Principles of the Theory of Dimensions

Abstract

The realm of mechanics consists of a variety of concepts such as energy, force, velocity, density, and so on. In the present book, the word ‘mechanics’ refers to classical or non-relativistic mechanics. No limit can be imposed upon the nature and the number of concepts; the progress of science means the birth of new ideas: the introduction of new concepts. On the other hand, each one of this unlimited number of concepts can be defined by means of only three independent entities—length, time and mass—referred to as fundamental entities. It is interesting to note that these three fundamental entities cannot themselves be defined. Nothing in the physical or external world is more obvious to us than what is implied by near and far, earlier and later, and lighter and heavier. We learn these notions, without being taught, simply by living in this world. A concept can be defined only by means of more familiar concepts, yet the forementioned terms remain obvious to us. Any entity that can be measured and expressed in numbers is a quantity. It follows that various mechanical quantities can be regarded merely as compositions of the same three measurable entities: length, time and mass. Let L, T and M be the units for length, time and mass. Since a mechanical quantity a can be considered as a composition of length, time and mass, the unit of a, denoted by [a], must be a function of the fundamental units, i.e.