XXXii INTRODUCTION 



possible of the complex relationships involving them. Further it seems desirable 

 that the units should be extensive in nature. It has been found possible to 

 express all measurable physical quantities in terms of five such units: ist, geo- 

 metrical considerations — length, surface, etc., — lead to the need of a length; 

 2nd, kinematical considerations — velocity, acceleration, etc., — introduce time; 

 3rd, mechanics — treating of masses instead of immaterial /oints — ■ intro- 

 duces matter with the need of a fundamental unit of mass; 4th, electrical, and 

 5th, thermal considerations require two more such quantities. The discov l ./ 

 of new classes of phenomena may require further additions. 



As to the first three fundamental quantities, simplicity and good use sanction 

 the choice of a length, L, a time interval, T, and a mass, M. For the measure- 

 ment of electrical quantities, good use has sanctioned two fundamental quan- 

 tities, — ■ the dielectric constant, A', the basis of the "electrostatic" system and 

 the magnetic permeability, \x, the basis of the "electromagnetic" system. Besides 

 these two systems involving electrical considerations, there is in common use a 

 third one called the "international" system which will be referred to later. For 

 the fifth, or thermal fundamental unit, temperature is generally chosen. 1 



Derived Units. — ■ Having selected the fundamental or basic units, — namely, 

 a measure of length, of time, of mass, of permeability or of the dielectric 

 constant, and of temperature, — it remains to express all other units for physi- 

 cal quantities in terms of these. Units depending on powers greater than unity 

 of the basic units are called " derived units." Thus, the unit volume is the volume 

 of a cube having each edge a unit of length. Suppose that the capacity of some 

 volume is expressed in terms of the foot as fundamental unit and the volume 

 number is wished when the yard is taken as the unit. The yard is three times 

 as long as the foot and therefore the volume of a cube whose edge is a yard is 

 3 X 3 X- 3 times as great as that whose edge is a foot. Thus the given volume 

 will contain only 1/27 as many units of volume when the yard is the unit of 

 length as it will contain when the foot is the unit. To transform from the foot 

 as old unit to the yard as new unit, the old volume number must be multiplied 

 by 1/27, or by the ratio of the magnitude of the old to that of the new unit of 

 volume. This is the same rule as already given, but it is usually more conven- 

 ient to express the transformations in terms of the fundamental units directly. 

 In the present case, since, with the method of measurement here adopted, a 

 volume number is the cube of a length-number, the ratio of two units of volume 

 is the cube of the ratio of the intrinsic values of the two units of length. Hence, 

 if I is the ratio of the magnitude of the old to that of the new unit of length, the 

 ratio of the corresponding units of volume is P. Similarly the ratio of two units 

 of area would be I 2 , and so on for other quantities. 



1 Because of its greater psychological and physical simplicity, and the desirability that the 

 unit chosen should have extensive magnitude, it has been proposed to choose as the fourth fun- 

 damental quantity, a quantity of electrical charge, e. The standard unit of electrical charge 

 would then be the electronic charge. For thermal needs, entropy has been proposed. While 

 not generally so psychologically easy to grasp as temperature, entropy is of fundamental im- 

 portance in thermodynamics and has extensive magnitude. (R. C. Tolman, The Measurable 

 Quantities of Physics, Physical Review, q, p. 237, 1017.) 



