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THE SCIENCE OF DYNAMICS 63 

for the sake of shortness, I shall do as the writers referred to, 
and speak of v as " the velocity." Equations in mechanics, 

such as "s = ;?L" are only possible if the left-hand side is of 
a 

the same kind as the right-hand side : we cannot equate spaces 
and times, for example. 

Suppose that we have fixed on the unit of length as one 
inch and the unit of time as one second. As unit of velocity 
we might choose the velocity with which, say, a inches are 
described uniformly in one second. If we did this, we should 
express the relation between the s units of space passed over 
by a body with a given velocity (v units) in a given time 
(t units) as s = avt ; whereas, if we defined the unit of velocity 
as the velocity with which the unit of length is travelled over 
in the unit of time, we should write s = vt. 

Among the units derived from the fundamental units such 
as those of length and time the simplest possible relations are 
made to hold. Thus, as the unit of area and the unit of 
volume, the square and the cube of unit sides are respectively 
used, the unit of velocity is the uniform rate at which unit of 
length is travelled over in the unit of time, the unit of 
acceleration is the gain of unit velocity in unit time, and so on. 

The derived units depend on the fundamental units, and the 
function which a given derived unit is of its fundamental units 
is called its "dimensions." Thus the velocity v is got by 
dividing the length s by the time t. The dimensions of a 
velocity are written 



and those of an acceleration denoted F 

w.m.ja. 

IT"'*! iT^H 2 

These equations are merely mnemonic ; the letters do not 
mean numbers. The mnemonic character comes out when we 
wish to pass from one set of units to another. Thus, if we 
pass to a unit of length 6 times greater and one of time c times 
greater, the acceleration / with the old units is related to that 
(/') with the new units by the equation