Two New Sciences

Galileo's final book, "Discourses and Mathematical Demonstrations Relating to Two New Sciences," published in 1638, served as his scientific testament covering 30 years of his physics work. Despite a ban from the Roman Inquisition, the book was published in Holland. It's written as a dialogue between three characters: Simplicio, Salviati, and Sagredo, representing different stages of Galileo's own beliefs. The book is divided into four 'days,' each addressing different areas of physics.

The first day discusses the resistance of bodies to separation, introducing the idea that scale matters—properties of objects change with size. It explores the strength of materials like hemp rope and copper wire, and the concept of a vacuum. Galileo also challenges Aristotle's belief that heavier bodies fall faster, asserting that in a vacuum, all bodies fall at the same speed. He uses pendulums to measure the speed of fall and discusses the vibration of strings.

The second day focuses on the cause of cohesion and the strength of materials. Galileo demonstrates how balances work with unequal arms and proves that the moment of a suspended weight is proportional to the square of its length. He analyzes the resistance to fracture of beams of various sizes and thicknesses, showing that larger animals require proportionately larger bones and that hollow cylinders are stronger than solid ones of the same weight.

The third day defines uniform and uniformly accelerated motion. Galileo explains that the velocity of falling bodies increases proportionally to time, not distance. He describes his famous experiment with a steel ball rolling down an inclined grooved piece of wood, using a water clock to accurately measure time and verify uniformly accelerated motion. He also considers descent along the chords of a circle, noting that descending along its arc is the fastest.

The fourth day explores projectile motion as a combination of uniform horizontal motion and accelerated vertical motion, resulting in a parabolic curve. Galileo details how to construct these parabolas and discusses the effects of air resistance. He explains that air resistance affects denser bodies less and offers greater resistance to faster bodies, and that speed reaches a maximum rather than increasing indefinitely.

Galileo's methodology, particularly for the law of falling bodies, was debated by contemporaries. He distinguished his arguments as natural suppositions, reinforced by observation and experimentation, making him both a rationalist and an empiricist. He accurately measured constant acceleration by slowing down falling bodies on an inclined plane. His experiments using a carefully constructed ramp and a water clock allowed him to meticulously measure time and confirm the law of falling bodies, even accounting for discrepancies due to friction and air resistance by presenting his ideas under ideal conditions.

Galileo refuted Aristotle's argument against Earth's motion by presenting the example of sailors on a moving boat. He explained that a dropped object would land at the base of the mast, not behind it, due to the simultaneous horizontal and vertical motion. This concept, the relativity of motions, became the basis for Newton's first law of inertia. He detailed how a ball dropped from a ship's mast experiences both accelerating vertical motion (gravity) and uniform horizontal motion (ship's movement), resulting in a parabolic trajectory, which the observer cannot distinguish due to sharing the horizontal motion.