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58 THE NATURE OF MATHEMATICS 

with a shorter string than the other half. This experiment 
showed that the bob of the pendulum rose, in virtue of the 
velocity acquired in its descent, just as high as it had fallen. 
This fact is in agreement with our instinctive knowledge of 
natural events ; for if a ball which falls down the length of an 
inclined plane could attain a greater velocity than one which 
falls through its height, we should only have to let the body 
pass with the acquired velocity to another more inclined plane 
to make it rise to a greater vertical height than that from 
which it had fallen. Hence we can deduce, from the accelera- 
tion on an inclined plane, the acceleration of free descent, for, 
since the final velocities are the same and s = %vt, the lengths 
of the sides of the inclined plane are simply proportional to 
the times taken by the ball to pass over them. 

The motion of falling that Galileo found actually to exist is, 
accordingly, a motion of which the velocity increases pro- 
portionally to the time. 

Like Galileo, we have started with the notions familiar to 
us (through the practical arts, for example), such as that of 
velocity. Let us consider this motion more closely. 

If a motion is uniform and c feet are travelled over in every 
second, at the end of t seconds it will have travelled ct feet. 
Put ct s for short. Then we call the " velocity " of the moving 

body the distance traversed in unit of time so that it is - units of 



length per second, the number which is the measure of the 
distance divided by the number which is the measure of the 
time elapsed. Galileo, now, attained to the conception of a 
motion in which the velocity increases proportionally to the 
time. If we draw a diagram and set off, from the origin O 
along the a;-axis OA, a series of abscissae which represent the 
times in length, and erect the corresponding ordinates to 
represent the velocities, the ends of these ordinates will lie on 
a line OB, which, in the case of the "uniformly accelerated 
motion" to which Galileo attained, is straight, as we have 
already seen. But if the ordinates represent spaces instead of 
velocities, the straight line OB becomes a curve. We see 
the distinction between the " curve of spaces " and " the curve 
of velocities," with times as abscissae in both cases. If the 
velocity is uniform, the curve of spaces is a straight line OB