GENERAL PROBLEMS OF METROLOGY AND MEASUREMENT TECHNIQUES

RATIONALIZATION OF A N G U L A R U N I T S

K. P. S h i r o k o v UDC 531.74(017).001.6

A Contribution to Discussion. The suggested rationalization of equations and units for measuring angles entails changes in a number of mathematical , mechanical, and electrotechnical equations (examples are provided in the article). It is only natural that suggestions for such a profound reorganization of theoretical relationships requires careful examination and clarification of all the arguments for and against it. However, a mere acquaintance with the contents of this article leads to the conclusion that the problem raised in it is of considerable interest. As distinct from previous suggestions, aU of which amounted to the adoption of new nonsystem units, the author of this article pointed out for the first t ime a theoretically substantiated manner of improving the SI system without disturbing the principles on which it is based.

The weak point of the International System of Units (SI) consists of the angular units, the radian and steradian*. Being based on purely mathemat ical considerations they are extremely inconvenient in practice. The ratio of the radian to a complete angle is equal to an irrational number 1/21r, and it is impossible to expect that goniometric in- struments and dials calibrated in radians would be used in practice, even if it were possible to organize their pro- duction. The steradian, equal to 1/4~r of a complete solid angle, is also inconvenient in practice.

Therefore, everywhere in practical measurements the angular units of degrees (...*), minutes (...'), and seconds (...") inherited by us from antiquity are still being used. The drawback of these units consists of their nondecimal partitioning, thus entailing in multiplication or division of angles the operations of fractioniziug and inverse trans- formation of composite concrete numbers.

At the t ime when Delambre and M~chaiu were determining the length of the meter by measuring the stretch of the paris meridian between Dunkirk and Barcelona, they used, for simplifying triangulation, decimal angular units, grads (1 grad -- lg -- 0.01 of a right angle)l" subdivided into metric minutes (1 metric minute = 1 c = 0.01g) and metric seconds (1 metric second = 1 cc -- 0.01c). Since then the competit ion between the new and old angular units is continuing. However, despite all the advantages of the decimal units, goniometers are not calibrated in them and the battlefield remains [nthehands of the old degrees.

This unsatisfactory situation prompts one to look for other solutions. Thus, the division of the circle into 3000 and6000 parts and other fractions is used, However, such units are employed only in certain narrow fields 9f technology, and not a single one of them has obtained general recognition. New suggestions of this kind have been recently made, whose authors claim universality for them.

~The angular units, radian and steradian, as well as units of other relative quantities do not depend on the dimension of the basic unit and, therefore, are not derived from it. On the other hand, they cannot be considered as funda- mental units, since relative units are always determined in terms of other quantities, i.e., they are not independent. Relative units remain the same in any system of units, and they can be considered as incorporated in all coherent systems. In accordance with existing international decisions, the radian and steradian are considered in the present articles as SI units. 1"In 1949 [9] it was suggested to cal l that unit a gon, and now there exist two nomenclatures- the grad and the gon. We shall use the first one.

Translated from Izmeri tel 'naya Tckhnika, No. 7, pp. 23-28, Iuly, 1972. Originalarticle submitted February 14, 1972.

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Fig. 1

r"

Fig, 2

According to the first such suggestion [2], the circle is divided into 600 parts and the new unit is called "corner" (with the notation . . c ) . A right angle is then equal to 150 c, 2/3 of a right angle to 100 c, 1/2 to 75c, and 1/3 to 50 c. The second suggestion [3] consists

of dividing the circle into 1000 parts,each of which is called by the author the metric de- gree (*lvI). The following relations will then hold: 90 ~ = 250~ 45 ~ = 125~ivI, 30~ 15 ~ = 41.666...*M, etc. The author sees the advantage of his proposal in "metrization" of the relationship between the unit and the circle, thus making it possible to express small angles in decimal fractions without complicated calculations.

Simultaneously with the metrization of angles it is suggested in [3] to adopt"metr ic" time, i.e., to divide 24 hours into 10 metric hours (MH), 1000 metric minutes (MM), and 100, 000 metric seconds (MS). This would provide a simple relationship between the value of angles on maps and the corresponding rotation t ime of the Earth (100*M ~ 1 MH, 10~vl

MM, I*M = 1 MM, 0.1~ ~ 10 MS, etc.).* It is true that the reduction of the size of a second by the factor of 1.157 (by 15.7%) would change the size of many SI units. In pard- cular, the retention of the existing size of other fundamental units, including the ampere, would increase the size of the units of velocity, viscosity, and frequency by a factor of

1.157; that of acceleration, force, energy, induction, magnetic flux, and inductance by a factor of 1.300; that of power, voltage, and resistance by a factor of 1.504; and it would reduce the units of charge by the factor of 1.157 and of capaci tance by the factor of 1.795. In essence it would produce a new system of units. Nevertheless, the idea of metrizing time is attractive, since we then deal with the el imination of the second important drawback of the International System of Units, namely the archaic subdivision of the units of t ime. Perhaps in due time human- ity will decide to adopt such a transformation.

However, let us return to angular units. The deficiency of all the above suggestions eonsistsof the fact that they are arbitrary and produce nonsystem units. The radian; which is unacceptable in practice, remains as before the coherent SI unit. In the present article we suggest a new solution of the problem of SI angular units, which is derived logically from modern concepts of the principles for developing systems of units and of the purpose of the equations' rationalization [1, 4 -6] .

It is known that relationships between physical quantities are used for forming derived units. Thus, the area of a geometrical figure is assumed to be proportional to the product of the lengths of mutually-perpendicular char- acteristic elements (sides, base and height, semiaxes, etc.). A body's density is assumed to be proportional to the ratio of its mass to volume, velocity to the ratio of the covered distance to time, etc. Various coefficients can be incorporated in the equations of the relationship between the quantities, depending on the shape of objects or the nature of the current phenomena. One of the shapes of the object (physical model) can be chosen as the principal one and its coefficient equated to unity in the corresponding relationship equation. For instance, the coefficient can be equated to unity in the equation representing the area of a rectangle

S~a = ab .

Then, in the area equations of other figures coefficients not equal to 1 will appear. For instance, for a tri- angle it will be

1 S t a = ~ ab;

and for an ellipse Sel = ~Ta b

etc. (a and b are characteristic elements of a figure).

In this case the principal model for determining area is a rectangle. A coherent unit of area can be formed in the simplest way (although this is not the only way) by providing the characteristic elements of the principal model with unit dimensions (a = 1 m and b = 1 m). We shall then obtain a model with a unit area (a square with its side equal to 1 m), It is possible to take an ellipse for the principal model. Then in the equation for the area of an ellipse the coefficient should be taken as 1, whereas in the equation for the area of a rectangle it would be necessary to use a coefficient equal to 1Dr. The unit of area in this case would be the area of an ellipse whose semiaxes are

denotes correspondence.

I011

o,.,~ e ~

, q

I,,

:~ ~ , ~ ~

o,..i ~

~ c q ~

~ . o . : o

~ ~~

0

;>

II II II

o ~ ~

~ " : : : : ~

~ o o

~ ~ % o o . . . . .

"~. i : : . ! "

-~ ,.,,., ~ ~ ~ . . . . . . ~ ~ _ ~ . ~ -~

0 0 II 11 I[ II [I [I . . . . . . . . . .

.,~

�9 ,',4 ~--4

o ~ o ~ I

g g g d g d d g g ~

~ ~ ~ ~ 1 7 6

~oo~~oo~

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TABLE 2. Solid Angle Values in

Steradians and Steroctants

Part of I r ra t ional /Dat ional sphere cut unit [unitt'" (s teroc- by solid (steradianl tant angle ]

1 4~ 8 1 /2 2~ 4 1 /4 g 2 1 /5 4 / 5 ~ 1 ,6 1 /8 1 / 2 ~ 1 III0 215~ 0 . 8 1 /100 1/25r~ 0 , 0 8

pig. 3 Fig. 4

equal to 1 m (i .e . , the area of a c i rc le with a radius of I m). It is obvious that such a unit would not be rat ional , since it would make the areas of a l l s imple figures expressed in i r ra t ional numbers.

In our example we have passed from ra t ional equations to i r rat ional ones. This has affected the size of the area unit, but i t has not changed the meaning assigned to the concept of an area, since every geomet r ica l figure can

have only one area. The ra t ional iza t ion of equations for other physical quantities is s imilar . Their concepts re- main unchanged, and i t is only the pr incipal mode l or the shape of the equations which re la te a given quantity with other quantit ies that change, thus producing variations in the size of the specific unit.

Let us now deal with angles. The angle is a measure of a body's rotat ion about a cer ta in axis. In the rotat ion of the body its points describe arcs of c i rcumferences . Therefore, i t would appear natural to express angles by the ratio of an arc to a given reference arc, since the values of angles which have a ra t ional ratio to a comple te angle would then be expressed by ra t ional numbers.

However, at present i t is customary to de termine angles by the ratio of an arc to the radius and, therefore, both the comple te angle and its fractions have an i rrat ional ratio to the coherent angular unit, the radian. The fact that the radian is an i r ra t ional unit has been noted in a number of works [1, 4]. However, these works do not dea l in de ta i l with the possibil i ty of ra t ional iz ing angular units. We shall demonstrate here that by ra t ional iz ing the equations which de termine angles i t becomes possible to form a coherent angular unit suitable for de c ima l fractions and convenient in p rac t ica l appl icat ions. It is shown in [4] that a ra t ional angular unit would consist of a comple te angle (860~ However, a unit more convenient for met r iza t ion would undoubtedly consist of a right angle, and it is selected for a l l subsequent considerations.

At present the sector of a c i rc le (Fig. 1) is taken as a pr incipal model, and the equation which represents the angte r has the form of

$

q~ = - - O) r

The length s of the arc and the radius r are the character is t ic e lements of this model . The select ion of the radius as one of the character is t ic elements , which is incommensurable with the length of the c i rcumference, makes this equation and also the angular unit, the radian, i rrat ional . In fact, the coherent angular unit, the radian, can be determined by providing the character is t ic e lements of the model in Fig. I with unit values. The right angle ex- pressed in these units wi l l then be re/2.

Let us ra t iona l ize (1). For this purpose let us se lec t as the character is t ic e l emen t instead of the radius, the length of the arc of a quadrant, i .e . , let us se lec t as a pr incipal model instead of a sector a quadrant of a c i rc le (Fig. 2). Then the equation defining an angle wil l have a ra t ional form (let us denote it by the subscript r):

or, i f the length of the arc is expressed by means of the radius, i t wil l assume the form

~ = ~, ~-~-/r

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TABLE 3. Certa in Mathemat i ca l Formulas

Functions

lira x - 0

Equivalent

in an i rrat ional

s i n x

form

X

Sirl X =

CoS X

t g x =

d -- - (sinx)= dx

d - -- Cos x) =

d x

d . . . . (tg x) = d x

~ s i n x d ~

j ' z:OS ,~ d x ~

S s c 2 x d x =

1

cos (-2- -- X) = sln (~ -- x)

. . .

in a ra t ional ized form

2

c o s ( 1 - - x ) = s i n ( 2 - - x )

sin (1 - - x ) = - - c o s (2 --x)

ctg ( - ~ - - - x) = - - tg ( g - x) etg (1 -- x) = - tg (2 - x)

CO~ X - -COS X 2

- - sinx - - - - - s i n x 2

1 n

CO.~ ~ X 2 COS 2 X

2 - - c o s x + C - - - - - c o s x + C

sin x+ C 2_ sin z + C

tgx+C 2"~tgx+C

which is also ra t ional . The select ion of a quadrant as the pr incipal model determines the ra t ional coherent angular unit equal to a right angle. In fact, by providing the character is t ic e lements with unit sizes, i .e . , by making s -- s o = 1 m, we obtain a right angle. Let us ca l l this unit a quadrant and denote it by d. It should be noted that sometimes right angles are denoted by L, yet, this sign is less convenient for printing and for forming f rac t ionaluni t notations. However, in this a r t i c le the problem of nomenclatures and notations of the suggested units is of secondary

signif icance.

It is possible to form,from the quadrant , f ract ional units convenient for p rac t ica l purposes, such as mi l l iqua- drant, md (1 m d = 0.001,d = 0.09 ~ = 10C),and microquadrant ,pd (1 ~d -- 10-6,d = 0.324" = 0.1co). It wi l l be seentha t

the new units have simple relationships to the grad and its fractions; they are a new version of angular metr iza t ion which has a l ready been carried out in establishing the metr ic systems of measures. The new version, as dist inct from

the or iginal one, is theore t ica l ly substantiated and leads to the formation of a coherent ra t ional unit of the inter-

nat ional system.

Let us now examine a table of angular values expressed indi f ferent units (Table 1). Rs upper part carries

values of angles expressed in a round number of existing degrees, minutes, and seconds. These values are repre- sented most s imply and lucidly by means of the suggested coherent ra t ional unit, the quadrant. All the angular values expressed in radians contain the number 7r, i .e . , they can only be evaluated approximate ly . The bottom part

of Table 1 carries d e c i m a l fractions of a c i rc le . Here the simplest form of expressions is also in quadrants, and this, by the way, is natural. Thus, i t wi l l be seen that the new plane angle unit s implif ies angular expressions. Moreover,

i t fac i l i ta tes mul t ip l i ca t ion and division of angles.

By adopting the quadrant as a plane angle unit, i t becomes possible to solve e legant ly the problem of thesol id angle. The same as in the case of a plane angle, in this ease the pr incipal model should consist of a spher ica loc tan t

with its character is t ic e lements , the surface area of the cu t -ou t sphere and the surface area of the octant (Fig. 4), instead of a cone with its character is t ic e lements , the surface area of the sphere cut out by the cone and the radius of the sphere (Fig. 3). Instead of the i r ra t ional equation which represents the first model,

S f l _ F2 '

where S is the surface area of a sphere with radius r cut out by the solid angle e, it is necessary to adopt the

ra t ional form

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TABLE 4. Certain Mechanica l Formulas

Quantity

Angle ,~ =

k ngular veloci ty m = ~/t =

Angular acce le ra t ion g = , xo /At =

Peripheral ve loc i ty v = s/t =

Peripheral acce le ra t ion a = Av/At =

Torque 2u = Fr = mar =

Power in rotation N = F s / t =

Work in rotat ion A = F s =

Cent r ipe ta l force z = m v ~ / r =

Centr ipe ta l acce le ra t ion ~ =

Moment of inert ia of a mass point J :

irratio na I

s

r

a

g

c 7 - ) = re0 =

A O g - - - ~ r { l )

At

m r 2 t o

- ( ( % = M o t

Frq~ = M~

Formula

r a tio na l ized

_ S ( ~ 2 S

s~ _ -• - 7 )

~ SO

a

So

Sor~ 7-) A ~ �9

So - - - ~ S0 ft} At

Fs0q~ ~- M~ t 2

Fsoq~ = -~- Mrp 2

t ' ~ 2

r n r u

ms~162176 ( - m r 4~x~

r \ -- - r ~-)

s~176 ( = - 4

msor ( = - 2- mr z)

where S O is the surface area of the spherical octant . Since S 0 = tlrZ/2, this expression can be represented in the form

2 S

k coherent unit of a solid angle can be formed by providing the pr incipal model ' s character is t ic e lements with unit dimensions, i .e . , by assuming that the areas in (4) are equal to 1 m 2 (8 = S O = 1 m2). It wil l be equal to the solid angle which cuts out on the surface of the sphere an area equal to 1/8 of the ent ire surface area of the sphere. Let us c a l l this unit the steroctant and denote it by "50," For comparing the solid angles thus obtained let us

examine Table 2. It wi l l be seen from it that the transition to the new unit e l imina tes i r ra t ional numer ica l values of angles and s implif ies the relationships between the spherical surface parts and the corresponding values o f solid angles.

It should also be noted here that changes in the angular units wil l en ta i l variations in the units of quanti t ies derived from angles, such as angular ve loc i ty and accelera t ion, as wel l as quantities re la ted to solid angles (for in-

stance, luminous flux). However, there are not many such quantit ies and their units can be changed wi th re la t ive ease.

The above ra t iona l i za t ion of equations which de termine the plane and solid angles makes it possible to incor- porate in the Internat ional System units sui table for p rac t ica l appl icat ion, and thus e l imina t e one of its major de-

f ic iencies . A ngular values with ra t ional relationships to comple te plane and solid angles e l imina te the i r ra t ional factor rr, and i t becomes possible to ca l ib ra te the scales and d ia l so fgon iomet r i c instruments direct ly inra t iona lunt t s .

However, a price must be paid for this convenient notation. If the ra t iona l iza t ion is comple ted , i .e . , not l imi ted only to p rac t ica l measurements, but extended to theore t ica l relationships, it wi l l be necessary to come to terms with the fact that some of these relationships wil l change their form. This, by the way, is natural, since the relat ionship between the c i rcumference and its radius is irrat ional , and the incorporation of the number v in the formulas which re la te arcs of a c i rc le to straight lines is more just if iable than its absence from them. In Tables 3- 5 provided below examples are given of formulas affected by the ra t ional iza t ion of the equat ion which determines the plane angle.

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TABLE 5. Certain Electrotechnical Formulas

Quantity

Frequency f =

Angular frequency o =

lt~tanta neous sinusoidal, current i = Ira s i n (tot q- ~) =

Effective sinusoidal current

I �9 , = l,,dt=

0 Mean sinusoidal current for half

a pe~od T/2

2 j" Ira s i n o~tdt = I raed = T 0

Impedance z =

Formula

irrational

1

T

2 ~ - - = 2uf

T

lm sin i -7 ' - + *) = lm sln (2a,t -}- *)

rationalized

1 T

4 - - - = 4f

Im sin ( - ~ - + * ) -- Im sin (4f' -F *)

I m

F f Ira

V2-

T/2 T/2 21 m ' 21m ] 2 8I m 2

41~ 2 Ira I - - ~ T - I - - - z r coscot [ ] - - cos o 3 t - - - - I r a

~ (:~ Y; ( ' ) : / : , ; _ o , ' + ~ L - ~ / ,oc ' / 2

It will be seen that the elimination of the number Tr from angular values leads to its appearance in certain formulas. By providing examples of such formulas, we pose the question of an immediate transition to their expres- sion in a rationalized form. It would probably be advisable to start with rationalizing angular units in practical measurements, and only later start adjusting formulas to correspond with these units. However, it should be stressed immediately that the new angular unWcan acquire all the rights of coherent derived SI units only by adopting the rationMized defining equations proposed in this article.

It goes without saying that in adopting the new plane and solid angle units it will be necessary to issue new tables for trigonometrical and other functions. However, tt is possible, even at an earlier date, to issue small tables for converting the new units to the existing ones (and vice versa), thus facilitating the utilization of existing tables in calculations with new units and making it possible to adopt them immediately in practice. This conversion would, undoubtedly, provide in future a considerable saving in the labor and time of many people.

L ITERATURE C I T E D

1. U. Stille, Messen und Rechnen in der Physik. Vieweg, Brunswick (1961). 2. A . I . Milovanov, Izmeritel'. Tekh., No. 7 (1970). 3. A.I . Mettler, Metric Time, a Practical Possibility. EdRed by Canadian Metric Association, June (1969). 4. J. Wallot, Groessengleichungan, Einheiten und Dimensionen. Barth Leipzig (195"/). 5. K.P. Shirokov, Izmeritel'. Tekh., No. 10 (1964). 6. K, P. Shirokov, Izmerkel ' . Tekh., No. 6 (1965).

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