We discussed (in a very sketchy way) the time between the Classical Greek period in which Archimedes lived and the Renaissance period in which Galileo Galilei lived. While the Romans tended toward more practical matters (architecture and engineering), the Muslims preserved and furthered the fields of mathematics and astronomy. We discussed the dominance of the Church in Europe during this period, and the pressure placed on it by the Protestant Reformation. We also considered the Church's adoption of Greek philosophy, especially Plato and Aristotle, for example by Augustine and Aquinas.

We mentioned two scientific contemporaries of Galileo -- Tycho Brahe (observer) and Johannes Kepler (calculator). Tycho had observed a super nova ("new star") and through parallex determined that it was far away where things were assumed to be unchanging. Kepler determined that orbits were elliptical rather than circular as had been assumed and was connected with the idea of the heavens being "perfect".

Another topic was Galileo's improvements on the telescope and his subsequent observations of the craters on the moon (suggesting that the moon was not a perfect sphere) and Jupiter's moons (suggesting there were bodies that revolved around something other than the Earth). Galileo saw these as supporting the heliocentric theory ("just a theory") of Copernicus (and perhaps Aristarchus) over the geocentric theory of Ptolemy.

Next we moved onto Galileo's studies of falling bodies. Experiments were not part of science (natural philosophy) during the time when Scholasticism was the major mode of thought. (Other names strongly associated with encouraging the "Scientific Method" are Roger Bacon and Francis Bacon.) Galileo's experiments, probably objects moving along an incline plane and not dropping objects from the Leaning Tower of Pisa, led him to the conclusion that objects of different mass move toward the Earth with the same acceleration.

cartoon: Galileo and others
Cartoon: Galileo
Cartoon: Galileo and Church
Cartoon: Gravitational mass
Video: Galileo's experiment on the moon

We returned to Galileo's study of falling bodies saying that objects of different mass experience the same acceleration. We said that acceleration involved a change in velocity. We distinguished velocity (which has a magnitide and direction) from a speed (which has just a magnitude). For example 50 mph is a speed, but 50 mph West is a velocity. We related velocity in turn to position noting that velocity involved a change in position.

We refined our definition of acceleration to note that it is a rate of change of velocity: a = (v₂-v₁)/(t₂-t₁).

Starting at t=0 we figured that the position at time t would be the initial position plus the average velocity multiplied by the time.

x=x₀ + v_ave t

where x₀ is the position at time t=0. Next, we brought in that for constant acceleration the average velocity is (v₀ + v)/2, where v is the velocity at time t and v₀ is the velocity at time t=0. Subsituting this into our position equation yields

x=x₀ + (v₀ + v) t /2

Again for constant acceleration the velocity depends on time through v = v₀ + a t, and we can substitute that into our position equation

x=x₀ + (v₀ + v₀ + a t ) t /2

x=x₀ + v₀ t + a t² /2

It was noted that we could arrive at the above expression by integrating a constant acceleration. However, since we are taking a quasi-historical approach we did not since calculus comes with Newton and Leibniz, and we are only up to Galileo.

In our discussion that Galileo was one of the first to understand the concept of acceleration, we mentioned Zeno's paradox to show how there had been some struggle to understand the notion of velocity and some even questioned the existence of "change". (In fact for the Greeks the idea of atoms arose as a synthesis of the thesis that everything is changing and the antithesis that nothing ever changes.)

Video: Zeno's Paradox The one with Achilles and the Tortoise

We ended by talking about measuring time in Galileo's time and mentioned the use of water clocks.