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THE SCIENCE OF DYNAMICS 57 

a certain instant represented by 0, and let the ordinates EF, 
CD, GH, and so on, corresponding to these abscissse represent 
in length the magnitude of the velocities acquired at the time 
represented by the respective abscissse. 

We observe, now, that, by our assumption, 0, F, D, H, lie 
in a straight line OB, and so : (1) At the instant C, at which 
one-half OC of the time of descent OA has elapsed, the velocity 
CD is also one-half of the final velocity AB ; (2) If E and Q 
are equally distant in opposite directions on OA from (7, the 
velocity GH exceeds the mean velocity CD by the same amount 
that the velocity EF falls short of it ; and for every instant 
antecedent to C there exists a corresponding one subsequent to 
C and equally distant from it. Whatever loss, therefore, as com- 
pared with uniform motion with half the final velocity, is suffered 
in the first half of the motion, such loss is made up in the second 
half. The distance fallen through we may consequently regard 
as having been uniformly described with half the final velocity. 

In symbols, if we call the number of units of velocity 
acquired in t units of time by the name v, and suppose that v 
is proportional to t, the number s of units of space descended 
through is proportional to |f 2 . In fact, s is given by ^vt, 
and, as v is proportional to t, s is proportional to ^ 2 . 

Now, Galileo verified this relation between s and t experi- 
mentally. The motion of free falling was too quick for 
Galileo to observe accurately with the very imperfect means 
such as water-clocks at his disposal. There were no 
mechanical clocks at the beginning of the seventeenth century ; 
they were first made possible by the dynamical knowledge of 
which Galileo laid the foundations. Galileo, then, made the 
motion slower, so that s and t were big enough to be measured, 
by rather primitive apparatus in which the moving balls ran 
down grooves in inclined planes. That the spaces traversed 
by the ball are proportional to the squares of the measures of 
the times in free descent as well as in motion on an inclined 
plane, Galileo verified by experimentally proving that a ball 
which falls through the height of an inclined plane attains the 
same final velocity as a ball which falls through its length. 
This experiment was an ingenious one with a pendulum whose 
string, when half the swing had been accomplished, caught on 
a fixed nail so placed that the remaining half of the swing was