Written as a letter to Galileo, an education graduate student narrates from course experiences that deepened her understandings of Galileo and natural and intellectual properties underlying his works. Having taken the Galileo-themed course with Elizabeth Cavicchi at MIT's Edgerton Center in three prior terms, on taking it a fourth time, this student, the paper's author, developed and documented perceptive observations of learning. Geometry instruction, as these classmates experienced in school, had equipped them with superficial, rigid access to similar triangles and ratio relations--the mathematical tools of Galileo. While applying these tools in class activities, the classmates faced confusion, dilemmas, and discrepancies. By giving themselves the space to work through these "troubles", classmates initiated their own "learning opportunities". New resilient, flexible understandings arose. Students were no longer confined by prior formulaic rules. The student author discerned a new Euclidean relationship of proportionalities among triangles constructed between parallel lines. These developments are articulated as the experiences arose, providing an analysis and immediacy that is rarely observed or documented frankly by educators and learners. A practitioner of the pedagogy critical exploration in the classroom based on researches of Eleanor Duckworth, Jean Piaget and Bärbel Inhelder, the teacher supported classmates in delving into their questions, which were not expected in advance. Class activities included: observing objects in perspective; collectively working through Euclid's definition of same ratio (in Elements Book 5) on their own terms; investigating errors arising in using a laser-cut prototype instrument based on Galileo's geometrical compass; and exploring a fulcrum balance. Learners' own curiosity opened up areas that they otherwise ignored. How is it ever possible to teach the spirit and method of scientific inquiry? How is it ever possible for anybody to learn it? What it takes for scientific inquiry to take root, develop, and grow? The details of a student/author finding her own ways into deep questions, and making sense of them, reveal the strengths of learning this way. In the background, we see the ways of the teacher. Both enhance our understanding of how Galileo understood learning.