DIMENSIONS OF UNITS. Measurable entities of different kinds cannot be compared directly. Each one must be specified in terms of a unit of its own kind; a single number attached to this unit forms its measure. Thus if the unit of length be taken to be L centimetres, a line whose length is 1 centimetres will be represented in relation to this unit by the number l/L; while if the unit is increased [L] times, that is, if a new unit is adopted equal to [L] times the former one, the numerical measure of each length must in consequence be divided by [L]. Measurable entities are either fundamental or derived. For example, velocity is of the latter kind, being based upon a combination of the fundamental entities length and time; a velocity may be defined, in the usual form of language expressive of a limiting value, as the rate at which the distance from some related mark is changing per unit time. The element of length is thus involved directly, and the element of time inversely in the derived idea of velocity; the meaning of this statement being that when the unit of length is increased [L] times and the unit of time is increased [T] times, the numerical value of any given velocity, considered as specified in terms of the units of length and time, is diminished [L]/[T] times. In other words, these changes in the units of length and time involve change in the unit of velocity determined by them, such that it is increased [V] times where [V]=[L][T]'. This relation is conveniently expressed by the statement that velocity is of + 1 dimension in length and of — 1 dimension in time. Again, acceleration of motion is defined as rate of increase of velocity per unit time; hence the change of the units of length and time will increase the corresponding or derived unit of acceleration [V]/[T] times; that is [L][T]2 times: this expression thus represents the dimensions (1 in length and 2 in time) of the derived entity acceleration in terms of its fundamental elements length and time. In the science of dynamics all entities are derived from the three fundamental ones, length, time and mass; for example, the dimensions of force (P) are those of mass and acceleration jointly, so that in algebraic form (P) = [M][L][T]z. This restriction of the fundamental units to three must therefore be applicable to all departments of physical science that are reducible to pure dynamics.
The mode of transformation of a derived entity, as regards its numerical value, from one set of fundamental units of reference to another set, is exhibited in the simple illustrations above given. The procedure is as follows. When the numerical values of the new units, expressed in terms of the former cries, are substituted for the symbols, in the expression for the dimensions of the entity under consideration, the number which results is the numerical value of the new unit of that entity in terms of the former unit: thus all numerical values of entities of this kind must be divided by this number, in order to transfer them from the former to the latter system of fundamental units.
As above stated, physical science aims at reducing the phenomena of which it treats to the common denomination of the positions and movements of masses. Before the time of Gauss it was customary to use a statical measure of force, alongside the kinetic measure depending on the acceleration of motion that the force can produce in a given mass. Such a statical measure could be conveniently applied by the extension of a spring, which, however, has to be corrected for temperature, or by weighing against standard weights, which has to be corrected for locality. On the other hand, the kinetic measure is independent of local conditions, if only we have absolute scales of length and time at our disposal. It has been found to be indispensable, for simplicity and precision in physical science, to express the measure of force in only one way; and statical forces are therefore now generally referred in theoretical discussions to the kinetic unit of measurement. In mechanical
engineering the static unit has largely survived; but the increasing importance of electrical applications is introducing uniformity there also. In the science of electricity two different systems of units, the electrostatic and the electrodynamic, still to a large extent persist. The electrostatic system arose because in the development of the subject statics came before kinetics; but in the complete synthesis it is usually found convenient to express the various quantities in terms of the electrokinetic system alone.
The system of measurement now adopted as fundamental in physics takes the centimetre as unit of length, the gramme as unit of mass, and the second as unit of time. The choice of these units was in the first instance arbitrary and dictated by convenience; for some purposes subsidiary systems based on multiples of these units by certain powers of ten are found convenient. There are certain absolute entities in nature, such as the constant of gravitation, the velocity of light in free space, and the constants occurring in the expression giving the constitution of the radiation in an enclosure that corresponds to each temperature, which are 'the same for all kinds of matter; these might be utilized, if known with sufficient accuracy, to establish a system of units of an absolute or cosmical kind. The wavelength of a given spectral line might be utilized in the same manner, but that depends on recovering the kind of matter which produces the line.
In physical science the uniformities in the course of phenomena are elucidated by the discovery of permanent or intrinsic relations between the measurable properties of material systems. Each such relation is expressible as an equation connecting the numerical values of entities belonging to the system. Such an equation, representing as it does a relation between actual things, must remain true when the measurements are referred to a new set of fundamental units. Thus, for example, the kinematical equation v2 = rtf 2l, if n is purely numerical, contradicts the necessary relations involved in the definitions of the entities velocity, acceleration, and length which occur in it. For on changing to a new set of units as above the equation should still hold; it, however, then becomes v2/[V]2=n .f 2/[F]2• l/[L]. Hence on division there remains a dimensional relation [V]2= [F]2[L], which is in disagreement with the dimensions above determined of the derived units that are involved in it. The inference follows, either that an equation such as that from which we started is a formal impossibility, or else that the factor n which it contains is not a mere number, but represents is times the unit of some derived quantity which ought to be specified in order to render the equation a complete statement of a physical relation. On the latter hypothesis the dimensions [N] of this quantity are determined by the dimensional equation [VP= [N ][F]2[L] where, in terms of the fundamental units of
length and time, [V] = [L][T]', [F] = [L][T]2; whence by
substitution it appears that [N] = [L]'[T]2. Thus, instead of being merely numerical, is must represent in the above formula the measure of some physical entity, which may be classified by the statement that it has the conjoint dimensions of time directly and of velocity inversely.
It often happens that a simple comparison of the dimensions of the quantities which determine a physical system will lead to important knowledge as to the necessary relations that subsist between them. Thus in the case of a simple pendulum the period of oscillation r can depend only on the angular amplitude a of the swing, the mass m of the bob considered as a point, and the length 1 of the suspending fibre considered as without mass, and on the value of g the acceleration due to gravity, which is the active force; that is, r=f(a, in, 1, g). The dimensions must be the same on both sides of this formula, for, when they are expressed in terms of the three independent dynamical quantities mass, length, and time, there must be complete identity between its two sides. Now, the dimensions
of g are [L][T]2; and when the unit of length is altered the
numerical value of the period is unaltered, hence its expression must be restricted to the form f(a, in, l/g). Moreover, as the period does not depend on the unit of mass, the form is further
X. SVII. 2,1reduced to f(a, l/g) ; and as it is of the dimensions + r in time, it must be a multiple of (l/g)i, and therefore of the form (ga) X1(1/g). Thus the period of oscillation has been determined by these considerations except as regards the manner in which it depends on the amplitude a of the swing. When a process of this kind leads to a definite result, it will be one which makes the unknown quantity jointly proportional to various powers of the other quantities involved; it will therefore shorten the process if we assume such an expression for it in advance, and find whether it is possible to determine the exponents definitely and uniquely so as to obtain the correct dimensions. In the present example, assuming in this way the relation r=Aapmsl'gs, where A is a pure numeric, we are led to the dimensional equation [T]=[a]P[M]s[L]r[LT2]3, showing that the law assumed would not persist when the fundamental units of length, mass, and time are altered, unless q=o, s= , r= z; as an angle has no dimensions, being determined by its numerical ratio to the invariable angle forming four right angles, p remains undetermined. This leads to the same result, r=4(a)l+1g i, as before.
As illustrating the power and also the limitations of this method of dimensions, we may apply it (after Lord Rayleigh, Roy. Soc. Proc., March 1900) to the laws of viscosity in gases. The dimensions of viscosity (p) are (force/area) = (velocity/length), giving [ML1T'] in terms of the fundamental units. Now, on the dynamical theory of gases viscosity must be a function of the mass m of a molecule, the number is of molecules per unit volume, their velocity of mean square i', and their effective radius a; it can depend on nothing else. The equation of dimensions cannot supply more than three relations connecting these four possibilities of variation, and so cannot here lead to a definite result without further knowledge of the physical circumstances. And we remark conversely, in passing, that wherever in a problem of physical dynamics we know that the quantity sought can depend on only three other quantities whose dynamical dimensions are known, it must vary as a simple power of each. The additional knowledge required, in order to enable us to proceed in a case like the present, must be of the form of such an equation of simple variation. In the present case it is involved in the new fact that in an actual gas the mean free path is very great compared with the effective molecular radius. On this account the mean free path is inversely as the number of molecules per unit volume; and therefore the coefficient of viscosity, being proportional to these two quantities jointly, is independent of either, so long as the other quantities defining the system remain unchanged. If the molecules are taken to be spheres which exert mutual action only during collision, we therefore assume
K cc ms~va',
which requires that the equation of dimensions [MLIT'] = [M]s[LTIlv[L]'
must be satisfied. This gives x =1, y =1, z= 2. As the temperature is proportional to mu , it follows that the viscosity is proportional to the square root of the mass of the molecule and the square root of the absolute temperature, and inversely proportional to the square of the effective molecular radius, being, as already seen, uninfluenced by change of density.
If the atoms are taken to be Boscovichian points exerting mutual attractions, the effective diameter a is not definite; but we can still proceed in cases where the law of mutual attraction is expressed by a simple formula of variation—that is, provided it is of type km2r•, where r is the distance between the two molecules. Then, noting that, as this is a force, the dimensions of k must be (M'L'+'T2] we can assume
µ cc m'iwk'°
provided [ML'T11=
[M]s[LTI]~[M'Ls+'T2]°, which demands and is satisfied by
x—W=I, y+2W=1, y+(s+I)w=—I,
so that w=? y=s+' x=s3 
s—I s—I' s—I. Thus, on this supposition,
S9 2 S+3 tl am2S2k SI 02S2
where 0 represents absolute temperature. (See DIFFUSION.)
When the quantity sought depends on more than three others, the method may often be equally useful, though it cannot give a complete result. Cf. Sir G. G. Stokes, Math. and Phys. Papers, v (1881) p. io6, and Lord Rayleigh, Phil. Mag. (1905), (I) p. 494, for examples dealing with the determination of viscosity from observations of the retarded swings of a vane, and with the formulation of the most general type of characteristic equation for gases respectively. As another example we may consider what is involved in Bashforth's experimental conclusion that the airresistances to shot of the same shane are proportional to the squa'es of their
linear dimensions. A priori, the resistance is a force which is determined by the density of the air p, the linear dimensions 1 of the shot, the viscosity of the air 1i, the velocity of the shot v, and the velocity of sound in air c, there being no other physical quantity sensibly involved. Five elements are thus concerned, and we can combine them in two ways so as to obtain quantities of no dimensions; for example, we may choose pvl/µ and v/c. The resistance to the shot must therefore be of the form jo pwu¢(pvl/p)f(v/c) this form being of sufficient generality, as it involves an undetermined function for each element beyond three. On equating dimensions we find x=2, y=–1, z=o. Now, Bashforth's result shows that cp(x)=x2. Therefore the resistance is pv2l2f (v/c), and is thus to our degree of approximation independent of the viscosity. Moreover, we might have assumed this practical independence straight off, on known hydrodynamic grounds; and then the argument from dimensions could have predicted Bashforth's law, if the present application of the doctrine of dimensions to a case involving turbulent fluid motion not mathematically specifiable is valid. One of the important results drawn by Osborne Reynolds from his experiments on the regime of flow in pipes was a confirmation of its validity: we now see that the ballistic result furnishes another confirmation.
In electrical science two essentially distinct systems of measurement were arrived at according as the development began with the phenomena of electrostatics or those of electrokinetics. An electric charge appears as an entity having different dimensions in terms of the fundamental dynamical units in the two cases: the ratio of these dimensions proves to be the dimensions of a velocity. It was found, first by W. Weber, by measuring the same charge by its static and its kinetic effects, that the ratio of the two units is a velocity sensibly identical with the velocity of light, so far as regards experiments conducted in space devoid of dense matter. The emergence of a definite absolute velocity such as this, out of a comparison of two different ways of approaching the same quantity, entitles us to assert that the two ways can be consolidated into a single dynamical theory only by some development in which this velocity comes to play an actual part. Thus the hypothesis of the mere existence of some complete dynamical theory was enough to show, in the stage which electrical science had reached under Gauss and Weber, that there is a definite physical velocity involved in and underlying electric phenomena, which it would have been hardly possible to imagine as other than a velocity of propagation of electrical effects of some kind. The time was thus ripe for the reconstruction of electric theory by Faraday and Maxwell.
The power of the method of dimensions in thus revealing general relations has its source in the hypothesis that, however complicated in appearance, the phenomena are really restricted within the narrow range of dependence on the three fundamental entities. The proposition is also therein involved, that if a changing physical system be compared with another system in which the scale is altered in different ratios as regards corresponding lengths, masses, and times, then if all quantities affecting the second system are altered from the corresponding quantities affecting the first in the ratios determined by their physical dimensions, the stage of progress of the second system will always correspond to that of the first; under this form the application of the principle, to determine the correlations of the dynamics of similar systems, originated with Newton (Principia, lib. ii. prop. 32). For example, in comparing the behaviour of an animal with that of another animal of the same build but on a smaller scale, we may take the mass per unit volume and the muscular force per unit sectional area to be the same for both; thus [LI, [MI, . . . being now ratios of corresponding quantities, we have [ML–3I= and [ML–lT–21–1, giving [LI = [TI; thus the larger animal effects movements of his limbs more slowly in simple proportion to his linear dimensions, while the velocity of movement is the same for both at corresponding stages.
But this is only on the hypothesis that the extraneous force of gravity does not intervene, for that force does not vary in the same manner as the muscular forces. The result has thus application only to a case like that of fishes in which gravity is equilibrated by the buoyancy of the water. The effect of the inertia of the water, considered as a perfect fluid, is included in this comparison; but the forces arisingfrom viscosity do not correspond in the two systems, so that neither system may be so small that viscosity is an important agent in its motion. The limbs of a land animal have mainly to support his weight, which varies as the cube of his linear dimensions, while the sectional areas of his muscles and bones vary only as the square thereof. Thus the diameters of his limbs should increase in a greater ratio than that of his body—theoretically in the latter ratio raised to the power 1, if other things were the same. An application of this principle, which has become indispensable in modern naval architecture, permits the prediction of the behaviour of a large ship from that of a mallscale model. The principle is also of very wide utility in unravelling the fundamental relations in definite physical problems of such complexity that complete treatment is beyond the present powers of mathematical analysis; it has been applied, for example, to the motions of systems involving viscous fluids, in elucidation of wind and waves, by Helmholtz (Akad. Berlin, 1873 and 1889), and in the electrodynamics of material atomic systems in motion by Lorentz and by Larmor.
As already stated, the essentials of the doctrine of dimensions in its .most fundamental aspect, that relating to the comparison of the properties of correlated systems, originated with Newton. The explicit formulation of the idea of the dimensions, or the exponents of dimension, of physical quantities was first made by Fourier, Theorie de la chaleur, 1822, ch. ii. sec. 9; the homogeneity in dimensions of all the terms of an equation is insisted on by him, much as explained above; and the use of this principle as a test of accuracy and precision is illustrated. (J. L.*)
End of Article: DIMENSIONS OF UNITS 

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