Galileo did not express his observations on the rate of speed of falling objects in a mathematical formula in the modern sense. He did, however, make significant contributions to understanding the relationship between distance, time, and acceleration, which laid the foundation for later mathematicians and physicists to express these relationships mathematically.

Here's why:

* Galileo's Approach: Galileo relied on observation, experimentation, and logical reasoning to understand how objects fall. He used inclined planes to slow down the acceleration of falling objects, making it easier to measure their motion. He observed that objects falling from the same height reach the ground at the same time, regardless of their mass, and that the distance traveled is proportional to the square of the time.

* Mathematical Formalization: The mathematical formulas we use today to describe freefall were developed later, primarily by Isaac Newton. He formulated the law of universal gravitation, which mathematically explains the force of gravity and its effects on falling objects. This law, along with Newton's laws of motion, provided the framework for describing falling objects in a precise mathematical language.

So, while Galileo did not express his observations in a mathematical formula, his work was crucial in establishing the empirical basis for the mathematical laws of motion and gravity that came later.