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Angle As a Fourth Fundamental Quantity

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Angle As a Fourth Fundamental Quantity JO URNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 66B, No. 3, JUly- September 1962 Angle as a Fourth Fundamental Quantity Jacques E. Romain* (April 18, 1962) The ad\:antage of co nsidering anglc as a. fourth fu ndamental quantity of geometry a~1 d physles IS stressed, a nd a n alternative approach is suggested to introduce an angular dimensIOn m such a \\"ay t hat t he physical laws, in a ny form, are dimensionally homogeneous. A few examples are descn bed to s how how so me eq uations of mathematics and mechanics should be revised to put t he new point of vic II' into p ractice. I, Introduction co mplex exponential formalism is used in physics i~ stead of trigonometric functions, Of course, the Chester H. Page 1 r ecently wrote in this JOUl'na] dIfficulty can be disposed of by stating that "this is ,l most interesting study about "physical entities and an illustration of the fact that equations used in their mn,thematical I' epl'esentn,tion ," One or the pllysics need not be dimensionally homogeneous ideas in that paper, which will be of intcrest h ere, is wh en terms <wise from the usc of an flJt iftce, The that advanLlge can be gn,ined rrom co nsidering an O'le fun ction eJO docs not <l.Ct ull.ll.\T occur in physics; it is a.s an add ~ tio r 1<tl ['undamen tal q uantit.L Dr. Pl~ge intl'Ochrccd with the co nveJi tion t lr at its real part nghtl.\· pOInts out tlr at, since the Jl1C1tSUl'e of an alone (or its irm1.g in nr,v part ,lIon c) represents n 'll1gle depend s 0 11 the chosen fll1g1e u nit, it wo uld be v1l.l'ia bl l' of in tercst. N onartiftcial ('xpon en tinls, such natul'<1.lnot to consider it 11 dinrensionless quantity. 11S e- lV/ k 1', IllLVe numeric ex ponent ." (Sec footnote 1. ) That point or view co uld lm l'clly be olTel'slressecl , and BuL it scems silllpler to dcflll e lhe angular dimension the. pl:csen ~ writer has the un pleasan t feeling 0[' an so tlll1t Euler's forrnula is dim ensioflnll y homogene­ nl'tlficwl tn ck about tlte gc ncl'all.\T11ccepted l'ule tb l1.t ous; tll11.t will be dOll e p rcsc rrn,\c. (Howevcr, tile angle is a. dime n s ion~ ess q uan tity, but ml1.,v only be possibility or r eco ncilin g Pa.ge's scheme with tb e measurcd 111 radmns If trouble is to be avoided. requirement of homogencity will be shown at the end Moreover, ignoring the dimension of nngle de­ of sec lion 3.) Moreover, the prcscn t tl'eatlll ent ])t'ives dimensional analysis oJ a part of its fruitfuln ess. seems to 11.Ilow a more SLJ'l1.ightfonvlU'cl extension to Although the number of ru n cb III en tal qUltntities in a solid a ngles, takcn ItS all 0 tlr er di sC i fl ct fundamental dimen ~io ~11.1 the? ry is not assigned by 11.ny logicnl quantity, on the same footing with p ln'll e anglc, by prescnptlOn, It IS well known 2 that the optimum sirnpl.v substituting "sLc radillil" for "n1 dia n" ,wd number of fun damental q uan liLies is t he smallest "'l.l'C'l" 1'01' "lenglh" in eq (l a). n umb? l' th n.t is enough to express evcry physical On t lte other hand, tllis attempt misses the pleas­ quantlt~T 111 the theo ry in tC l' ms or properly chosen an t feflture, in Pnge's work, that torque and energy fun~ l alIle r ~tal qwmtities , without arbitrarily ltssignin g acquire different dimensions (as do action and a ll gu­ adlmenslOnless Stl1.tuS to an." physical co nstant. In lar momentum, or areal velocity tl.lld kinematic vis­ VIeW of these argumenLs, it seems mthel' sllrprisin O' cosity coeffi cient). But there really is no rea on, in that eflrlier atternpts 3, 1, 5,6 did not seem to meet the theory of dimensions, to demand that no two n ll~ ' widespread aueliell ce. different p hysical quantities should have the same This paper describes an aHem a ti ve approach to the dimensional formula; and it seems hardly possible to introduction of a n angular dimension, It mllst not devise a dimensional system in which no such dupli­ be co nstrued as polernics n.ga,inst Page's app)'oach : cation would happen anywhere: Jor instance, even actually both systems Me equally consistent. How­ in Page's system, a frequency and a velocity gradient ever, let it be noted that the pr:esent point of view have t he same dimensions. docs nwar with the unappealing reltture, in Pflge's s.l"stem, thflt Euler's formuln 2. Basic Definitions W= - 1) The appeamnce of angles in an equ ation can often be traced to the relation is clilll err sionally nonhornoge rrc ous, and that therefore dimensionfll homogell city is Hot preserved when the s= ()R (1) ·Prcscnt address, OO Il (, l' al Dynamics, Vort " ·ol'th. rr cxas. I C. H . Page, J. Research ;.rBS 65 8 (M atb. and _Vl aLh. Phys.) No, 4 227-23,; (1961). ' between an arc s of a circle, its radius R and the ~ J. Pa l acio ~ , Analyse dimcn sionncllc (Gauthier-Villars, Va ris, 1960). 3 \V. C. MIchels and A. L. Patterson, Elements of ~\Io clc rn l'hysics (Van central angle () measured in radians. The nece sity Xostl'and, l-'rincctoll , N.J., 195 1L namely sectioll 13.4. 'F. H aberli, Schwriz. Arch. ange\\,. \\'iss. T echn , 1 5,343 (1949). of dimensional homogeneity of both sides of eq (1) ' P . Moon and I). E. Spe ncer,.T. Fran klin InsL. 24-8,495 (l949). (in which sand R are lengths) seems to imply that Ii Internationa.l O r g: ~ln i zHLio n for Standardization Doc rSO/TC 1<) 131 E, D eeember, l954. , . -, () is dimensionless. 97 It has been tried (see footnote 6) to a,void contra­ eq (1). The reader should turn back to the elemen­ diction by taking into consid eration two kinds of tary demonstration of eq (4); the general scheme is: "a,ngles": the "geometrical angle" which is the usual (1 ) show that t he length of an arc e of a circle lies one, a, quan tity related to the portion of the plane between those of two line segments which are pro­ contained between two concurrent straight lines, and portionftl to sin e and tan e; (2) divide by the radius t he " analytical angle" which is the e to be found in H: the two segments yield sin e and tan e, while t Il e eq (1). But the so-called "all alytical ft ngle" is no arc becomes siR, or < e> on accoun t of eq (3). Thus, new notion indeed: it is plainly the measure of the the true form of eq (4), valid for a,ny unit of angle, is angle in radians. This can be seen immediately from eq (1) , which originates in a theorem statin g that in IU110-)o(sin e) /<e>= 1. (5) a circle two arcs are proportion al to t heir cen tral angles. When one of the ftrcs is equal to tbe radius A similar equation would be obtained with tan e its central angle is one radian, an d the proportion in stead of sin e. becomes The equations showing how to replace the sine s= (e/1 racl )R , (I a) ancl tangent o[ a smftll angle by t he angle itself follow in the correc:t form which is the correct form of eq (1), with e a true "geometrical" angle. (6) How is it t hat one should think himself obliged to particularize the measures of angles under the naJne As a consequence, every equation derived from eq of "analytical a,ngles," while not feeling any need (3) or (5) should be recalcula,tecl. The details of whatsoever of "anaJytical lengths," "analytical t lte calculations are left to t he reader. masses" or "analytical time"? The explanation , Here are a few results: and the reason why mechanics could be developed (a) D erivatives of trigo nometric Junctions: without recognizing the fundamen tal dimension of angle, is that angles often ftppear in mechanics d (s in x) /dx= (1/1 md) cos x, (7) t hrough their measure in radians, which indeed can be seen to be derived from eq (1a) . Something simi­ or lar happens in other chapters of physics: [or instance d(sin x) = cos x cl< x> , (8) it is well-known that the dimensional analysis in photometry is less complete if the further funda­ and anftlogous expression s for the derivatives or cos mental dimension of solid angle is not taken into x, ti1l1 x, . .. An appli cfttion of eq (7) which will account; actually what is here being said of angles be of further use is can be extended to solid angles. The basic angular quantity will be taken here to d(sin wt )/clt=<w> cos wt. (9) mean the ordinary "geometrical" angle, cer tainly the most primftry notion in any study of angles. Trig­ (b) By co mparing t he power expansions of ei X to onometric functions will be defined in the elemen­ those of cos x, sin x derived from eq (7), namely tary way as ratios of lengths, i.e., dimensionless quan ti ties.
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