(Part 1)

Newton's Principia was not the first work in physics, nor was it the first mathematical description of trajectories.

So, lets sketch out a brief history of ballistics before Newton. In fact, lets back up to the late medieval theory of physics. In the 14th century, people were still using the Aristotelian theory of physics. This was not a mathematical physics, but a purely philosophical one. A body can have two kinds of motion; natural or artificial. But not at the same time. So consider the example of throwing a stone in the air, say at a 45 degree angle. It would start out with an artificial motion, namely the motion you gave the stone. Somehow, this motion gets expended or lost along the trajectory, and at some point the stone runs out of artificial motion. Then, it stops for a brief moment, after which it begins its natural motion straight down. The ballistics of this looked like this:

https://i.imgur.com/kf3Q6TG.jpg

Now the biggest physics conundrum of that time was the following: how does artificial motion get imparted to an object? Somehow your hand gives the stone a motion, but this motion persists for some time before it runs out. One big theory was that the air is causing the motion to continue. Imagine this as follows: the stone pushes the air in front of it to the sides, and also creates a gap behind itself. So air rushes back in behind the stone, and this rush of air pushes the stone forward again. Like a cycle. I don't have time to look for any medieval drawings of this, but my paint skills are unrivalled so here it goes:

https://i.imgur.com/3l7G6BX.png

Of course, this has some serious problems. For example, people realized that the arrows they used for archery have a very aerodynamic shape in the back, so there's not a lot of area for the air to push on. How is it then that they move so fast and for so long? It almost runs completely contrary to what daily experience tells us, namely that broad flat objects fly less far than narrow objects. Which was a major reason why people were not happy with this theory.

The other theory is what we call 'impetus' theory. Consider it the forerunner of the Newtonian concept of inertial force. According to this theory, every object has like a fuel tank on board, for artificial motions. Your hand fills up this tank with some artificial motion, and the body gradually expends it. Once it's expended, it returns to its natural motion. We can find this for example in a commentary by Jean Buridan in the 14th century. The question he is answering is whether the motion of an arrow is faster halfway through the shot than at its beginning:

Question XIII (whether the projectiles move swifter at half-way than at the beginning or at the end): "And you see that the projector who moves the projectile is for some time tied with the projectile, continuously pushing the projectile before its ejection; like this, a man who casts a stone moves his hand with the stone, and also in shooting an arrow the string moves for some time with the arrow pushing it; and same also is for the sling which throws the stone, or for the machines which throw much bigger stones. And then, as long as the projector pushes the projectile which exists together with him, the motion is slower at the beginning, since only then the extrinsic mover moves the stone or the arrow; but during the movement an impulse (impetus) is acquired continuously, which combined with the extrinsic mover moves the stone or the arrow, which for this move swifter. But after the ejection from the projector, the projector does not move anymore, but only the acquired impulse (does move), as we shall see elsewhere; and that impulse, because of the resistance of the medium, weakens continuously, for which the motion gets continuously slower. And therefore, one must understand that the violent motions, i.e. those of projectiles, are swifter at the beginning than half-way or at the end, of course excluding that part of motion when the projector is together with the projectile; in fact, considering the remaining motion as a whole, the greatest velocity occurs at the beginning. And in this way the authoritative opinions of Aristotle and of the others must be reconciled. It is true that on this regard I have a doubt, since some say that the arrow thrown by the bow would be more perforating at a distance of twenty foot than at a distance of two foot, and therefore after the ejection from the bow the greatest velocity would be not yet at the beginning. And I have not experienced this, therefore I don’t know if it is true; but if it were true, some say that the impetus is not immediately generated by motion, but continuously as a consequence of motion; and then it is not completely generated at the ejection from the bow, but is accomplished in some time, as the rarefaction and the evaporation follow the heating, but not perfectly at once; indeed, once the heating has ceased, for the water is removed from the fire, yet for some time rarefaction and evaporation are seen to continue. And thus it is clear." (taken from Bocalleti, Galileo and the Equations of Motion, p. 41)

(If you are surprised (or in a state of disbelief) that people seriously thought trajectories looked like triangles, you are not alone. As my supervisor once said with some exasperation "Surely these people have seen fountains before! How can they not have realized that's what trajectories look like?")

After some time we do find so-called "mixed motion" depictions of trajectories. In these, the change from artificial to natural motion is not as abrupt, as there is a transitional period of mixed motion. Trajectories on this account begin to take on a more familiar shape:

https://i.imgur.com/5osU68D.jpg

So here you can nicely see the stages; first 'violent' (i.e. against its nature, artificial) motion, then mixed motion, then natural motion.