This picture was first published in 1638! It is from Galileo’s great work Two New Sciences, that he smuggled out of his home imprisonment in Florence, when he was 72 years old and effectively blind. Though famous for his telescope and the first images of the surface of the moon, he had not before published his seminal work on motion. It is easy today to gloss over his extraordinary achievements, even being called the “father of science,” and to make him a caricature battling for the Copernican theory. But this view of the renegade, the persistent critic who fathered three illegitimate children, fails to recognize his profound contribution to humanity. Galileo invented the experiment. Before him people “observed” nature. They developed instruments that simulated phenomena like the motion of the planets. But they did not “experiment.” They did not ask “What if…?

An experiment is a process in which we can change not only inputs, but the rules connecting inputs to outputs as well. Galileo’s classic experiment, the motion of objects sliding down inclined planes, enabled him to dilute gravity, to slow down the motion of a falling body, so that he could measure the distance traveled in each time unit. To do this: he built the first accurate way of measuring short periods of time, he constructed an inclined plane so he could ask “What if I change its slope?” He developed the concept of repeated trials to measure, re-measure, and measure again so fundamental to experimental science.

In the fourth chapter, which he called the fourth day, in Two New Sciences (the first new science being the science of proportion) on the science of motion, he showed how projectile motion, the motion of objects shot out of cannon, thrown, or dropped, can be envisioned as the composition of two motions horizontal and vertical. The horizontal motion of a projectile is constant, it goes the same distance in every unit of time. The vertical motion of a projectile, like the motion of any falling body is accelerated, the distance it travels increases as the square of the time. Added together the motions to produce the path of the projectile and that path is a parabola. We would call the graph that Galileo drew a distance/distance graph (both axes are distance measures). At each point in time, we move across and down. This compounding of motions like the compounding of functions in the Parametric Equations Lab enables us to put together two separate functions linked by a parameter (a parametric variable) to model motion.

As you experiment with the Parametric Equations Lab, imagine you are Galileo, experimenting with inclined planes, dreaming of dropping balls from the Leaning Tower, and explaining why, if Copernicus is right that the earth rotates, we don’t feel ourselves moving. And as you experiment replicate has graph, the first ever drawn and imagine how he would have used it to explain the motion of projectiles.