# Part II: Galileo's Analysis of Projectile Motion

Galileo brought his lifetime of insight as an experimenter -- and mathematician -- to a conclusion in his greatest work, published in 1638, the Dialogues of the Two New Sciences. Here, in the second half of the book, he took up the question of projectile motion. This illustration

reflects the general opinion before Galileo which followed largely Aristotelian lines but incorporating as well a later theory of "impetus" -- which maintained that an object shot from a cannon, for example, followed a straight line until it "lost its impetus," at which point it fell abruptly to the ground. Later, simply by more careful observation, as this illustration from a work by Niccolo Tartaglia clearly shows,

it was realized that projectiles actually follow some sort of a curved path, but what sort of curve? No one knew until Galileo.

It was another essential insight that led Galileo, finally, to his most remarkable conclusion about projectile motion. First of all, he reasoned that a projectile shot from a cannon is not influenced by only one motion, but by two -- the motion that acts vertically is the force of gravity, and this pulls the projectile down by the times-squared law. But while gravity is pulling the object down, the projectile is also moving forward, horizontally at the same time. And this horizontal motion is uniform and constant according to his principle of inertia. But could he demonstrate this? In fact, by using his inclined plane again, Galileo was indeed able to demonstrate that a projectile is subject to two independent motions, and these combine to provide a precise sort of mathematical curve.

What would happen if, instead of rolling along the horizontal plane, the ball were now allowed to simply fall freely once it got to the bottom of the plane? If Galileo were correct about the horizontal and vertical motions being independent, it would still continue to move horizontally with a uniform, constant speed, but gravity would now begin to pull it down vertically at the same time, the distance increasing porportionally to the square of the time elapsed... and this is exactly what Galileo found.

You can see the experiment simulated in the

computer animation linked to the picture above.

You will notice how the path of the ball traces an exact curve like the one below.

Here is a page from one of Galileo's manuscripts in which he writes down the figures he obtained in performing this experiment himself.

What he actually comes to see is that, in fact, the curve has an exact mathematical shape -- it is one the Greeks had already studied and called the parabola. The extraordinary conclusion Galileo reached in this book on the Two New Sciences is that the path any projectile follows is a parabola, and he drew exact consequences from this discovery which, as he said, could only have been achieved by the sort of exacting analysis that mathematics made possible.

Go on the next section: Conclusion: Part I.