As we have seen, instruments of measurement are only means of comparison between one magnitude and another, and as a general rule we must assume some one arbitrary magnitude, in terms of which all results of measurement are to be expressed. Mere ratios between any series of objects will never tell us their absolute magnitudes; we must have at least one ratio for each, and we must have one absolute magnitude. The number of ratios n are expressible in n equations, which will contain at least n + 1 quantities, so that if we employ them to make known n magnitudes, we must have one magnitude known. Hence, whether we are measuring time, space, density, mass, weight, energy, or any other physical quantity, we must refer to some concrete standard, some actual object, which if once lost and irrecoverable, all our measures lose their absolute meaning. This concrete standard is in all cases arbitrary in point of theory, and its selection a question of practical convenience.
There are two kinds of magnitude, indeed, which do not need to be expressed in terms of arbitrary concrete units, since they pre-suppose the existence of natural standard units. One case is that of abstract number itself, which needs no special unit, because any object which exists or is thought of as separate from other objects (p. 157) furnishes us with a unit, and is the only standard required.
Angular magnitude is the second case in which we have a natural unit of reference, namely the whole revolution or perigon, as it has been called by Mr. Sandeman.1 It is a necessary result of the uniform properties of space, that all complete revolutions are equal to each other, so that we need not select any one revolution, but can always refer anew to space itself. Whether we take the whole perigon, its half, or its quarter, is really immaterial; Euclid took the right angle, because the Greek geometers had never generalised their notions of angular magnitude sufficiently to treat angles of all magnitudes, or of unlimited quantity of revolution. Euclid defines a right angle as half that made by a line with its own continuation, which is of course equal to half a revolution, but which was not treated as an angle by him. In mathematical analysis a different fraction of the perigon is taken, namely, such a fraction that the arc or portion of the circumference included within it is equal to the radius of the circle. In this point of view angular magnitude is an abstract ratio, namely, the ratio between the length of arc subtended and the length of the radius. The geometrical unit is then necessarily the angle corresponding to the ratio unity. This angle is equal to about 57°, 17′, 44″·8, or decimally 57°·295779513... .2 It was called by De Morgan the arcual unit, but a more convenient name for common use would be radian, as suggested by Professor Everett. Though this standard angle is naturally employed in mathematical analysis, and any other unit would introduce great complexity, we must not look upon it as a distinct unit, since its amount is connected with that of the half perigon, by the natural constant 3·14159... usually denoted by the letter π.
When we pass to other species of quantity, the choice of unit is found to be entirely arbitrary. There is absolutely no mode of defining a length, but by selecting some physical object exhibiting that length between certain obvious points—as, for instance, the extremities of a bar, or marks made upon its surface.