UNITÀ E DIMENSIONI FISICHE. - The notion of measuring certain quantities, particularly lengths, areas, and weights, and hence of choosing their respective units of measurement, dates back to the most ancient civilizations. These units, naturally homogeneous with the quantities to be measured, were, from that time until relatively recent times, chosen with reference to the dimensions of familiar objects—arm, foot, palm for lengths; some conventional model table for areas; a certain weight for weights—and differed from country to country. The first attempt at unification and rationalization was made by the First French Republic through 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 examined scientifically in the first decades of the twentieth century. It was then established that the general fact—previously glimpsed in particular cases—is that, for measuring physical quantities, it is not always necessary to choose a unit for each one of them, but that it is sufficient to choose a few fundamental units, given that natural relations exist among the various quantities. This is ultimately a generalization of what had already been established for the measurement of geometrical quantities, all of which can be referred to the unit chosen for measuring lengths. Indeed, it is known from Euclid’s book on the Similarity of Elements that if two similar geometrical 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 expressed by the so-called dimensional definitions: [length] = [L], [area] = [S] = [L²], [volume] = [V] = [L³]. For example, considering two cubes with sides of 1 metre and 1 decimetre, respectively, the ratio λ of the sides is 10, and therefore the ratio of the corresponding areas must be, and is, 10² = 100, while that of the volumes is 10³ = 1000.
The extension of this theory to broader fields of physics is made by considering 2, 3, or 4 fundamental quantities for successively broader fields. Thus, in passing from geometry to kinematics, in addition to L one must consider a kinematic quantity, for which time T is conventionally chosen. The definitions are immediately obtained: [time] = [T], [velocity] = [v] = [LT⁻¹], [acceleration] = [a] = [v/T] = [LT⁻²].
The transition to the consideration of dynamical quantities is less immediate (v. MECCANICA), because there is no intuitive indication of which new quantity should be preferred as fundamental in addition to L and T, and because the natural relation among the various quantities of dynamics entails its fundamental law. Taking it in Newton’s formulation—force = mass × acceleration—there are evidently two possibilities: to take either force or mass as fundamental. In general, the latter is chosen, and thus the series of dimensional definitions of dynamical quantities begins with the following: [mass] = [M], [force] = [LT⁻² M], and proceeds by formulating the definitions of all the other quantities by means of these and of the preceding geometrical and kinematic definitions. Thus, for example: [density] = [mass/volume] = [L⁻² M], [work] = [force × displacement] = [LT⁻² M, L] = [L²T⁻² M], etc.
These few notions are already sufficient to provide exact guidance in questions concerning changes of units of measurement, and to characterize dimensionally—and in certain cases to determine outright—important dynamical laws.
1. Change of units
It is sufficient to take account of the dimensional ratios between the fundamental units being substituted, as the following simple examples demonstrate: near the surface of the earth, the acceleration of gravity, referred to the metre and the second, has approximately the value a = 10 m sec.⁻², which is inexactly expressed as 10 m per sec.What is its value: a) in feet per second; b) in metres per minute; c) in feet per minute? Taking the dimensions of acceleration into account, observe: a) since approximately 1 metre = 3 feet, it follows that a = 30 feet sec.⁻²; b) since 1 sec. = 1/60 of a minute and therefore 1 sec.⁻² = 3600 min.⁻², it follows that a = 36000 m min.⁻²; c) a = 108,000 feet min.⁻².
2. Characterization of laws
Since it is logically possible to 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 makes it possible to reduce this homogeneity to that relating to the fundamental quantities. This simple fact always makes it possible to characterize, and sometimes even to determine, the still unknown form of a law when one knows only which quantities enter into II. Example: Galileo had established that the duration t of the 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 l'½ g'²/³, the law of the pendulum can only have the form t = n. √lig, in which n is a purely numerical constant that need only be determined once and for all, and which is famously equal to π.Turning now to the more complex field of electricity, it should be recalled that among the fundamental quantities one must also assume any electrical quantity, and one resorts to some known electrological law to connect it with the previously established L, T, and M. The existing arbitrariness led, from the outset, to the consideration of various dimensional systems, formally different but substantially equivalent; the best known are the so-called electrostatic and electromagnetic systems (v. ELETTRICITÀ; ELETTROLOGIA). Several others of greater technical interest have recently been added to these, among them the Giorgi system, which the International Electrotechnical Committee decided to adopt universally.
Finally, in passing to the thermal field, it should be recalled that temperature is generally assumed as the fourth quantity (v. TERMODINAMICA).
In these two fields as well, the theory of dimensions is the indispensable instrument for any changes in units of measurement.
The use of dimensional theories for characterizing electrological and thermological laws, on the other hand, has been made possible only by recent studies by P. Straneo.