UNITY AND PHYSICAL DIMENSIONS. - The notion of the measurement of certain quantities, in particular of lengths, areas, weights, and hence of the choice of the respective units of measurement, goes back to the most ancient civilizations. These units, naturally homogeneous with the quantities to be measured, from then until relatively recent times were chosen with reference to the dimensions of familiar objects — cubit, foot, palm for lengths — to some conventional standard board for areas, to a certain weight for weights, and they differed from country to country. The first attempt at unification and rationalization was made by the first French Republic with the imposition of the decimal metric system.
With the rapid multiplication of the quantities to be considered, especially following the great development of physics, the problem of their measurement was considered scientifically in the first decades of the 20th century. It was then established that the fact, already glimpsed previously in particular cases, is general: that for the measurement of physical quantities it is not always necessary to choose a unit for each of them, but that it is sufficient to choose a few fundamental units, since natural relations exist among the various quantities. It is fundamentally a generalization of what had already been established for the measurement of geometric quantities, which can all be referred to the unit chosen for the measurement of lengths. It is known in fact from Euclid's book on the Similitude of elements that if two similar geometric figures have corresponding lengths in the ratio λ, they must necessarily have corresponding areas in the ratio λ² and corresponding volumes in the ratio λ³. And this, and nothing other than this, is what is customarily expressed by the so-called dimensional definitions: [length] = [L], [area] = [S] = [L²], [volume] = [V] = [L³]. For example, considering two cubes with sides of 1 meter and 1 decimeter, the ratio λ of the sides is 10 and therefore the ratio of the corresponding areas must be, and is, 10² = 100, and that of the volumes 10³ = 1000.
The extension of this theory to broader fields of physics is carried out by considering 2, 3, or 4 fundamental quantities of successively broader fields. Thus, to pass from geometry to kinematics, it is necessary to consider, besides L, a kinematic quantity for which one customarily chooses time T. The definitions are immediately obtained: [time] = [T], [velocity] = [v] = [LT⁻¹], [acceleration] = [a] = [v/T] = [LT⁻²].
Less immediate is the transition to the consideration of dynamic quantities (v. MECCANICA), because there is no intuitive indication as to the new quantity to be preferred as fundamental besides L and T, and because the natural relation among the various quantities of dynamics implies its fundamental law. Assuming it in Newton's formulation: force = mass × acceleration, the two possibilities of assuming either force or mass as fundamental become evident. In general the latter is chosen, and therefore the series of dimensional definitions of dynamic quantities begins with the statements: [mass] = [M], [force] = [LT⁻² M], and continues by formulating the definitions of all the other quantities by means of these and of the preceding geometric and kinematic definitions. Thus, for example: [density] = [mass/volume] = [L⁻² M], [work] = [force × displacement] = [LT⁻² M, L] = [L²T⁻² M], etc.
Already these few notions are sufficient to guide one exactly in questions of changing units of measurement and to characterize dimensionally, and in certain cases to determine outright, important dynamic laws.
1. Change of units
It is enough to take account of the dimensional ratios between the fundamental units that are substituted, as the following simple examples demonstrate: near the surface of the earth the acceleration of gravity, referred to the meter and the second, has approximately the value a = 10 m sec.⁻², which is inexactly stated as 10 m per sec.
What is its value: a) in feet per second; b) in meters per minute; c) in feet per minute? Taking account of the dimensions of acceleration, observe: a) that since approximately 1 meter = 3 feet, one deduces a = 30 feet sec.⁻²; b) that since 1 sec. = 1/60 of a minute and therefore 1 sec.⁻² = 3600 min⁻², one deduces a = 36000 m min⁻²; c) a = 108,000 feet min⁻².
2. Characterization of laws
Since logically one can compare only things that are homogeneous with one another, the formal expressions of physical laws must always consist of equations whose members are homogeneous; however, the theory of dimensions allows this homogeneity to be reduced to that relative to the fundamental quantities. This simple fact always makes it possible to characterize and sometimes even to determine the still unknown form of the law when only one knows which quantities enter into it. Example: Galileo had established that the duration t of a pendulum's oscillation depends only on its length l and on the local acceleration of gravity g. The duration t must therefore be equal to a combination of l and g homogeneous with a time. But since the only possible combination satisfying this condition is l½ g'²/³, the law of the pendulum can only be of the form t = n. √lig, in which n is a purely numerical constant that need be determined once and for all, and which is notoriously equal to π.
Passing now to the more complex field of electricity, it is recalled that one must assume, among the fundamental quantities, also any electrical quantity whatsoever, and one has recourse to some well-known electrodynamic law to connect it to the preceding L, T, M. The existing arbitrariness has led from the outset to the consideration of various dimensional systems, formally different but substantially equivalent; the so-called electrostatic and electromagnetic systems are very well known (v. ELETTRICITÀ; ELETTROLOGIA). To these, several others of greater technical interest have recently been added, among which is the Giorgi system, which the International Electrotechnical Committee decided to adopt universally.
Passing finally to the thermal field, it is recalled that as the fourth quantity one generally assumes temperature (v. TERMODINAMICA).
Also in these two fields the theory of dimensions is the indispensable instrument for any changes of units of measurement.
The utilization of dimensional theories for the characterization of electrodynamic and thermodynamic laws has instead been made possible only by recent studies of P. Straneo.
Bibliography
Paolo Straneo