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Differing Angles on Angle

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Differing Angles on Angle 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.
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