Tunnland, Sweden, 44
U
Uncia, ancient, 34
Un^, Switzerland, 61
Unit, circular, defined, 41
Unit current-turn, 133
do. per centimeter, 137
Unit pole, magnetic, 138
Unit-pole-centimeter unit, 142
Units, see under respective names
absolute system, 11
absolute vs. concrete, 12
C.G.S. system, 11
changing — in formulas, 6
concrete vs. absolute, 12
compound names of, 4
inter-relaticm of, 1
three groups of, 1
vs. quantities, 4
U.S. standards, 29, 45, 56, 98, 108,
112, 115, 117, 119, 122, 124, 166
U.S. to British, volumes, 46
Unze, Germany, 60
Useful numbers, 170
Valency, 125. 126. 129
Vara, California, 31
Argentine, Cent. America, Chile,
Cuba, Curagoa, Mexico, 172
Paraguay, Peru, Venezuela, 173
Spain. 34, 173
Varying functions, 97
Vector potential, 22, 24
Vector quantities 96
Vedro, Russia, 173
Velocity:
angular, 86
do., physical, 19, 23
do., frequency, 121
do., rate of increase of, 88
concrete units, 86
light, 86
do. as a relation, 11, 14, 21, 24,
25, 96
linear, 85
do., physical, 19
do., rate of increase of, 87
molecules, 86
Velte, France, 54
Venezolano, Venezuela, 166
Vergees, Isle of Jersey, 172
Verst, Russia, 33
Vierling, Germany, 60 .
Viertel, Antwerp, 55
VioUe. 146
defined, 145
Vis- viva, 72
do., physical, 19
INDEX.
195
Vlocka, Russian Poland, 173
Volt:
tables, 109
text, 108
applied, 110
electro-chemical energy, 129
induced. 110
international, 110
do., defined, 108
legal, 109
Beichsanstalt, 110
relations to other units, 110
standards, defined, 108
true, 110
do., defined, 109
to calories, 129
-ampere, 80
-coulomb (joule), 74
Voltage, 108
of decomposition, 128
do., calculation of, 129
Vorktum, Sweden, 33
Volume:
table, 46
text. 45
fundamental standards, 45
digit conversion tables, 51
physical, 18
foreign, 54
U.S. to British, 46
water, 69, 71
and mass, 67
and time, 95
weights, 67
from specific gravities, 69
W
Water, flow of, 95
foot of, pressure, 65
meter of, pressure, 65
pressures of, 63
specific heat of, 171
volimie of, 71
weights, 70
TFatt:
table, 80, 125
defined, 124 •
relations to other units, 125
magnetic power 144
per candle, 149
-hour, 76. 123
defined, 122
per gram, 126
per minute. 81
per second, 82
-second (joule), 74
Wave-length 31
Waves, periodicity, 86
Weber, 138
Weddras, Russia, 55
Wedro, Russia, 55
Week, 94
Weight:
table, 57
text, 56
fundamental standards, 56
digit conversion tables, 59
Ehysical, 18
ars^ 62
coatmgs, 63. 126
forces, 83
deposits, 63, 126
foreign, 60, 172
materials, 67
rails, 62
relative, chemical, 60
sheets, 63
water, 69-71
wires, 62
and length, 62
and measures, tables, 30-173
do., text, 27
and surface, 63
and money, 167
and volume, 67
from specific gravities, 69
Werschock, Russia, 33
Werst, Russia, 33
Weston cells, defined, 108
do., voltage of, 110
do., temperature correction, 111
Wey, 54
Winchester bushel, 53
Wires, weights of, 62
Wispel, Germany, 54
Work:
tables, 74
digit conversion table, 77
text, 72
electrical, 122
do., units, 74, 123
magnetic, 143
physical, 19
rate of doing, 79
Tard: table, 30
to meters, digit table, 39
cubic, 50
do., to cb. meters, digit table, 51
solid, 54
square, 42
do., to sq. met., digit table, 44
Year (solar), 94
calendar, civil, common Julian,
lunar sidereal, 94
an.-imalistic, Gregorian, legal, nat-
ural, tropical, 95
Yen, Japan, 165
196
INDEX.
Z
ZehnXing, Germany 60
Zent, Germany, 60
Zoll, Austria, Germany, Switzer-
land, 33
Zorzec, Poland, 55
X, as an angle, 89
ic, useful functions of, 169
10 to the nth power, 8
% defined, 7
% grades. 90
o/oo defined, 8
Voo grades, 90
- (hyphen) in names of unitSi 4
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