History of Western Philosophy - Bertrand Russell [301]
Galileo first discovered the importance of acceleration in dynamics. 'Acceleration' means change of velocity, whether in magnitude or direction; thus a body moving uniformly in a circle has at all times an acceleration towards the centre of the circle. In the language that had been customary before this time, we might say that he treated uniform motion in a straight line as alone 'natural', whether on earth or in the heavens. It had been thought 'natural' for heavenly bodies to move in circles, and for terrestrial bodies to move in straight lines; but moving terrestrial bodies, it was thought, would gradually cease to move if they were let alone. Galileo held, as against this view, that every body, if left alone, will continue to move in a straight line with uniform velocity; any change, either in the rapidity or the direction of motion, requires to be explained as due to the action of some 'force'. This principle was enunciated by Newton as the 'first law of motion'. It is also called the law of inertia. I shall return to its purport later, but first something must be said as to the detail of Galileo's discoveries.
Galileo was the first to establish the law of falling bodies. This law, given the concept of 'acceleration', is of the utmost simplicity. It says that, when a body is falling freely, its acceleration is constant, except in so far as the resistance of the air may interfere; further, the acceleration is the same for all bodies, heavy or light, great or small. The complete proof of this law was not possible until the air pump had been invented, which was about 1654. After this, it was possible to observe bodies falling in what was practically a vacuum, and it was found that feathers fell as fast as lead. What Galileo proved was that there is no measurable difference between large and small lumps of the same substance. Until his time it had been supposed that a large lump of lead would fall much quicker than a small one, but Galileo proved by experiment that this is not the case. Measurement, in his day, was not such an accurate business as it has since become; nevertheless he arrived at the true law of falling bodies. If a body is falling freely in a vacuum, its velocity increases at a constant rate. At the end of the first second, its velocity will be 32 feet per second; at the end of another second, 64 feet per second; at the end of the third, 96 feet per second; and so on. The acceleration, i.e. the rate at which the velocity increases, is always the same; in each second, the increase of velocity is (approximately) 32 feet per second.
Galileo also studied projectiles, a subject of importance to his employer, the duke of Tuscany. It had been thought that a projectile fired horizontally will move horizontally for a while, and then suddenly begin to fall vertically. Galileo showed that, apart from the resistance of the air, the horizontal velocity would remain constant, in accordance with the law of inertia, but a vertical velocity would be added, which would grow according to the law of falling bodies. To find out how the projectile will move during some short time, say a second, after it has been in flight for some time, we proceed as follows: First, if it were not falling, it would cover a certain horizontal distance, equal to that which it covered in the first second of its flight. Second, if it were not moving horizontally, but merely falling, it would fall vertically with a velocity proportional to the time since the flight began. In fact, its change of place is what it would be if it first moved horizontally for a second with the initial velocity, and then fell vertically for a second with a velocity proportional to the time during which it has been in flight. A simple calculation shows that its consequent course is a parabola, and this is confirmed by observation except in so far as the resistance of the air interferes.
The above gives a simple instance