When was science born? And guess what: the Catholic Church was the midwife.

Pierre Duhem (fromWikipedia)

"The history of science alone can keep the physicist from the mad ambitions of dogmatism as well as the despair of pyrrhonian scepticism. " Pierre Duhem, The Aim and Structure of Physical Theory

INTRODUCTION

Some atheists and materialists would say that science arose in the 16th and 17th centuries, fully mature,  like Botticelli's Venus arising from the ocean.   According to them it arose then because Europe had shaken off the limiting bonds of Catholic doctrine.   Pierre Duhem's historical studies of science  shows that this is not so.*

Botticellis' "Venus arising from the ocean"

THE EDICTS OF PARIS, 1277

Rather, Duhem dates the birth of science as 1277, the year the Bishop of Paris, Etienne Tempier, condemned a number of errors from astrology and from the Peripatetic philosophers (those following Aristotle).    The condemned articles contended that the earth could not move, that worlds other than earth could not exist, that empty space (a vacuum) was impossible, and proposed principles of motion that were shown later to be false.

Bishop Tempier condemned the articles not because of scientific errors, but because they apparently limited God's power.   Once these Peripatetic dicta were declared non-binding, scholars--almost all of them clerics--were able to explore new ways to explain the world around us, ways that would grow into the scientific method.   Bertrand Russell's comment about the two books of Aristotle that embodied the condemned articles, "The Physics" and "On The Heavens" (which he greatly admired), is appropriate:  “there is hardly a sentence in either that is not contradicted by modern science”. 

Two areas of science. cosmology and dynamics (the physics of motion),  then grew from babyhood to adolescence in the period from 1277 to the 17th century. The growth in cosmology culminated in the Copernican Revolution, the idea that the earth was no longer the center of the universe but revolved around the sun.   But there was much work preliminary to that--the notion did not spring full-blown  to Copernicus.    I'll give a brief summary of Duhem's account of the development of dynamics below;  for a more complete discussion about this and the growth of cosmology please go to History of Physics before Einstein .

PHYSICS OF MOTION: 1277 To the 15th Century

The two Aristotelian principles of physical motion that were totally incorrect had to do with the effects of gravity and momentum--what kept moving bodies moving.   According to Aristotle, gravity manifested itself in the following ways:  heavy elements wanted to move to the center of the universe, the center of the  earth; the heavier the element (the more massive), the faster it would move;  lighter elements (air or fire) would move away from the center of the earth.  

Aristotle's theory of motion for projectiles was even stranger: 

"He held that the projectile was moved by the fluid medium, whether air or water,  through which it passed and this, by virtue of the vibration brought about in the fluid at the moment of throwing, and spread through it [the vibration through the medium]".  Pierre Duhem, History of Physics before Einstein, location 305

William of Occam (he of "Occam's Razor") argued against Aristotle's theory, as did other scholastics.   Jean Buridan (1295-1363), Rector of the University of Paris, gave  what is essentially the modern interpretation for projectile motion:

Buridan gave the name impetus [emphasis added] to the virtue or power communicated to the projectile by the hand or instrument throwing it; he declared that in any given body in motion this impetus was proportional to the velocity, and that in different bodies in motion propelled by the same velocity, the quantities of impetus were proportional to the mass or quantity of matter defined as it was afterward defined by Newton."  ibid, location 311

Thus "impetus" is that which we now term "momentum", defined by p = mv, where p is momentum, m is mass, and v is velocity.     The effects of air resistance and gravity (for a projectile thrown upwards) in slowing the motion, "destroying impetus", were taken into account by Buridan.   He thus analyzed the motion of the pendulum, the mechanism for impact and rebound, and the deformation of elastic bodies.

Albert of Saxony (1320-1390, Bishop of Halberstadt),  Buridan's pupil added on to Buridan's kinematic theory the following:

 "the velocity of a falling body must be proportional either to the time elapsed from the beginning of the fall or to the distance travelled during this time."  ibid, location 322.

This implies that gravity is a force that increases the impetus (momentum) of the falling body.     If these dynamic laws applied to motion of bodies on earth, they should also apply to the "heavenly" bodies, despite the restriction laid down by Aristotle that the motion of the heavenly spheres was governed by different laws than those of earthly bodies.   If there is nothing to reduce impetus--no air or medium to give rise to friction--then the heavenly bodies can keep moving without ceasing;  the initial impetus is given "by God at the moment of creation".    Thus the notion of inertia came into being.

THE LAWS OF FALLING BODIES, 15TH-16TH CENTURY

Although Leonardo Da Vinci is better known for his art and work with  military armaments and flying machines, he did significant work in physics.    He understood the principle of conservation of energy and composition of forces in statics.   Although he recognized that the velocity of a freely-falling body is proportional to the time the body has fallen, he did not recognize the importance of increasing impetus due to gravity.

The significant law for falling bodies,  was proposed by Nicole Oresme (1320-1385, Bishop of Lisieux)

"...in a uniformly varied motion, the space traversed by the moving body is equal to that which it would traverse in a uniform motion whose duration would be that of the preceding motion, and whose velocity would be the same as that which affected the preceding motion at the mean instant of its duration."  ibid. location 571

Note that "uniformly varied motion" is equivalent to uniform acceleration. (These propositions were put forth before acceleration was understood as a property of motion.)  This means that the distance travelled in such motion during an interval of time would be that given by the average velocity for the interval (mean of beginning and ending velocity) multiplied by the time interval.

Oresme's proposition was modified by  Domingo Soto (1494-1560, professor of theology at Salamanca:

"The velocity of a falling body increases proportionally to the time of the fall.

The space traversed in a uniformly varied motion is the same as in a uniform motion occupying the same time, its velocity being the mean velocity of the former." ibid.

Soto's formulation leads to Galileo's  Law of Falling Bodies, that the distance fallen is proportion to the square of the time of fall.   This can be seen diagrammatically (as shown by Oresme) or algebraically** .

All the work described above sets the stage for Galileo--his important contribution was to introduce experimental tests of the mathematical hypotheses, inclined  plane experiments (done in my first year physics lab at Caltech), telescopic observations to confirm the Copernican hypothesis,   It should be noted that Galileo's ideas about dynamics did not yet reach the stage taught in first year physics classes today.

"He then taught that the motion of a freely falling body was uniformly accelerated; in favour of this law, he contented himself with appealing to its simplicity without considering the continual increase of impetus under the influence of gravity. Gravity creates, in equal periods, a new and uniform impetus which, added to that already acquired, causes the total impetus to increase in arithmetical progression according to the time occupied in the fall; hence the velocity of the falling body." ibid. location 970.

So we see that although Galileo achieved much, in different areas of physics and astronomy, he still followed in the tradition of his predecessors.   The relation of acceleration to force had to wait for Newton's Law, F= m a.

HISTORICAL CONSIDERATIONS 

Why did this development of physics and cosmology occur begin and grow in Medieval Christendom, but not in the ancient Hellenistic worlds or other civilizations?    Excellent answers have been given, in some detail,  by Fr. Jaki and Dr. Trasancos in the references listed below, but I want to add my own opinion.   First, there was a world view, founded on Judaeo-Christian theology, that God was good and created a universe that was good and meant to be intelligible to mankind.   Second, as Duhem points out, the Medieval scholars were freed in 1277 from erroneous restrictions they would have had to follow if Aristotle's principles were to be a compulsory base for theories.    Third, in the earliest part of this growth they were priests;   this meant that they, as do academics today, had time to do scholarly work and did not have to worry about earning a living from non-scholarly pursuits.  

Finally, I want to emphasize again: there was a continuity of development from the 13th century to Galileo;  science did not spring full-blown with Galileo, despite his critical introduction of empirical verification of theoretical ideas and his insights in astronomy and dynamics.

NOTES AND REFERENCES

*Duhem has written several histories of science.   The one linked above is the most easily (and cheaply) available.  Fr. Stanley Jaki has commented about Duhem's work and philosophy of science in several biographies, as has Dr. Stacy Trasancos in her book "Science was born of Christianity", which proceeds from Fr. Jaki's works.

**Consider the time t1=0 and time t2 =t (end of fall).   At t1=0, the initial velocity is 0.   At time t2, the velocity is v.   Soto's first proposition would say v = kt.
Soto's second proposition would give   d=v' t  where v' is the mean velocity =kt/2
(since v=0 at t=0).   Thus  d= (kt/2)xt = (k/2) t^2.     We now recognize the proportionality constant k as the acceleration, a.