INSTITUTE OF PHYSICS PUBLISHING METROLOGIA Metrologia 42 (2005) L23–L26 doi:10.1088/0026-1394/42/4/L02

SHORT COMMUNICATION Differing angles on angle

W H Emerson

Le Trel, 47140 Auradou, France

Received 14 March 2005 Published 10 May 2005 Online at stacks.iop.org/Met/42/L23 Abstract Values of plane angles are expressed with a choice of several units. Historically the quantity needed a unit because it was, and still is, used as a base quantity. ISO/TC 12 deﬁnes it as a derived, dimensionless quantity, and the International System of Units (SI) gives it the ‘dimensionless unit’ radian, which now means no more than ‘one’. This paper discusses the confusion that arises from the dual uses of these terms. The author argues that solid angle is logically a two-dimensional angle derived from plane angle, and has units naturally derived from units of plane angle. ISO/TC 12, however, deﬁnes the quantity independently of plane angle as a dimensionless quantity to which, nevertheless, the SI assigns a ‘unit’.

1. Plane angle it as the amount of turning of one of those lines that would be necessary to make it coincident with or parallel to the The word ‘angle’, when used to designate a quantity, means other. The quantity thus deﬁned has all the hallmarks of a base different things to different people. Essentially it is a quantity in any system of quantities; it is not derived from any phenomenon at a point; it has no linear dimension, and other quantity and is indeed difﬁcult to deﬁne without some historically it has been regarded as a difference of direction as circularity; and, as with the other ancient base quantities of viewed from a point. Babylonian astronomers needed a way of our modern systems, nobody really needs to ponder the term’s expressing differences between the directions of ‘points’ in the meaning. heavens as viewed from Earth. They invented a unit by which Some two centuries ago mathematicians started using a such a difference, an angle, could be given a value: the degree. new unit of angle. Instead of dividing a complete reversal of They made a degree one 360th of a full turn around the sky, direction by 180, which gives the unit ‘degree’, they divided it or one 180th of the greatest amount by which the directions of by the number π and called the unit ‘radian’. It is deﬁned as two stars could differ. the angle subtended at the centre of a circle by an arc whose Navigators needed a way of expressing changes of heading length equals that of the circle’s radius. That is, of course, a of their ships. They divided their compass cards to indicate deﬁnition of a particular unit of angle, not of the quantity angle, 32 evenly-spaced directions, called points of the compass, the which is not associated by deﬁnition with circular geometry. total amounting to a change that would bring the ship back to The new unit, more difﬁcult to realize than degree, was not and 1 its original heading. A point is thus equal to 11 4 degrees, and is not used to express the results of measurements of angles; nowadays navigators express changes of heading in degrees. it was adopted because it simpliﬁed certain mathematical They formerly expressed the altitude of a star above the horizon expressions. For example, it is impossible to deﬁne the in units of degrees, minutes and seconds of arc; now they centripetal acceleration of a point on a rotating shaft, radius r, state it as a decimal number of degrees. The degree, the without stating its angular velocity ω using a relative angle, Babylonian astronomers’ unit, is still the universally accepted that is angle relative to a reference angle, a unit. If that unit 2 2 unit of angle for anyone who measures angles, whether or not is degree, the acceleration is (π/180) ωdegr, where ωdeg is the he calls himself a metrologist, or who uses the results of such shaft’s angular velocity with the changing angle expressed as measurements. a number of degrees. With radian as the unit of angle the All the dictionaries that I have consulted [1] deﬁne angle angular velocity is, say, ωrad and the centripetal acceleration 2 in terms of the differences of direction of two intersecting becomes simply ωradr. By putting π into a deﬁnition of lines. The larger, internationally respected dictionaries deﬁne a unit of angle the constant π may be eliminated from the

0026-1394/05/040023+04$30.00 © 2005 BIPM and IOP Publishing Ltd Printed in the UK L23 Short Communication deﬁnitions of many derived quantities that are functions of and c (c being the hypotenuse), the ratio a/c was the sine of the angle. Mathematicians, physicists and engineers who derive angle opposite the side of length a, by deﬁnition. Now it is said and manipulate expressions for quantities that describe cyclic to be the sine of the ratio of the length of an arc, bounded by phenomena use radian as their unit of angle as a matter of the lines b and c and centred on their point of convergence, to course, and when they give an angle a value, the numerical the arc’s radius. We are asked to accept that those are identical, part of the value is nearly always a multiple or a fraction of π. trigonometric functions, yet sin θ now deﬁned by equation (3), Indeed, the presence of π in the expression of a value is usually with θ as a number between −2π and +2π, rather than as a seen as sufﬁcient to indicate that the unit is radian, and the unit trigonometric function. is not then stated. I have suggested elsewhere [1] that the quantity ϑ, deﬁned However, not stating the unit can lead to anomalous as the ratio of an arc of a circle to its radius, is akin to the statements. For example, the sine of the angle θ can be trigonometric functions sine, cosine and tangent, a function of represented by a summed inﬁnite series: an angle θ, though, unlike the others, it is a linear function. It, too, deserves a name that is not ‘angle’, like the other θ = ϑ − 1 ϑ3 1 ϑ5 −···, sin 3! + 5! (1) trigonometric functions of angle. When, in 1960, the International System of Units (SI) [2], where θ was given that name by the Conference´ Gen´ erale´ des Poids ϑ = . (2) rad et Mesures (CGPM) there were six base units for six base Many texts, however, show that equation with the symbol quantities (now seven). Those base quantities did not and do θ on both sides of the equation, that is with θ replacing ϑ not include plane angle. That quantity was not deﬁned (the by omission of the unit radian, and without mentioning that SI does not deﬁne kinds of quantities or cite deﬁnitions; it omission, thus: only deﬁnes their units), but the name radian was adopted for its unit. Nor, exceptionally, was that unit deﬁned. Unlike all = − 1 3 1 5 −··· other SI units (except steradian, for solid angle) it was listed sin θ θ 3! θ + 5! θ . (3) neither as a base unit nor a derived unit. It, with steradian, was That transformed equation can be satisﬁed dimensionally only called a ‘supplementary’ unit. Radian is, of course, deﬁned in if θ is made a dimensionless quantity, without a unit. By the almost any dictionary, but the Comite´ International des Poids et deﬁnition of radian, ϑ is the ratio of the length of an arc of a Mesures (CIPM) did not adopt the lexicographers’ universally circle subtending an angle at the circle’s centre, to that of the agreed deﬁnition, nor any other. radius of the circle. That ratio is, of course, a number that is In 1969, the CIPM interpreted the decision of the CGPM, proportional to the angle at the centre, but most ordinary people in 1960, which classed the units radian and steradian as would not regard it as the same thing, as being that angle. They supplementary units, as allowing the freedom of treating those would argue that an arc subtending an angle cannot at the same units as base units [2]. In 1980, the CIPM (following a time be that angle, even if the arc is normalized by dividing resolution adopted by ISO/TC 12 in 1978) decided that they its length by that of its radius. An angle is a phenomenon must be treated as ‘dimensionless, derived units’. No other at a point, but clearly an arc of a circle is not ‘at’ a point, dimensionless quantities were considered to have need of particularly a point that is outside itself. units. In 1995, the CGPM, acknowledging that the status If an angle is an arc of a circle made dimensionless by of the supplementary units in relation to the base units and dividing it by its radius of curvature ρ, then the absolute values derived units of the System ‘[gave] rise to debate’, decided of the lengths of the arc and its radius of curvature are irrelevant. to do away with the name ‘supplementary’ and to make those = An elemental angle may be written as dθ ds/ρ where ds is units ‘dimensionless, derived units’ [derived from existing base an elemental length of a curved line in the neighbourhood of a units]. That ‘debate’ was supposedly between the proponents point where the radius of curvature is ρ and may be a function of the historical, classical view of angle as effectively a base of s. If angles, like elements of arc length, are additive, then quantity, and generally accepted as such by metrologists, and logically the angle corresponding to the length s is those of the practice of mathematical analysts of deﬁning angle s ds as a ratio of lengths. The analysts prevailed and, supposedly, θ = . (4) the debate was ofﬁcially at an end. The proposals of the ρ 0 CIPM are not open to general debate in the manner of a draft That seems to be a legitimate representation of the quantity, but International Standard. an angle is supposed to be ‘at’ a point, and there is no single The deﬁnition of angle now adopted by ISO/TC 12 and point that can be assigned to the quantity expressed thus as an the CGPM leaves it dimensionless and thus without need of integral. a unit. Yet the name ‘angle’ has always been associated The substitution of the symbol for angle for that of a pure with units. Furthermore, the quantity angle is used to deﬁne number, and the adoption of the name angle for that number, derived quantities such as ‘angular momentum’ and their units, changes the meaning of the term. It is no longer a difference of in which a unit of angle commonly appears. The CGPM’s direction or an amount of turning from one direction to another, decision of 1995 means that an angular velocity that was deﬁned independently of all other quantities; it becomes a ratio formerly stated with rad s−1 as its unit has now, logically, the of two lengths, like a trigonometric function. Moreover, if θ unit s−1. It has been necessary to invent a ‘dimensionless unit’ is no longer a symbol for an angle as the term is traditionally with a borrowed name, with no value other than unity, so that understood, what is the sine of θ if the argument is the ratio of a unit of angular velocity may still be called rad s−1. The two lengths? Formerly, in a right-angled triangle of sides a, b unit degree is no longer permitted to be that deﬁned by the

L24 Metrologia, 42 (2005) L23–L26 Short Communication Babylonians and universally still in use; it is now deﬁned by Equation (6) deﬁnes the quantity solid angle. ISO/TC 12 [3] as π/180. An element of solid angle may be visualized by A derivation of a quantity normally either assigns a name considering a rectangular, elemental area da on the surface to a relationship between two base quantities of the system of a sphere centred on the point. It subtends1 the element (e.g. an area or a velocity), or to a dimensionless relationship dΩ of solid angle at the centre. Let c be the length of the such as a Reynolds number, or it expresses an observed law circumference of a great circle on the sphere, p be a unit of of nature (e.g. a force). It contains no arbitrary element plane angle and n the number of such units in a full turn. The unless, rarely, it be a numerical coefﬁcient. By contrast the rectangle’s length on a great circle through the point where deﬁnition of a unit of a quantity is always arbitrary, normally the chosen axis intersects the sphere is d(cθ/np); its width is by consensus. sin θ d(cφ/np); its area is, using equation (5), There is nothing arbitrary about the classical concept of angle, the quantity; but that is not true of the mathematical dφ dΩ da = c2 dθ sin θ = c2 , (7) analysts’ concept. In the geometry of the circle the length of np 2 np 2 an arc may be divided by any of several characteristic lengths, the circle’s radius, diameter, circumference or any constant whence fraction or multiple of any of them, to give a ratio that is a(np)2 Ω = . (8) always proportional to certain angles subtended by the arc. c2 Those ratios are proportional not only to the angle subtended = = by the arc at the centre, but also to angles subtended anywhere If p a complete turn (tr), n 1 and on the circumference. In the mathematicians’ deﬁnition of a Ω = tr2. (9) ‘angle’ the radius is chosen arbitrarily as the characteristic (c)2 dimension, and ISO/TC 12’s deﬁnition makes the centre of the circle the point to which the quantity is supposed to have If p = deg, n = 360 and relevance. There is no doubt that mathematicians, scientists and engineers ﬁnd important uses for the quantity they call 3602a Ω = deg2. (10) ‘angle’, but it is not the concept of that name that has been (c)2 recognized for some millennia. That historical concept is still recognized throughout the world, by surveyors, navigators, If p = rad, n = 2π and astronomers, architects, industrial technicians, ordinary people and all who measure angles or use the results; but not by the (2π)2a Ω = rad2, (11) mathematicians and scientists who are represented in ISO/TG (c)2 12 or the CIPM. or, if r is the radius of the sphere, 2. Solid angle a Ω = rad2. (12) 2 The SI concept of solid angle (whose undeﬁned unit was r the other former ‘supplementary’ unit) is equally arbitrary in None of the relations (8) to (12) deﬁnes solid angle. As with deﬁnition. It is deﬁned by ISO/TC 12 [3] as the ratio of two plane angles, a solid angle is a phenomenon at a point, not on areas on or characteristic of a sphere: the ratio of an area on the the surface of a sphere. Equation (8) and those following it sphere’s surface to that of a square of side equal to the sphere’s deﬁne the relations between units of solid angle and units of radius. The SI’s solid angle is not derived from plane angle, plane angle. yet a solid angle at a point, as naturally conceived, is clearly a If s is the length of an arc, of a great circle on the sphere, two-dimensional angle, not deﬁned by one of many ratios of p that subtends a unit p of plain angle at the centre, equation (8) spherical areas that are proportional to it. may be written as A section through a solid angle at a point, and containing a a line through the point that we may call its axis, is a plane angle Ω = p2. (13) 2 bounded by a pair of half lines both terminating at the point. sp One of those lines is at an angle θ to the axis. The plane of In words: the section is at an angle φ to a plane of reference through the same axis. The angle θ is a function of φ. In any system of units where p is a base unit of plane An element of the solid angle Ω at the point is angle, the coherent unit of solid angle [p] is the solid = angle subtended at the centre of a sphere by an area dΩ dθ sin θ dφ (5) 2 ap on the surface equal to sp, the square of the length (the contribution associated with dφ increases with sin θ), sp of an arc of a great circle that subtends a unit p of whence plane angle at the centre, times p2, the square of that θ unit of plane angle. = = − Ω sin θ dθ dφ (1 cos θ)dφ (6) 1 0 For brevity I use here the word ‘subtend’ extended to two dimensions. An area on a sphere ‘subtends’ a solid angle at the centre equal to the solid angle the integration with respect to φ being through a complete at the apex of a cone, having its apex at the centre, that cuts that area on the rotation. sphere.

Metrologia, 42 (2005) L23–L26 L25 Short Communication 3. Conclusion spheres and their radii, but from plane angle, as in equation (6). It is a two-dimensional angle with units derived from units of The two concepts of plane angle, one treating it as a base plane angle. If radian is a unit of plane angle, a base unit as it is quantity, the other deﬁning it as a dimensionless ratio of two commonly regarded, the coherent unit of solid angle is radian2. lengths, are incompatible. The former, historical concept ﬁts The numerical part of the value of a rationally deﬁned, solid consistently into a system of quantities that accepts it as a angle expressed with radian2 as unit is the quantity called a base quantity. The second is of a quantity that should not be ‘solid angle’ in the SI, without a unit but whose value may confused with the ﬁrst; and both suffer from a dual use of be followed by the word ‘steradian’. The addendum means terminology. no more than ‘this number is a value of a solid angle as ISO Solid angle and its units are currently deﬁned without deﬁnes it’. reference to and independently of plane angle. The SI ‘unit’ of solid angle bears the name steradian, which contains the References name radian but has no deﬁned relationship with that unit of plane angle. Solid angle was even allowed formerly to be [1] Emerson W H 2002 Metrologia 39 105–9 [2] 1998 The International System of Units 7th edn (Sevres:` Bureau considered a base quantity and steradian a base unit. Such International des Poids et Mesures) practices overlook the fundamental relationship between the [3] 1993 Quantities and Units: ISO Standards Handbook 3rd edn quantities. Solid angle is a quantity derived, not from areas on (Geneva: International Organisation for Standardisation)

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