UNITY AND PHYSICAL DIMENSIONS. – The notion of measuring certain quantities, particularly lengths, areas, weights, and thus the choice of their respective units of measurement, dates back to the earliest civilizations. These units, naturally homogeneous to the quantities to be measured, were chosen from the dimensions of common objects—arm, foot, palm for lengths; some conventional standard board for areas; a certain weight for weights—and varied from country to country. The first attempt at unification and rationalization was made by the first French Republic with the imposition of the metric decimal system.
With the rapid multiplication of quantities to be considered, especially following the great development of physics, the problem of their measurement was addressed scientifically in the first decades of the 19th century. It was then observed that the fact—already glimpsed in particular cases—was general: for measuring physical quantities, it is not always necessary to choose a unit for each one, but it is sufficient to choose a few fundamental units, given that natural relationships exist among the various quantities. This is essentially a generalization of what had already been observed for the measurement of geometric quantities, all of which can be referred to the unit chosen for measuring lengths. Indeed, as Euclid’s *Elements* on Similarity states, if two similar geometric figures have corresponding lengths in the ratio λ, their corresponding areas must necessarily be in the ratio λ² and their corresponding volumes in the ratio λ³. 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 meter and 1 decimeter, the ratio λ of the sides is 10, and thus the ratio of corresponding areas must be 10² = 100, and that of volumes must be 10³ = 1000.
The extension of this theory to broader fields of physics is achieved by considering 2, 3, or 4 fundamental quantities in successively broader fields. Thus, to move from geometry to kinematics, it is necessary to consider, in addition to L, a kinematic quantity for which time T is usually chosen. The following definitions are immediately obtained: [time] = [T], [velocity] = [v] = [LT⁻¹], [acceleration] = [a] = [v/T] = [LT⁻²].
The transition to the consideration of dynamic quantities (see MECHANICS) is less immediate because there is no intuitive indication of which new quantity to prefer as fundamental beyond L and T, and because the natural relationships among dynamic quantities imply its fundamental law. Assuming it in Newton’s formulation—force = mass × acceleration—two possibilities become evident: choosing force or mass as the fundamental quantity. In general, the latter is chosen, and thus the series of dimensional definitions for dynamic quantities begins with the positions: [mass] = [M], [force] = [MLT⁻²], and proceeds by formulating the definitions of all other quantities by means of these and the previous geometric and kinematic definitions. Thus, for example: [density] = [mass/volume] = [ML⁻³]; [work] = [force × displacement] = [MLT⁻² × L] = [ML²T⁻²], etc.
These few notions are already sufficient to guide the exact handling of questions regarding changes of units of measurement and to dimensionally characterize—and in certain cases even directly determine—important dynamic laws.
1. Change of units
It is sufficient to take into account the dimensional ratios between the fundamental units being substituted, as shown by the following simple examples: near the surface of the Earth, the acceleration due to gravity, referred to the meter and the second, has approximately the value a = 10 m sec.⁻² (often imprecisely 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 into account the dimensions of acceleration, it is observed that: a) since approximately 1 meter = 3 feet, it follows that a = 30 feet sec.⁻²; b) since 1 sec. = 1/60 of a minute and thus 1 sec.⁻² = 3600 min⁻², it follows that a = 3600 m min⁻²; c) a = 108,000 feet min⁻².
2. Characterization of laws
Since only homogeneous things can be logically compared, the formal expressions of physical laws must always be considered in equations whose members are homogeneous; however, dimensional theory allows this homogeneity to be reduced to that relating to fundamental quantities. This simple fact always permits the characterization—and sometimes even the direct determination—of the still unknown form of a law when only the quantities involved in it are known. For example, Galileo observed that the period of a pendulum depends only on its length and the local acceleration due to gravity. The period t must therefore be equal to a combination of l and g that is homogeneous with respect to time. But since the only possible combination satisfying this condition is l¹ᐟ² g⁻¹ᐟ², the law of the pendulum can only be of the form t = n√(l/g), where n is a purely numerical constant that need only be determined once and for all, and which is known to be equal to π.Turning now to the more complex field of electricity, it is noted that it is necessary to assume, among the fundamental quantities, also some electrical quantity and to relate it to the previous L, T, M by means of some known electrophysical law. The arbitrariness that exists has led, from the beginning, to the consideration of various dimensional systems that are formally different but substantially equivalent; the well-known so-called electrostatic and electromagnetic systems (see ELECTRICITY; ELECTROLOGY) are examples. To these have recently been added several others of greater technical interest, among which the Giorgi system, which the International Electrotechnical Committee decided to adopt universally.
Finally, in the thermal field, it is noted that temperature is generally assumed as the fourth fundamental quantity (see THERMODYNAMICS).
In these two fields as well, dimensional theory is the indispensable tool for any changes of units of measurement.
The use of dimensional theories for the characterization of electrophysical and thermal laws has only been made possible by recent studies by P. Straneo.