NBS MONOGRAPH 56

Systems Of

Electrical Units

U.S. DEPARTMENT OF COMMERCE

HATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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ERRATA TO ACCOMPANY NATIONAL BURBIAU OF STANDARDS

MONOGRAPH 56, SYSTEMS OF ELECTRICAL UNITS

Page Column Line Now reads in part Should read

15

14

16

22

2 ft • « « « •

2

1

1

eQ 4»4»1» a • «

(25.....

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v.27

{A^}p=2,657W2

the henry/ampere ....

two times

four times

quantities

physical

the henry/meter

34.5

February 15,
I
UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS . A. V. Astin, Director

Systems of

Electrical Units

Francis B. Silsbee

Reprinted from the Journal of Research of the National

Bureau of Standards^—C. Engineering and Instrumentation

Vol. 66C, No. 2, April-June 1962

National Bureau of Standards Monograph 56

Issued September 20, 1962

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 30 cents

Foreword

As part of its mission to provide a basis for exact measurements in science

and industrj', the National Bureau of Standards from time to time publishes

survey material on the status of important areas of measurement. This publication sm'veys the various systems of units used in the hterature on electricity

and magnetism and briefly traces their historical development.

In electricit}- there has been a gradual evolution over the past half-century

both in the definition of units and in the experimental methods for realizing and

maintaining them in the laboratory. A significant change was the establishment by international agreement of the "absolute" system of electrical units on January 1, 1948 as announced in NBS Circular 459, Announcement of Changes

in Electrical and Photometric Units. Additional details regarding the change

were given in XBS Cii'culars C475, Establishment and Alaintenance oj the Electrical Units, and C531, Extension and Dissemination of the Electrical and Magnetic Units by the National Bureau of Standards. However, since these

pubHcations were issued, fm'ther uuprovements have been made m experimental procedures. Furthermore, smce the publication in 1916 of NBS Circular

60, Electric Units and Standards, and NBS Scientific Paper 292, International

System of Electric and Magnetic Units, preferences and plailosophies regarding

alternative systems of units have changed substantially. These latter developments are currently under discussion by such international standardizing bodies

as the International Electrotechnical Commission, the International Union of Pure and Applied Physics, and the International Organization for Standardization.

It is hoped that the survey included in the present pubhcation will serve

as a stimulus to the further development, understanding, and utilization of systems of electrical units.

A. V. AsTiN, Director.

iii

Systems of Electrical Units

Francis B. Silsbee

(January 19, 1962)

The various sj-stems of measurement, with their respective sets of units, used in the

literature on electricity and magnetism are described in detail. Their historical development is summarized. The manner in which each is derived from either of the two alternative points of view of the experimentalist and the theoretician is compared and contrasted. The desirability of recognizing both points of view in international standardization, particularly when discussing rationalization, is pointed out. The present status of the absolute measurements on which all electrical units are based is reported, and tables are included for the conversion of equations and numerical values from one sj^stem to another.

Contents

1. Introduction 1 2. Historical summary 2

3. Fundamental philosophies 4

4. Experimental approach 6

4.1. Nomenclature of units and standards 6

4.2. Measurement 9

4.3. Present experimental basis for electrical units 11

4.4. Experimental estabhshment of physical laws 13

4.5. Derivation of germane systems of measure- ment by the Realist 16

5. Theoretical approach 17

5.1. The mathematical model 17

5.2. Coherent abstract units 19

1. Introduction

This paper has been prepared with several objectives in mind. The first is to provide a definite

account of the authorities and procedures on which

measurements of electrical and magnetic quantities are currenth' based. A second is to ofi^er a nomenclature in the field of units and standards which is fairly consistent with current usage and which if generally adopted would minimize semantic confusion in the field. A thu'd is to provide a brief

historical survey to record the successive steps in

the evolution of the various systems of electrical

units, together with a systematic tabulation for converting equations and data from one system to another. A fourth objective is to reconcile the

current controversy which was triggered by the

1950 decision of the International Electrotechnical

Commission to recommend the use in the future

of a "rationalized" system of measurement.

The sharpness of this conflict is illustrated concisely by comparing the following statements:

(a) "1 oersted = 1,000 ampere-turns/meter."

(b) "The number of ampere-turns per meter =

1,000/4t times the number of oersteds."

Each of these statements has been made frequently

by scientists and engineers of recognized standing.

Their apparent contradiction is, in the author's

6. Systems of measurement and representation 20

6.1. Sizes of units 20

6.2. Nomenclature of systems of measurement. - 21

6.3. Development of MKSA systems 21

6.4. CGS systems 23

6.5. Practical and "international" systems 25

6.6. IMiscellaneous svstems 25

6.7. The "fourth unit" problem 26

7. Dimensions 27

8. Rationalization 29

9. Summary and conclusions 32

10. Appendixes 34

10.1. Tables (for table 6 see p. 29) 34

10.2. Chronology 38

10.3. Notation, glossary, and organizations 39

10.4. References 41

opinion, merely one particularly striking indication

of a very deep-seated difference in the points of view and resulting philosophies of two major classes of workers and thinkers in the field of physics.

Hence, the fourth objective of this monograph is to reconcile these contrasting philosophies by disentangling them as completely as possible, even at the

expense of a possibly excessive amount of circumlocution and repetition in the text. In the development of this paper, a brief historical

summary will meet the third objective; the major

division of the two philosopMes will then be outlined, the basic prmciple of the first (experimental)

philosophy being also appropriate for introducing

digi-essions to cover the first and second objectives.

The contrastmg theoretician's philosophy will then

be described on the foundation of the quantity cal- culus. The various systems of equations and imits

in the electrical field" will then be listed and compared. This will be followed by a brief discussion

of the subject of dimensions from both points of

^-iew. The process of rationalization as seen from

the theoretician's point of view will contrast with

that described earliei-, and the suggestions of various

other writers who have recently attempted to cor- relate or reconcile the philosophies will be discussed.

Because of semantic pitfalls, the glossary (sec. 10.3)

gives in extenso the particular meanings with which

certain terms are used in this monograph.

1
2. Historical Summary

The concepts of quantities, units, standards (see

glossary), and their names and symbols constitute

in efYect an international language by means of which

workers in different countries or in different branches

of science exchange and compare their ideas and

experimental findings. Therefore, it is natural that a history of the development of these subjects

should consist mainly of a chronology of proposals

by individual workers, or by small groups, of s3'-stems

oi' units and names therefor and of actions by larger

national and international organizations accepting

or rejecting such proposals as parts of an international

language. In section 10.2 is given a sketchy chronology listing various milestones in these developments.

The early workers in the electrical field, especiaU}^

the telegraph engineers, made frequent use of ex- temporized standards of resistance and of voltage.

A table published in 1864 lists the conversion factors

between units defined by standards which range

from "25 feet of copper wire weighing 345 grains"

to "1 German mile (8,238 yards) of iron wire

inch in diameter," and include Siemen's "column of

mercury 1 meter long and 1 sq mm in cross section,"

as well as units defined in absolute terms as "10'

feet/second" or as "10 meters/second." The Daniell

cell was widel}'^ used as a standard of voltage until

the Clark cell appeared in 1872.

However, in his studies of terrestrial magnetism,

Gauss in 1833 had realized the possiblity and desirability of tying his results into the more permanent

and widely recognized system of mechanical units.

He invented "absolute methods" (see glossarj", sec. 10.3) for measuring magnetic moment and magnetic

field intensity. His colleague at Gottingen, W.

Weber, in 1840 extended the work to the measure- ment of current by the tangent galvanometer and

later by the electrodynamometer and in 1851 to the measurement of resistance [21].^ Gauss and

Weber used the millimeter, milligram, and second

as basic imits.

A major influence in the development of systems

for electrical measurement was exerted for almost

half a century by the Committee on Electrical Standards appointed first in 1861 by the Britisli Association for the Advancement of Science. It was active

from 1861 to 1870 and was reactivated from 1881

until it turned over its apparatus and responsibilities to the British National Phj'^ical Laboratory in 1912

[1]. Under the leadership of Professor William

Thomson (later Lord Kelvin), this group contributed

both experimental and theoretical points of view to the problem. In its first report, 1862, it recognized

as desirable qualities in the units that they:

(1) be of convenient size.

(2) bear a definite relation to the unit of work,

"the great connecting link between all physical measurements."

(3) bear a definite relation to other electrical units.

(4) be perfectly definite and not likely to require

correction or alteration from time to time.

1 Figures in brackets Liidicate the literature references at the end of this paper.

(5) be reproducible (a) in case the original standard were injured or (b) so that an observer

unable to obtain copies might be able to manufacture them.

They also were confronted with the following ex- perimental "facts of life" in the field of electrical

measurement:

(a) no artificial reference standard (see glossary)

is truly pennanent.

(b) errors in reproducing a prototype standard

(see glossary) are materially greater than the

errors arising in comparing two reference

standards of the same nominal value.

(c) errors in calibrating a reference standard by

an absolute measurement are usually even

greater than those encountered in reproducing a prototype standard.

(d) electrical units germane (see glossary) to either the meter, gram, and second or the

foot, grain, and second were very dift'erent in magnitude from the electrical quantities

of engineering interest.

The committee's response to this situation set the

pattern for all future developments. To secure

point (3) they immediately stated that "the material

relations between these units are, clearly, that a

unit electromotive force maintained between two

points of a conductor separated by the unit of

resistance shall produce unit current, and that this

current shall in the unit of time convey the unit

quantity of electricity." Also to secure the advantage of point (2) they unmediately recognized

the immense value of the work of Gauss and Weber

and set up basic absolute definitions initially germane

to the meter, gram, and second. They also initiated a program of absolute measurements, first of re- sistance and much later of current. The "BA unit

of resistance" which resulted in 1864 corresponded

to a mercury column 1 sq mm in cross section and

104.8 cm long, and hence was about 0.986 ohm as we now know it. To meet point (1) they recognized

a practical system purely for electrical quantities

defined as decimal multiples of the MGS units

which they fu'st used. In more modern language,

they chose 10^ CGS electromagnetic units of elec- tromotive force as the practical unit because it was

approximately equal to that of the Daniell cell, and

suggested the name volt for it. They chose 10^ CGS electromagnetic units of resistance with the

name ohm as the practical unit because it was

approximately equal to the Siemens Unit defined by

a column of mercury 1 m long and of 1 sq mm

cross section. They thus could meet requirement

(5) by specif3ang the proper length of such a column.

To meet point (4) in spite of fact (a) they initiated a program of studies on the stability of the resistance

of alloys. In recognition of (c) and (b) they constructed a considerable number of standard resistors

of the best known construction adjusted as closely as feasible to their "BA imit" and distributed

them internationally and by sale to the public.

(Faraday in 1865 was their first paying customer).

Another BA Committee on "The Selection and

Nomenclature of Dynamical and Electrical Units"

2
in 1873 decided to base theoretical definitions in both dynamics and electricity on the centimetergram-second (CGS) system rather than the metergram-second (MGS) system, mainly because in the

former the density of water is substantially unity.

It also urged the merits of the dynamical over the

gravitational units in mechanics, thus making the

gram primarily a unit of mass and not of force. They proposed the names dyne and erg and defined

the horsepower as approximately "7.46 erg-nines ^ per second." The reporter of this committee,

Professor J. D. Everrett, published in 1875 a little book "lUustrations of the C.G.S. System of Units."

This timely committee action gave such impetus to the CGS system that it has since come to have

widespread application in all branches of science

and engineering. It has almost met the pious hope

of its originators that their selection should "be so made that there will be no subsequent necessity

for amending it." The 3^ear 1881 saw the first of a series of international electrical congresses (see sec. 10.2) which for the next quarter of a century served as forums for the discussion of nomenclature, units, etc., and as authorities for the approval and promulgation of those ideas which proved acceptable. The 1881

meetmg in Paris approved of the basic status of the

CGS units, and of the parallel practical set with the names ohm, volt, ampere, coulomb, and farad. It

also set up a Commission which in 1884 recom- mended a legal ohm defined by a prototype mercury

column 106 cm long and 1 sq mm in cross section at 0 °C (i.e., approximately 0.9973 ohm).

By 1893 the 4th International Electrical Congress

at Chicago was able to crystallize the situation fur- ther by defining the ohm, ampere, and volt in terms

of both the decimal multiples of the CGS electro- magnetic units and also in terms of prototype standards. It passed a series of resolutions addressed to the various governments represented, urging them

to "formally adopt [them] as legal units of electrical measure." The prototype for the ohm was lengthened to 106.3 cm (equivalent to about 1.000 5 ohm).

The prototype for the volt was the Clark Zn-Hg cell to which was then assigned the value 1.434 v. The 6th International Electrical Congress in St. Louis in 1904 recognized the distinction between two

aspects of these developments. On the one hand,

there was an overriding necessity for prompt official and universal conformity in the sizes of the units used

in commercial measurements. This could best be

secured by cooperative governmental actions. On

the other hand, the improvement and invention of new and more useful nomenclatures and concepts

could best be fostered by providing a forum where

they could be discussed freely and by which the best

usages could be recognized and coordinated. Accordingly two separate resolutions were passed suggesting these two parallel lines of progress. In sequel

the first led through several intermediate steps to the

inclusion in 1921 of electrical units in the scope of

the International Committee on Weights and Meas2 See p. 20, footnote 13.

ures (ICWM) (see glossary), while the second led to the organization during the next few years on a per- manent basis of the International Electrotechnical

Commission (lEC).

The next major step in the first line of progress was the International Conference on Electrical Units

and Standards at London in 1908, attended by

official delegates from 24 countries. It recognized

the basic importance of the CGS systems of units

and their decimal multiples but also recognized that

their experimental realization by absolute measure- ment could not then be attained with the accul-acies

desirable for much engineering work. It therefore

recommended as representing these and "sufficiently near to them to be adopted for the purposes of

electrical measurements and as a basis for legislation" a separate system of "International Electrical

Units." The International Ohm defined as the re- sistance, at 0 °C, of a column of mercury 106.300 cm

long and weighing 14.4521 g, and the International

Ampere defined as the current which would deposit

silver from an aqueous solution of silver nitrate at a rate of 0.00111800 g/sec were basic units of this

system. In 1910 delegates from the British, German, and French national laboratories met at the

Bureau of Standards in Washington and experimentally intercompared their respective national standard

cells and resistors. From the results, values on a

unified basis were assigned to the various standards

and the units then arrived at were maintained [44, 46] as closely as practicable, disseminated throughout

the civilized world, and used in commerce, industry,

and science until January 1, 1948.

Pursuant further to the first St. Louis resolution,

the 6th International Conference on Weights and

Measures in 1921 voted to amend the Convention

of the Meter (of 1875) ^ to assume authority over

electric and photometric units.

In 1927 an Advisory Committee on Electricity was established to advise the International Committee on electrical problems, and the facilities at the

International Bureau were enlarged to enable them

to make precise comparisons of electrical standards.

Since 1931 (except in time of war) the International

Bureau has made intercomparisons of standard cells and standard resistors submitted periodically by the

various national laboratories. This enables each

laboratory to know how its units as mamtained

compare with those of the other nations, and to make

adjustments on the rare occasions when such may

become necessary to restore uniformity.

By 1928 many experimenters felt that the situation had changed since 1908. The availability of

calibration services from national laboratories had

eliminated the need for convenient reproducibility in protoptyes (desideratum (5)). Also, experience had

3 This multi-lateral international treaty establislied a self-perpetuating Inter- national Committee on Weights and Measures consisting of 18 scientists ap- pointed by reason of their individual competence but with the proviso that only one member be appointed from any one nation. This Committee supervises the work of the International Bureau of Weights and Measures which occupies

laboratories on a plot of internationalized territory in Sevres near Paris. The

operations of the Committee are reviewed and given formal approval by an

International Conference on Weights and Measures which normally meets

every six years and on which all nations signatory to the Convention of the Meter

I are represented.

3
shown that with modern techniques "fact (c)" was

no longer true * and that the errors in the absolute

measurement of resistance and current probably did

not then exceed those of reproducing the units by

using the prototype standards. By 1933 the 8th

General Conference approved in principle the change

back to absolute units and authorized the Inter- national Committee to proceed as fast as reliable

data became available. World War II, however,

intervened and it was not until October 1946 that

the International Committee voted to make the

change efl'ective January 1, 1948 [43]. Since January 1, 1948, the various national laboratories have continued to maintain their units by

groups of standard resistors and standard cells with

very satisfactory results, on the newly assigned basis,

and with the expectation of occasional revision in the

basis as better absolute determinations become available. The comparisons of 1957 at the International

Bureau after the lapse of almost a decade showed

that the units as maintained in Germany, the United

States, France, Canada, Great Britain, Japan, and

Russia (see glossary) all still lie within a range of a few microvolts and microhms.

The responsibility for the standardization of defi- nitions and nomenclature covered by the second

resolution of the St. Louis Congress has been borne

mainly by the lEC. The work has proceeded since

1904 at a necessarily more leisurely tempo and with

less precise discussions of detail. The two classic

CGS electrostatic and electromagnetic systems sufficed for Maxrwell's immortal Treatise of 1873, but

long before 1904 a number of improvements had been

proposed.

In 1882 Heaviside had complained of the presence

of a factor "47r"in many formulas as due to an unwise definition of the unit magnetic pole and in 1891

[111, 112] he initiated a vigorous campaign for the use of what he called a more "rational" system (see

sec. 8). His theoretically very elegant remedy

would have involved changing the legalized units by

factors involving "V47r" and therefore proved unacceptable to the practical engineer. Alternative

partial systems which avoided changes in the units

of voltage, current, or resistance, but at the expense

of changing the simple choice of unity for the permeabilit}^ of space were suggested, by Perrj^, Baily,

Flemming, Fessenden, and others [113, 114, 115,

116]. Kennell}' has used the adjective "subrationalized" to denote such schemes.

Another improvement on the CGS system which

results in a desirable symmetry in the coefficients of

electric and magnetic quantities is usually called the

"Gaussian" system and was used by Foppl in 1894.^

In theoretical developments it is often desirable to express the dimensions (see sec. 7) of electrical quantities in terms of four basic dimensions rather than

three. Consideration of these possibilities led Giorgi to

offer, initially in 1901 [51, 53], a "package deal" in

* It Is Interesting to note that "fact (c)" still holds for temperature measure- ment and that the theoretical Kelvin Thermodynamic Scale still has to be supple- mented by the more reproducible International Practical Temperature Scale.

* Lorentz [52] refers to this as "associated with Gauss, Helmholtz, and Hertz."

the form of the MKS system. This gives rationalization, symmetry, four basic units (to which dimensions can be assigned), freedom from memorizing a large

number of decimal exponents 10^, 10*, 10"^ etc., and

the possibility of a single system applicable to all branches of science while retaining the firmly en- trenched practical electrical units (ohm, volt, . . . ). Unfortunately the cost of the package includes using

germane units of density and permeability in terms

of which water has a density of 1,000 and air a magnetic permeability of 47r-10~^. This major proposal

naturally stimulated a great deal of discussion and

during the ensuing half century received a gradually increasingly favorable response, primarily in the field of electricity.

At its 1930 Oslo meeting the lEC indulged in a very protracted discussion, apparently resulting from

a confusion between the "dimensions" of mathematical variables and the inherent "kinds" of pln'sical

quantities. It ended by voting that B and H are

different in nature and that Tm (see glossary), the

"permeability of space," has physical dimensions.

In 1935 it voted "that the system with four fundamental units, comprising the three units: metre,

kilogramme, second and a fourth fundamental unit

to be chosen later be adopted under the name Giorgi

system." In 1938 the lEC recommended "as the

connecting link between the electrical and mechanical

units, the permeability of free space with the value

1X0=10"^ in the unrationalized system, or /io=47r-10~^

in the rationalized system."

In 1950 the lEC took the final step and recom- mended the use of the MKS system with the equations in their rationalized form as suggested by

Giorgi. It also resolved "that for the purpose of

developing the definitions of the units the fourth

principal unit should preferably be the ampere."

The 1950 action of the lEC served to trigger off a further protracted discussion on the proper interpretation of rationalization. The S3rmbols, Units,

and Nomenclature (SUN) (see glossary) Committee

of the International Union of Pure and Applied

Physics (lUPAP), consisting as it does largely of

theoreticians, promptly (1951) voted that "in the

case that the equations are rationalized, the rationalization should be effected by the introduction of

new quantities" [8]. In the lEC, however, the

experimentalists who prefer to change units are also

represented and long arguments in Philadelphia

(1954) [11], Opatija (1956), Stockholm (1958), and

Madrid (1959) have failed to bring agreement. It

is the hope of the author that this paper may contribute to the reconciliation of the two groups.

3. Fundamental Pliilosophies

The quantitative development of electromagnetism, like that of any other branch of science, has

been marked by the interaction of two distinct,

though complementary, kinds of work: experunental

operations in the laboratory and theoretical studies

applying mathematical reasoning. The interplay

4
between these processes has been very close and has

proved very fruitful. The experknentalist has hit

upon new phenomena and recognized the need for new concepts in terms of which to describe them.

The theoretician thus stimulated has sharpened the

definitions of his concepts, discovered possible rela- tions between them, and suggested further experiments to confirm and extend such predicted relations.

During the development of the science each type of

worker has evolved an ever more useful and powerful

set of tools both in the form of laboratory apparatus

and of mathematical methods. In this process even

the basic concepts have been modified, not only by

the inclusion of new ones, but also by changes in the

definitions of certain old ones. A major step suggested long ago by Heaviside but only recently

receiving official recognition in this evolution is called "rationalization" and involves the deliberate

changing of the coefficients conventionally used in

certain equations of electromagnetism. Unfortunately it is often described by the misleading phrase

"use of rationalized units." It is this step which

has brought into prominence a situation which has

existed throughout the development of the science

but which has hitherto been safely disregarded.

This situation is that the experimenter and the

theoretician, in spite of their effective cooperation,

have each developed his own specialized nomenclature which is different in some of its connotations

from that of the other although he uses the same

words. To explain the semantic situation more

clearly the following sections will expand in more

detaO. the two distinct points of view and their

resulting connotations.

To apply the power of mathematics to any branch

of science, the physical relationships involved are best put into the form of equations. There are two

ways of doing this. The first way starts with measurement. The

natural phenomena are conceived as describable in

terms of a number of definable and measurable

physical quantities. These taken together constitute

what may be called a physical model of Nature. A

particular sample of each kind of quantity (see

glossary) involved in the phenomena under study is selected arbitrarily as a physical unit. Operations

are developed by which other examples of the same

kind of physical quantity can be compared with the

physical unit. The result of this operation is a number called the "measure" or the "numerical

value" of the physical quantity in terms of the

physical unit. The numbers thus obtained by

measurement are then written into equations which

express the way in which the measures of certain

dependent variable physical quantities depend on

the measiires of other independently controlled

physical quantities. By the algebraic manipulation

and combination of such measure equations a complete science can be built up. The second way is to construct a mathematical

model which has a certain correspondence at many

points with the phenomena studied. The model

consists of a number of kinds of mathematical

elements which will here be called "symbolic quantities" (see glossary). One element of each kind is assigned a measure 1 and called a "symbolic unit."

The equations relating these symbolic quantities in general look like and correspond to the measure equations obtained in the first way, but the letter symbols in the equations represent the symbolic quantities

themselves. Such equations are called "quantity

equations" and have much to offer in mathematical

elegance and convenience.

At first sight there appears to be little difference

between these two ways of introducing mathematics.

In any one system of units and equations, the rela- tion between each symbolic quantity and unit of the

mathematical model and the corresponding physical

quantity and unit in the physical system being

studied is indeed very close. As a result both the

physical quantity and its mathematical model are customarily given the same name (e.g., "electric

current") and their units are given the same name

(e.g., "ampere"). In a great many circumstances

there is no occasion to distinguish between them.

However, when, as in this paper, one is concerned

with more than one set of equations or of units, the

correspondence between the model and the reality

is in general different for the different models.

Failure to distinguish between the mathematical

model and the physical model in such cases has been

the basis of a great deal of confusion and misunderstanding.

For this reason in this paper the distinction between

the two "levels of abstraction" will be carried to an

extreme, and probably unnecessary, extent by the

frequent insertion of the adjectives "physical" or "symbolic" (see glossary) to designate respectively

the actual physical quantity and its corresponding

element in the mathematical model. Also following

Konig [88], who early realized this basic cause of

confusion, the words "Kealist" and "Synthetiker"

(see glossary) will be used to emphasize the distinction

in the two philosophies. The words as here used

represent the extreme ends of the spectrum. Any

living scientist or engineer thinks and speaks sometimes like a Realist and sometimes like a Synthetiker.

No harm results even if he applies both types of

thinking to the same problem, provided that at each instant he is aware of which type he is using.

However, when he slips unconsciously from one type

of thinking to the other or when two members of an

international committee are simultaneously thinking

in different types, then trouble is sure to develop.

The Realist who thinks only in terms of physical

quantities and units and considers aU his equations

to be measure equations, in general exemplifies the

operating engineer, tester of materials, writer of

specifications, metrologist, laboratory experimenter,

or measurer of the constants of nature. The Synthetiker who thinks only in terms of symbolic quantities and units and considers all his equations to be

647596—62 2 5
quantity equations, in general exemplifies the college

professor, textbook writer, or theoretical physicist.

It is interesting to contrast the backgrounds and

motivations of the men who hold these contrasting

points of view. The Realists deal experimentally

with energized electrical apparatus in tlie laboratory,

the powerhouse, and the industrial plant. Through

long familiarity they come to attribute to properties

like current, inductance, magnetic field strength as much realit}'' as to their machinery and raw materials.

They quite overlook the fact that these electrical

quantities are in truth only artificial concepts in- vented for convenience in describing the natural

phenomena concerned. They must deal with a wide range of magnitudes, from microvolts to megavolts, and find it convenient to use a plurality of

non-germane physical units in expressing their

measvu'ed results. Also, in English-speaking countries they must frequently shift results between the

British and the Metric systems. Hence they very

frequently apply the basic principle that; "the measure is always inversely as the unit" and have come to regard it as fundamental in the science of

measurement. They therefore cling to it not only

when the change is (1) from one non-germane (see

glossary) unit to another in the same system or (2) between the germane units of two systems in which

the equations are identical but the basic units differ,

but also (3) even when the change involves a change

in the coeffi-cients in their measure equations.

As they are content to write only measure equations, they are quite willing to forego the use of

quantity equations and the use of letter sjonbols to denote their physical quantities. These are small

prices to pay for the universality of the principle

that the measure is inversely as the miit, and for the

comfort of thinking (albeit mistakenl}") that they

deal with "real" quantities.

In contrast, the Synthetikers realize that both

they and the Realists are dealing with conceptual

artifacts. With their mathematical background

they readily conceive of their symbolic quantities as defined by the equations of the system. They

seldom have any use for units, but when they do

they recognize the neatness of a set of symbolic

coherent units (see glossary), each defined merely

by the dimensions (see glossary) of the quantity

involved, together with a few basic symbolic units.

They rarely use noncoherent units and rarely have

occasion to translate a measure from one set of units

to another. The sacrifice usually made of the

universality of tlie inverse law relating measure to

unit, is a very small price to i^ny for the elegance of

the quantity-calculus with its complete independence

of units.

As the Sjmthetiker group is the more articulate of

the two and has already provided most of the litera- ture on systems of electrical units, the arrangement

of the present paper has been to give first the whole

picture from the side of the Realist to illustrate how

complete and effective his approach can be. Then,

in the interest of fairness, the Synthetiker's side with

its neat elegance is given as a climax.

4. Experimental Approach

In presenting the situation from the point of view

of the Realist, it seems advisable first to review in some detail the language of the laboratory. Using

the terms there defined, the logical basis of experimental measurement will then be sketched and

illustrated, with a detailed digression to give an

up-to-date picture of the current basis for electrical

measurement. To demonstrate the basic logic of

the Realist, his process for establishing phj^sical

laws by purely experimental methods is then illus- trated. In tables 1, 2, and 3 (sec. 10.1) the overall

results of such operations are fonnally tabulated.

Certain warnings as to the mathematical handling

of a Realist's results are followed by an outline of the

Realist's process for deriving formal definitions for any of his germane systems of measurements.

4.1. Nomenclature of Units and Standards

Before outlining the point of view of the Realist,

let us first review the vocabulary he uses to describe

his operations by using words such as those italicized

in the following paragraphs. He thinks of a physical

quantity as an example of a measurable (and therefore definable to some desired degi'ee of precision)

physical property which possesses the attribute of

magnitude as well as of kind. The unit ("physical"

in our nomenclature) is a sample of a physical

quantity selected arbitraril}^, but usually not capriciously, for the purpose of measuring other

physical quantities of the same kind. Measurement

is the act of comparing the magnitude of the measurand (the physical quantity the magnitude of which

is to be measured) with the magnitude of the unit.

The number resulting from this act is the measure

(or numerical value) of the measurand in terms of

the unit and is always a numeric.

A physical standard is a physical system of such

a nature that it embodies in definite and usualh'' convenient form one or more examples of one kind

of physical quantity, and to which a value (or values)

has been assigned to indicate the measure of the embodied quantity in terms of some appropriate

specified unit.

For any given physical quantity there is usually

a large assortment of dift'erent units. This situation

is the result of many factors including convenience,

historical accident, the particulate nature of matter,

and especially the numerous different attempts

which have been made to secure the advantages

which result from the existence of simple systematic

relations among the units of different kinds of quantities. In the various proposed logical systems of

measurement, the units of a few quantities are

selected as basic units and defined in terms of

artificial or natural standards. The units of the

remaining quantities are called derived units and

are defined by operational procedures by which a value in terms of each new derived unit is assigned

to each standard embodying one of the remaining

quantities.

6
Thus as units of electric charge we have among

others the statcoulomb (esu). the coulomb, the

ab coulomb (emu), the milhcoulomb. the electronic

charge, the f araday, and the ampere-hour. The rela- tions between the magnitudes of these units are known in some cases by definition and in others as the result of experiment. For each unit there is one

ideal magnitude fL^ed by reference to the definition

of the unit. However, in actual laboratory operations this ideal is approached only asymptoticaUy

as experimental methods are refmed. One must

therefore recognize the existence at any particular

time in any given laboratory of a unit maintained

in that laboratory at that time, which in general is not exactly equal to the ideal. Thus, in 1950. the

magnitude of the ohm as maintained by the British

Xational Physical Laboratory was smaUer by 2.2

li ohm than the magnitude of the ohm as maintained

at the U.S. Xational Bureau of Standards. The

resistors used in the comparisons between the two

laboratories showed no difference as great as 0.1 jLtohm between their values before and after their two

crossings of the Atlantic for the comparison. Hence

the observed difference in the two units is probably

real, but who is to say which magnitude is closer to the ideal ohm? The units as maintained at some

small laboratory in a imiversity or industrial factory

may weU depart much more widely from the ideal. In addition to the differences resulting from the

unknown and unavoidable inaccuracies in measurement, other, and usuaUy larger, differences in units

have been produced on certain occasions by the

formal actions of iatemational standardizing bodies.

An example is the decrease in the ideal magnitude

of the ohm by 490 /xohm effected January 1, 1948 on

the recommendation of the General Conference on

Weights and Measures. Such dehberate changes

are made only at relatively long intervals and are usuaUy signaled by a change in an adjective in the

formal name of the unit. Thus the "Legal Ohm"

of 1884 was followed in 1893 by the "Ohm/' in 1908

bv the "International Ohm." and in 1948 bv the

"(Absolute) Ohm."

The word standard is also used with a variety of

meanings both as a notin and as an adjective. Its use as a noun to designate a physical standard (as

distinct from printed standards of practice or of safety) should preferably be limited to physical

objects or systems which are used or intended for use in the definition or maintenance of a unit and for the cahbration of other instruments or measuring

devices in terms of that unit. A shop or laboratory

instrument, even though of very high accuracy, h

used in everyday operations to measure physical

quantities should not be designated a standard.

However, a measuring device may compare an un- known measurand with some known quantity in a

physical system which temporarily serves as a standard. Also a term like standard resistor is preferable

to st-andard of resistance or resistance standard,

because it stresses the fact that the complete physical

structure (ahoy wu-e, terminals, supports, etc.) is meant.

Physical standards are used for a variety of ptu-poses and a correspondingly large variety of adjectives are apphed to the noun standard to describe

these uses. The adjective prototype designates

members of that very small group of standards which

serve to define the basic units of a system of measure-

: nient (see glossary). On the assumption that; the

I whole world now uses only measuring systems based

i on the 'Tnternational System of Units" ° fLxed in

i 19oS by the International Committee for Weights

and ]NIeasiu-es, there currently exist prototype

standards for only five kinds of quantity. These

include one individual artifact, tlie International

I Kilogram preserved at Sevres, to the mass of which

' is assigned the value 1 kg in the International

System: the wavelength in vacuo of the orange-red

hue of ki'ypton S6. to which is assigned the value

1/1.650,763.7.3 m: the tropical year of the earth-sun

svstem, to which for 12h Ephemeris Time of January

O', 1900' is assigned the value 31,556,925.9747 sec: the

temperature of the triple point of water, to which is assigned the value 273.16 °K: and the luminous

intensity per scpare centimeter of a blackbody at

; the melting point of platinum, to which is assigned

the value 60 candelas. There is obviously only one

prototype standard each of mass and of time, while

there are in existence as many protot^'pe standards

of length, temperature, and luminous intensity as may happen to be set up and used for standardizing

purposes at any giA^en time. Of coiu'se if some

measurement laboratory is operating in such complete isolation that it is obliged to establish its units

cpite independently of the present group of cooperating national and international laboratories, the

standards which define its basic units will also be

properly designated as "prototypes." Huntoon

and Fano [45] have suggested the possibility that all prototype standards may ultimately be selected

properties of atoms or molecules rather than of macroscopic bodies.

It may be noted here that except for the special case of the prototype kilogram the value assigned to a standard need not be 1 unit and may be very

different. Even when the standard is constructed

with the intention that its nominal value shaU be one

unit and hence that it should embody a quantity the measure of which is exactly 1. errors in manufacture

or subsequent changes usuaUy cause its measure to depart slightly from unity. Of com'se, when the

definition of a unit is changed, as in 194S, the

assigned values of all standards of that kind should be

5 Care must be taten to distinguish for example between (1) tbe "ampere" or "absolute ampere" introduced efiecHvely Jan. 1, 19iS, defined by an electro- mechanical esperiment, and constituting one of the basic units of rhis SI (System

Internationale') and (2) the older and now obsolete "International Ampere"

defined bv the Ix)ndon Conference of 190S by means of the silver conlometer

(see p. 25)". A similar distinction is needed for the other electrical units.

7
changed to correspond to the new unit even though

there has been no change in the magnitude of the

quantity enihodied hy the standard. When the

magnitude of the quantity embodied in a standard

has been found to have drifted with time, the

standard must be assigned a new value.

The process of making appropriate measurements on a standard on which to base a correct assignment

of a value is called a calibration of the standard. All

standards except prototype standards must be

calibrated in some way.

This need has caused the development throughout

the civilized world of a hierarchy of standardizing

laboratories. Each national laboratory (see glossary)

maintains a set of units by means of its national

standards. Periodic intercomparisons at the Inter- national Bureau of Weights and Measures help

coordinate the activities of the national laboratories

and enable them to achieve close agreement among

the electrical units as maintained by them. Within

each nation other laboratories have their standards

calibrated at the national laboratory and in turn use

their standards to calibrate other standards and

measuring equipment.

Within any one laboratory there is also a hierarchy

of standards. The highest in rank are preferably

called "reference standards" and serve to maintain

the corresponding unit in the laboratory. Calibrated by reference to them are the working standards which are regularly used to calibrate the shop

instruments and measuring devices used in the everyday work of the main organization. Another categor}^ is that of interlab oratory standards, which are those sent periodically to the national laboratory or other source of high accuracy and which then serve

to bring the magnitude of each unit to the given

laboratory. In some cases some of the reference

standards are used as interlaboratory standards,

but in other cases it is best to spare the reference

standards from the disturbances incident to trans- portation and to count on the statistical accmnulation of data by the repeated round-trip shipments

of a rugged interlaboratory standard to build up a high accuracy in the final assignment of a value to the undisturbed reference standard. The adjectives

traveling (voyageur) and sedentary (sedentaire) are used by the International Bureau and sometimes by

others to designate these two uses of standards.

In another categorj^ of standards are the transfer

standards, which are of specialized construction so that under widely var3dng conditions of use they

continue to meet the criteria required for defining

the quantities which they embody; or, alternatively,

experience only a definite and known change in value for which an accurate correction may be made.

The most common example is the standard transfer

wattmeter, which is so constructed that its deflection

for a given active power is the same on alternating

current as on direct current. Other examples are

resistors which have the same resistance on alternating and on direct current; attenuators which can be

calibrated by d-c resistance measurements and used

to produce known attenuation in a-c circuits; and

resistors capable of carrjang very large currents. Many standards embod}^ only a single example of

the quantity concerned and are called single-valued;

examples are gage-blocks, standard cells, most

standard resistors. Others embody a plurality of

examples of quantities of the same kind and are

called multi-valued standards. Examples are graduated scales, decade-type resistance boxes, or capacitors. Still other standards like continuously adjustable air capacitors or inductors may be set to embody any desired value of the quantity within

their range mth a precision limited only by the

readability of their scale and mechanical imperfections in their construction. These are preferably

called continuously adjustable'' standards.

The word standard is also conveniently applied

either as a noun or as an adjective to a class of usually more complex measuring devices often called standard

instruments which are used in much the same way

as simpler physical standards. Typical examples

of such instrumental standards are thermometers,

floating hydrometers, and electrical indicating in- struments such as ammeters, voltmeters, wattmeters,

etc. Like any other standard (prototypes excepted)

they have to be calibrated by some operation higher

in the hierarchy. By a slight extension of our concepts each can be said to embody a range of magnitudes of one kind of quantity. Thus when the

ammeter is deflected to its 5-amp scale mark a current of 5 amp does then exist in its cu'cuit. Similarly the hydrometer float embodies a definite mass and the measure of this mass if divided by the measure of the immersed volume equals the measure

of the density of the liquid in which it floats. The

thermometer indicates a particular temperature

when its bulb embodies that temperature.

Physical standards of still another type consist of

samples of particular materials which embody

measurable properties to which definite values have

been assigned. One subclass of this type consists of

what may be called standard reference materials.

Each such material embodies some ph3"sical quantity,

not significantly dependent on its geometrical shape,

which has been measured and which can therefore

be used for the calibration of measuring devices.

A standard reference sample of highly purified

benzoic acid offers an almost unique example of a standard embodying three different lands of quantity: (1) its temperature of melting, (2) its heat of

combustion, and (3) its specific heat.

Examples in the electrical field are liquids of measured volume resistivity or dielectric constant.

If such a standard reference material is used in a

" The use of "continuously variable" is to be deprecated as it implies the occur- rence of variations which are not under the control of the operator.

8
test cell, a calibration factor is obtained by which the

corresponding properties of other liquids used subsequently in the same cell can be computed. Other

examples are bars or strips of ferromagnetic materials. The magnetic flux (usually expressed as a

flux density using conventionally assumed cross- sectional dimensions) corresponding to a succession

of acciu"ately measured applied magnetizing forces

is measured in one laboratory. The specimens are then used to verify the calibration of permeameters

in other laboratories lower in the hierarchy. Standard reference materials are widely used as standards

of viscosity, temperature, refractivity, and chemical

composition.

A somewhat different subclass of standard materials, preferably called standard ingredients, includes

samples of substances prepared in large uniform

batches for use as ingredients in other materials

(e.g., standard fillers for rubber compounds) to eliminate certain manufacturing variables when

studying the effects of others.

4.2. Measurement

Having established a vocabulary let us now

develop the Realist's approach by considering his

major operation, which is measurement.

Measurement has been defined as "the assignment

of numerals to represent properties in accordance

with physical laws." In the present connection we

are concerned with a somewhat more specialized

operation which can establish what Stevens [3] has

classified as a "ratio scale" for each measurable

physical property.

To qualify as "measurable" a property must be

recognized as having two aspects, both of which must

be definite: first its particular physical nature (e.g., electric current, resistance, energy) ; and second its magnitude. This means that there is an experimental operation for determining quantitatively its relation as smaller than, equal to, or larger than other

examples of the same kind of property and by what

ratio. Because of this latter feature measurable

physical properties are usually called "physical

quantities."

To be measurable to a given degree of accuracy

the physical quantity must first of all be identifiable

by particular defining operations, of at least that ac- curacy which can discriminate between it and other

similar but different phenomena. A major feature

in the development of any branch of science is the

successive recognition of such physical quantities

and the continuing improvement in the scope and

incisiveness of their definitions.

The process by which a particiilar concept has

been successively refined is exemplified by the concept

of electrical resistance. In a general way this was

early recognized as that property of a part of an

electric circuit by reason of which the current produced by a given voltage is limited in magnitude.

In many cases the ratio of the measure of the voltage

at the terminals of the circuit element to the measure

of the resulting current in it was found to be substantially constant, over a very wide range of currents. This fact justifies the recognition of the ratio as a measurable physical quantitj^. It was christened

resistance and chcuit elements exhibiting this property prominently are called resistors. Further

studies showed that the method of measiirement

should be lunited to the use of unvarying current

in order to separate out an extraneous efl'ect which

is now recognized as a quantity of a different kind

called reactance. Later extensions of the concept

of resistance restored the possibility of measure- ment using alternating current, provided observations based on phase relations served to discriminate

between the a-c resistance and the reactance. The

extension to radiation resistance has made the

quantity a property of antennas as well as of resistors. To insure that the cm-rent resulted only from the

applied voltage, procedures such as taking the mean

of values before and after reversing the polarity were

specified and an additional new concept of internal

parasitic emf (electrochemical, thermoelectric, etc.) was invented to complete the description. If the

voltage used was so high that corona discharge

caused the current to be different in different parts

of the resistor, a frnther specification had to be

included to bar observations under such conditions.

When the current was so large as to change the

temperature materially a similar limitation had to be imposed. This was usually expressed by stating

that the measure of the resistance was defined as the limit of the ratio of the measure of the applied

voltage to the measm'e of the resulting current as both approached zero. Even with these limitations,

results may be found to be different at different

ambient temperatures or with different conditions

of mechanical strain. Therefore new additional

concepts of temperature coefficient and strain coeffi- cient have to be included in the picture to preserve

the desired definiteness of the concept of resistance.

In addition to the basic requirement of definite

identifiability just discussed, many physical quantities possess the further useful attribute which we

may call additivity (see glossary). This permits

their use in the direct establishment of a ratio scale. Addivity means that if two examples of the quantity

are properly combined the measure of the resultant

in any unit must equal the sum of their separate measures in that unit. Many physical quantities

have the attribute of additivity. For the simple

concept of length the existence of this attribute is almost intuitive, provided that the combination

rule is to put the components end to end in the same

straight line. For volumes of liquid the rule involves

pouring the contents of small containers into a

larger one, and must be limited by a clause that no

9
mutual solution of niisciblo li([uids is permitted.

Similar additivity is found, for example, in noninductive resistances connected in series, in direct currents toward (or from) a branch point, and in direct voltages in a series circuit. For any additive physical quantity, if there is also

available some form of hidicator or detector which can

show, with the needed sensitivity, whether or not

two examples of the quantity arc equal, and if not

which is the greater, then it is possible to construct a ratio scale for that kind of physical quantity.

This procedure can best be imderstood by again

considering a particular example, say electrical

resistance. For the detector we use the classic Wheatstonc bridge cu-cuit with ratio arms A and B

across the batter}', an adjustable but uncalibrated arm Ei, and a fourth arm X. Here the letters serve merely to identily the four examples of resistance

embodied in the four resistors. For simplicity let us assume that resistance B has been adjusted so that the bridge remains balanced when A and B

are interchanged. Then if the galvanometer shows

a balance it insures (1) that the cm'rent in A' is the same as the current in Ri and (2) that the voltage

drop in X is equal to tha t in A'l (and in A and in B

also), and hence that the resistances X and Ri are

of equal magnitude. Initially', of course, no numerical values have been assigned to any of the resistances

and the scale on which the adjustable contactor

of Ri moves is unmarked. Now, ^\dth the galvanometer on balance, mark the contactor position "x". Replace X with another resistor Y and adjust I" imtil the bridge is again balanced, thus making the

magnitude of Y equal to that of X. Connect A' and Fin series in the A" arm, and restore the balance

by sliding the contactor to a new position R2. Mark

the new position "2x". By repetitions of this

process a true scale of resistance, in which the

resistance of X serves as a temporary unit, can be

laid out on R. It should be noted that nothing has

been said as to the linearity or otherwise of the

resulting spacing of the marks along R. It is necessar}^ merely that for each marked setting the

resistance of the arm shall be definite and reproducible enough for use in the applications of the scale

in futm-e measurements. With the scale of resist- ance once obtained it may be applied to the ratio arms of the Kelvin double bridge and thus extended

to low values of resistance defined by resistors of

four-terminal construction.

When a scale of say 10 equal steps has been

established, the total resistance, lOx, can be used

as the basis for building up a second decade the

elements of which each have magnitude 10 times

those of the preceding decade. The combination

of n such decades in series yields a multivalued

standard resistor having 10" discrete values. Assuming the individual elements to have adequate stability

this yields a scale precise to 1 part m 10" (see sec.

6). It then remains only to assign arbitrarily to the

resistance x a permanent numerical value to fix

the unit of resis(.ance. A consideration of the

factors involved in such arbitrary assignments

throughout the field of electromagnetics is a major

purpose of this paper.

Of com-se, an additive scale could be estabhshed

with a mmimum of operations but with less con- venience by building up a series of components each

having only two elements so that n components

yield 2" discrete values. For ph3'sical quantities

such as voltage, mutual inductance, or mass (using an equal arm balance), which can be either added

or subtracted, the scale need only contain powers

of 3 (i.e., 1, 3, 9, . . . units). Such schemes

requu-e the adjustment of fewer components.

If an experimental situation can be set up in which some quantity, for which an adequate ratio scale has

been established, can be made proportional by a loiown factor to some other ph^^sical quantit}', which

itself may not be additive, then the latter can be

measured directly. A simple example is the measure- ment of the specific volume (which is not additive)

of a liquid by the method of balancing columns. In

this method the liquid to be measured and a standard

liquid are placed in adjacent open containers. A

long inverted U-tube is placed so that one open end

is immersed in each liquid. Suction at the bend in the U draws up a column of each liquid. The

heights of the columns are measured. Here by

definition the specific volume is proportional to the

height of liquicl column supported by a given dif- ference in pressure. The height for the same

pressure difterence of the column of standard liquid

of Imown density fixes the factor of proportionality.

The heights are directly measurable on the basic

scale of length. This principle is the basis for the

potentiometer and the voltage divider which measure

voltages by use of the scale of resistance. The

calibration of a direct reading indicating ammeter

or voltmeter establishes a similar proportionality

between the reading (not necessarih^ the deflection)

and the current or voltage.

The measurement of a physical quantity by direct

reference to its own appropriate ratio scale or some

scale arranged to be proportional to it is called a

direct or comparative measurement.

Other definable properties such, for mstance, as density and resistivity do not have the attribute of

additivit}^ and it is sometimes not easy to set up a simple proportionality between them and some

additive property. However, enough properties are additive so that the magnitudes of the other pi'operties can be compared by the indirect process of

measurmg a pluralit}^ of component quantities in terms of which each non-additive quantity is defined

and combining their individual measures in accord- ance with the definition of the new quantity to obtain the measure of the new quantity by what

may be called an indirect, derivational, or absolute

measurement. Thus measurement of the mass and

10
volume of a body permits the computation of the measure of its density; measurements of resistance,

length, and cross section yield a measure of resistivity, etc. Moreover, often a quantity like electric current, although it is additive, as combined at the

branch points of a cu'cuit, is in practice often measured indirectly by using the ratio scale built up by

resistances in combination with a standard voltage.

Examples of indirect measurements are the

measurement of energy in terms of current, voltage,

and time; magnetic induction in terms of flux and

area; capacitance in terms of resistance and fre- quency; inductance in terms of capacitance and

resistance, etc. The adjective absolute is usually

applied only to those operations in which a quantity

is measured indirectly and in terms of the ultimate

basic units (usually length, mass, and time) of the

system of units used.

4.3. Present Experimental Basis for Electrical Units

Each national standardizing laboratory endeavors

to maintain a set of electrical ph^'sical units which

is constant in time and in agreement with the magnitudes recommended in 1946 by the International Committee on Weights and Aleasm'es [41]. This Committee had based its recommendation on a careful

consideration of all available experimental data

obtained by absolute measurements of resistance

and of current in terms of the units of length, mass, and time and of the postulated value of 47r-10~" for the magnetic constant r„, in the rationalized

MKSA system of measurement.

The basis for the maintenance of the electrical units at XBS involves the construction and preser- vation of a group of refei'ence standards of the highest

equality; the assumption that their secular drifts in

magnitude tend on the average to cancel; and then- periodic use in the precise measurement of some

constant of nature as a check on possible di'ifts. (For more detail see [44] also.)

For the ohm there is used a group of about 20

standard resistors, made of annealed manganin

wire, each mounted in a sealed container of the

double-walled t^'pe [42]. These are stored in thermostat ed oil baths and are intercompared annualty

by a substitution method with a precision of 1 in

10''. The resulting measures are examined on the

assimiption that the mean of the magnitudes of a subgroup of 10 of these standards has not changed

since the preceding intercomparison. If any individual resistor of the 10 originally chosen for the

subgroup is found to show a change considered

large compared to those of its fellows, it is rejected

and another member of the larger group is used to carry the unit forward. If the measure of each

resistor differs from the mean of the subgroup by

about the same amount as at the previous intercomparison, a new value is then assigned to each

standard in the group. In this new assignment the

mean resistance of the ten resistors is assumed to be

the same as it w-as at the preceding intercomparison.

The newly assigned value for each individual resistor

then differs from this mean by the newdy measured

amount.

The volt is maintained in much the same way by

using a group of 40 cadmium standard cells of the

saturated type. Cells of several different forms

(i.e., acid and neutral) are included in the group.

Comparisons are made to 0.1 ixv. It is seen that the prmiar}' reason for expecting

the standards and the units based on them to remain

constant is merely the simple assumption that

examples of these particiilar physical systems (i.e., pieces of alloy wire, and electrochemical cells) if stored under reasonably constant conditions will not change their physical properties. The basis for confidence in this assumption is found in the rea- sonably satisfactory, though far from perfect, record

of comparisons of groups of such s^'stems during the

past half century [44]. This record, as derived from

international intercomparisons among the six cooperating national laboratories, between 1910 and 1948

shows that after the lapse of about 20 years the

standard resistors of two laboratories had drifted

hj about 30 m ohm and had increased by this amount

the units they were maintaining. These laboratories

then assigned new values to theh standards to recover the old unit. SimUarh^, after 25 years two

laboratories found it deshable to increase their

volts b}' about 80 y-Y to restore their units. Since

the reassignment of values for the national standards

in 1948 the performance has been better. In 1957,

almost a decade after they had been reassigned values on a uniform basis, the units both of resistance and

of emf of the six national laboratories compared at the International Bureau of Weights and Pleasures

feU W'ithin a range of ±6-10"'^ from the mean of all. To obtain an independent alternative basis for maintaining the electrical units over long intervals,

two t}-pes of project are currenth' under waj' at XBS. The fu'st is to redetermine at desu-ed intervals some "constant of nature" in terms of the units as maintained. If the same measme is obtained at each later periodic redetermination, it gives a strong

confirmation that the units have not changed during

the interval. Two such constants are the gyromagnetic ratio of the proton and the electrochemical

equivalent of silver. The first [47] involves primarily

the measurement of electric current, frequency, and

the pitch of a winding on a single-layer solenoid.

The second [48] involves prunarily the measurement

of electric current, time interval, and the mass of

the electrochemically corroded silver. Frequency

and tune can be measxu"ed with ample accuracj^.

The other variables in the g}Tomagnetic experiment

may introduce a random uncertainty of 1 or 2 in 10^. Although in a determination of the gyromagnetic

1 1
ratio possible sj'stematic errors may exxeed this estimate, a repetition of the experiment under the same eonditions after the lapse of 10 years would

suffice nevertheless to detect a steady drift in tlie groups of standards equivalent to only 2 in 10^ per

year. The electrochemical experiment is not quite

as reproducible (perhaps by a factor of 4) but offers an entirely independent and therefore very valuable

backstop to detect drifts. The precise measurement

of current in either of these experiments involves

the standards for both the ohm and the volt. It

is possible but very improbable that separate drifts in the magnitudes of the two types of standard

should be such as to compensate exactly.

Basic projects of the second type include the absolute measurement of resistance and of current. These projects have two objectives: first, the determination of any difference which may exist

between the unit as maintained by the national

reference standards and the ideal absolute unit.

The second objective is to detect any change in the

unit as maintained since the previous absolute

measurement. The accuracy with which the first objective can be attained is currently perhaps not

much better than 10 in 10^, largely because of the

possible presence of systematic errors which are not

eliminated by using detectors of extreme sensitivity nor by accumulating data through many repetitive

observations. However, to the extent that such

systematic errors remain truly constant from one

use of the apparatus to a subsequent use, they do

not limit the accuracy in attaining the second

objective. The ability to repeat an absolute meas- urement after a lapse of 10 years may be as high as

1 in 10«. Since the International Committee on Weights

and Measures made its decision in 1946 on the

recommended values of the units, a number of

additional absolute measurements have been made.

For the ampere the only recent work published is that by Driscoll and coworkers at NBS [28, 29].

When using a current balance and measuring the

force between coaxial single-layer helical coils, they

obtained in 1957 0.999992 as the measure of 1 absolute ampere in terms of the volt and the ohm

as currently maintained at NBS. When using an electrodynamometer of the PeUat type and measuring the torque between two concentric single-layer

helical coils with their axes at right angles, Driscoll

obtained in 1957 0.999987 for the NBS measure of

1 absolute ampere. The agreement between the two

methods is very gratifjdng, because it is unlikely

that many sources of systematic error would be

present to an equal extent in both of two pieces of

apparatus which are so different mechanicaU}^.

However, one somrce of uncertainty is common to both, namely, the local value of gravity, g. The

measures here given are based on Dryden's [22] estimate from his revision of the Potsdam data.

More work has been done on the ohm. In 1949

Thomas, Peterson, Cooter, and Kotter [23] using the

Wenner method obtained 1.000006±0.000010 as the measure of an absolute ohm in terms of the unit

preserved at NBS with 1-ohm standards since

January 1, 1948. In this measurement the biggest

single source of error was probabl}^ the uncertainty

in the distribution of current in the primary winding

of the mutual inductor. Current distribution is affected by resistivity-stress relationship in the

copper wire. The current-distribution correction

used in 1949 was based on resistivit^^-stress studies

made by Kotter in 1940. Later studies made by

Wells in 1956 [26] gave additional data which, had

they been available in 1949, would have resulted in a value of 1.000003 for the measure of the absolute

ohm in terms of the unit maintained at NBS.

During the decade 1950-1960 the latter unit agreed

with the unit maintained by the International

Bureau of Weights and Measures within 1 /xohm.

In 1953 Rayner [24] of the British National Physical

Laboratory, using the Campbell method, reported

0.999996 ±0.00008 for the measiu-e of an absolute

ohm when reduced to the international basis.

In 1957 Romanowski and Olson [27] of the National

Research Council of Canada reported a result equivalent to 1.000003 ±0.000020 for the measure of the

absolute ohms in terms of the units of the International Bureau.

In 1956 Thompson and Lampard of the Australian

National Standards Laboratory discovered a new

theorem in electrostatics [25] which can be applied

to the computation with very high accuracy of the

capacitance of small three-terminal capacitors.

Cutkosky [31] in 1960 completed a measurement

using such a capacitor and obtained 0.9999977 for the measure of the absolute ohm in terms of the unit

then maintained at NBS. If there has been no

relative drift between the units of NBS and of the

International Bureau, this means a measure of

0.9999987 on the international basis. Cutkosky's

method involves stepping up in four decimal stages

from 1 pf to 0.01 ^f ;the comparison at that level and at

1,592 c/s (aj=10'' radian/sec) of the admittances of a pair of capacitors with the conductances of a pair of

10,000-ohm resistors; and the further stepping down

in four more decimal stages to 1 ohm. Nevertheless

the extreme simplicity of the computable capacitor

and the simple self-checking featiu^es available in the 10:1 steps limited the uncertainty to ±3 in 10® (50 percent confidence interval). To this estimate an uncertainty of ±1 in 10® in the speed of light

makes a significant contribution. This method

evidently constitutes a significant "breakthrough"

in the field of absolute electrical measurement.

It is of course the intent of the International

Committee on Weights and Measures to keep the

electrical units as close as practicable to their ideal

12
values as defined. The adjustment of 1948 must be

considered as merely the latest in a series of such

adjustments which began with the change from the

BA unit to the legal ohm in 1884. The adjustment

of 1948 seems to have been chosen very wisely. In

fact, the data just quoted suggest that no further

change is to be expected for a long time, that is until (a) a materially more accurate value for g becomes available, (b) measuring techniques in

science and industry increase materially in their

requirements for acciiracy, and (c) some further

revolutionary increases in the accuracy in absolute

measurements become attainable which wUl reduce

the present limits of uncertainty materially below

the small apparent discrepancies between the ideal

and the maintained values. The standardizing

laboratory is still confronted with the "fact of life" (b) (p. 2), namely that simple comparison methods

will always outstrip absolute methods in accuracy.

4.4, Experimental Establishment of Physical Laws

The modern student, in a world well supplied with

calibrated apparatus and recognized systems of

measurement, naturally considers experimental re- search to involve the operations of making measure- ments on imknown quantities and then expressing

their relations to known quantities by appropriate

equations. However, a perusal of the writings of

the earlier classical workers in any field shows instead

that their results were usually stated merely as proportionalities. Thus Newton wrote: "The alteration

of motion is ever proportional to the motive force

impressed"; Coulomb: "The repulsive force ... is in the inverse ratio of the square of the distances";

Faraday: "The chemical power of a current of elec- tricity is in direct proportion to the absolute quantity of electricity which passes." The writing of a measiu-e equation or a quantity equation always

involves an additional conventional operation.

The nature of these steps by which physical laws

are discovered and demonstrated experimentally by

the Realist as relations between the measures of

physical quantities can perhaps best be understood

by considering a couple of examples.

The first extremely simple case illustrates the basic

principles and by its contrast with the usual theoretical procedure serves to emphasize the differences

between the two philosophies. The second some- what complex example is offered because it applies

to the ciu-rently moot question of rationalization.

First let us consider the measurement of area. A

Realist supplied with a scale for measming length,

graduated in any arbitrary equal intervals (say for example in iBches and 16ths), and a sheet of cross- section paper of any mesh (say for example millimeters) could study the measurement of area as a purely empirical matter without regard to geometry.

He would draw various geometric figures of various

sizes and measure their dimensions in his scale units

(say in inches). He would also count the number of

squares of his cross-section paper enclosed by their

perimeters. For each rectangle, triangle, circle, and

regular hexagon, respectively, he woiild express his

data by the experimental measure equations ^

{^,},=322{6 ]s{h},

{An},=2,Q57{l}l (4.4.1)

or in a more general literal form

{A^],=K,sA<^}Ab]s (4.4.2)

where a and b are appropriate orthogonal dimensions.

Here the subscripts p and s denote the use of the

arbitrary paper and scale units respectively, and the

subscript n, which may take on the values r, t, c, or

h, indicates the shape of the area measured.

He notes that the experimental coefficients are very nearly in the ratios:

K^sr -.K.st: K^sc : i^p.^ : : 1 : K : TT : 3 ^- (4.4.3)

The theorems of plane geometry derived independently by the Synthetiker also show that for these shapes the coefficients K„ would be in these same ratios. The usual textbook also goes on to state dogmatically "Ar =wl," thus making the additional tacit and arbitrary assmnption that Kr for a

rectangle (rather than Kt for a triangle or K,. for a

circle) is to be set equal to unity or, in other words,

that the unit of area shall be chosen as being equal

to the area of a square which has each side of unit

length. Our Realist following this suggestion can make his measure equation look like the Synthetiker's

quantity equation Ar—wl by arbitrarily choosing 645

of his preliminary square units, as the physical unit

of area which is germane (see glossary) both to his

physical unit of length, and to the geometric measure

equations with their coefficients Kpsr='i-, Kpsc=Tr,

etc. (i.e., if his scale unit were 1 in. he would find

that his germane unit of area was the square inch).

In strict analogy to the foregoing consider now the more ambitious program of a Realist studying magnetism. He has both a graduated scale to measure

lengths in a recognized unit, say the meter, and apparatus for measuring current in a recognized unit,

say the ampere. Let a subscript a designate the use

of a set of physical units germane to the meter, the

ampere, and to the equations defining the ampere.

* The reader will appreciate that these particular numerical values will result

if the units of the scale and paper happen to be those suggested parenthetically

in the text.

647596—62 3 13
The Realist also has several short magnetized needles

each suspended by a silk fiber, and a stopwatch by

which he can measure, in cycles per second (i.e., also system a), the frequency of small oscillations of the

needle. He observes both the rest position of the

needle and its frequency of oscillation when dis- placed therefrom, when it is suspended in various

definite locations near each of three systems of ciu:- rent-carrying conductors. These locations are (1) at a distance r from a long straight conductor; (2) at the center of each of a set of circles of radii r; and (3) at the center of one of several uniformly Avound sole- noids of pitch, r,, and of such length that their open

ends subtend an angle 2e at the center. The subscripts I, c, and s respectively denote quantities pertaining to these three kinds of geometric arrangement

of conductor.

In an}^ one experiment he finds that the squares of

the measures of the frequencies are proportional to the measures of the current. By analogy with a pendulum in the gravitational field of the earth he

postulates the existence of a magnetic force field. For each of the needles he plots the squares of the measures of the frequencies against the quotient of

the measure of the current by the measure of the dis- tance, radius, or pitch. He finds these graphs to be

straight lines, the slopes of which he designates by

Snai, Snac, and Snas for the long wire, the circle, and

the solenoid respectively. The subscript n here

designates the particular needle and the subscript a

indicates that the standard germane units of current

and distance were used.

Hence he can write a set of equations of the fonn

{fn.,n =S„aAI.}a/{r,}a (4.4.4)

where the subscript g indicates the possible substitution of I, c, or s to get the measure equation for any

of the thi'ee geometries used. He finds that the slopes

Snag can be arranged in an array which has very

nearly the form shown below where S denotes the

slope found with the first needle and the long straight

conductor, SM2 that with the second needle, etc.

\^ Needle,

Geom- \, etry, \

1 2 n

Straight, I Shi S-Mrl . . . . . . S-M„l

Circle, c S-Mz-TT . . . . . . S-M„-7r

Solenoid, s S-l-27r COS e S-M2-27C cos « . . . . . S-M„-2Tr COS c

(4.4.5)

This shows that for any needle, n, the slopes in a given colmnn are in the ratios

Snal-S„ac'S„as--'i--Tr-2Tr COS € (4.4.6)

and for any geometry, g, the slopes in a given row

are in the ratios

Sui-Siag . . . S„ag--l--M2. . . . Mn. (4.4.7)

From these facts he infers that the measures of

the squares of the frequency are proportional to both the measure of a new phj^sical quantity, M,

which depends only on the needle, and to the measure

of a new physical quantity, H, which depends only on the geometry and size of the circuit and the

magnitude of the current. He can factor each slope

into a constant S (i.e., the first member), a part

Kn which depends only on the needle, and a part

Kg which depends only on the geometry, thus

getting for any slope

Sna,= S-Kn-Kg. (4.4.8)

Combining eq (4.4.8) with eq (4.4.4) gives the set

of equations

{fn.,]VKn=SKAI,]a/[r,]a. (4.4.9)

The right member of each of the eq (4.4.9), although

it involves the constant S and hence the strength

of one needle, is independent of n. Hence the left member must be also independent. Therefore each

left member can be considered as an appropriate measure {Hg] p of the new physical quantity H

which depends on g, in terms of a preliminary

physical unit, j,Ujf. The operational definition

for measuring the physical quantity H is that, when

the oscillation frequency is four times as high, H is to be considered two times as large. Also the preliminary physical unit of H is the sample of the

physical quantity H existing at 1 m from the straight

wire when the measm-e of the current is 1/S amp.

The use of this preliminary unit gives the set of three measure equations

{Hc}p=Kpac{Ic]a/{rc}a

{Hs}p=Kpas{Is}a/{rs}a (4.4.10)

where

Kpai =S

Kpac —T'S Kpas^'^irS COS €. We need not follow the Realist further in Ms study

of the needles or the relation of Mwith their magnetic

moments and moments of inertia. Instead we see that liis colleague the Synthetiker from eq (4.4.10)

is led to recognize the more general law of Ampere '

d{H^]p=KpaA^^A^ sin e (4.4.11)

X'Ala

where

KpaA=S/2. (4.4.12)

» Here, of course, the geometry subscript "A" designates that the current Ta

is in an elementary length dU at a distance Ta from the point at which the field strength (Ha is measured, and 0 is the angle between the directions of r and dlA,

14
From this he can proceed to deduce equations

appropriate to still other geometrical arrangements.

For example the measure of the magnetic field strength {H^} j, at the center of a hexagon of side h

will be

{Hn},=K,an{h]al{h]a (4.4.13)

provided

K^an=^Sj^[^. (4.4.14)

The Realist's next step is to eliminate the individualistic factor S due to his preliminary unit for H

symbolized by the subscripts p. He does this by

(1) choosing some readily described geometry which we may designate by d and for which he writes

{Ha]a=KMa}al{r,]a (4.4.15)

and (2) assigning some smiple coefficient Kad to this

particular geometry. He then (3) derives a new

physical unit germane to the basic units of system a and to the chosen coefficient Kad- This means that

aUH=vUH--^- (4.4.16)

This step reduces the data obtained with the uncalibrated needles and the temporary preliminary

imit pUa to measures based on a single physical

unit aUn, which is germane to the more basic

units aUi and aUi and to the simple arbitrary

coefficient Kad-^^

It should be noted here, however, that the change

in units indicated by eq (4.4.16) is not the only

possible procedure for obtaining the desired value

of Kad- An alternative would have been to consider

that the revised measure equation (4.4.15) and the desired coefficient Kad gave the measure in the old provisional unit pUa of a newly conceived physical

quantity H' of the same kind as Hhut of a magnitude

related to the physical quantity H as formerly

conceived by the relation

{H']=§^{H]. (4.4.17)

This alternative usually seems repugnant to the

Realist.

It is by processes of the general nature here illus- trated that the Realists have built up the whole

discipline of electromagnetics into a collection of measure equations. Table 1 (see sec. 10.1 for all tables) lists a niunber of these equatons in wdiich

by common agreement the coefficient Kaa is unity.

Table 2 lists other equations in which the coefficients are different in different systems of measurement.

In both tables to economize space the { }'s are omitted but each quantity symbol should be regarded

as merely a measure as long as we are viewing the

equations merely as established experimentally by

'« The process here outlined is not so very unlike the actual historical sequence which started with Biot's experimental proportionality like the first eq (4.4.10). Laplace proceeded to eq (4.4.11) choosing KaA equal to 1. In 1893 Heaviside

suggested that in effect it was more "rational" to set KaA^yjiTt. This change

in the choice of this and of certain other proportionality factors constitutes the

act of "rationalization."

the Realist. Later we shall see that the identical

equations without the { }'s are used by the Synthetiker to show relations between his symbolic

quantities. In table 2 the equations are wi-itten in

column 2 with a number of arbitrary parameters in their coefficients. By this device [57, 58, 60] it is possible to assign various sets of parameters in such

a way that each set yields the set of equations to which one of the many alternative proposed systems

of measurement is germane. The correlation between the sets of parameters and the systems of

measurements is indicated in table 3. It will be

noted that (column 4) serves to distinguish symmetrical from unsymmetrical systems (see sec. 6.2). Fr (column 5) serves to distinguish unrationalized

from rationalized systems.The reasons for particular

choices of coefficients are primarily of concern to only the Syiithetiker and will be mentioned specifically

in section 6. The Realist woi'king in any one system

with one set of parameters and a small set of basic

units proceeds to define and realize experimentally

his derived germane physical units for each of the

other quantities involved as shown in section 4.5

below.

After the Realist has expressed his experimental

laws as measure equations, he is free to combine

them by any desired mathematical operations,

because the sjonbols in the ec^uations represent

numbers for which such operations are permissible.

This is the rigorous basis for his algebra and he should

preferably stick to it. However as a short cut he

often finds it desirable to use mathematical phraseology to obtain conciseness in describing his experimental operations. When he combines the lengths

of two gage blocks by wringing them together he says

he has "added" them. When by measurement he

has ascertained that the length of his desk is three

feet, instead of the rigorous measure equation

"{-Z^desk}ft=3" he writes "Zdesk=3 ft." His replacement of the verb "is" by the symbol "=" is more

than a mere substitution, and introduces mathematical connotations. It leads him to call his

abbreviated statement an "equation" and to say

that he has "multiplied" the physical unit "foot"

by the number "3." Such a "multiplication" of a symbolic unit by a number lies at the very heart of

the Synthetiker's quantity-calculus, but to a pure

Realist it means primarily that the noun "feet" is modified by the adjective "three." He may also winte "L (in inches) = 12Z (in feet)" and may generalize the combined information by writing "1 ft=

12 in.," and "the ratio of 1 ft to 1 in. is 12." This

leads some writers to state as useful principles:

(1) two of the Realist's physical quantities, if of the same kind, may be "added" or "subtracted"; (2) one physical quantity may be divided by another of

the same kind; (3) a physical c^uantity may be multiplied by a number. To this extent physical quantities can be said to be amenable to some of the principles used by the Synthetiker in the quantitycalculus of symbolic quantities, and presented in section 5. However the Realist must stop at this point. He cannot multiply together two physical

15
hantities (even if of the same kind) nor divide one

puysical quantity by another of a different kind.

He can, and should perforin these hitter operations

only on the measures of his physical quantities. On

the other hand the Synthetiker (as will be seen) can perform these operations on his symbolic quantities.

The temptation to describe the Realist's operations

in such mathematical sounding language is very great

and yielding to it often saves words and space. However it is just the possibility of doing this to a limited extent which has led many wTiters (-svith their readers) to slip unconsciousl}'- beyond the pale

from being Realists to being SjTithetikers. This

usage hides the fundamental distinction between

physical quantities and S5Tnbolic quantities. Either

the measures of the Realist or the s3'nibolic quantities

of the S3-nthetiker are amenable to all the familiar

operations of algebra. For the Realist to consider

that he is applying some of them to his physical

quantities, although it may be justifiable, is dangerous, because, like an alcoholic, he may not realize

when to stop. The wise Realist considers each

letter symbol used in describing his phj'sical operation to be either a measure (i.e., a number) in a measure equation or an abbreviation in a sentence.

If he wants to play with mathematical operations

he should go the whole way, become a Synthetiker,

and realize that he is using only s^aiibolic quantities

and not physical quantities.

4.5. Derivation of Germane Systems of Measurement

by the Realist

To describe and predict phenomena on the basis

of the proportionalities discovered expermientalh'

between various measurable physical quantities b}^

the method exemplified in the preceding section

4.4, the Realist must set up measure equations. For

this purpose he must have chosen and defined

operation all 3^ (a) a set of dift'erent kinds of phj-sical

quantities and also (b) a set of N ph3-sical units, one

for each kind of physical quantit3^

The choice of the units in the set might conceivably be entirely capricious. In this case each of the

resulting measure equations would contain an experimentall3' deteimined numerical coefficient. Many

equations in engineering handbooks are the result

of this process, particularly where the sizes of the

units have been selected to be of the same order of

magnitude as the quantities concerned.

In most scientific work, however, it has been found

much more desirable to make the choice of the

ph3^sical units for quantities of different kinds in a s3'Stematic fashion. To do this the Realist selects a small number, p, (usually three or four in electromagnetism) of basic physical units. Each is defined

hy reference to a prototype standard. He then

%vrites a set of n{n=N—p) independent measure

equations, each of which is based on a proportionality

established experimentall3^ as illustrated in section

4.4, and each of which contains a proportion alit3' constant Kg. The n values of Kg can be chosen

arbitrarily and are usually taken as unity for some

simple geometric arrangement. Historically this choice of the Kg's has been the work of the Synthetiker and is accepted without challenge by the

Realist. The complete process has involved a

total of 2N arbitrar3' choices, namel3'^ A'' operationally

defined ph3^sical quantities, p basic ph3-sical units,

and n=N—p coefficients. Kg. Then b3^ the process

exemplified in eq (4.4.16) above the Realist defines

the set of n germane derived ph3^sical units for the

N—p remaining ph3'sical quantities. He also, as needed, defines other non-germane units of his S3'stem

as specified multiples or fractions of each germane

unit. The entire ensemble of four sets of components,

namely (1) A^^ physical quantities, (2) p basic pl^^si- cal units, (3) n independent measure equations, and

(4) n derived germane ph3^sical units is called a measurement system. To this may be added any

convenient non-germane physical units.

As we shall see in section 5, the S3mthetiker in his mathematical model also constructs a complete

measurement system with four sets of components plus

a set of N dimensions. However, the sequence and

conception of his quantity equations and S3Tnbolic

units and quantities is essentially different from those

of the Realist.

A convenient way by which the Realist can be

assured of the independence of his n measure equations and define the n derived germane ph3'sical

units of a system is to use a sequential procedure.

He starts with one of the n equations which involves measures in terms of two or more of the p basic physical units of the system, together with the measure

of only one new physical quantit3'- (i.e., one of the n ciuantitles the units of which are to be derived).

For example:

{x],^Kg{y]a-{z]a. (4.5.1)

Then for the geometry appropriate to Kg the ger- mane unit of X (i.e., aUx) is l/i^^ times the example

of X present when {y] = l and {z] = l. By selecting a sequence of measure equations at each of which

a single new physical quantity is introduced, a complete S3^stem of measurement with its germane

set of ph3'sical units can be built up. This is further

exemplifi_ed in section 6.3. If in this sequence of operations an equation is in- troduced which involves two new physical quantities,

A'^ is thereby raised by 2 whUe n is raised hj oi\\j 1

.

Therefore p must be increased by 1 also, and the

Realist must select an additional basic physical unit

for one of the two new physical quantities with an

appropriate protot3"pe standard to define it. The initial choice of the number, p, of basic physical units is somewhat arbitrary. Even in mechanics there is no particular "magic" in the use of the

usual three, length, mass, and time. This can be seen by considering the Newtonian equation for gravitation

{F]=G{m,}{m,}l{rY. (4.5.1)

On a three-basic S3^stem {p =Z) the coefficient, G, ap- pears as an experimental "constant of Nature" which

16
has been found to have the magnitude 6.670 10"^^ newton (m)^ per (kg)-.

As a second alternative, however, it would be

entirely possible to set up a consistent set of mechanical units %vith onl}- the meter and the second

as basic imits together vrith the choice of 6.670-10"^^

as the niunber assigned to 6^ in eq (4.5.1). Then

the derived germane physical imit of mass (kilogram) would be defined as the mass which when

placed 1 m from an equal mass experiences a gravitational acceleration of 6.670-10"^^ m/(sec)-.

Such a system with onty two basic units is occasionally used in astronomy. Its rejection in physics

stems, of course, from the low accuracy obtainable

in experimentally assigning values to mass standards

in terms of the units thus defined.

A third alternative should not be overlooked.

Empty space might be considered to have a gravitational property and to constitute a prototype standard to which there might be assigned the value

6.670-10~" in terms of a third basic unit of gravitation. This unit, together with the meter and the

second, would be the three basic units of this system.

The kilogram would again be a derived unit defined

by eq (4.5.1), but the system would have three in- stead of two basic units. The last two alternative

interpretations of eq (4.5.1) appear whimsical in the

field of mechanics but have been mentioned here

because they are strictly analogous to parallel rela- tions which have been seriously discussed in the

electrical field (see pp. 21-22).

5. Theoretical Approach

In section 4 procedures have been described by

which the Realist can develop a number of complete

germane systems of measurement using physical

units and the resulting measures related by measure

equations. Each S3'stem is characterized by its set of coefficients and its set of basic imits. Any one

such development satisfies the needs of the experimenter, the engineer, and the businessman. In the

present section the alternative development, which is preferred by the mathematician and by writers con- cerned with theoretical relations in electromagnetics,

is set forth.

5.1. The Mathematical Model

The Synthetiker, being aware of the concepts in- vented by his coUeague, the Realist, to describe the

properties of the latter's physical systems, and knowing the proportionalities found experimentally between the measures of these properties, sets up for each of the Realist's measurement systems a mathematical model which he so designs that the model

for each particular system bears a one-to-one corre- spondence with the Realist's system and thus with

the actual physical universe. The correspondence

may be dift'erent for different sj'stems of measurements.

For each of the N kinds of physical quantity conceived by the Realist, the Synthetiker sets up, for

each particular measurement system, a class of physico-mathematical quantities or mathematical

elements which for contrast wiU. herein be called

"symbolic quantities." The members of any one of

these A'' classes are characterized by having a common

dimensionality (i.e., the quotient of any two elements

of the same class is a numeric), and a magnitude

relative to the other members of the same class. This means that any one of the symbolic mathematical elements may be written

Q-{Q}a{Q)a (5.1.1)

where {Q)a is that member of the class Q to which

is assigned unit magnitude (i.e., it is the symbolic

unit of Q in the unit system identified by the subscript a), while {Q,}a is the number which is the measure of Q in terms of the symbolic unit (Q}aThe SynthetUver then proceeds to -write equations

which express the desired relations between his

mathematical elements (i.e., his sjmibolic quantities).

He can write n such quantity equations each cor- responding to, and being identical in appearance to, one of the n measure equations which the Realist

has developed experimentaUy as described^in section

4.4. The Synthetiker, however, regards the letter symbols in his equations as denoting not the numerical measures but the complete mathematical

elements. Such equations are caUed "quantity equations," and their use "quantity calculus." Justifi- cation for their use may be traced back to D. Gregory,

Boole," and Maxwell [81, 82, 83, 84]. Theu use has been revived by Wallot [85], Landolt [86], Page

[89, 90], and others in recent decades but is stiU not

often explicitly stated or widely appreciated in en- gineering circles. The inherent elegance and simplicity of this approach can be illustrated by writing Ohm's Law

fu'st as a measui'e equation

{V}a={I}a-{R}a (5.1.2)

which is equivalent to the statement that the measure of voltage, {V}a, in a particular set of units a is numericaU}^ equal to the product of the measures of the current and of the resistance in the same germane

set of units. By contrast the quantity equation

V=IR, (5.1.3)

which might also be written in greater detail, as

{V}AV}a={I}.{r),-{R}c{R)c (5.1.4)

is true regardless of the units employed. Thus

(5.1.3) makes the general statement that "the mathematical element which corresponds to the potential

difference at the terminals of a resistor is equal to

II Thus in "The Mathematical Analysis of Logic" (Oxford, 1847) Boole writes: "They who are acquainted with the present state of the theory of Symbolical

Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon

the laws of their combination. Every system of interpretation which does not

affect the truth of the relations supposed, is equally admissible, and it is thus

that the same process may, under our scheme of interpretation represent the

solution of a question on the properties of numbers, under another that of a geo- metrical problem, and under a third, that of a problem in dynamics or optics. . . ."

17
the product of the elements which correspond re- spectively to the current in the resistor and to the

resistance." The sets of units indicated by the subscripts a, b, and c, can be quite unrelated. To give a

typical, more specific interpretation of (5.1.4), it may

be considered for example as equivalent to the fol- lowing statement, "The measure of the voltage in

kilovolts times one kilovolt equals the measure of the

cmTcnt in milliamperes times one milliampere multiplied by the measure of the resistance in ohms multiplied by one ohm." As the names "current" and

"resistance" here denote sjmibolic quantities, the

Synthetiker is quite agreeable to postulating that

their product results in a voltage. In contrast the

Realist dealing with the ph5'Sical quantities finds it meaningless to talk about multiplying a procession

of electrons by a property of some alloy which resists such a procession.

The Synthetiker normally selects the same N—n

quantities for which the Realist has chosen basic

physical units and regards the corresponding N—n

classes of mathematical elements as being basic symbolic quantities. From this base he defines in

succession the other derived symbolic quantities

(elements in his mathematical model) which cor- respond to the physical quantities of the Realist.

As an example, suppose y and z are two of the Synthetiker's basic quantities and (?/> and (2) are particular samples of each quantitj^' to which he arbitraril}^ assigns the measure 1 and which therefore are two of the symbolic basic units of his S3^stem. For some particular experimental or geometrical

situation, here denoted by a subscript g, he notes

that experiment has shown the proportionalities in- dicated by the measure equation

{x]=K{y].{z]. (5.1.5)

The Synthetiker then writes the quantity equation

x=K,yz, (5.1.6)

choosing a convenient coefficient Kg which he considers appropriate to the geometry, g. He also writes another quantity equation

(x}={y)-{2} (5.1.7)

in which (x) symbolizes the coherent sjmibolic unit

of X. This is commonly called a "unit equation."

Equations (5.1.6) and (5.1.7) together serve to define the meaning of the operation of multiplication of an element of 77 by an element of s. The result of this

operation is the creation of an element of x. Also

inserting (y) and (s) in place of y and 2 in eq (5.1.6)

shows that the product of a unit of y by a unit of s

in geometry g produces an amount of x to which is to be assigned the measure Kg. Tlois joint action of the

two equations defines both x and (x). Of course,

either equation establishes the dimensionality of x

12 See however the exceptional departui-e from this simplicity in the MKSA

System, third interpretation (sec. 6.3).

and the dimensional equation (see sec. 7)

[x] = [ym = [y-z]. (5.1.8)

By successive applications of processes similar to

this, the Synthetiker builds up the complete set of

N sjmibolic quantities, one corresponding to each of

the physical quantities of the Realist. The key to the correspondence is the parallelism in form between the Realist's measure eq (5.1.5) and the

Sj^ithetiker's quantit}^ eq (5.1.6). The value of K

in (5.1.5) is immaterial as it can be readily adjusted

by the Realist to be equal to Kg hy his choice of liis plwsical unit of x. It should be noted that the usual correspondence of

basic sjaiibolic quantities with basic physical units

is purely a matter of convenience and not a logical

necessity. Also one or more of the sjnnbolic quantities which the Sjmthetiker prefers to consider

"basic" may correspond to a phj^sical quantity

which the Realist measures b}^ using a derived (i.e., nonbasic) physical unit (e.g., the ampere in the MKS

systems)

.

The Synthetiker, using ciuantity equations, in the

establishment of which the concept of "units" entered

only briefi.}', can combine, extend, and manipulate

his initial n defining equations to deduce new and

valuable relationships between elements in his mathematical model. Most theoretical textbooks present

such developments first and introduce a chapter

entitled "Units and Dimensions" only somewhere in the last quarter of the volume, if at all. Although the Synthetiker has infrequent need of

units as such, he is much concerned by the dift'er- ences between the various measurements systems

listed in section 6 because these S3'stems diff'er in the

coefficients in their n defining equations (as tabulated

in table 3), as well as in the size of their basic units.

Thus, as usuall}'^ treated, electric current in the CGS

electrostatic system (symbolized by Is) is related to

electric current in the CGS electromagnetic sj^stem

(symbolized by 1^), in the Heaviside-Lorentz S3^stem

(sjmibolized hyln), and in the unrationalized MKSA

system (symbolized by „//) by

Is =cIm =I„/i4,T =CnIf\ n^m. (5.1.9)

He must be careful to distinguish, by using subscripts or a similar device, between these four different

s3'mbolic quantities all labeled "electric current,"

and all corresponding in different S3'stems to a single

phj'sical quantit3'', also called "electric current" for which the Realist uses the abbreviation "/." Similar

relations involving positive and negative powers of

c, Tm, and \l4ir relate the other s3axibolic quantities

used in the various systems. Thus

Vs^VJc^^l4^V>,=nV,/c^\T^ (5.1.10)

and

Es=RJc'=iirE,=„R_r/c\T„ (5.1.11)

and so on.

18
Page [90] has suggested an alternative procedure

by which the Synthetilcer may avoid changing his

symbolic quantities when rationalizing or shifting

from an electrostatic to an electromagnetic system.

This procedure is to introduce dimensions in the

"geometric factor," Kg, in the quantity equation

and thus adjust the symbolic unit in terms of which

the measure of the symbolic quantity is computed.

It is unfortunate that the names of the various

measurement systems are prunarily based on the

sizes of the basic units (e.g., CGS, MKSA, etc.)

when the features which are really more unportant,

at least from the point of view of the Synthetiker,

are the coefficients in the n equations. The Realist

dealing with physical quantities and units can happily

lump all changes between systems as changes in

"unit" but the Synthetiker must discriminate between effects of the coefficients in changing his

"quantities" and the effects of choices of basic units

which change his symbolic coherent units.

5.2. Coherent Abstract Units

The sequence of "unit equations," of which (5.1.7)

is an example, when each is combined with the

quantity equation like (5.1.6) appropriate to some

geometry g serves to define a sequence of symbolic

units, {x), which are independent not only of the

particular geometry, g, chosen in the defining

process but also of the particular coefficients. Kg,

used in the quantity equations of the system. As

an example of such a step in mechanics the quantity

equation for the constant linear acceleration which

causes a point to move a distance s in time t is

{a]{a)=2{s}{s)l[t}\tr-. (5.2.1)

The measm-e equation is

{a]n=2{s]nl{t}l. (5.2.2)

As noted above, (5.2.1) is true for any assortment

of units, but (5.2.2) must have consistent units here

indicated by the subscript n, in all terms. Dividing

(5.2.1) by (5.2.2) yields the unit equation

{0)n ={s)nl{t)l. (5.2.3)

Equation (5.2.3) is a quantity equation as is eq

(5.2.1) but unlike (5.2.1) it must not be interpreted

as meaning that a point having constant unit ac- celeration will move a distance s in time t. It does

state that if, for instance, the unit length in system n is the meter and that of time is the second, then

the coherent unit of acceleration is the meter/(sec)^.

(Williams [68] has referred to this type of treatment

as "an algebra of names.")

An alternative statement is that (5.2.1) defines the

mathematical operation of dividing a symbolic

quantity, distance, s, by the square of the symbolic

quantity, time, t, as creating such an amount, a/2, of the symbolic quantity, acceleration, that if it

were doubled it would correspond to the physical

acceleration which does, if maintained constant,

move a point a physical distance whose measure is [s] in a physical time whose measure is {i}. In

contrast with this (5.2.3) states that the operation

of division of a symbolic unit of distance, {s)n, in

system n by the square of a symbolic unit of time,

{t)n, in the same system produces that amount of

the symbolic quantity, acceleration, which is to be

taken as the coherent symbolic unit in system n, namely (a)„.

The application of this process to the n defining

equations yields n independent unit equations. The

Synthetilcer also selects, as a symbolic unit for each

of the N—n basic symbolic quantities, an example

which corresponds in the model to the basic physical

unit of the Realist. The n unit equations then suffice to deduce formally the n derived symbolic coherent

units for the other mathematical elements.

Inspection of eq (5.2.3) shows that this, and any

of the n unit equations, might have been written

by inspection by assigning to each basic symbolic

unit factor on the right side an exponent equal to the dunensional exponent (see sec. 7) appropriate

to the quantity whose symbolic unit appears on the

left side. The coefficients of all such unit equations

are necessarily 1. While this procedure suffices for the formal writing

of the symbols for the symbolic coherent units by

the Synthetiker, it appears to the Realist as an unsatisfactory guide for any analogous steps in the

laboratory. The Synthetiker therefore offers an

alternative procedure for defining the same symbolic

coherent units. This alternative is to go back to an

equation such as (5.2.2), set {t]n=l and {s}„=l/2

for the particular case of constant linear acceleration of a point for which (5.2.1) is appropriate.

The example of the symbolic quantity a (i.e., the

mathematical element) which corresponds to the

physical acceleration then existing constitutes by

definition the symbolic coherent unit of a. Although this type of definition, like the corresponding definition (sec. 5) of a germane physical unit,

refers to some particular case, it also involves the

coefficient (in this case 2) for the same case; and

hence the resulting symbolic unit will be independent

of what particular case is chosen. This independence

of the particular case is true of germane physical

units also. As examples in the electrical field let us consider

the symbolic units of charge in the classic CGS

electrostatic and electromagnetic systems. In the

former we write the quantity equation for two equal

point charges at a separation r in vacuo F=% (5.2.4)

and the measure equation

{F].= '^- (5.2.5)

19
Dividing eq (5.2.4) by (5.2.5) and rearranging gives

{Qs}s={r}s^s=cm"Y'' sec-'. (5.2.6)

In the electromagnetic system we ^vrite F=-^ (5.2.7)

and

Dividing eq (5.2.7) by (5.2.8) and rearranging now

gives

{Qm)m={t)n.W)'..= ^T^"Y"- (5-2.9)

In terms of the alternative form of definition of symbolic units, {Qm)m is one of the two equal ex- amples of Qm present when r=l cm and 7^=9-10^°

dynes. However, from (5.2.4) and (5.2.7) we find

that in general for the same phj^sical situation

Qs=cQ^; (5.2.10)

also from (5.2.5) and (5.2.8) for the same situation

iQs].= {c},,m{Q„,}m. (5.2.11)

Dividing (5.2.10) by (5.2.11) yields, since {v) is the same in systems s and m,

{Qs)M^)s.m{Qm)m. (5.2.12)

In other words, the two symbolic units, each coherent

with the dimensions of its symbolic quantity and

both coherent with the same basic CGS units, differ hy («)cGs, that is by having dimensions which difi^er by the dimension of velocity. This is to be expected

because each of the units is an example of the corre- sponding kind of symbolic quantity. The two

symbolic quantities also differ by the numerical

factor 3-10'° (approx) while the two symbolic units

differ only by the factor 1 cm/sec.

These relations are in marked contrast to those

which exist between the physical quantities and

units of the Realist. Both philosophies agree on

the measure eq (5.2.11), but the Realist considers

only the single physical quantity Q and measures

it by either of two physical units ^Uq or which

differ by a numerical factor of 3 10'°. Thus, in the

electrostatic system, the Realist regards {Q}s as the measure of a physical quantity Q in terras of

sUq, while the Synthetiker gets {Qs}s as the measure

of a symbolic quantity Qs in terms of {Q^s- In the

electromagnetic system, the Realist regards the

smaller as the measure of the same Q in terms

of the larger unit nJJQ. To the Sjmthetiker the

smaller {Qm]m is the measure of a symbolic quantity

which differs from Q, by a factor 3-10'° cm/sec,

in terms of a symbolic unit which differs from

(Qs)» by a factor of only 1 cm/sec. A similar shift

in the correspondence between the mathematical

models and the physical quantities is found in the

shift from an unrationalized to a rationalized system

(see sec. 8).

6. Systems of Measurement and

Representation

As outlined in sec. 2, the progressive improvements in experimental procedures by the Realist

and the invention of more useful concepts by the

Sjmthetiker have led to the use dming the past 100

3^ears in the technical literature of an unfortunately

large variety of dift'erent systems of measurement.

The more significant of these are described or listed in this section. Although systems based on the

centimeter, the gram, and the second as basic

mechanical units were historically the first to come

into use and are still widely used in manj^ branches

of science, systems based on the meter, the kilogram,

and the second are currently favored in electrical engineering and are gaining favor in physics. In

this section the latter group will be described first to exemplify the alternative modes of developing

a system of measm-ement.

6.1. Sizes of Units

For commercial and engineering purposes it is very

desirable that the measures dealt with in daUy

operations be numbers not too far removed from

unity. Hence units should be available of roughly

the same order of magnitude as the quantities to be

measured. It usually matters very little in commercial transactions whether there is a simple relation

between the units for physical quantities of different

kinds. In practice no one carries in his head or cares

to know how the inch, the mile, the acre, the gallon, or the kilowatthour are related. On the other hand,

in scientific work a very great convenience and reduction in burden on the memory is obtained if units are

related in systematic fashion. Hence a unique set of units germane to the equations to be used and to a few arbitrary basic units is the primary desideratum

and the insertion where needed of integral powers of ten as factors is no hardship.

In the past a great deal of effort has been wasted

in attempts to satisfy both sets of requirements by

the same set of units. A much wiser procedure is to start with a germane set as a basis. The needs of the

engineer and the marketplace can then be met by

applying decimal factors as needed to create an

assortment of non-germane units. An internationally

recognized set of prefixes for such decimally related

units is given in table 5." The prefixing of these

" The prefixes from "micro" to "mpija" seem to have been proposed at the initial Invention of the metric system. In 1870 the B A Committee on the Nomenclature

of Dynamical and Electrical Units approved a system suggested by Dr. G.

Johnstone Stoney for higher decimal mviltiples. In this system the cardinal number of the exponent of 10 is added after the name of the germane miit for positive exTionents and the ordinal number is prefixed to the name of the unit If the exponent is negative. Thus ' '10' prrams" is written as " 1 gram-nine" and 10"" gram is written as "1 eleventh-gram." This logical system was used very little and has been replaced by the additional prefixes nano, pico, giga, and tera.

20
syllables to the names of a germane unit is widely

recognized as producing the name of a decimally

related larger or smaller unit of the same kind.

Exceptions to these arguments are found in certain

specialized fields such as atomic physics. Here

measurement systems have been proposed in which

certain atomic constants, e.g., electron charge,

proton mass, Bohr magneton, etc., have been chosen

as basic units of a system [45, 61]. Similarly, in

astronomy the mean radius of the earth's orbit, the mass and luminosity of the sun, and the speed of

light have been used. The motive in these cases has

been not so much to avoid large decimal factors as to

correlate directly similar measurements on different

atoms or celestial objects without any reference to standards of human dimensions.

6.2, Nomenclature of Systems of Measurement

Systems of measurement may be classified in

various ways. If the sequence of derivation is started by assuming a conventional value for a magnetic quantity such as the system is called

electromagnetic. In this case becomes a constant

of nature to be determined experimentally. If the

sequence is started by assuming a conventional value

for an electrical quantity, the system is called

electrostatic. If the coefficients in the defining equations for geometrical configurations having spherical

or cylindrical symmetry involve explicit factors of

47r and of 27r respectively, while such explicit factors are absent in those equations pertinent to rectilinear

geometries, the system is said to be rationalized. A

system is symmetrical if the coefficients in the equations are such as to exhibit a symmetry between

electric quantities on the one hand and magnetic

quantities on the other. For example energy density

in the symmetrical Heaviside-Lorentz system is

while in the unsymmetrical, unrationalized CGS

electromagnetic system

The adjective absolute is often applied to the term

"system of measurement" or "set of units" to indicate that the units are chosen systematicaUy

and based on the units of length, mass, and time, to distinguish it from a system in which the units are based on more arbitrary prototype standards such

as the properties of particular materials, e.g., the

resistivity of mercury and the electrochemical

equivalent of silver. A system is complete or comprehensive if it is designed to be extended to cover the whole range of physical quantities by a single logical system. It

is partial or incomplete if its systematic use is limited

to only a portion of the entire field.

6.3. Development of MKSA Systems

The process by which a Realist buUds up a systematic set of physical units in the cm'rently popular

rationalized MKSA system is as follows. The

process starts by selecting the meter, the kilogram,

and the second as the three basic mechanical units,

each being defined by means of the prototype

standards listed in section 4.1 above. A set of germane

mechanical units is derived from them by choosing

values (usually unity) for the K's in the experimental measure equations of mechanics.

from {^rect}=-^i{w} {Z} -The unit of area (square

meter) is the area of a

rectangle of which the

product of the measures

of the sides is 1 {K-i = \).

from [v}=K2{l]l[t] The unit of velocity (meter/

second) is the velocity of a uniformly moving point

which traverses a distance whose measure is 1 m in a time whose measure is 1 sec {K2=\).

from {a]=Kz[l]l{tY The unit of acceleration

(meter/ (second)^) is the

uniform acceleration

which moves a point ini- tially at rest a distance

whose measure is }{ m in a time whose measure is

1 sec (^^3=2).

from [F}=Ki[m]{a} The unit of force (newton)

is that force which imparts to a mass whose

measure is 1 kg an acceleration whose measure

is 1 m/(sec)' {Ki=\).

It may be noted that the Realist is free to use any

one of the many possible measure equations to define a unit. He might have used {^circle] =7r{r}^, or

{a] = {^.v]l{M] equally well. The next step in buUding up the MKSA electro- magnetic system is to select an equation involving

both mechanical and electrical effects. The usual

choice is eq (12) of table 2 and to write

{/r}=ilMLQ!ill (6.3.1)

27r{r}

for the measure of the force in vacuo between elements

of length I of two infinitely long parallel conductors

spaced r meters apart, and carrying a current /. This step is an example of the case mentioned

above, in that the measures of two new physical

quantities / and jVm have been introduced simultaneously. As in the case of the gravitational

constant, 6 (see sec. 4.5), the magnetic constant,

rVm., introduced in eq (6.3.1) has at least three possible

interpretations.

21
On the first interpretation (which is analogous

with the second alternative in the gravitational case)

the Realist replaces {rT„i} by the numeric 47r-10~". Thus he defines the germane physical unit of current (ampere) as that constant current which, "if maintained in two straight parallel conductors of infinite length, of negligible circular section, and placed 1 m

apart in a vacuum will produce between these

conductors a force equal to 2X10"^ MKS units of

force per meter of length." On this first interpretation the ampere is a derived physical unit and the use of {rTm} involves no increase in the number (3) of basic units. The choice of a conventional (as

opposed to an experimental) number for the coefficient

in an electromagnetic (opposed to an electrostatic)

equation entitles the s^^stem to be called electromagnetic. The factor {rTm] serves as the primary

link between electromagnetic and mechanical units

in the system. This was emphasized in the lEC

resolution of 1938. This interpretation is tabulated

in row Ic of table 3. A second interpretation (which is analogous to the

third alternative in the gravitational case) is to consider that empty space is a prototype standard

embodying the property of magnetic permeability

and having a measure 47r l0~^ in terms of a fourth

basic unit (viz, the henry/ampere) of the system.

This second interpretation is the one often given by

Synthetikers and is tabulated in row la of table 3. However, the Realist has no reason to prefer one or the other. The desire of many writers to introduce

a fourth unit as basic really stems from a confusion

between the basic units of a set of physical units

and the basic generators of a system of physical di- mensions (see sec. 7). What is in effect a third interpretation of eq (6.3.1)

and the significance of r„, is embodied in Giorgi's

early proposals of the MKSA system [51, 53]. Approaching the problem from the Synthetiker's point

of view (in which "dimensions" and "units" are

closely linked), Giorgi pointed out the desirability

of considering his system as based on four basic units

and suggested that either the ampere or the ohm

might be chosen as the "fourth unit." Presumably

he thought of the silver coulometer or the column of

mercury as possible prototype standards on which to base the expermiental realization of his set of units.

Later Campbell [59], realizing that the stability of

high-grade alloy resistors exceeded the reproducibility of the mercury column, lu-ged the adoption of

what he called the "Definitive S3^stem of Units" in which the basic units were the meter, Idlogram,

second, and an ohm defined and maintained by a prototype standard resistor which would be kept in the custody of the International Bureau of Weights

and Measures. This proposal is listed in row 11 of table 3. The "definitive coulomb" would be

derived from this and the mechanical unit of energy

(joule) by setting Ki = l in the equation

{w}=KAQy{R}l{t}. (6.3.2)

'< This is a translation of the wording of the International Committee on Weights and Measures.

Campbell's suggestion was welcomed by Giorgi as a desirable modification of his earlier suggestions.

However, it was opposed by the national standardizing laboratories because they realized the impracticability of constructing a prototype standard of

resistance which would be adequately stable, and of

measuring power or energ}^ by mechanical means

with adequate accurac3^ The relegation of the magnetic constant to the status of an experimentally

determined quantity (in analogy to the fii-st interpretation of G) in what was otherwise an electromagnetic system was a further objection. In 1938

the lEC definitely rejected the "Definitive Sj^stem"

in favor of using a conventional value of as the

"link" between mechanical and electrical units.

Giorgi's espousal of this third interpretation has

created an ambiguity in the name "Giorgi System"

and thus has led many writers to prefer the less ambiguous name "MKSA System." Even the name

"MKSA" is a bit misleading because while the meter,

kilogram, and second are basic physical units of the

sj'Stem, the ampere is a derived physical unit on any

of the thi'ee interpretations set forth above. The

lEC in urging tliis name in 1950 was perhaps swa3^ed

by the feeling that current was a convenient fourth

dimension and that therefore its unit, the ampere,

should appear in the name of the system. In this

third interpretation (listed in row lb of table 3) the

Synthetiker chooses the ampere as a basic coherent

symbolic unit of current and defines it, not by a prototype standard, but by specifying that its magnitude

is such as to make the measure of by the rationalized eq (6.3.1) exacth^ equal to 47r-10~''. Thus in his

2{N—n) arbitrary choices of basic quantities and

units he chooses three mechanical quantities each

with its own unit, but he selects electric current as one

basic quantity but the unit of permeability as the

basic unit. It seems to the writer that the choice of

both the quantity and the unit of permeability as in the second interpretation (as on line la, table 3) is the more elegant. An alleged objection to this is that the use of permeability (or of resistance) as a

basic dimension leads to fractional dimensional ex- ponents, which are avoided by using current or charge as basic. However in the practical application of dimensional analysis the user is free to use any

set of dimensions he may choose regardless of those

used as basic in defining s3mibolic coherent units.

Having defined the ampere by eq (6.3.1), the next

steps are to define the other germane physical units,

viz, volt, ohm, coulomb, farad, henry, weber, and

tesla, by using in sequence eqs (6), (7), (8), (9), and

(10) of table 1, eq (14), column 3 of table 2, and eq (12)

of table 1 in that order. This process leads to the

sequence of definitions reading:

The Volt —The volt is the difference of electric potential between two points of a conducting wire carrjdng a constant current of 1 amp, when

the power dissipated between

these points is equal to 1 w. The Ohm —The ohm is the electric resistance

between two points of a conduc22
tor when a constant difference

of potential of 1 v, applied between these two points, produces

in this conductor a current of 1 amp, this conductor not being

the seat of any electromotive

force. The Coulomb—The coulomb is the quantity of

electricity transported in 1 sec by a current of 1 amp.

The Farad —The farad is the capacitance of a

capacitor between the plates of

which there appears a difference

of potential of 1 v when it is charged b^^ a quantity of elec- tricity equal to 1 coulomb.

The Henry •—The henry is the inductance of a

closed circuit in which an elec- tromotive force of 1 V is produced when the electric current

in the cu'cuit varies uniformly

at a rate of 1 amp/sec.

The Weber —The weber is the magnetic flux

which, linking a circuit of 1 turn,

produces in it an electromotive

force of 1 V as the flux is reduced

to zero at a uniform rate in 1 sec. The Tesla —The tesla is a flux density of one

weber/m-.

It must be kept in mind that neither the choice of

equation (those hsted from table 1 have K=l, but

this is immaterial) on which to base the new germane

unit for each new quantity, nor even the sequence

in which the units are developed is of importance.

If the equations are mutually consistent the same

germane physical unit will result from any sequence.

The names of these germane physical units are listed

in column 3 of table 4. The choice of a sequence for

any particular purpose depends largely on that purpose. That listed here was chosen by the International Committee on Weights and Measures because

of its convenience for concise legal wording. A quite

different sequence might be prepared by a teacher in his first explanation to a student. The Realist is guided largely by the attainable accuracy, convenience, or availability of particular apparatus in his

experimental realization of a derived unit in terms

of others.

The Synthetiker, starting with the symbols m, k,

s, and a for his basic symbolic units, can use the

dimensional exponents present in the same ecpations

listed in table 1 and table 2, column 2 and Avrite down by inspection the symbols for the derived co- herent symbolic units as listed in column 9 of table 4. Each abstract unit listed in column 9 corresponds in the rationalized system to the physical unit listed in the same row in column 3 and also in the unrationalized system to the physical unit listed in column 4. Although Giorgi initially urged the use of rationalized equations, the delay in the lEC between their

acceptance of his basic units in 1935 and their advocacy of rationahzation in 1950 has permitted the

accumulation of a considerable literature expressed

in a nonrationalized MKSA System. The measure

equations of this system are obtained by inserting

the values of the parameters in row 2 of table 3 in the appropriate places in column 2 of table 2. A

development in sequence similar to that in the ra- tionalized case will yield the appropriate set of ger- mane physical units. These will be the same as in

the rational system except that as shown in column

4, table 4 the physical units of D, i/', H, (R are smaller

in the unrationalized system by the factor 47r, while

that of magnetic polarization, J, is greater by this

factor. Also the constants ^r^^lO"^ and nTe=

107{c}^=l. 11-10"^° approximately in the unrationalized system. The measures of the electric and

magnetic susceptibilities of any given substance are smaller in the nonrationalized system by the factor

47r, but because these physical properties are defined

by simple numerics, even the Realist is content to consider that he is describing this property by dif- ferent physical quantities in the two systems.

6.4. CGS Systems

The impetus given to the CGS systems of measure- ment by the British Association Committee in 1873 was so great that they have received a justified

worldwide recognition and use. The equations for

the two classic systems based on the centimeter,

gram, and second as basic units are listed in table 1 and in columns 4 and 6 of table 2. The values of the parameters as given in rows 3 and 4 of table 3, when substituted in column 2 of table 2, wiU also give the equations for the CGS electrostatic and for the CGS electromagnetic systems respectively. It should be pointed out here that, in his Treatise,

Maxwell used a definition for electric displacement

density, D, which was smaller by a factor Aw than

the value fixed by the more symmetrical definition

used by most of the other early writers. The coefficients of D in his equations are therefore always

greater by 47r than in the classic equations.

To derive the physical units germane to the equations of the electrostatic system the Realist begins

with eq (10) of table 2, which, with the appropriate

parameters is

{F}=- 'Qi]{Q2

e r (6.4.1)

For empty space he sets e=l and also for an unrationalized electrostatic system sets {rf;} = l and

derives the unit of charge as that charge which when

placed 1 cm from another equal charge in vacuum

experiences a repulsive force of 1 d}me. The derivation of the units of current, electric field strength,

voltage, capacitance, etc., then can follow from eqs

(8), (17), (20), (9), etc., of table 1. The names of

resulting germane physical units are listed in column

5 of table 4. KenneUy suggested prefixing the syllable "stat-" to the names of units of the practical

(i.e., now MKSA) system to obtain "statcoulomb,

statvolt . . ." as names for the units thus defined.

This practice is widely used in the United States but not in Europe.

23
Similarly for the CGS electromagnetic system the

Realist starts historicall}^ with eq 11, table 2 for the

force between two equal magnetic poles and sets /x and Tm each equal to 1 in vacuo thus defining a physical unit of magnetic pole strength. Equation

(18) of table 1 then defines a unit of H. The third

step is then to uee the equation

{H]=kJ^^ (6.4.2)

for the magnetic field strength at the center of the

circular conducting loop in which there is a current whose measure is {/}. Setting Km=2iv for this sys- tem gives the electromagnetic unit of current Jji as germane to eq (6.4.2) and the centimeter, gram,

and second.

Using eqs (6), (7), (8), (9), and (10) of table 1, eq (14) of table 2, and eq (12) of table 1 in sequence

then yields the germane phj^sical CGS electro- magnetic units for voltage, resistance, charge, ca- pacitance, inductance, flux, and flux density,

respectively.

The names of the resulting germane physical

units are listed in column 6 of table 4. The prefix

"ab-" has been suggested for application to the names of the units of the MKSA system to give names appropriate to the electrical units of the

CGS electromagnetic system. Thus one obtains

abampere, abvolt, abohm, abcoulomb, abfarad,

abhenry, etc. These names are commonly used

in the United States but not in Europe.

Unfortunately this notation has not been extended

to magnetic units. In 1900 the AIEE had suggested

for consideration by the Paris Congress as names

of the CGS units the "gilbert" for magnetomotive

force, "oersted" for reluctance, "maxwell" for flux,

and "gauss" for flux density. However, the Paris

Congress of 1900 instead reported only two names,

viz, "maxwell" for flux and "gauss" for field intensity.

In 1930 the lEC confirmed the name "maxwell"

for flux but shifted the name "gauss" to flux density,

the "oersted" to magnetic field strength, and approved the "gilbert" for magnetomotive force. These assignments of names to the CGS magnetic

units broke down the earlier system by which the

units named after scientists had all been in the

"practical" system. One suggested way to remedy

this was to use the names "pra-maxwell" and

"pra-gilbert" for the practical units of flux and

magnetomotive force. However, in 1935 the lEC

adopted the name "weber" for the MKSA unit of

flux (10^ maxwells) and in 1954 it adopted "tesla"

for 10*^ gausses. The actions in 1900 also had the effect of favoring

the use of magnetic units which were not germane

to the practical (or International) system. Hence

CGS electromagnetic units are still widely used in specifying the properties of ferromagnetic materials

and the introduction of MKSA units for this purpose

has been retarded.

The 1930 lEC action had assigned different

dimensions (sec. 7) to flux density and magnetic

field strength and this was the motive for giving

different names to the units for these two quantities.

In terrestrial magnetism the name "gamma"

(symbol 7) is applied to a unit equal to 10~^ oersted

and is widel}^ used.

The discussions since 1930 on the theoretical advantages of basing a system of measurement on four

rather than three dimensions have led some writers to advocate modifying the two classic 3-dimensional

CGS systems by introducing what they usually

call a "fourth unit" as basic. Guggenheim [63, 109]

and Fleury [64] have suggested the name "fraiildin"

for a basic CGS electrostatic unit of charge, and

deBoer [118] has suggested "biot" as the name for a

basic CGS electromagnetic unit of current. To the

Realist these are merely synonyms for "statcoulomb"

and "abampere" (or "dekaampere") respectively

but to the Synthetiker they are very convenient

as building blocks for forming the names of two

complete modern sets of coherent symbolic electrical units on the basis of the universal and time-honored

CGS mechanical foundation. The equations for these systems when written with the constants

and Te appearing explicitly are symmetrical in form

like the MKSA equations. The Realist however

must distinguish the CGS-F, as an electrostatic

and the CGS-B as an electromagnetic system.

Fortunately the high accuracy to which c is currently

known makes this distinction rather academic

(see sec. 6.2).

Suggestions have also been made to rationalize

the 3-dimensional CGS systems but this step is usually combined with the introduction of symmetry

as in the Heaviside-Lorentz system.

The original pair of CGS electrostatic and electro- magnetic systems each had the very great convenience that either the permittivity, or alternatively

the permeability, of space (and also of many real materials) was assumed to be unity. This makes

each sj^stem very useful for certain problems but

very unhandy for others. Many textbooks use both systems, shifting from one to the other as needed.

Helmholtz and Lorentz attribute to Gauss the credit

for realizing the logic of assigning the same physical

dimensions to electric charge and to magnetic pole

strength because both Q^jr and rn^lr represented work.

Maxwell showed by combining eqs 13 and 14

(table 2) that electromagnetic phenomena may be

propagated in space by waves having the speed

given by

If we set {r„} = l and {r,} = l and {V,} = {c]=Z-W

approximately, we get the list of parameters in row

5 of table 3. If these are inserted in the equations

in column 2 of table 2 the resulting equations (given

in column 5) will be found to have various analogous

electric and magnetic quantities appear in symmetrical fashion. These equations are usually called

"Gaussian." In tliis system the units for electrical

quantities are the same as those of the CGS electrostatic system while those for magnetic quantities are

24
the same as those of the CGS electromagnetic sys- tem. The values of these physical units are identified

in columns 5 and 6 of table 4 by the notation

"(gaussian)."

About 1900 H. A. Lorentz took the further step

of applying the rationalization as suggested by

Heaviside to the symmetrical Gaussian equations

and used the resiilting "Heaviside-Lorentz'' equations and units germane to them in his \rritings.

His example has been foUowed in the textbooks of many Synthetikers. Most of the physical units in

this system differ from those in the other CGS

systems by factors involving powers of -\47r. Xo

individual names have been assigned to these units,

but the magnitudes of the germane physical units are hsted in column 7 of table 4.

6.5. Practical and International Systems

In contrast to the Gaussian and Lorentz systems,

which are of great convenience and elegance for

theoretical work but which are never used directly

in experimental operations, there are the "practical"

and the "International" systems invented for and

used by the electrical engineers. The six electrical imits (volt, ohm, ampere, coulomb, farad, henry)

and the two mechanical units, joule and watt, of the

practical system were defined as exact decimal

multiples of the germane units first (in 1S62) of an MGS electromagnetic system and later (since 1873)

of the CGS electromagnetic system. In the resolutions of the 1893 Chicago Congress and in the British

and .American legislation which immediately fol- lowed, the practical units were assumed to be

indistinguishable from imits defined by the mercury

column and the silver co\ilometer. However, the

London Conference of 1908 definitely restored the

distinction. The practical system was always recognized as being lirnited in applicability to electrical quantities. If extended in logical fashion retaining

the magnetic constant r„=l, the mechanical units

germane to it are found to be 10' m, 10~" g, and 1 sec and are seen to be very "impractical.'" It was therefore occasionally referred to as the "QuadrantEleventh-gram-Second (QES) System." TTith the

coming into use of the MKSA systems the use of the

name "practical" has faded out, but the same physical units, to which have been added the "weber" for

magnetic flux and the "tesla" for magnetic induction,

continue in constant use. ! From the Realist's point of view the germane

physical units of the MKSA system are identical in kind, magnitude, and name with those of the old

practical set. The Synthetiker dealing with coherent

sjmbohc units is careful to note that the practical

units being defined in terms of the CGS electromagnetic system must be considered as 3-dimensional

while the MKSA symbolic units are considered

4-dimensional.

The imits of the "International" set recognized

exphcitly by the London Conference of 1908 differed

in magnitude from the corresponding practical units

only by the smaU discrepancies present in the results

of the absolute measurements available at the turn

of the century. With the benefit of later determinations the International Committee on Weights and

Measures in 1946 [41] decided that the mean magnitudes of the International ohm and volt as then

maintained at the six cooperating national laboratories were related to the absolute (i.e., practical)

units as follows:

"1 mean International ohm = 1.00049 absolute ohms

1 mean International volt = 1.00034 absolute volts."

In the L'nited States the units as previously maintained and certified by the National Bureau of Standards had dift'ered slightly from the mean of the units

of all the national laboratories. Hence the changes

made January 1, 194S [43] to pass from the International to the absolute (practical or jSIKSA) units in

the United States were:

1 International ohm, or henry =1.000495

absolute ohms, or henrys

1 International volt or" weber= 1.000330

absolute volts, or webers

1 International ampere, or coulomb =

0.999835 absolute ampere, or coulomb

1 International farad=0. 999505 absolute

farad

1 Liternational watt, or joule=l. 000165

absolute watts, or joules.

Although the "Liternational" units were usually

considered as limited in apphcation to electric and

magnetic measurements it is quite possible to consider them as part of a complete system in which the

basic units are the centimeter, the second, the "International ampere," and the "International ohm,"

The unit of mass germane to these imits and the

usual electromagnetic equations is approximately

10' grams and the xmit of force is approximately 10' dynes. Because of the convenience and accuracy in

measuring power and energy by electrical means, this

international system did in effect constitute the

basis for practically ah precise scientific and industrial measurements for half a century.

6.6. Miscellaneous Systems

Li addition to the systems discussed in sections 6.3,

6.4, and 6.5, many others have been suggested and in some cases used to a limited extent. In his widely

used textbooks Karapetoft' [54, 55] used what he

called the "Ampere-Ohm System of Units." The

parameters of the equations of this system are listed in row 12 of table 3. It used rationalized equations.

In 1916 DeUinger [56] pointed out explicitly that

the engineering fraternity were in effect using the

complete system of "International Electrical Units"

as hsted in row 10 of table 3. He also pointed out

the desirability of rationalization, and being a Eeahst

suggested that the desirable rationalized measure

equations relating magnetic field strength and cm-rent could easily be obtained by using the ampere-turn as a non-germane unit of magnetomotive force in place

of the gilbert. However he was obliged in conse- quence to write {5}gauss==j^M{-S'}a-t/cm. He also

25
wrote {*}maxweii=lOj { E} yond{t] second bccausG the

germane "International" unit is the volt-second and

not the maxwell.

A more recent proposal which makes rather

fundamental changes in the electromagnetic equations has been made by Lobl [G9]. He suggests

removing r„j and Te from their usual positions in the

relations between B and H and D and E respectively.

Instead he introduces a corresponding pair of

dimensional constants c,„ and Ce into Maxwell's

circuital equations. He thus gets curl H=J+Ce

and curl E=—c,„-^- If equations with these

coefficients, which Lobl has called "Paritatische,"

are used with the meter, kilogram, and second as basic units, the concrete units of this system are

identical with those of the rationalized MKSA

system. The dimensions of several s^^mbolic quantities and of the corresponding units are however

different from those of the symbolic quantities in the

usual MKSA system. This proposal offers certain

advantages but it remains to be seen whether it will be adopted by the Synthetikers of the future.

6.7. The "Fourth Unit" Problem

In his original "Treatise" Maxwell had found it desirable to introduce as separate concepts and

therefore as distinct kinds of ph^^sical quantities the

members of the pairs (1) magnetic induction, B, and

magnetic field strength, H, and (2) displacement

density, D, and electric field strength, E. He wrote

B =juH and D=€E/47r, but later writers to preserve

the analogy between electric and magnetic equations

wrote D=eE. In the classic CGS electromagnetic

and electrostatic systems, the coefficients m and e respectively were assumed to be numerics and to have

in vacuo the measure 1. This meant that the

symbolic quantities B and H were of the same

dimensions. Hence their symbolic units were identical and were both called "gauss." A similar

situation existed for D and E. It became customary

to write

B=H-t-47rI (6.7.1)

and to state that H was that part of B produced by

the kno\\m macroscopic currents in the system, while

the intensity of magnetization, I, was the effect of

"concealed" Amperian currents. Riicker [102] in 1889 suggested that the classic

systems of measurement previously considered as 3-dimensional could be extended to become 4-dimensional by attributing dimensions other than numeric

to permittivity or to permeability. Although he was apparently motivated by a mistaken belief that

there was some "mystic" inherent connection between dimensions and Idnds of physical quantities

his suggestion aroused considerable interest. Other

early writers, notably Heaviside, also were careful

to discriminate between "absolute" and "relative"

permeability and permittivity, and to regard only

the latter as a pure numeric. It was included b}''

Giorgi in his early advocacy of the MKSA sj'stem. In 1930 the lEC discussed these ideas at great

length. The discussion was unusually acrimonious

because of the (at that time unrecognized) dift'er- ences in the habits of thought of the Realists and the

Synthetikers who participated. The latter finally

prevailed and voted officially "that the formula

B = MoH represents the modern concept of the physical relations for magnetic conditions in vacuo, it being understood that, in this expression, /xq possesses physical dimensions."

This action really involved more than a mere

choice of a convenient dimensional label, but was meant to recommend the practice of regarding the

physical quantities "magnetic induction" and "magnetic-field strength" as differing in kind. Hence

their physical units were entitled to distinguishing names. It also required, as implied by the resolution, that new coefficients and (in their notation "mo" and "eo") should be written explicitly in all appropriate equations. Because of lack of appreciation of the distinction betw'een units and dimensions,

this action also initiated a demand for the official adoption of a "fourth basic unit." In 1938 the lEC

reconunended that the assumption of lO"'^ in the

unrationalized and 47r-10~' in the rationalized MKSA

system as the, not necessarily dimensionless, value

for Fm gave a sufficient link between electrical and

mechanical units. In spite of this, the mistaken

demand for an official selection of a particular "fourth

unit" continued until 1950 when the lEC recom- mended "that, for the purpose of developing the

definitions of the units, the fourth principal unit

should preferably be the ampere as defined by the

General Conference on Weights and Measures."

This is a minor convenience if the ampere is considered merely as a fourth basic symbolic unit from

which to derive mathematically the other symbolic

units of the MKSA system. It is, however, from

the point of view of the Realist definitely erroneous

to consider the ampere as a basic physical unit

because it must be experimentally derived from the

basic mechanical units using an arbitrarily assigned

value for r„j. However, the shift in the chosen number of basic

dimensions from three to four involved no change

in the coefficients in the equations, nor in the magnitudes of the physical units, germane to any given

set of basic units. Hence the International Conference on Weights and Measures was careful later to avoid any reference to dimensions in its announce- ment of the shift in 1948 from the "International"

to the "absolute" set of units. The only immediate

effect on the Realist in 1930 was the change in the

name of the physical unit of magnetic field strength

from "gauss" to "oersted," and the discontinuance

of the previous unofficial use of "oersted" as the

name of the CGS unit of magnetic reluctance.

The eft'ects of the change in dimensions on the

Synthetiker are much greater and more complex

than might be thought at first sight. In the first place the distinction between the concepts of

"relative" and "absolute" permittivit}^ and perme26
ability had to be explicitly recognized in all pertinent

equations. New sjonbols tr and /x^- were introduced

to denote the relative quantities and eo and mo to denote the particular values of e and ii applicable

in vacuo. This system of symbols has not proved

entirely satisfactory because many subscripts other

than r are needed to denote particular states or components of physical systems to which the values

of permeability or permittivity apply. Even the

subscript o is used to denote initial permeability of

ferromagnetic materials. Also the concepts denoted

by €o and juq owe their primary significance to the

fact that in any system of measurement they are conventionally chosen constants characteristic of the

system. An international movement has therefore

started to give eo and /zq the names "electric constant"

and "magnetic constant" respectively. In further- ance of this change the new distinctive symbols

Ve and Vm have been used in this paper and elsewhere.

Further effects of the explicit recognition of these

distinctions can be seen by considering the eq (6.7.1)

relating the magnetic flux density, B, in a material

to the magnetic field strength, H, and the intensity

of magnetization, I, at any point. On the new basis

in a rationalized system we must write either

B=Mr„H=r„,H+J (6.7.2) or B=Mr.H=r„(H+M). (6.7.3)

The lEC in 1954 instead of choosing between (6.7.2)

and (6.7.3) preferred to recognize both J and M as useful concepts. They have been named "magnetic

polarization" and "magnetization" respectively. In

a rationalized system J is identical with the "intrinsic induction" usually denoted by Bj, while in an

vmrationalized system

Bi =47rJ. (6.7.4)

In all cases J = r^M. (6.7.5)

In the older 3-dimensional system the volume

integral of the intensity of magnetization taken over

a magnetized body was defined as the "magnetic

moment." In the 4-dimensional system this concept

also becomes bivalent. Thus the volume integral

of M has been caUed the "area moment" of a magnet

or of a current loop, and for a plane loop is equal to the product of the current by the area. The volume

integral of J has been called the dipole moment, and

in the case of a long, slender permanent magnet is equal to the product of its pole strength by its length.

It would, of course, be possible to push this duality a

bit further and define two kinds of magnetic poles.

However, this step has not received any formal support. A more recent proposal is to call the volume

integral of M the "electromagnetic moment" and to ignore dipole moment. The torque on a magnetized

body would be the product of this "electromagnetic

moment" by the induction, B. A similar duality of course exists in the electrostatic case. Usually one writes in the rationalized

system

D=r,eE = r,E +P, (6.7.6)

where P is called "electric polarization." The name

"electrization" has been suggested [65, 66] for the

quantity P/r^ but no formal action has been taken as

yet.

Still another effect of the use of four basic dimensions is, of course, to introduce difi^erent dimensional

labels for many other quantities. This is a matter

of slight importance to the Kealist. For the Synthetiker, however, it means he must discriminate

between the various mathematical elements (symbolic quantities) which in the various systems cor- respond with a single given physical quantity. 'Also,

the symbolic cohereiat units for these symbolic quantities will change in dimensionality and symbolism

though not in magnitude.

In 1930 the lEC had in fact formally assigned a 4-dimensional nature to the CGS electromagnetic

system, prior to its adoption of the MKS system

(1935) and its adoption of subrationalization (1950).

However, it had not explicitly amended or rescinded

any actions of earlier organizations which had

clearly recognized the classic systems as 3-dimensional. To minimize ambiguity, the introduction of a pair of 4-dimensional CGS systems, one electro- static and the other electromagnetic, has been urged

to replace the classic CGS systems. The names

"franldin" for the 4-dimensional basic symbolic unit

of charge in the electrostatic system and "biot" for the 4-dimensional basic symbolic unit of current in the electromagnetic system have been proposed (see also p. 24). On this basis in 1951 the SUN Committee of the International Union of Pure and

Applied Physics [8] recommended the introduction

of such a pair of systems, though not of the particular new unit names. The sizes of the franklin

and the biot are chosen so that Ve=l in the CGSFranklin system, while Vm,= l in the CGS-Biot

system. Hence 1 biot={c} franldins/sec. In other

words, these systems are identical to the systems

proposed by Riicker except for the choice of which

units are called "basic." They are symmetrical

but not rationalized, although they might be. The choice of equations in the CGS Biot and

Franklin systems makes all the symbolic quantities

in them identical with the corresponding quantities

in the unrationalized MKSA system. The differ- ences in the measures in the CGS-Biot and the

unrationalized MKSA systems arise only from the

decimal difl^erences in the sizes of their basic units.

This produces corresponding decimal differences in both the coherent symbolic and the germane ph^'sical

units. When compared with the rationalized MKSA

(Giorgi) system, the symbolic quantities in some

cases differ by a factor of 47r as well as by decimal

factors. This is also true of the germane physical

units of the CGS-Biot system. The coherent symbolic units however dift'er only by decimal factors

from those of the rationalized MKSA system.

7. Dimensions

The concept of dimensions initiated by Fourier in 1822 [101] is so closely related to and so often con- fused with that of units that a brief discussion of

27
this concept from our two alternative points of

view seems desirable. A dimension ma}'' be described

as a label of convenience attached to a symbolic

quantity to give some, but not complete, information about its relations to other quantities. The

name "dimension" originated from the elementary

application of the concept in geometry in which

surfaces and volumes, being quantities measured by

multiplying together two or thi-ee lengths measured

in mutually orthogonal directions, were said to have "dimensions" "2" or "3" with respect to length. This meant that if the symbolic unit of

length were imagined to decrease by, say, 1 percent,

the measures in terms of the resulting decreased

coherent symbolic units of area and of volume would

be increased by 2 percent and 3 percent respectively, to a first approximation.

These notions are readily extended to all symbolic

quantities. It can be shown (Bucldngham [103],

Bridgman [104]) that any of the n quantity equations

by wiiich a derived symbolic quantity, Q, is defined

in terms of some or all of the N—n basic symbolic

quantities A, B . . ., present in some particular

system can be put in the form

Q=KA"B^ .... (7.1) Here X is a constant not afl^ected by any change in the basic symbolic units of the system provided

coherence is maintained. {K may, of course, depend upon the relative magnitudes of all quantities

of each kind in the system, e.g., the shape of parts,

the ratios of resistances in various arms of a network,

etc. When rewritten as a measure equation, (7.1)

becomes

{Q]a=K{An.{B]^, . . . (7.2)

where the subscripts a denote that these measures

are in terms of a particular coherent set of units.

Let us now assume a shift to a new set of coherent

units, denoted by b, in which the new basic symbolic

unit of A is decreased by a factor X while the other

basic symbolic units are unchanged. Then

{A},=X{A},, (7.3)

but since eq (7.1) is still true regardless of arbitrary

changes in sizes of basic units we must also have

{Q},=K{A}UB}t .... (7.4) By (7.3)

{^}?=X''{^}«, (7.5)

hence for (7.4) to remain true

{Q},=X"{Q}, (7.6) or {Q),=X-"{Q\. (7.7) Hence the exponent a indicates the relative rates at which the measures and inversely the coherent

symbolic units of Q and A must vary.

It is customarj'' to summarize the relations of one

quantity, Q, to the group of basic quantities A, B,

etc., by \vriting

[Q]=[A"B» . . .]. (7.8)

This is often called a "dimensional equation" and

is in effect a concise form for encoding the dimensional exponents a, /S, ... in relation to the quantities each connects. Either member of eq (7.8) is called "the dimension of Q." In the particular case where a=^— . . . =0,Q is said to "have the

dimension of a numeric" or in common parlance to be "dimensionless." It will be noted that the information contained in a dimensional equation such

as (7.8) is illustrated in section 4.4 by its prediction

of the form of the experimental eq (4.4.1) but that it fails to give the information in the proportionality

4.4.3 (p. 13). It places no limitation on the value

of KpsnAlthough (7.1) was assumed to relate Q only to basic quantities A, B, etc., of the system, this limitation is not necessary and A, B, etc., can equally

well be members of any other convenient alternative

group of independent quantities not normally

considered basic. The resulting dimensional ex- ponents are then equally useful in checking for

blunders in algebraic manipulations and in dimensional analysis. The symbolic units listed in columns

9, 10, 11, 13, and 15 of table 4 show the exponents

for a number of quantities in reference to several

measurement systems and to alternative sets of

basic dimensions.

Buckingham's [103] Il-theorem shows that with

certain restrictions any complete phj^sical equation

relating symbolic quantities can be put in the form

;A(ni, Hs, . . .n,) =0 (7.9)

where each of the n's is a product of powers of some

of the symbolic quantities involved, raised to such

exponents that the entire product has the dimension

of a numeric. Here \p indicates any function of the

independent arguments IIi, 112, . . . and i is the

maximum number of independent dimensionless

products which can be found by combining in various

ways the AT' quantities involved in the particular

problem. Tliis number of dimensionless products

(or independent arguments of the function \p) is equal to the excess of the number of quantities

involved in the particular problem over the number

N—n of basic dimensions of the system. The smaller

the number of n's the more definite is the information that can be obtained by dimensional analysis.

It is partly for this reason that systems of measure- ment considered to involve four rather than three

basic dimensions are much preferred by Synthetikers.

The other practical application of dimensions

(i.e., to the detection of blunders) is of interest to the Kealist as well as the Synthetiker. A measure

equation must remain true if expressed in a set either of germane physical or of coherent symbolic

units, even though the sizes of the basic units are changed. Hence in an3^ equation as a check one

substitutes for each quantity or measure the dimen28
sion of its corresponding symbolic unit as given in

table 4. In the resulting dimensional equation the

dimensional exponents for each of the basic dimensions will be foimd to be the same in aU the terms. A failure to meet this test indicates an error in the

original equation. Unfortunately the converse

is not true and this dimensional check does not

guarantee the correctness of the nirmerical coeffi- cients of the terms. In past literature much confusion will be found

which has originated in the unwarranted assmiiption

that in some mystic fashion the dimensions assigned

to a quantity were related to the physical natiire of the quantity. This is not necessarily true. In

terms of the dimensional system commonly used

it appears that, for example, both rationalized

and unrationalized magnetic field strength have

the same dimensions [11"^], although the true Synthetiker regards them as different quantities.

Resistance and reactance as weU as magnetomotive

force and current illustrate other pairs of quantities

usually considered as isodimensional but wliich are considered by the Realist to be physicaUj^ quite

different in nature. There is therefore no direct

general connection between dimensions and physical

imits and quantities. A recent suggestion by Page

[90] which assigns to plane angles a dimension

different from that of numerics offers an escape from

these apparent inconsistencies. The physical quantity, electric charge, though conceived as unique by

the Realist, was assigned different dimensions (i.e., [U'^AfH-'] and [V-M^^]) in the two classic CGS

systems.

On the other hand the dunensional exponents for a given symbolic quantity', relative to a set of more

basic quantities, are identical with the exponents

in the unit equation which relates the coherent symbohc

unit of that quantity to the s^mibohc imits of the more basic quantities. Hence the dimension of a given quantity can be thought of as a sort of generalized sjTnbohc unit which retains some of the information specified by the latter but which is not limited to any particular choice of the sizes of the basic imits.

To surmnarize these relations, we see that the

dimensional exponents (a, ^, etc.) appearing in

either an experimental measure equation or the

corresponding pahs of symbolic quantities, form

the label ([(?] = [A" . . .]), or dimension appropriate to aU examples of that kind of quantity.

For any given system of basic symbolic imits the

insertion of the same dimensional exponents gives

the corresponding unit equation {Q)a={A)a-

(M (e.g., (W)cGs=cm2 gi sec Corresponding to each symbolic unit we have a germane

physical unit defined as that example of the physical

quantity existing when it has the measure "1" in terms of the more basic physical units.

As an example of the application of dimensional

analysis consider the braking action of a drag magnet

on the rotating disk of a watthour meter. We may

assume that for a series of geometrically similar

combinations of magnet and disk, the retarding

torque, r, depends only on the angular speed, w, of

the disk, the average fiux density, B, under the magnet poles, the resistivity, p, of the disk, and some

linear dimension, D, which fixes the mechanical size of the structure. Attacking the problem first with

three basic dimensions which we choose as force,

length, and time we write in column 2 of table 6

the dimension of each of the five symbolic quantities. Since the number of variables, 5, exceeds the

number of basic dimensions, 3, by 2, we fmd by the

methods of Buckingham or Bridgman [103, 104]

that the situation is describable by an equation of the form

i/'(ni,n2)=o

with two dimensionless products. These are Ui=D-'B-'t

and

(7.10)

(7.11)

(7.12)

Hence we can write without loss of generahty

ni=i/'2(n2) or

T=BW^2{pB-'<^-'). (7.13)

Only by using additional information, such as ex- perimental data showing that t varies as co+\ can we infer that ip2{^)='^~^ and find how t varies with

CO and D.

Table 6

Quantity Dimension in

F, L, T F, L, I, T

Torane . . _ _ r [FL]

[T->]

[F'i2L-i] m

[FL]

[T-q [F [FLn-iT->]

[L]

Anenlar sDeed - . _ _ w

Flux density B..

Resistivity ^ p.. Size.. D..

In contrast to this let us use an analysis employing

four dimensions, choosing F, L, I, and T as basic.

These yield the dimensions in the last column of

table 6. Since the number of variables is greater by

only 1 than the number of basic dimensions, there

exists only the single dimensionless product

II=tD-'B-'po:-\

Hence we get directly

T=KD'B'o}/p.

(7.14)

(7.15)

As a means of obtaining a more satisfying symmetry and also perhaps in order to get more effective- ness in dimensional analysis some writers have

proposed the use of five basic dimensions in defining

sets of symbolic quantities and units. The present

status of these suggestions is summarized by StUle

[10]. 8. Rationalization

A major cause of the proliferation of the unduly

large number of alternative systems of measurement

29
in the field of electricity has been the quest for what

Heaviside called "rationalization." The underlying

ideas can best be illustrated by considering the concepts used in electrostatics. The results of experiments using arbitrar}' preliminary phj'sical units can be expressed by the equation

{F,},=KaQ,},{Q,},IU}[r]l (8.1)

which introduces the measures of a new^ quantity,

electric charge, Q. A symbolic quantity, Qs, which

will correspond to Q is then postulated, as satisfying

the quantity equation analogous to (8.1) and a numerical factor e which woidd depend on an intervening isotropic medium. A further step is to postulate an electric field as a symbolic vector quantitj^ E

defined by the equation

Fi=/^2QiE (8.2)

where Fj is the force on charge in field E. Maxwell also conceived of another sjanbolic vector

quantity which corresponded to a postulated

outward displacement caused by the presence of the

concentrated charge Q2 and was related to it by

^T),^ndA=K^Q, (8.3)

where the integral is taken over a closed surface

surrounding ^2- On a spherical surface of radius r, centered on Qo

B^=K^Q^x,IATrr\ (8.4)

Maxwell showed that there must be the further

relation

Y)=Ki€E (8.5)

for isotropic media. Regarding (S.l) as a quantity

equation it can be factored and combined with the

otliers to give

K2'E,^¥,|Q,=K^Q2Vx|ir^ =^^^K^Y)2|K^e =4.^^K,K^Y.|K^

(8.6)

whence

K,=A-kK,K^K,. (8.7) With three new symbolic quantities Q, E, D to define,

the Synthetiker is free to assign any values he desires

to threeof the K's, the fourth tlien being fixed by (8.7).

The classic choice as Maxwell himself wrote "miless an absurd and useless coefficient be introduced" was to make Kx = \. Also Ko is universally taken

as 1. Most classical writers also chose — 1 and

hence Kz='^-k in both the electrostatic and the

analogous magnetostatic equations. Maxwell liimself wrote B =iuH like the others but wrote D =eE/47r,

thus introducing a partial rationalization in the

equations in his treatise. As can be seen from columns 4, 5, and 6 of table 2

this classic choice of the K's, leads to the appearance

of an explicit factor "4Tr" in many equations where

it would not be expected, such as the field equations

13 and 16 and the capacitance of a rectangidar plate

capacitor (eq (22)). On the other hand, 4ir does

not appear in the formulas such as that for the

capacitance between concentric spheres (eq 23)

where it would be expected from the spherical

symmetry. Heaviside in the 1880's called attention

to this "disease" which he called an "eruption of

47r's" and vigorously urged tlie use of tlie alternative

choices 1^3=1 and Xi = l/47r as the basis for devising

a more "rational" set of "units." His continual

reference to a change of units indicates that, although

a theoretician par excellence, he had the habit of a Realist in thinking of electromagnetic equations as measure equations, the coefficients of which can be

changed by a new choice of physical units. The

changes in the physical units chosen by him for Q

and for magnetic pole strength propagate throughout the rest of the rationalized SA^stem of measure- ment so that practically all the physical units are

affected. Colmnn 7 of table 4 indicates the physical

units Heaviside proposed and their relation to those

of the classic systems listed in columns 5 and 6. The insertion of the parameters listed in row 6 of

table 3 into the equations of column 2 of table 2

will give the rationalized equations which he preferred. This system was used by Lorentz [52] and

other theoretical writers but the concrete physical

units of the "practical" system had become embodied

in so many standard instruments that a shift to the

Heaviside system was quite impractical.

Although the expression "rationalized units" has

been used almost universally in the literature when

referring to the Heaviside-Lorentz system, it would

have been much more logical to consider the rationalized equations as being also quantity equations. As

such they serve to define a new set of rationalized

symbolic quantities. It is mathematical elements

thus defined with which Lorentz constructed his

mathematical model of the electron. In columns 12

and 14 of table 4 the symbols with subscript h denote

the symbolic quantities rationalized in accordance

with Heaviside's equations which correspond to the

phj'sical quantity listed in the same row of columns

1 and 2. Column 13 or 15 gives the coherent symbolic unit appropriate to each of the rationalized symbolic quantities in column 12 or 14. The greatest inconvenience from the 47r's occurs in magnetic measurements. Remedial changes in the

definition of magnetomotive force, and of H were

suggested by Perry [113] in 1891 and Baily [114] in

1895. A more complete system to which Kennelly

has given the name "subrationalization" was proposed by Fessenden [116] in 1900. These ideas were

incorporated by Giorgi [53] in his proposals of 1901

and similar choices of coefficient were urged later by

Karapetoff [54, 55], Dellinger [56], Darrieus, and

others.

The Fessenden scheme when combined with the use of four basic symbolic quantities involves changing the electric and magnetic constants so that

,r,=„r,/47r (8.8) and

,r„=47r„r„, (8.9)

30
but keeps

In electrostatics one wiites

Ai=i/47r.re=iAr, (8.10) and. sets A4=.r,=„r,/4x. (8.11)

Here the net effect is to make no net change in Coulomb's law and thus to leave the s}Tnbolic quantities

Qs, Is, and many other quantities unaltered. However, the change in A4 reduces D, Te, and \p by a factor

of 47r. In magnetostatics one writes

K^=\/4wrT„ (8.12)

and sets K,=,T^^4t„T^. - (8.13)

This increases the denominator of the Coulomb's law

expression by (47r)- and defines a new rationalized

magnetic pole larger by 4x than the classic. To re- tain K2=l the clefinitions of H and must be

changed also, giving

rH=^H/4^ and =„J^/47r (8.14)

and making corresponding changes in J and (R. However, the changed value of has a compensating

effect so that B, and M are not affected.

If the Fessenden rationalized choice of coefficients

is considered as leading to a change in the germane

physical units used to measure certain physical quantities, its effect is seen by reference to column 4 of table 4. Here are hsted the unrationalized physical

units for the six physical quantities affected out of the total Ust of 26 for which the rationalized units are

listed in column 3. Alternatively from the Sjmthetiker's point of view, the Fessenden subrationalization has changed the definitions of the six symbolic

quantities indicated in column 8 of table 4. Here

the subprefLxes n and r denote the unrationalized and

rationalized SATiibolic quantities respective!}'. The

corresponding symbolic units in the 4-dimensional

electromagnetic system are listed in colunuis 9 and

10 and in the 4-dimensional electrostatic system in

column 11. This change in the s3Tnbolic quantities is also tabulated in table 7, which is in a form to be used

when translating a quantity equation in an unrationalized system to the corresponding equation

in a subrationalized system (i.e., Fessenden or Giorgi rationahzation) or vice versa. The fact that subrationalization affected only a

fraction of the various quantities and the more

important fact that the quantities affected and

their physical units were not such as are usually

embodied in physical standards, made its mtroduction far more practicable than Heaviside's

earlier proposal. All the quantities listed in table

7 are of the nature of auxiliary concepts to some

extent removed from direct experimental operations

and their measures are always postulated or computed from those of other more tangible quantities.

These facts doubtless account for the gradually

increasing acceptance of subrationalization.

The further fact that the adoption of rationalization by the lEC in 1950 occurred soon after the

renaissance of "quantity calculus" had led Synthetikers to regard the process of rationalization

merely as a change in the coefficients of certain

equations without changing any dimensional ex- ponents. The}^ thus conclude, logically, that the

coherent s^nnbolic imits are not affected by the

change which is therefore to be considered -to be

mereh^ the use of a new set of rationalized symbolic

quantities. Apparently, it was on such a basis that

the SUN Committee of the lUPAP voted that

rationalization should be regarded as a change only

of quantities and not of units. This action completely ignores the other side of the coin and the

fact that the Realist usually prefers to use changed

germane phj'sical units to measure unchanged

ph3'sical quantities. In the councils of the lEC,

both points of view are represented but until recentlj' the protagonists of each have faUed to appreciate the advantages of the alternative

approach.

Konig [88] was one of the first to realize the

existence of the two points of view of the "Realist"

and the "Synthetiker" and to distinguish between

two "levels of abstraction," experimental and

dimensional, which correspond to physical and

S3Tnbolic quantities respectivel}'. He has also made

a valiant attempt to develop a complete new specialized algebra designed to handle mathematically

the relations between physical quantities considered

as mathematical variables. In this modified quantity calculus, the Realist finds preserved his fond

tradition that the quantitj- remains invariant even though the ec^uations are rationalized. The required

departures from the rules of ordinary algebra,

however, are so serious as to probablj^ discourage

the tj-pical Realist, who is normally content to be

limited to measm'e equations. Hence, there seems

little to be gained by creating stiLl another mathematical model inteiTnediate between those here

called phA^sical and s^mibolic.

JMost writers have followed the historical sequence

in which the science was confronted with a change.

The}' describe rationalization as a process by which

an older sj'stem of measm'ement is changed to a newer one. The Realist sees it as a change in imits

and the Synthetiker as a change in quantities. Both

consider that the other's process must lead either to noncoherence or to a situation where the manner of

describing a physical situation changes the situation

itself. Either is anathema.

If, on the other hand, one considers that the science

is confronted with a choice between two alternative

systems each of which is internally logical and consistent, the appearance of paradox is largely avoided.

In an}^ single complete system, either rationalized or unrationalized, there exist both (1) a pair of sets of

phj^sical and of symbolic quantities, the members of

which correspond in a manner dependent on the

chosen equations of the particular system and also

(2) a pair of sets of germane phj^sical and of coherent

31
sjmibolic units which also correspond in accordance

with the same equations as well as with the choice

of basic units. In either system the correspondence

between the two sets of each pair is complete and

self-consistent. The 1950 decision of the lEC was

merely to recommend the future use of a particular

measurement system with its particular correspondences. The chosen correspondences are neither more

nor less self-consistent than those previously used.

9. Summary and Conclusions

It wUl be seen from the foregoing that there has

been a long evolution of the systems of units and

measiu'ements in the electrical field in which an initial very excellent start has been successively improved upon but unavoidably at the cost of accu- mulating complexity and confusion in the literature.

This evolution is perhaps nearing its end, and the

complexity may be reduced shortly to the "peaceful

coexistence" of two systems. The MKSA system in

its rationalized form has now won almost universal

acceptance in electrical engineering and its use seems

to be spreading in physics and in other branches of

engineering. The older CGS system still holds undisputed sway in many other branches of science. In

electrophysics it is stUl widely used either in the symmetrical Gaussian form, or by the practically equivalent process of using the CGS electrostatic system

for electrostatic pi'oblems and the CGS electromagnetic system for magnetic problems. It seems questionable whether the 4-dimensional CGS systems will be much used as alternatives during an interim period.

The practical line of development which leads to the experimental definition, establishment, maintenance, and dissemination of the physical electrical

units seems to be in very satisfactory shape. The

national standardizing laboratories, coordinated by

the services of the International Bureau of Weights

and Measures, are continually gaining in the scope

and accuracy of their facilities. The lower echelons

of the hierarchy of standardizing laboratories are rapidly increasing in niunbers and in their recognition by industry and commerce as essential links in the interdependent network of modern manufacturing. The next step in the series of adjustments of

the maintained electrical units closer to their ideal

value will surely amount to only a very few parts

per million and may not be needed for a long time.

The theoretical line of development of measure- ment systems and nomenclature is temporarily

bogged down in the discussions of various international organizations by what superficially seem to be

semantic difficulties, i.e., the use of words like "unit"

and "quantity" each with two different meanings.

However, this is merely a sjanptom of the still deeper

difi^erence in the habits of thought of the two classes of workers in the electrical field. These difficulties can be largely avoided by the careful explicit recognition, as exemplified in this paper, of the two dis- tinct ways of looking at the S3^stems of measurement

and their equations. The results of this distinction can be seen by the following summarization.

The Realist deals only with the concepts of

physical quantities which are characterized qualitatively by "kind" and quantitatively by "magnitude," which he regards as fixed by nature and as independent of the units in terms of which they are measured and the equations used to relate the

results of such measurements. He uses only physical units, i.e., specified samples of each kind of

physical quantity to which he has assigned the measure "1". He deals only with measure equations

in which the literal symbols represent the numerical measures of his physical quantities. He commonl}^,

but by no means universally, prefers to use a set of

pliysical units defined by a choice of (a) a small

nimiber of basic units, (b) a set of equations with

generally recognized simple coefficients, and (c) a

set of derived physical units which are germane

both to the basic units and to the equations. However he often for convenience uses other non-germane

units, defined as numerical multiples of the normal

gemiane unit and simultaneously he modifies accordingly the coefficient in the equations concerned to restore germaneness. He is therefore constantly aware that his equations are true only in a set of

consistent (i.e., geiTnane) units. Hence he fre- quently writes "in units this equation

becomes ." In all operations he trusts the principle that the measure of a given quantity

varies inversely as the unit used to measiu-e it, regardless of whether the change in the unit is the

result of a change in a basic unit of the system, of a change in the coefficient in an equation (e.g., rationalization), or of theuseof anon-genuane unit. Hence

the conversion factors for measures given in table 8 are the reciprocals of the ratios of his corresponding

physical units. He therefore, for example, writes

as quoted at (b) on p. 1 when comparing the measures of a particular magnetic field in terms of

two alternative physical units "the number of

ampere-turns per meter=1000/47r times the number

of oersteds."

On the other hand, the Synthetiker deals only

with sjTTibolic quantities (i.e., mathematical ele- ments) which are defined by a set of quantity equations. Symbolic quantities are characterized qualitatively by "dimensionality" and quantitatively by

"magnitude." His equations are identical in form

to the systematic measure equations of the Realist,

but the letter sjrmbols in the Synthetiker's quantity

equation represent the complete concept of symbolic

quantity both qualitative and quantitative. From

the parallel between his quantity equations and the

Realist's measure equations, he sets up a correspond- ence between his symbolic quantities and the

Realist's physical quantities, giving them the same

name. During the evolution of the science different

coefficients have been used in certain equations.

Each of the resulting sets of equations has in general

constituted a new and different mathematical model

with new and difl'erent correspondences between

the symbolic and the physical quantities of the same name. Hence two sjonbolic quantities which

in different models correspond to the same physical

32
quantity may be of different magnitude and even

of different dimensionality. The Sjoathetiker's quantity equations are true regardless of units and in his

operations he has very little use for the concepts of

miit and of measure. However, for completeness

he conceives of a symbolic unit for each symbolic

quantity. This is sjnnbolized by writing the product of a set of basic symbolic units each raised to the same dimensional exponent as in the expression

for the dimension of the symbolic quantity of which

it is the unit. Changes in the coefficients in the

equations do not change the dimensional exponents

and therefore do not change the coherent symbolic

units of any quantity. The Synthetiker has little use for noncoherent units ; hence his units are invariant to changes in the coefficients in the equations.

They are changed only by changes in the sizes of his basic imits. Hence the conversion factors in table

8 are not the reciprocals of the corresponding symbolic imits except when the change in measures

results solely from a change in the basic units.

The Synthetiker for example therefore wiites as quoted at (a) on p. 1 when comparing the symbohc units of two alternative measurement systems

"1 oersted=1000 ampere-turns per meter."

A person working only in one system of measure- ment can continue to think sometimes as a Realist

and at other times as a Synthetiker. He can use

the same words as names for both kinds of quantities

and units, but would be wise to be aware at all times

which role he is playing. The writer who is concerned with the relatively rare paper which involves

the comparison or discussion of more than one measurement system has a much greater need to be

constantly alert as to his role and should for clarity

indicate to his reader by the appropriate use of

adjectives, such as "physical" or "symbolic" or

their equivalents, just what level and type of concept he is discussing in any particular paragraph.

A material help could be secured by the consistent use of the unit names as listed in column 3, 5, 6, and

7 of table 4 and combinations of these names when

designating physical units only; and in contrast the use of the symbols such as those listed in

column 9, 10, 11, 13, and 15 of table 4 and other

combinations of basic unit names when designating

symbolic units.

Further problems confronting the national and

international standardizing bodies include the following. Shall a second quantity "electrization" be

recognized to correspond to "magnetization"? Shall

two types of magnetic moment (and of electric

moment also) be recognized and provided with

names, units, and symbols, or will one suffice? How

can a more satisfactory system of names and symbols

be invented to denote the different aspects of permeability and of permittivity (i.e., relative versus

absolute, a-c versus d-c, initial versus cyclic, dif- ferential versus normal). Here a plethora of electric terms (specific inductive capacity, dielectric constant, electric constant, real component of phasor

dielectric constant, permittivity, capacitivity) contrasts with a paucity of magnetic terms (permeability, inductivity).

A major cause of the present impasse in international standardization in the field of electrical systems of measurement has been the failure of

many disputants to recognize the equal validity

of the two habits of thought set forth in this paper.

Energy has been wasted in attempts either to decide

in favor of one as against the other, or failing this, to formulate some particular, and necessarily ambiguous, wording which would receive the formal

approval of both groups, because the two groups

gave two different meanings to certain key words.

Instead let us hope that steps will be taken soon

to recognize officially both habits of thought as equally valid. Each is to be preferred in its own

field but the Realist and the Synthetiker should

tolerate the usage and appreciate the effectiveness

of the other's concepts for particular purposes.

In some distant future, a single measurement

system may win universal acceptance. Then there

automatically will be one, and only one, correspondence between each symbolic and its corresponding

physical quantity or unit. Until that Utopia is reached, and the literature of the past has been

forgotten, the coexistence of the two habits of thought must be recognized.

The writer expresses his gratitude to the many

fellow members of standardizing committees and

to his colleagues at the National Bureau of Standards

whose patience during protracted discussions of

this elusive subject have contributed so much to the

clarification of the concepts. In addition to C. C.

Murdock and C. H. Page who have so many times

corrected my erring logic, and F. L. Hermach, who

so meticulously scrutinized and improved the equations and tables, I cannot refrain from also listing

gratefully F. Avcin, C. C. Chambers, F. K. Harris,

E. I. Hawthorne, H. Konig, F. R. Kotter, M.

Landolt, C. Peterson, S. A. Schelkunoff, J. J. Smith,

C. Stansbury, U. StiUe, and S. R. Warren, Jr.

33
No.

10. Appendixes

10.1. Tables

Table 1. Some equations having the same coefficients in all usual systems of measurement

Relation

uniform velocity constant acceleration from rest

force to accelerate mass

work, energy power -- —-

voltage for given power and current (d-c) resistance (d-c) —

electric charge..- capacitance inductance --

electric flux magnetic flux — electric current.. conductance

resistivity

Equation

v=s/<

a=v/(=2s/(s V=ma

W=f¥-ds = mv^l2 = feidt P=d Wjdt V=PII R= VII Q=fTdt

C=QIV

V=LdIldt 4'=ffD-ndA

<l>=ffBndA

I=ff\ nd. 1 G= \IR = IIV p=RAIl

No. Relation

conductivity

force on charge in electric field.. force on pole in magnetic field.

.

flux of B through closed surface potential difference ..- magnetomotive force ..-

reluctance

Equation

F = QE F=mH

//Bnrf.4=0

Notes for Table 1 This table lists a number of the equations frequently used in electrical engi- neering in which the coefficient is the same (and usually 1) in all the various systems of measurement thus far proposed. From the point of view of the

Realist, each symbol should have been enclosed in ( |'s, since it represents to him the measure of a physical quantity. The Realist uses some of the equations in column 2 to fix the size of the germane physical units for the quantities

listed in column 1. From the point of view of the Synthetiker, column 2 lists quantity equations, which he uses to define the symbolic quantities listed in column 1. The sequence of listing is convenient for either purpose but is not

Important. The sequence is not logically continuous because in any system some

steps involve equations of table 2 (e.g., equation 12, table 2) in which the coefficient

is different in different systems.

Table 2. Some equations having different coefficients in different systems of measurement

Relation

Electric flux from charge Q

Magnetic flux from pole m

Permittivity, «r.

Electric polarization, P«

Permeability, ixV^ Magnetic polarization, J Magnetization, M

Electric susceptibility, x« Magnetic susceptibility, xm

Coulomb's Law

Coulomb's Law (magnetostatic) Force on length I of long

parallel current-carrying wires at separation r Magnetic field from currents

Electric field from flux change Speed of propagation, v Divergence of displacement Poynting vector Energy density, electric Energy density, magnetic

Magnetic field of current element

Inductance of slender toroid Capacitance of plate capacitor Capacitance of spherical capacitors Force on current element

Force on moving charge

Equations (parametric

form)

*=r,Q

<i>=r,-7« D= er,E D=r,E-t-r,p, B=Air„H=r,„H+Bi

B=r„H-f-rrJ

B=r„(H-i-r,M)

x„=(,.-i)/r.= -=—

4irer,r2 r rmimi

27rr,'r

r,^H-(/s=rrW/-|-'^

~ dl r.5>E-rfs=

div D = r,(7 s=r.ExH/rr

TF, = E-D/2rr = r,eE2/2r, T7„ = H-B/2rr=r„MH2/2r,

.„ r. Ids sin a "H=—;

TT,

C=T,tAIT,d

4neT,a-b (6-a)r, F= HsXB/r,

F=QvXB/r.

MKSA (Giorgi) rationalized

D= err,E D=rr,E-|-p.

B=;irr„H = rr„H-|-B,

B=,r^H+J

B=rr„(H-f-M) ''•^'-^^"^^ xTM=M-l = M/H

Q1Q2 F=

F=

2wr ^H.ds=NI+^

-d$

dl fE-dsdiv D= q S = EXH

ir,=E-n/2=,r,fE2/2

TF„ = H-B/2=Mrr„H2/2

Ids sin a

C=rT,tAld

^ Eb—a F=/dlXB

F=QtXB

Electrostatic

*=47rQ

<t=47r7n D= eE D = E+4n^P,

B=MH/c2=H/c3-|-Bi B = H/c2+47rJ B=(H-|-47rM)/c! X,= (e-l)/4,r = P,/E

Xm=(M-l)Mjr = M/H

Q1Q2 F=-

er2

Mr- 2m /iW

fH-ds=i7rNI+~

div D=47r? S = EXH/47r

TF,=E-D/87r=6E=/87r >Fm=H-B/87r dH= Ids sin airL=iirixN'-Am

C=tAliTrd

b—a F= ZdlXB F=QvXB

Gaussian

*=47rQ

<f' = 4;rm D= eE D=E-f4irP, B=ixH = H+Bi

B=H-|-47rJ B=H-f47rM

X.= (f-l)47r = P./E Xm=iM~Dlin = MIH

F= Q1Q2

er2

F=

d*

cj-a.-ds=i-,rNI-^

c<PE-ds=—

dl

div D=47r? S = cEXH/47r

TK, = E-D/87r = eE'/87r

dH = 7ds sin a/cr^

'"L=iTrnN'Al3 C=tAHird

b—a

F=/dlXB/c

F=QtXB/c

Electromagnetic

*= 47rQ *=4?rTO D=eE/c2 D= E/c2-|-47rP, B= mH = H+B,

B= H-f4irJ B= H-|-4,rM X.= («-l)/4,r = PeC!/E

Xm=(M-l)/47r = M/H

F=-

«r2

F=2nhhl

d<S! fn-da=ixNI+

H.ds-^

div D= 4ir? S = EXH/47r

Tl',= E-D/87r=£E!/87rc2 Tr;„=H-B/87r=MH2/87r dH= Ida sin a/r^

C=tAliwc-d

ta-b (6-a)c2 F=7dlXB

F=QtXB

Notes for Table 2

In columns 3, 4, 5, and 6 are given for the four most commonly used systems of measurement some of the equations in which some of the coefficients are different

in the different systems. Column 2 gives the same equations in a more generalized parametric form. To obtain the equations appropriate to still other measurement systems the parameters as listed in the appropriate row of table 3 can be substituted in the equations of column 2. It should be noted that a Realist would write all of these as measure equations enclosing each letter symbol in ( )'s. To economize on space these have been omitted and the equations appear only in the form of the Synthetiker's quantity equations.

34
Table 3. Parameters and basic units for certain systems

Parameters for equations, table

2, column 2 Postiilated imits for basic quantities

10

Measurement system

MKSA (rationalized) (2d In- terpretation) MKSA (rationalized) (3d in- terpretation) MKSA (rationalized) (1st in- terpretation MKSA (unrationalized) (2d interpretation) CGS electrostatic C OS electromagnetic Gaussian

Heaviside-Lorentz GGS-Franklm (unrationalized) CGS-Biot (tmrationalized)

Practical

= International

Definitive (Campbell 1933) Ampere-Ohm (Karapetoff 1911)

rr„

47r-10-^ nTm

1/C2

1

11nr„

nr„

1

" 1

rr„

4ir-10»

1074-n-c2

nr.

1

1/C2 1

1nr.

„r,

1/C2

« 1/C2

J,

10»/4tc2

Length Mass Time Permeability Current Charge Resistance

1

47r

4ir

i-w1 a 4ir d47r

47r 4ir

1

1

meter

meter

meter

centimeter

centimeter

centimeter

centimeter

centimeter

centimeter

10" meter

centimeter

meter

centimeter

kilogram

kilogram

kilogram

kilogram gram

gram

gram

gram gram

gram

10-" gram

kilogram

second

second

second

second

second

second

second

second second

second second

second

second

second

• henry/ meter ampere

henry/ meter

biot

franklin

Interna- national ampere

International ampere

International ohm

Definitive ohm

International ohm

Notes for Table 3 General: The spaces left blank in columns 7, 9, 10, 11, and 12 correspond to derwei, not basic, quantities or units. The speed of light denoted by c has the value 2.997925- 10' meter/second in row Ic: 2.997925- 10'" centimeter/second in rows

3, 4, 5, 6, 7, 8, 10, and 12; and 30 quadrants/second in row 9.

Specific Xotes:

» Items in row la indicate the basis of the rationalized MKSA system as rec- ognized by the IEClnl938. This corresponds to the " second interpretation" (see p. 22), namely that space constitutes a prototype standard of magnetic permeabUity to which is assigned the conventional value of 47r-10-'. This interpretation

is satisfactory both to the Realist who is thereby given an experimentally reaUzable fourth basic physical unit and to the Synthetiker who is given a fourth independent symbolic quantity, permeability, on which to base his set of dimensions. Items in row lb indicate the basis implied by the IE C in 1950 that the "ampere"

be regarded as the fourth unit. This corresponds to the third interpretation (see p. 22). This is satisfactory to the Synthetiker, to whom it is immaterial which

of the mutually coherent units of current and of permeabiUty is regarded as the

basic one. It is imsatisfactory to the Realist because no prototype standard is currently recognized for defining the ampere independently as a physical unit

except by first defining something equivalent to a physical unit of permeability. The items in row Ic correspond to the 1st interpretation (see p. 22) of Tm as a numerical coefficient. This is satisfactory to the Realist, who derives the same

set of physical units from row Ic as from row la. It is unsatisfactory to the Synthetiker because it, like rows 3, 4, 5, and 6, yields a set of only 3-dimensional

sjTnbolic quantities and units.

^ The physical units for most electrical quantities derived on the system listed in row 9 are identical with those of rows la, lb, and Ic and differ by only a few

parts in 10,000 from those in row 10.

» In the Internationa] system TTM and r, were experimentally measured constants of nature equal to 0.99951 and 1.00049/c2 respectively. In practice these departures from 1.0000 were usually ignored.

<i The 4-dimensional CGS systems are sometimes used with the equations

rationalized by setting r, equal to 1.

e The "henry/meter" is a convenient equivalent of the more logical "kilogrammeter2/ampere2 second2" as a name for the imit of permeability.

35
Table 4. Quantities, units, and their correspondences

Physical quantities and their germane physical units

Quiintity Abbreviation

Rationalized (Fessenden) MKSA

XJnrationalized

MKSA CGS-ESU CGS-EMU

Rationalized

(Heaviside)

Heaviside-Lorentz

Names of the physical units

Length Mass Time

Force Work

Power

Current

Voltage

Electric gradient Charge

Electric flux

Electric displacement

Electric polarization

Capacitance

Resistance • Conductance ^ Resistivity Conductivity

Inductance

Magnetic flux

Magnetic induction Magnetic polarization Magnetic field strength. Magnetization

Magnetomotive force

Reluctance

IM

T

F

IK P

IV

E

B

JH

M

(R

meter (m)

kilogram (kg) second (s) newton (n) joule (j) watt (w) ampere (a) volt (v) volt/meter coulomb (c)

coulomb (c)

coulomb/meter! coulomb/meter'

farad (f) ohm (D) mho (Siemens) ohm-m (n-m) mho/m

henry (h)

weber (wb)

tesla (t) tesla (t) ampere-turn/

meter, ampere-turn/

meter, ampere-turn

ampere-turn/

weber.

coulomb/47r

coulomb/

4jr meter!.

47r tesla ampere-turn/

iir meter.

ampere-turn/

ampere-turn/

weber.

centimeter gram

second dyne

erg

erg/second statampere, esu (gaussian)°

statvolt esu, (gaussian) statvolt/cm, esu (gaussian). statcoulomb (or franklin), esu (gaussian).

statcoulomb/47r, esu (gaussian) . statcoulomb/4ircm2, esu

(gaussian). statcoulomb/cm2, esu (gaussian). statfarad, esu (gaussian) statohm, esu (gaussian) statmho, esu (gaussian) statohm-cm, esu (gaussian) statmho/cm, esu (gaussian)

stathenry,*! esu (gaussian)

statweber, esu

stattesla, esu

iir stattesla, esu

statampere-turn/4ir cm, esu

statampere turn/4ir, esu

centimeter gram

second dyne

erg

erg/second abampere (or biot), emu

abvolt, emu

abvolt/cm, emu

abcoulomb, emu

abcoulomb/4jr, emu

abcoulomb/4ir cm', emu

abcoulomb/cm2, emu

abfarad, emu

abohm, emu

abmho, emu

abohm-cm, emu

abmho/cm, emu

abhenry,"! emu (gaussian)

maxwell, line, emu (gaussian). o

gauss, emu (gaussian)

iir gauss, emu (gaussian)

oersted, emu (gaussian)

gilbert, emu (gaussian)

maxwell/gilbert, emu

(gaussian)

.

centimeter gram

second dyne

erg

erg/second statampere/ V4'r V47r statvolt

^fi^r statvolt/cm

statcoulomb/4 -^It!

statcoulomb/

statcoulomb/

Viir cm2. statcoulomb/

-v/4ir cm'.

statfarad/4ir

4jr statohm

statmlio/4jr

i-TT statohm-cm

statmho/47r cm

iw abhenry,

4ir stathenry*.

V47r maxwell

V4x gauss

-v/4£ gauss

Vlir oersted

V4t gilbert

maxwell/gilbert

Notes on Table 4 General: Table 4 shows In each row for some particular physieal quantity the correspondences between it, with its physical units used by the Realist, and the symbolic quantities and units used by the Synthetiker. Column 1 contains the name of the physical quantity and column 2 the abbreviation for the quantity

used by the Realist when for example he writes" (H)" for the measure of magnetic

field strength. Columns 3, 4, 5, 6, and 7 contain merely the names of the corre- sponding germane physical units used in the 6 different measurement systems by

the Realist to measure the quantities listed in column 1. In column 4 only those germane physical units have been listed which are different from the corresponding physical units of the rationalized system. Unfortunately the complete

definition of any one of these physical units is impossibly long to use in a Table. Even a name such as "ampere-turn/47r meter" in column 4 row 23 should be

considered merely as an abbreviation for " that sample of magnetic field strength present In a long slender solenoid when the excitation is caused by a current

sheet having 1 ampere for each ijr meters of axial length." The name should not

be considered a quotient obtained by dividing separate factors.

In contrast the letter symbols in columns 8, 12, and 14 are those used by the Synthetiker to designate the symbolic quantities which he uses in the six systems and the entries in columns 9, 10, 11, 13, and 15 are his symbolic units. The subscripts n—f, r—f, s, m, q, ft for the symbolic quantities denote that they are used

respectively in the unrationalized (nX/), and rationalized (rX/) 4-dimensional systems, the classic 3-dimensional COS electrostatic (AT,), electromagnetic (Xm), Gaussian (X,), and Heaviside-Lorentz (Xa) systems (see also column 13, table 3). The correspondences can be seen by following any row. Thus in row 23 to the Realist's single physical quantity H of column 2, the Synthetiker may set up a correspondence with either „H/ or rH/ of column 8 or H. of column 12 or Hm, or Hft of column 14 (H,is identical with H„) depending upon which measure- ment system and set of cqutaions he prefers to use. However the Synthetiker can use the single symbolic unit cm-i/^gj^s-' in column 15 to measure either Hm

in the unrationalized C G S electromagnetic system or H * in the Heaviside-Lorentz

system. The Realist uses the oersted (column 6) in the former and a nameless

unit (column 7) larger by -sli-K In the latter system.

36
Table 4. Quantities, units, and their correspondences—Continued

Symbolic quantities and their coherent symbolic units

4-dimenslonal 3-dimensional

Symbolic

MKSA CGS-Bi CGS-Fr CGS--ESU CGS-EMU

Row quantity Symbolic units Symbolic

quantity Symbolic

unit SymboUcquantity Symbolicunit

Row

8 9 10 11 12 13 14 15

1 L m cm cm L cm L cm 1 2 M kg g g M g M g 2 3 T s s s T s T s 3

4 F m kg s-2 cm g s-2 cm g s-2 F cm gs-2 F cm gs-2 4

5 W m^g s~2 cm^g cm^g W cm2gs-2 W om2gs-2 5

6 P m^kg S-' cm2g s-3 cm g S-' P om2gs-3 P cm^gs-' 6

7 if a bi s-ifr prn3/2gI/2s-2 Im (jjQl/2gl/2s-l 7 8 Vf m^kg s~3a~' cm^g s-'bi-i cm2g s-2fr-i V.,V„Vk omi'2gi/2s-i v„ Cm3/2gl/2s-2 8

9 E/ i-Ll ii-^ o d E.,Ej,Ei, cm-i/2gi/2s-i E„ cmi«gi/2s-2 9

10 Q/ sa s bi fr cm3/2gi/2s-i Q» cm'/2gi/2 10

11 sa s bi fr em^/sg'/V cmi/2gi/2 11

12 nDf, rD/ m-2sa cm-'s bi cm-2fr D„D„D), cm-i/2gi/2s"' -Dm cm-3/2gi/2 12

13 „P/,rP/ m-2sa cm-2s bi cm-2fr P.,Pj,Pa cm-i/2gi/2s-' Pm cm-3/2gi/2 13

14 m-2kg-is*a2 cm-2g-is<bi2 cm-2g-is2frs Ct, Cg, C\ cm Cm cm~lfi2 14

15 Ri,Xf, Zj « m!kg s-3a-2 cm'g s-3bi-2 cm2g s-ifr"2 PtjRgflih * cm~^s Pm * cm s~' 15

16 G/,B/, V/ b m-2kg-is3a2 cm-2g-'s3bi2 cm-2g-'s frz 0.,Og,G\ b cm S-' cm-'s 16

17 Pf mskg s-3 a-2 cm3gs-3bi-2 cm3gs-ifr-2 pt,Pg,i>h s Pm em's-' 17 18 yi m-3kg-is'a2 cm-3g-is3bi2 cm-'g-'s fr' yi,ye,yh s-i 7m cm-2s 18

19 L, m^kg s-2a-2 cm2g s-2bI-2 cm2gfr-2 Ti3, 'Lgf^Tin cm-'s2 cm 19

20 */ m^kg s-2a-i cm'gs-^bi-i cm2gs-'fr-' *. cmi/2gi/2 '^m,'5>g,'t'h cm3/2gi/2s-i 20

21 n

.

t>/ kg s-2a-i gs-2bi-i gs->fr-> 21 22 J/ kg s-%-1 gs-2bi-i gs-ifr-i J. cm-3/2gi/2 Jm,Jg,J„ om-U2gi/2s-i 22 23 „H/„H/ m-'a cm-'bi cm-'s-ifr H. cmi/2gi/2s-2 H„HgHk cm-i/2gi/2s-i 23

24 M/ m-'a cm-ibi cm-'s-ifr M. cmi«gi/2s-2 Mm cm-i/2gi/2s-i 24

25 a bi s-ifr em3/2gi/2s-2 cmi/2gi/2s-' 25

26 m-2kg-is2a2 cm-2g-is2bl2 cm-2g-ifr2 0t. cm s-2 ^mM„^. cm-i 26

Specific Notes:

» The names of the units shown for resistance are also used to express reactance and impedance.

b The names of the units shown for conductance are also used to express sus- ceptance and admittance.

= The notation "(gausslan)" applied to certain unit names in columns 5 and

6 Indicates that these constitute the set of physical units used In the symmetrical

CGS or Gaussian system.

d In the symmetrical systems Inductance may be regarded either as an electric quantity (symbol 'Z) or as a magnetic quantity (symbol "i). The physical

units appropriate to these two cases are the stathenry and the abhenry respectively. In the Heavlside-Lorentz system either unit is greater by a factor of

4x. The corresponding symbolic quantities are listed in column 12 and 14 and

are defined by equation 10, table 1, and equation 21, table 2, respectively.

37
Table 5. Prefixes for decimal multiples

Prefix Abbreviation Factor

pico P

10-12 10"^ micro io-« milli m 10-3 centi c 10-2 deci d 10-1 dcka da 10+' hecto h 10+» kilo k 10+3 mega M 10+« giga O 10+» tera T 10+12

Tlie prefix "myria" is sometimes used for 10+' and "lakh" for 10+5.

Note.—-Table 6 is located on page 29.

Table 7. Conversion of symbolic quantities in quantity

equations

Relations between unrationalized symbolic quantities " nXi" and symbolic quantities "rAT/" rationalized in the Fessenden-Giorgi manner used in quantity

equations in systems based on four basic symbolic units.

1 Quantity2 3

Rationalized =Unrationalized

Electric flux .-- 4)rrD/=nD/

47rr4'/=n'I'/ 4jre,r, = e„r.

'X'/4ir=„Xe 47r,H/=„H/ 47r,<^=„<^

,J//4^=„J/

rXm/47r=„X". rM/=„M/ (or I)

Absolute permittivity Magnetic field strength Magnetomotive force Magnetic polarization

Reluctance

Absolute permeability Magnetic susceptibility --. Magnetization

To change an unrationalized equation to the rationalized form substitute the corresponding item in column 2 for each item in column 3 which appears in the equation; and conversely.

Table 8. Conversion of measures

Multiply the measure in germane or coherent units of the system listed at the top of the column by the factor listed in the table

to obtain the measure in the MKSA rationalized system. Here c is 2.997925- lO'".

Row MKSA COS-ESU CGS-EMU Gaussian

Unrationalized Rationalized

1 Length ...i.- 10-2 10-2 10-2 10-2 2 Mass M.. 10-3 10-3 10-3 10-3 3 Time T.. 1 1 1 1 4 Force _.F._ 10-5 10-5 10-5 10-5

5 Work W.. 10-" 10-' 10-' 10-'

6 Power P.- 10-' 10-' 10-' 10-'

7 Current /._ 10/c 10 10/c 101 8 Voltage - v.. lO-^C 10-8 lO-Sc 9 Elec. gradient E._ 10-«c io-« 10-6C 10-«-v/4irC 10 Charge - Q- 10/c 10 lO/c 10/-v/4^c

11 Elec. flux 1/477 10/4irC 10/4ir 10/47rc 12 Elec. displacement __ D_. l/iTT 105/47rC 105/47r 105/47rc 105/-v/47rc 13 Elec; polarization P... lO'C 105 lOSc 105/-v/4ic 14 Capacitance _ C._ 10»/C2 10» 10«/C2 10»/47rC2 15 Resistance _ R.. 10-V2 io-« 10-9C2 10-» 47rc2

16 Conductance (?_. 109/C2 10» 10»/C2 10«/47rC2 17 Resistivity —P-- 10-1IC2 10-11 10-11C2 10-11 i^ci

18 Conductivity - T-- 10I1/C2 10" 1011/C2 19 Inductance . . . i._ 10-»C2 io-« 10-»c2, 10-»

»

10-«47rC2, IO-Htt ' 20 Mag. flux 10-?c 10-8 10-8 10-s -v/47r

21 Mag. induction B.. 1 lO-'c 10-' 10-1 22 Mag. polarization J-. iir 4ir-10-'c 47r-10-4 10-1 V4t

23 Mag. field strength ..H.- l/47r 103/47rC 103/4ir 103/47r 24 Magnetization M_- 1 103/c 103 103 103/ -v/4^ 25 Magnetomotive force __ __ jr_ l/4ir 10/47rC 10/47r 10/47r

26 Reluctance..- (R- l/47r 109/4tC2 10»/4ir 10V47r 10«/4ir 27 Permittivity tP... l/47r 10ll/4jrc2 10ii/47r 10ii/47rc2 10ii/47rc2 28 Permeabilitv--- nv„.. iw 47r-10-'c2 4jr-10-' 4ir-10-' 47r-10-'

» Inductance may be measured either in COS electrostatic or electromagnetic units in the Gaussian System and in units

greater by a factor of iir in the Heaviside-Lorentz System.

10.2. Chronology—Electrical Units

1791 —Commission on the Meter received by Louis

XVI.

1799 —Metric System legal in France by "Law of 18 Germinal, year 3." 1822 —J. B. J. Fourier published his "Theory of Heat"

with discussion of physical dimensions.

1822 —A. M. Ampere suggested concepts of "electric tension" and "electric current." 1827 —G. S. Ohm published his "Law."

1833 —K. F. Gauss introduced absolute measurements

in terrestrial magnetism.

1840 —W. Weber introduced absolute meas. of current, tangent galvanometer.

1851 —W. Weber introduced absolute meas. of resist- ance. 1860 —W. Siemens used Hg column as standard of resistance 1 m X 1 sq mm.

1862-67 —Brit. Assoc. Committee on Electrical Standards

active, ohm = 10' MGS unit, "Weber" =

unit of charge.

38
1867 —BA Comm. 5th Report used "farad" for 10""

MGS electromagnetic unit (i.e., 1 microfarad).

1867-73 —Brit. Assoc. Committee on the Selection and

Nomenclature of Dynamical and Electrical Units—CGS system, dyne, erg, prefixes from

micro to mega-, horsepower = 7.46-10^ erg/sec, volt = 10+8 CGS em, farad = IQ-^ CGS em,

B. A. ohm = 109 CGS em = 104.8 cm X

1 sq mm Hg column.

1872 —Clark Zn-Hg standard cell proposed as 1.457 (!) volt. 1875 —Convention of the Meter established Internat. Committee on Weights and Measures.

1881 1st Internat. Elec. Congress (Paris) 1 ampere =1

volt/1 ohm, 1 farad=l coulomb/1 volt. 1882 —0. Heaviside first suggested rationalization. 1884 —Committee of 1st Int. Cong, reports 1 "legal

ohm" = resist. of 106 cmXl sq mm Hg.

1887 —Physikalisch-technische Reichsanstalt founded

in Berlin. 1889 —2d Internat. Elec. Cong. (Paris) joule, watt and

quadrant (i.e., 10" meters) as units of energy, power, and inductance. 1891 —3d Internat. Elec. Cong. (Frankfort).

1892 —Informal Elec. Cong. (Edinburgh).

1893 —Weston Cd-Hg standard cell= 1.018 volt. 4th

Internat. Elec. Cong. (Chicago) confirmed

decimal multiple CGS basic for joule, watt,

volt, coulomb, farad and henry, but offered "equal" alternative ampere — 0.00118 g/sec

Ag; ohm= 106.3 cm Hg, volt= 1/1.434X Clark

ceU. 1894 —Above alternative units made legal in US and

UK.

1894 —A.I.E.E. proposed gilbert, weber, oersted, gauss

as names for CGS electromagnetic units of magnetomotive force, flux, reluctance, and

induction, respectively.

1896 —Alternative units made legal in France. 1898 —Alternative units made legal in Germany.

1899 —Natl. Physical Lab. founded in Teddington,

England.

1900 •—^Fessenden and others suggested subrationalization, 5th Internat Elec. Cong. (Paris) maxwell

for unit of flux; gauss for unit of "magnetic

intensity" (taken by some as H, by others as B). 1901 —Giorgi suggested subrationalized, 4-dimensional,

MKS system, Natl. Bur. of Standards founded

at Washington.

1904 -—6th Internat. Elec. Congress (St. Louis) set up

permanent Internat. Electrotech. Commission. 1905 —Conf. of Nat. Std. Labs. (Berlin) ohm and

ampere to be basic, volt derived, Weston cell substituted for Clark cell as reference standard. 1908 -—Internat. Conf. on Electrical Units and Standards (London). Distinguished between (1) "practical" decimal multiples of CGS em and

(2) "International" ohm and ampere defined by Hg and Ag.

1910-11 —Internat. Technical Comm at NBS set consistent International "Washington Unit" values for

coils and cells. Weston normal cell derived

at 1.0183 volt. 1911 —7th Internat. Elec. Cong. (Turin) definitions and

symbols; / (not C) for current; R+jX for inductive resistor. 1921 —6th Internat. General Conf. on Weights and

Measures extended scope of ICWM to cover

Electricity and Photometry.

1927 —Consultative Committee on Electricity set up by

ICWM.

1928 —lEC (Bellagio) sets up subcommittee on Magnetic Units. 1928 —Int. Bur. of Weights and Measures with enlarged

buildings and staff began periodic intercompari- son of electrical standards. AIEE Stds, Comm. urged shift from "Inter- national" to "absolute" units.

1930 —lEC (Stockholm and Oslo) voted B = mo// with

Ale having dimensions; confirmed CGS units of: flux = maxwell; flux density = gauss; field strength = oersted, magnetomotive force =

gilbert. Proposed units of frequency = hertz, of reactive power = var. 1931 —lEC divided TC No. 1 to create Subcommittee

on Electric and Magnitudes and Units

(EMMU)

lUPAP created Committee on Symbols,

Units, and Nomenclature (SUN)

1932 —EMMU (Paris) proposed "weber = 10^ maxwells; Siemens = mho. 8th Gen. Conf. Weights

and Measures authorized change to "absolute"

electrical units at discretion of Internat. Committee.

1935 —EMMU (Scheveningen) adopted Giorgi (MKS)

system with 4th unit left open; confirmed

hertz and Siemens, confirmed weber = 10' maxwells; Consultative Comm of ICWM

advocated yuo as basis for "absolute" units. 1938 —EMMU (Torquay) recommended na as link to mechanics; proposed newton = 10' dynes. 1946 —Internat. Comm Weights and Measures set values for new absolute units and date for adoption. 1948 —Jan. 1, absolute electrical units effective. 1950 —U.S. Congress passed Public Law 617 fixing

electrical units. EMMU (Paris) selected ampere as 4th unit; confirmed newton; recom- mended total rationalization of Giorgi (MKSA) System; appointed Comm of Experts to interpret "rationalization." 1951 —lUPAP recommended "that in the case that the equations are rationalized, the rationalization should be effected by the introduction of new

quantities."

SUN in Doc. 51-5 proposed 4-dimensional

CGS system. 1952 —I.S.O. Tech Comm No. 12 issued draft table of quantities and units. 1954 —EMMU (now Tech Comm No. 24) (Philadelphia)

approved rationalized equations; proposed

tesla = 1 weber/meter 2. 10th Gen. Conf. on Weights and Measures established "System

International" (SI) of units based on meter, kilogram, second, ampere, candela, and degree

Kelvin. 1956 —lEC (Munich) confirmed tesla. 1956 —SUN issued Document 56-7 [13] defining susceptibility; questioning current symbols for permeability and permittivity; stating status

of symbols for electromagnetic and dipole moments.

1958 —T. C. 24 (of lEC) (Stockholm) discussed rationalization, revised convention on sign of reactive power. 1959 —T. C. 24 (Madrid) discussed rationalization; pro- posed lenz = 1 ampere-turn/meter. 1960 —11th Gen. Conf. on Weights and Measures defined meter by wave length of Kr*^; defined second by tropical year; confirmed lEC name

tesla. 10.3. Notation, Glossary, and Organizations

a. Notation X Generalized abbreviation to identify any

physical quantity (see column 2, table 4). Xi General symbol for a symbolic quantity which

serves as an element in a mathematical

model. {X]a Measure of X in system a. aUx Germane physical unit of X in system a. {Xi)a Coherent symbolic unit of Xi in system a. \Xi\a Measure of Xi in system a. [Xi] Dimension of A',.

e Relative permittivity. n Relative permeability.

Te, Tm, Vr, Parameters. See table 3 and "Constant,

magnetic . . ." below. 39
b. Glossary

Absolute—An adjective applied to "method" or "measure- ment" to designate an operation in which a quantity is measured, usually indirectly and in terms of the ultimate

basic units (usually those of length, mass and time) of the measurement system used. Additivity—That attribute of a physical quantity as a result of which the measure of the conventional resultant combination of two or more examples of the quantity is equal to the sum of the measures of the component physical

quantities.

Basic—An adjective applied to "unit" or "quantity" to identify members of the small group of p=N—n units or quantities from which the Realist derives his other physical

units and the Synthetiker his other symbolic quantities.

Other writers use "fundamental."

Coherent—An adjective applicable to a symbohc unit which

indicates that it is related simply and consistently to (1) the basic units of the system and (2) the dimensions of the symbolic quantity of \vhich it is a unit. Note: when

coherent symbolic units are used the measure equation in terms of them is identical in form with the quantity equation. (Guggenheim uses "germane" in this sense.) Constant, electric, Te—The factor of proportionality which

relates electric charges to the electrostatic forces they

produce. In an electrostatic sj-stem it has a conventionally

chosen value characteristic of the measurement system. In an electromagnetic system it has a value derived experimentally from a conventionally chosen magnetic constant. Most writers hitherto have called this quantity "permittivity (or capacitivity) of space" with the symbol to or e„. Constant, magnetic, Fm—The factor of proportionality which

relates electric currents to the electrodynamic forces they

produce. In an electromagnetic sy.stem it has a conventionally chosen value characteristic of the measurement

system. In an electrostatic system it has a value derived

experimentally from a conventionally chosen electric constant. Most writers hitherto have called this quantity

"permeability of space" with the symbol fio or Mo- Dimension—A label of convenience which indicates for a derived symbolic quantity the relative rate at which it would vary with virtual variations in the basic symbolic

quantities of the system. By extension, to a similar encoded relation to other symbolic quantities not necessarily basic. Dimensions form elements of a multiplicative group. Hence the product of any pair of dimensions

is a dimension. The unit element of the group is "numeric"

or "pure number" which is therefore a dimension.

Dimensional exponent—The exponent relating the relative rates of change of a derived symbolic quantity and a more

basic symbolic quantity in a measurement system. For example if X=f(Y,Z) the dimensional exponent of .Y Y c>X

relative to Y is ''^~'x"^y' ^^^^^ writers use "dimension"

(see sec. 7). Germane—An adjective applicable to a physical unit which

indicates that it is related simply and consistently to (I) the basic units of the system and (2) the coefficients in the equations of the system.

Kind—That attribute of a phj'sical quantity which distin- guishes it qualitatively in regard to its physical nature, its relation to the phenomena, etc. from quantities of other

kinds. Two physical quantities are of the same kind if operational methods are available for the meaningful

comparison of their relative magnitudes.

Magnitude—That attribute of a quantity which distin- guishes it quantitatively in regard to size, extent, intensity,

etc., relative to other quantities of the same kind. Measure—The number obtained by either (1) measuring a

physical quantity by comparing it experimentally with a

physical unit of the same kind; or (2) by dividing a symbolic

quantity by a symbolic unit of the same kind. Other

writers have also used "magnitude," "value," "numerical

value." Quantity, physical—Any example of a "real" physical entity, as conceived by the experimenter for the precise descrip- tion of a phenomenon and operationally defined so as to be

measurable. It is characterized by its kind and magnitude.

Other writers have also used "entity," "physical entity,"

"magnitude," "quantity," "experimental quantity," "concrete quantity" for this concept.

Quantity, symbolic—-Any example of an element which, in a mathematical model, corresponds to some physical quantity in nature. Other writers have also used "concrete quantity" (Maxwell), "abstract quantity," "mathematical

variable," "magnitude," "idon" for this concept. Rationalization—A name given by Oliver Heaviside to the

use of a (in his opinion) more rational set of coefficients in the electromagnetic equations. He assumed this to be

secured by the use of a set of rationalized derived units. In a set of rationalized equations the factor "Air" is made

to appear only in those equations involving geometric

arrangements having spherical symmetry.

Realist—A fictitious character postulated to perform experimental measurement operations and to use mathematical

manipulation on measure equations only. He therefore

deals only with physical quantities, physical units, and

measure equations.

Standard, physical—A physical system some property of which embodies an example of a physical quantity to which

a value has been assigned to indicate its supposed measure

in terms of some physical unit. Standard, prototype—A standard which serves to define a

basic physical unit of a measurement system by fixing

independently an essential feature of its definition. Some

writers (e.g., A. G. McNish [14, 15]) limit this adjective to standards which are entirely independent of values assigned

to all other prototype standards. Standard, reference—The standard or group of standards of highest rank in a given laboratory which serve to maintain

in that laboratory the unit of some physical quantity.

Synthetiker—A fictitious character postulated to use only

quantity equations which express the relations among

symbolic quantities. He derives symbolic units in terms

of which he can formally write measures for symbolic

quantities.

Unit—A particular sample of a quantity either physical or symbolic in terms of which the quantity can be measured

or expressed quantitatively.

Unit, physical—A particular sample of a physical quantity

of such magnitude that it is assigned the measure "1."

Unit, symbolic—A particular sample of a symbolic quantity

of such magnitude that it is assigned the measure "1."

c. Organizations

ICWM —International Committee on Weights and Measures.

Pavilion Breteuil, Sevres, France (French ini- tials CIPM). See footnote 3, p. 3. IBWM —International Bureau of Weights and Measures

(French initials BIPM). See footnote 3, p. 3. lEC —International Electrotechnical Commission (French

initials CEI) founded 1904. Serves as organization of UNESCO in field of electrical engineering.

EMMU—Electric and Magnetic Magnitudes and Units. Former name of lEC Technical Committee TC

24 dealing with this subject.

lUPAP —International Union of Pure and Applied Physics. Serves as organization of UNESCO in field of physics.

SUN —Symbols, Units and Nomenclature. Committee

of lUPAP on this subject.

ISO —International Standards Organization. Is the branch of UNESCO for standardization. Its Technical Committee TC 12 cooperates very

closely with TC 24 of lEC in field of electrical engineering.

ASA —Am. Standards Association. Coordinates standardization activities of professional societies in the U.S. and internationally. Its Committee

C61 cooperates with TC 24 of lEC.

d. National Standardizing Laboratories NBS —National Bureau of Standards, Washington 25, D.C. and Boulder, Colorado. NPL —National Physical Laboratory, Teddington, England.

40
PTB —Physikalisch-technische Bundesanstalt, Braunschweig, West Germany.

IM —Institute of Metrology, Leningrad, USSR.

LCIE —Laboratoire Centrale des Industries Electriques,

Fontenay-aux-Roses, France. BFPM —Bureau Federale des Folds et Mesures, Berne,

Switzerland. ETL -—Electrotechnical Laboratory, Tokyo, Japan.

NSL —National Standards Laboratory, Chippendale,

NSW, Australia. NRC —National Research Council of Canada, Ottawa,

Ontario.

10.4. References

The literature dealing with systems of units and dimensions is extremely voluminous. A valuable and extensive

list of such references in all branches of physical measurements

is given by Stille [10]. The references listed below have been

selected largely to document particular points stressed in the

text, or to refer to items of current or historical interest or to valuable summary papers in English.

a. General

[1] Reports of the Comm. on Electrical Standards of the

Brit. Assoc. for Adv. of Science (reprinted by Cambridge Univ. Press, 1913).

[2] A. E. Kennelly, Historical outline of the electrical units,

J. Eng. Education 19, 229 (1928)

.

[3] S. S. Stevens, On the theory of scales of measurements,

Science 103,677, (1946).

[4] N. Campbell and L. Hartshorn, The experimental basis

of electromagnetism, Proc. Physical Soc, (London)

58, 634 (1946) ; 60, 27 (1948) ; 62, 422, 429 (1949).

[5] W. R. Varner, The Fourteen Systems of Units (Book,

Vantage Press, Inc., New York, 1961).

[6] Public Law 617—81st Congress, An Act to Redefine the Units and Establish the Standards of Electric and

Photometric Measurements, 15 USCA 221-222 (July

21, 1950).

[7] Coulomb's Law Comm. of the AAPT, The teaching of

electricity and magnetism at the college level, Am. J. of Physics 18, 1 and 69, (1950).

[8] Int. Union of Pure and Applied Physics, Document

UIP 4 (Nov. 1951).

[9] E. Bradshaw, Electrical Units, London, (Chapman

Hall, Ltd. 1952).

[10] U. Stille, Measurement and Calculation in Physics

(in German) Braunschweig (F. Vieweg und Sohn,

1955).

[11] J. J. Smith, Recommendations of I.E.C. TC 24, Elec. Eng. 74, 406 (May 1955).

[12] S.U.N. Comm., Symbols and units U.I.P. 6, Physics

Today 9, 23, (Nov. 1956).

[13] S.U.N. Comm., Document SUN 56-7, Physics Todav

10,30, (Mar. 1957).

[14] A. G. McNish, Classification and nomenclature for standards of measurement, I.R.E. Trans, on Instru- mentation, p. 371, Dec. 1958.

[15] A. G. McNish, The basis of our measuring system,

Proc. I.R.E. 47, 637, May 1959.

b. Absolute Measurements

[21] W. Weber, Measurements of electrical resistance according to an absolute standard. Phil. Mag. [4], 33, 226, 261 (1861).

[22] H. L. Dryden, A reexamination of the Potsdam absolute determination of gravity, J. Research NBS 39, 303 (1942) RP1502.

[23] J. L. Thomas, C. Peterson, I. L. Cooter, and F. R.

Kotter, An absolute measurement of resistance by

the Wenner method, J. Research NBS 43, 291 (1949)

RP2029.

[24] G. H. Rayner, An absolute measurement of resistance by Albert Campbell's bridge method, Proc. I.E.E.

(Brit.) 100, 250 (1953) Part IV, Mono. No. 95.

[25] A. M. Thompson and D. G. Lampard, A new theorem

in electrostatics and its application to calculate standards of capacitance. Nature 177, 888 (1956).

[26] T. E. Wells, Measurement of the resistance-strain

relation and Poisson's ratio for copper wires, Proc.

I.S.A. 11 (1956) No. 56-8-3.

[27] M. Romanowski and N. Olson, An absolute determination of resistance. Can. J. Phvs. 35, 1312 (1957)

NRC No. 4513.

[28] R. L. Driscoll and R. D. Cutkosky, Measurement of current with the NBS current balance, J. Research

NBS 60, 297 (1958) RP2846.

[29] R. L. Driscoll, Measurement of current with the Pellat electrodynamometer, J. Research NBS 60, 287 (1958)

RP2845.

[30] M. C. McGregor, J. F. Hersh, R. D. Cutkosky, F. K.

Harris, and F. R. Kotter, New apparatus at NBS for absolute capacitance measurement, I.R.E. Trans, on

Instrumentation 1-7, 233 (1958).

[31] R. D. Cutkosky, Evaluation of the NBS unit of resist- ance based on a computable capacitor, J. Research

NBS 65A (Phys. and Chem.) No. 3, 147 (1961).

c. Maintenance of Units

[41] Int. Comm. on Weights and Measures, Proces. Verbaux.

(in French) (1946).

[42] J. L. Thomas, Stability of double-walled manganin resistors, J. Research NBS 36, 107 (1946) RP1692.

[43] N.B.S., Announcement of changes in electrical and

photometric units, NBS Circ. 459 (1947).

[44] F. B. Silsbee, Establishment and maintenance of the

electrical units, NBS Circ. 475 (1949).

[45] R. D. Huntoon and U. Fano, Atomic definition of primary standards. Nature 166, 167 (1950).

[46] F. B. Silsbee, Extension and dissemination of the electrical andmagnetic units by the NBS, Circ. 531 (1954).

[47] P. L. Bender and R. L. Driscoll, A free precession de- termination of the proton gyromagnetic ratio, IRE

Trans, on Instrumentation, 1-7, 176 (1958).

[48] D. N. Craig, J. I. Hoffman, C. A. Law, and W. J. Hamer, Determination of the value of the faraday

with a silver-perchloric acid coulometer, J. Research

NBS 64A (Phys. and Chem.), 381 (1960).

d. Systems of Units

[51] G. Giorgi, Rational units of electromagnetism (in Italian), Atti Dell A.E.I., p. 402 (1901).

[52] H. A. Lorentz, Maxwell's electromagnetic theory (in German), Ency. Math. Wissen 5, 83 (1904).

[53] G. Giorgi, Proposals concerning electrical and physical

units. Trans Elec. Cong. St. Louis, 1, 136 (1904).

[54] V. Karapetoff, The Magnetic Circuit, New York

(McGraw Hill, 1911).

[55] V. Karapetoff, The Electric Circuit, New York

(McGraw Hill, 1912).

[56] J. H. Dellinger, The international system of electric and magnetic units, B.S. Sci. Pap. 13, 599 (1916) S292.

[57] V. P. Hessler and D. D. Robb, Generalized electrical formulas, Elec. Eng. 70, 332 (1951).

[58] V. Karapetoff, A general theory of systems of electric and magnetic units, AIEE Trans. 51, 715 (1932).

[59] G. A. Campbell, A definitive system of units. Bull. Nat.

Res. Council No. 93, p. 48 (1933).

[60] L. Page, Mathematical considerations underlying the

formulation of the electromagnetic equations and

the selection of units. Bull. Nat. Res. Council No.

93, p. 39 (1933).

[61] E. U. Condon, Where do we live? Reflections on

physical units and the universal constants. Am.

Physics Teacher 2, 63 (1934).

[62] R. T. Birge, On the estabhshment of fundameni,al and

derived units with special reference to electrical units. Am. Physics Teacher 3, 102, 171 (1935).

[63] E. A. Guggenheim, Names of electric units, Nature

148, 751 (1941).

[64] P. Fleury, Coordination of the mechanical and electrical units in an international system: Formulae

41
classic and rationalized (in French), J. Phys. 3, 33

(1948) . [65] P. Cornelius, Proposals and recommendations concerning the definitions and units of electromagnetic

quantities, Philips Research Reports 4, 232 (1949).

[66] P. Cornelius, The rationalized Giorgi system and its consequences, Philips Research Reports 4, 123

(1949) . [67] J. DeBoer, Quantities and units (in Dutch), Ned. T. Natuurk'de 16, 293 (1950).

[68] L. H. A. Carr and others, Symposium on the MKS

system of units, Trans I.E.E. (Brit.) 97, 1, 235 (19.50). [69] 0. Lobl, Memorandum on electric and magnetic units

(in German), E.T.Z. 72, 455 (1951).

e. Quantity Calculus

[81] D. F. Gregory, On the re^l nature of symbolical algebra. Trans. Roy. Soc. Edinburgh 14, 208 (1838).

[82] A. Lodge, The multiplication and division of concrete

quantities. Nature 38, 181 (1888).

[83] J. C. Maxwell, A Treatise on Electricity and Magnetism, 3d ed., Oxford (Clarendon Press, 1904), Vol. 1, p. 1. [84] J. C. Maxwell, Dimensions, Ency. Britt. 9th Ed. 7, 241.

[85] J. Wallot, Dimensions, units and measurement sys- tems (in German), Handbuch d. Physik, Vol. 2

(J. Springer, Berhn, 1926).

[86] M. Landolt, Quantity, Measure and Unit (in German)

(Rascher Verlag, Zurich 1943).

[87] A. Hochrainer, Short lessons in quantities and quantity

equations (in German), Electrotechnik u. Maschinenbau 63, 350 (1944).

[88] H. Konig, On the multiple significance of the concept

of quantity (in German), Bull. Assoc. Suisse des

Elec'n 41, 625 (1950).

[89] C. H. Page, Units and dimensions in physics, Am. J. Phys. 20, 1 (1952).

[90] C. H. Page, Physical entities and mathematical

representation. J. Research NBS 65B (Math, and

Math. Phys.), No. 4, 227 (1961).

f. Dimensions

[101] J. B. J. Fourier, Theory of Heat (in French) (1822)

(G. E. Stechert).

[102] A. W. Rvicker, Suppressed dimensions, Phil. Mag. [5], 27, 104 (1889).

103] E. Buckingham, On physically similar systems; Illus- trated by the use of dimensional equations, Phys. Rev. 4, 345 (1914).

104] P. W. Bridgman, Dimensional Analysis, New Haven

(Yale Univ. Press, 1922).

105] R. T. Birge, On electric and magnetic units and dimensions. Am. Physics Teacher 2, 41 (1934). 106] A. Sommerfeld, On the dimensions of the electromagnetic quantities (in German) Z. Tech. Phys. 16, 420

(1935).

107] N. Campbell, Dimensions and the facts of measurement,

Phil. Mag. [7] 33, 761 (1942).

108] H. Dingle, On the dimensions of physical magnitudes,

Phil. Mag. [7] 33, 321, 692, (1942); 34, 588 (1943)

109] E. A. Guggenheim, Units and dimensions, Phil. Mag.

[7], 33, 479 (1942).

g. Rationalization

111] 0. Heaviside, The Electrician (London) 10, 8, (1882).

112] O. Heaviside, Electromagnetic Theory (1891) (reprinted

by Dover Publications, New York, 1950) Sec. 90,

p. 31. 113] J. Perry, Practical electro-magnetic units, Electrician (London) 27, 355 (1891).

114] F. B. Baily, Electrician (London) 35, 449, (1895).

115] J. A. Fleming, Electrician (London) 44, 324 (1900).

116] R. A. Fessenden, A proposed system of units, Elec. World 35, 282 (1900).

117] L. Bouthillon, On the rationalization of the units and

the formulas of electricity; pro and con (in French),

Bull. Soc. Fran. Electriciens [6], 9, 351 (1949).

118] J. deBoer, Rationalization (in Dutch), Ned. T. Natuurk'de 17, 5 (1951). 119] R. Vermeulen, Dimensional analysis, units and rationalization. Philips Research Reports 7, 432 (1952). 120] J. R. Barker and R. O. Davies, Calculus of measure and its application to the relations between rationalized and unrationalized electromagnetic systems,

Am. J. Phys. 21, 281 (1953).

121] F. B. Silsbee, Does rationalization change units?, Elec. Eng. 76, 296 (1957).

122] E. Perucca, Solid angle in electromagnetism, rationali- zation, calculus of quantities. La Ricerca Scientifica, Anno. 30, Supp. M 12, Dec. (1960). Washington, D.C.

42 U.S GOVERNMENT PRINTrNG OFFICE: 1962


U.S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineering in the field indicated by

its t itle. A brief description of the activities, and of the resultant publications, appears on the inside of the front

cover. WASHINGTON, D.C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements.

Dielectrics. High Voltage.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering

Metrology. Mass and Scale. Volimaetry and Densimetry. Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment. Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Reference Materials. Applied Analjrtical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics, Rheology. Combustion Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry, Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research.

Metallurgy, Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal Physics. Electrolysis and Metal Deposition.

Inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth, Physical Properties.

Crystallography,

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research,

Data Processing Systems. Components and Techniques. Computer Technology. Measurements Automation.

Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy, Solid State Physics. Electron Physics, Atomic

Physics.

Instrumentation, Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation.

Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy.

Molecular Kinetics. Mass Spectrometry.

OfSce of Weights and Measures. BOULDER, COLO.

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Cryogenic Technical Services. CENTRAL RADIO PROPAGATION LABORATORY

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