5

CHAPTER I.

GENERAL THEORY OF UNITS.

Units and Derived Units.

1. THE numerical value of a concrete quantity is its ratio to a selected magnitude of the same kind, called the unit.

Thus, if L denote a definite length, and the unit

L

length, is a ratio in the strict Euclidian sense, and is ī

called the numerical value of L.

The numerical value of a concrete quantity varies directly as the concrete quantity itself, and inversely as the unit in terms of which it is expressed.

2. A unit of one kind of quantity is sometimes defined by reference to a unit of another kind of quantity. For example, the unit of area is commonly defined to be the area of the square described upon the unit of length; and the unit of volume is commonly defined as the volume of the cube constructed on the unit of length. The units of area and volume thus defined are called derived units, and are more convenient for calculation than indepenIdent units would be. For example, when the above

definition of the unit of area is employed, we can assert that [the numerical value of] the area of any rectangle is equal to the product of [the numerical values of] its length and breadth; whereas, if any other unit of area were employed, we should have to introduce a third factor which would be constant for all rectangles.

3. Still more frequently, a unit of one kind of quantity is defined by reference to two or more units of other kinds. For example, the unit of velocity is commonly defined to be that velocity with which the unit length would be described in the unit time. When we specify a velocity as so many miles per hour, or so many feet per second, we in effect employ as the unit of velocity a mile per hour in the former case, and a foot per second in the latter. These are derived units of velocity.

Again, the unit acceleration is commonly defined to be that acceleration with which a unit of velocity would be gained in a unit of time. The unit of acceleration is thus derived directly from the units of velocity and time, and therefore indirectly from the units of length and time.

4. In these and all other cases, the practical advantage of employing derived units is, that we thus avoid the introduction of additional factors, which would involve needless labour in calculating and difficulty in remembering.*

5. The correlative term to derived is fundamental. Thus, when the units of area, volume, velocity, and

* An example of such needless factors may be found in the rules commonly given in English books for finding the mass of a body when its volume and material are given. "Multiply the volume in cubic feet by the specific gravity and by 62·4, and the product will be the mass in pounds;" or "multiply the volume in cubic

acceleration are defined as above, the units of length and time are called the fundamental units.

Dimensions.

6. Let us now examine the laws according to which derived units vary when the fundamental units are changed.

Let V denote a concrete velocity such that a concrete length L is described in a concrete time T; and let v, l, t denote respectively the unit velocity, the unit length, and the unit time.

The numerical value of V is to be equal to the numerical value of L divided by the numerical value of T. But

This equation shows that, when the units are changed (a change which does not affect V, L, and T), v must vary directly as l and inversely as t; that is to say, the unit of velocity varies directly as the unit of length, and inversely as the unit of time.

Equation (1) also shows that the numerical value

V

a given velocity varies inversely as the unit of length, and directly as the unit of time.

inches by the specific gravity and by 253, and the product will be the mass in grains." The factors 62-4 and 253 here employed would be avoided-that is, would be replaced by unity, if the unit volume of water were made the unit of mass.

7. Again, let A denote a concrete acceleration such that the velocity V is gained in the time T', and let a denote the unit of acceleration. Then, since the numerical value of the acceleration A is the numerical value of the velocity V divided by the numerical value of the time T', we have

This equation shows that when the units a, l, t are changed (a change which will not affect A, L, T or T'), a must vary directly as 1, and inversely in the duplicate

A

ratio of t; and the numerical value will vary inversely

a

as 7, and directly in the duplicate ratio of t. In other words, the unit of acceleration varies directly as the unit of length, and inversely as the square of the unit of time; and the numerical value of a given acceleration varies inversely as the unit of length, and directly as the square of the unit of time.

It will be observed that these have been deduced as

direct consequences from the fact that [the numerical value of] an acceleration is equal to [the numerical value of] a length, divided by [the numerical value of] a time, and then again by [the numerical value of] a time.

The relations here pointed out are usually expressed by

saying that the dimensions of acceleration* are

length

or

(time)

that the dimensions of the unit of acceleration* are

unit of length

(unit of time)2

8. We have treated these two cases very fully, by way of laying a firm foundation for much that is to follow. We shall hereafter use an abridged form of reasoning, such as the following :-

Such equations as these may be called dimensional equations. Their full interpretation is obvious from what precedes. In all such equations, constant numerical factors can be discarded, as not affecting dimensions.

9. As an example of the application of equation (2) we shall compare the unit acceleration based on the foot and second with the unit acceleration based on the yard and minute.

Let 7 denote a foot, L a yard, t a second, T a minute, Ta minute. Then a will denote the unit acceleration based on the foot and second, and A will denote the unit

* Professor James Thomson ('Brit. Assoc. Report,' 1878, p. 452) objects to these expressions, and proposes to substitute the following:

This is very clear and satisfactory as a full statement of the meaning intended; but it is necessary to tolerate some abridgment of it for practical working.