Galileo (1564–1642, Italy) was the first person in human history who understood that a physical science can only be founded upon the dual act of mathematical modelings and physical experiments or observations. Modeling the formal world in the fashion of Plato is not enough, one has to simultaneously meddle in the affairs of the physical world in the fashion of Aristotle.

Galileo’s mathematics obviously lacked our modern methods and notations. Galileo never used algebraic symbols or equations, trigonometry, negative numbers, decimals and calculus. His calculations were based on ratios and proportions as propounded by Euclid.

Galileo’s first important theorem was about uniform motion, motion at constant velocity. If an object travels for $t=2$ h at a speed of $v=60$ km h$^{-1},$ then the distance covered $s=vt=120$ km. Galileo did not use this algebraic language, but the language of ratios. If two distances are given as $s_1=vt_1$ and $s_2=vt_2$, then dividing one by the other gives

$$ \frac{t_1}{t_2} = \frac{s_1}{s_2} $$

or the ratio of the times is equal to the ratio of the distances if the velocity between the two distances and time intervals is constant. If the velocity is also different at the two different times, then it can be shown that

$$ \frac{t_1}{t_2} = \frac{s_1}{s_2} \frac{v_2}{v_1} $$

or the ratio of the times is equal to the product of the direct ratio of the distances and the inverse ratio of the velocities.

What if the velocity is not just different at the two instances, but changes continually during the interval, that is if the velocity is subjected to a constant acceleration $a$? Galileo understood that in this case, $v=at$ and

$$ s = \frac{1}{2} at^2 $$

if an object is accelerated from rest for a period of time. If an object is falling freely under gravity on the surface of the Earth, it is subjected to a constant acceleration $g$ and the equations become $v=gt$ and $s=gt^2/2$. Galileo did not use this notation, but he did discover the ‘square’ relationship between distance and time:

$$ \frac{s_1}{s_2} = \frac{t_1^2}{t_2^2}. $$

Galileo combined the concepts of motion at constant velocity and at constant acceleration in his model of a projectile. He proved that the trajectory of a projectile is a parabola and that the vertical motion is accelerated whereas the horizontal motion has a constant speed. This recognition paved way for the formulation of vectors, things that can be resolved into multiple (vertical and horizontal) components. Using the ancient geometric methods of Euclid, he created a table of parabola dimensions for angles of elevation ranging from 1 degree to 89 degrees.

His mathematical methods looked back to Euclid, but his experimental and observational methods looked forward to our modern age. His system of units was as modern as us. His unit of length was called punto where 1 punto = 0.094 cm. And for time he used 1 tempo = 0.01 s. For weighing he used 1 grain = 0.065 g. His units were so small that his experiments always yielded large numbers, therefore he could get many significant figures without resorting to decimals.

His celestial observations set the standard for astronomical observations once and for all.

Galileo’s other great achievement was a formulation of the principle of inertia: an object moving at a constant speed will keep moving at that speed forever unless an external push or pull acts on it. He did not use the concepts of force and energy which are essential for giving a more precise formulation of this principle.

Galileo died in 1642 in Italy, and in that same year Isaac Newton was born in England.