Book: Discourses and Mathematical Demonstrations Relating to Two New Sciences

Overview
"Discourses and Mathematical Demonstrations Relating to Two New Sciences" distills Galileo's mature thinking on motion and material strength into a compact, mathematically driven account. It isolates two broad problems: the laws governing moving bodies and the behavior of solid bodies under stress, and treats them with experiments, geometric proofs, and careful thought experiments. The work marks a decisive shift from qualitative Aristotelian explanations to quantitative, predictive science.

Form and Characters
The treatise unfolds as a sustained dialogue among three interlocutors: Salviati, who speaks for the new mathematical approach; Simplicio, who defends traditional Aristotelian views; and Sagredo, an intelligent layman who evaluates arguments and experiments. This conversational form allows conflicting hypotheses to be compared directly, with Salviati advancing precise demonstrations while Simplicio raises classical objections. The rhetorical device creates a pedagogical drama in which empirical evidence and geometric reasoning gradually win acceptance.

Motion and Kinematics
A central achievement is the clear statement and geometric demonstration of uniformly accelerated motion. Using experiments such as the inclined plane and pendulum timing, Galileo shows that distances traveled under constant acceleration grow as the square of elapsed time and that instantaneous speed is proportional to time. These results lead to a unified treatment of free fall, inclined motion, and the composition of motion. Projectile trajectories are analyzed by decomposing motion into independent uniform horizontal motion and uniformly accelerated vertical motion, yielding parabolic paths as an approximation that captures observed behavior.

Principles and Concepts
Galileo articulates early versions of principles that later matured into Newtonian mechanics. He challenges Aristotelian ideas about natural rest and motion, emphasizes the role of inertia, and introduces thought experiments that anticipate the relativity of uniform motion. The careful use of limiting procedures and idealized experiments , such as imagining frictionless planes or infinitely small time intervals , allows mathematical laws to be extracted from imperfect real-world apparatus.

Strength of Materials and Scaling
The first "science" investigates the resistance and fracture of solid bodies, producing striking conclusions about how strength does not scale linearly with size. Galileo reasons that weight increases with volume while structural strength depends on cross-sectional area, which leads to the square-cube principle: larger bodies are proportionally weaker. He applies this idea to beams, animal anatomy, and architectural design, explaining why structures must be built with changing proportions at larger scales. Detailed geometric arguments and simple experiments on beams and plates underpin his conclusions about breaking strength, flexure, and the role of material geometry.

Method and Legacy
Mathematical demonstration combined with careful experiment defines the book's method: precise measurements, reduction to geometric laws, and elimination of unsupported qualitative causes. The dialogues crystallize a scientific ethos that prizes predictive formulae and reproducible tests. The work became a cornerstone for classical mechanics, influencing later scientists who formalized dynamics and structural mechanics, and it established enduring standards for the mathematical treatment of natural phenomena.